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https://www.mersenneforum.org/showthread.php?s=4b0ff12d3b286326285621090a639c61&t=2150&page=2 | 1,603,624,798,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107888931.67/warc/CC-MAIN-20201025100059-20201025130059-00618.warc.gz | 817,730,419 | 12,436 | Register FAQ Search Today's Posts Mark Forums Read
2005-03-16, 22:40 #12 Kosmaj Nov 2003 2·1,811 Posts Larry, 8331405*2-1 = 16662809 is also prime. Thus, the 100th prime occurs for n=45052 while in this respect the best among "old" candidates is k=1581823815 where the 100th prime occurs for n=38374. Very close! More such results can be found here.
2005-03-16, 23:45 #13 robert44444uk Jun 2003 Oxford, UK 7×271 Posts 100 primes For the groups interest, the best +1 number to get to a hundred primes got there with n=3258 with k = 1108828374241*M(59), where M(59) is a certain multiple of small primes p up to 59, excluding those with order base 2 of less than p-1. This is a Payam number, par excellence. It may well be that the - series produced better, but I don't know for sure. In any case there are better numbers out there, although Phil Carmody thought that this was close to the ideal number at to that (i.e. 100 primes) range. For your interest, this number was checked up to n=100000 with "only" 140 primes, so we stopped checking it in earnest - ther are plenty with greater than that at that level, but software does not allow for fast checking of theses k, because k is rather large. Regards Robert Smith Ps this series is prime at n= (note the huge gap at 4495-7544) 3 11 13 14 18 19 23 25 26 29 31 42 45 50 64 65 71 86 98 101 116 119 134 191 194 195 212 257 259 269 276 284 285 294 307 324 396 403 406 420 447 449 459 480 486 540 545 602 607 625 654 659 703 742 780 827 828 838 848 867 923 960 976 982 1005 1011 1021 1037 1049 1081 1100 1130 1145 1192 1254 1327 1427 1488 1548 1599 1647 1663 1684 1818 1844 1861 1880 1892 1946 1951 1971 2044 2130 2150 2227 2328 2393 3215 3224 3258 no 100 3289 3405 3436 3450 3722 3833 4172 4227 4337 4495 7544 8666 9037 9263 9758 10493 10628 12817 12959 15351 15762 16529 18995 19249 21117 22601 25769 27486 32648 33107 38099 40751 52586 53046 53060 53105 54241 60535 75013 85111 102912 126708
2005-03-16, 23:49 #14 robert44444uk Jun 2003 Oxford, UK 111011010012 Posts n=10000 PS A good count for primes at 10000, on the + side would be greater than 116. The candidate at that level was 2158430601663 *M(67), whose 116th prime is at n=9971 Regards Robert Smith
2005-04-17, 09:06 #15 gribozavr Mar 2005 Internet; Ukraine, Kiev 11×37 Posts Primes for 15k=187466565 with n from 1 to 100000 Here are primes I've found for 15k=187466565 with n from 1 to 100000: Code: 187466565 7 187466565 8 187466565 11 187466565 13 187466565 25 187466565 26 187466565 36 187466565 42 187466565 43 187466565 44 187466565 49 187466565 54 187466565 57 187466565 61 187466565 73 187466565 78 187466565 81 187466565 95 187466565 97 187466565 99 187466565 125 187466565 128 187466565 129 187466565 137 187466565 159 187466565 167 187466565 187 187466565 189 187466565 224 187466565 234 187466565 235 187466565 320 187466565 395 187466565 440 187466565 491 187466565 546 187466565 561 187466565 602 187466565 621 187466565 684 187466565 694 187466565 703 187466565 788 187466565 803 187466565 1158 187466565 1298 187466565 1422 187466565 1561 187466565 1727 187466565 1920 187466565 1946 187466565 2265 187466565 2675 187466565 2990 187466565 2994 187466565 3150 187466565 3296 187466565 3453 187466565 3525 187466565 3608 187466565 4006 187466565 4391 187466565 4688 187466565 4866 187466565 6842 187466565 6996 187466565 7014 187466565 7115 187466565 7393 187466565 7796 187466565 8505 187466565 9722 187466565 10587 187466565 11144 187466565 13574 187466565 13629 187466565 13793 187466565 13837 187466565 14678 187466565 15213 187466565 16210 187466565 16578 187466565 17355 187466565 19121 187466565 20215 187466565 21542 187466565 22635 187466565 28375 187466565 37256 187466565 41362 187466565 41785 187466565 41888 187466565 63163 187466565 65203 187466565 74889 187466565 78879 187466565 80097 187466565 92598 I will continue testing this 15k with n up to 200000.
2005-04-21, 03:46 #16 VBCurtis "Curtis" Feb 2005 Riverside, CA 104508 Posts Primes for k=3803443215: n=1 through n=9 already on 15k.org site. 19,20,28 (PRPs), 53,56,58,112,235,313,339,367,372,376,397,413,454,525,543,576, 847,850,1027,1048,1061,1125,1160,1301, 1318, 1329,1503,1874, 2033,2536,2544,2553,2709,2953,3352,3649,3726,3887,4709,5075, 5167,5175,7092,10902,11933,12333,19168,19822,20276,21919, 22792,23567,24845,25793,39906,40936,42092,42258,51165,66033, 71287,72993,85641,90943,119027,159420,161247,167492,169409, 178053. 82 primes so far. I'm at roughly n=188000 currently, planning to process to n=250,000, then decide whether to push the low-weight number beyond 1M or this number to higher powers. What's the current cutoff for top-5000 entry? n=220,000? -Curtis
2005-04-21, 04:06 #17 lsoule Nov 2004 California 32508 Posts The top-5000 cutoff is just below n=189,000 now. -Larry
2005-04-25, 18:47 #18 gribozavr Mar 2005 Internet; Ukraine, Kiev 11·37 Posts Primes for 15k=187466565 with n from 100000 to 150000 The range 100000-150000 was sieved by me, gribozavr, and LLR'ed mostly by my friend yasya. So, please, give us both a credit for theese primes (write something like gribozavr and yasya): Code: 187466565 107376 187466565 109098 187466565 111081 187466565 127062 187466565 130833 187466565 134459 n=150000-200000 in progress
2005-09-16, 17:49 #19
Cruelty
May 2005
31228 Posts
Attached see primes for k=736320585 for 1<n<250000 (total of 100 primes incl. 2 Sophie Germain).
Continuing sieving and testing.
Attached Files
PRIMES_736320585_250k.txt (2.0 KB, 215 views)
2005-09-24, 20:10 #20 gribozavr Mar 2005 Internet; Ukraine, Kiev 11·37 Posts I have finished testing k=187466565 for n=1-200k. I have found some more primes: Code: 187466565 160453 187466565 165138 187466565 178879 187466565 187871 Whom do I send the lresults.txt? Can any of theese primes go into Top-5000? The biggest one is 56564 digits long.
2005-09-24, 20:31 #21
paulunderwood
Sep 2002
Database er0rr
65668 Posts
Quote:
Can any of theese primes go into Top-5000? The biggest one is 56564 digits long.
because they are too small according to:
http://primes.utm.edu/primes/submit.php which states:
Quote:
Currently primes must have 60222 or more digits to make the list
In a years time the bar will have been raised to 70k
2005-09-28, 12:49 #22 grobie Sep 2005 Raleigh, North Carolina 337 Posts Please dont tell me I have been wasting my time I just noticed the posts from gribozavr and he has already been testing my reserved K-187466565
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Sun Oct 25 11:19:58 UTC 2020 up 45 days, 8:30, 0 users, load averages: 1.58, 1.55, 1.51 | 2,664 | 6,898 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2020-45 | latest | en | 0.702508 |
https://physicshelpforum.com/threads/medical-physics-correcting-presbyopia.15058/ | 1,579,985,659,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251681412.74/warc/CC-MAIN-20200125191854-20200125221854-00436.warc.gz | 608,292,152 | 14,881 | # Medical physics - Correcting presbyopia
#### SassyBean
Hi all,
I have another problem, this time involving the lens equation.
(1/f= 1/v+ 1/u)
v and u are in Meters and 1/f is the power in diopters (D)
I have solved some problems where the near point of an eye is correct and the far point needs correcting and where the far point is correct (infinite) and the near point needs adjusting, however the last question has me stumped.
It seems as though I need to modify both the near point and far point simultaneously but this also seems strange as it specifies I need not change the near point but instead maintain it and bring objects into focus at that distance.
My working so far has been to calculate the difference from a healthy eye in the near and far points separately and combine the answer to give an overall lens strength adjustment, but I don't connect this with answering the near point issue correctly and would therefore like some guidance:
An elderly person with presbyopia has a near point of 0.40m and a far point of 4.0m
Calculate the power of spectacle lens required to enable objects at the near point to be seen clearly
My working :
Near point:
Actual ( 1 / 0.4 ) + ( 1 / 0.02 ) = 52.5 D
Required ( 1 / 0.25 ) + ( 1 / 0.02 ) = 52 D
Difference = 0.5 D
+
Far point:
Actual ( 1 / 4 ) + ( 1 / 0.02 ) = 50.25 D
Required ( 1 / ∞ ) + ( 1 / 0.02 ) = 50 D
Difference = -0.25 D
0.5 – 0.25 = 0.25 D
Lens power required = 0.25 D | 405 | 1,449 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2020-05 | latest | en | 0.958095 |
https://www.transtutors.com/questions/a-quality-control-inspector-is-examining-newly-produced-items-for-faults-the-inspect-1598789.htm | 1,582,469,142,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875145774.75/warc/CC-MAIN-20200223123852-20200223153852-00111.warc.gz | 928,083,561 | 16,132 | # A quality control inspector is examining newly produced items for faults. The inspector searches... 1 answer below »
A quality control inspector is examining newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration. Given that a flaw is actually present, let p denote the probability that the flaw is detected during any one fixation (this model is discussed in “Human Performance in Sampling Inspection,” Human Factors, 1979: 99–105).
a. Assuming that an item has a flaw, what is the probability that it is detected by the end of the second fixation (once a flaw has been detected, the sequence of fixations terminates)?
b. Give an expression for the probability that a flaw will be detected by the end of the nth fixation.
c. If when a flaw has not been detected in three fixations, the item is passed, what is the probability that a flawed item will pass inspection?
d. Suppose 10% of all items contain a flaw [P(randomly chosen item is flawed) 5 .1]. With the assumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not flawed, but could also pass if it is flawed)?
e. Given that an item has passed inspection (no flaws in three fixations), what is the probability that it is actually flawed? Calculate for p = .5.
Mohit J | 300 | 1,394 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2020-10 | latest | en | 0.932904 |
https://dabangnewslive.in/laser-beams-solution-codechef/ | 1,638,244,894,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964358903.73/warc/CC-MAIN-20211130015517-20211130045517-00275.warc.gz | 261,799,959 | 13,678 | # Laser Beams solution codechef
Ira is developing a computer game. This game features randomized generation and difficulty of levels. To achieve randomized difficulty, some enemies in each level are randomly replaced with stronger ones.
To describe how do the levels in the game look, let’s introduce a coordinate system in such a way that OxOx axis goes from left to right, and OyOy axis goes from bottom to top. A level is a rectangle with opposite corners in points (0,0)(0,0) and (a,b)(a,b). Each enemy’s position is a point in this rectangle.
As for now, Ira has implemented one type of enemy in the game, in two different versions — basic and upgraded. Both versions of enemies Ira has implemented fire laser rays in several directions:
## Laser Beams solution codechef
• basic enemies fire four laser rays in four directions: to the right (in the same direction as the vector (1,0)(1,0)), to the left (in the same direction as the vector (1,0)(−1,0)), up (in the same direction as the vector (0,1)(0,1)), and down (in the same direction as the vector (0,1)(0,−1));
• upgraded enemies fire eight laser rays in eight directions: four directions listed for basic enemies, and four directions corresponding to vectors (1,1)(1,1)(1,1)(1,−1)(1,1)(−1,1)(1,1)(−1,−1).
Laser rays pass through enemies and are blocked only by the borders of the level (sides of the rectangle that denotes the level). Enemies are unaffected by lasers.
### Laser Beams solution codechef
The level Ira is working on has nn enemies. The ii-th enemy is in the point (xi,yi)(xi,yi), and it has a probability of pipi to be upgraded (it’s either upgraded with probability pipi, or basic with probability 1pi1−pi). All these events are independent.
Ira wants to estimate the expected difficulty. She considers that a good way to evaluate the difficulty of the level is to count the number of parts in which the level is divided by the laser rays. So, she wants to calculate the expected number of these parts.
Help her to do the evaluation of the level!
### Input Laser Beams solution codechef
The first line contains three integers nnaa and bb (1n1001≤n≤1002a,b1002≤a,b≤100) — the number of enemies in the level and the dimensions of the level.
Then nn lines follow, the ii-th of them contains three integers xixiyiyi and pipi′ (1xia11≤xi≤a−11yib11≤yi≤b−11pi9999991≤pi′≤999999), meaning that the ii-th enemy is located at (xi,yi)(xi,yi) and has a probability of pi106pi′106 to be upgraded.
No two enemies are located in the same point.
### Output Laser Beams solution codechef
Print one integer — the expected number of parts in which the lasers divide the level, taken modulo 998244353998244353 (i. e. let the expected number of parts be xyxy as an irreducible fraction; you have to print xy1mod998244353x⋅y−1mod998244353, where y1y−1 is a number such that yy1mod998244353=1y⋅y−1mod998244353=1).
Examples
input
### Copy Laser Beams solution codechef
1 2 2
1 1 500000
output
### Copy Laser Beams solution codechef
6
input
### Copy Laser Beams solution codechef
2 3 2
1 1 500000
2 1 500000
output
### Copy Laser Beams solution codechef
499122187
### Note Laser Beams solution codechef
Explanation to the first example:
With probability 1212, the only enemy is not upgraded and the level looks like this (44 parts):
With probability 1212, the only enemy is upgraded and the level looks like this (88 parts):
So, the expected number of parts is 412+812=64⋅12+8⋅12=6. | 968 | 3,469 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2021-49 | latest | en | 0.89334 |
http://mathforum.org/library/drmath/view/56937.html | 1,519,297,713,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891814101.27/warc/CC-MAIN-20180222101209-20180222121209-00499.warc.gz | 207,317,605 | 3,436 | Associated Topics || Dr. Math Home || Search Dr. Math
### From Infinite Decimals to Mixed Fractions
```
Date: 10/09/98 at 19:00:50
From: Jessi Michelle
Subject: A challenge from a teacher
I was given a challenging problem by my math teacher just to see if I
could find a way to figure it out. The problem is to write this
fraction as a mixed number:
.131313... + .555...
--------------------
.161616... - .222...
I think that the top can first be simplified to .6868.... After that
I'm stuck because of the 2 away from 1 thing in a repeating decimal.
It seems like it would be easier to change the decimals to fractions,
but I don't know how. Help!
```
```
Date: 10/13/98 at 11:10:28
From: Doctor Nick
Subject: Re: A challenge from a teacher
Hi Jessi -
Yes, converting the decimals to fraction is the way to go. There is a
neat trick you can use to convert repeating fractions like this. I'll
give you a couple of examples.
The main trick is to multiply the number by a power of 10 (10, 100,
1000, 10000, etc.) that you pick to get the right effect. For
instance, if x = 0.55555..., then 10x = 5.5555555..., and so
10x - x = 5. That is, 9x = 5, so x = 5/9. The trick is to pick a power
of 10 that makes the repeating pattern in the decimal expansion "line
up" so it disappears when you subtract. Here is another example: if
x = 0.1616161616..., then 100x = 16.16161616161616..., and so
100x - x = 16, Then 99x = 16, which implies x = 16/99.
This works even with decimals that don't repeat right away:
x = 0.71232323232323...
100x = 71.23232323232323...
100x - x = 99x = 71.23 - 0.71 = 70.52
x = 70.52/99 = 7052/9900 = 1763/2475
That's a little more complicated than the other cases, but the method
still works.
Now, convert all the decimals in your problem to fractions, and
simplify to a single fraction, and you'll have it.
Have fun,
- Doctor Nick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series
Middle School Fractions
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#### hifum
##### Active Member
wow how did you get that i was so confused
D?
#### dem()sthenes
##### Member
the other options don't work.
#### idkkdi
##### Well-Known Member
You guys are playing chess, while I'm playing Death Note.
D?
F
#### idkkdi
##### Well-Known Member
You boys got no chance against this question. ))))))
#### hifum
##### Active Member
B. Ngl, I don't know how it works.
i would faint if that was in the test
#### idkkdi
##### Well-Known Member
This question doesn't seem legit. What was the answer again?
#### dem()sthenes
##### New Member
Yeah should be D.
#### idkkdi
##### Well-Known Member
You could easily make the argument that a filled-in heart is not possible as it is nowhere else on the grid, then you would be between b and c which can't be easily differentiated to get an answer.
But sure, D it is, filled in heart is allowed.
##### New Member
You could easily make the argument that a filled-in heart is not possible as it is nowhere else on the grid, then you would be between b and c which can't be easily differentiated to get an answer.
But sure, D it is, filled in heart is allowed.
Well KIRA it's because each column is mirrored (so notice there is a star on top there is also a star on the bottom) so that leaves B or D because of the green circle. But also each column must have at least one of each shape (so the middle row can't have another unfilled green circle and must have a heart).
#### idkkdi
##### Well-Known Member
Well KIRA it's because each column is mirrored (so notice there is a star on top there is also a star on the bottom) so that leaves B or D because of the green circle. But also each column must have at least one of each shape (so the middle row can't have another unfilled green circle and must have a heart).
One of each shape works, but you could argue that the red heart is not an allowed shape in the first place. Your mirrored statement is just wrong. Take the middle column for instance.
#### idkkdi
##### Well-Known Member
u sound like me trying to argue with my teacher for more marks on the essay i completed the night before
Except I get high enough marks on essays I complete the night before that I don't have to argue for marks .
It would have made the student's life much easier if a filled-in heart was somewhere else in the grid, so that its existence could be confirmed.
#### hifum
##### Active Member
Except I get high enough marks on essays I complete the night before that I don't have to argue for marks .
It would have made the student's life much easier if a filled-in heart was somewhere else in the grid, so that its existence could be confirmed.
well these questions weren't made to be easy, your supposed to think outside the box *literally*
#### idkkdi
##### Well-Known Member
well these questions weren't made to be easy, your supposed to think outside the box *literally*
No lol, that question isn't hard, it's just badly written. | 703 | 2,987 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2022-27 | latest | en | 0.960135 |
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This week, we have a pretty detailed topic, so expect to spend a little extra time working on the discussion this week. As always, feel free to explore this topic more deeply through the conversation.
Again, like we have been doing, at the end of the discussion, you will be asked to craft a 150 word reflection on what you have learned through this conversation and post it to the Reflection Journal. Dive in, be active, help out your classmates when they need it, and, because we could never say this enough, enjoy the conversation.
XYZ Corporation produces a commercial product that is in great demand by consumers on a national basis. Unfortunately, near the plant where it is produced there is a large population of dove-tail turtles who are adversely affected by contaminants from the plant. XYZ has a filtering process that is expensive and any increase in filtering effectiveness reduces their profit. Dove tailed turtles are not a protected species hence there are no environmental rules regulating XYZ’s level of contaminants.
Clearly, no filtering at all would maximize XYZ’s profitability. However, a local environmental group monitors XYZ’s contaminant level and maintains a website showing the percentage mortality rate of the dove tailed turtles due to XYZ’s contaminants. XYZ has noticed that the higher the mortality percentage, the less items are bought and the lower their profitability.
Their Marketing Department and Research Group has established the following Revenue Function, R(x), as a function of Dove Tail Turtle Mortality expressed as decimal between 0 to 1, representing mortality rate:
R(x) = 1 + x – x2; 0 < x ≤ 1
R(x) is expressed in billions of dollars and represents the revenue generated. Since XYZ has fixed operating costs of one billion dollars, the profit function, P(x) is given by P(x) = R(x) – 1.
You have been hired as the mathematical consultant for XYZ Corporation. They have asked you to help them optimize their profit and request that you:
1. Find the optimal dove-tail turtle mortality rate 2. State what the maximum profit will be.
Discussion:
As a successful mathematics student, you easily do the math and come up with the solution. Please include the answers to questions 1 and 2 in your discussion post.
But don’t collect your money yet. You also understand that mathematics is a human activity and carries with it a certain social responsibility. Are there any ethical considerations that you would bring to your client’s attention?
References
Henrich, D. (2011, November 2). Mathematical ethics: A problem based approach.
http://www.fields.utoronto.ca/programs/mathed/meetings/minutes/12-13/HenrichSept2012.pdf
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Customers referred by a friend | 1,260 | 5,588 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2023-06 | latest | en | 0.953263 |
https://tutorialsprime.net/200170357/ | 1,620,525,171,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988953.13/warc/CC-MAIN-20210509002206-20210509032206-00024.warc.gz | 579,656,707 | 13,780 | ## (Solved) Question 3 - Cost Behaviour Megan Supplies Ltd is a wholesaler for a large variety of plumbing supplies. The company's accountant, Hannah, has...
Thank you!
Question 3 – Cost Behaviour
Megan Supplies Ltd is a wholesaler for a large variety of plumbing supplies.
The company's
accountant, Hannah, has recently completed a cost study of the firm's
Shipping Department in which he used the kilograms of supplies loaded or
Washer compiled the following data. Required:
1. Draw a scatter diagram of the cost data for the Shipping Department.
2. Estimate the Shipping Department's cost behaviour using the high–low
method. Use an
equation to express the results of this estimation method.
3. Using the equations estimated in Part 2 predict the Shipping Department's
costs for a
month when loads totalling 4500 kilograms are moved.
4. Prepare least squares regression analysis to estimate the variable and fixed
components
for the Shipping Department costs.
5. Based on your spreadsheet write the least squares regression equation for
the
Department's costs.
6. Using the equation from Part 5, predict the firm's Shipping Department's
costs for a month
when loads totalling 4500 kilograms are moved.
7. Why do these cost predictions estimated using the high–low and regression
methods
Question 4 – Cost-Volume-Profit Analysis
The Kingswood Company produces thin limestone sheets that are used for the
facings on
buildings. As can be seen in the contribution margin statement, last year the
company had a net profit of \$157 500, based on sales of 1800 tonnes. The
manufacturing capacity of the firm's facilities is 3000 tonnes per year. Required:
1. Calculate the company's break-even volume, in tonnes, for the most recent
year. (Ignore
income taxes.)
2. If the sales volume is estimated to be 2100 tonnes in the next year, and if
the prices and
costs stay at the same levels and amounts, what net profit can management
expect next year?
3. The company has an overseas customer who has offered to buy 1500
tonnes at \$450 per
tonne. Assume that all the firm's costs would be at the same levels and rates
as in the year
just ended. What net profit would the firm earn if it took this order and
from local customers so as not to exceed production capacity?
4. Kingswood plans to market its product in a new territory. Management
estimates that an
advertising and promotion program costing \$61 500 per year would be
needed for the next
two or three years. In addition, a \$25 per tonne sales commission to the sales
force in the new
territory, over and above the current commission, would be required. How
many tonnes
would need to be sold in the new territory to maintain the firm's current net
profit? Assume
that sales and costs will continue as in the year just ended in the firm's
established territories.
5. Management is considering replacing its labour-intensive production
process with an
automated production system. This would result in an increase of \$58 500
annually in fixed
manufacturing costs. The variable manufacturing costs would decrease by
\$25 per tonne.
Calculate the new break-even volume in tonnes and in sales dollars.
6. Ignore the facts presented in requirement 5. Assume that management
estimates that the
selling price per tonne will decline by 10 per cent next year. Variable costs will increase by
\$40 per tonne, and fixed costs will not change. What sales volume in dollars
would be
required to earn a net profit of \$94 500?
Question 5 – Activity-Based Costing and Management
Opsonin Inc manufactures three products for the pharmaceuticals industry:
• product P : annual sales, 8000 units.
• product Q : annual sales, 15 000 units.
• product R : annual sales, 4000 units.
The company uses a traditional, volume-based product costing system with
manufacturing
overhead applied on the basis of direct labour dollars. The product costs have
been calculated
Machinery
\$1 225 000
Machine setup
5 250
Inspection
525 000
Material handling
875 000
Engineering
344 750
Total
\$2 975 000
Opsonin Inc's pricing method has been to set a budgeted setting price equal
to 150 per cent of full product cost. However, only the Product Q have been
selling at their budgeted price. The budgeted and actual current prices for all
three products are the following: Opsonin Inc has been forced to lower the price of product P in order to get
orders. In contrast,
Opsonin Inc has raised the price of product R several times, but there has
been no apparent loss of sales. Opsonin Inc has been under increasing
pressure to reduce the price even further on gismos. In contrast, Opsonin Inc's
competitors do not seem to be interested in the market for Product R. Opsonin
Inc apparently has this market to itself.
Required:
1. Is product P the company's least profitable product? Explain your answer.
2. Is product R a profitable product for Opsonin Ltd? Explain your answer.
3. Comment on the reactions of Opsonin Inc's competitors to the firm's pricing
strategy. What
dangers does Opsonin Inc face?
4. Opsonin Inc's financial controller, Obrien Sally, recently attended a
conference at which
activity-based costing systems were discussed. She became convinced that
such a system
would help Opsonin Inc's management to understand its product costs better. She obtained
top management's approval to design an activity-based costing system, and
an ABC project
team was formed. In Stage 1 of the ABC project, each of the overhead items
listed in the
overhead budget was placed into its own activity cost pool. Then an activity
driver was
identified for each activity cost pool. Finally, the ABC project team compiled
data showing
the percentage of each activity driver that was consumed by each of Opsonin
Inc's product
lines. These data are summarised as follows: Show how the financial controller determined the percentages given above for
raw material
costs. (Round to the nearest whole per cent.)
5. Develop product costs for the three products on the basis of a simple
activity-based product costing system. (Round to the nearest cent.)
6. Calculate a budgeted price for each product, using Opsonin Inc's pricing
formula. Compare
the new budgeted prices with the current actual selling prices and previously
reported
product costs.
7. Refer to the new budgeted prices for Opsonin Inc's three products, based
on the new activitybased costing system. Write a memo to the company
managing director commenting on the situation Opsonin Inc has been facing
regarding the market for its products and the actions of its competitors.
Discuss the strategic options available to management. What do you
recommend, and why?
8. Refer to the product costs developed in requirement 5. Prepare a table
showing how Opsonin Inc's traditional, volume-based product costing system
distorts the product costs of product P, Q and R.
Solution details:
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Tutor | 1,683 | 7,363 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2021-21 | latest | en | 0.900047 |
https://graph-inequality.com/graph-inequalities/function-domain/algebra-ii-math-cheat-test.html | 1,558,467,014,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232256546.11/warc/CC-MAIN-20190521182616-20190521204616-00229.warc.gz | 499,613,403 | 10,912 | Try the Free Math Solver or Scroll down to Tutorials!
Depdendent Variable
Number of equations to solve: 23456789
Equ. #1:
Equ. #2:
Equ. #3:
Equ. #4:
Equ. #5:
Equ. #6:
Equ. #7:
Equ. #8:
Equ. #9:
Solve for:
Dependent Variable
Number of inequalities to solve: 23456789
Ineq. #1:
Ineq. #2:
Ineq. #3:
Ineq. #4:
Ineq. #5:
Ineq. #6:
Ineq. #7:
Ineq. #8:
Ineq. #9:
Solve for:
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• {searchTerms} | 984 | 4,165 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2019-22 | latest | en | 0.915345 |
http://www.romannumerals.co/numerals-converter/dcclxxxvi-in-numbers/ | 1,721,087,760,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514724.0/warc/CC-MAIN-20240715224905-20240716014905-00175.warc.gz | 57,696,986 | 14,061 | ## What number is "DCCLXXXVI"?
### A: 786
DCCLXXXVI = 786
Your question is, "What is DCCLXXXVI in Numbers?". The answer is '786'. Here we will explain how to convert, write and read the Roman numeral letters DCCLXXXVI in the correct Arabic number translation.
## How is DCCLXXXVI converted to numbers?
To convert DCCLXXXVI to numbers the translation involves breaking the numeral into place values (ones, tens, hundreds, thousands), like this:
Place ValueNumberRoman Numeral
Conversion700 + 80 + 6DCC + LXXX + VI
Hundreds700DCC
Tens80LXXX
Ones6VI
## How is DCCLXXXVI written in numbers?
To write DCCLXXXVI as numbers correctly you combine the converted roman numerals together. The highest numerals should always precede the lower numerals to provide you the correct written translation, like in the table above.
700+80+6 = (DCCLXXXVI) = 786
## More from Roman Numerals.co
DCCLXXXVII
Now you know the translation for Roman numeral DCCLXXXVI into numbers, see the next numeral to learn how it is conveted to numbers.
Convert another Roman numeral in to Arabic numbers. | 274 | 1,080 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.4375 | 3 | CC-MAIN-2024-30 | latest | en | 0.799858 |
https://blog.csdn.net/whitesun123/article/details/79972696 | 1,547,634,345,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583657151.48/warc/CC-MAIN-20190116093643-20190116115643-00369.warc.gz | 429,411,606 | 30,455 | # Implement strStr
输入: haystack = "hello", needle = "ll"
public int strStr_2(String haystack, String needle) {
if(haystack.length() < needle.length())
return -1;
if(needle.length() == 0)
return 0;
int h = haystack.length(),n = needle.length();
for(int i = 0; i <= h-n; i++){//不需要遍历完,因为到后面长度就不符合了
int j = 0;
for(j = 0; j < n; j++){
if(haystack.charAt(i+j) != needle.charAt(j))
break;
}
System.out.println(j);
if(j == n)
return i;
}
return -1;
}
public int strStr_3(String haystack, String needle) {
if(haystack.length() < needle.length())
return -1;
if(needle.length() == 0)
return 0;
int h = haystack.length(),n = needle.length();
for(int i = 0; i <= h-n; i++){
if(haystack.substring(i, i+n).equals(needle)){
return i;
}
}
return -1;
}
private void GetNext(String needle,int next[])
{
int pLen = needle.length();
next[0] = -1;
int k = -1;
int j = 0;
while (j < pLen - 1)
{
//p[k]表示前缀，p[j]表示后缀
if (k == -1 || needle.charAt(j) == needle.charAt(k))
{
++k;
++j;
next[j] = k;
}
else
{
k = next[k];
}
}
}
public int KmpSearch(String haystack, String needle) //kmp
{
if(haystack.length() < needle.length())
return -1;
if(needle.length() == 0)
return 0;
int i = 0;
int j = 0;
int[] next = new int[needle.length()];
GetNext(needle, next);
int sLen = haystack.length();
int pLen = needle.length();
while (i < sLen && j < pLen)
{
//①如果j = -1，或者当前字符匹配成功（即S[i] == P[j]），都令i++，j++
if (j == -1 || haystack.charAt(i) == needle.charAt(j))
{
i++;
j++;
}
else
{
//②如果j != -1，且当前字符匹配失败（即S[i] != P[j]），则令 i 不变，j = next[j]
//next[j]即为j所对应的next值
j = next[j];
}
}
if (j == pLen)
return i - j;
else
return -1;
}
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120 | 609 | 1,637 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2019-04 | latest | en | 0.235266 |
https://www.usu.edu/physics/assessments/applied-professional-emphasis/course_objective_mapping | 1,713,149,343,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296816939.51/warc/CC-MAIN-20240415014252-20240415044252-00370.warc.gz | 983,713,526 | 8,350 | #### Relevant Courses (PHYS)
Physics Knowledge
Demonstrate understanding of how science and physics work in practice. 1020, 1040, 1080, 1100, 1200, 1800, 2110, 2120, 2210, 2310, 2220, 2320, 2215, 2225, 3870, 3880
Explain how experimental evidence can falsify scientific hypotheses and how it can contribute to acceptance of scientific concepts. 1040, 1200, 1800, 2110, 2120, 2210, 2310, 2220, 2320, 2215, 2225, 2710, 3710
Distinguish physics from other sciences by explaining the differences in focus on subject matter, kinds of questions, kinds of explanations, and techniques. 1200, 2110, 2120, 2210, 2310, 2220, 2320
Identify main points of scientific ethics and responsibility relating to laboratory practice, work with students and collaborators, co-authorship, publication and public advocacy. 2215, 2225, 4900
Explain how science is a community effort and argue both the necessity of scientific cooperation and the advantages and disadvantages of solitary science. 1020, 1200, 2110, 2120, 2210, 2310, 2220, 2320, 2710, 3710
Identify and relate the major historical threads in the development of physics. Identify major contemporary issues in physics and a range of applications of physics in today’s economy. 1020, 1200, 2110, 2120, 2210, 2310, 2220, 2320, 2710, 3710
Solve correctly algebraic and calculus problems from typical bachelor’s degree physics texts. 2110, 2120, 2210, 2310, 2220, 2320, 2710
Interpret the meaning of the mathematics that occurs in a particular physics context from typical bachelor’s degree physics texts. 2110, 2120, 2210, 2310, 2220, 2320
Estimate orders of magnitude of physics quantities; estimate orders of magnitude of solutions to physics problems; explain how to identify quickly whether a problem solution or other physics quantity is of reasonable magnitude. 1800, 2110, 2120, 2210, 2310, 2220, 2320, 3700
Graph related physics quantities in ways that illuminate underlying physical interpretations; interpret graphs from typical bachelor’s degree physics texts. 1800, 2110, 2120, 2210, 2310, 2220, 2320, 3700
Build and work with mathematical models. 2110, 2120, 2210, 2310, 2220, 2320, 3600, 3700, 4600, 4710, 4720, 4900
Give examples of physics problems with similar mathematics but different physics. 2110, 2120, 2210, 2310, 2220, 2320, 3600, 3700, 4600, 4710, 4720 , 4900
Organize a problem from a typical bachelor’s degree physics text by identifying the relevant physics principles, identifying relevant vs. irrelevant quantities, and making appropriate diagrams. 2110, 2120, 2210, 2310, 2220, 2320
Organize quantitative information in a problem from a typical bachelor’s degree physics text by clearly stepping through the mathematics of the problem solution. 2110, 2120, 2210, 2310, 2220, 2320
Understanding and application of the fundamental notions of force and energy. 1020, 1200, 1800, 2110, 2210, 2310
Understanding of the use of energy considerations to study complex systems from a thermodynamic viewpoint. 2110, 2210, 2310, 3700
Understanding the fundamental interactions of nature, principally electromagnetism. 1200, 1800, 2120, 2220, 2320, 3600, 3710, 4600
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Understanding of the myriad phenomena of light, its propagation, and its interaction with various physical media. 1200, 2120, 2220, 2320, 4600, 4650, 4680
Laboratory and Computer Skills
Follow practices necessary for safety in using undergraduate research or teaching laboratory equipment. Explain these practices to others, including identifying both potential dangers and ethical issues. Suggest how safety could be improved in a particular undergraduate research or teaching laboratory. 2110, 2120, 2215, 2225, 3870, 3880, 4900
Carry out error analysis on laboratory data; explain what the errors mean for data interpretation. 2215, 2225, 3870, 3880, 4900
Evaluate the quality of laboratory data; explain the importance of such evaluation. 3870, 3880, 4900
Design a laboratory measurement to answer a physics question on the level of typical bachelor’s degree physics texts. 4900
Analyze the connections between what one measures and how one infers the physics interpretation of the measurements. 2110, 2120, 2215, 2225, 3870, 3880, 4900
Outline ethical laboratory practices and make arguments for their importance. Include ethics of reporting laboratory procedures and results as well as ethical practices in carrying out an experiment and reporting data. 4900
Apply critical analysis of the generation and collection of data to computer experiments. 4900
Research and Communication
Demonstrate physics research skills and understanding. 4900
Demonstrate the ability to communicate about science. 3870, 3880, 4900
Outline ethical research practices 3870, 3880, 4900 | 1,445 | 5,211 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2024-18 | latest | en | 0.762785 |
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# There are three states of matter;Solids, gases, and liquids
Sep 29, 2009 #1
i know i am really horrible writter....:(
And I also need help for making good last sentence.
------------------------------------------------------------ ----------
States of Matter
I went to Great America with my sister on Sunday, and she bought me an ice cream. It was quite hot day, and my ice cream started to melt slowly. My cold ice cream (solid) melted and it turned into a liquid.
There are three states of matter: solids, liquids, and gases. Our world is filled with substances which could be classified as liquids, gases, and solids. For example, air is a gas, your pencil case is a solid, and water is a liquid. Solids, gases, and liquids can be classified by their volume, shape, or elements, compound, or mixture.
A solid is a substance that keeps its shape and volume. For example, ice, rock, and diamond are solids. It has a distinct shape and volume, and also the particles in a solid are very close, and the atoms in a solid are not allowed to move too much. For this reason, a solid has a distinct shape and volume.
There are two types of solids: crystalline solids and amorphous solids. There are many solids that the particles form a regular, repeating pattern. This means that the atoms in these solids are arranged in an orderly manner. You can easily find the examples of crystalline solids, for example, sugar, salt, snow and ice .Also, crystalline solids are melted at a certain temperature .
Another type of solids- amorphous solids- does not have repeating or regular pattern, and it is not melted at a specific temperature. Plastics, rubber, and glass are the examples of amorphous solids.
One of the states of matter is a liquid. Unlike a solid, a liquid does not have certain shape , but it has definite volume, because its particles are allowed to move freely. A liquid takes the shape of its container.
Surface tension is one of characteristic property of liquid, which is the 'result of an inward pull among the molecules of a liquid that brings the molecules on the surface closer together'. (Physical Science, page74) Another property of a liquid is called, 'viscosity' which means 'a liquid's resistance to flowing' (Physical Science, page74) It depends of the size and the attract
of the particles. A liquid with high viscosity flows slowly, for example, honey.
As I have mentioned, the world is filled with states of matter. The last state of matter is gases. There is one thing you do every day, even though you do not really recognize. Yes, you do breathe in and out the air every day. A gas does not have definite shape, but unlike a liquid , it changes its volume very easily because the ' particles of a gas can be pressed and squeezed into a small volume'. (Physical Science, page75) As a gas moves, its particles fill the whole space available.
Therefore, a gas does not have definite volume and shape.
I've been talking about three states of matter; liquids, solids, and gases. But here is another question. Are there any other states of matter? How about plasma? Can't plasma be a one of the states of matter? Science keeps giving us questions. Now, it's your turn to answer my question!
Sep 30, 2009 #2
This is not bad at all for an eighth grade essay! I like the ending very much.
The first thing you must do is stop calling yourself a bad writer. | 774 | 3,468 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2017-39 | latest | en | 0.955373 |
https://www.physicsforums.com/threads/initial-and-final-energy-problem.810742/ | 1,508,274,959,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187822488.34/warc/CC-MAIN-20171017200905-20171017220905-00179.warc.gz | 986,974,995 | 16,365 | # Initial and Final Energy problem
1. Apr 26, 2015
### YamiBustamante
1. The problem statement, all variables and given/known data
A 2.4 kg piece of wood slides on the surface shown in the figure . The curved sides are perfectly smooth, but the rough horizontal bottom is 31m long and has a kinetic friction coefficient of 0.27 with the wood. The piece of wood starts from rest 4.0m above the rough bottom.
a) Where will this wood eventually come to rest?
b) For the motion from the initial release until the piece of wood comes to rest, what is the total amount of work done by friction?
2. Relevant equations
E_k = (1/2)mv^2
E_f = mg(u_k)d
E_g = mgh
3. The attempt at a solution
I think I have an idea for part a but I'm not sure. Would this be correct?
Initial Energy = Final Energy
Potential Gravitational Energy = Friction Energy
mgh = Ugdm
(2.4kg)(9.8m/s^2)(4.0m) = 0.27(9.8m/s^2)(2.4kg)d
And then I figured that I would solved for d...
Would that be correct?
And for part b would I use the work kinetic energy theorem even though it has friction?
#### Attached Files:
• ###### 869d1c00-750f-4ceb-acf9-02091ff0b81a.jpeg
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2. Apr 26, 2015
### paisiello2
I think you got it. | 367 | 1,220 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2017-43 | longest | en | 0.904395 |
https://www.assignmentexpert.com/homework-answers/physics/mechanics-relativity/question-126774 | 1,675,713,301,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500357.3/warc/CC-MAIN-20230206181343-20230206211343-00315.warc.gz | 644,314,276 | 66,495 | 107 149
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# Answer to Question #126774 in Mechanics | Relativity for Manish deore
Question #126774
consider a cylinder of a square section 10cm side which strouhal number is about 0.13 it has mass of 1 kgand mounted on a spring of stiffness 1 N/m give value of flow velocity at which viv occur in m/s
1
2020-07-21T12:38:06-0400
"Sr=\\frac{f\\cdot l}{v}"
"f=\\frac{1}{2\\pi\\sqrt{m\/k}}=\\frac{1}{2\\pi\\sqrt{1\/1}}=0.16Hz"
"v=\\frac{f\\cdot l}{Sr}=\\frac{0.16\\cdot 0.1}{0.13}=0.12m\/s"
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APPROVED BY CLIENTS | 253 | 743 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2023-06 | latest | en | 0.812818 |
https://mathematica.stackexchange.com/questions/98679/how-to-solve-or-plot-roots-of-the-equation-involves-bessel-function-of-first-and | 1,632,483,925,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057524.58/warc/CC-MAIN-20210924110455-20210924140455-00448.warc.gz | 420,879,709 | 39,939 | # How to solve or plot roots of the equation involves Bessel function of first and second kind?
Here is my equation
x^2 + BesselJ[m,k*x^2]*x + k*BesselK[m,k]==0.
I would like to solve this equation for different initial guesses of x_0 like x_0 = 1,2 etc, where m is a constant, and is 2, (m =2) and k is a variable in the range of 1 to 5.
Here I want to plot the roots of x over a range of k (from 1 to 5).
• Check out FindRoot Nov 5 '15 at 7:16
• The answers below both note the lack of real roots. There may be complex roots, but you'll need to do some more work... but, how exactly did you encounter this function? Nov 5 '15 at 8:18
If you plot the left-hand part of your equation for m=2 and play with the kvalues:
Manipulate[
Plot[x^2 + BesselJ[m, k*x^2]*x + k*BesselK[m, k], {x, -5, 5},
PlotRange -> {0, pr}],
{k, 0.1, 5}, {pr, 0.1, 5}]
you will see something like this
and playing with the PlotRange fixed by pryou will see that the equation is likely to have no solutions, at least in the range of parameters I have chosen. So the question to answer here is, if this equation has any solution at all.
Have fun!
• It is an equation which we arrived after solving few equations and we are looking for its solutions. I am not interested in using manipulating solution plot, I just want to find a root by giving an initial guess to x. Nov 5 '15 at 8:49
• @Vemula Ramakrishna Reddy You probably did not understand me. It is not manipulation that I demonstrate, but manipulation helps one to make sure that your equation likely has no solutions. Or at least I would make a special search for conditions at which such a solution might exist. Nov 5 '15 at 10:12
• Dear Alexei Boulbitch, thank you for your valuable help. I just want to ask you one thing can I collect roots of this equation, hope it has roots. Can you tell me how to solve this one using NSolve numerically Nov 6 '15 at 7:34
• @Vemula Ramakrishna Reddy NSolve has nothing to do with your equation. You may use FindRoot to look for numerical solutions. The latter, however, requires an initial guess as an input. To give such, it would be useful to know, where approximately lies the root. For that you typically plot the left-hand part of the equation and see where does it cross the x axis. That's what I did above. As I see at k<=5 it does not. I would check for larger k at your place. Further, I see that the point of minimum of the left-hand part is at r=0. See the continuation Nov 6 '15 at 8:14
• Continuation: Then I would check at which k this point touches the Ox axis. This yields the following equation: D[x^2 + BesselJ[m, k*x^2]*x + k*BesselK[m, k], x] == 0 // FullSimplify . It should be solved together with your original one. Usually, this is easier than to solve the original one. Judging by eye, the minimum point is at x=0. If so, you do not need to use the second equation, but simply substitute x=0 into the first one. This yields k BesselK[2, k] which is positive at any k. So this also tells me that there is likely to be no solution. Nov 6 '15 at 8:23
By varying the controls, from the Plot you can see that there are no real roots in the interval.
Manipulate[
f[m_, k_, x_] = x^2 + BesselJ[m, k*x^2] x + k BesselK[m, k];
Column[{
NMinimize[{f[m, k, x], 0 <= x <= 5}, x],
Plot[f[m, k, x], {x, 0, 5}]}],
{{m, 2}, Range[0, 5]},
{{k, 5}, 1, 5, Appearance -> "Labeled"}] | 973 | 3,376 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2021-39 | latest | en | 0.950175 |
https://stat.ethz.ch/pipermail/r-help/2014-June/375579.html | 1,582,837,310,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875146809.98/warc/CC-MAIN-20200227191150-20200227221150-00330.warc.gz | 563,404,691 | 2,099 | # [R] finding average of entire rows to equal values in a vector
arun smartpink111 at yahoo.com
Sat Jun 7 10:39:10 CEST 2014
```Hi,
This is not clear.
If this is a combination of rows using a specific formula as you showed, use ?combn
dat <- read.table(text="0.7, 0.3, 0.6, 0.9
1.0, 0.1, 0.4, 0.7
1.2, 0.8, 0.3, 0.1",sep=",",header=FALSE)
indx <- combn(seq(dim(dat)[1]), 2)
vec1 <- c(0.86, 0.46, 0.5, 0.63)
indx1 <- sapply(seq_len(ncol(indx)), function(i) {
ind <- indx[, i]
val <- unlist(round((2 * dat[ind[1], ] + dat[ind[2], ])/3, 2))
tol <- 0.015
all(abs(val - vec1) <= tol)
})
dat[indx[, indx1], ]
# V1 V2 V3 V4
#1 0.7 0.3 0.6 0.9
#3 1.2 0.8 0.3 0.1
A.K.
Hi,
I'm looking for a way to find the average of several rows, of a 4 x 7 matrix, that will eventually reach a fixed set of values in a vector while always averaging the entire row from the 4x7 to get there. So... in a matrix of 4 columns and 7 rows with values like
0.7, 0.3, 0.6, 0.9
1.0, 0.1, 0.4, 0.7
1.2, 0.8, 0.3, 0.1
...
..
.
how can i write it to find the possible combinations of rows 1, 2, and 3 so that it matches the vector: 0.86, 0.46, 0.5, 0.63 with some small margin of error?
in this case the answer is (2x(row 1)+(row 3))/3
best,
-sunny0
``` | 517 | 1,235 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2020-10 | latest | en | 0.655594 |
https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Laws+of+Motion&q_type=&q_topic=Conservation+of+Momentum+&q_category=&question_id=PHENJE11155357 | 1,548,288,280,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547584431529.98/warc/CC-MAIN-20190123234228-20190124020228-00411.warc.gz | 996,326,088 | 15,584 | A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms−1.The magnitude of its momentum is recorded as from Physics Laws of Motion Class 11 Manipur Board
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A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms−1.The magnitude of its momentum is recorded as
• 17.6 kg ms−1
• 17.565 kg ms−1
• 17.56 kg ms−1
• 17.57 kg ms−1
A.
17.6 kg ms−1
P = mv = 3.513 × 5.00 ≈ 17.6
P = mv = 3.513 × 5.00 ≈ 17.6
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1233 Views
What is Aristotle’s law of motion?
Aristotle’s law of motion states that an external force is required to keep the body in motion.
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When the branches of an apple tree are shaken, the apples fall down. Why?
The apple fall from an apple tree when it shaken because of inertia of rest. Apple is in a state of rest and when the tree is suddenly shaken, apples still tends to remain in it's same state of rest whereas branches move.
So, the apples fall down.
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Define inertia.
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1332 Views | 449 | 1,673 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2019-04 | latest | en | 0.902483 |
https://finance.yahoo.com/news/good-terex-corporation-nyse-tex-121707208.html | 1,571,103,938,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986655735.13/warc/CC-MAIN-20191015005905-20191015033405-00385.warc.gz | 470,019,829 | 105,569 | U.S. Markets closed
# How Good Is Terex Corporation (NYSE:TEX), When It Comes To ROE?
One of the best investments we can make is in our own knowledge and skill set. With that in mind, this article will work through how we can use Return On Equity (ROE) to better understand a business. We'll use ROE to examine Terex Corporation (NYSE:TEX), by way of a worked example.
Over the last twelve months Terex has recorded a ROE of 11%. One way to conceptualize this, is that for each \$1 of shareholders' equity it has, the company made \$0.11 in profit.
### How Do You Calculate ROE?
The formula for ROE is:
Return on Equity = Net Profit ÷ Shareholders' Equity
Or for Terex:
11% = US\$97m ÷ US\$861m (Based on the trailing twelve months to June 2019.)
Most readers would understand what net profit is, but it’s worth explaining the concept of shareholders’ equity. It is the capital paid in by shareholders, plus any retained earnings. Shareholders' equity can be calculated by subtracting the total liabilities of the company from the total assets of the company.
### What Does Return On Equity Mean?
ROE measures a company's profitability against the profit it retains, and any outside investments. The 'return' is the amount earned after tax over the last twelve months. The higher the ROE, the more profit the company is making. So, all else being equal, a high ROE is better than a low one. That means ROE can be used to compare two businesses.
### Does Terex Have A Good ROE?
By comparing a company's ROE with its industry average, we can get a quick measure of how good it is. However, this method is only useful as a rough check, because companies do differ quite a bit within the same industry classification. If you look at the image below, you can see Terex has a similar ROE to the average in the Machinery industry classification (14%).
That's not overly surprising. ROE tells us about the quality of the business, but it does not give us much of an idea if the share price is cheap. For those who like to find winning investments this free list of growing companies with recent insider purchasing, could be just the ticket.
### Why You Should Consider Debt When Looking At ROE
Companies usually need to invest money to grow their profits. The cash for investment can come from prior year profits (retained earnings), issuing new shares, or borrowing. In the case of the first and second options, the ROE will reflect this use of cash, for growth. In the latter case, the use of debt will improve the returns, but will not change the equity. That will make the ROE look better than if no debt was used.
### Terex's Debt And Its 11% ROE
Terex clearly uses a significant amount of debt to boost returns, as it has a debt to equity ratio of 1.57. Its ROE is quite good but, it would have probably been lower without the use of debt. Investors should think carefully about how a company might perform if it was unable to borrow so easily, because credit markets do change over time.
### The Bottom Line On ROE
Return on equity is one way we can compare the business quality of different companies. A company that can achieve a high return on equity without debt could be considered a high quality business. If two companies have the same ROE, then I would generally prefer the one with less debt.
But when a business is high quality, the market often bids it up to a price that reflects this. The rate at which profits are likely to grow, relative to the expectations of profit growth reflected in the current price, must be considered, too. So you might want to take a peek at this data-rich interactive graph of forecasts for the company.
Of course, you might find a fantastic investment by looking elsewhere. So take a peek at this free list of interesting companies.
We aim to bring you long-term focused research analysis driven by fundamental data. Note that our analysis may not factor in the latest price-sensitive company announcements or qualitative material.
If you spot an error that warrants correction, please contact the editor at editorial-team@simplywallst.com. This article by Simply Wall St is general in nature. It does not constitute a recommendation to buy or sell any stock, and does not take account of your objectives, or your financial situation. Simply Wall St has no position in the stocks mentioned. Thank you for reading. | 952 | 4,385 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2019-43 | latest | en | 0.95105 |
https://biology.stackexchange.com/questions/46343/mechanism-by-which-water-flows-through-xylem/48412 | 1,632,396,596,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057421.82/warc/CC-MAIN-20210923104706-20210923134706-00176.warc.gz | 186,785,088 | 37,958 | # Mechanism by which water flows through xylem
I was doing a Cambridge iGCSE past paper when I came across the question:
Describe the mechanism by which water flows through the xylem
I thought the correct answer would revolve around the xylem being composed of dead cells, thus allowing water to use capillary action to flow upwards, but it wasn't.
Instead, the mark scheme referred only to transpiration causing tension and so forth, not even mentioning capillary action.
Is capillary action not a valid answer? And if not, why not?
Thanks
- NO,for capillary action 2 things are of importance 1) the radius of capillary tube[R] ,2)the tension force between molecules at surface of liquid [T] (for water=72.8 dynes/cm= 728 X 10^-6 N)
- height to which the fluid will rise due to cap.effect, H=(2T cosα)÷(ρgR)
- [ρ=density of liq,here water =1000kg/m³]
- numer.=constant(for given liq) denomin.=variable R
- so if capillary was in effect for trees 100s of meters tall the deno. wud have to be small while the ρg is a fixed value here for water = 10000 units the R wud have to be impossibly small
• WHAT HAPPENS IS the transpiration of water droplets from the stomata causes a suction as no air can enter to fill in for the water lost the water as to rise up taking more of it from the soil | 335 | 1,300 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2021-39 | latest | en | 0.931014 |
https://www.physicsforums.com/threads/average-crush-force-exerted-on-a-car.295038/ | 1,521,772,151,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257648113.87/warc/CC-MAIN-20180323004957-20180323024957-00450.warc.gz | 854,652,766 | 14,259 | # Average crush force exerted on a car
1. Feb 24, 2009
### Mathbather
1. The problem statement, all variables and given/known data
2. Relevant equations
Kinetic energy = Ke = (1/2)(mv^2) (joules - J)
where m = mass (kg)
v = velocity (m/s)
3. The attempt at a solution
What I was thinking was that the kinetic energy is divided uniformly over the distance d and hence
Average crush force = mv^2 / 2d
But I don't know how to prove that. Of course I could try to explain it with words, but I don't feel that it is enough.
2. Feb 24, 2009
### Delphi51
All of the KE is converted to work in crushing things: W = Fd = 1/2*m*v^2
3. Feb 24, 2009 | 201 | 647 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2018-13 | longest | en | 0.956572 |
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# Math Nation Geometry Answer Key Section 8
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# Trivia: What's the lowest possible GMAT score?
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Trivia: What's the lowest possible GMAT score? [#permalink]
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27 Nov 2006, 21:39
Can you still get 200 if you took a nap and didn't answer any question during the test?
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28 Nov 2006, 03:29
As I understand it, 200 is the minimum. According to the most recent official score report I have seen, 0% of test-takers scored between 200-220 and 1% scored 230-250.
28 Nov 2006, 03:29
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# Trivia: What's the lowest possible GMAT score?
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Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 609 | 2,155 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2017-09 | longest | en | 0.89849 |
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Can someone explain what's happening in this image?
How come when it's covered quickly, the fire still burns, but when it's covered slowly the fire is extinguished?
Note by Calvin Lin
5 years, 9 months ago
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Smothering an oil fire in a frying pan is not something that works best in haste. Firemen routinely give demonstrations on how it's supposed to be done, and one of the worst ways to do is is to "slap" the top on. Slow is better. Now, the physics: The ignition temperature of cooking oil can be as low as 300 to 350 degrees, which is easily reached with a frying pan on the stove. Hence even if you were smother the oil fire, if the temperature is not allowed to drop, it can spontaneously re-ignite.
Let's look at the video. The timing of it appears to have been perfected by a magician for effect. Slapping the lid on top doesn't hardly even give the fire enough time to use up all the oxygen when the lid is slapped on, so when the lid is pulled off suddenly, the fire billows out to great effect! Then he "carefully smothers" the fire by sliding the lid over it, so that apparently putting out the fire seems successful. But of course, the video is cut short (after which it loops) before it shows the moment the oil spontaneously re-igniting. A longer time is needed to let the oil cool down below ignition temperature.
In contrast, even paper has a far higher ignition temperature, so that once a paper fire is smothered, and there are no embers, it's quite difficult to make it re-ignite spontaneously. Here's where "one's intuitive experience with ordinary burning paper" serves poorly in dealing with cooking oil fires.
- 5 years, 9 months ago
But of course, the video is cut short (after which it loops) before it shows the moment the oil spontaneously re-igniting.
I don't think the fire re-ignites. Here is what seems to be the source video for the gif.
We can see that the fire does stay out for 5 seconds before the video ends. Also, it seems to be a firefighting demonstration (we can see a fire extinguisher in the back and the guy seems to be wearing a firefighting uniform). So it seems unlikely this is a trick.
- 5 years, 9 months ago
True. The frying pan is on a table, not on the stove.
Nevertheless, never trust an oil fire not to spontaneously re-ignite. Like a trick birthday cake candle.
- 5 years, 9 months ago
As for me this sounds like a clearly performed trick. When the lid is closed suddenly,it cuts off the oxygen supply only for a few fractions of second but when the lid slides over slowly, it prevents the supply for few seconds. As a result the fire sets off and it is once turned off it cannot revive again.
- 5 years, 9 months ago
Yes, that is related to what's happening. As Michael said
Slapping the lid on top doesn't hardly even give the fire enough time to use up all the oxygen
Though, if you consider the "slow close" scenario, the supply was completely cut off only for a split second. It's about the amount of oxygen that is left in the pan that can fuel the fire.
Staff - 5 years, 9 months ago
When it is covered quickly the time period of covering fire is reduced due to which the gas such as carbon dioxide is not produced in that much amount to extinguish but when dragged slowly produces such gasaes which extinguis fire by cutting supply of oxygen
- 5 years, 9 months ago
There's is very less oxygen in the pan, When the red man put the pan onto the fire,fire extinguishes due to the sudden loss of contact from air(oxygen).
- 5 years, 9 months ago
How come when it's covered quickly, the fire still burns, but when it's covered slowly the fire is extinguished?
Staff - 5 years, 9 months ago
Even I had first attempted that and then I realized that its not what we are requiring.
- 5 years, 9 months ago
Because...u fast forward and repeated the clip when he is about to cover up the lid...?
- 5 years, 9 months ago | 1,394 | 5,584 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2021-31 | latest | en | 0.884202 |
https://ecologyinfo.com/how-much-electricity-does-a-top-loader-washing-machine-use | 1,685,992,576,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224652161.52/warc/CC-MAIN-20230605185809-20230605215809-00718.warc.gz | 257,927,925 | 8,938 | # How much electricity does a top loader washing machine use?
8
Date created: Thu, Apr 29, 2021 1:27 AM
Date updated: Wed, Sep 7, 2022 1:32 PM
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## Top best answers to the question «How much electricity does a top loader washing machine use»
Non-ENERGY STAR top-loading clothes washing machines can use between 400 and 1,300 watts. Newer ENERGY STAR clothes washing machines use an average of 500 watts. One common and overlooked fact is that roughly 90% of the energy used to clean our clothes with washing machines is to heat water.
Washing machine: around 16p per one hour wash On average, a 6 litre load washing machine takes around 1 unit of electricity – roughly about 16p – to run an hour-long wash. Does front loader use less water?
Older top loader washing machines generally use more water and energy, this is why more modern and energy efficient designs are front loaders. A washing machine will use 400 to 1300 watts, with modern Energy Star rated models using about 500 watts.
3) Even though most washing machines only use 50-150 Wh/load (from our measurements of a few models), it still might need quite a large inverter, say 1200W, to power it for the few minutes while it is in spin dry mode, etc. 4) The Fischer & Paykel Smart Drive top loaders use a DC type of motor which soft starts.
The average power consumption of a washing machine is 400 to 1300 watts but it also depends on the size of the machine and brand like Samsung washing, Bosch, whirlpool washing, LG washing machine, etc.
Your washing machine consumes 12,000W electricity every month. 1 electricity unit = 1 kilowatt of electricity used per hour. To calculate the kilowatts, divide the total amount of watts by 1000. 12000 W / 1000 = 12 Kwh.
Front-loaders use less water than top-loaders, and thus require less energy to heat it, but it's still around 85% of energy going to heat the water even in a front-loader. Here's how energy is used depending on the temperature selected: Price per load (electricity), based on water temperature. Wash/Rinse Setting.
Depending on the type, capacity, efficiency and cycle setting of your washing machine, it can cost anywhere from 5 cents to 67 cents per load of washing. This estimation assumes an electricity usage rate of 27.5/kWh and a gas usage rate of 2.07MJ (for homes with gas heaters). | 557 | 2,334 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2023-23 | latest | en | 0.906158 |
https://programmer.group/balanced-binary-tree.html | 1,652,863,783,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662521883.7/warc/CC-MAIN-20220518083841-20220518113841-00156.warc.gz | 545,251,378 | 7,368 | # Balanced binary tree
Keywords: data structure
In extreme cases, a binary search tree evolves into a tree with only one child on one side, such as a left or right child for each non-leaf. When searching, you need to traverse the tree to find the target value. Its fast search value is not reflected. If the search tree balances the height difference between left and right subtrees when it is built, If the height difference between left and right subtrees is less than 1, then its search efficiency is O(lgn). Balanced binary tree is such a tree. It has the following characteristics:
1. The height difference between left and right subtrees should not exceed 1
2. Left and right subtrees are balanced binary trees, and empty nodes consider them a balanced binary tree.
The key feature of balanced binary trees is that the height difference between left and right subtrees whose roots are any one of the nodes cannot exceed 1. Because of this feature limitation, when creating a balanced binary tree, it is necessary to rotate the tree to restore its balance. There are several specific cases of rotation:
1. LL type rotation (one-way right): In this case, the left subtree of the left child of a node causes the node to be unbalanced, and the node needs to be rotated to the right, as shown in the following figure:
As shown in the figure above, a node with a value of 3, when it has only one left child, the height difference between the left and right children is 1. When 1 is inserted again, the height difference between the left and right children becomes 2 and needs to be rotated to the right:
LL_Rotate(node):
leftNode=node.left;// Left child of node to rotate
leftRightNode=leftNode;// Right child of left child to rotate node
leftNode.right=node;// Update the left child node and make yourself the right node of the left child, that is, the right child of 2 in Figure 3 above
node.height=max(node.left.height,node.right.height)+1;// Update Height
leftNode.height=max(leftNode.left.height,leftNode.right.height)+1;// With the new root node height, the node node needs to be updated first...
return leftNode; // Return to the left child's place to replace yourself, e.g. in the picture above, 2 replaces 3
2. RR type rotation (one-way left): In this case, the right subtree of a node's right child causes the node to become unbalanced and needs to be rotated to the left, as shown in the following figure:
As shown above, because 3 inserts, the balance factor of root node 1 becomes 2, requiring 1 to rotate left in the opposite direction to LL:
RR_Rotate(node):
rightNode=node.right;// Get Right Child
rightLeftNode=rightNode.left;// Get the left child of the right child
rightNode.left=node;// Update Right Child
node.right=rightLeftNode;
node.height=max(node.left.height,node.right.height)+1;
rightNode.height=max(rightNode.left.height,right.right.height)+1;
; return rightNode; // Return
3. LR type rotation (first left, then right): In this case, the right subtree of a left child of a node causes the node to be unbalanced, so the left child of the node needs to be rotated left, then rotated right, as shown in the following figure:
As shown in the figure above, the left child of a node with a node value of 8, because the insertion of 6 causes an imbalance of 8 nodes, needs to rotate the left child 4 first, replace the position of 4 with 6, then rotate 8 with right, and replace 8 with 6:
Left_Right_Rotate(node):
node.left=LL_Rotate(node.left); // Turn left for left child
return RR_Rotate(node);// Turn yourself right
4. RL type rotation (first right, then left): In this case, the left subtree of the right child of a node causes the node to be unbalanced, so the right child of the node needs to be rotated right, and then left, as shown in the following figure:
As shown in the figure above, the balance factor of 4 is broken by adding a left child node 6 to its right child 7, which requires that node 7 be rotated right first and then left-rotated to 4, as opposed to LR, with the pseudocode as follows:
Right_Left_Rotate(node):
node.right=RR_Rotate(node.right);// Turn right for right child
return LL_Rotate(node);// Turn left on yourself
Getting the height of a node:
Get_Height(node):
if node==null:
return 0;
leftHeight=Get_Height(node.left);
rightHeight=Get_Height(node.right);
return max(leftHeight,rightHeight)+1;
Obtaining the balance factor, the balance factor of a node is determined by the height difference between left and right subtrees:
Get_Factor(node):
return Get_Height(node.left)-Get_Height(node.right);
The balance adjustment of nodes involves left, right, left and right rotation, right and left rotation:
balance(node):
if node==null
return null;
//Left subtree causes imbalance
if getFactor(node)==2 :
//Left child's left subtree causes imbalance
if getFactor(node.left)==1:
node=LL_Rotate(node)
else:
//Left child's right subtree causes imbalance
node=Left_Right_Rotate(node);
//Right subtree causes imbalance
else if getFactor(node)==-2:
The right child of the right subtree causes imbalance
if getFactor(node.right)==-1:
node=RR_Rotate(node);
else
//Right child's left subtree causes imbalance
node=Right_Left_Rotate(node);
//Return rotated nodes
return node;
Insertion of tree:
insert_val(node,val):
if node==null:
return new Node(val);
//Here is a recursive creation process
if val<node.val:
node.left=insert_val(node.left,val);
node.left.parent=node;
else
node.right=insert_val(node.right,val);
node.right.parent=node;
return balance(node);
Tree deletion operation, deletion operation is a bit complex, you need to find the node to replace your own, its method is to find the maximum value of the left subtree of the deleted node; If the node to be deleted does not have a left subtree, it needs to be found
The minimum value of the right subtree, if the node to be deleted is a leaf node, it can be deleted directly:
delete_node(node,val):
if node==null:
if val < node.val:
delete_node(node.left,val);
else if val > node.val:
delete_node(node.right,val);
else: //Target node found
//The target node is a leaf node
if node.left==null && node.right==null:
parent=node.parent;
//The node to be deleted is the root node
if parent==null:
root=null;
else if parent.left==node:
parent.left=null;
Other: //Delete Nodes
parent.right=null;
else if node.left!=null:
left=node.left
maxNode=left
while left!=null:
maxNode=right;
left=left.right;
node.val=maxNode.val
delete_node(node.left,node.val);
Other: //with node.left above!= Null, instead, changes the left to right
balance(node);// Adjust the balance of trees
The above AVL tree is built and analyzed. The key point is the balanced operation of nodes. Recursive operation is used when creating and deleting nodes. It is a design technique, as follows: Complete sample code:
package avl;
import java.util.ArrayList;
import java.util.List;
/**
* AVL Definition of tree
*/
public class AVLTree {
//root node
Node root=null;
/**
* Insert a new value
* @param val
* @return
*/
public AVLTree insertVal(int val){
if(root==null){
root=new Node(val);
}else{
insertVal(root,val);
}
return this;
}
/**
* Insertion of Nodes
* @param node
* @param val
* @return
*/
private Node insertVal(Node node,int val){
if(node==null){
node=new Node(val);
return node;
}
if(val<node.val){
Node left=insertVal(node.left,val);
node.left=left;
left.parent=node;
}else{
Node right=insertVal(node.right,val);
node.right=right;
right.parent=node;
}
node=balance(node);
return node;
}
/**
* Delete Node
* @param val
*/
public void deleteVal(int val){
deleteVal(root,val);
}
private void deleteVal(Node node,int val){
if(node==null){
return;
}else if(val<node.val){
deleteVal(node.left,val);
balance(node);
}else if(val>node.val){
deleteVal(node.right,val);
balance(node);
}else{
//Leaf node, delete directly
if(node.left==null && node.right==null){
Node parent=node.parent;
if(parent==null){
root=null;
}
if(parent.left==node){
parent.left=null;
}else{
parent.right=null;
}
}else{
//If the left subtree is not empty, find its largest successor node
if(node.left!=null){
Node left=node.left;
Node maxNode=left;
//Notice how this finds the largest successor node
while(left!=null){
maxNode=left;
left=left.right;
}
node.val=maxNode.val;
deleteVal(node.left,maxNode.val);
balance(node);
}else{
Node right=node.right;
Node maxNode=right;
while(right!=null){
maxNode=right;
right=right.left;
}
node.val=maxNode.val;
deleteVal(node.right,maxNode.val);
balance(node);
}
}
}
}
/**
* Action of balancing nodes
* @param node
* @return
*/
private Node balance(Node node){
if(node==null){
return null;
}
if(getFactor(node)==2){
if(getFactor(node.left)==1){
node= LL_Rotate(node);
}else{
node= LR_Rotate(node);
}
}else if(getFactor(node)==-2){
if(getFactor(node.right)==-1){
node= RR_Rotate(node);
}else{
node= RL_Rotate(node);
}
}
return node;
}
/**
* Get the height of the node
* @param node
* @return
*/
private int getHeight(Node node){
if(node==null){
return 0;
}
int left=getHeight(node.left);
int right=getHeight(node.right);
int max=Math.max(left,right);
return max+1;
}
/**
* Get the Balance Factor of Nodes
* @param node
* @return
*/
private int getFactor(Node node){
if(node==null){
return 0;
}
return getHeight(node.left)-getHeight(node.right);
}
/**
* First right, then left
* @param node
* @return
*/
private Node RL_Rotate(Node node){
Node right=LL_Rotate(node.right);
node.right=right;
right.parent=node;
return RR_Rotate(node);
}
/**
* Left then Right
* @param node
* @return
*/
private Node LR_Rotate(Node node){
Node left=RR_Rotate(node.left);
node.left=left;
left.parent=node;
return LL_Rotate(node);
}
/**
* One-way Left Rotation
* @param node
* @return
*/
private Node RR_Rotate(Node node){
Node right=node.right,parent=node.parent;
Node rightLeft=right.left;
right.left=node;
node.parent=right;
node.right=rightLeft;
if(rightLeft!=null){
rightLeft.parent=node;
}
right.parent=parent;
if(parent!=null){
if(parent.left==node){
parent.left=right;
}else{
parent.right=right;
}
}else{
root=right;
}
return right;
}
/**
* One-way Right Rotation
* @param node
* @return
*/
private Node LL_Rotate(Node node){
Node left=node.left,parent=node.parent;
Node leftRight=left.right;
left.right=node;
node.parent=left;
node.left=leftRight;
if(leftRight!=null){
leftRight.parent=node;
}
left.parent=parent;
if(parent!=null){
if(parent.left==node){
parent.left=left;
}else{
parent.right=left;
}
}else{
root=left;
}
return left;
}
/**
* Preorder traversal
* @param node
*/
public void preOrder(Node node){
if(node!=null){
System.out.print(node);
preOrder(node.left);
preOrder(node.right);
}
}
/**
* Intermediate traversal
* @param node
*/
public void inOrder(Node node){
if(node!=null){
inOrder(node.left);
System.out.print(node);
inOrder(node.right);
}
}
/**
* Post-order traversal
* @param node
*/
public void postOrder(Node node){
if(node!=null){
postOrder(node.left);
postOrder(node.right);
System.out.print(node);
}
}
/**
* Traversing trees layer by layer
*/
public void printByLevel(){
System.out.println("=========================");
List<Node> temp=new ArrayList<>();
if(root!=null){
}
while(temp.size()>0){
List<Node> nodes=new ArrayList<>();
temp.stream().forEach(obj-> {
System.out.print(obj);
if(obj.left!=null){
}
if(obj.right!=null){
}
});
System.out.println();
temp.clear();
}
}
public static void main(String[] args) {
AVLTree tree=new AVLTree();
tree.insertVal(1).insertVal(2).insertVal(3).insertVal(4).insertVal(5).insertVal(7).insertVal(6);
tree.printByLevel();
tree.deleteVal(6);
tree.printByLevel();
tree.deleteVal(4);
tree.printByLevel();
}
}
class Node{
public int val;
public Node left,right,parent;
public Node(int val){
this.val=val;
}
@Override
public String toString() {
return val+" ";
}
}
Posted by bben95 on Wed, 01 Dec 2021 00:31:56 -0800 | 2,878 | 11,865 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2022-21 | longest | en | 0.893325 |
http://icpc.njust.edu.cn/Problem/Zju/3487/ | 1,604,185,756,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107922463.87/warc/CC-MAIN-20201031211812-20201101001812-00589.warc.gz | 44,297,321 | 9,917 | # Ordinal Numbers
Time Limit: Java: 2000 ms / Others: 2000 ms
Memory Limit: Java: 65536 KB / Others: 65536 KB
## Description
Ordinal numbers refer to a position in a series. Common ordinals include zeroth, first, second, third, fourth and so on. Ordinals are not often written in words, they are written using digits and letters. An ordinal indicator is a sign adjacent to a numeral denoting that it is an ordinal number, rather than a cardinal number. In English, the suffixes -st (e.g. 21st), -nd (e.g. 22nd), -rd (e.g. 23rd), and -th (e.g. 24th) are used. The rules are as follows:
• If the tens digit of a number is 1, then write "th" after the number. For example: 13th, 19th, 112th, 9311th.
• If the tens digit is not equal to 1, then use "st" if the units digit is 1, "nd" if the units digit is 2, "rd" if the units digit is 3, and "th" otherwise: For example: 2nd, 7th, 20th, 23rd, 52nd, 135th, 301st.
## Input
There are multiple test cases. The first line of input is an integer T ≈ 1000 indicating the number of test cases.
Each test case consists of a cardinal number 0 ≤ n < 1,000,000,000.
## Output
For each test case, output the corresponding ordinal number.
## Sample Input
5
1
2
3
4
1024
## Sample Output
1st
2nd
3rd
4th
1024th
References
http://en.wikipedia.org/wiki/Names_of_numbers_in_English
http://en.wikipedia.org/wiki/Ordinal_number_(linguistics)
None
## Source
The 8th Zhejiang Provincial Collegiate Progr | 444 | 1,447 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2020-45 | latest | en | 0.828525 |
https://www.homeownershub.com/maintenance/how-large-a-copper-water-line-for-150-run-to-a-guest-house-529086-.htm | 1,516,328,873,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887692.13/warc/CC-MAIN-20180119010338-20180119030338-00328.warc.gz | 921,838,691 | 12,863 | # How large a copper water line for 150' run to a guest house? Pressure regulator?
My water pressure is 80 psi at the 1.5" line service entrance to the main house.
I want to run a branch line 150' to the guest house which has clothes washer, kitchen sink, bathroom, exterior hose bib. What size copper pipe should I use for this 150' run for good results?
What is a good brand of pressure regulator? I want to reduce the pressure to 50 lbs at the guest house entrance.
Thanks for any information.
Ted in So. Calif.
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
John Brown wrote:
I would think that 3/4" should handle it. However, I strongly suggest you check the local authorities before beginning as they may have specifications for this.
--
Joseph E. Meehan
26 + 6 = 1 It's Irish Math
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
You wont reduce the available pressure by using a smaller pipe, you will only reduce the available flow volume in Gallons per minute.
Static pressure will be at the 80 psi from the time you turn on the faucet until the flow reaches a constant volume, dependant on pipe friction loss and supply pressure--then and only then will it drop to a lower pressure at the other end.
IOW, you will turn on the faucet and get a big spurt at 80 psi for a few seconds, and then it will equalize at a lower pressure and flow.
To reduce pressure you would put a pressure regulator on the line, suggest at the end of the run.
--
SVL
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Rule of thumb pressure drops, PVC pipe (psi/100 ft): 1/2 3/4 1" 3 gpm 8.6 2.2 0.7 5 gpm 22.2 5.7 1.7 10 gpm 80.5 20.4 6.3
Copper pipe will be about 0.2 to 0.5 psi/100 ft more loss.
You could get by with 3/4" but I'd favor 1"
RB
PrecisionMachinisT wrote:
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
I would use 1 inch pipe and probably pvc schedule 40. I would not reduce the pressure by mechanical means unless your going to be using CPVC inside. The pressure and volume will be enough reduced by adding the 150' of pipe into the system.
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
hear in the us we or i should say most would use black pollyelean ken uhrick
wrote:
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
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# EngageNY
This collection contains Common Core Aligned materials including full curricula, units, lessons, and close reading activities developed by New York State Education Department (NYSED) to support educators in reaching the State’s vision for a college and career ready education for all students.
134 affiliated resources
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"Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Algebra
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
05/14/2013
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Module 2 builds on students' previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry. The heart of the module is the study of precise definitions of sine and cosine (as well as tangent and the co-functions) using transformational geometry from high school Geometry. This precision leads to a discussion of a mathematically natural unit of rotational measure, a radian, and students begin to build fluency with the values of the trigonometric functions in terms of radians. Students graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions. The study of the properties culminates in the proof of the Pythagorean identity and other trigonometric identities.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Algebra
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
08/15/2014
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"In this module, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). Students identify appropriate types of functions to model a situation. They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as, the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions, is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics."
Subject:
Algebra
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
09/16/2014
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Students build a formal understanding of probability, considering complex events such as unions, intersections, and complements as well as the concept of independence and conditional probability. The idea of using a smooth curve to model a data distribution is introduced along with using tables and technology to find areas under a normal curve. Students make inferences and justify conclusions from sample surveys, experiments, and observational studies. Data is used from random samples to estimate a population mean or proportion. Students calculate margin of error and interpret it in context. Given data from a statistical experiment, students use simulation to create a randomization distribution and use it to determine if there is a significant difference between two treatments.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Algebra
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
03/24/2016
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In this module, students reconnect with and deepen their understanding of statistics and probability concepts first introduced in Grades 6, 7, and 8. Students develop a set of tools for understanding and interpreting variability in data, and begin to make more informed decisions from data. They work with data distributions of various shapes, centers, and spreads. Students build on their experience with bivariate quantitative data from Grade 8. This module sets the stage for more extensive work with sampling and inference in later grades.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Mathematics
Statistics and Probability
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
08/01/2013
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In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Algebra
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
09/17/2013
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In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Algebra
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
09/17/2013
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Making Evidence-Based Claims ELA/Literacy Units empower students with a critical reading and writing skill at the heart of the Common Core: making evidence-based claims about complex texts. These units are part of the Developing Core Proficiencies Program. This unit develops students' abilities to make evidence-based claims through activities based on a close reading of the Nobel Peace Prize Speeches of Rev. Dr. Martin Luther King, Jr. and President Barack Obama.
Subject:
Arts and Humanities
Literature
Material Type:
Primary Source
Teaching/Learning Strategy
Unit of Study
Provider:
New York State Education Department
Provider Set:
EngageNY
04/04/2013
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Making Evidence-Based Claims ELA/Literacy Units empower students with a critical reading and writing skill at the heart of the Common Core: making evidence-based claims about complex texts. These units are part of the Developing Core Proficiencies Program. This unit develops students' abilities to make evidence-based claims through activities based on a close reading of the first chapter of W.E.B. Du Bois' The Souls of Black Folk.
Subject:
Arts and Humanities
Material Type:
Primary Source
Teaching/Learning Strategy
Unit of Study
Provider:
New York State Education Department
Provider Set:
EngageNY
04/04/2013
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Making Evidence-Based Claims ELA/Literacy Units empower students with a critical reading and writing skill at the heart of the Common Core: making evidence-based claims about complex texts. These units are part of the Developing Core Proficiencies Program. This unit develops students' abilities to make evidence-based claims through activities based on a close reading of President Ronald Reagan's First Inaugural Address and Secretary Hillary Clinton's 2011 APEC Address.
Subject:
Arts and Humanities
Material Type:
Primary Source
Teaching/Learning Strategy
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Provider:
New York State Education Department
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EngageNY
04/04/2013
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The goal of the Listening and Learning Strand is for students to acquire language competence through listening, specifically building a rich vocabulary, and broad knowledge in history and science by being exposed to carefully selected, sequenced, and coherent read-alouds. The 9 units (or domains) provide lessons (including images and texts), as well as instructional objectives, core vocabulary, and assessment materials. The domain topics include: Different Lands, Similar Stories; Fables and Stories; The Human Body; Early World Civilizations; Early American Civilizations; Astronomy; Animals & Habitats; Fairy Tales; and History of the Earth.
Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .
Subject:
English Language Arts
Language, Grammar and Vocabulary
Material Type:
Diagram/Illustration
Teaching/Learning Strategy
Unit of Study
Provider:
New York State Education Department
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EngageNY
04/04/2013
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The Skills Strand teaches the mechanics of reading. Students are taught systematic and explicit phonics instruction as their primary tool for decoding written English. By the end of grade 2, students have learned all of the sound spelling correspondences in the English language and are able to decode written material they encounter. In addition to phonics, students also are taught spelling, grammar, and writing during the Skills Strand. A downloadable story "Kits Hats" with illustrations is provided for instruction.
Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .
Subject:
Arts and Humanities
English Language Arts
Language, Grammar and Vocabulary
Material Type:
Diagram/Illustration
Provider:
New York State Education Department
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EngageNY
01/31/2023
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This module ensures that students read, write, listen and speak to learn the history and contributions of Native Americans in New York State, particularly the Iroquois Confederacy. It focuses on reading and listening to primary and secondary sources to gather specific details and determine central ideas, and to reinforce reading fluency and paragraph writing. Students will read literature to develop an understanding of setting, characterization and theme, and informational writing.
Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .
Subject:
Arts and Humanities
English Language Arts
Language, Grammar and Vocabulary
Material Type:
Lesson Plan
Teaching/Learning Strategy
Unit of Study
Provider:
New York State Education Department
Provider Set:
EngageNY
04/04/2013
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Making Evidence-Based Claims ELA/Literacy Units empower students with a critical reading and writing skill at the heart of the Common Core: making evidence-based claims about complex texts. These units are part of the Developing Core Proficiencies Program. This unit develops students' abilities to make evidence-based claims through activities based on a close reading of the Commencement Address Steve Jobs delivered at Stanford University on June, 2005.
Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .
Subject:
Arts and Humanities
Material Type:
Primary Source
Teaching/Learning Strategy
Unit of Study
Provider:
New York State Education Department
Provider Set:
EngageNY
04/04/2013
Conditional Remix & Share Permitted
CC BY-NC-SA
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0.0 stars
The goal of the Listening and Learning Strand is for students to acquire language competence through listening, specifically building a rich vocabulary, and broad knowledge in history and science by being exposed to carefully selected, sequenced, and coherent read_alouds. The 9 units (or domains) provide lessons (including images and texts), as well as instructional objectives, core vocabulary, and assessment materials. The domain topics include: Nursery Rhymes and Fables; Five Senses; Stories; Plants; Farms; Kings and Queens; Seasons and Weather; Colonial Towns; and Taking Care of the Earth.
Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .
Subject:
Arts and Humanities
Ecology
English Language Arts
Language, Grammar and Vocabulary
Life Science
Literature
Material Type:
Diagram/Illustration
Teaching/Learning Strategy
Unit of Study
Provider:
New York State Education Department
Provider Set:
EngageNY
04/04/2013
Conditional Remix & Share Permitted
CC BY-NC-SA
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The Skills Strand teaches the mechanics of reading. Students are taught systematic and explicit phonics instruction as their primary tool for decoding written English. By the end of grade 2, students have learned all of the sound spelling correspondences in the English language and are able to decode written material they encounter. In addition to phonics, students also are taught spelling, grammar, and writing during the Skills Strand. A downloadable story "Kits Hats" with illustrations is provided for instruction.
Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .
Subject:
Arts and Humanities
English Language Arts
Language, Grammar and Vocabulary
Material Type:
Diagram/Illustration
Provider:
New York State Education Department
Provider Set:
EngageNY
01/31/2023
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CC BY
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EngageNY.org is developed and maintained by the New York State Education Department (NYSED) to support the implementation of key aspects of the New York State Board of Regents Reform Agenda. This is the official web site for current materials and resources related to the Regents Reform Agenda. The agenda includes the implementation of the New York State P-12 Common Core Learning Standards (CCLS), Teacher and Leader Effectiveness (TLE), and Data-Driven Instruction (DDI). EngageNY.org is dedicated to providing educators across New York State with real-time, professional learning tools and resources to support educators in reaching the StateŰŞs vision for a college- and career-ready education for all students.
Subject:
Arts and Humanities
Material Type:
Provider:
New York State Education Department
Provider Set:
EngageNY
06/26/2013
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Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss similarity, students must first have a clear understanding of how dilations behave. This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria. An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Geometry
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
07/03/2014
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Module 3, Extending to Three Dimensions, builds on students understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids. The student materials consist of the student pages for each lesson in Module 3. The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Geometry
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
07/03/2014
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This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year. It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story. This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Geometry
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY | 3,963 | 18,788 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2023-23 | latest | en | 0.875048 |
https://www.math.toronto.edu/~jkamnitz/courses/mat247_2014/index.html | 1,723,358,634,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640975657.64/warc/CC-MAIN-20240811044203-20240811074203-00730.warc.gz | 686,607,898 | 2,481 | ## Algebra II
Instructor: Joel Kamnitzer, jkamnitz@math.toronto.edu
Office: 6110 Bahen 416-978-5163
Office Hours: Mondays 10:00 am - 12:00 pm
Office Hours: Monday 2-3 pm, Friday 10-11 am in 215 Huron 10th floor.
Lectures: Monday 1-2 pm in FG 103, Tuesday 2-3 pm in RW 117, Thursday 1-2 pm in FG 103
Tutorial: Tuesday 1-2 pm, RW 143 (last names A-M) and RW117 (last names N-Z)
Midterm: Tuesday March 4, 1:15-3 pm, EX 300. Here is the review sheet.
Final: Here is the review sheet and here are the definitions from the course, part 1,part 2 and part 3 (thanks to Alexandra, Eric, and Angela).
• Homework (10 assignments) 20%
• Quizes (February 4 and March 25) 10%
• Midterm test (March 4) 20%
• Final exam 50%
• Text: We won't have any specific textbook for the course. The following are all helpful references and I believe that Freidberg, Insel, and Spence will be closest to what we do in class.
• Axler, Linear Algebra done right,
• Freidberg, Insel, and Spence, Linear algebra
• Hoffman and Kunze, Linear algebra
• Treil, Linear algebra done wrong
• Charles W. Curtis, Linear algebra: an introductory approach
• Tim Gowers, How to lose your fear of tensors.
• Kevin Purbhoo, Notes on tensor products and the exterior algebra
• Notes on symmetric bilinear forms
• Syllabus:
Jordan canonical form (Axler 8 and 9, Curtis 7, Freidber Insel Spence 7)
Bilinear forms (Friedberg Insel Spence 6.8, Curtis 27 and 31, Treil 7)
Duals and tensor products (Curtis 8, Treil 8)
Homework:
Assignments are to be handed in at the beginning of tutorial, before 1:10 pm (note change from last semester).
Assignment 1 (pdf), due Tuesday January 14
Assignment 2 (pdf), due Tuesday January 21.
Assignment 3 (pdf), due Tuesday January 28.
Assignment 4 (pdf), due Tuesday February 4.
Assignment 5 (pdf), due Tuesday February 11.
Assignment 6 (pdf), due Tuesday February 25.
Assignment 7 (pdf), due Tuesday March 11.
Assignment 8 (pdf), due Tuesday March 18.
Assignment 9 (pdf), due Tuesday March 25.
Assignment 10 (pdf), due Tuesday April 1.
Working Together: I encourage you to work together on homework assignments. Your peers are your allies, and you should feel free to learn from them. However, you should always write up solutions on your own. Working together on quizzes and exams is forbidden.
Calculator: Calculators are not allowed on any quiz or exam.
Absences: For excused absences, documentation must be provided. For planned absences, documentation should be provided no later than two weeks prior to the absence. In general, for excused absences, any missed assignments will simply not be used in the calculation of nal course grades. For unexcused absences, missed assignments will receive a grade of zero.
Cheating: The University takes cheating very seriously. Please see http://www.utoronto. ca/academicintegrity/Academic_integrity.pdf. To quote this document: "Ignorance of the rules does not excuse cheating or plagiarism". | 798 | 2,935 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2024-33 | latest | en | 0.852584 |
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Damped Simple Harmonic Motion
A mass of 25g is attached to a vertical spring with a spring constant k = 3 dyne/cm. The surrounding medium has a damping constant of 10 dyne*sec/cm. The mass is pushed 5 cm above its equilibrium position and released. Find (a) the position function of the mass, (b) the period of the vibration, and (c) the frequency of the vibration.
Solution Preview
Please see the attached file for detailed solution.
Since we adjust the coordinate system so that x = 0 corresponds to the spring being unstretched, then the ...
Solution Summary
The solution is comprised of detailed explanations of damped simple harmonic motion. By solving the differential equation, the motion of the mass attached to a spring in the damping medium is described step-by-step.
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https://byjus.com/question-answer/f-x-6x-5-has-a-maximum-at-x-0-true-false-1/ | 1,638,348,646,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964359976.94/warc/CC-MAIN-20211201083001-20211201113001-00553.warc.gz | 225,948,023 | 31,240 | Question
# f(x)=6x5 has a maximum at x = 0. True False
Solution
## The correct option is B False If f(x) has derivative upto nth order and f′(c)=f”(c)..fn−1(c)=0, then A) n is even, fn(c)<0=>x=c is a point of maximum. B) n is even, fn(c)>0=>x=c is a point of minimum. C) n is odd, fn(c)<0=>f(x) is decreasing about x=c D) n is odd, fn(c)>0=>f(x) is increasing about x=c So, we will differentiate until we get a non-zero value at x=0 f(x)=6x5 f′(x)=30x4 f′(0)=0 f”(x)=120x3 f”(0)=0 f”′(x)=f3(x)=360x2 f3(0)=0 f””(x)=f4(x)=720x f4(x)=0 f””′(x)=f5(x)=720 f5(0)=720 which is non –zero. So, we get n = 5, which is odd. n is odd,n is odd, fn(c)>0=>f(x) is increasing about x=c That is f(x) is increasing at x=0. It does not have a maximum or minimum at x=0
Suggest corrections | 315 | 790 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2021-49 | latest | en | 0.852084 |
https://rnasystemsbiology.org/and-pdf/2293-doubling-and-halving-worksheets-pdf-216-265.php | 1,628,220,925,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046152112.54/warc/CC-MAIN-20210806020121-20210806050121-00432.warc.gz | 495,655,653 | 6,245 | # Doubling and halving worksheets pdf
File Name: doubling and halving worksheets .zip
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Published: 17.04.2021
## Doubling And Halving Worksheets Grade 5: Compound Shapes
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Some of the worksheets displayed are Multiplication strategy doubling and halving, Doubling and halving, Mathsphere, Multiplication division, Doubling halving, Maths work from mathsphere mathematics, Doubling and halving opposites, Downsend school year 5 easter revision booklet. Doubling and halving We are learning to simplify multiplications by doubling and halving numbers. Exercise 1: doubling and halving What to do Use the strategy of doubling and halving to rewrite these multiplications as simpler problems, then answer the problem. Some of the worksheets for this concept are Multiplication strategy doubling and halving, Doubling and halving, Mathsphere, Multiplication division, Doubling halving, Maths work from mathsphere mathematics, Doubling and halving opposites, Downsend school year 5 easter revision booklet. Year 5 Doubling And Halving Multiplication.
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## (Online Math Practice)
With a creative twist, our resources are designed and created by real teachers around the world to meet the countries national curriculum standards. Can be timed or untimed Ideal for starter and Read More … Read more. One child asked questions, one answered, and the other two kept score with tallies on their whiteboards to make sure they got a matching, accurate final score. Multiplication Facts. Teaching KS1 doubling and halving skills early lays an important foundational knowledge that will become the building blocks to learning more complicated skills and principles down the line. Author: Created by heath
### Doubling And Halving Worksheets Grade 5: Compound Shapes
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This simple game has been designed to help your child to practice their doubling and halving with a range of different numbers. Please give feedback on our Doubling and Halving Practice Zone at the bottom of the page. In our practice zones, you get the chance to practice your math skills online with instant feedback.
• #### Marc C. 22.04.2021 at 00:32
Includes vertical and horizontal problems, as well as math riddles, task cards, a … We offer word problems, color by numbers, and even subtraction crosswords, so your child has the necessary tools to succeed.
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We can measure the length, weight and capacity of objects.
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07.04.2021 at 12:26 | 821 | 3,778 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2021-31 | latest | en | 0.906723 |
http://sourceforge.net/p/matplotlib/mailman/message/28472175/ | 1,438,556,637,000,000,000 | text/html | crawl-data/CC-MAIN-2015-32/segments/1438042989301.17/warc/CC-MAIN-20150728002309-00104-ip-10-236-191-2.ec2.internal.warc.gz | 224,468,040 | 6,736 | ## [Matplotlib-users] patch for basemap when using cyclic or polycyclic projections in celestial coordinates
[Matplotlib-users] patch for basemap when using cyclic or polycyclic projections in celestial coordinates From: Molly Swanson - 2011-11-29 22:38:57 ```I would like to suggest the following fixes for the mpl_toolkits.basemap module to improve its treatment of celestial (rather than geographic) coordinates. The first one, posted at https://github.com/mollyswanson/basemap/commit/23db4bbebf4d7fe6ca202b5dad50b6a2054dd685 changes the call function in basemap's init.py to correctly transform lat/lon values into xy map coordinates in the case of a cyclic or polycyclic projection with lon_0 not equal to 0. The second, posted at https://github.com/mollyswanson/basemap/commit/35470b51523e9429d26cefc911dca843264581b9 changes one line in the drawparallels. This line is to avoid drawing lines between points on the parallel that span the whole map. However, the old version uses a fixed value for the distance between the points rather than scaling it to the radius of the sphere used in the projection, so if you use a non-default radius (such as 180/pi, so your x-y values are in degrees on the sky instead of meters on the earth) it won't work. This fix scales the cutoff value to the radius of the projection sphere. The following example illustrates the issues that are addressed here: from mpl_toolkits.basemap import Basemap import matplotlib.pyplot as plt figure(1) #make a basemap centered on longitude of 90 m=Basemap(celestial=False,lon_0=90,projection='hammer') #draw map boundary and grid m.drawmapboundary() m.drawparallels(np.arange(-90.,91.,30.),labels=[1,0,0,0]) m.drawmeridians(np.arange(-90.,271.,30.),labels=[0,0,0,0]) #define a test polygon - a triangle with corners at [lon,lat]=[90,30],[120,60],[120,30] polygon=array([[90,30],[120,60],[120,30],[90,30]]) #convert to map coordinates polyxy=m(polygon[:,0],polygon[:,1]) plt.plot(polyxy[0],polyxy[1]) plt.savefig('basemap1.png') figure(2) #make a celestial basemap centered on longitude of 90 m=Basemap(celestial=True,lon_0=90,projection='hammer',rsphere=180./pi) #draw map boundary and grid m.drawmapboundary() m.drawparallels(np.arange(-90.,91.,30.),labels=[1,0,0,0]) m.drawmeridians(np.arange(-90.,271.,30.),labels=[0,0,0,0]) #define a test polygon - a triangle with corners at [lon,lat]=[90,30],[120,60],[120,30] polygon=array([[90,30],[120,60],[120,30],[90,30]]) #convert to map coordinates polyxy=m(polygon[:,0],polygon[:,1]) plt.plot(polyxy[0],polyxy[1]) plt.savefig('celestial_basemap1.png') Thank you! Molly Swanson ```
[Matplotlib-users] patch for basemap when using cyclic or polycyclic projections in celestial coordinates From: Molly Swanson - 2011-11-29 22:38:57 ```I would like to suggest the following fixes for the mpl_toolkits.basemap module to improve its treatment of celestial (rather than geographic) coordinates. The first one, posted at https://github.com/mollyswanson/basemap/commit/23db4bbebf4d7fe6ca202b5dad50b6a2054dd685 changes the call function in basemap's init.py to correctly transform lat/lon values into xy map coordinates in the case of a cyclic or polycyclic projection with lon_0 not equal to 0. The second, posted at https://github.com/mollyswanson/basemap/commit/35470b51523e9429d26cefc911dca843264581b9 changes one line in the drawparallels. This line is to avoid drawing lines between points on the parallel that span the whole map. However, the old version uses a fixed value for the distance between the points rather than scaling it to the radius of the sphere used in the projection, so if you use a non-default radius (such as 180/pi, so your x-y values are in degrees on the sky instead of meters on the earth) it won't work. This fix scales the cutoff value to the radius of the projection sphere. The following example illustrates the issues that are addressed here: from mpl_toolkits.basemap import Basemap import matplotlib.pyplot as plt figure(1) #make a basemap centered on longitude of 90 m=Basemap(celestial=False,lon_0=90,projection='hammer') #draw map boundary and grid m.drawmapboundary() m.drawparallels(np.arange(-90.,91.,30.),labels=[1,0,0,0]) m.drawmeridians(np.arange(-90.,271.,30.),labels=[0,0,0,0]) #define a test polygon - a triangle with corners at [lon,lat]=[90,30],[120,60],[120,30] polygon=array([[90,30],[120,60],[120,30],[90,30]]) #convert to map coordinates polyxy=m(polygon[:,0],polygon[:,1]) plt.plot(polyxy[0],polyxy[1]) plt.savefig('basemap1.png') figure(2) #make a celestial basemap centered on longitude of 90 m=Basemap(celestial=True,lon_0=90,projection='hammer',rsphere=180./pi) #draw map boundary and grid m.drawmapboundary() m.drawparallels(np.arange(-90.,91.,30.),labels=[1,0,0,0]) m.drawmeridians(np.arange(-90.,271.,30.),labels=[0,0,0,0]) #define a test polygon - a triangle with corners at [lon,lat]=[90,30],[120,60],[120,30] polygon=array([[90,30],[120,60],[120,30],[90,30]]) #convert to map coordinates polyxy=m(polygon[:,0],polygon[:,1]) plt.plot(polyxy[0],polyxy[1]) plt.savefig('celestial_basemap1.png') Thank you! Molly Swanson ``` | 1,418 | 5,106 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2015-32 | latest | en | 0.689457 |
https://www.cuemath.com/ncert-solutions/q-7-exercise-11-4-mensuration-class-8-maths/ | 1,620,436,581,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988831.77/warc/CC-MAIN-20210508001259-20210508031259-00635.warc.gz | 760,374,690 | 17,884 | # Ex.11.4 Q7 Mensuration Solution - NCERT Maths Class 8
Go back to 'Ex.11.4'
## Question
If each edge of a cube is doubled,
(i) How many times will its surface area increase?
(ii) How many times will its volume increase?
Video Solution
Mensuration
Ex 11.4 | Question 7
## Text Solution
What is Known?
Initial surface area and volume of a cube.
What is unknown?
Increased surface area and volume of a cube.
Reasoning:
Whenever sides are doubled in any structure then area becomes four times the original structure and the volume becomes eight times the original structure.
Steps:
It the initial edge of the cube is $$l \rm\,cm.$$
If each edge of the cube is doubled, then it becomes $$2l\rm\, cm.$$
(i) Initial surface area $$=$$ $$6{l^2}$$
New surface area
$\, = 6{(2l)^2} = 6 \times 4{l^2} = 24{l^2}$
Ratio $$=$$ $$6{l^2}:24{l^2}=1:4$$
(ii) Initial volume of the cube $$= \,{l^3}$$
New volume$$\, = {(2l)^3} = 8 \times {l^3}$$
Ratio $$=$$ $${l^3}$$: $$8{l^3}$$$$= 1 :8$$
Thus, the surface area will be increased by four times and volume of the cube will be increased by eight times.
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• Personalized curriculum to keep up with school | 381 | 1,323 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.40625 | 4 | CC-MAIN-2021-21 | latest | en | 0.859949 |
https://sonichours.com/how-fast-is-136-km-in-mph/ | 1,695,699,865,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510130.53/warc/CC-MAIN-20230926011608-20230926041608-00676.warc.gz | 582,042,131 | 20,891 | General
# How Fast Is 136 Km In Mph
How fast is 136 km in mph? It is easy to convert 136 km into mph. Using the formula below, you can convert 136 KM to mph. It is important that you note that the speed limit of a vehicle can be higher or lower than 136 km/h. If you are planning on traveling by car, you must keep in mind that a single kilometer equals one mile.
You traveled approximately 136 km in a single day. This is the same distance that you drive in a day. You would drive about 50 kilometers per hour, but your car might have traveled some parts of the route faster or slower. Multiplying the distance by 10 will convert 136 km into mph.
Calculate how fast 136 km is in miles per hour by dividing the distance between one mile (or one kilometer) One mile is 1.01 miles and is equal to about 2:15 minutes. Twenty miles per hour equals approximately 235.125 miles per gallon. One kilometer is approximately 138.6 miles. The Naismiths Rule is the basis for the conversion factor, formula and metric system.
The most widely used system of measurement is the metric system. The United States, Canada, and Europe use mph as the primary units for measuring speed. Canada uses km/h and miles. One statute mile is the distance you travel in an hour. This measurement may be different in other countries. A vehicle may go 136 km in one hour while another may go a hundred and eighty kilometers. | 323 | 1,399 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.15625 | 3 | CC-MAIN-2023-40 | longest | en | 0.93452 |
https://www.askassignmenthelp.com/connect-managerial-accounting-chapter-2.html | 1,582,345,737,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875145648.56/warc/CC-MAIN-20200222023815-20200222053815-00408.warc.gz | 646,400,767 | 12,799 | # Connect Managerial Accounting Chapter 2
Q1. As of the end of June, the job cost sheets at Racing Wheels, Inc., show the following total costs accumulated on three custom jobs.
Job 102 was started in production in May and the following costs were assigned to it in May: direct materials, \$14,000; direct labor, \$3,800; and overhead, \$1,748. Jobs 103 and 104 were started in June. Overhead cost is applied with a predetermined rate based on direct labor cost. Jobs 102 and 103 were finished in June, and Job 104 is expected to be finished in July. No raw materials were used indirectly in June. Using this information, answer the following questions. (Assume this company’s predetermined overhead rate did not change across these months.)
1&2. Complete the table below to calculate the cost of the raw materials requisitioned and direct labor cost incurred during June for each of the three jobs?
3. Using the accumulated costs of the jobs, what predetermined overhead rate is used?
4. How much total cost is transferred to finished goods during June?
1&2. Complete the table below to calculate the cost of the raw materials requisitioned and direct labor cost incurred during June for each of the three jobs?
3. Using the accumulated costs of the jobs, what predetermined overhead rate is used?
4. How much total cost is transferred to finished goods during June?
Q2. Starr Company reports the following information for August.
Raw materials purchased on account \$88,200 Direct materials used in production \$54,600 Factory wages earned (direct labor) \$17,750 Overhead rate 120 % of direct labor cost
Prepare journal entries to record the following events.
1. Raw materials purchased.
2. Direct materials used in production.
3. Direct labor used in production.
Q3. In December 2016, Custom Mfg. established its predetermined overhead rate for jobs produced during 2017 by using the following cost predictions: overhead costs, \$700,000, and direct materials costs, \$500,000. At year-end 2017, the company’s records show that actual overhead costs for the year are \$785,900. Actual direct material cost had been assigned to jobs as follows.
Jobs completed and sold \$420,000 Jobs in finished goods inventory 78,000 Jobs in work in process inventory 57,000 Total actual direct materials cost \$555,000
1. Determine the predetermined overhead rate for 2017.
2&3. Enter the overhead costs incurred and the amounts applied during the year using the predetermined overhead rate and determine whether overhead is overapplied or underapplied.
4. Prepare the adjusting entry to allocate any over- or underapplied overhead to Cost of Goods Sold.
1. Determine the predetermined overhead rate for 2017.
2&3. Enter the overhead costs incurred and the amounts applied during the year using the predetermined overhead rate and determine whether overhead is overapplied or underapplied.
4. Prepare the adjusting entry to allocate any over- or underapplied overhead to Cost of Goods Sold.
## Connect Managerial Accounting Chapter 2 Quiz
Q1. Kayak Company uses a job order costing system and allocates its overhead on the basis of direct labor costs. Kayak Company’s production costs for the year were: direct labor, \$30,000; direct materials, \$50,000; and factory overhead applied \$6,000. The overhead application rate was:
• 5.0%
• 12.0%
• 20.0%
• 500.0%
• 16.7%
OH rate = OH applied/Direct Labor Costs = \$6,000/\$30,000 = 20%
Q2. An example of direct labor cost is:
• Supervisor salary
• Maintenance worker wages
• Janitor wages
• Product assembler wages
• Accountant salary
Q3. Copy Center pays an average wage of \$12 per hour to employees for printing and copying jobs, and allocates \$18 of overhead for each employee hour worked. Materials are assigned to each job according to actual cost. Jobs are marked up 20% above cost to determine the selling price. If Job M-47 used \$350 of materials and took 20 hours of labor to complete, what is the selling price of the job?
• \$852
• \$1,140
• \$456
• \$720
• \$708
Direct materials \$350 + Direct labor (\$12 * 20) + Factory overhead (\$18 * 20) = \$350 + \$240 + \$360 = \$950 Total cost
\$950 * 120% = \$1,140
Q4. The balance in the Work in Process Inventory at any point in time is equal to:
• The costs for jobs finished during the period but not yet sold.
• The cost of jobs ordered but not yet started into production.
• The sum of the costs for all jobs in process but not yet completed.
• The costs of all jobs started during the period, completed or not.
• The sum of the materials, labor and overhead costs paid during the period.
Q5. Lowden Company has an overhead application rate of 160% and allocates overhead based on direct material cost. During the current period, direct labor cost is \$50,000 and direct materials used cost \$80,000. Determine the amount of overhead Lowden Company should record in the current period.
• \$31,250
• \$50,000
• \$80,000
• \$128,000
• \$208,000
\$80,000 direct materials * 1.60 = \$128,000
Q6. Portside Watercraft uses a job order costing system. During one month Portside purchased \$173,000 of raw materials on credit; issued materials to production of \$164,000 of which \$24,000 were indirect. Portside incurred a factory payroll of \$95,000, of which \$25,000 was indirect labor. Portside uses a predetermined overhead rate of 170% of direct labor cost. The journal entry to record the issuance of materials to production is:
• Debit Raw Materials Inventory \$153,000; credit Accounts Payable \$153,000.
• Debit Work in Process Inventory \$140,000; debit Factory Overhead \$24,000; credit Raw Materials Inventory \$164,000.
• Debit Raw Materials Inventory \$195,000; credit Work in Process Inventory \$195,000.
• Debit Work in Process Inventory \$140,000; debit Raw Materials Inventory \$24,000; credit Materials Inventory \$164,000.
• Debit Finished Goods Inventory \$140,000; credit Raw Materials Inventory \$140,000.
Q7. A company has an overhead application rate of 125% of direct labor costs. How much overhead would be allocated to a job if it required total labor costing \$20,000?
• \$5,000
• \$16,000
• \$25,000
• \$125,000
• \$250,000
\$20,000 * 1.25 = \$25,000
Q8. A job order costing system would best fit the needs of a company that makes:
• Shoes and apparel.
• Paint
• Cement
• Custom machinery.
• Pencils and erasers.
Q9. A job cost sheet includes:
• Direct materials, direct labor, operating costs. | 1,535 | 6,461 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2020-10 | longest | en | 0.962086 |
http://www.ehow.co.uk/how_7783943_measure-noise-electrical-circuit.html | 1,490,258,086,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218186841.66/warc/CC-MAIN-20170322212946-00415-ip-10-233-31-227.ec2.internal.warc.gz | 506,579,683 | 18,218 | # How to Measure Noise in an Electrical Circuit
Written by natasha parks
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Electrical "noise" covers a range of natural and artificial electrical changes or disturbances from analogue equipment, such as amplifiers. The goal for engineers is to reduce this excessive, unwanted noise to its lowest possible point for any given circuit. Knowing how to do this means knowing how much additional, unnecessary noise there is at the present time. When the current noise level is depleted as close to zero as possible, you will have a circuit that can process electronic signals at optimal efficiency for your given set-up. Every circuit has such a "noise figure": it defines the lowest limit of a reliable signal.
Skill level:
Easy
### Things you need
• UUT e.g. linear amplifier
• Noise source
• Noise figure meter
• Calculator
## Instructions
1. 1
Specify the bandwidth over which you will measure unwanted, excess noise. Set it to a measurable range you are comfortable with. Ignore all other frequencies during the calculation to make the process clearer and more straightforward. A reasonable frequency range to measure is 1 Hertz.
2. 2
Set up or prepare your existing circuit using a linear amplifier. Measure the amplifier's gain over the bandwidth of interest using the figure meter. Measure the signal at the starting point and then the finishing point of the amplifier and calculate the gain, which is the output signal minus the input signal. Record the value.
3. 3
Introduce a noise source such as thermal noise, a temperature-dependent "white" noise --- also called "Johnson Noise" --- that is produced from unpredictable fluctuations in voltage or current, into your unit under test (UUT), such as the amplifier. Ensure the noise source produces a signal in the required range of frequencies, or you will not be able to detect it.
4. 4
Calculate the amplifier's noise in volts from its gain value using a calculator and the equation: voltage is equal to the square root of four times the Boltzmann's constant --- 1.374 multiplied by 10 to the minus 23 Joules per Kelvin --- multiplied by the temperature of the set-up, the resistance of the amplifier in ohms and the bandwidth you selected.
5. 5
Use a noise figure meter to measure random noise within the circuit at the frequency you defined for yourself. This noise figure is the total power output minus the known output from the noise source. The calculated figure represents the amount of unwanted noise being generated internally through the circuit as a direct result of the amplifier. Noise power is measured in decibel minutes.
#### Tips and warnings
• Take care with units and powers during the calculations. Room temperature --- temperature of the circuit set-up --- measures about 290 degrees Kelvin, so use this as a basis for the measurement.
• Make sure voltages are safely insulated and earthed before switching circuits on. Use a transformer to step down any high voltages, such as 120-volt household power supplies. Electricity is dangerous when used improperly.
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By using the eHow.co.uk site, you consent to the use of cookies. For more information, please see our Cookie policy. | 692 | 3,328 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2017-13 | latest | en | 0.90696 |
https://www.geeksforgeeks.org/average-odd-numbers-till-given-odd-number/?ref=lbp | 1,653,506,586,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00429.warc.gz | 892,192,849 | 28,847 | # Average of odd numbers till a given odd number
• Last Updated : 24 Mar, 2021
Given an odd number n, find the average of odd numbers from 1 to n.
Examples:
```Input : n = 9
Output : 5
Explanation
(1 + 3 + 5 + 7 + 9)/5
= 25/5
= 5
Input : n = 221
Output : 111```
Method 1 We can calculate average by adding each odd numbers till n and then dividing sum by count.
Below is the implementation of the approach.
## C++
`// Program to find average of odd numbers``// till a given odd number.``#include ` `// Function to calculate the average``// of odd numbers``int` `averageOdd(``int` `n)``{`` ``if` `(n % 2 == 0) {`` ``printf``(``"Invalid Input"``);`` ``return` `-1;`` ``}` ` ``int` `sum = 0, count = 0;`` ``while` `(n >= 1) {` ` ``// count odd numbers`` ``count++;` ` ``// store the sum of odd numbers`` ``sum += n;` ` ``n = n - 2;`` ``}`` ``return` `sum / count;``}` `// driver function``int` `main()``{`` ``int` `n = 15;`` ``printf``(``"%d"``, averageOdd(n));`` ``return` `0;``}`
## Java
`// Program to find average of odd numbers``// till a given odd number.``import` `java.io.*;` `class` `GFG {`` ` ` ``// Function to calculate the average`` ``// of odd numbers`` ``static` `int` `averageOdd(``int` `n)`` ``{`` ``if` `(n % ``2` `== ``0``) {`` ``System.out.println(``"Invalid Input"``);`` ``return` `-``1``;`` ``}`` ` ` ``int` `sum = ``0``, count = ``0``;`` ``while` `(n >= ``1``) {`` ` ` ``// count odd numbers`` ``count++;`` ` ` ``// store the sum of odd numbers`` ``sum += n;`` ` ` ``n = n - ``2``;`` ``}`` ``return` `sum / count;`` ``}`` ` ` ``// driver function`` ``public` `static` `void` `main(String args[])`` ``{`` ``int` `n = ``15``;`` ``System.out.println(averageOdd(n));`` ``}``}` `/*This code is contributed by Nikita tiwari.*/`
## Python3
`# Program to find average``# of odd numbers till a``# given odd number.` `# Function to calculate``# the average of odd``# numbers``def` `averageOdd(n) :` ` ``if` `(n ``%` `2` `=``=` `0``) :`` ``print``(``"Invalid Input"``)`` ``return` `-``1`` ` ` ``sm ``=` `0`` ``count ``=` `0` ` ``while` `(n >``=` `1``) :` ` ``# count odd numbers`` ``count ``=` `count ``+` `1` ` ``# store the sum of`` ``# odd numbers`` ``sm ``=` `sm ``+` `n` ` ``n ``=` `n ``-` `2`` ` ` ``return` `sm ``/``/` `count`` ` `# Driver function``n ``=` `15``print``(averageOdd(n))` `# This code is contributed by Nikita Tiwari.`
## C#
`// C# Program to find average ``// of odd numbers till a given``// odd number.``using` `System;` `class` `GFG {`` ` ` ``// Function to calculate the`` ``// average of odd numbers`` ``static` `int` `averageOdd(``int` `n)`` ``{`` ``if` `(n % 2 == 0) {`` ``Console.Write(``"Invalid Input"``);`` ``return` `-1;`` ``}`` ` ` ``int` `sum = 0, count = 0;`` ``while` `(n >= 1) {`` ` ` ``// count odd numbers`` ``count++;`` ` ` ``// store the sum of odd numbers`` ``sum += n;`` ` ` ``n = n - 2;`` ``}`` ``return` `sum / count;`` ``}`` ` ` ``// driver function`` ``public` `static` `void` `Main()`` ``{`` ``int` `n = 15;`` ``Console.Write(averageOdd(n));`` ``}``}` `/*This code is contributed by vt_m.*/`
## PHP
`= 1)`` ``{` ` ``// count odd numbers`` ``\$count``++;` ` ``// store the sum of`` ``// odd numbers`` ``\$sum` `+= ``\$n``;` ` ``\$n` `= ``\$n` `- 2;`` ``}`` ``return` `\$sum` `/ ``\$count``;``}` ` ``// Driver Code`` ``\$n` `= 15;`` ``echo``(averageOdd(``\$n``));`` ` `// This code is contributed by vt_m.``?>`
## Javascript
``
Output:
`8`
Method 2
The average of odd numbers can find out only in single steps
by using the following formula
[n + 1 ] / 2
where n is last odd number.
How does this formula work?
```We know there are (n+1)/2 odd numbers till n.
For example:
There are two odd numbers till 3 and there are
three odd numbers till 5.
Sum of first k odd numbers is k*k
Sum of odd numbers till n is ((n+1)/2)2
Average of odd numbers till n is (n + 1)/2```
Below is the implementation of the approach.
## C
`// Program to find average of odd numbers``// till a given odd number.``#include ` `// Function to calculate the average``// of odd numbers``int` `averageOdd(``int` `n)``{`` ``if` `(n % 2 == 0) {`` ``printf``(``"Invalid Input"``);`` ``return` `-1;`` ``}` ` ``return` `(n + 1) / 2;``}` `// driver function``int` `main()``{`` ``int` `n = 15;`` ``printf``(``"%d"``, averageOdd(n));`` ``return` `0;``}`
## Java
`// Program to find average of odd``// numbers till a given odd number.``import` `java.io.*;` `class` `GFG``{`` ``// Function to calculate the`` ``// average of odd numbers`` ``static` `int` `averageOdd(``int` `n)`` ``{`` ``if` `(n % ``2` `== ``0``)`` ``{`` ``System.out.println(``"Invalid Input"``);`` ``return` `-``1``;`` ``}`` ` ` ``return` `(n + ``1``) / ``2``;`` ``}`` ` ` ``// driver function`` ``public` `static` `void` `main(String args[])`` ``{`` ``int` `n = ``15``;`` ``System.out.println(averageOdd(n));`` ``}``}` `// This code is contributed by Nikita tiwari.`
## Python3
`# Program to find average of odd``# numbers till a given odd number.` `# Function to calculate the``# average of odd numbers``def` `averageOdd(n) :`` ``if` `(n ``%` `2` `=``=` `0``) :`` ``print``(``"Invalid Input"``)`` ``return` `-``1`` ` ` ` ` ``return` `(n ``+` `1``) ``/``/` `2`` ` `# driver function``n ``=` `15``print``(averageOdd(n))` `# This code is contributed by Nikita tiwari.`
## C#
`// C# Program to find average``// of odd numbers till a given``// odd number.``using` `System;` `class` `GFG``{`` ``// Function to calculate the`` ``// average of odd numbers`` ``static` `int` `averageOdd(``int` `n)`` ``{`` ``if` `(n % 2 == 0)`` ``{`` ``Console.Write(``"Invalid Input"``);`` ``return` `-1;`` ``}`` ` ` ``return` `(n + 1) / 2;`` ``}`` ` ` ``// driver function`` ``public` `static` `void` `Main()`` ``{`` ``int` `n = 15;`` ``Console.Write(averageOdd(n));`` ``}``}` `// This code is contributed by vt_m.`
## PHP
``
## Javascript
``
Output:
`8`
My Personal Notes arrow_drop_up | 2,422 | 6,738 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2022-21 | latest | en | 0.448158 |
https://fdocument.org/document/example-of-hessenberg-reduction.html | 1,685,550,178,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224646937.1/warc/CC-MAIN-20230531150014-20230531180014-00430.warc.gz | 293,303,090 | 27,626 | of 21 /21
Eigenvalues In addition to solving linear equations another important task of linear algebra is finding eigenvalues. Let F be some operator and x a vector. If F does not change the direction of the vector x, x is an eigenvector of the operator, satisfying the equation F (x)= λx, (1) where λ is a real or complex number, the eigenvalue corresponding to the eigenvector. Thus the operator will only change the length of the vector by a factor given by the ei- genvalue. If F is a linear operator, F (ax)= aF (x)= x, and hence ax is an eigenvector, too. Eigenvectors are not uniquely determined, since they can be multiplied by any constant.
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Example of Hessenberg Reduction
### Transcript of Example of Hessenberg Reduction
Eigenvalues
In addition to solving linear equations another important task of linear algebra is findingeigenvalues.
Let F be some operator and x a vector. If F does not change the direction of the vectorx, x is an eigenvector of the operator, satisfying the equation
F (x) = λx, (1)
where λ is a real or complex number, the eigenvalue corresponding to the eigenvector.Thus the operator will only change the length of the vector by a factor given by the ei-genvalue.
If F is a linear operator, F (ax) = aF (x) = aλx, and hence ax is an eigenvector, too.Eigenvectors are not uniquely determined, since they can be multiplied by any constant.
In the following only eigenvalues of matrices are discussed. A matrix can be consideredan operator mapping a vector to another one. For eigenvectors this mapping is a merescaling.
Let A be a square matrix. If there is a real or complex number λ and a vector x suchthat
Ax = λx, (2)
λ is an eigenvalue of the matrix and x an eigenvector. The quation (2) can also be writ-ten as
(A− λI)x = 0.
If the equation is to have a nontrivial solution, we must have
det(A− λI) = 0. (3)
When the determinant is expanded, we get the characteristic polynomial, the zeros ofwhich are the eigenvalues.
If the matrix is symmetric and real valued, the eigenvalues are real. Otherwise at leastsome of them may be complex. Complex eigenvalues appear always as pairs of complexconjugates.
For example, the characteristic polynomial of the matrix
(
1 41 1
)
is
det
(
1− λ 41 1− λ
)
= λ2 − 2λ− 3.
The zeros of this are the eigenvalues λ1 = 3 and λ2 = −1.
Finding eigenvalues usign the characteristic polynomial is very laborious, if the matrixis big. Even determining the coefficients of the equation is difficult. The method is notsuitable for numerical calculations.
To find eigenvalues the matrix must be transformed to a more suiteble form. Gaussianelimination would transform it into a triangular matrix, but unfortunately the eigenva-lues are not conserved in the transform.
Another kind of transform, called a similarity transform, will not affect the eigenvalues.Similarity transforms have the form
A → A′ = S−1AS,
where S is any nonsingular matrix, The matrices A and A′ are called similar.
QR decomposition
A commonly used method for finding eigenvalues is known as the QR method. The met-hod is an iteration that repeatedly computes a decomposition of the matrix know as itsQR decomposition. The decomposition is obtained in a finite number of steps, and it hassome other uses, too. We’ll first see how to compute this decomposition.
The QR decomposition of a matrix A is
A = QR,
where Q is an orthogonal matrix and R an upper triangular matrix. This decompositionis possible for all matrices.
There are several methods for finding the decomposition
1) Householder transform
2) Givens rotations
3) Gram–Schmidt orthogonalisation
In the following we discuss the first two methods with exemples. They will probably beeasier to understand than the formal algorithm.
Householder transform
The idea of the Householder transform is to find a set of transforms that will make allelements in one column below the diagonal vanish.
Assume that we have to decompose a matrix
A =
3 2 11 1 22 1 3
We begin by taking the first column of this
x1 = a(:, 1) =
311
2
and compute the vector
u1 = x1− ‖ x1 ‖
101
0 =
−0.741657411
2
This is used to create a Householder transformation matrix
P1 = I− 2u1u
T1
‖ u1 ‖2=
0.8017837 0.2672612 0.53452250.2672612 0.6396433 −0.72071350.5345225 −0.7207135 −0.4414270
It can be shown that this is an orthogonal matrix. It is easy to see this by calculating thescalar product of any two columns. The products are zeros, and thus the column vectorsof the matrix are mutually orthogonal.
When the original matrix is multiplied by this transform the result is a matrix with zerosin the first column below the diagonal:
A1 = P1A =
3.7416574 2.4053512 2.93987370. 0.4534522 −0.61559270. −0.0930955 −2.2311854
Then we use the second column to create a vector
x2 = a(2 : 3, 1) =
(
0.4534522−0.0930955
)
,
from which
u2 = x2− ‖ x2 ‖
(
10
)
=
(
−0.0094578−0.0930955
)
.
This will give the second transformation matrix
P2 = I− 2u2u
T2
‖ u2 ‖2=
1 0 00 0.9795688 −0.20110930 −0.2011093 −0.9795688
The product of A1 and the transformation matrix will be a matrix with zeros in the se-cond column below the diagonal:
A2 = P2A1 =
3.7416574 2.4053512 2.93987370 0.4629100 −0.15430330 0 2.3094011
Thus the matrix has been transformed to an upper triangular matrix. If the matrix isbigger, repeat the same procedure for each column until all the elements below the diago-nal vanish.
Matrices of the decomposition are now obtained as
Q = P1P2 =
0.8017837 0.1543033 −0.57735030.2672612 0.7715167 0.57735030.5345225 −0.6172134 0.5773503
R = A2 = P2P1A =
3.7416574 2.4053512 2.93987370 0.4629100 −0.15430330 0 2.3094011
.
The matrix R is in fact the Ak calculated in the last transformation; thus the originalmatrix A is not needed. If memory must be saved, each of the matrices Ai can be sto-red in the area of the previous one. Also, there is no need to keep the earlier matrices Pi,but P1 will be used as the initial value of Q, and at each step Q is always multiplied bythe new tranformation matrix Pi.
As a check, we can calculate the product of the factors of the decomposition to see thatwe will restore the original matrix:
QR =
3 2 11 1 22 1 3
.
Orthogonality of the matrix Q can be seen e.g. by calculating the productQQT:
QQT =
1 0 00 1 00 0 1
In the general case the matrices of the decomposition are
Q = P1P2 · · ·Pn,
R = Pn · · ·P2P1A.
Givens rotations
Another commonly used method is based on Givens rotation matrices:
Pkl(θ) =
1
cos θ · · · sin θ... 1
...− sin θ · · · cos θ
1
.
This is an orthogonal matrix.
Finding the eigenvalues
Eigenvalues can be found using iteratively the QR-algorithm, which will use the previousQR decomposition. If we started with the original matrix, the task would be computa-tionally very time consuming. Therefore we start by transformin the matrix to a moresuitable form.
A square matrix is in the block diagonal form if it is
T11 T12 T13 · · · T1n
0 T22 T23 · · · T2n
0 0...
0 0 0 · · · Tnn
,
where the submatrices Tij are square matrices. It can be shown that the eigenvalues ofsuch a matrix are the eigenvalues of the diagonal blocks Tii.
If the matrix is a diagonal or triangular matrix, the eigenvalues are the diagonal ele-ments. If such a form can be found, the problem is solved. Usually such a form cannotbe obtained by a finite number of similarity transformations.
If the original matrix is symmetric, it can be transformed to a tridiagonal form withoutaffecting its eigenvalues. In the case of a general matrix the result is a Hessenberg mat-rix, which has the form
H =
x x x x xx x x x x0 x x x x0 0 x x x0 0 0 x x
The transformations required can be accomplished with Householder transforms or Gi-vens rotations. The method is now slightly modified so that the elements immediatelybelow the diagonal are not zeroed.
Transformation using the Householder transforms
As a first example, consider a symmetric matrix
A =
4 3 2 13 4 −1 22 −1 1 −21 2 −2 2
We begin to transform this using Householder transform. Now we construct a vector x1
by taking only the elements of the first column that are below the diagonal:
x1 =
321
Using these form the vector
u1 = x1− ‖ x1 ‖
101
0 =
−0.741657421
and from this the matrix
p1 = I− 2u1uT1 / ‖ u1 ‖=
0.8017837 0.5345225 0.26726120.5345225 −0.4414270 −0.72071350.2672612 −0.7207135 0.6396433
and finally the Householder transformation matrix
P1 =
1 0 0 00 0.8017837 0.5345225 0.26726120 0.5345225 −0.4414270 −0.72071350 0.2672612 −0.7207135 0.6396433
Now we can make the similarity transform of the matrix A. The transformation matrixis symmetric, so there is no need for transposing it:
A1 = P1AP1 =
4 3.7416574 0 03.7416574 2.4285714 1.2977396 2.1188066
0 1.2977396 0.0349563 0.29521130 2.1188066 0.2952113 4.5364723
The second column is handled in the same way. First we form the vector x2
x2 =
(
1.29773962.1188066
)
and from this
u2 = x2− ‖ x2 ‖ (1, 0)T =
(
−1.18690722.1188066
)
and
p2 =
(
0.5223034 0.85275970.8527597 −0.5223034
)
and the final transformation matrix
P2 =
1 0 0 00 1 000 0 0.5223034 0.85275970 0 0.8527597 −0.5223034
Making the transform we get
A2 = P2A1P2 =
4 3.7416574 0 03.7416574 2.4285714 2.4846467 0
0 2.4846467 3.5714286 −1.87082870 0 −1.8708287 1
Thus we obtained a tridiagonal matrix, as we should in the case of a symmetric matrix.
As another example, we take an asymmetric matrix:
A =
4 2 3 13 4 −2 12 −1 1 −21 2 −2 2
The transformation proceeds as before, and the result is
A2 =
4 3.4743961 −1.3039935 −0.47767383.7416574 1.7857143 2.9123481 1.1216168
0 2.0934787 4.2422252 −1.03077590 0. −1.2980371 0.9720605
,
which is of the Hessenberg form.
QR-algorithm
We now have a simpler tridiagonal or Hessenberg matrix, which still has the same eigen-values as the original matrix. Let this transformed matrix be H. Then we can begin tosearch for the eigenvalues. This is done by iteration.
As an initial value, take A1 = H. Then repeat the following steps:
– Find the QR decomposition QiRi = Ai.
– Calculate a new matrix Ai+1 = RiQi.
The matrix Q of the QR decomposition is orthogonal and A = QR, and so R = QTA.Therefore
RQ = QTAQ
is a similarity transform that will conserve the eigenvalues.
The sequence of matrices Ai converges towards a upper tridiagonal or block matrix, fromwhich the eigenvalues can be picked up.
In the last example the limiting matrix after 50 iterations is
7.0363389 −0.7523758 −0.7356716 −0.38026314.265E − 08 4.9650342 −0.8892339 −0.4538061
0 0 −1.9732687 −1.32342020 0 0 0.9718955
The diagonal elements are now the eigenvalues. If the eigenvalues are complex numbers,the matrix is a block matrix, and the eigenvalues are the eigenvalues of the diagonal 2× 2submatrices. | 3,187 | 10,808 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.40625 | 4 | CC-MAIN-2023-23 | latest | en | 0.879856 |
https://forum.linuxcnc.org/38-general-linuxcnc-questions/35332-pairing-up-mesa-7i76e-with-vfd-delta-c2000?start=30 | 1,553,405,171,000,000,000 | text/html | crawl-data/CC-MAIN-2019-13/segments/1552912203326.34/warc/CC-MAIN-20190324043400-20190324065400-00085.warc.gz | 497,062,964 | 11,044 | # Pairing up Mesa 7i76e with VFD Delta C2000
09 Jan 2019 12:45 #123825 by andypugh
A line driver into a high-impedence oscilloscope might give a higher than specified voltage. You could try loading it with a resistor. It is rated 50mA, so I would first try a 1k resistor (5mA) and see if that squares-up and reduces the peak and noise in the signal.
To figure out the resolution, just watch the counts in halmeter, rotate the spindle 10x by hand, and divide the change in counts by 10.
09 Jan 2019 19:02 #123852 by vmihalca
I did some more research being helped by a friend that's more into electronics.
We have discovered that the board outputs differential outputs, and there is no common ground between the phases.
Also we discovered that the voltage is from -7.5 to +7.5 and not from 0 to 15
We created a voltage divider using two 500ohm resistors, and filtered the noise with a 1nF capacitor. We ended up with a 4.8v peak to peak.
But this voltage is from -2.4 to +2.4 and not from 0-4.8v.
Can I feed this signal into the mesa board? Or do I need a 0-5v signal.
09 Jan 2019 19:47 #123861 by andypugh
The Mesa inputs are selectable TTL or Differential, IIRC.
What you have is differential. (which is a good thing, it is more noise-resistant)
manual: www.mesanet.com/pdf/parallel/7i76man.pdf
Page 2 W4 W5 W6
09 Jan 2019 20:21 #123867 by vmihalca
So you're saying its ok to feed into the mesa board a signal that' from - to + and not from 0 to +?
I would not like to get my board fried.
I thought I need to supply a signal between 0 and 5V
09 Jan 2019 20:34 - 09 Jan 2019 20:35 #123868 by PCW
7I76E differential encoder inputs have a common mode range of -7 to +12V
and a differential sensitivity of 200 mV, and a maximum differential voltage of 5V,
So the will take positive and negative inputs as long as they are in that range
You must make sure that no input exceeds +13V or -9V or you may damage the input receiver chip. If you have large signals and large noise spikes, you should use your input divider. Note that the 7I76E encoder inputs are terminated with a 120 Ohm resistor across the differential inputs pairs, so you may need to adjust your divider accordingly. As long as you get about 2V across the inputs its good.
Last edit: 09 Jan 2019 20:35 by PCW.
10 Jan 2019 21:52 - 10 Jan 2019 21:54 #123924 by vmihalca
I have made today some measurements and some tests. I have discovered that the encoder board has a SG pin that I assumed it comes from 'signal ground'.
I have connected the ground of the probes to the ground, and then connected the probes to the A and A' with a 110ohm resistor between them to simulate the internal resistor that mesa has.
See the measurements in the pics below.
I am wondering if the noise that appears in the signal can get out of the -7/+12v range and if this could affect the mesa board.
Given how the signal looks now, do you think its safe to connect the wires to the Mesa's encoder input?
Also, do I need to connect the ground from the encoder to the GND encoder pin on the Mesa?
##### Attachments:
Last edit: 10 Jan 2019 21:54 by vmihalca.
10 Jan 2019 22:30 #123926
I would not connect that. You are using non shielded cable for signalling, shielding is a must in such machines, to many devices making to much electrical noise.
BTW, cable shielding should be connected only on one side, not both. Usually on the electronics side.
10 Jan 2019 23:08 #123931 by vmihalca
Do you think a double shielded FTP cable would do it?
You said about connecting only one side; here the sides I have are the Mesa board and this board that converts from the resolver signal.
To which of them should I connect the shielding? Where should I connect the shielding? To ground?
Sorry if I'm asking stupid questions, I'm a very beginner with the electronics.
11 Jan 2019 14:33 #123975
Electronics side.
Encoders or resolvers should be grounded at the Mesa input side ( or other board ), motors should be grounded at the driver side, sensors and switches also at the board side, etc | 1,068 | 4,025 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2019-13 | longest | en | 0.953484 |
https://blog.newtum.com/fibonacci-series-in-c-using-while-loop/ | 1,725,776,961,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700650960.90/warc/CC-MAIN-20240908052321-20240908082321-00619.warc.gz | 127,609,203 | 64,152 | # Fibonacci series in C using while loop
The Fibonacci series, a key mathematical sequence, holds profound importance in both mathematics and computer science. This blog explores its significance while delving into the fundamentals of the C programming language. Understanding loops, particularly the while loop, becomes pivotal in unraveling the magic behind the generation of the Fibonacci series in C.
```// Fibonacci series in C using while loop
#include <stdio.h>
int main()
{
int n1 = 0, n2 = 1, n3, count;
printf("Enter the limit \n");
scanf("%d", &count);
printf("\n%d\n%d\n", n1, n2);
count = count - 2;
while(count)
{
n3 = n1 + n2;
printf("%d\n", n3);
n1 = n2;
n2 = n3;
count = count - 1;
}
return 0;
}```
Know the Fibonacci Series in Python Using For Loop here!
Explanation of the code:
This C program generates a Fibonacci series using a while loop. It begins by initializing the first two numbers, n1 and n2, as 0 and 1, respectively.Â
The user is prompted to input the limit of the series, stored in the ‘count’ variable.Â
The initial values are then printed. Using a while loop, the program calculates the next Fibonacci number (n3) by adding the previous two (n1 and n2).Â
This process continues until the desired count is reached, with each new Fibonacci number being printed.Â
The loop iterates, updating variables to progress the series, and the program concludes after generating the specified Fibonacci sequence.
#### Output:
Explore the fascinating world of For Loop in C Check out!
``````Enter the limit 8
0
1
1
2
3
5
8
13``````
## Practical Applications of the Fibonacci Series
The Fibonacci series is not just a theoretical concept but has numerous practical applications across various fields:
#### a. Nature
• Phyllotaxis: The arrangement of leaves on a stem, the pattern of florets in a flower, and the spiral shells of mollusks often follow the Fibonacci sequence.
• Tree Branching: The branching patterns of trees and plants, as well as the arrangement of leaves around a stem, often adhere to the Fibonacci sequence to optimize sunlight exposure and nutrient absorption.
• Fruit Sprouts: Pineapples, pinecones, and other fruits exhibit Fibonacci patterns in their spirals and seed arrangements.
#### b. Arts and Architecture
• The Golden Ratio: Derived from the Fibonacci sequence, the Golden Ratio is used extensively in art and architecture to create aesthetically pleasing compositions. This ratio appears in the proportions of the Parthenon, the pyramids, and many Renaissance paintings.
• Spiral Galaxies: The shape of spiral galaxies in space follows the logarithmic spiral, closely related to the Fibonacci sequence.
#### c. Algorithm Design
• Dynamic Programming: Algorithms to compute Fibonacci numbers efficiently often use dynamic programming to store and reuse previously computed values, reducing computational time.
• Recursive Solutions: Recursive algorithms to generate Fibonacci numbers provide a clear example of recursion, demonstrating how functions can call themselves with modified parameters.
• Data Structures: Fibonacci heaps, a type of priority queue, are used in network optimization algorithms, such as Dijkstra’s shortest path algorithm.
#### d. Finance
• Stock Market Analysis: Fibonacci retracement levels are used in technical analysis to predict potential reversal points in the financial markets. These levels are based on key Fibonacci ratios derived from the sequence.
The widespread occurrence of the Fibonacci sequence in natural phenomena, art, and algorithm design underscores its fundamental importance. Understanding and applying this sequence can provide insights and solutions in diverse fields, showcasing the interplay between mathematics and the world around us.
In conclusion, the Fibonacci series, a mathematical marvel, finds applications across diverse fields. Its beauty lies in its recurrence and influence on nature, art, and algorithms. This exploration highlights the while loop’s pivotal role, showcasing its importance in unraveling the mesmerizing patterns of the Fibonacci sequence.
Newtum Online Compiler is a valuable resource offering quality education and support for programming and tech enthusiasts, facilitating effective learning. | 873 | 4,262 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2024-38 | latest | en | 0.857183 |
https://www.wikiwand.com/zh-sg/%E5%9B%9B%E7%B6%AD%E9%9B%BB%E6%B5%81%E5%AF%86%E5%BA%A6 | 1,624,592,749,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488567696.99/warc/CC-MAIN-20210625023840-20210625053840-00251.warc.gz | 951,034,421 | 28,165 | # 四维电流密度
## 维基百科，自由的百科全书
${\displaystyle J^{a}=(c\rho ,\mathbf {j} )=(c\rho ,j_{x},j_{y},j_{z})}$
${\displaystyle j}$是一般的电流密度，${\displaystyle \rho }$是电荷密度，${\displaystyle c}$是光速。
${\displaystyle D\cdot J=\partial _{a}J^{a}={\frac {\partial \rho }{\partial t))+\nabla \cdot \mathbf {j} =0}$
${\displaystyle J^{a}{}_{,a}=0\,}$
${\displaystyle J^{a}{}_{;a}=0\,}$广义相对论
${\displaystyle {\frac {\partial F^{\alpha \beta )){\partial x^{\alpha ))}=\mu _{0}J^{\beta ))$ | 220 | 466 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.046875 | 3 | CC-MAIN-2021-25 | longest | en | 0.222412 |
https://www.bbc.com/news/education-40826391 | 1,597,386,233,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439739177.25/warc/CC-MAIN-20200814040920-20200814070920-00260.warc.gz | 586,316,478 | 40,122 | # GCSE results: How the new grading system works
From this year, some GCSE papers in England are being graded numerically on a scale from 9 to 1.
Remember politicians saying GCSEs were too easy and needed to be made more rigorous?
Well that's what has been happening, as new exams have been developed and are being phased in over a number of years.
This year's students are the first to sit exams in the new GCSES in English (literature and language) and maths - arguably the biggest subjects in the curriculum.
Another 20 subjects will have 9 to 1 grading in 2018, with most others following in 2019.
## New numbering system
Contrary to what you might expect, 1 is not the highest grade, 9 is.
However, 9 will be awarded to fewer pupils than A* is currently.
In fact, three number grades, 9, 8 and 7, correspond to the current top grades of A* and A.
This is designed to give more differentiation at the top end.
A grade 6 is a bit higher than the old B grade.
## Two pass marks?
And - unlike the current system, under which a C is seen as a "good" pass - the new system has a "standard pass", a grade 4, and a "strong pass", grade 5.
The Department for Education has stressed that the old and new grading systems cannot be directly compared.
But there are points where they align.
The bottom of the current grade C and the bottom of the new grade 4 is probably the key point of alignment.
This has prompted some employers to say the system is confusing.
## Measuring schools
Schools will be measured on the proportion of pupils achieving grade 4 and above.
But they will also be measured on the proportion of pupils achieving grade 5 and above.
And this grade 5 is being described as the benchmark, in line with the expectations of the strong performing education systems around the world.
Education secretary Justine Greening said she expected more pupils to get a grade 5 over time as England's education system improved.
Meanwhile, at the bottom, the new system has less detail, with grades D, E and F corresponding to grades 1 and 2, and the bottom of a grade 1 corresponding to the bottom of a grade G.
## Harder GCSEs?
The short answer is yes.
The Department for Education has deliberately required exam boards to make exam content more "challenging".
The way the questions are set out in maths, for example, is very different to what has come before.
And the way the qualifications are taken, with exams at the end, rather than with modules - tested in stages along the way at different parts of the year - is also seen as more demanding.
But, and it is a big but, exam boards will use statistical measures to ensure that standards remain the same.
So despite what are really quite significant changes, broadly the same proportion of students that received a grade C or above in 2016 will receive a grade 4 and above this year.
## Is it fair?
And the same share of students will receive a grade 7 as received a grade A last year.
However, if candidates find an exam paper harder than expected the grade boundaries will be set a little lower.
So they may need a few less marks to get a certain grade than they would have done previously.
Thus, continuity of standards is maintained so that the first crop of students to take these exams are not disadvantaged.
However, some teachers' leaders have complained about the level of difficulty in the new exams.
Maths has been particularly tricky - with pupils from top sets of their schools coming out of exams saying they could not understand whole sections of the exam paper.
And head teachers say they are concerned about the scale of pressure being placed on pupils, due to the high stakes nature of these new exams.
## The rest of the UK
This year also marks a divergence in qualifications between the nations, with candidates in England, Wales and Northern Ireland now all studying different exams.
In Wales, exams in English, Welsh and maths (six GCSEs in total) have also been toughened, but the qualification is still taken in units. New GCSEs in other subjects are being phased in.
In Northern Ireland, pupils are generally sitting old-style GCSEs in all subjects this year, but changes are ahead.
Scottish students already study completely different qualifications.
## Related Internet links
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PGE 310 - HW6 -Solution
PGE 310 - HW6 -Solution - The University of Texas at Austin...
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1 The University of Texas at Austin PGE 310: Formulation and Solution in Geosystems Engineering Homework #6: Root Finding (Secant Method) and Solving Linear System of Equations SOLUTION By Hossein Roodi Questions 1 is solved for both cases: using global variables and not using global variables Question #1 Solution: Not Using Global Variables: 1. Root Finding: Using Secant Method (MATLAB PROGRAMMING) The Colebrook equation is used for friction factor in fully developed pipe flow: ! " # \$ ) Re 51 . 2 7 . 3 / log( 2 1 f D f % 0 ) Re 51 . 2 7 . 3 / log( 2 1 \$ " " \$ f D f y % Where f is friction factor, D / % is the relative roughness of the pipe material, and Re is Reynolds number based on pipe diameter D . You are supposed to solve this equation for f using secant method . a) Write a MATLAB function called “ Colebrook.m” that when sent Darcy factor ( f ), Relative roughness ( eD ) and Reynolds number ( Re ), it returns the value of function. - Note: In MATLAB, log (x) is reserved for natural log of x . For logarithm in base of 10 you are supposed to use log10(x).So in this problem if you write log instead of log10 , your root will be 0.0108 instead of 0.0572 and the number of iteration will change. b) Write a MATLAB function called “ secant.m” that when sent a function ( func ), initial guesses ( 0 x & 1 x ), it returns the root of equation ( root ) and the number of iterations. In your function file, choose a tolerance that you think is appropriate . Recall that secant method formulation is: ) ( ) ( ) )( ( 1 1 1 i i i i i i i x f x f x x x f x x # # # \$ # # " So, in each step two estimations are needed, like x i and x i-1 and the next guess for root is obtained by the above mentioned formulation. (x i+1 is obtained based on x i and x i-1 ) In the following function secant.m x0 and x1 are assigned as the two estimation values using in secant method formulation. Aftereach iteration x0 and x1 are updated.
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2 c) Write a script file called “ frictionfac.m ” to calculate the value of friction factor for the following values " Set 6 10 Re \$ , 03 . 0 / \$ D % , 006 . 0 0 \$ x and 007 . 0 1 \$ x Results: Print all your files and the results. Upload function files and script file on the Blackboard. (35 %)
3 Question #1 Solution Using Global Variables: If global variables are used for eD and Re , then it is not required to declare them as input variables neither in Colebrooks.m nor in secant.m . MATLAB assumes them global and recognizes them within all functions. To make them global, only thing you need to do is writing command“ global eD Re” at the beginning of your script file and your function files. Also, since Colebrook.m will have only one input, you need to erase them from the end of feval functions in secant.m .
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4 2. Developing System of Equations for a Real World Problem: (By HAND) An oil company called OILHW6 wants to make decision about pumping from three oil wells in one of its reservoirs in Texas. Unknowns are oil outputs from each well: Q1, Q2 and Q3 Q1 contains 30% Gasoline , 30% Diesel Fuel and 40% Kerosene Q2 contains 60% Gasoline , 20% Diesel Fuel and 20% Kerosene Q3 contains 50% Gasoline , 20% Diesel Fuel and 30% Kerosene In order to separate these chemicals, OILHW6 transfers Q1, Q2
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{[ snackBarMessage ]} | 980 | 3,767 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2018-17 | latest | en | 0.836214 |
http://www.lmfdb.org/Character/Dirichlet/65/47 | 1,571,780,648,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987824701.89/warc/CC-MAIN-20191022205851-20191022233351-00218.warc.gz | 282,930,036 | 6,483 | # Properties
Conductor 65 Order 4 Real No Primitive Yes Parity Even Orbit Label 65.f
# Related objects
Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(65)
sage: chi = H[47]
pari: [g,chi] = znchar(Mod(47,65))
## Basic properties
sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 65 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 4 Real = No sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Even Orbit label = 65.f Orbit index = 6
## Galois orbit
sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
## Values on generators
$$(27,41)$$ → $$(i,i)$$
## Values
-1 1 2 3 4 6 7 8 9 11 12 14 $$1$$ $$1$$ $$-1$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$
value at e.g. 2
## Related number fields
Field of values $$\Q(i)$$
## Gauss sum
sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 65 }(47,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{65}(47,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(47,r) e\left(\frac{2r}{65}\right) = -2.069323049+7.792169282i$$
## Jacobi sum
sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 65 }(47,·),\chi_{ 65 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{65}(47,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(47,r) \chi_{65}(1,1-r) = 1$$
## Kloosterman sum
sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 65 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{65}(47,·)) = \sum_{r \in \Z/65\Z} \chi_{65}(47,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right) = 2.2444794081i$$ | 777 | 1,918 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2019-43 | latest | en | 0.293755 |
https://docs.scipy.org/doc/numpy-1.9.3/reference/generated/numpy.argpartition.html | 1,601,125,247,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400241093.64/warc/CC-MAIN-20200926102645-20200926132645-00293.warc.gz | 342,338,740 | 3,307 | # numpy.argpartition¶
numpy.argpartition(a, kth, axis=-1, kind='introselect', order=None)[source]
Perform an indirect partition along the given axis using the algorithm specified by the kind keyword. It returns an array of indices of the same shape as a that index data along the given axis in partitioned order.
New in version 1.8.0.
Parameters: a : array_like Array to sort. kth : int or sequence of ints Element index to partition by. The kth element will be in its final sorted position and all smaller elements will be moved before it and all larger elements behind it. The order all elements in the partitions is undefined. If provided with a sequence of kth it will partition all of them into their sorted position at once. axis : int or None, optional Axis along which to sort. The default is -1 (the last axis). If None, the flattened array is used. kind : {‘introselect’}, optional Selection algorithm. Default is ‘introselect’ order : list, optional When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. Not all fields need be specified. index_array : ndarray, int Array of indices that partition a along the specified axis. In other words, a[index_array] yields a sorted a.
partition
Describes partition algorithms used.
ndarray.partition
Inplace partition.
argsort
Full indirect sort
Notes
See partition for notes on the different selection algorithms.
Examples
One dimensional array:
>>> x = np.array([3, 4, 2, 1])
>>> x[np.argpartition(x, 3)]
array([2, 1, 3, 4])
>>> x[np.argpartition(x, (1, 3))]
array([1, 2, 3, 4])
>>> x = [3, 4, 2, 1]
>>> np.array(x)[np.argpartition(x, 3)]
array([2, 1, 3, 4])
numpy.partition
numpy.argmax | 429 | 1,706 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2020-40 | latest | en | 0.715504 |
https://learnhtml.foobrdigital.com/the-relationship-between-path-difference-and-phase-difference/ | 1,702,218,615,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679102469.83/warc/CC-MAIN-20231210123756-20231210153756-00359.warc.gz | 400,051,866 | 42,471 | Categories
# The relationship between Path Difference and Phase Difference
Consider a progressive wave motion advancing in the positive direction of the x-axis
Path Difference vs Phase Difference
Let A and B be two points in the medium through which the wave passes.
The path difference between A and B is, x = x2 – x1
By the time the wave reaches B from A the phase of vibration of A has changed. The difference between the states of vibration of A and B is called phase difference (ΔO).
From this wave motion, if we consider any two consecutive crests c1 and c2, the path difference between them is λ, the time difference is T and the phase difference is 2π.
A path difference of (λ) corresponds to a phase difference of 2π, thus, a path difference (x) corresponds to the phase difference 2πr/λ.
Δϕ = (2πx)/λ = 2π/λ (path difference)
Where k = 2π/λ is called wave number or propagation constant of the wave motion.
A path difference (λ) corresponds to a time difference (T), therefore, a path difference (x) corresponds to a time difference of (x/λ)T.
The relations connecting the path difference, phase difference and time difference are given in the below table. | 283 | 1,178 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2023-50 | latest | en | 0.917501 |
https://ask.learncbse.in/t/use-the-binomial-series-to-expand-the-function-as-a-power-series/50692 | 1,600,632,903,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400198652.6/warc/CC-MAIN-20200920192131-20200920222131-00276.warc.gz | 298,453,228 | 3,088 | Use the binomial series to expand the function as a power series
use the binomial series to expand the function as a power series. What is the radius of convergence?
(8+x)^(1/3) | 45 | 178 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2020-40 | latest | en | 0.90608 |
https://www.jiskha.com/display.cgi?id=1320592695 | 1,516,369,706,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887981.42/warc/CC-MAIN-20180119125144-20180119145144-00571.warc.gz | 903,377,152 | 4,203 | # Science (Chemistry)
posted by .
I'm not sure how to do a couple of problems on my chemistry worksheet. Can you please help me?
You dissolve 0.00902 g of naOH in enough water to make 1000 mL. What is the Ph of this soultion.
What is the molarity, pH, and pOH of 5.00 of naOH in 750.0 mL of soultion. I know how to do the pH and the pOH usaully but how do I do it with the g?
Last question:
You take 2 mL of 18.0 M H2SO4 and add it to 998 mL of wate4r. What is the pH and pOH?
Can you please show me how to do these... I'm lost. Thanks
• Science (Chemistry) -
convert the grams to moles.
Molarity=mass/molmass / volumesolution
= .00902/40*1 = you do it.
Now the dissociation equation:
NaOH>>Na+ + OH-
completely dissociated, so the moles of NaOH become the concentration of Na+, and OH-
pOH= -log(molarity above)
pH= 14-pOH
same thing on the sulfuric acid, except the molarity determination.
moles of H2SO4=.002*18=.036
Molarity= .036/1.0=/036
or another way:
you are diluting it 1000/2=500 times, or new molarity= 18/500=.036M
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More Similar Questions | 902 | 2,979 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2018-05 | latest | en | 0.887519 |
https://studybay.com/blog/molar-mass-of-benzene/ | 1,718,578,782,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861671.61/warc/CC-MAIN-20240616203247-20240616233247-00831.warc.gz | 497,801,517 | 28,117 | # Benzene Molar Mass - How to Calculate Values?
By: Angelina Grin
7 min
0
10.04.2022
Chemistry belongs to the natural sciences and studies substances, their structure, properties, chemical compound, and their results, the laws by which all reactions and transformations operate. Since chemistry is a science unlimited, students have difficulty calculating atomic mass, molecular mass, number of moles. To show how to do the calculations quickly, we will look at the molar mass of benzene. However, the algorithm below can be applied to any substance.
## Benzene - What Is This Substance?
Chemical formula - C6H6
Molecular formula - C6H6
Empirical formula - CH
Molar mass is 78,11 g/mol.
Atomic weight is 78,11 g/mol.
Number of atoms: benzene is a hexagonal molecule of six carbon atoms and six hydrogen atoms.
Boiling point is + - 80,1 °C.
Melting point is + - 5,5 °C.
Benzene (phenyl hydrogen) is an organic compound in an uncolored liquid with a slightly pungent sweet odor. Benzene is a hydrocarbon from a number of the simplest arenes. The attention of benzene is toxic, carcinogenic, and contaminant.
It is generally accepted that benzene is a component of gasoline. Many companies actively use it in various industrial sectors. For example, scientists create medicinal and veterinary drugs, a wide range of plastics, synthetic rubber, dyes, etc. Although this substance is present in crude oil, it is synthesized from its other industry components.
### Physical Properties of Benzene
Benzene is a colorless, transparent, highly mobile liquid with aromatic compounds that burns with a bright smoky flame and forms flammable vapors. In the cold, it solidifies into a crystalline mass that melts at 6 ° C.
This compound is perfectly miscible with ether, chloride, cyclohexane, ethanol, methane, octane, oxide, toluene, xylene gasoline, and other organic solvents. Benzene is practically immiscible with water: no more than 0.08% dissolves at 22 ° C. Benzene is an excellent solvent for fats, resins, oils, asphalt, alkaloids, sulfur, phosphorus, iodine, etc. With some solvents, benzene forms azeotropic mixtures.
### Chemical Properties of Benzene
We want to pay special attention to the chemical structure of benzene. Each of the six carbon atoms in its molecule is in a state of sp2 hybridization and is bonded to two adjacent carbon atoms and a hydrogen atom by three chemical structure -bonds. The bond angles between each pair of π-bonds are equal to 120 °. Thus, the skeleton of ð-bonds is a regular hexagon, in which all carbon atoms and all ð-bonds C-C and H-H lie in the same plane:
Substitution reactions are inherent in benzene. It reacts with nitrate acid, olefins, chloroalkanes, halogens, and sulfate acid. The rupture of the benzene ring is carried out at rigid parameters of temperature and pressure.
When interacting with olefins, substances homologous to benzene are formed, in particular ethylbenzene and isopropyl benzene. Reaction with Cl2 and Br2 in the presence of a catalyst component gives phenyl chloride. Other reactions typical of this reagent include formylation, sulfonation, nitration, catalytic hydrogenation, oxidation, ozonolysis.
## How to Calculate the Molar Mass of Benzene?
It is effortless to calculate benzene molar mass. To do this, you can follow these steps:
1. Prepare the periodic table in advance. You can find this table in any chemical book or open it on the Internet. With this table, you can write out the valencies of chemical elements and the atomic mass unit.
2. The next step is to formulate a chemical formula. If you know what elements are included in the substance, you can draw up a formula using the periodic table. In our case, the elements form the following formula: C6H6.
3. Now it's time to start working with the periodic table. Open the table and write down each element's valence - C, H. You can also use the valence table to simplify the task. However, it is often not in chemistry books, so you will have to use the periodic table during the lesson.
4. Having written out the valencies of each element, you can start calculating. However, before doing this, write down the weight of each chemical. The weight is in the periodic table. The atomic weight of C is 12.0107, and H is 1.00784. The formula also includes numbers that need to be multiplied along with the mass. You should get the following formulas:
Mr (C6H6) = 6 · Ar (C) + 6 · Ar (H).
Mr (C6H6) = 6 · 12 + 6 · 1 = 72 + 6 = 78. Based on these formulas, you can see that to calculate the molar mass of ethanol, you need to add all the masses together.
5. You are now close to the final step. Determine the mass of one molecule. To do this, you need to use the Avogadro number and the total molar mass of benzene. As a result, you can get the following formula:
m (C6H6) = Mr (C6H6) / NA = 78 / 6,02 · 1023 = 12,9 · 1023g
It is essential to end the task with these calculations so that the assignment looks complete.
## Examples of Solving Tasks with Molar Mass of Benzene
In chemistry, you can find more complex tasks using the molar mass of benzene. Let's take a look at these tasks and examples of solutions.
Task: Calculate the mass of benzene formed due to dehydrogenation of 184.8 g of cyclohexane.
Decision: First, we need to draw up the reaction equation - C6H14 = C6H6 + 4H2. Now we need to find ν of cyclohexane. We divide the mass by the molar mass: 184,8 / 86 = 2,15 mol.
Since the ratio of cyclohexane and benzene in the equation is the same, then benzene will also have ν equal to 2,15 mol. Then we find the mass of benzene, according to the formula. We need to multiply ν by the molar mass: 2,15 · 78 = 167,7g.
Answer: The mass of benzene is 167,7 g.
Task: What mass of benzene (C6H6) was chlorinated in the light if 11,35 grams of hexachlorocyclohexane (C6H6Cl6) was obtained by chlorination of benzene which is 65% yield?
Decision: Let's write the reaction equation for benzene chlorination:
Recall that the reaction product's yield is understood as the ratio of the mass (number of moles) of the practically obtained substance to the mass theoretically calculated using the reaction equation:
Let's calculate the theoretical mass of hexachlorocyclohexane (C6H6Cl6) by the formula:
m (t) (C6H6) = 11,35 ⋅ 100/65 = 17,46 (g).
Taking into account that the molar mass of benzene (C6H6) and hexachlorocyclohexane (C6H6Cl6) is 78 g/mol and 220 g/mol, respectively, we find the mass of benzene (C6H6) using the reaction equation:
78 g of C6H6 participates in the formation of 220 g of C6H6Cl6
x g C6H6 participates in the formation of 17,46 g C6H6Cl6
From there, we get the following calculations:
Consequently, 6,19 grams of benzene was chlorinated.
Answer: Chlorination of benzene weighing 6,19 grams.
## How Is Benzene Mined?
The main ways to obtain pure benzene are:
• coking of coal raw materials;
• catalytic aromatization of petroleum gas fractions;
• pyrolysis of oil fractions (both gasoline and heavier);
• trimerization of ethine.
## Where Is Benzene Used?
Currently, benzene is mainly used in the production of ethylbenzene, cumene, and cyclohexane. Based on benzene, various intermediate products are still obtained, which are used in the future to get synthetic rubbers, plastics, synthetic fibers, dyes, insecticides, medicinal substances, etc.
Scientists also use benzene to produce some insecticides, blowing agents, and flame retardants. People use benzene as a solvent and extractant to manufacture varnishes and paints or as a highly active additive to motor fuels.
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Angelina Grin
Creative Writer and Blog Editor
Despite my relatively young age, I am a professional writer with more than 14 years of experience. I studied journalism at the university, worked for media and digital agencies, and organized several events for ed-tech companies. Yet for the last 6 years, I've worked mainly in marketing. Here, at Studybay, my objective is to make sure all our texts are clear, informative, and engaging. | 2,033 | 8,063 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2024-26 | latest | en | 0.873081 |
https://www.physicsforums.com/threads/strong-force-as-exchange-of-mesons-or-of-quark-and-antiquark.950414/ | 1,618,236,942,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038067400.24/warc/CC-MAIN-20210412113508-20210412143508-00324.warc.gz | 1,007,777,903 | 19,628 | # Strong force as exchange of mesons, or of quark and antiquark
• I
The (residual) strong force between nucleons can be desribed as
- The exchange of a meson (from a nucleon to the other), as in picture b)
- The exchange of a quark and an antiquark: in picture a) one nucleon "gives" a quark and receive an antiquark and it's the opposite for the other
I do no see how these two description are consistent with each other since in picture b) the meson (a quark + an antiquark) goes from one nucleon to the other, while in a) there is an exchange. So are these two interpretation equivalent? The nucleon that gives the quark also gives the antiquark (and therefore a meson) or receives it instead?
#### Attachments
• 2mK5f.png
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mfb
Mentor
Same thing. You don't have an actual time axis anyway in these diagrams, only initial and final states.
Here is your meson going from left to right in the left diagram:
#### Attachments
• meson.png
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crick
Same thing. You don't have an actual time axis anyway in these diagrams, only initial and final states.
Here is your meson going from left to right in the left diagram:
View attachment 227384
But there is actually a time axis and it is vertical (going from down to up). I'm aware that antiparticles move backwards in time (in the picture it moves downward a bit) but the fact is that it moves from left to right "in space".
Is this wrong? So actually (besides moving backwards in time) the direction in space indicated by the arrow in the feynman diagram should be reversed for an antiparticle?
Staff Emeritus
I'm aware that antiparticles move backwards in time
No they don't.
Klystron
mfb
Mentor
But there is actually a time axis and it is vertical (going from down to up).
The only meaningful times in a Feynman diagram are the initial and final states. Everything inside doesn't have a time ordering, and in fact you have to consider all times for all vertices for calculating such a diagram. | 497 | 1,993 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2021-17 | longest | en | 0.916165 |
https://analyticsindiamag.com/when-to-apply-l1-or-l2-regularization-to-neural-network-weights/?utm_source=rss&utm_medium=rss&utm_campaign=when-to-apply-l1-or-l2-regularization-to-neural-network-weights | 1,639,013,216,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363641.20/warc/CC-MAIN-20211209000407-20211209030407-00020.warc.gz | 170,271,366 | 32,817 | # When to Apply L1 or L2 Regularization to Neural Network Weights?
In the procedure of regularization, we penalize the coefficients or restrict the sizes of the coefficients which helps a predictive model to be less biased and well-performing. When we talk about neural networks, we can also apply the same procedure of regularization on the weights of the neural networks to make them efficient and robust. In this article, we will be discussing how we can perform L1 and L2 regularization of neural network weights and the effect of regularization on the neural networks. The major points that we will discuss here are listed below.
1. Problem with Larger Weights
2. Benefits of Small Weights
3. Penalizing Large Weights
1. Weight Size Calculation
2. Determine Amount of Attention
4. Where to Use Regularization?
Let’s begin the discussion by understanding the problems with larger weights in neural networks.
Problem with Larger Weights
In most of the neural networks, we see that we start the learning procedure of the neural networks using the parameters, especially the network weights. We use training data and stochastic gradient descent to make the network learned for a given task. So as much as we train the network we create more weights on the training data and this causes the overfitting of the network on the training data. Because as the training procedure increases the size of weights we used to learn increases so that the network can learn the example in the training data more efficiently.
The main motive behind any type of modelling procedure is to make the model that has large variance and less bias to a particular class in the training data. A network becomes more complex as the weight increases which can be considered as the sign of a network that is overly specialized in training data. Also, the overweight causes change in the outcome more than expected, if the changes occur in the input. That is why we are required to merge less weight in the network to make it more simple and task-oriented.
Another problem with the large weight is that the large weights are incompatible with the interrelationship between the variables. Many times we see that we have multiple input variables where every input variable has some relevance with the output variable. The relevance can be higher or lesser and a network with fewer weights are more capable of learning the relevance than the network with larger weights.
Benefits of Small Weights
By understanding the problem with large weights we got to know about the importance of the small weights. We can make the network use smaller weights in the learning. A simple way to perform this is to make the optimization process of the network consider the weight size with the loss in the calculation. We can also use the regularization methods which are focused on limiting the capacity of the models by adding penalties to the objective function.
More formally we can say that there will be a larger loss score because of a larger penalty if the weights used in the network are larger. After capturing the larger loss score the optimization process can make the model use smaller weights.
As we know that the small weights are more regular and not biased with any class this makes the model be benefitted from the performance and we can say that the penalties we are using for optimization are weights regularization. In the article, we have seen that this approach of penalizing model coefficients used in statistic models is considered shrinkage. Using penalties in such a model coefficient, we make the coefficient shrink during the optimization process.
When using the penalties with the coefficient of the neural network, we make the network pay less attention to the irrelevant input variables with the addition of less generalization error.
Penalizing Large Weights
As of now, we have seen in the above sections how a neural network with small weights is a more efficient way of modelling. But when talking about the training, increments in size of the weights are not in our hands but by penalizing them we can control the size. The procedure of penalizing the model can be completed into two parts where the first part belongs to the calculation of the size of the weights and in the second part, we define the amount of attention for optimization procedure that can be paid to the penalty. Both of the parts are necessary to be followed during the modelling of networks.
Weight Size Calculation
In neural networks, the weights are the real values that can be of any nature either positive or negative. So simply adding weights to them is not an appropriate approach. Ultimately here we are talking about the regularization process which can be defined as the process of performing restriction or regularization of the estimates where the features are estimated using coefficients of the models. We can say that the regularization in machine learning is a way to penalize the complex model which helps in reducing the overfitting and increases the performance of the models for new inputs by deploying the small weights into the modem or network. So the addition of weights or regularization can be performed by two main approaches that are listed as follows:
• L1 Regularization: Using this regularization we add an L1 penalty which is an absolute value of the magnitude of the coefficient or weights using which we restrict the size of coefficients. In regression analysis we mostly see the L1 penalty in the Lasso regression.
The above image is a mathematical representation of the lasso function where the function under the box is a representation of the L1 penalty.
• L2 Regularization: Using this regularization we add an L2 penalty which is basically square of the magnitude of the coefficient of weights and we mostly use the example of L2 penalty in the ridge regression.
the above image is a mathematical representation of the ridge function where the function under the box is a representation of the L2 penalty.
By the above definitions we can say that the L1 penalty tries to make the weights near to zero or zero if possible. This outcomes as the more sparse weights in the networks. Using L2 we can perform slight changes in the weights and using these penalties in regularization we can penalize large weights more severely.
In the field of neural networks we can say that the L2 penalty is used for decaying the weights that is why it is the most used approach for regularization or weight size reduction. There is one more approach we can use for weight addition or regularization in which we include both kinds of penalties which we see in elastic net regression. After this calculation we can proceed for the next part where we need to determine the amount of attention for the optimization process.
Determine Amount of Attention
###### The Rise Of An Empire: GEForce & The Dawn Of A New Decade 2021
In the above, we have seen how the L1 and L2 regularization helps in the calculation of the weights. After the calculation, we can add the size of weights to the function that we are using for optimization of the loss of the network and we can call this function a loss objective function.
Normally we don’t add each weight of the penalty directly. Before adding them we optimize them using the alpha or lambda parameters. Using these hyperparameters we control the learning process to give attention to the penalty
The value of the alpha hyperparameter varies between zero to one and if the value is in zero we say it as the no penalty and the value is in one we can say it as the full penalty and using the values the hyperparameters controls the model to being biased form a low amount bias to high amount bias.
Using The size of the penalty, the model can be controlled for performing the underfitting or overfitting of the training data. So if the size of the penalty is low we can allow the model to overfit and if the penalty is strong we can allow the model to underfit the training data.
To choose the type of regularization to use in the network we calculate the vector norm of the weights on each layer and this calculation makes the process of choosing alpha value flexible. Either we can use the alpha value as default for each layer or we can choose a different alpha value for each layer.
Where to Use Regularization
In the above section, we have seen how we can regularize the weights of the neural networks but we all know there are such conditions where we may need to apply the regularization techniques. Some of the conditions are listed below.
• We can use them with any neural network because it is a generic approach for making the model performance higher. But it is suggested to use especially with the LSTM models. It can be mostly used with sequential input and such connections which are recurrent.
• If in any situation the scale of input values are not similar we can use the regularization because of its great ability to update the input variable to have the same scale.
• We normally see that the large networks mostly become overfitted to the training data we can use for regularization with the large networks.
• Pre-trained neural networks are better with those data only on which they are trained and to use with newer data or different inputs we can use the regularization. It helps the network to perform a variety of data that are irrelevant to each other.
As we have seen in the article that L1 and L2 both of the regularization approach is useful and also we can apply both of them rather than choosing between them. In the regression procedure, we have seen the success of the elastic net where both of the penalties are used. We can also try this approach in neural networks. Also, we use the small values of the hyperparameter in the regularization that helps in controlling the contribution of each weight to the penalty. Manually assigning value to the hyperparameter we can use the grid search method for choosing the right hyperparameter for better performance.
Final Words
In this article, we have got an introduction to the problem of the neural networks in the context of the weights of the coefficients and we saw how the L1 and L2 regularization techniques from the regression analysis help us to find an optimal solution. Along with that we also saw the procedure of how we can perform the regularization and improve the neural network with situations where we can use the regularization to enhance the performance of the neural network.
What Do You Think?
`Join our Telegram Group. Be part of an engaging community` | 2,036 | 10,618 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2021-49 | latest | en | 0.939832 |
https://philpapers.org/browse/quantities/ | 1,713,027,703,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296816820.63/warc/CC-MAIN-20240413144933-20240413174933-00276.warc.gz | 440,984,406 | 47,822 | # Quantities
Edited by Zee R. Perry (New York University, Shanghai)
Summary Two characteristics distinguish quantities from non-quantitative properties and relations. First, every quantity is associated with a class of determinate “magnitudes” or “values” of that quantity, each member of which is a property or relation itself. So when a particle possesses mass or charge, it always instantiates one particular magnitude of mass or charge -- like 2.5 kilograms or 7 Coulombs. Second, the magnitudes of a given quantity (alternatively, the particulars which instantiate those magnitudes) exhibit “quantitative structure”, which comprises things like: ordering structure, summation/concatenation structure, ratio structure, directional structure, etc. We often represent quantities using similarly-structured mathematical entities, like numbers, vectors, etc. Classic debates about quantities concern attempts to give a metaphysical account of quantitative structure without appealing to mathematical entities/structures. Other questions include: How do quantities play the roles they do in measurement, the laws of nature, etc? Are a quantity's magnitudes fundamentally absolute (like 2.5 kilograms) or comparative (like twice-as-massive-as)?
Key works Mundy 1987 is a seminal paper in this area. Field 1980 and Field 1984 are not directly concerned with the metaphysics of quantity proper, but represent an early and very influential attempt to account for quantitative structure without relying on mathematics. The exchange between Bigelow et al 1988 and Armstrong 1988 is an influential treatment of the absolute/comparative debate in the metaphysics of quantity.
Introductions Eddon 2013 provides a very useful opinionated introduction.
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1. Nominalism and Immutability.Daniel Berntson - manuscript
Can we do science without numbers? How much contingency is there? These seemingly unrelated questions--one in the philosophy of math and science and the other in metaphysics--share an unexpectedly close connection. For as it turns out, a radical answer to the second leads to a breakthrough on the first. The radical answer is new view about modality called compossible immutabilism. The breakthrough is a new strategy for doing science without numbers. One of the chief benefits of the new strategy is (...)
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2. In this paper, I define and study an abstract algebraic structure, the dimensive algebra, which embodies the most general features of the algebra of dimensional physical quantities. I prove some elementary results about dimensive algebras and suggest some directions for future work.
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3. Comparativist Theories or Conspiracy Theories: the No Miracles Argument Against Comparativism.Caspar Jacobs - forthcoming - Journal of Philosophy.
Although physical theories routinely posit absolute quantities, such as absolute position or intrinsic mass, it seems that only comparative quantities such as distance and mass ratio are observable. But even if there are in fact only distances and mass ratios, the success of absolutist theories means that the world looks just as if there are absolute positions and intrinsic masses. If comparativism is nevertheless true, there is a sense in which it is a cosmic conspiracy that the world looks just (...)
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4. In Defence of Dimensions.Caspar Jacobs - forthcoming - British Journal for the Philosophy of Science.
The distinction between dimensions and units in physics is commonplace. But are dimensions a feature of reality? The most widely-held view is that they are no more than a tool for keeping track of the values of quantities under a change of units. This anti-realist position is supported by an argument from underdetermination: one can assign dimensions to quantities in many different ways, all of which are empirically equivalent. In contrast, I defend a form of dimensional realism, on which some (...)
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5. The Π-Theorem as a Guide to Quantity Symmetries and the Argument Against Absolutism.Mahmoud Jalloh - forthcoming - In Karen Bennett & Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics. Oxford: Oxford University Press.
In this paper a symmetry argument against quantity absolutism is amended. Rather than arguing against the fundamentality of intrinsic quantities on the basis of transformations of basic quantities, a class of symmetries defined by the Π-theorem is used. This theorem is a fundamental result of dimensional analysis and shows that all unit-invariant equations which adequately represent physical systems can be put into the form of a function of dimensionless quantities. Quantity transformations that leave those dimensionless quantities invariant are empirical and (...)
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6. Trope Bundle Theories of Substance.Markku Keinänen & Jani Hakkarainen - 2024 - In A. R. J. Fisher & Anna-Sofia Maurin (eds.), The Routledge Handbook of Properties. London: Routledge. pp. 239-249.
In this chapter, we provide an opinionated introduction to contemporary trope bundle theories of substance. We assess different trope bundle theories on the grounds of their two main aims: to provide an adequate account of substances or objects by means of tropes and a reductive analysis of inherence, that is, object's having tropes as their properties. Our discussion begins by a presentation of Donald C. Williams’ and Keith Campbell's paradigmatic trope theories, which maintain that tropes are independent existents. After highlighting (...)
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7. Ethics without numbers.Jacob M. Nebel - 2024 - Philosophy and Phenomenological Research 108 (2):289-319.
This paper develops and explores a new framework for theorizing about the measurement and aggregation of well-being. It is a qualitative variation on the framework of social welfare functionals developed by Amartya Sen. In Sen’s framework, a social or overall betterness ordering is assigned to each profile of real-valued utility functions. In the qualitative framework developed here, numerical utilities are replaced by the properties they are supposed to represent. This makes it possible to characterize the measurability and interpersonal comparability of (...)
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8. Dimensional Analysis: Essays on the Metaphysics and Epistemology of Quantities.Mahmoud Jalloh - 2023 - Dissertation, University of Southern California
This dissertation draws upon historical studies of scientific practice and contemporary issues in the metaphysics and epistemology of science to account for the nature of physical quantities. My dissertation applies this integrated HPS approach to dimensional analysis—a logic for quantitative physical equations which respects the distinct dimensions of quantities (e.g. mass, length, charge). Dimensional analysis and its historical development serve both as subjects of study and as a sources for solutions to contemporary problems. The dissertation consists primarily of three related (...)
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9. The nature of the physical and the meaning of physicalism.Mahmoud Jalloh - 2023 - Theoria. An International Journal for Theory, History and Foundations of Science 38 (2):205-223.
I provide an account of the physical appropriate to the task of the physicalist while remaining faithful to the usage of “physical” natural to physicists. Physicalism is the thesis that everything in the world is physical, or reducible to the physical. I presuppose that some version of this position is a live epistemic possibility. The physicalist is confronted with Hempel’s dilemma: that physicalism is either false or contentless. The proposed account of the physical avoids both horns and generalizes a recent (...)
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10. On the Challenges of Measurement in the Human Sciences.Cristian Larroulet Philippi - 2023 - Dissertation, University of Cambridge
Measurement practices are central to most sciences. In the human sciences, however, it remains controversial whether the measurement of human attributes—depression, happiness, intelligence, etc.—has been successful. Are, say, widely used depression questionnaires valid measuring instruments? Can we trust self-reported happiness scales to deliver quantitative measurements as it is sometimes claimed? These and related questions are till today hotly disputed. There are two main frameworks under which human measurements are studied and criticized. One is the so-called construct validity framework. Here, criticisms (...)
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11. Degrees of Consciousness.Andrew Y. Lee - 2023 - Noûs 57 (3):553-575.
Is a human more conscious than an octopus? In the science of consciousness, it’s oftentimes assumed that some creatures (or mental states) are more conscious than others. But in recent years, a number of philosophers have argued that the notion of degrees of consciousness is conceptually confused. This paper (1) argues that the most prominent objections to degrees of consciousness are unsustainable, (2) examines the semantics of ‘more conscious than’ expressions, (3) develops an analysis of what it is for a (...)
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12. On Mereology and Metricality.Zee R. Perry - 2023 - Philosophers' Imprint 23.
This article motivates and develops a reductive account of the structure of certain physical quantities in terms of their mereology. That is, I argue that quantitative relations like "longer than" or "3.6-times the volume of" can be analyzed in terms of necessary constraints those quantities put on the mereological structure of their instances. The resulting account, I argue, is able to capture the intuition that these quantitative relations are intrinsic to the physical systems they’re called upon to describe and explain.
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13. Aggregation Without Interpersonal Comparisons of Well‐Being.Jacob M. Nebel - 2022 - Philosophy and Phenomenological Research 105 (1):18-41.
This paper is about the role of interpersonal comparisons in Harsanyi's aggregation theorem. Harsanyi interpreted his theorem to show that a broadly utilitarian theory of distribution must be true even if there are no interpersonal comparisons of well-being. How is this possible? The orthodox view is that it is not. Some argue that the interpersonal comparability of well-being is hidden in Harsanyi's premises. Others argue that it is a surprising conclusion of Harsanyi's theorem, which is not presupposed by any one (...)
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14. Intrinsicality and determinacy.Erica Shumener - 2022 - Philosophical Studies 179 (11):3349-3364.
Comparativism maintains that physical quantities are ultimately relational in character. For example, an object’s having 1 kg rest mass depends on the relations it stands in to other objects in the universe. Comparativism, its advocates allege, reveals that quantities are not metaphysically mysterious: Quantities are reducible to familiar relations holding among physical objects. Modal accounts of intrinsicality—such as Lewis’s duplication account or Langton and Lewis’s combinatorial account—are popular accounts preserving many of our core intuitions regarding which properties are intrinsic. I (...)
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15. Some Consequences of Physics for the Comparative Metaphysics of Quantity.David John Baker - 2020 - In Karen Bennett & Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics Volume 12. Oxford University Press. pp. 75-112.
According to comparativist theories of quantities, their intrinsic values are not fundamental. Instead, all the quantity facts are grounded in scale-independent relations like "twice as massive as" or "more massive than." I show that this sort of scale independence is best understood as a sort of metaphysical symmetry--a principle about which transformations of the non-fundamental ontology leave the fundamental ontology unchanged. Determinism--a core scientific concept easily formulated in absolutist terms--is more difficult for the comparativist to define. After settling on the (...)
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16. A puzzle about rates of change.David Builes & Trevor Teitel - 2020 - Philosophical Studies 177 (10):3155-3169.
Most of our best scientific descriptions of the world employ rates of change of some continuous quantity with respect to some other continuous quantity. For instance, in classical physics we arrive at a particle’s velocity by taking the time-derivative of its position, and we arrive at a particle’s acceleration by taking the time-derivative of its velocity. Because rates of change are defined in terms of other continuous quantities, most think that facts about some rate of change obtain in virtue of (...)
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17. Heat in Renaissance Philosophy.Filip Buyse - 2020 - Encyclopedia of Renaissance Philosophy.
The term ‘heat’ originates from the Old English word hǣtu, a word of Germanic origin; related to the Dutch ‘hitte’ and German ‘Hitze’. Today, we distinguish three different meanings of the word ‘heat’. First, ‘heat’ is understood in colloquial English as ‘hotness’. There are, in addition, two scientific meanings of ‘heat’. ‘Heat’ can have the meaning of the portion of energy that changes with a change of temperature. And finally, ‘heat’ can have the meaning of the transfer of thermal energy (...)
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18. Measures of Similarity.Karin Enflo - 2020 - Theoria 86 (1):73-99.
This article analyses the relationship between the concept of single aspect similarity and proposed measures of similarity. More precisely, it compares eleven measures of similarity in terms of how well they satisfy a list of desiderata, chosen to capture common intuitions concerning the properties of similarity and the relations between similarity and dissimilarity. Three types of measures are discussed: similarity as commonality, similarity as a function of dissimilarity, and similarity as a joint function of commonality and difference. Relative to the (...)
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19. Riemann’s Scale: A Puzzle About Infinity.Øystein Linnebo - 2020 - Erkenntnis 88 (1):189-191.
Ordinarily, the order in which some objects are attached to a scale does not affect the total weight measured by the scale. This principle is shown to fail in certain cases involving infinitely many objects. In these cases, we can produce any desired reading of the scale merely by changing the order in which a fixed collection of objects are attached to the scale. This puzzling phenomenon brings out the metaphysical significance of a theorem about infinite series that is well (...)
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20. Newton on active and passive quantities of matter.Adwait A. Parker - 2020 - Studies in History and Philosophy of Science Part A 84:1-11.
Newton published his deduction of universal gravity in Principia (first ed., 1687). To establish the universality (the particle-to-particle nature) of gravity, Newton must establish the additivity of mass. I call ‘additivity’ the property a body's quantity of matter has just in case, if gravitational force is proportional to that quantity, the force can be taken to be the sum of forces proportional to each particle's quantity of matter. Newton's argument for additivity is obscure. I analyze and assess manuscript versions of (...)
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21. Boyle, Spinoza and Glauber: on the philosophical redintegration of saltpeter—a reply to Antonio Clericuzio.Filip A. A. Buyse - 2019 - Foundations of Chemistry 22 (1):59-76.
The so-called ‘redintegration experiment’ is traditionally at the center of the comments on the supposed Boyle/Spinoza controversy. A. Clericuzio influentially argued in his publications that, in De nitro, Boyle accounted for the ‘redintegration’ of saltpeter on the grounds of the chemical properties of corpuscles and “did not make any attempt to deduce them from mechanical principles”. By way of contrast, this paper argues that with his De nitro Boyle wanted to illustrate and promote his new corpuscular or mechanical philosophy, and (...)
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22. Dispositional and categorical properties, and Russellian Monism.Eric Hiddleston - 2019 - Philosophical Studies 176 (1):65-92.
This paper has two main aims. The first is to present a general approach for understanding “dispositional” and “categorical” properties; the second aim is to use this approach to criticize Russellian Monism. On the approach I suggest, what are usually thought of as “dispositional” and “categorical” properties are really just the extreme ends of a spectrum of options. The approach allows for a number of options between these extremes, and it is plausible, I suggest, that just about everything of scientific (...)
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23. Quantity Tropes and Internal Relations.Markku Keinänen, Antti Keskinen & Jani Hakkarainen - 2019 - Erkenntnis 84 (3):519-534.
In this article, we present a new conception of internal relations between quantity tropes falling under determinates and determinables. We begin by providing a novel characterization of the necessary relations between these tropes as basic internal relations. The core ideas here are that the existence of the relata is sufficient for their being internally related, and that their being related does not require the existence of any specific entities distinct from the relata. We argue that quantity tropes are, as determinate (...)
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24. The Problem of Fregean Equivalents.Joongol Kim - 2019 - Dialectica 73 (3):367-394.
It would seem that some statements like ‘There are exactly four moons of Jupiter’ and ‘The number of moons of Jupiter is four’ have the same truth-conditions and yet differ in ontological commitment. One strategy to resolve this paradoxical phenomenon is to insist that the statements have not only the same truth-conditions but also the same ontological commitments; the other strategy is to reject the presumption that they have the same truth-conditions. This paper critically examines some popular versions of these (...)
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25. A Trope Theoretical Analysis of Relational Inherence.Markku Keinänen - 2018 - In Jaakko Kuorikoski & Teemu Toppinen (eds.), Action, Value and Metaphysics - Proceedings of the Philosophical Society of Finland Colloquium 2018, Acta Philosophica Fennica 94. Helsinki: Societas Philosophica Fennica. pp. 161-189.
The trope bundle theories of objects are capable of analyzing monadic inherence (objects having tropes), which is one of their main advantage. However, the best current trope theoretical account of relational tropes, namely, the relata specific view leaves relational inherence (a relational trope relating two or more entities) primitive. This article presents the first trope theoretical analysis of relational inherence by generalizing the trope theoretical analysis of inherence to relational tropes. The analysis reduces the holding of relational inherence to the (...)
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26. Trooppiteoriat ja relaatiossa olemisen analyysi.Markku Keinänen - 2018 - Ajatus 75 (1):121-150.
Trope theories aim to eschew the primitive dichotomy between characterising (properties, relations) and characterized entities (objects). This article (in Finnish) presents a new trope theoretical analysis of relational inherence as the best way out of the impasse created by the alleged necessity to choose between an eliminativist and a primitivist ("relata-specific") view about relations in trope theory.
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27. Kinds of Tropes without Kinds.Markku Keinänen, Jani Hakkarainen & Antti Keskinen - 2018 - Dialectica 72 (4):571-596.
In this article, we propose a new trope nominalist conception of determinate and determinable kinds of quantitative tropes. The conception is developed as follows. First, we formulate a new account of tropes falling under the same determinates and determinables in terms of internal relations of proportion and order. Our account is a considerable improvement on the current standard account (Campbell 1990; Maurin 2002; Simons 2003) because it does not rely on primitive internal relations of exact similarity or quantitative distance. The (...)
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28. Metaphysics of Quantity and the Limit of Phenomenal Concepts.Derek Lam - 2018 - Inquiry: An Interdisciplinary Journal of Philosophy (3):1-20.
Quantities like mass and temperature are properties that come in degrees. And those degrees (e.g. 5 kg) are properties that are called the magnitudes of the quantities. Some philosophers (e.g., Byrne 2003; Byrne & Hilbert 2003; Schroer 2010) talk about magnitudes of phenomenal qualities as if some of our phenomenal qualities are quantities. The goal of this essay is to explore the anti-physicalist implication of this apparently innocent way of conceptualizing phenomenal quantities. I will first argue for a metaphysical thesis (...)
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29. Is There a Humean Account of Quantities?Phillip Bricker - 2017 - Philosophical Issues 27 (1):26-51.
Humeans have a problem with quantities. A core principle of any Humean account of modality is that fundamental entities can freely recombine. But determinate quantities, if fundamental, seem to violate this core principle: determinate quantities belonging to the same determinable necessarily exclude one another. Call this the problem of exclusion. Prominent Humeans have responded in various ways. Wittgenstein, when he resurfaced to philosophy, gave the problem of exclusion as a reason to abandon the logical atomism of the Tractatus with its (...)
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30. Macroscopic Metaphysics: Middle-Sized Objects and Longish Processes.Paul Needham - 2017 - Cham: Springer.
This book is about matter. It involves our ordinary concept of matter in so far as this deals with enduring continuants that stand in contrast to the occurrents or processes in which they are involved, and concerns the macroscopic realm of middle-sized objects of the kind familiar to us on the surface of the earth and their participation in medium term processes. The emphasis will be on what science rather than philosophical intuition tells us about the world, and on chemistry (...)
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31. The Metaphysics of Quantities and Their Dimensions.Bradford Skow - 2017 - In Karen Bennett & Dean W. Zimmerman (eds.), Oxford Studies in Metaphysics: Volume 10. Oxford University Press. pp. 171-198.
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32. Quantity of Matter or Intrinsic Property: Why Mass Cannot Be Both.Mario Hubert - 2016 - In Laura Felline, Antonio Ledda, F. Paoli & Emanuele Rossanese (eds.), New Developments in Logic and Philosophy of Science. London: College Publications. pp. 267–77.
I analyze the meaning of mass in Newtonian mechanics. First, I explain the notion of primitive ontology, which was originally introduced in the philosophy of quantum mechanics. Then I examine the two common interpretations of mass: mass as a measure of the quantity of matter and mass as a dynamical property. I claim that the former is ill-defined, and the latter is only plausible with respect to a metaphysical interpretation of laws of nature. I explore the following options for the (...)
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33. What Are Quantities?Joongol Kim - 2016 - Australasian Journal of Philosophy 94 (4):792-807.
ABSTRACTThis paper presents a view of quantities as ‘adverbial’ entities of a certain kind—more specifically, determinate ways, or modes, of having length, mass, speed, and the like. In doing so, it will be argued that quantities as such should be distinguished from quantitative properties or relations, and are not universals but are particulars, although they are not objects, either. A main advantage of the adverbial view over its rivals will be found in its superior explanatory power with respect to both (...)
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34. The Distinction between Primary Properties and Secondary Qualities in Galileo's Natural Philosophy.F. Buyse - 2015 - Cahiers du Séminaire Québécois En Philosophie Moderne / Working Papers of the Quebec Seminar in Early Modern Philosophy 1:20-45.
In Il Saggiatore (1623), Galileo makes a strict distinction between primary and secondary qualities. Although this distinction continues to be debated in philosophical literature up to this very day, Galileo's views on the matter, as well as their impact on his contemporaries and other philosophers, have yet to be sufficiently documented. The present paper helps to clear up Galileo's ideas on the subject by avoiding some of the misunderstandings that have arisen due to faulty translations of his work. In particular, (...)
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35. Maudlin on the Triangle Inequality.Marco Dees - 2015 - Thought: A Journal of Philosophy 4 (2):124-130.
Tim Maudlin argues that we should take facts about distance to be analyzed in terms of facts about path lengths. His reason is that if we take distances to be fundamental, we must stipulate that constraints like the triangle inequality hold, but we get these constraints for free if we take path lengths to be prior. I argue that Maudlin is mistaken. Even if we take path lengths as primitive, the triangle inequality follows only if we stipulate that the fundamental (...)
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36. Uninstantiated Properties and Semi-Platonist Aristotelianism.James Franklin - 2015 - Review of Metaphysics 69 (1):25-45.
A problem for Aristotelian realist accounts of universals (neither Platonist nor nominalist) is the status of those universals that happen not to be realised in the physical (or any other) world. They perhaps include uninstantiated shades of blue and huge infinite cardinals. Should they be altogether excluded (as in D.M. Armstrong's theory of universals) or accorded some sort of reality? Surely truths about ratios are true even of ratios that are too big to be instantiated - what is the truthmaker (...)
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37. A Trope Nominalist Theory of Natural Kinds.Markku Keinänen - 2015 - In Ghislain Guigon & Gonzalo Rodríguez Pereyra (eds.), Nominalism About Properties: New Essays. New York, NY: Routledge. pp. 156-174.
In this chapter, I present the first systematic trope nominalist approach to natural kinds of objects. It does not identify natural kinds with the structures of mind-independent entities (objects, universals or tropes). Rather, natural kinds are abstractions from natural kind terms and objects belong to a natural kind if they satisfy their mind-independent application conditions. By relying on the trope theory SNT (Keinänen 2011), I show that the trope parts of a simple object determine the kind to which it belongs. (...)
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38. Heavy Duty Platonism.Robert Knowles - 2015 - Erkenntnis 80 (6):1255-1270.
Heavy duty platonism is of great dialectical importance in the philosophy of mathematics. It is the view that physical magnitudes, such as mass and temperature, are cases of physical objects being related to numbers. Many theorists have assumed HDP’s falsity in order to reach their own conclusions, but they are only justified in doing so if there are good arguments against HDP. In this paper, I present all five arguments against HDP alluded to in the literature and show that they (...)
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39. Magnitudes: Metaphysics, Explanation, and Perception.Christopher Peacocke - 2015 - In Danièle Moyal-Sharrock, Volker Munz & Annalisa Coliva (eds.), Mind, Language and Action: Proceedings of the 36th International Wittgenstein Symposium. Boston: De Gruyter. pp. 357-388.
I am going to argue for a robust realism about magnitudes, as irreducible elements in our ontology. This realistic attitude, I will argue, gives a better metaphysics than the alternatives. It suggests some new options in the philosophy of science. It also provides the materials for a better account of the mind’s relation to the world, in particular its perceptual relations.
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40. Properly Extensive Quantities.Zee R. Perry - 2015 - Philosophy of Science 82 (5):833-844.
This article introduces and motivates the notion of a “properly extensive” quantity by means of a puzzle about the reliability of certain canonical length measurements. An account of these measurements’ success, I argue, requires a modally robust connection between quantitative structure and mereology that is not mediated by the dynamics and is stronger than the constraints imposed by “mere additivity.” I outline what it means to say that length is not just extensive but properly so and then briefly sketch an (...)
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41. Anaxagorae Homoeomeria.David Torrijos-Castrillejo - 2015 - Elenchos 36 (1):141-148.
Aristotle introduced in the history of the reception of Anaxagoras the term “homoiomerous”. This word refers to substances whose parts are similar to each other and to the whole. Although Aristotle’s explanations can be puzzling, the term “homoiomerous” may explain an authentic aspect of Anaxagoras’ doctrine reflected in the fragments of his work. Perhaps one should find a specific meaning for the term “homoiomerous” in Anaxagoras, somewhat different from the one present in Aristotle. This requires a review of the sense (...)
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42. Intrinsic Explanations and Numerical Representations.M. Eddon - 2014 - In Robert M. Francescotti (ed.), Companion to Intrinsic Properties. De Gruyter. pp. 271-290.
In Science Without Numbers (1980), Hartry Field defends a theory of quantity that, he claims, is able to provide both i) an intrinsic explanation of the structure of space, spacetime, and other quantitative properties, and ii) an intrinsic explanation of why certain numerical representations of quantities (distances, lengths, mass, temperature, etc.) are appropriate or acceptable while others are not. But several philosophers have argued otherwise. In this paper I focus on arguments from Ellis and Milne to the effect that one (...)
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43. An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structure.James Franklin - 2014 - London and New York: Palgrave MacMillan.
An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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44. Galileo and Spinoza.F. Buyse (ed.) - 2013 - Routledge.
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45. Absolutism vs Comparativism About Quantity.Shamik Dasgupta - 2013 - Oxford Studies in Metaphysics 8:105-150.
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46. Fundamental Properties of Fundamental Properties.M. Eddon - 2013 - In Karen Bennett Dean Zimmerman (ed.), Oxford Studies in Metaphysics, Volume 8. pp. 78-104.
Since the publication of David Lewis's ''New Work for a Theory of Universals,'' the distinction between properties that are fundamental – or perfectly natural – and those that are not has become a staple of mainstream metaphysics. Plausible candidates for perfect naturalness include the quantitative properties posited by fundamental physics. This paper argues for two claims: (1) the most satisfying account of quantitative properties employs higher-order relations, and (2) these relations must be perfectly natural, for otherwise the perfectly natural properties (...)
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47. Quantitative Properties.M. Eddon - 2013 - Philosophy Compass 8 (7):633-645.
Two grams mass, three coulombs charge, five inches long – these are examples of quantitative properties. Quantitative properties have certain structural features that other sorts of properties lack. What are the metaphysical underpinnings of quantitative structure? This paper considers several accounts of quantity and assesses the merits of each.
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48. Quantity and number.James Franklin - 2013 - In Daniel Novotný & Lukáš Novák (eds.), Neo-Aristotelian Perspectives in Metaphysics. London: Routledge. pp. 221-244.
Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.
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49. Multi‐track dispositions.Barbara Vetter - 2013 - Philosophical Quarterly 63 (251):330-352.
It is a familiar point that many ordinary dispositions are multi-track, that is, not fully and adequately characterisable by a single conditional. In this paper, I argue that both the extent and the implications of this point have been severely underestimated. First, I provide new arguments to show that every disposition whose stimulus condition is a determinable quantity must be infinitely multi-track. Secondly, I argue that this result should incline us to move away from the standard assumption that dispositions are (...)
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50. The ontological distinction between units and entities.Gordon Cooper & Stephen M. Humphry - 2012 - Synthese 187 (2):393-401.
The base units of the SI include six units of continuous quantities and the mole, which is defined as proportional to the number of specified elementary entities in a sample. The existence of the mole as a unit has prompted comment in Metrologia that units of all enumerable entities should be defined though not listed as base units. In a similar vein, the BIPM defines numbers of entities as quantities of dimension one, although without admitting these entities as base units. (...)
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1 — 50 / 126 | 7,488 | 34,810 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2024-18 | latest | en | 0.878706 |
https://robotics.stackexchange.com/questions/2068/my-raspberry-pi-is-losing-power-in-a-surge | 1,696,068,078,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510671.0/warc/CC-MAIN-20230930082033-20230930112033-00327.warc.gz | 523,540,145 | 44,800 | My Raspberry Pi is losing power in a surge
I have an RC car. The battery provides power to the ESC and then the ESC provides 6 V back out to the receiver. Instead of the receiver I have a Raspberry Pi, which uses the 6 V, steps it down to 5 V and provides power to the Raspberry Pi.
The problem
Every time we go full power*, there is a lack of voltage and the Raspberry Pi seems to hard reset.
* By full power we mean direct to 100% and not ranging from 0-100
I am not an expert in electrical circuits, but some of the suggestions are to use a capacitor to provide the missing 5 V in the interim. How do I prevent the Raspberry Pi from dying in the event of full power?
• Suspect the problem is that "full power" is using too much current, therefore the supply voltage is collapsing, causing a brown out. You don't say what you are using ti "step down to 5V" either. Jan 14, 2014 at 19:55
• But in either case, this is very borderline (if not over it) for being Off-Topic here...! Jan 14, 2014 at 19:55
• The simplest solution really is use a separate battery for the PI
– dm76
Jan 15, 2014 at 12:23
Since you're running directly from a battery I would say it's safe to just add as much decoupling (in other words caps across your input power) as possible, since the only real downside (that I think is relevant to your setup) to adding a lot of capacitance is increased in-rush current (since the capacitor naturally acts as a short-circuit during charge-up).
In certain circumstances however the high current draw associated with excessive decoupling during initial start-up should be avoided. An example is when you use a switching converter to step a voltage up/down. If the converter has built in over current protection (and no slow start feature), the short-circuit caused by the uncharged caps will cause the converter to stagger (starting up, over current and starting up again), and never fully reach its target voltage. However, since you're running directly from a battery this shouldn't be a problem, since a battery can be driven way above it's rated current capacity (for short periods).
Another thing to also remember is that since there exists a large energy store across your power rails (the capacitors), the system might take a while to discharge (after being powered off). In other words, your Pi will probably run for another 30 seconds (depending on how much capacitance you add) or so after you disconnect your main battery.
Finally, always try to add capacitors that's rated at at least twice your operating voltage (for instance, if you have 6V batteries try to get 16V caps). Motors that reverses their direction very quickly may induce sufficiently large voltage spikes back into your system, and cause your caps to explode (hopefully your motor driver has sufficient clamping diodes).
I would say a single 1000 uF electrolytic cap would be more than enough decoupling. If your Pi continues to brown-out, I guess the more appropriate cause would be your batteries not being able to supply the required current. Remember, the reason you Pi is restarting (or browning out) is due to the supply voltage dipping because the batteries are unable to supply the current the motors require. Adding capacitors will help with surges in currents (such as motors accelerating), but obviously won't solve long term high current draw.
• Hey thanks! Would a 1000uF 50V capacitor be good? I know you said get a double but I assume getting double or more is OK, and if so would the charge not stack up till it reaches 50V and wouldnt that then discharge at 50V or would it stick to 6V given the source is only 6V. Nov 15, 2013 at 14:36
• Sure, the larger the voltage rating the better. The voltage rating would have an effect on the physical dimensions of the cap, the larger the voltage rating the larger the cap. The capacitor voltage rating indicates up to what voltage the capacitor can safely be operated, not to what voltage the capacitor will be charged (that is dependent on your input voltage). In other words, if your supply voltage is nominally 6V, the capacitor will not charge up beyond 6V (ie, not to its rated voltage). Nov 15, 2013 at 21:40
• Using just a capacitor em parallel, it will discharge with the motor load. It will probably get better results putting a diode and the the capacitor. But this will add the voltage drop to the capacitors, so a low forward voltage drop is important. Jan 12, 2014 at 1:55
It sounds like you're experiencing a "brown out" caused when the excessive current draw from the battery causes a drop in the supply voltage. This is due to the fact that batteries have internal resistance (a.k.a output impedance).
In this example, if the load drops to $0.2\Omega$, the internal resistance of the battery will cause the output voltage to be divided equally with the external resistance -- the load will only see 5V. It's possible that your 6V supply is dropping to 3V (or less) under load, in much the same way.
A capacitor will delay this effect, but what you really need is a voltage regulator -- many are available in an integrated circuit (IC) package. Some voltage regulators will step up voltage when insufficient, but most simply lower a supply voltage to the voltage needed by a component like your RPi (in which case, your battery voltage would have to be increased so that it never dipped below 5V under full motor load).
Alternatively, you could use a separate battery pack for the RPi. This is a common solution for mobile robots, because it ensures that when the robot is immobilized due to lack of power, radio communication with the onboard PC will not be lost.
• IC From? sorry not good with abbreviations and google spurts out alot of stuff for "IC" Nov 15, 2013 at 23:03
• Integrated circuit, or chip etc. Texas Instruments' TPS family is an example of a popular switching converter available on the market. Nov 16, 2013 at 6:33
• We already have a Voltage Step down (goo.gl/8YviO4), which converts the input of 6V to 5V. So the V going into the Pi is more than enough it is still sucking more than it needs, Maybe replacing the ESC to better handle full power surges might be a good idea. Or would a high V battery help? Nov 17, 2013 at 14:41
• I've updated the question with a bit on "internal resistance", which should help answer your questions. A higher voltage battery may help prevent the problems you see, if the problem is based on heavy load. Note that you get maximum power when the internal and external impedances are equal, so you should never have a reason to cause your battery voltage to drop to less than half of its nominal voltage. Some batteries are rated for even less load than that.
– Ian
Nov 17, 2013 at 23:14
I must agree with the other two answers, however the main issue is that you do not have enough voltage into your regulator (I see from your comment to Ian that you are using a Pololu D15V35F5S3 Regulator). If you refer to the Pololu D15V35F5S3 Product Description, down at the bottom you will find the following graph:
Looking at the red line for 5V output: Note for all currents greater than zero, the dropout voltage is greater than 1V. (The minimum input voltage necessary to achieve 5V output is 5V + dropout voltage.) The more current used by your 5V loads (Pi), the greater the dropout. The problem is compounded by any voltage drop in your 6V source due to current surges (see Ian's answer).
You either need a higher input voltage, a lower dropout regulator (this may be difficult and insufficient), a different regulator (buck-boost), or a different power source for the Pi.
• Ah, very true, didn't see the reference to the Pololu converter (I originally thought he was referring to the on-board LDO). Although, it does seem strange that the Pi is even starting up, since the board supposedly consumes 500 mA at 5V with no external peripherals (and according to the graph 500 mA requires about a 1.5V drop, but I guess the converter is right at its limit). Nov 18, 2013 at 14:09
• Then again, I'm guessing the 500 mA refers to peak current requirements, which is probably rare enough. Nov 18, 2013 at 14:19
• @EDDY74 I think the main thing to notice here is there is no headroom. Any dip in input voltage or fluctuation in output current will likely result in a voltage change. Without an increase in input voltage, it's not really working as a voltage regulator.
– Tut
Nov 18, 2013 at 14:35
• Hey thanks, This helps alot, I can try increasing V that should be the easiest solution currently Nov 19, 2013 at 0:35 | 1,998 | 8,550 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2023-40 | latest | en | 0.952237 |
https://jackuldrich.com/blog/future/take-a-swim-in-the-lake-of-tomorrow/ | 1,597,071,417,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439736057.87/warc/CC-MAIN-20200810145103-20200810175103-00442.warc.gz | 295,312,177 | 15,798 | Take a Swim in the Lake of Tomorrow | Jack Uldrich
Take a Swim in the Lake of Tomorrow
Posted in Analogy, Books, Future, Futurist
One of my favorite stories about the power of exponential growth is a story about the lake and the water lily, and it can help anyone that needs to be jolted out of their current—or what I called their linear—mode of thinking.
In a nutshell, here’s the story: Imagine a lake that sprouts a single lily on June 1 and its splits into equal size lilies every day for a month. Further assume that the lilies of are such a size that at the end of the month the entire lake is covered with the pesky aquatic plants.
Under such a scenario what percentage of the lake do you imagine would be covered on June 20th—or two-thirds of the way into this exercise? One percent? Five percent? Ten percent? Perhaps higher?
I am sorry to say that not only are all of the above guesses wrong, they are in the words of my old teacher, Mr. Senta, “dangerously wrong.” By Day 20, lilies cover roughly 0.01 of the lake—a wee one-tenth of one percent.
What transpires in the next 10 days, though, is nothing short of transformational. Here’s the math (some of the numbers have been rounded slightly):
Day 20: .01%
Day 21: .02%
Day 22: .04%
Day 23: .078%
Day 24: 1.56%
Day 25: 3.125%
Day 26: 6.25%
Day 27: 12.5%
Day 28: 25%
Day 29: 50%
Day 30: 100%
I recount this story because it reveals a common misunderstanding about exponential trends. In the beginning, most people don’t even recognize the trend as exponential. For instance, a single lily growing to cover one-tenth of one percent of a lake hardly seems noteworthy, let alone deserving of special attention.
The problem with this negligence is that it can cause people to ignore or dismiss some very big and significant trends. All the while exponential math continues to weave its inextricable magic. Unfortunately, all too often, by the time people finally do grasp how fast things are progressing–say on Day 28 of the pond example–and hope to either capitalize on its explosive growth or, alternatively, avoid being overwhelmed by its growing power, it is too late.
Here’s the point. The forces that I mentioned earlier: computers, data storage, the sequencing of the human, brain scanning, artificial intelligence; genetic algorithms, robotics, nanotechnology and knowledge itself, have been and are all continuing to advance at astounding rates; yet, today, they cover only one-tenth of one percent of the proverbial lake.
It is essential, therefore, that the “Exponential Executive” think of today as being the metaphorical equivalent of “Day 20” in the pond analogy. The really big developments are still a few years off in the future, but they are coming fast and the time to begin preparing yourself and your organization for this is now.
To survive, you will need to learn how to “Jump the Curve.”
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A Futurist’s First Lesson
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# the value of $\underset { n\rightarrow \infty }{ lim } \sum _{ r=1 }^{ n }{ \frac { 1 }{ \sqrt { { n }^{ 2 }-{ r }^{ 2 }{ } } } }$
${ sin }^{ -1 }x$
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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
#### Realted Questions
Q1 Subjective Medium
Evaluate: $\displaystyle\int {\dfrac{1+\cos 4x}{\cot x-\tan x}dx}$
1 Verified Answer | Published on 17th 09, 2020
Q2 Subjective Medium
Evaluate
$\int { \dfrac { x+1 }{ { x }^{ 2 }+3x+12 } } dx$
1 Verified Answer | Published on 17th 09, 2020
Q3 Single Correct Medium
Evaluate : $\displaystyle\int^2_0\sqrt{6x+4}dx$
• A. $\dfrac{64}{9}$
• B. $7$
• C. $\dfrac{60}{9}$
• D. $\dfrac{56}{9}$
1 Verified Answer | Published on 17th 09, 2020
Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle\int_{0}^{1}\sqrt x \ dx$
Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$ | 424 | 1,110 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2021-25 | latest | en | 0.516735 |
https://www.hackmath.net/en/calculator/mixed-fractions?input=2+1%2F2+%2B+4+2%2F3 | 1,713,347,255,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817146.37/warc/CC-MAIN-20240417075330-20240417105330-00739.warc.gz | 710,001,777 | 7,714 | # Calculator addition of two mixed numbers
This calculator performs basic and advanced operations with mixed numbers, fractions, integers, and decimals. Mixed numbers are also called mixed fractions. A mixed number is a whole number and a proper fraction combined, i.e. one and three-quarters. The calculator evaluates the expression or solves the equation with step-by-step calculation progress information. Solve problems with two or more mixed numbers fractions in one expression.
### 21/2 + 42/3 = 43/6 = 7 1/6 ≅ 7.1666667
Spelled result in words is seven and one sixth (or forty-three sixths).
### Calculation steps
1. Conversion a mixed number 2 1/2 to a improper fraction: 2 1/2 = 2 1/2 = 2 · 2 + 1/2 = 4 + 1/2 = 5/2
To find a new numerator:
a) Multiply the whole number 2 by the denominator 2. Whole number 2 equally 2 * 2/2 = 4/2
b) Add the answer from the previous step 4 to the numerator 1. New numerator is 4 + 1 = 5
c) Write a previous answer (new numerator 5) over the denominator 2.
Two and one half is five halfs.
2. Conversion a mixed number 4 2/3 to a improper fraction: 4 2/3 = 4 2/3 = 4 · 3 + 2/3 = 12 + 2/3 = 14/3
To find a new numerator:
a) Multiply the whole number 4 by the denominator 3. Whole number 4 equally 4 * 3/3 = 12/3
b) Add the answer from the previous step 12 to the numerator 2. New numerator is 12 + 2 = 14
c) Write a previous answer (new numerator 14) over the denominator 3.
Four and two thirds is fourteen thirds.
3. Add: 5/2 + 14/3 = 5 · 3/2 · 3 + 14 · 2/3 · 2 = 15/6 + 28/6 = 15 + 28/6 = 43/6
It is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common multiple of both denominators - LCM(2, 3) = 6. It is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 2 × 3 = 6. In the following intermediate step, it cannot further simplify the fraction result by canceling.
In other words - five halfs plus fourteen thirds is forty-three sixths.
## What is a mixed number?
A mixed number is an integer and fraction whose value equals the sum of that integer and fraction. For example, we write two and four-fifths as . Its value is . The mixed number is the exception - the missing operand between a whole number and a fraction is not multiplication but an addition: . A negative mixed number - the minus sign also applies to the fractional . A mixed number is sometimes called a mixed fraction. Usually, a mixed number contains a natural number and a proper fraction, and its value is an improper fraction, that is, one where the numerator is greater than the denominator.
## How do I imagine a mixed number?
We can imagine mixed numbers in the example of cakes. We have three cakes, and we have divided each into five parts. We thus obtained 3 * 5 = 15 pieces of cake. One piece when we ate, there were 14 pieces left, which is of cake. When we eat two pieces, of the cake remains. | 815 | 3,012 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.3125 | 4 | CC-MAIN-2024-18 | latest | en | 0.867173 |
http://learnmech.com/introduction-to-governors-types-of-governor/ | 1,547,910,587,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583668324.55/warc/CC-MAIN-20190119135934-20190119161934-00082.warc.gz | 118,602,477 | 22,988 | Introduction To Governors
Flywheel which minimizes fluctuations of speed within the cycle but it cannot minimize fluctuations due to load variation. This means flywheel does not exercise any control over mean speed of the engine. To minimize fluctuations in the mean speed which may occur due to load variation, governor is used. The governor has no influence over cyclic speed fluctuations but it controls the mean speed over a long period during which load on the engine may vary.
The function of governor is to increase the supply of working fluid going to the primemover when the load on the prime-mover increases and to decrease the supply when the load decreases so as to keep the speed of the prime-mover almost constant at different loads.
Fig. Governor
Example: when the load on an engine increases, its speed decreases, therefore it becomes necessary to increase the supply of working fluid. On the other hand, when the load on the engine decreases, its speed increases and hence less working fluid is required. When there is change in load, variation in speed also takes place then governor operates a regulatory control and adjusts the fuel supply to maintain the mean speed nearly constant. Therefore, the governor automatically regulates through linkages, the energy supply to the engine as demanded by variation of load so that the engine speed is maintained nearly constant.
CLASSIFICATION OF GOVERNORS
The broad classification of governor can be made depending on their operation.
(a) Centrifugal governors
(b) Inertia and flywheel governors
(c) Pickering governors.
Centrifugal Governors
In these governors, the change in centrifugal forces of the rotating masses due to change in the speed of the engine is utilized for movement of the governor sleeve. One of this type of governors is shown in Figure. These governors are commonly used because of simplicity in operation.
Fig. Centrifugal Governor
Inertia and Flywheel Governors
In these governors, the inertia forces caused by the angular acceleration of the engine shaft or flywheel by change in speed are utilized for the movement of the balls. The movement of the balls is due to the rate of change of speed in stead of change in speed itself as in case of centrifugal governors. Thus, these governors
are more sensitive than centrifugal governors.
Pickering Governors
This type of governor is used for driving a gramophone. As compared to the centrifugal governors, the sleeve movement is very small. It controls the speed by dissipating the excess kinetic energy. It is very simple in construction and can be used for a small machine.
Types of Centrifugal Governors
Depending on the construction these governors are of two types :
(a) Gravity controlled centrifugal governors, and
(b) Spring controlled centrifugal governors.
Gravity Controlled Centrifugal Governors
In this type of governors there is gravity force due to weight on the sleeve or weight of sleeve itself which controls movement of the sleeve. These governors are comparatively larger in size.
Spring Controlled Centrifugal Governors
In these governors, a helical spring or several springs are utilised to control the movement of sleeve or balls. These governors are comparatively smaller in size. | 667 | 3,246 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2019-04 | longest | en | 0.909719 |
http://www.jiskha.com/display.cgi?id=1334792658 | 1,493,614,719,000,000,000 | text/html | crawl-data/CC-MAIN-2017-17/segments/1492917127681.50/warc/CC-MAIN-20170423031207-00141-ip-10-145-167-34.ec2.internal.warc.gz | 568,596,711 | 3,685 | # statistics
posted by on .
The ages (in years) of 10 infants and the number of hours each slept in a day
Age, x: 0.1, 0.2, 0.4, 0.7, 0.6, 0.9, 0.1, 0.2, 0.4, 0.9
Hours slept, y: 14.9, 14.5, 13.9, 14.1, 13.9, 13.7, 14.3, 13.9, 14.0, 14.1
Find the equation of the regression line. Then use the regression equation to predict the value of y for the given x, if meaningful. If it is not meaningful, explain why.
a. X-0.3years b. X=3.9years c. X=0.6years d. X=0.8years | 206 | 466 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2017-17 | latest | en | 0.742919 |
https://www.testingdocs.com/study/courses/flowgorithm-beginner/lesson/flowgorithm-assign-statement/ | 1,723,444,650,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641036271.72/warc/CC-MAIN-20240812061749-20240812091749-00625.warc.gz | 777,321,458 | 55,535 | Course Content
Introduction to Computer System
Introduction to Computer System
0/5
Computer Software
Computer Software
0/3
Introduction to Flowcharts
Introduction to Flowcharts
0/5
Introduction to Flowgorithm
Introduction to Flowgorithm
0/5
Flowgorithm Installation
Flowgorithm Installation on Windows
0/5
Flowgorithm Features
Flowgorithm Features
0/1
Flowgorithm User Interface
Flowgorithm User Interface
0/3
Flowgorithm Flowchart Shapes
Flowgorithm Flowchart Shapes
0/1
Flowgorithm Data Types
Flowgorithm Data Types
0/1
0/1
Flowgorithm I/O
Flowgorithm Input/Output
0/3
Flowgorithm Variables
Flowgorithm Variables
0/6
Flowgorithm Numbers
Flowgorithm Numbers
0/1
Flowgorithm Strings
Flowgorithm Strings
0/1
Flowgorithm File Format
Flowgorithm File Format
0/1
Flowgorithm Program Attributes
Flowgorithm Program Attributes
0/1
IPO Chart
IPO Chart ( Input, Process and Output Chart )
0/2
First Flowgorithm Flowchart
First Flowgorithm Flowchart
0/2
0/2
Flowgorithm Lab
This is lab activity to create and run flowcharts using Flowgorithm Software.
0/6
Flowgorithm Quiz
Flowgorithm Quiz
0/1
Flowgorithm Beginner Exercises
Flowgorithm Beginner Exercises
0/1
Flowgorithm Beginner
## Assign Shape
The Assign shape or symbol assigns data to the variables in the flowchart. The Assign symbol is the rectangle shape.
The Assign symbol is the process block. You perform calculations and can assign value to the variable. It is where a variable is given a new value.
## Assignment Properties window
Give the variable on the right-hand side Variable: box.
Specify the expression or data in the Expression: left-hand side box.
Click on the OK button.
Join the conversation | 398 | 1,664 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2024-33 | latest | en | 0.60013 |
http://www-math.mit.edu/~djk/calculus_beginners/chapter04/section01.html | 1,620,415,670,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988802.93/warc/CC-MAIN-20210507181103-20210507211103-00186.warc.gz | 56,808,149 | 3,135 | 4.1 More Complicated Functions
Differential calculus is about approximating more complicated functions by linear functions. We now address the question, what more complicated functions do we want to deal with?
Most all of the functions we will talk about can be formed by starting with three basic functions, and applying the operations of addition, subtraction, multiplication, division, inversion (like in going from the square to the square root) and substitution to copies of them.
We can define even more functions by using calculus, but these need not be investigated now.
The three basic functions are the identity function, the sine function and the exponential function. For the moment we will start with only the first, the identity function.
If we multiply copies of the identity function together, we get powers of it, like $$x * x$$ (which is $$x$$ squared), or $$x * x * x$$, which is $$x$$ cubed, and so on. Any function consisting of a positive power multiplied by a constant is called a monomial. If we add or subtract a finite number of these, we get what are called polynomials.
The simplest polynomials are the linear functions we have already mentioned. The next more complicated ones are quadratic functions; these have the form, $$ax^2 + bx + c$$, where $$a, b$$ and $$c$$ are numbers. Cubic functions have a cube term in the, quartic functions a term like $$dx^4$$, and so on.
We can evaluate and plot quadratic functions with very little more effort than we expended on linear functions. The only difference is that we should add a quadratic coefficient say in B6, and enter =B$6*A10*A10+B$2*A10+B\$3 into B10 (and then copy this down column B.)
For example, try this, putting $$1$$ in B6. After entering the instruction above in A10, you have to copy it into B11 through B500, and you can now plot any quadratic by changing your parameters.
When you do this you will find something that is sort of nice, all quadratics look more or less alike except that some are upside down.
That is, if you plot a quadratic and don't pay attention to the scales of your graph or which end is up, and where its peak or valley is, you cannot tell them apart. Quadratics with a given sign for the quadratic coefficient, are all alike except for scale and location of their high and low points.
A second nice fact about quadratics is that we know how to solve some equations of the form $$f(x) = 0$$, when $$f$$ is quadratic.
What equations are those?
Well, we know how to solve equation
$x^2 = A$ which means the same thing as: $x^2 - A = 0$
when A is a positive number. We can solve them because a solution is, by definition, the square root of A.
Actually we define $$\sqrt{A}$$ (also written as $$A^{\frac{1}{2}}$$) to be the positive number whose square is $$A$$, when $$A$$ is positive, and the two solutions to this equation are $$\sqrt{A}$$ and $$-\sqrt{A}$$.
By arithmetic manipulations you can reduce any quadratic to this solvable form, and solve it, and you will get the famous quadratic formula for solutions.
How is that and what is that?
The equation $$ax^2 + bx + c = 0$$ can be rewritten (when $$a$$ is not $$0$$, after dividing by $$a$$) as
$x^2 + \frac{bx}{a} + \frac{c}{a} = 0$
which is the same as
$(x + \frac{b}{2a})^2 = \frac{b^2-4ac}{4a^2}$
Thus, the square root of the left hand side is plus or minus the square root of the right hand side here.
$x + \frac{b}{2a} = \frac{\sqrt{(b^2 - 4ac)}}{2a}$
or
$x + \frac{b}{2a} = -\frac{\sqrt{b^2 - 4ac}}{2a}$
This is a peculiar way to write the standard quadratic formula.
Exercise 4.1 Find two solutions to each of the following equations:
$x^2 - 3x - 4 = 0$
$4x^2 - 3x - 1 = 0$ | 953 | 3,682 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.59375 | 5 | CC-MAIN-2021-21 | latest | en | 0.925728 |
https://digitalfinancelearning.com/sikuli-variables/ | 1,669,525,463,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710192.90/warc/CC-MAIN-20221127041342-20221127071342-00268.warc.gz | 245,743,483 | 30,182 | # Sikuli Variables
## How Do I Use Variables in Sikuli?
Sikuli works just the same as python with variables. If you do not know what variables are, they are “containers” that can store numbers, letters, symbols, spaces, etc. You can store any type of data to use later on in the code as a reference. Think of variables as ways of saving time and not having to repeat yourself. Variables in Sikuli follow the same rules and syntax as Python so if you have some experience with Python, using Sikuli should be a piece of cake. We do not have to define what the data type is and Python automatically assumes what it is based on what is entered.
Here is a finance-related scenario that is handling budget information using variables.
Mark has a budget of \$5,910,149 for his business. He spends \$92,000 per month and wants to find out how much he has left in his budget in month 24.
Solving this problem using variables is very easy.
First, we create a variable called “Budget” and store “5910149” by making it equal to Budget.
Budget = 5910149
Note: When you are defining a variable, you will need to use one = sign. Using two == is for when applying logic to define a statement as true or false.
Now every time we reference the variable ‘Budget‘, it should return ‘5910149’.
You can easily check this by using the print command.
print(Budget) would print the number out in the message log when the script has been run.
Since we know the variable can be referenced, we can create some calculations to see how much money Mark has left in month 24. We know he spends \$92,000 per month.
Since we have to reference month 24, we can make a month variable by defining it as
month = 24
Since we know Mark plans on spending \$92,000 per month, we can also store that number in a variable.
monthspend = 92000
The last thing we need to do is create a variable to tell us what is remaining in the budget after x months.
We can create a variable called ‘RemainingBudget’ and equal it to Budget – (monthspend * month).
This translates to 5910149 – (92000*24)
If we print RemainingBudget, the answer should be 3702149.
We can make this even more sophisticated by making the variables be defined by whatever is entered in an input box or even ask the user to select the month and year from the dropdown menu. Variables save a lot of time by not having to repeat or calculate multiple times. Check out our section on Sikuli input boxes and dropdown menus if you are interested in learning more about defining variables with user interaction.
## How to Automate Account Reconciliations?
Automate your Accounting Account Reconciliations Here is a scenario of Jill automating her account reconciliations. She goes through the process step by step by using…
## Python If/Elif/Else Statements
If/Else Decision Making With Python Python is a programming language created by Guido van Rossum in 1991. It is one of the most widely used…
## Editing Word Documents and PDF Files with Python
Working with DOCX and PDF files in Python PDF (portable document format) and DOCX (Office Open XML format) are two of (probably) the most common…
## Sikuli Input Boxes
Sikuli User Input Boxes Sikuli has many advantages what automating Digital Finance processes. When building automations, the main objective is to make the user be… | 747 | 3,323 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2022-49 | longest | en | 0.932393 |
https://masm32.com/board/index.php?PHPSESSID=75444qmh7odun47gq7b76agk4o;topic=11502.msg125435#msg125435 | 1,725,792,358,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700650976.41/warc/CC-MAIN-20240908083737-20240908113737-00724.warc.gz | 379,425,968 | 12,855 | ## News:
Message to All Guests
NB: Posting URL's See here: Posted URL Change
## Python anyone?
Started by raymond, November 28, 2023, 11:30:19 AM
#### raymond
Got into a brief discussion recently in another forum about possibilities of assembly vs other languages. Following is one answer I got:
Quoterayfil wrote:
"Show me any other language which could extract the square root of any number with a precision of 1000 decimal digits within a split second."
Following Python snippet can extract the square root with a precision of 10000 decimal digits in 0.01 second.
CODE
from decimal import *
from time import time
def sq(n, pr):
start = time()
getcontext().prec = pr
print(Decimal(n).sqrt()
print(time() - start)
sq(5, 10000)
Mathematica is even faster than Python.
I would be VERY curious to see a finished program which could deliver such a feat, i.e. 0.01 second for 10,000 digits.
Is there anyone
a) sufficiently familiar with Python to understand the offered snippet,
b) capable of generating a working exe to confirm its speed and its output
If proven, its algo would be extremely interesting.
Thanks
Whenever you assume something, you risk being wrong half the time.
https://masm32.com/masmcode/rayfil/index.html
#### jack
hi Raymond
on my PC python takes 0.035 seconds
I have been playing with MP floating point arithmetic for a while in FreeBasic, my algorithms are very simple, the following little program builds an mp float from a 10000 characters long string and and takes the square root, squares it and gets the relative error, the average time per loop is 0.011 seconds
mind you that there's overhead in building the number from string and then squaring it to get the relative error
the same algorithms in C took 0.0054 seconds
dim as decfloat x, y
dim as long i
dim as double t
t=timer
for i=1 to 9
x=trim(str(i))+string(NUMBER_OF_DIGITS-1, trim(str(i))) ' build a 10000 digits number from string
y=sqr(x) 'square root
y=(y*y-x)/x 'calculate the relative error
print fp2str(y, 20)
next
t=timer-t
print t/9
<obviously the decimal floating point package is not included, if you are interested I can upload it and post a link>
#### TimoVJL
QuoteStefan Krah's mpdecimal package (libmpdec): a complete implementation of the General Decimal Arithmetic Specification that will – with minor restrictions – also conform to the IEEE 754-2008 Standard for Floating-Point Arithmetic. Starting from Python-3.3, libmpdec is the basis for Python's decimal module.
mpdec
This fixed version run, but don't print nothing
from decimal import *
from time import time
def sq(n, pr):
start = time()
getcontext().prec = pr
print(Decimal(n).sqrt())
print(time() - start)
sq(5, 10000)
With x64 version:
0.18700003623962402
May the source be with you
#### jack
for people on Windows I recommend WinPython the download is huge but includes the most popular libraries like SciPy and many others
#### jj2007
Quote from: jack on November 28, 2023, 06:43:20 PMI recommend WinPython
See Using a Masm DLL from Python or this:
Py_Initialize PROTO C
Py_Finalize PROTO C
PyRun_SimpleString PROTO C :DWORD
Init
call Py_Initialize
fn PyRun_SimpleString, "print('Hello World')"
call Py_Finalize
Exit
end start
That was over 8 years ago, and now intense googling has not produced any python*.lib. They seem to fumble a lot with this language; version 3.8, for example, is the last one that works on Windows 7: "Last WinPython version that is said to still work on Windows 7 should be WinPython64-3.8.9.0"
#### HSE
In the phone program is a little slow:
0.221851
Equations in Assembly: SmplMath
#### GoneFishing
#6
python 3.9(64-bit)
Quote0.04675626754760742
Quote from: jj2007 on November 28, 2023, 08:34:51 PMThat was over 8 years ago, and now intense googling has not produced any python*.lib.
python*.lib resides in libs folder
#### jj2007
python*.lib resides in libs folder
I've done that in the meantime, using this archive. There's a problem, though:
include \masm32\include\masm32rt.inc
.data
hDll dd ?
hPyList_New dd ?
.code
start:
print hex\$(eax), 9, "LoadLib", 13, 10
print hex\$(rv(GetLastError)), 9, "last error", 13, 10
print hex\$(hPyList_New), 9, "PyList_New", 13, 10
exit
end start
Output:
000000C1 last error
00000000 PyList_New
Googling 0xc1 loadlibrary python yields thousands of hits. Error 0xC1 = dec 193 means bad exe format. So it loads the library correctly as 32-bit code but fails to get the address of PyList_New, which is according to Timo's TLPEView, part of python3.dll
You can solve the problem with
invoke SetCurrentDirectory, chr\$("\Python\")
Apparently, python3.dll loads libraries not from its own folder but rather 1. from the executable's folder and then 2. from Windows\System32, where it finds 64-bit DLLs. It's probably a feature
#### GoneFishing
all worked fine:
00000000 last error
715A3A30 PyList_New
#### jj2007
Quote from: GoneFishing on November 29, 2023, 02:53:00 AMI copied python*.dlls to my test project folder
And all worked fine, sure. But you would have to do that for all your projects, thus wasting an awful lot of disk space.
Next version of MasmBasic will have this macro, so that you can access the DLL from any folder:
SetDllFolder MACRO arg:=<0>
ifndef sdfOld\$
.DATA?
sdfOld\$ db 260 dup(?) ; MAX_PATH
.CODE
endif
ifdif <arg>, <0>
.if 1
push repargA(arg)
call SetCurrentDirectory
else
.endif
endif
ENDM
Usage:
SetDllFolder "\Python"
...
SetDllFolder
#### GoneFishing
I prefer pure MASM32.
And you're right python wants to know the path to its folder which contains not only dlls but also python*.zip with all needed modules.
Your MasmBasic HelloWorld proggie outputs a lot of internal configuration info when it doesn't find any modules:
QuotePython path configuration:
PYTHONHOME = (not set)
PYTHONPATH = (not set)
program name = 'python'
isolated = 0
environment = 1
user site = 1
import site = 1
sys._base_executable = 'C:\\masm32\\projects\\python\\39x32\\t2.exe'
sys.base_prefix = ''
sys.base_exec_prefix = ''
sys.platlibdir = 'lib'
sys.executable = 'C:\\masm32\\projects\\python\\39x32\\t2.exe'
sys.prefix = ''
sys.exec_prefix = ''
sys.path = [
'C:\\masm32\\projects\\python\\39x32\\python39.zip',
'.\\DLLs',
'.\\lib',
'C:\\masm32\\projects\\python\\39x32',
]
#### raymond
Quote from: jack on November 28, 2023, 12:55:11 PMhi Raymond
on my PC python takes 0.035 seconds
Many thanks to all those having shown interest in this subject.
Jack indicates that the quoted 0.01 second may be quite possibly achievable.
My main interest was to find out if there is a significantly different way of extracting a square root compared to all those I have learned in the past.
The most promising one for speed would have to be using logarithms. But then, are there means to convert those rapidly back and forth with sufficient precision? Or is it something totally different? A self-contained working program would certainly be useful to run under ollydbg and get some of the details.
Whenever you assume something, you risk being wrong half the time.
https://masm32.com/masmcode/rayfil/index.html
#### jack
#12
my algorithm uses log and exp evaluated using double precision as a first approximation, thereafter I use the Newton-Raphson method, something like this
x0 = x scaled so it's between 1 and 10 but less than 10
ex = the exponent in base 10 of x
y0 = first approximation
y0 = exp(log(x0)/2)*exp(2.302585092994046*ex/2) ; the second exp is mp-exp evaluated only to 16 digits
prec=32
then do the Newton-Raphson calculations in a loop doubling the precision each time in the loop
the calculations inside the loop are only evaluated to prec precision
Paul Dixon posted an implementation of the square root in PowerBasic with inline asm https://forum.powerbasic.com/forum/user-to-user-discussions/programming/816462-arbitrary-length-number-math?p=816479#post816479
but the Newton-Raphson method is several orders of magnitude faster than his approach
#### jack
just in case that someone would like to have a look at my humble implementation of multiple precision decimal floating-point I attach the code in FreeBasic DecFloat-FB
#### GoneFishing
hi Raymond,
QuoteA self-contained working program would certainly be useful
In theory the small python script in your post can be converted to exe format.
I've used Pyinstaller for that purpose but the result doesn't look "debuggable" because of its size :
exe = 1018KB + dependency folder = 10.2MB
BTW performance improved to 0.031305789947509766 | 2,245 | 8,533 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2024-38 | latest | en | 0.856793 |
https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/?replytocom=233364 | 1,579,778,307,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250610004.56/warc/CC-MAIN-20200123101110-20200123130110-00259.warc.gz | 676,668,385 | 138,520 | Suppose one is given a ${k_0}$-tuple ${{\mathcal H} = (h_1,\ldots,h_{k_0})}$ of ${k_0}$ distinct integers for some ${k_0 \geq 1}$, arranged in increasing order. When is it possible to find infinitely many translates ${n + {\mathcal H} =(n+h_1,\ldots,n+h_{k_0})}$ of ${{\mathcal H}}$ which consists entirely of primes? The case ${k_0=1}$ is just Euclid’s theorem on the infinitude of primes, but the case ${k_0=2}$ is already open in general, with the ${{\mathcal H} = (0,2)}$ case being the notorious twin prime conjecture.
On the other hand, there are some tuples ${{\mathcal H}}$ for which one can easily answer the above question in the negative. For instance, the only translate of ${(0,1)}$ that consists entirely of primes is ${(2,3)}$, basically because each translate of ${(0,1)}$ must contain an even number, and the only even prime is ${2}$. More generally, if there is a prime ${p}$ such that ${{\mathcal H}}$ meets each of the ${p}$ residue classes ${0 \hbox{ mod } p, 1 \hbox{ mod } p, \ldots, p-1 \hbox{ mod } p}$, then every translate of ${{\mathcal H}}$ contains at least one multiple of ${p}$; since ${p}$ is the only multiple of ${p}$ that is prime, this shows that there are only finitely many translates of ${{\mathcal H}}$ that consist entirely of primes.
To avoid this obstruction, let us call a ${k_0}$-tuple ${{\mathcal H}}$ admissible if it avoids at least one residue class ${\hbox{ mod } p}$ for each prime ${p}$. It is easy to check for admissibility in practice, since a ${k_0}$-tuple is automatically admissible in every prime ${p}$ larger than ${k_0}$, so one only needs to check a finite number of primes in order to decide on the admissibility of a given tuple. For instance, ${(0,2)}$ or ${(0,2,6)}$ are admissible, but ${(0,2,4)}$ is not (because it covers all the residue classes modulo ${3}$). We then have the famous Hardy-Littlewood prime tuples conjecture:
Conjecture 1 (Prime tuples conjecture, qualitative form) If ${{\mathcal H}}$ is an admissible ${k_0}$-tuple, then there exists infinitely many translates of ${{\mathcal H}}$ that consist entirely of primes.
This conjecture is extremely difficult (containing the twin prime conjecture, for instance, as a special case), and in fact there is no explicitly known example of an admissible ${k_0}$-tuple with ${k_0 \geq 2}$ for which we can verify this conjecture (although, thanks to the recent work of Zhang, we know that ${(0,d)}$ satisfies the conclusion of the prime tuples conjecture for some ${0 < d < 70,000,000}$, even if we can’t yet say what the precise value of ${d}$ is).
Actually, Hardy and Littlewood conjectured a more precise version of Conjecture 1. Given an admissible ${k_0}$-tuple ${{\mathcal H} = (h_1,\ldots,h_{k_0})}$, and for each prime ${p}$, let ${\nu_p = \nu_p({\mathcal H}) := |{\mathcal H} \hbox{ mod } p|}$ denote the number of residue classes modulo ${p}$ that ${{\mathcal H}}$ meets; thus we have ${1 \leq \nu_p \leq p-1}$ for all ${p}$ by admissibility, and also ${\nu_p = k_0}$ for all ${p>h_{k_0}-h_1}$. We then define the singular series ${{\mathfrak G} = {\mathfrak G}({\mathcal H})}$ associated to ${{\mathcal H}}$ by the formula
$\displaystyle {\mathfrak G} := \prod_{p \in {\mathcal P}} \frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}}$
where ${{\mathcal P} = \{2,3,5,\ldots\}}$ is the set of primes; by the previous discussion we see that the infinite product in ${{\mathfrak G}}$ converges to a finite non-zero number.
We will also need some asymptotic notation (in the spirit of “cheap nonstandard analysis“). We will need a parameter ${x}$ that one should think of going to infinity. Some mathematical objects (such as ${{\mathcal H}}$ and ${k_0}$) will be independent of ${x}$ and referred to as fixed; but unless otherwise specified we allow all mathematical objects under consideration to depend on ${x}$. If ${X}$ and ${Y}$ are two such quantities, we say that ${X = O(Y)}$ if one has ${|X| \leq CY}$ for some fixed ${C}$, and ${X = o(Y)}$ if one has ${|X| \leq c(x) Y}$ for some function ${c(x)}$ of ${x}$ (and of any fixed parameters present) that goes to zero as ${x \rightarrow \infty}$ (for each choice of fixed parameters).
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then the number of natural numbers ${n < x}$ such that ${n+{\mathcal H}}$ consists entirely of primes is ${({\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}$.
Thus, for instance, if Conjecture 2 holds, then the number of twin primes less than ${x}$ should equal ${(2 \Pi_2 + o(1)) \frac{x}{\log^2 x}}$, where ${\Pi_2}$ is the twin prime constant
$\displaystyle \Pi_2 := \prod_{p \in {\mathcal P}: p>2} (1 - \frac{1}{(p-1)^2}) = 0.6601618\ldots.$
As this conjecture is stronger than Conjecture 1, it is of course open. However there are a number of partial results on this conjecture. For instance, this conjecture is known to be true if one introduces some additional averaging in ${{\mathcal H}}$; see for instance this previous post. From the methods of sieve theory, one can obtain an upper bound of ${(C_{k_0} {\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}$ for the number of ${n < x}$ with ${n + {\mathcal H}}$ all prime, where ${C_{k_0}}$ depends only on ${k_0}$. Sieve theory can also give analogues of Conjecture 2 if the primes are replaced by a suitable notion of almost prime (or more precisely, by a weight function concentrated on almost primes).
Another type of partial result towards Conjectures 1, 2 come from the results of Goldston-Pintz-Yildirim, Motohashi-Pintz, and of Zhang. Following the notation of this recent paper of Pintz, for each ${k_0>2}$, let ${DHL[k_0,2]}$ denote the following assertion (DHL stands for “Dickson-Hardy-Littlewood”):
Conjecture 3 (${DHL[k_0,2]}$) Let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then there are infinitely many translates ${n+{\mathcal H}}$ of ${{\mathcal H}}$ which contain at least two primes.
This conjecture gets harder as ${k_0}$ gets smaller. Note for instance that ${DHL[2,2]}$ would imply all the ${k_0=2}$ cases of Conjecture 1, including the twin prime conjecture. More generally, if one knew ${DHL[k_0,2]}$ for some ${k_0}$, then one would immediately conclude that there are an infinite number of pairs of consecutive primes of separation at most ${H(k_0)}$, where ${H(k_0)}$ is the minimal diameter ${h_{k_0}-h_1}$ amongst all admissible ${k_0}$-tuples ${{\mathcal H}}$. Values of ${H(k_0)}$ for small ${k_0}$ can be found at this link (with ${H(k_0)}$ denoted ${w}$ in that page). For large ${k_0}$, the best upper bounds on ${H(k_0)}$ have been found by using admissible ${k_0}$-tuples ${{\mathcal H}}$ of the form
$\displaystyle {\mathcal H} = ( - p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, - p_{m+1}, -1, +1, p_{m+1}, \ldots, p_{m+\lfloor (k_0+1)/2\rfloor - 1} )$
where ${p_n}$ denotes the ${n^{th}}$ prime and ${m}$ is a parameter to be optimised over (in practice it is an order of magnitude or two smaller than ${k_0}$); see this blog post for details. The upshot is that one can bound ${H(k_0)}$ for large ${k_0}$ by a quantity slightly smaller than ${k_0 \log k_0}$ (and the large sieve inequality shows that this is sharp up to a factor of two, see e.g. this previous post for more discussion).
In a key breakthrough, Goldston, Pintz, and Yildirim were able to establish the following conditional result a few years ago:
Theorem 4 (Goldston-Pintz-Yildirim) Suppose that the Elliott-Halberstam conjecture ${EH[\theta]}$ is true for some ${1/2 < \theta < 1}$. Then ${DHL[k_0,2]}$ is true for some finite ${k_0}$. In particular, this establishes an infinite number of pairs of consecutive primes of separation ${O(1)}$.
The dependence of constants between ${k_0}$ and ${\theta}$ given by the Goldston-Pintz-Yildirim argument is basically of the form ${k_0 \sim (\theta-1/2)^{-2}}$. (UPDATE: as recently observed by Farkas, Pintz, and Revesz, this relationship can be improved to ${k_0 \sim (\theta-1/2)^{-3/2}}$.)
Unfortunately, the Elliott-Halberstam conjecture (which we will state properly below) is only known for ${\theta<1/2}$, an important result known as the Bombieri-Vinogradov theorem. If one uses the Bombieri-Vinogradov theorem instead of the Elliott-Halberstam conjecture, Goldston, Pintz, and Yildirim were still able to show the highly non-trivial result that there were infinitely many pairs ${p_{n+1},p_n}$ of consecutive primes with ${(p_{n+1}-p_n) / \log p_n \rightarrow 0}$ (actually they showed more than this; see e.g. this survey of Soundararajan for details).
Actually, the full strength of the Elliott-Halberstam conjecture is not needed for these results. There is a technical specialisation of the Elliott-Halberstam conjecture which does not presently have a commonly accepted name; I will call it the Motohashi-Pintz-Zhang conjecture ${MPZ[\varpi]}$ in this post, where ${0 < \varpi < 1/4}$ is a parameter. We will define this conjecture more precisely later, but let us remark for now that ${MPZ[\varpi]}$ is a consequence of ${EH[\frac{1}{2}+2\varpi]}$.
We then have the following two theorems. Firstly, we have the following strengthening of Theorem 4:
Theorem 5 (Motohashi-Pintz-Zhang) Suppose that ${MPZ[\varpi]}$ is true for some ${0 < \varpi < 1/4}$. Then ${DHL[k_0,2]}$ is true for some ${k_0}$.
A version of this result (with a slightly different formulation of ${MPZ[\varpi]}$) appears in this paper of Motohashi and Pintz, and in the paper of Zhang, Theorem 5 is proven for the concrete values ${\varpi = 1/1168}$ and ${k_0 = 3,500,000}$. We will supply a self-contained proof of Theorem 5 below the fold, the constants upon those in Zhang’s paper (in particular, for ${\varpi = 1/1168}$, we can take ${k_0}$ as low as ${341,640}$, with further improvements on the way). As with Theorem 4, we have an inverse quadratic relationship ${k_0 \sim \varpi^{-2}}$.
In his paper, Zhang obtained for the first time an unconditional advance on ${MPZ[\varpi]}$:
Theorem 6 (Zhang) ${MPZ[\varpi]}$ is true for all ${0 < \varpi \leq 1/1168}$.
This is a deep result, building upon the work of Fouvry-Iwaniec, Friedlander-Iwaniec and BombieriFriedlanderIwaniec which established results of a similar nature to ${MPZ[\varpi]}$ but simpler in some key respects. We will not discuss this result further here, except to say that they rely on the (higher-dimensional case of the) Weil conjectures, which were famously proven by Deligne using methods from l-adic cohomology. Also, it was believed among at least some experts that the methods of Bombieri, Fouvry, Friedlander, and Iwaniec were not quite strong enough to obtain results of the form ${MPZ[\varpi]}$, making Theorem 6 a particularly impressive achievement.
Combining Theorem 6 with Theorem 5 we obtain ${DHL[k_0,2]}$ for some finite ${k_0}$; Zhang obtains this for ${k_0 = 3,500,000}$ but as detailed below, this can be lowered to ${k_0 = 341,640}$. This in turn gives infinitely many pairs of consecutive primes of separation at most ${H(k_0)}$. Zhang gives a simple argument that bounds ${H(3,500,000)}$ by ${70,000,000}$, giving his famous result that there are infinitely many pairs of primes of separation at most ${70,000,000}$; by being a bit more careful (as discussed in this post) one can lower the upper bound on ${H(3,500,000)}$ to ${57,554,086}$, and if one instead uses the newer value ${k_0 = 341,640}$ for ${k_0}$ one can instead use the bound ${H(341,640) \leq 4,982,086}$. (Many thanks to Scott Morrison for these numerics.) UPDATE: These values are now obsolete; see this web page for the latest bounds.
In this post we would like to give a self-contained proof of both Theorem 4 and Theorem 5, which are both sieve-theoretic results that are mainly elementary in nature. (But, as stated earlier, we will not discuss the deepest new result in Zhang’s paper, namely Theorem 6.) Our presentation will deviate a little bit from the traditional sieve-theoretic approach in a few places. Firstly, there is a portion of the argument that is traditionally handled using contour integration and properties of the Riemann zeta function; we will present a “cheaper” approach (which Ben Green and I used in our papers, e.g. in this one) using Fourier analysis, with the only property used about the zeta function ${\zeta(s)}$ being the elementary fact that blows up like ${\frac{1}{s-1}}$ as one approaches ${1}$ from the right. To deal with the contribution of small primes (which is the source of the singular series ${{\mathfrak G}}$), it will be convenient to use the “${W}$-trick” (introduced in this paper of mine with Ben), passing to a single residue class mod ${W}$ (where ${W}$ is the product of all the small primes) to end up in a situation in which all small primes have been “turned off” which leads to better pseudorandomness properties (for instance, once one eliminates all multiples of small primes, almost all pairs of remaining numbers will be coprime).
— 1. The ${W}$-trick —
In this section we introduce the “${W}$-trick”, which is a simple but useful device that automatically takes care of local factors arising from small primes, such as the singular series ${{\mathfrak G}}$. The price one pays for this trick is that the explicit decay rates in various ${o(1)}$ terms can be rather poor, but for the applications here, we will not need to know any information on these decay rates and so the ${W}$-trick may be freely applied.
Let ${w}$ be a natural number, which should be thought of as either fixed and large, or as a very slowly growing function of ${x}$. Actually, the two viewpoints are basically equivalent for the purposes of asymptotic analysis (at least at the qualitative level of ${o(1)}$ decay rates), thanks to the following basic principle:
Lemma 7 (Overspill principle) Let ${F(w,x)}$ be a quantity depending on ${w}$ and ${x}$. Then the following are equivalent:
• (i) For every fixed ${\epsilon>0}$ there exists a fixed ${w_\epsilon > 0}$ such that
$\displaystyle |F(w,x)| \leq \epsilon + o(1)$
for all fixed ${w \geq w_\epsilon}$.
• (ii) We have
$\displaystyle F(w,x) = o(1)$
whenever ${w = w(x)}$ is a function of ${x}$ going to infinity that is sufficiently slowly growing. (In other words, there exists a function ${w_0: {\bf R}^+ \rightarrow {\bf N}}$ going to infinity with the property that ${F(w,x)=o(1)}$ whenever ${w = w(x)}$ is a natural number-valued function of ${x}$ is such that ${w(x) \rightarrow \infty}$ as ${x \rightarrow \infty}$ and ${w(x) \leq w_0(x)}$ for all sufficiently large ${x}$.)
This principle is closely related to the overspill principle from nonstandard analysis, though we will not explicitly adopt a nonstandard perspective here. It is also similar in spirit to the diagonalisation trick used to prove the Arzela-Ascoli theorem.
Proof: We first show that (i) implies (ii). By (i), we see that for every natural number ${n}$, we can find a real number ${x_n}$ with the property that
$\displaystyle |F(w,x)| \leq \frac{2}{m}$
whenever ${1 \leq m \leq n}$, ${1 \leq w \leq n}$, and ${x \geq x_n}$ are such that ${w \geq w_{1/m}}$. By increasing the ${x_n}$ as necessary we may assume that they are increasing and go to infinity as ${n \rightarrow \infty}$. If we then define ${w_0(x)}$ to equal the largest natural number ${n}$ for which ${x \geq x_n}$, or equal to ${1}$ if no such number exists, then one easily verifies that ${F(w,x)=o(1)}$ whenever ${w= w(x)}$ goes to infinity and is bounded by ${w_0}$ for sufficiently large ${x}$.
Now we show that (ii) implies (i). Suppose for contradiction that (i) failed, then we can find a fixed ${\epsilon>0}$ with the property that for any natural number ${n}$, there exist ${w_n \geq n}$ such that ${|F(w_n,x_n)| \geq \epsilon}$ for arbitrarily large ${x_n}$. We can select the ${w_n}$ to be increasing to infinity, and then we can find a sequence ${x_n}$ increasing to infinity such that ${|F(w_n,x_n)| \geq \epsilon}$ for all ${n}$; by increasing ${x_n}$ as necessary, we can also ensure that ${w_0(x) \geq w_n}$ for all ${x \geq x_n}$ and ${n}$. If we then define ${w(x)}$ to be ${w_n}$ when ${x_n \leq x < x_{n+1}}$, and ${w(x)=1}$ for ${x < x_1}$, we see that ${|F(w,x)| \geq \epsilon}$ whenever ${x=x_n}$, contradicting (ii). $\Box$
Henceforth we will usually think of ${w}$ as a sufficiently slowly growing function of ${x}$, although we will on occasion take advantage of Lemma 7 to switch to thinking of ${w}$ as a large fixed quantity instead. In either case, we should think of ${w}$ as exceeding the size of fixed quantities such as ${k}$ or ${h_k-h_1}$, at least in the limit where ${x}$ is large; in particular, for a fixed ${k_0}$-tuple ${{\mathcal H}}$, we will have
$\displaystyle \nu_p = k_0 \hbox{ for all } p > w \ \ \ \ \ (1)$
if ${x}$ is large enough. A particular consequence of the growing nature of ${w}$ is that
$\displaystyle \sum_{p > w} \frac{1}{p^2} = o(1) \ \ \ \ \ (2)$
as this follows from the absolutely convergent nature of the sum ${\sum_{n=1}^\infty \frac{1}{n^2}}$ and hence also ${\sum_p \frac{1}{p^2}}$. As a consequence of this, once we “turn off” all the primes less than ${w}$, any errors in our sieve-theoretic analysis which are quadratic or higher in ${1/p}$ can be essentially ignored, which will be very convenient for us. In a similar vein, for any fixed ${k_0}$-tuple ${{\mathcal H}}$, one has
$\displaystyle \prod_{p>w} \frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}} = 1+o(1) \ \ \ \ \ (3)$
which allows one to truncate the singular series:
$\displaystyle {\mathfrak G} = \prod_{p \leq w} \frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}} + o(1). \ \ \ \ \ (4)$
In order to “turn off” all the small primes, we introduce the quantity ${W}$, defined as the product of all the primes up to ${w}$ (i.e. the primorial of ${w}$):
$\displaystyle W := \prod_{p \leq w} p.$
As ${w}$ is going to infinity, ${W}$ is going to infinity also (but as slowly as we please). The idea of the ${W}$-trick is to search for prime patterns in a single residue class ${b \hbox{ mod } W}$, which as mentioned earlier will “turn off” all the primes less than ${w}$ in the sieve-theoretic analysis.
Using (4) and the Chinese remainder theorem, we may thus approximate the singular series as
$\displaystyle {\mathfrak G} = \frac{|C(W)|}{W} (\frac{\phi(W)}{W})^{-k_0} + o(1) \ \ \ \ \ (5)$
where ${\phi(W)}$ is the Euler totient function of ${W}$, and ${C(W) \subset {\bf Z}/W{\bf Z}}$ is the set of residue classes ${b \hbox{ mod } W}$ such that all of the shifts ${b+h_1,\ldots,b+h_{k_0}}$ are coprime to ${W}$. Note that if ${n+{\mathcal H}}$ consists purely of primes and ${n}$ is sufficiently large, then ${n}$ must lie in one of the residue classes in ${C(W)}$. Thus we can count tuples with ${n+{\mathcal H}}$ all prime by working in each residue class in ${C(W)}$ separately. We conclude that Conjecture 2 is equivalent to the following “${W}$-tricked version” in which the singular series is no longer present (or, more precisely, has been replaced by some natural normalisation factors depending on ${W}$, such as ${(\phi(W)/W)^{-k_0}}$):
Conjecture 8 (Prime tuples conjecture, W-tricked quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Assume ${w}$ is a sufficiently slowly growing function of ${x}$. Then for any residue class ${b \hbox{ mod } W}$ in ${C(W)}$, the number of natural numbers ${n < x}$ with ${n=b \hbox{ mod } W}$ such that ${n+{\mathcal H}}$ consists entirely of primes is ${(\frac{1}{W} (\frac{W}{\phi(W)})^{k_0} + o(1)) \frac{x}{\log^{k_0} x}}$.
We will work with similarly ${W}$-tricked asymptotics in the analysis below.
— 2. Sums of multiplicative functions —
As a result of the sieve-theoretic computations to follow, we will frequently need to estimate sums of the form
$\displaystyle S_{0,R,I}( f, g ) := \sum_{d \in {\mathcal S}_I} \frac{f(d)}{d} g( \frac{\log d}{\log R} )$
and
$\displaystyle S_{1,R,I}( f, g ) := \sum_{d \in {\mathcal S}_I} \mu(d) \frac{f(d)}{d} g( \frac{\log d}{\log R} )$
where ${f: {\bf N} \rightarrow {\bf C}}$ is a multiplicative function, the sieve level ${R>0}$ (also denoted ${D}$ in some literature) is a fixed power of ${x}$ (such as ${x^{1/4}}$ or ${x^{1/4+\varpi}}$), ${\mu}$ is the Möbius function, ${g: {\bf R} \rightarrow {\bf C}}$ is a fixed smooth compactly supported function, ${I}$ is a (possibly half-infinite) interval in ${(w,+\infty)}$, and ${{\mathcal S}_I}$ is the set of square-free numbers that are products ${p_1 \ldots p_j}$ of distinct primes ${p_1,\ldots,p_j}$ in ${I}$. (Actually, in applications ${g}$ won’t quite be smooth, but instead have some high order of differentiability (e.g. ${k_0+l_0-1}$ times continuously differentiable for some ${l_0>0}$), but we can extend the analysis of smooth ${g}$ to sufficiently differentiable ${g}$ by various standard limiting or approximation arguments which we will not dwell on here.) We will also need to control the more complicated variant
$\displaystyle S_{2,R,I}(f,g_1,g_2) := \sum_{d_1,d_2 \in {\mathcal S}_I} \frac{\mu(d_1) \mu(d_2) f([d_1,d_2])}{[d_1,d_2]} g_1( \frac{\log d_1}{\log R} ) g_2( \frac{\log d_2}{\log R} )$
where ${g_1,g_2:{\bf R} \rightarrow {\bf C}}$ are also smooth compactly supported functions. In practice, the interval ${I}$ will be something like ${(w, x^{1/4+\varpi})}$, ${(w, x^\varpi)}$, ${[x^\varpi,x^{1/4+\varpi}]}$. In particular, thanks to the ${W}$-trick we will be able to turn off all the primes up to ${w}$, so that ${I}$ only contains primes larger than ${w}$, allowing us to take advantage of bounds such as (2).
Once ${d}$ is restricted to ${{\mathcal S}_I}$, the quantity ${f(d)}$ is determined entirely by the values of the multiplicative function ${f}$ at primes in ${I}$:
$\displaystyle f(d) = \prod_{p \in {\mathcal P} \cap I: p | d} f(p).$
In applications, ${f}$ will have the size bound
$\displaystyle f(p) = k + O( \frac{1}{p} ) \ \ \ \ \ (6)$
for all ${p \in I}$ and some fixed positive ${k}$ (note that we allow the implied constants in the ${O()}$ notation to depend on quantities such as ${k}$); we refer to ${k}$ as the dimension of the multiplicative function ${f}$. Henceforth we assume that ${f}$ has a fixed dimension ${k}$. We remark that we could unify the treatment of ${S_{0,R,I}}$ and ${S_{1,R,I}}$ in what follows by allowing multiplicative functions of negative dimension, but we will avoid doing so here. In our applications ${k}$ will be an integer; one could also generalise much of the discussion below to the fractional dimension case, but we will not need to do so here.
Traditionally the above expressions are handled by complex analysis, starting with Perron’s formula. We will instead take a slightly different Fourier-analytic approach. We perform a Fourier expansion of the smooth compactly supported function ${e^x g(x)}$ to obtain a representation
$\displaystyle e^x g(x) = \int_{\bf R} e^{-itx} \hat g(t)\ dt \ \ \ \ \ (7)$
for some Schwartz function ${\hat g}$; in particular, ${\hat g}$ is rapidly decreasing. (Strictly speaking, ${\hat g}$ is the Fourier transform of ${g}$ shifted in the complex domain by ${i}$, rather than the true Fourier transform of ${g}$, but we will ignore this distinction for the purposes of this discussion.) In particular we have
$\displaystyle g(\frac{\log d}{\log R}) = \int_{\bf R} \frac{1}{d^{\frac{1+it}{\log R}}} \hat g(t)\ dt$
for any ${d}$. By Fubini’s theorem, we can thus write ${S_{0,R,I}}$ as
$\displaystyle S_{0,R,I}(f,g) = \int_{\bf R} \sum_{d \in {\mathcal S}_I} \frac{f(d)}{d^{1+\frac{1+it}{\log R}}} \hat g(t)\ dt,$
which factorises as
$\displaystyle S_{0,R,I}(f,g) = \int_{\bf R} (\prod_{p \in I} (1 + \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})) \hat g(t)\ dt.$
Similarly one has
$\displaystyle S_{1,R,I}(f,g) = \int_{\bf R} (\prod_{p \in I} (1 - \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})) \hat g(t)\ dt.$
and
$\displaystyle S_{2,R,I}(f,g_1,g_2) = \int_{\bf R} \int_{\bf R} (\prod_{p \in I} (1 - \frac{f(p)}{p^{1+\frac{1+it_1}{\log R}}} - \frac{f(p)}{p^{1+\frac{1+it_2}{\log R}}} + \frac{f(p)}{p^{1+\frac{1+it_1+1+it_2}{\log R}}} ))$
$\displaystyle \hat g_1(t_1) \hat g_2(t_2)\ dt_1 dt_2.$
In order to use asymptotics of the Riemann zeta function near the pole ${s=1}$, it is convenient to temporarily truncate the above integrals to the region ${|t| \leq \sqrt{\log R}}$ or ${|t_1|, |t_2| \leq \sqrt{\log R}}$:
Lemma 9 For any fixed ${A>0}$, we have
$\displaystyle S_{0,R,I}(f,g) = \int_{|t| \leq \sqrt{\log R}} (\prod_{p \in I} (1 + \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})) \hat g(t)\ dt + O( \log^{-A} R)$
and
$\displaystyle S_{1,R,I}(f,g) = \int_{|t| \leq \sqrt{\log R}} (\prod_{p \in I} (1 - \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})) \hat g(t)\ dt + O( \log^{-A} R)$
and
$\displaystyle S_{2,R,I}(f,g) = \int_{|t_1|, |t_2| \leq \sqrt{\log R}}$
$\displaystyle (\prod_{p \in I} (1 - \frac{f(p)}{p^{1+\frac{1+it_1}{\log R}}} - \frac{f(p)}{p^{1+\frac{1+it_2}{\log R}}} + \frac{f(p)}{p^{1+\frac{1+it_1+1+it_2}{\log R}}} ))$
$\displaystyle \hat g_1(t_1) \hat g_2(t_2)\ dt_1 dt_2 + O(\log^{-A} R).$
Also we have the crude bound
$\displaystyle S_{0,R,I}(f,g) = O( \log^k R ).$
Proof: We begin with the bounds on ${S_{0,R,I}}$. From (6) we have
$\displaystyle \log |1 + \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})| \leq k p^{-1-\frac{1}{\log R}} + O( p^{-2} )$
for ${p \in I}$ (which forces ${p>w}$, so there is no issue with the singularity of the logarithm) and thus
$\displaystyle \prod_{p \in I} (1 + \frac{f(p)}{p^{1+\frac{1+it}{\log R}}}) = O( \exp( k \sum_p p^{-1-\frac{1}{\log R}} ) ).$
Since
$\displaystyle \prod_p (1-\frac{1}{p^{1+1/\log R}}) = \frac{1}{\zeta(1+1/\log R)} = \frac{1}{\log R + O(1)}$
we see on taking logarithms that
$\displaystyle \sum_p p^{-1-\frac{1}{\log R}} = \log\log R + O(1)$
and thus
$\displaystyle \prod_{p \in I} (1 + \frac{f(p)}{p^{1+\frac{1+it}{\log R}}}) = O( \log^k R ).$
The bounds on ${S_{0,R,I}(f,g)}$ then follow from the rapid decrease of ${\hat g}$. The bounds for ${S_{1,R,I}}$ and ${S_{2,R,I}}$ are proven similarly. $\Box$
From (6) and the restriction of ${I}$ to quantities larger than ${w}$, we see that
$\displaystyle (\prod_{p \in I} (1 + \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})) = (1+o(1)) \zeta_I(1+\frac{1+it}{\log R})^k$
and
$\displaystyle (\prod_{p \in I} (1 - \frac{f(p)}{p^{1+\frac{1+it}{\log R}}})) = (1+o(1)) \zeta_I(1+\frac{1+it}{\log R})^{-k}$
and
$\displaystyle (\prod_{p \in I} (1 - \frac{f(p)}{p^{1+\frac{1+it_1}{\log R}}} - \frac{f(p)}{p^{1+\frac{1+it_2}{\log R}}} + \frac{f(p)}{p^{1+\frac{1+it_1+1+it_2}{\log R}}} ))$
$\displaystyle = (1+o(1)) \zeta_I(1+\frac{1+it_1}{\log R})^{-k} \zeta_I(1+\frac{1+it_2}{\log R})^{-k}$
$\displaystyle \zeta_I(1-\frac{1+it_1+1+it_2}{\log R})^{k}$
where ${\zeta_I}$ is the restricted Euler product
$\displaystyle \zeta_I(s) := \prod_{p \in I} (1-\frac{1}{p^s})^{-1},$
which is well-defined for ${\hbox{Re}(s)>1}$ at least (and this is the only region of ${s}$ for which we will need ${\zeta_I}$).
We now specialise to the model case ${I = (w,+\infty)}$, in which case
$\displaystyle \zeta_I(s) = \zeta(s) \prod_{p \leq w} (1 - \frac{1}{p^s})$
where ${\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}}$ is the Riemann zeta function. Using the basic (and easily proven) asymptotic ${\zeta(s) = \frac{1}{s-1} + O(1)}$ for ${s}$ near ${1}$
$\displaystyle \zeta_I(s) = (1+o(1)) \frac{\phi(W)}{W} \frac{1}{s-1}$
for ${s = 1+O(1/\sqrt{\log R})}$, if ${w}$ is sufficiently slowly growing (this can be seen by first working with a fixed large ${W}$ and then using Lemma 7). Note that because of the above truncation, we do not need any deeper bounds on ${\zeta}$ than what one can obtain from the simple pole at ${s=1}$; in particular no zero-free regions near the line ${\{ 1+it: t \in {\bf R} \}}$ are needed here. (This is ultimately because of the smooth nature of ${g}$, which is sufficient for the applications in this post; if one wanted rougher cutoff functions here then the situation is closer to that of the prime number theorem, and non-trivial zero-free regions would be required.)
We conclude in the case ${I = (w,+\infty)}$ that
$\displaystyle S_{0,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^k \int_{|t| \leq \sqrt{\log R}} (1+o(1)) (1+it)^{-k} \hat g(t)\ dt$
$\displaystyle + O( \log^{-A} R)$
and
$\displaystyle S_{1,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^{-k} \int_{|t| \leq \sqrt{\log R}} (1+o(1)) (1+it)^k \hat g(t)\ dt$
$\displaystyle + O( \log^{-A} R)$
and
$\displaystyle S_{2,R,I}(f,g_1,g_2) = (\frac{\phi(W)}{W} \log R)^{-k} \int_{|t_1|, |t_2| \leq \sqrt{\log R}}$
$\displaystyle (1+o(1)) (1+it_1)^k (1+it_2)^k (1+it_1+1+it_2)^{-k} \hat g_1(t_1) \hat g_2(t_2)\ dt_1 dt_2$
$\displaystyle + O(\log^{-A} R);$
using the rapid decrease of ${\hat g, \hat g_1, \hat g_2}$, we thus have
$\displaystyle S_{0,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^k (\int_{\bf R} (1+it)^{-k} \hat g(t)\ dt + o(1))$
and
$\displaystyle S_{1,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^{-k} (\int_{\bf R} (1+it)^k \hat g(t)\ dt + o(1))$
and
$\displaystyle S_{2,R,I}(f,g_1,g_2) = (\frac{\phi(W)}{W} \log R)^{-k} (\int_{\bf R} \int_{\bf R}$
$\displaystyle (1+it_1)^k (1+it_2)^k (1+it_1+1+it_2)^{-k}$
$\displaystyle \hat g_1(t_1) \hat g_2(t_2)\ dt_1 dt_2 + o(1)).$
We can rewrite these expressions in terms of ${g}$ instead of ${\hat g}$. Using the Gamma function identity
$\displaystyle (1+it)^{-k} = \int_0^\infty e^{-x(1+it)} \frac{x^{k-1}}{(k-1)!}\ dx$
and (7) we see that
$\displaystyle \int_{\bf R} (1+it)^{-k} \hat g(t)\ dt = \int_0^\infty g(x) \frac{x^{k-1}}{(k-1)!}\ dx$
whilst from differentiating (7) ${k}$ times at the origin (after first dividing by ${e^x}$) we see that
$\displaystyle \int_{\bf R} (1+it)^k \hat g(t)\ dt = (-1)^k g^{(k)}(0).$
Combining these two methods, we also see that
$\displaystyle \int_{\bf R} \int_{\bf R} (1+it_1)^k (1+it_2)^k (1+it_1+1+it_2)^{-k} \hat g_1(t_1) \hat g_1(t_2)\ dt_1 dt_2$
$\displaystyle = \int_0^\infty g^{(k)}_1(x) g^{(k)}_2(x) \frac{x^{k-1}}{(k-1)!}\ dx.$
We have thus obtained the following asymptotics:
Proposition 10 (Asymptotics without prime truncation) Suppose that ${I = (w,+\infty)}$, and that ${f}$ has dimension ${k}$ for some fixed natural number ${k}$. Then we have
$\displaystyle S_{0,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^k (\int_0^\infty g(x) \frac{x^{k-1}}{(k-1)!}\ dx + o(1))$
and
$\displaystyle S_{1,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^{-k} ((-1)^k g^{(k)}(0) + o(1))$
and
$\displaystyle S_{2,R,I}(f,g) = (\frac{\phi(W)}{W} \log R)^{-k}$
$\displaystyle (\int_0^\infty g^{(k)}_1(x) g^{(k)}_2(x) \frac{x^{k-1}}{(k-1)!}\ dx + o(1)).$
These asymptotics will suffice for the treatment of the Goldston-Pintz-Yildirim theorem (Theorem 4). For the Motohashi-Pintz-Zhang theorem (Theorem 5) we will also need to deal with truncated intervals ${I}$, such as ${(w,x^{1/\varpi})}$; we will discuss how to deal with these truncations later.
— 3. The Goldston-Yildirim-Pintz theorem —
We are now ready to state and prove the Goldston-Yildirim-Pintz theorem. We first need to state the Elliott-Halberstam conjecture properly.
Let ${\Lambda: {\bf N} \rightarrow {\bf R}}$ be the von Mangoldt function, thus ${\Lambda(n)}$ equals ${\log p}$ when ${n}$ is equal to a prime ${p}$ or a power of that prime, and equal to zero otherwise. The prime number theorem in arithmetic progressions tells us that
$\displaystyle \sum_{n < x: n = a \hbox{ mod } q} \Lambda(n) = (1 + o(1)) \frac{x}{\phi(q)}$
for any fixed arithmetic progression ${a \hbox{ mod } q}$ with ${a}$ coprime to ${q}$. In particular,
$\displaystyle \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{n < x: n = a \hbox{ mod } q} \Lambda(n) - \frac{1}{\phi(q)} \sum_{n < x} \Lambda(n)| = o( \frac{x}{\phi(q)} )$
where ${({\bf Z}/q{\bf Z})^\times}$ are the residue classes mod ${q}$ that are coprime to ${q}$. By invoking the Siegel-Walfisz theorem one can obtain the improvement
$\displaystyle \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{n < x: n = a \hbox{ mod } q} \Lambda(n) - \frac{1}{\phi(q)} \sum_{n < x} \Lambda(n)| = O( \frac{x}{\phi(q) \log^A x} )$
for any fixed ${A>0}$ (though, annoyingly, the implied constant here is only ineffectively bounded with current methods; see this previous post for further discussion).
The above error term is only useful when ${q}$ is fixed (or is of logarithmic size in ${x}$). For larger values of ${q}$, it is very difficult to get good error terms for each ${q}$ separately, unless one assumes powerful hypotheses such as the generalised Riemann hypothesis. However, it is possible to obtain good control on the error term if one averages in ${q}$. More precisely, for any ${0 < \theta < 1}$, let ${EH[\theta]}$ denote the following assertion:
Conjecture 11 (${EH[\theta]}$) One has
$\displaystyle \sum_{1 \leq q \leq x^\theta} \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{n < x: n = a \hbox{ mod } q} \Lambda(n) - \frac{1}{\phi(q)} \sum_{n < x} \Lambda(n)|$
$\displaystyle = O( \frac{x}{\log^A x} )$
for all fixed ${A>0}$.
This should be compared with the asymptotic ${\sum_{1 \leq q \leq x^\theta} \frac{x}{\phi(q)} = (C+o(1)) x \log x^\theta}$ for some absolute constant ${C>0}$, as can be deduced for instance from Proposition 10. The Elliott-Halberstam conjecture is the assertion that ${EH[\theta]}$ holds for all ${0 < \theta < 1}$. This remains open, but the important Bombieri-Vinogradov theorem establishes ${EH[\theta]}$ for all ${0 < \theta < 1/2}$. Remarkably, the threshold ${1/2}$ is also the limit of what one can establish if one directly invokes the generalised Riemann hypothesis, so the Bombieri-Vinogradov theorem is often referred to as an assertion that the generalised Riemann hypothesis (or at least the Siegel-Walfisz theorem) holds “on the average”, which is often good enough for sieve-theoretic purposes.
We may replace the von Mangoldt function ${\Lambda(n)}$ with the slight variant ${\theta(n)}$, defined to equal ${\log p}$ when ${n}$ is a prime ${p}$ and zero otherwise. Using this replacement, as well as the prime number theorem (with${O(x / \log^A x)}$ error term), it is not difficult to show that ${EH[\theta]}$ is equivalent to the estimate
$\displaystyle \sum_{1 \leq q \leq x^\theta} \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{n < x: n = a \hbox{ mod } q} \theta(n) - \frac{x}{\phi(q)}| = O( \frac{x}{\log^A x} ). \ \ \ \ \ (8)$
Now we establish Theorem 4. Suppose that ${EH[\theta]}$ holds for some fixed ${1/2 <\theta < 1}$, let ${k_0}$ be sufficiently large depending on ${\theta}$ but otherwise fixed, and let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. We would like to show that there are infinitely many ${n}$ such that ${n + {\mathcal H}}$ contains at least two primes. We will begin with the ${W}$-trick, restricting ${n}$ to a residue class ${b \hbox{ mod } W}$ with ${b \in C(W)}$ (note that ${C(W)}$ is non-empty because ${{\mathcal H}}$ is admissible).
The general strategy will be as follows. We will introduce a weight function ${\nu: {\bf Z} \rightarrow {\bf R}^+}$ that obeys the upper bound
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W} \nu(n) \leq (\alpha+o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0} R} \ \ \ \ \ (9)$
and lower bound
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W} \theta(n+h) \nu(n)$
$\displaystyle \geq (\beta-o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0-1} R} \ \ \ \ \ (10)$
for all ${h \in {\mathcal H}}$ and some fixed ${\alpha,\beta > 0}$, where ${R}$ is a fixed power of ${x}$ (we will eventually take ${R = x^{\theta/2}}$). (The factors of ${W, \phi(W)}$, ${x}$, and ${\log R}$ on the right-hand side are natural normalisations coming from sieve theory and the reader should not pay too much attention to them.) Informally, (9) says that ${\nu}$ has some normalised density at most ${\alpha}$, and then (10) roughly speaking asserts that relative to the weight ${\nu}$, ${n+h}$ has a probability of at least ${\beta/\alpha -o(1)}$ of being prime. If we sum (10) for all ${h \in H}$ and then subtract off ${\log 3x}$ copies of (9), we conclude that
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W} (\sum_{h \in {\mathcal H}} \theta(n+h) - \log 3x) \nu(n)$
$\displaystyle \geq (k_0\beta - \alpha \frac{\log x}{\log R}-o(1)) (\frac{W}{\phi(W)})^{k_0} \frac{x}{W \log^{k_0-1} R}.$
In particular, if we have the crucial inequality
$\displaystyle k_0 \beta > \alpha \frac{\log x}{\log R} \ \ \ \ \ (11)$
we conclude that
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W} (\sum_{h \in {\mathcal H}} \theta(n+h) - \log 3x) \nu(n) \gg \frac{x}{W \log^{k_0-1} x}$
and so ${\sum_{h \in {\mathcal H}} \theta(n+h) - \log 3x}$ is positive for at least one value of ${n}$ between ${x}$ and ${2x}$. This can only occur if ${n+{\mathcal H}}$ contains two or more primes. Thus we must have ${n+{\mathcal H}}$ containing at least two primes for some ${n}$ between ${x}$ and ${2x}$; sending ${x}$ off to infinity then gives ${DHL[k_0,2]}$ as desired.
It thus suffices to find a weight function ${\nu}$ obeying the required properties (9), (10) with parameters ${\alpha,\beta,R}$ obeying the key inequality (11). It is thus of interest to make ${R}$ as large a power of ${x}$ as possible, and to minimise the ratio between ${\alpha}$ and ${\beta}$. It is in the former task that the Elliott-Halberstam hypothesis will be crucial.
The key is to find a good choice of ${\nu}$, and the selection of this weight is arguably the main contribution of Goldston, Pintz, and Yildirim, who use a carefully modified version of the Selberg sieve. Following (a slight modification of) the Goldston-Pintz-Yildirim argument, we will take a weight of the form ${\nu(n) = \lambda(n)^2}$, where
$\displaystyle \lambda(n) := \sum_{d \in {\mathcal S}_I: d|P(n)} \mu(d) g( \frac{\log d}{\log R} ) \ \ \ \ \ (12)$
where ${g: {\bf R} \rightarrow {\bf R}}$ is a fixed smooth non-negative function supported on ${[-1,1]}$ to be chosen later, ${I := (w,+\infty)}$, and ${P(n)}$ is the polynomial
$\displaystyle P(n) := \prod_{h \in {\mathcal H}} (n+h).$
The intuition here is that ${\lambda}$ is a truncated approximation to a function of the form
$\displaystyle \lambda_a(n) := \sum_{d \in {\mathcal S}_I: d|P(n)} \mu(d) \log^a \frac{P(n)}{d}$
for some natural number ${a}$, which one can check is only non-vanishing when ${P(n)}$ has at most ${a}$ distinct prime factors in ${I}$. So ${\nu(n)}$ is concentrated on those numbers ${n}$ for which ${n+h}$ already has few prime factors for ${h \in {\mathcal H}}$, which will assist in making the ratio ${\alpha/\beta}$ as small as possible.
Clearly ${\nu}$ is non-negative. Now we consider the task of estimating the left-hand side of (9). Expanding out ${\nu = \lambda^2}$ using (12) and interchanging summations, we can expand this expression as
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W: d_1,d_2 | P(n)} 1.$
The constraint ${d_1,d_2 | P(n)}$ is equivalent to requiring that for each prime ${p}$ dividing ${[d_1,d_2]}$, ${n}$ lies in one of the residue classes ${-h_i \hbox{ mod } p}$ for ${i=1,\ldots,k_0}$. By choice of ${I}$, ${p > w}$, so all the ${h_i}$ are distinct, and so we are constraining ${n}$ to lie in one of ${k_0}$ residue classes modulo ${p}$ for each ${p|[d_1,d_2]}$; together with the constraint ${n = b \hbox{ mod } W}$ and the Chinese remainder theorem, we are thus constraining ${n}$ to ${k_0^{\Omega([d_1,d_2])}}$ residue classes modulo ${W [d_1,d_2]}$, where ${\Omega([d_1,d_2])}$ is the number of prime factors of ${[d_1,d_2]}$. We thus have
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W: d_1,d_2 | P(n)} 1 = k_0^{\Omega([d_1,d_2])} \frac{x}{W[d_1,d_2]} + O( k_0^{\Omega([d_1,d_2])} ).$
Note from the support of ${g}$ that ${d_1,d_2}$ may be constrained to be at most ${R}$, so that ${d_1d_2}$ is at most ${R^2}$. We can thus express the left-hand side of (9) as the main term
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) k_0^{\Omega([d_1,d_2])} \frac{x}{W[d_1,d_2]}$
plus an error
$\displaystyle O( R^2 \sum_{d_1,d_2 \in {\mathcal S}_I} g(\frac{\log d_1}{\log R}) g(\frac{\log d_2}{\log R}) \frac{k_0^{\Omega(d_1)} k_0^{\Omega(d_2)} }{d_1 d_2} ).$
By Proposition 10, the error term is ${O( R^2 \log^{2k_0} R )}$. So if we set
$\displaystyle R := x^{\theta/2}$
then the error term will certainly give a negligible contribution to (9) with plenty of room to spare. (But when we come to the more difficult sum (10), we will have much less room – only a superlogarithmic amount of room, in fact.) To show (9), it thus suffices to show that
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) \frac{k_0^{\Omega([d_1,d_2])}}{[d_1,d_2]}$
$\displaystyle \leq (\alpha+o(1)) (\frac{W}{\phi(W)})^{k_0} \log^{-k_0} R.$
But by Proposition 10 (applied to the ${k_0}$-dimensional multiplicative function ${k_0^{\Omega([d_1,d_2])}}$) and the support of ${g}$, this bound holds with ${\alpha}$ equal to the quantity
$\displaystyle \alpha = \int_0^1 g^{(k_0)}(x)^2 \frac{x^{k_0-1}}{(k_0-1)!}\ dx.$
Now we turn to (10). Fix ${h \in {\mathcal H}}$. Repeating the arguments for (9), we may expand the left-hand side of (10) as
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R})$
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W: d_1,d_2 | P(n)} \theta(n+h).$
Now we consider the inner sum
$\displaystyle \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W: d_1,d_2 | P(n)} \theta(n+h).$
As discussed earlier, the conditions ${n=b \hbox{ mod } W}$ and ${d_1,d_2 | P(n)}$ split into ${k_0^{\Omega([d_1,d_2])}}$ residue classes ${n = a \hbox{ mod } W [d_1,d_2]}$. However, if ${n = -h \hbox{ mod } p}$ for one of the primes ${p}$ dividing ${[d_1,d_2]}$, then ${\theta(n+h)}$ must vanish (since ${R = x^\theta}$ is much less than ${n+h}$). So there are actually only ${(k_0-1)^{\Omega([d_1,d_2])}}$ residue classes ${a \hbox{ mod } W[d_1,d_2]}$ for which ${a+h}$ is coprime to ${W[d_1,d_2]}$. We thus have
$\displaystyle \sum_{x \leq n \leq 2x: n = a \hbox{ mod } W[d_1,d_2]} \theta(n+h) = (k_0-1)^{\Omega([d_1,d_2])} \frac{x}{\phi(W[d_1,d_2])}$
$\displaystyle + O( (k_0-1)^{\Omega([d_1,d_2])} E(x; W[d_1,d_2]) )$
where ${E(x;q)}$ denotes the quantity
$\displaystyle E(x;q) := \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\sum_{x \leq n \leq 2x: n = a \hbox{ mod } q} \theta(n) - \frac{x}{\phi(q)}|. \ \ \ \ \ (13)$
Remark 1 There is an inefficiency here; the supremum in (13) is over all primitive residue classes ${a \hbox{ mod } q}$, but actually one only needs to take the supremum over the ${(k_0-1)^{\Omega(q)}}$ residue classes ${a \hbox{ mod } q}$ for which ${P_h(a) = 0 \hbox{ mod } q}$, where ${P_h(a) := \prod_{h' \in {\mathcal H}:h' \neq h} (a+h'-h)}$. This inefficiency is not exploitable if we insist on using the Elliott-Halberstam conjecture as the starting hypothesis, but will be used in the arguments of the next section in which a more lightweight hypothesis is utilised.
The left-hand side of (10) is thus equal to the main term
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) (k_0-1)^{\Omega([d_1,d_2])} \frac{x}{\phi(W[d_1,d_2])}$
plus an error term
$\displaystyle O( \sum_{d_1,d_2 \in {\mathcal S}_I} g(\frac{\log d_1}{\log R}) g(\frac{\log d_2}{\log R}) (k_0-1)^{\Omega([d_1,d_2])} E(x; W[d_1,d_2]) ).$
We first deal with the error term. Since ${[d_1,d_2]}$ is in ${{\mathcal S}_I}$ and is bounded by ${R^2}$ on the support of this function, and each ${d \in {\mathcal S}_I}$ has ${3^{\Omega(d)}}$ representations of the form ${d = [d_1,d_2]}$, we can bound this expression by
$\displaystyle O( \sum_{d \in {\mathcal S}_I: d \leq R^2} 3^{\Omega(d)} (k_0-1)^{\Omega(d)} E(x;Wd) ).$
Note that we are assuming ${g}$ to be a fixed smooth compactly supported function and so it has magnitude ${O(1)}$. On the other hand, from Proposition 10 and the trivial bound ${E(x;Wd) = O( \frac{x \log x}{Wd} + \frac{x}{\phi(W) \phi(d)} )}$ we have
$\displaystyle \sum_{d \in {\mathcal S}_I: d \leq R^2} 3^{2\Omega(d)} (k_0-1)^{2\Omega(d)} E(x;Wd) = O( x \log^{O(1)} x )$
while from (8) (and here we crucially use the choice ${R = x^{\theta/2}}$ of ${R}$) one easily verifies that
$\displaystyle \sum_{d \in {\mathcal S}_I: d \leq R^2} E(x;Wd) = O( x \log^{-A} x )$
for any fixed ${A}$. By the Cauchy-Schwarz inequality we see that the error term to (10) is negligible (assuming ${w}$ sufficiently slowly growing of course). Meanwhile, the main term can be rewritten as
$\displaystyle \frac{x}{\phi(W)} \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) \frac{f([d_1,d_2])}{[d_1,d_2]}$
where ${f}$ is the ${k_0-1}$-dimensional multiplicative function
$\displaystyle f(d) := \prod_{p|d} (k_0-1) \frac{p}{p-1}.$
Applying Proposition 10, we obtain (10) with
$\displaystyle \beta = \int_0^1 g^{(k_0-1)}(t)^2 \frac{t^{k_0-2}}{(k_0-2)!}\ dt.$
To obtain the crucial inequality (11), we thus need to locate a fixed smooth non-negative function supported on ${[-1,1]}$ obeying the inequality
$\displaystyle k_0 \int_0^1 g^{(k_0-1)}(t)^2 \frac{t^{k_0-2}}{(k_0-2)!}\ dt > \frac{2}{\theta} \int_0^1 g^{(k_0)}(t)^2 \frac{t^{k_0-1}}{(k_0-1)!}\ dt . \ \ \ \ \ (14)$
In principle one can use calculus of variations to optimise the choice of ${g}$ here (it will be the ground state of a certain one-dimensional Schrödinger operator), but one can already get a fairly good result here by a specific choice of ${g}$ that is amenable for computations, namely a polynomial of the form ${g(t) := \frac{1}{(k_0+l_0)!} (1-t)^{k_0+l_0}}$ for ${t \in [0,1]}$ and some integer ${l_0>0}$, with ${g}$ vanishing for ${t>1}$ and smoothly truncated to ${[-1,1]}$ somehow at negative values of ${t}$. Strictly speaking, this ${g}$ is not admissible here because it is not infinitely smooth at ${1}$, being only ${k_0+l_0-1}$ times continuously differentiable instead, but one can regularise this function to be smooth without significantly affecting either side of (14), so we will go ahead and test (14) with this function and leave the regularisation details to the reader. The inequality then becomes (after cancelling some factors)
$\displaystyle k_0 \int_0^1 (1-t)^{2l_0+2} \frac{t^{k_0-2}}{(k_0-2)!}\ dt > \frac{2}{\theta} \int_0^1 (l_0+1)^2 (1-t)^{2l_0} \frac{t^{k_0-1}}{(k_0-1)!}\ dt .$
Using the Beta function identity
$\displaystyle \int_0^1 (1-t)^a t^b\ dt = \frac{a! b!}{(a+b+1)!}$
we have
$\displaystyle \alpha = \frac{(2l_0)!}{(l_0!)^2 (k_0+2l_0)!} \ \ \ \ \ (15)$
and
$\displaystyle \beta = \frac{(2l_0+2)!}{((l_0+1)!)^2 (k_0+2l_0+1)!} \ \ \ \ \ (16)$
and the preceding equation now becomes
$\displaystyle k_0 \frac{(2l_0+2)!}{(2l_0+k_0+1)!} > \frac{2}{\theta} (l_0+1)^2 \frac{(2l_0)!}{(2l_0+k_0)!}$
which simplifies to
$\displaystyle 2\theta > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}).$
Actually, the same inequality is also applicable when ${l_0}$ is real instead of integer, using Gamma functions in place of factorials; we leave the details to the interested reader. We can then optimise in ${l_0}$ by setting ${2l_0+1 = \sqrt{k_0}}$, arriving at the inequality
$\displaystyle 2\theta > (1 + \frac{1}{\sqrt{k_0}})^2.$
But as long as ${\theta > 1/2}$, this inequality is satisfiable for any ${k_0}$ larger than ${(\sqrt{2\theta}-1)^{-2}}$. This concludes the proof of Theorem 4.
Remark 2 One can obtain slightly better dependencies of ${k_0}$ in terms of ${\theta}$ by using more general functions for ${g}$ than the monomials ${\frac{1}{(k_0+l_0)!} (1-x)^{k_0+l_0}}$, for instance one can take linear combinations of such functions. See the paper of Goldston, Pintz, and Yildirim for details. Unfortunately, as noted in this survey of Soundararajan, one has the general inequality
$\displaystyle k_0 \int_0^1 g^{(k_0-1)}(t)^2 \frac{t^{k_0-2}}{(k_0-2)!}\ dt \leq 4 \int_0^1 g^{(k_0)}(t)^2 \frac{x^{k_0-1}}{(k_0-1)!}\ dx \ \ \ \ \ (17)$
which defeats any attempt to directly use this method using only the Bombieri-Vinogradov result that ${EH[\theta]}$ holds for all ${\theta < 1/2}$. We show (17) in the case when ${k_0}$ is large. Write ${f(t) := g^{(k_0-1)}(t) t^{k_0/2-1}}$, then (17) simplifies to
$\displaystyle \frac{k_0 (k_0-1)}{4} \int_0^1 f(t)^2\ dt \leq \int_0^1 (f'(t) - (k_0/2-1) t^{-1} f(t))^2 t\ dt.$
The right-hand side simplifies after some integration by parts to
$\displaystyle \int_0^1 f'(t)^2 t + \frac{(k_0-2)^2}{4} f(t)^2 t^{-1}\ dt.$
Subtracting off ${\int_0^1 \frac{(k_0-2)^2}{4} f(t)^2\ dt}$ from both sides, one is left with
$\displaystyle \frac{3k_0-4}{4} \int_0^1 f(t)^2 \leq \int_0^1 f'(t)^2 t + \frac{(k_0-2)^2}{4} f(t)^2 (t^{-1} - 1)\ dx.$
From the fundamental theorem of calculus and Cauchy-Schwarz, one has the bound
$\displaystyle |f(y)|^2 \leq (\int_0^1 f'(y)^2 y\ dy) (\log(1/y)).$
Using this bound for ${y}$ close to ${1}$ and dominating ${\frac{3k_0-4}{4}}$ by ${\frac{(k_0-2)^2}{4} (y^{-1} - 1)}$ for ${y}$ far from ${1}$, we obtain the claim (at least if ${k_0}$ is large enough). There is some slack in this argument; it would be of interest to calculate exactly what the best constants are in (17), so that one can obtain the optimal relationship between ${\theta}$ and ${k_0}$.
To get around this obstruction (17) in the unconditional setting when one only has ${EH[\theta]}$ for ${\theta<1/2}$, Goldston, Pintz, and Yildirim also considered sums of the form ${\sum_{x \leq n \leq 2x: n = b \hbox{ mod } W} \theta(n+h) \nu(n)}$ in which ${h}$ was now outside (but close to) ${{\mathcal H}}$. While the bounds here were significantly inferior to those in (10), they were still sufficient to prove a variant of the inequality (11) to get reasonably small gaps between primes.
— 4. The Motohashi-Pintz-Zhang theorem —
We now modify the above argument to give Theorem 5. Our treatment here is different from that of Zhang in that it employs the method of Buchstab iteration; a related argument also appears in the paper of Motohashi and Pintz. This arrangement of the argument leads to a more efficient dependence of ${k_0}$ on ${\varpi}$ than in the paper of Zhang. (The argument of Motohashi and Pintz is a bit more complicated and uses a slightly different formulation of the base conjecture ${MPZ[\varpi]}$, but the final bounds are similar to those given here, albeit with non-explicit constants in the ${O()}$ notation.)
The main idea here is to truncate the interval ${I}$ of relevant primes from ${(w,\infty)}$ to ${(w,x^\varpi)}$ for some small ${\varpi}$. Somewhat remarkably, it turns out that this apparently severe truncation does not affect the sums (9), (10) here as long as ${k_0 \varpi}$ is large (which is going to be the case in practice, with ${k_0}$ being comparable to ${\varpi^{-2}}$). The intuition is that ${\nu}$ was already concentrated on those ${n}$ for which ${P(n)}$ had about ${O(k_0)}$ factors, and it is too “expensive” for one of these factors to as large as ${x^\varpi}$ or more, as it forces many of the other factors to be smaller than they “want” to be. The advantage of truncating the set of primes this way is that the version of the Elliott-Halberstam conjecture needed also acquires the same truncation, which gives that version a certain “well-factored” form (in the spirit of the work of Bombieri, Fouvry, Friedlander, and Iwaniec) which is essential in being able to establish that conjecture unconditionally for some suitably small ${\varpi}$.
To make this more precise, we first formalise the conjecture ${MPZ[\varpi]}$ for ${0 < \varpi < 1/4}$ mentioned earlier.
Conjecture 12 (${MPZ[\varpi]}$) Let ${{\mathcal H}}$ be a fixed ${k}$-tuple (not necessarily admissible) for some fixed ${k \geq 2}$, and let ${b \hbox{ mod } W}$ be a primitive residue class. Then
$\displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} \sum_{a \in ({\bf Z}/q{\bf Z})^\times: P(a) = 0 \hbox{ mod } q} |\Delta_{b,W}(\theta; q,a)| = O( x \log^{-A} x)$
for any fixed ${A>0}$, where ${I = (w,x^{\varpi})}$ and ${\Delta_{b,W}(\theta;q,a)}$ is the quantity
$\displaystyle \Delta_{b,W}(\theta;q,a) := | \sum_{x \leq n \leq 2x: n=b \hbox{ mod } W; n = a \hbox{ mod } q} \theta(n) \ \ \ \ \ (18)$
$\displaystyle - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x: n = b \hbox{ mod } W} \theta(n)|.$
and
$\displaystyle P(a) := \prod_{h \in {\mathcal H}} (a+h).$
This is the ${W}$-tricked formulation of the conjecture as (implicitly) stated in Zhang’s paper, which did not have the restriction ${n = b \hbox{ mod } W}$ present (and with the interval ${I}$ enlarged from ${(w,x^\varpi)}$ to ${(1,x^\varpi)}$, and ${{\mathcal H} \cup \{0\}}$ was required to be admissible). However the two formulations are morally equivalent (and Zhang’s arguments establish Theorem 6 with ${MPZ[\varpi]}$ as stated). From the prime number theorem in arithmetic progressions (with ${O( x \log^{-A} x)}$ error term) together with Proposition 10 we observe that we may replace (18) here by the slight variant
$\displaystyle \Delta'_{b,W}(\theta;q,a) := | \sum_{x \leq n \leq 2x: n=b \hbox{ mod } W; n = a \hbox{ mod } q} \theta(n) \ \ \ \ \ (19)$
$\displaystyle - \frac{1}{\phi(Wq)} x|$
without affecting the truth of ${MPZ[\varpi]}$.
It is also not difficult to deduce ${MPZ[\varpi]}$ from ${EH[1/2 + 2 \varpi]}$ after using a Cauchy-Schwarz argument to dispose of the ${k^{\Omega(d)}}$ residue classes ${a}$ in the above sum (cf. the treatment of the error term in (10) in the previous section); we leave the details to the interested reader. Note however that whilst ${EH[1/2+2\varpi]}$ demands control over all primitive residue classes ${a}$ in a given modulus ${q}$, the conjecture ${MPZ[\varpi]}$ only requires control of a much smaller number of residue classes (roughly polylogarithmic in number, on average). Thus ${MPZ[\varpi]}$ is simpler than ${EH[1/2+2\varpi]}$, though it is still far from trivial.
We now begin the proof of Theorem 5. Let ${0 < \varpi < 1/4}$ be such that ${MPZ[\varpi]}$ holds, and let ${k_0}$ be a sufficiently large quantity depending on ${\varpi}$ but which is otherwise fixed. As before, it suffices to locate a non-negative sieve weight ${\nu}$ that obeys the estimates (9), (10) for parameters ${\alpha,\beta,R}$ that obey the key inequality (11), and with ${g}$ smooth and supported on ${[-1,1]}$. The choice of weight ${\nu}$ is almost the same as before; it is also given as a square ${\nu(n) = \lambda(n)^2}$ with ${\lambda}$ given by (12), but now the interval ${I}$ is truncated to ${(w,x^\varpi)}$ instead of ${(x,\infty)}$. Also, in this argument we take
$\displaystyle R = x^{1/4 + \varpi}$
We now establish (9). By repeating the previous arguments, the left-hand side of (9) is equal to a main term
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) k_0^{\Omega([d_1,d_2])} \frac{x}{W[d_1,d_2]} \ \ \ \ \ (20)$
plus an error term which continues to be acceptable (indeed, the error term is slightly smaller than in the previous case due to the truncated nature of ${I}$). At this point in the previous section we applied Proposition 10, but that proposition was only available for the untruncated interval ${[w,+\infty)}$ instead of the truncated interval ${[w,x^\varpi)}$. One could try to adapt the proof of that proposition to the truncated case, but then one is faced with the problem of controlling the truncated zeta function ${\zeta_I}$. While one can eventually get some reasonable asymptotics for this function, it seems to be more efficient to eschew Fourier analysis and work entirely in “physical space” by the following partial Möbius inversion argument. Write ${J := [x^\varpi,\infty)}$, thus ${I \cup J = [w,+\infty)}$. Observe that for any ${d \in {\mathcal S}_{I \cup J}}$, the quantity ${\sum_{a \in {\mathcal S}_J: a|d} \mu(a)}$ equals ${1}$ when ${d}$ lies in ${{\mathcal S}_I}$ and vanishes otherwise. Hence, for any function ${F(d_1,d_2)}$ of ${d_1}$ and ${d_2}$ supported on squarefree numbers we have the partial Mobius inversion formula
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} F(d_1,d_2) = \sum_{a_1, a_2 \in {\mathcal S}_J} \mu(a_1) \mu(a_2) \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}} F(a_1 d_1, a_2 d_2)$
and so the main term (20) can be expressed as
$\displaystyle \sum_{a_1, a_2 \in {\mathcal S}_J} \mu(a_1) \mu(a_2) \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}} \mu(a_1 d_1) g(\frac{\log a_1d_1}{\log R}) \mu(a_2 d_2) g(\frac{\log a_2 d_2}{\log R}) \ \ \ \ \ (21)$
$\displaystyle k_0^{\Omega([a_1d_1,a_2d_2])} \frac{x}{W [a_1d_1,a_2d_2]}.$
We first dispose of the contribution to (21) when ${a_i,d_j}$ share a common prime factor ${p_* \in J}$ for some ${i,j=1,2}$. For any fixed ${i,j}$, we can bound this contribution by
$\displaystyle \ll \frac{x}{W} \sum_{p_* \in J} \sum_{a_1,a_2 \in {\mathcal S}_J} \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}} 1_{p^2_*|a_id_j} (a_1 d_1 a_2 d_2)^{-1/\log R} \frac{k_0^{\Omega([a_1d_1,a_2d_2])}}{[a_1d_1,a_2d_2]}.$
Factorising the inner two sums as an Euler product, this becomes
$\displaystyle \ll \frac{x}{W} \sum_{p_* \in J} \frac{1}{p_*^2} ( \prod_{p \in I \cup J} 1 + O(\frac{1}{p^{1+1/\log R}}) ).$
[UPDATE: The above argument is not quite correct; a corrected (and improved) version is given at this newer post.] The product is ${O(\log^{O(1)} R)}$ by e.g. Mertens’ theorem, while ${\sum_{p_* \in J} \frac{1}{p_*^2} \ll x^{-\varpi}}$. So the contribution of this case is negligible.
If ${a_i,d_j}$ do not share a common factor ${p_* \in J}$ for any ${i,j=1,2}$, then we can factor ${[a_1d_1,a_2d_2]}$ as ${[a_1,a_2][d_1,d_2]}$. Rearranging this portion of (21) and then reinserting the case when ${a_i,d_j}$ have a common factor ${p_* \in J}$ for some ${i,j}$, we may write (21) up to negligible errors as
$\displaystyle \frac{x}{W} \sum_{a_1, a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}} \mu(d_1) g(\frac{\log d_1}{\log R} + \frac{\log a_1}{\log R})$
$\displaystyle \mu(d_2) g(\frac{\log d_2}{\log R} + \frac{\log a_2}{\log R}) \frac{k_0^{\Omega([d_1,d_2])}}{[d_1,d_2]}.$
Note that we can restrict ${a_1,a_2}$ to be at most ${R}$ as otherwise the ${g}$ factors vanish. The inner sum
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_{I \cup J}} \mu(d_1) g(\frac{\log d_1}{\log R} + \frac{\log a_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R} + \frac{\log a_2}{\log R}) \frac{k_0^{\Omega([d_1,d_2])}}{[d_1,d_2]}$
is now of the form that can be treated by Proposition 10, and takes the form
$\displaystyle (\frac{\phi(W)}{W} \log R)^{-k_0} (\int_0^\infty g^{(k_0)}(x + \frac{\log a_1}{R}) g^{(k_0)}(x + \frac{\log a_2}{R}) \frac{x^{k_0-1}}{(k_0-1)!}\ dx$
$\displaystyle + o(1)).$
Here we make the technical remark that the translates of ${g}$ by shifts between ${0}$ and ${1}$ are uniformly controlled in smooth norms, which means that the ${o(1)}$ error here is uniform in the choices of ${a_1, a_2}$.
Let us first deal with the contribution of the ${o(1)}$ error term. This is bounded by
$\displaystyle o( \frac{x}{W} (\frac{\phi(W)}{W} \log R)^{-k_0} \sum_{a_1,a_2 \in {\mathcal S}_{(x^\varpi, R]}} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} ).$
The inner sum factorises as
$\displaystyle \prod_{x^\varpi < p \leq R} (1 + \frac{3 k_0}{p})$
which by Mertens’ theorem is ${O(1)}$ (albeit with a rather large implied constant!), so this error is negligible for the purposes of (9). Indeed, (9) is now reduced to the inequality
$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]}$
$\displaystyle \int_0^\infty g^{(k_0)}(x + \frac{\log a_1}{\log R}) g^{(k_0)}(x + \frac{\log a_2}{\log R}) \frac{x^{k_0-1}}{(k_0-1)!}\ dx \ \ \ \ \ (22)$
$\displaystyle \leq \alpha+o(1).$
Note that the factor ${\frac{x^{k_0-1}}{(k_0-1)!}}$ increases very rapidly with ${x}$ when ${k}$ is large, which basically means that any non-trivial shift of the ${g^{(k_0)}}$ factors to the left by ${\frac{\log a_1}{\log R}}$ or ${\frac{\log a_2}{\log R}}$ will cause the integral in (22) to decrease dramatically. Since all the ${a_1,a_2}$ in ${{\mathcal J}}$ are either equal to ${1}$ or bounded below by ${x^\varpi}$, this will cause the ${a_1=a_2=1}$ term to dominate in the regime when ${k_0 \varpi}$ is large (or more precisely ${k_0 \varpi \gg \log k_0}$), which is the case in applications.
At this point, in order to perform the computations cleanly, we will mimic the arguments from the previous section and take the explicit choice
$\displaystyle g(x) := \frac{1}{(k_0+l_0)!} (1-x)_+^{k_0+l_0}$
for some integer ${l_0>0}$ and ${x>0}$ (and some smooth continuation to ${[-1,1]}$ for negative ${x}$, and so
$\displaystyle g^{(k_0)}(x) = (-1)^{k_0} \frac{1}{l_0!} (1-x)^{l_0}_+$
for positive ${x}$. (Again, this function is not quite smooth at ${1}$, but this issue can be dealt with by an infinitesimal regularisation argument which we omit here.) The left-hand side of (22) now becomes
$\displaystyle \frac{1}{(l_0!)^2} \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \int_0^\infty (1-x-\frac{\log a_1}{\log R})_+^{l_0} (1-x-\frac{\log a_2}{\log R})_+^{l_0}$
$\displaystyle \frac{x^{k_0-1}}{(k_0-1)!}\ dx.$
The integral here is a little bit more complicated than a beta integral. To estimate it, we use the beta function identity to observe that
$\displaystyle \int_0^\infty (1-x-\frac{\log a_1}{\log R})_+^{2l_0} \frac{x^{k_0-1}}{(k_0-1)!}\ dx = (1 - \frac{\log a_1}{\log R})_+^{k_0+2l_0} \frac{(2l_0)!}{(k_0+2l_0)!}$
and
$\displaystyle \int_0^\infty (1-x-\frac{\log a_2}{\log R})_+^{2l_0} \frac{x^{k_0-1}}{(k_0-1)!}\ dx = (1 - \frac{\log a_2}{\log R})_+^{k_0+2l_0} \frac{(2l_0)!}{(k_0+2l_0)!}$
and hence by Cauchy-Schwarz
$\displaystyle \int_0^\infty (1-x-\frac{\log a_1}{\log R})_+^{l_0} (1-x-\frac{\log a_2}{\log R})_+^{l_0} \frac{x^{k_0-1}}{(k_0-1)!}\ dx$
$\displaystyle \leq (1 - \frac{\log a_1}{\log R})_+^{k_0/2+l_0} (1 - \frac{\log a_2}{\log R})_+^{k_0/2+l_0} \frac{(2l_0)!}{(k_0+2l_0)!}.$
This Cauchy-Schwarz step is a bit wasteful when ${a_1,a_2}$ are far apart, but this does seems to only lead to a minor loss of efficiency in the estimates. We have thus bounded the left-hand side of (22) by
$\displaystyle \frac{(2l_0)!}{(l_0!)^2 (k_0+2l_0)!} \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]}$
$\displaystyle (1 - \frac{\log a_1}{\log R})_+^{k_0/2+l_0} (1 - \frac{\log a_2}{\log R})_+^{k_0/2+l_0}.$
It is now convenient to collapse the double summation to a single summation. We may bound
$\displaystyle (1 - \frac{\log a_1}{\log R})_+^{k_0/2+l_0} (1 - \frac{\log a_2}{\log R})_+^{k_0/2+l_0} \leq (1 - \frac{\log [a_1,a_2]}{\log R^2})_+^{k_0/2+l_0}$
(since ${\frac{\log [a_1,a_2]}{\log R^2}}$ is less than the greater of ${\frac{\log a_1}{\log R}}$ and ${\frac{\log a_2}{\log R}}$) and observe that each ${a \in {\mathcal S}_J}$ has ${3^{\Omega(a)}}$ representations of the form ${a = [a_1,a_2]}$, so we may now bound the left-hand side of (22) by
$\displaystyle \frac{(2l_0)!}{(l_0!)^2 (k_0+2l_0)!} \sum_{a \in {\mathcal S}_J} \frac{(3k_0)^{\Omega(a)}}{a} (1 - \frac{\log a}{\log R^2})_+^{k_0/2+l_0}.$
Note that an element ${a}$ of ${{\mathcal S}_J}$ is either equal to ${1}$, or lies in the interval ${[x^{n\varpi}, x^{(n+1)\varpi})}$ for some natural number ${n \geq 1}$. In the latter case, we have
$\displaystyle (1 - \frac{\log a}{\log R^2})_+ \leq (1 - \frac{2n \varpi}{1 + 4\varpi})_+.$
In particular, this expression vanishes if ${n \geq 2 + \frac{1}{2\varpi}}$. We can thus bound the left-hand side of (22) by
$\displaystyle \frac{(2l_0)!}{(l_0!)^2 (k_0+2l_0)!} (1 + \sum_{1 \leq n < 2 + \frac{1}{2\varpi}} (1 - \frac{2n \varpi}{1 + 4\varpi})^{k_0/2 + l_0}$
$\displaystyle \sum_{a \in {\mathcal S}_J: a < x^{(n+1)\varpi}} \frac{(3k_0)^{\Omega(a)}}{a} ).$
If we introduce the quantity
$\displaystyle \Phi_{3k_0}(z,y) := \sum_{j=0}^\infty (3k_0)^j \sum_{y \leq p_1 < \ldots < p_j: p_1 \ldots p_j < z} \frac{1}{p_1 \ldots p_j} \ \ \ \ \ (23)$
then we have thus bounded the left-hand side of (22) by
$\displaystyle \frac{(2l_0)!}{(l_0!)^2 (k_0+2l_0)!} (1 + \sum_{j=1}^\infty (1 - \frac{2j \varpi}{1 + 4\varpi})_+^{k_0/2 + l_0} \Phi_{3k_0}( x^{(j+1)\varpi}, x^\varpi )).$
We observe that
$\displaystyle \Phi_{3k_0}(z,y) = 1 \ \ \ \ \ (24)$
when ${y \geq z}$, while in general we have the Buchstab identity
$\displaystyle \Phi_{3k_0}(z,y) \leq 1 + 3k_0 \sum_{y \leq p < z} \frac{1}{p} \Phi(\frac{z}{p}, p) \ \ \ \ \ (25)$
as can be seen by isolating the smallest prime ${p_1}$ in all the terms in (23) with ${j \geq 1}$. (This inequality is very close to being an identity, the only loss coming from the possibility of the prime factor ${p}$ being repeated in a term associated to ${\frac{1}{p} \Phi(\frac{z}{p},p)}$.) We can iterate this identity to obtain the following conclusion:
Lemma 13 For any ${n \geq 1}$, we have
$\displaystyle \Phi_{3k_0}(z,y) \leq \prod_{j=1}^{n-1} (1 + 3k_0 \log(1+\frac{1}{j})) + o(1)$
whenever ${z \leq y^n}$ and ${y \geq x^\varpi}$ for some fixed ${\varpi > 0}$, with the error term being uniform in the choice of ${z,y}$.
Proof: Write ${A_n := \prod_{j=1}^{n-1} (1 + 3k_0 \log(1+\frac{1}{j}))}$. We prove the bound ${\Phi_{3k_0}(z,y) \leq A_n + o(1)}$ by strong induction on ${n}$. The case ${n=1}$ follows from (24). Now suppose that ${n>1}$ and that the claim has already been proven for smaller ${n}$. Let ${z \leq y^n}$ and ${y > x^\varpi}$. Note that ${\frac{z}{p} \leq p^j}$ whenever ${p \geq z^{\frac{1}{j+1}}}$. We thus have from (25) and the induction hypothesis that
$\displaystyle \Phi_{3k_0}(z,y) \leq 1 + 3k_0 \sum_{j=1}^{n-1} \sum_{z^{\frac{1}{j+1}} \leq p < z^{\frac{1}{j}}} \frac{1}{p} (A_j + o(1) );$
applying Mertens’ theorem (or the prime number theorem) we have
$\displaystyle \sum_{z^{\frac{1}{j+1}} \leq p < z^{\frac{1}{j}}} \frac{1}{p} ( A_j + o(1) ) = A_j \log(1 + \frac{1}{j}) + o(1)$
and the claim follows from the telescoping identity
$\displaystyle A_n = 1 + 3k_0 \sum_{j=1}^{n-1} A_j \log(1+\frac{1}{j}).$
$\Box$
Applying this inequality, we have established (22) with
$\displaystyle \alpha := \frac{(2l_0)!}{(l_0!)^2 (k_0+2l_0)!} (1 + \kappa) \ \ \ \ \ (26)$
where
$\displaystyle \kappa := \sum_{1 \leq n < 2 + \frac{1}{2\varpi}} (1 - \frac{2n \varpi}{1 + 4\varpi})^{k_0/2 + l_0} \prod_{j=1}^{n} (1 + 3k_0 \log(1+\frac{1}{j})) ). \ \ \ \ \ (27)$
We remark that as a first approximation we have
$\displaystyle \prod_{j=1}^{n} (1 + 3k_0 \log(1+\frac{1}{j})) ) \approx \frac{(3k_0)^{n}}{n!}$
and
$\displaystyle (1 - \frac{2n \varpi}{1 + 4\varpi})^{k_0/2 + l_0} \approx \exp( - n k_0 \varpi )$
so in the regime ${k_0 \varpi \gg \log k_0}$, ${\kappa}$ is roughly ${3k_0 \exp( - k_0 \varpi )}$, which will be negligible for the parameter ranges of ${k_0,\varpi}$ of interest. Thus the ${\alpha}$ in this argument is quite close to the ${\alpha}$ from (15) in practice.
Now we turn to (10). Fix ${h \in {\mathcal H}}$. As in the previous section, we can bound the left-hand side of (10) as the sum of the main term
$\displaystyle \sum_{d_1,d_2 \in {\mathcal S}_I} \mu(d_1) g(\frac{\log d_1}{\log R}) \mu(d_2) g(\frac{\log d_2}{\log R}) (k_0-1)^{\Omega([d_1,d_2])} \frac{x}{\phi(W[d_1,d_2])}$
plus an error term
$\displaystyle O( \sum_{d_1,d_2 \in {\mathcal S}_I} g(\frac{\log d_1}{\log R}) g(\frac{\log d_2}{\log R}) (k_0-1)^{\Omega([d_1,d_2])} E'(x; [d_1,d_2]) )$
where ${E'(x;q)}$ is the quantity
$\displaystyle E'(x;q) := \sum_{a \in ({\bf Z}/q{\bf Z})^\times: P_h(a) = 0 \hbox{ mod } q} |\Delta'_{b,W}(\theta; q,a)|,$
${P_h}$ is the polynomial ${P_h(a) := \prod_{h' \in {\mathcal H} \backslash \{h\}} (n+h'-h)}$, and ${\Delta'_{b,W}}$ was defined in (19). Using the hypothesis ${MPZ[\varpi]}$ and Cauchy-Schwarz as in the previous section we see that the error term is negligible for the purposes of establishing (10). As for the main term, the same argument used to reduce (9) to (22) shows that (10) reduces to
$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{(k_0-1)^{\Omega([a_1,a_2])}}{\phi([a_1,a_2])} \int_0^\infty g^{(k_0-1)}(x + \frac{\log a_1}{\log R}) g^{(k_0-1)}(x + \frac{\log a_2}{\log R}) \frac{x^{k_0-2}}{(k_0-2)!}\ dx$
$\displaystyle \geq \beta-o(1).$
Here, we can do something a bit crude; with our choice of ${g}$, the integrand is non-negative, so we can simply discard all but the ${a_1=a_2=1}$ term and reduce to
$\displaystyle \int_0^\infty g^{(k_0-1)}(x) g^{(k_0-1)}(x) \frac{x^{k_0-2}}{(k_0-2)!}\ dx \geq \beta$
(The intuition here is that by refusing to sieve using primes larger than ${x^\varpi}$, we have enlarged the sieve ${\nu}$, which makes the upper bound (9) more difficult but the lower bound (10) actually becomes easier.) So we can take the same choice (16) of ${\beta}$ as in the previous section:
$\displaystyle \beta := \frac{(2l_0+2)!}{((l_0+1)!)^2 (k_0+2l_0+1)!}.$
Inserting this and (26) into (11) and simplifying, we see that we can obtain ${DHL[k_0,2]}$ once we can verify the inequality
$\displaystyle 1+4\varpi > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1+\kappa).$
As before, ${l_0}$ can be taken to be non-integer if desired. Setting ${k_0}$ to be slightly larger than ${(\sqrt{1+4\varpi}-1)^{-2} \approx (2\varpi)^{-2}}$ we obtain Theorem 5.
— 5. Using optimal values of ${g}$ (NEW, June 5, 2013) —
We can do better than given above by using an optimal value of ${g}$. The following result was obtained recently by Farkas, Pintz, and Revesz, and independently worked out by commenters on this blog:
Theorem 14 (Optimal GPY weight) Let ${k_0 > 2}$ be an integer. Then the ratio
$\displaystyle \frac{\int_0^1 f'(x)^2 x^{k_0-1}\ dx}{\int_0^1 f(x)^2 x^{k_0-2}\ dx}$
where ${f: [0,1] \rightarrow {\bf R}}$ is a smooth function with ${f(1)=0}$ that is not identically vanishing, has a minimal value of
$\displaystyle \lambda := \frac{j_{k_0-2}^2}{4}$
where ${j_{k_0-2}}$ is the first zero of the Bessel function ${J_{k_0-2}}$. Furthermore, this minimum is attained if (and only if) ${f}$ is a scalar multiple of the function
$\displaystyle f_0(x) = x^{1-k_0/2} J_{k_0-2}(2\sqrt{x\lambda}).$
Proof: The function ${J_{k_0-2}}$, by definition, obeys the Bessel differential equation
$\displaystyle x^2 \frac{d}{dx^2} J_{k_0-2} + x \frac{d}{dx} J_{k_0-2} + (x^2 - (k_0-2)^2) J_{k_0-2} = 0$
and also vanishes to order ${k_0-2}$ at the origin. From this and routine computations it is easy to see that ${f_0}$ is smooth, strictly positive on ${[0,1)}$, and obeys the differential equation
$\displaystyle \frac{d}{dx} (x^{k_0-1} \frac{d}{dx} f_0(x)) + \lambda x_0^{k-2} f_0(x) = 0. \ \ \ \ \ (28)$
If we write ${g_0(x) := \frac{f'_0}{f_0}(x)}$, which is well-defined away from ${1}$ since ${f_0}$ is non-vanishing on ${[0,1)}$, then ${g_0}$ obeys the Ricatti-type equation
$\displaystyle (k_0-1) g_0(x) + x g'_0(x) + x g_0(x)^2 + \lambda = 0. \ \ \ \ \ (29)$
$\displaystyle Q( f ) := \int_0^1 f'(x)^2 x^{k_0-1}\ dx - \lambda \int_0^1 f(x)^2 x^{k_0-2}\ dx$
for smooth functions ${f: [0,1] \rightarrow {\bf R}}$ with ${f(1)=0}$. A calculation using (29) and integration by parts shows that
$\displaystyle \int_0^1 (f'(x)-g_0(x)f(x))^2 x^{k_0-1}\ dx = Q(f)$
and so ${Q(f) \geq 0}$, giving the first claim; the second claim follows by noting that ${f'-g_0 f}$ vanishes if and only if ${f}$ is a scalar multiple of ${f_0}$. (Note that the integration by parts is a little subtle, because ${f_0}$ vanishes to first order at ${x=1}$ and so ${g_0}$ blows up to first order; but ${g_0 f^2}$ still vanishes to first order at ${x=1}$, allowing one to justify the integration by parts by a standard limiting argument.) $\Box$
If we now test (14) with a function ${g: [0,1] \rightarrow {\bf R}}$ which is smooth, vanishes to order ${k_0}$ at ${x=1}$, and has a ${(k_0-1)^{th}}$ derivative equal to ${f_0}$, we see that we can deduce ${DHL[k_0,2]}$ from ${EH[\theta]}$ whenever
$\displaystyle 2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}.$
Using the known asymptotic
$\displaystyle j_n = n + c n^{1/3} + O( n^{-1/3} )$
for ${c := 1.8557571\ldots}$ and large ${n}$ (see e.g. Abramowitz and Stegun), this is asymptotically of the form
$\displaystyle 2 \theta > 1 + 2c k_0^{-2/3} + O( k_0^{-1} )$
or
$\displaystyle k_0 > (2c (2\theta-1))^{-3/2},$
thus giving a relationship of the form ${k_0 \sim (\theta-1/2)^{-3/2}}$ that is superior to the previous relationship ${k_0 \sim (\theta-1/2)^{-2}}$.
A similar argument can be given for Theorem 5, using ${g}$ of the form above rather than a monomial ${\frac{(1-x)^{k_0+l_0}}{(k_0+l_0)!}}$ (and extended by zero to ${[1,+\infty)}$). For future optimisation we consider a generalisation ${MPZ[\varpi,\delta]}$ of ${MPZ[\varpi]}$ in which the interval ${I}$ is of the form ${[w,x^\delta)}$ rather than ${[w,x^\varpi)}$, so that ${J}$ is now ${[x^\delta,\infty)}$ rather than ${[x^\varpi,\infty)}$. As before, the key point is the estimation of ${\alpha}$. The arguments leading to (22) go through for any test function ${g}$, so we have to show
$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \int_0^\infty g^{(k_0)}(x + \frac{\log a_1}{\log R}) g^{(k_0)}(x + \frac{\log a_2}{\log R}) \frac{x^{k_0-1}}{(k_0-1)!}\ dx \leq \alpha+o(1).$
As ${g}$ has ${(k_0-1)^{th}}$ derivative equal to ${f_0}$, this is
$\displaystyle \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} \int_0^{1-\frac{\log \max(a_1,a_2)}{\log R}} f'_0(x + \frac{\log a_1}{\log R}) f'_0(x + \frac{\log a_2}{\log R}) \ \ \ \ \ (30)$
$\displaystyle \frac{x^{k_0-1}}{(k_0-1)!}\ dx \leq \alpha+o(1).$
We need some sign information on ${f'_0}$:
Lemma 15 On ${[0,1)}$, ${f_0}$ is positive, ${f'_0}$ is negative and ${f''_0}$ is positive.
Proof: From (28) we have
$\displaystyle x f''_0(x) + (k_0-1) f'_0(x) + \lambda f_0(x) = 0.$
From construction we already know that ${f_0}$ is positive on ${[0,1)}$. The above equation then shows that ${f'_0}$ is negative at ${x=0}$, and that ${f_0}$ cannot have any local minimum in ${(0,1)}$, so ${f'_0}$ is negative throughout. To obtain the final claim ${f''_0>0}$ we use an argument provided by Gergely Harcos in the comments: from the recursive relations for Bessel functions we can check that ${f''_0}$ is a positive multiple of ${x^{-k_0/2} J_{k_0}(2 \sqrt{x\lambda})}$, and the claim then follows from the interlacing properties of the zeroes of Bessel functions. $\Box$
Write ${A := \int_0^1 f_0(x)^2 \frac{x^{k_0-2}}{(k_0-2)!}}$, so ${A}$ is positive and by Theorem 14 we have
$\displaystyle \int_0^1 f'_0(x)^2 \frac{x^{k_0-1}}{(k_0-1)!} = \frac{j_{k_0-2}^2}{4(k_0-1)} A.$
If ${a_1 < R}$, then as ${f'_0}$ is negative and increasing we have
$\displaystyle -f'_0(x + \frac{\log a_1}{\log R}) \leq -f'_0( x / (1-\frac{\log a_1}{\log R}) )$
for ${0 \leq x \leq 1 - \frac{\log a_1}{\log R}}$, and thus by change of variable
$\displaystyle \int_0^{1-\frac{\log a_1}{\log R}} f'_0(x + \frac{\log a_1}{\log R})^2 \frac{x^{k_0-1}}{(k_0-1)!} \leq (1-\frac{\log a_1}{\log R})^{k_0} \frac{j_{k_0-2}^2}{4(k_0-1)} A$
for ${a_1 < R}$, and thus
$\displaystyle \int_0^{1-\frac{\log a_1}{\log R}} f'_0(x + \frac{\log a_1}{\log R})^2 \frac{x^{k_0-1}}{(k_0-1)!} \leq (1-\frac{\log a_1}{\log R})_+^{k_0} \frac{j_{k_0-2}^2}{4(k_0-1)} A$
for all ${a_1}$. Similarly
$\displaystyle \int_0^{1-\frac{\log a_2}{\log R}} f'_0(x + \frac{\log a_2}{\log R})^2 \frac{x^{k_0-1}}{(k_0-1)!} \leq (1-\frac{\log a_2}{\log R})^{k_0}_+ \frac{j_{k_0-2}^2}{4(k_0-1)} A$
for all ${a_2}$. By Cauchy-Schwarz we can thus bound the integral in (30) by
$\displaystyle (1-\frac{\log a_1}{\log R})_+^{k_0/2} (1-\frac{\log a_2}{\log R})_+^{k_0/2} \frac{j_{k_0-2}^2}{4(k_0-1)} A$
and so (30) reduces to
$\displaystyle \frac{j_{k_0-2}^2}{4(k_0-1)} A \sum_{a_1,a_2 \in {\mathcal S}_J} \frac{k_0^{\Omega([a_1,a_2])}}{[a_1,a_2]} (1-\frac{\log a_1}{\log R})_+^{k_0/2} (1-\frac{\log a_2}{\log R})_+^{k_0/2} \leq \alpha + o(1).$
Repeating the arguments of the previous section, we can reduce this to
$\displaystyle \frac{j_{k_0-2}^2}{4(k_0-1)} A \sum_{a \in {\mathcal S}_J} \frac{(3k_0)^{\Omega(a)}}{a} (1-\frac{\log a}{\log R})_+^{k_0/2} \leq \alpha + o(1)$
and by further continuing the arguments of the previous section we end up being able to take
$\displaystyle \alpha = \frac{j_{k_0-2}^2}{4(k_0-1)} A (1 + \kappa)$
where
$\displaystyle \kappa := \sum_{1 \leq n < \frac{1+4\varpi}{2\delta}} (1 - \frac{2n \delta}{1 + 4\varpi})^{k_0/2} \prod_{j=1}^{n} (1 + 3k_0 \log(1+\frac{1}{j})) ). \ \ \ \ \ (31)$
Also, the previous arguments allow us to take
$\displaystyle \beta = A.$
The key inequality (11) now becomes
$\displaystyle 1 + 4\varpi > \frac{j_{k_0-2}^2}{k_0(k_0-1)} (1+\kappa), \ \ \ \ \ (32)$
thus ${MPZ[\varpi,\delta]}$ implies ${DHL[k_0,2]}$ whenever (32) is obeyed with the value (31) of ${\kappa}$. | 27,507 | 77,275 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1564, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.359375 | 3 | CC-MAIN-2020-05 | latest | en | 0.886023 |
https://pt.scribd.com/presentation/329706198/Sonico | 1,579,397,051,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250594101.10/warc/CC-MAIN-20200119010920-20200119034920-00241.warc.gz | 630,038,074 | 68,873 | Você está na página 1de 14
# REGISTRO SNICO
SONIC LOG
Through the Sonic log we calculate the
porosity of the rock.
The principle of this tool is simple. Through
sound wave we calculate the porosity. As
much pores present in the rock the travel
time will be greater and in less porous rock
the travel time will be little. Because the
speed of sound wave in different medium is
different.
First arrival in the recipient is the P wave
(compretional), which travels through the
rock and fluid. The S wave (Shear) that
travels only through the rock, comes later.
Finally comes the Stoneley wave, which is
sensitive to permeability and fractures.
## In clay formations transit
time (Dt) is higher, the sonic
log provides high values.
The sonic log responds only
to the primary porosity
(matrix).
As density tool measures
the total porosity, a
difference between the two
measurements may indicate
the presence of secondary
porosity.
sec = D S
## ABREVIATURAS WELL CONTROL
http://perfob.blogspot.pe/2013/02/conceptos-y-abreviaturas-decontrol-de.html
Hay una serie de conceptos y siglas que se deben saber entender y manejar cuando se habla de Control de Pozos, owell control,en Ingls. Algunos
de los mismos son los siguientes:
## ACF = Annular Capacity Factor (Factor de Capacidad Anular)
BHP = Bottom Hole Pressure (Presin de Fondo del Pozo)
BOPE = Blow Out Preventer Equipment (Equipo de Preventores de Arremetidas)
BPUTS = Bring Pumps Up To Speed (Aumentar Velocidad de las Bombas a...)
CLF = Choke Line Friction (Friccin de la Linea del Estrangulador )
CMW / OMW = Current Mud Weight / Original Mud Weight (Peso Actual del Lodo / Peso Original del Lodo)
CP = Casing Pressure (Presin en el Revestidor)
DPP = Drill Pipe Pressure (Presin en la Tubera)
ECD = Equivalent Circulating Density (Densidad Equivalente de Circulacin)
EOB = End of Build (Seccin Final de Construccin)
FCP = Final Circulating Pressure (Presin de Circulacin Final)
FD / MW = Fluid Density / Mud Weight (Densidad del Fluido / Peso del Lodo)
FIT = Formation Integrity Test (Prueba de Integridad de la Formacin)
FOSV = Full Opening Safety Valve (Apertura Total de la Vvula de Seguridad)
FP / PP = Formation Pressure / Pore Pressure (Presin de la Formacin / Presin de Poro)
FrP = Friction Pressure (Presin de Friccin)
HCR = Hight Closing Ratio (Vlvula Hidrulica de Alto Radio de Cierre)
HP = Hydrostatic Pressure (Presin Hidrostatica)
IBOP = Inside Blow Out Preventer (Interior de la Preventora de Arremetida)
ICP = Initial Circulating Pressure (Presin de Circulacin Inicial)
ISICP = Initial Shut-in Casing Pressure (Presin de Cierre Inicial en el Revestidor)
ABREVIATURAS 2
RESUMEN DE REGISTROS | 725 | 2,667 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2020-05 | latest | en | 0.551648 |
https://homework.cpm.org/category/CON_FOUND/textbook/gc/chapter/9/lesson/9.2.2/problem/9-70 | 1,726,393,246,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651622.79/warc/CC-MAIN-20240915084859-20240915114859-00844.warc.gz | 263,847,696 | 15,795 | ### Home > GC > Chapter 9 > Lesson 9.2.2 > Problem9-70
9-70.
Find the area of the Marina’s drugstore (FIDO) in problem 9-69. Show all work.
Draw a diagram that will help you solve for the area. Notice the figure being split into smaller more manageable figures, especially right triangles.
Find the height of the trapezoid.
Note that the green line that has been drawn has created a special $30°\text{-}60°\text{-}90°$ right triangle. Use this information to solve for $h$.
After solving for $h$, use the formula $\left(\frac{1}{2}\right)\left(b_1+b_2\right)h$ to find the area
$\text{Area}\approx265\text{ m}^2$ | 184 | 619 | {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2024-38 | latest | en | 0.788176 |
https://worldwidescience.org/topicpages/s/shell+finite+elements.html | 1,597,385,431,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439739177.25/warc/CC-MAIN-20200814040920-20200814070920-00397.warc.gz | 557,081,603 | 199,482 | #### Sample records for shell finite elements
1. A finite element for plates and shells
Muller, A.; Feijoo, R.A.; Bevilacqua, L.
1981-08-01
A simple triangular finite element for plates and shells, is presented. Since the rotation fields are assumed independent of the displacement fields, the element allows one to solve thick shells problems. In the limit for thin shell, the Kirchoff-Love hypothesis is automatically satisfied, thus enlarging its range of application. (Author) [pt
2. Finite rotation shells basic equations and finite elements for Reissner kinematics
Wisniewski, K
2010-01-01
This book covers theoretical and computational aspects of non-linear shells. Several advanced topics of shell equations and finite elements - not included in standard textbooks on finite elements - are addressed, and the book includes an extensive bibliography.
3. A study on the nonlinear finite element analysis of reinforced concrete structures: shell finite element formulation
Lee, Sang Jin; Seo, Jeong Moon
2000-08-01
The main goal of this research is to establish a methodology of finite element analysis of containment building predicting not only global behaviour but also local failure mode. In this report, we summerize some existing numerical analysis techniques to be improved for containment building. In other words, a complete description of the standard degenerated shell finite element formulation is provided for nonlinear stress analysis of nuclear containment structure. A shell finite element is derived using the degenerated solid concept which does not rely on a specific shell theory. Reissner-Mindlin assumptions are adopted to consider the transverse shear deformation effect. In order to minimize the sensitivity of the constitutive equation to structural types, microscopic material model is adopted. The four solution algorithms based on the standard Newton-Raphson method are discussed. Finally, two numerical examples are carried out to test the performance of the adopted shell medel.
4. A study on the nonlinear finite element analysis of reinforced concrete structures: shell finite element formulation
Lee, Sang Jin; Seo, Jeong Moon
2000-08-01
The main goal of this research is to establish a methodology of finite element analysis of containment building predicting not only global behaviour but also local failure mode. In this report, we summerize some existing numerical analysis techniques to be improved for containment building. In other words, a complete description of the standard degenerated shell finite element formulation is provided for nonlinear stress analysis of nuclear containment structure. A shell finite element is derived using the degenerated solid concept which does not rely on a specific shell theory. Reissner-Mindlin assumptions are adopted to consider the transverse shear deformation effect. In order to minimize the sensitivity of the constitutive equation to structural types, microscopic material model is adopted. The four solution algorithms based on the standard Newton-Raphson method are discussed. Finally, two numerical examples are carried out to test the performance of the adopted shell medel
5. Finite element model for nonlinear shells of revolution
Cook, W.A.
1979-01-01
Nuclear material shipping containers have shells of revolution as basic structural components. Analytically modeling the response of these containers to severe accident impact conditions requires a nonlinear shell-of-revolution model that accounts for both geometric and material nonlinearities. Existing models are limited to large displacements, small rotations, and nonlinear materials. The paper presents a finite element model for a nonlinear shell of revolution that will account for large displacements, large strains, large rotations, and nonlinear materials
6. A finite-element for the analysis of shell intersections
Koves, W.J.; Nair, S.
1994-01-01
The analysis of discontinuity stresses at shell intersections is a problem of great importance in several major industries. Some of the major areas of interest are pressure-containing equipment, such as reactors and piping, in the nuclear and fossil power industry; pressure vessels and heat exchangers in the petrochemical industry; bracing in offshore oil platforms; and aerospace structures. A specialized shell-intersection finite element, which is compatible with adjoining shell elements, has been developed that has the capability of physically representing the complex three-dimensional geometry and stress state at shell intersections. The element geometry is a contoured shape that matches a wide variety of practical nozzle configurations used in ASME Code pressure vessel construction, and allows computational rigor. A closed-form theory of elasticity solution was used to compute the stress state and strain energy in the element. The concept of an energy-equivalent nodal displacement and force vector set was then developed to allow complete compatibility with adjoining shell elements and retain the analytical rigor within the element. This methodology provides a powerful and robust computation scheme that maintains the computational efficiency of shell element solutions. The shell-intersection element was then applied to the cylinder-sphere and cylinder-cylinder intersection problems
7. Extensions to a nonlinear finite-element axisymmetric shell model based on Reissner's shell theory
Cook, W.A.
1981-01-01
Extensions to shell analysis not usually associated with shell theory are described in this paper. These extensions involve thick shells, nonlinear materials, a linear normal stress approximation, and a changing shell thickness. A finite element shell-of-revolution model has been developed to analyze nuclear material shipping containers under severe impact conditions. To establish the limits for this shell model, the basic assumptions used in its development were studied; these are listed in this paper. Several extensions were evident from the study of these limits: a thick shell, a plastic hinge, and a linear normal stress
8. Extensions to a nonlinear finite element axisymmetric shell model based on Reissner's shell theory
Cook, W.A.
1981-01-01
A finite element shell-of-revolution model has been developed to analyze shipping containers under severe impact conditions. To establish the limits for this shell model, I studied the basic assumptions used in its development; these are listed in this paper. Several extensions were evident from the study of these limits: a thick shell, a plastic hinge, and a linear normal stress. (orig./HP)
9. r-Adaptive mesh generation for shell finite element analysis
Cho, Maenghyo; Jun, Seongki
2004-01-01
An r-adaptive method or moving grid technique relocates a grid so that it becomes concentrated in the desired region. This concentration improves the accuracy and efficiency of finite element solutions. We apply the r-adaptive method to computational mesh of shell surfaces, which is initially regular and uniform. The r-adaptive method, given by Liao and Anderson [Appl. Anal. 44 (1992) 285], aggregate the grid in the region with a relatively high weight function without any grid-tangling. The stress error estimator is calculated in the initial uniform mesh for a weight function. However, since the r-adaptive method is a method that moves the grid, shell surface geometry error such as curvature error and mesh distortion error will increase. Therefore, to represent the exact geometry of a shell surface and to prevent surface geometric errors, we use the Naghdi's shell theory and express the shell surface by a B-spline patch. In addition, using a nine-node element, which is relatively less sensitive to mesh distortion, we try to diminish mesh distortion error in the application of an r-adaptive method. In the numerical examples, it is shown that the values of the error estimator for a cylinder, hemisphere, and torus in the overall domain can be reduced effectively by using the mesh generated by the r-adaptive method. Also, the reductions of the estimated relative errors are demonstrated in the numerical examples. In particular, a new functional is proposed to construct an adjusted mesh configuration by considering a mesh distortion measure as well as the stress error function. The proposed weight function provides a reliable mesh adaptation method after a parameter value in the weight function is properly chosen
10. Nonlinear Finite Element Analysis of Shells with Large Aspect Ratio
Chang, T. Y.; Sawamiphakdi, K.
1984-01-01
A higher order degenerated shell element with nine nodes was selected for large deformation and post-buckling analysis of thick or thin shells. Elastic-plastic material properties are also included. The post-buckling analysis algorithm is given. Using a square plate, it was demonstrated that the none-node element does not have shear locking effect even if its aspect ratio was increased to the order 10 to the 8th power. Two sample problems are given to illustrate the analysis capability of the shell element.
11. Explicit Dynamic Finite Element Method for Predicting Implosion/Explosion Induced Failure of Shell Structures
Jeong-Hoon Song
2013-01-01
Full Text Available A simplified implementation of the conventional extended finite element method (XFEM for dynamic fracture in thin shells is presented. Though this implementation uses the same linear combination of the conventional XFEM, it allows for considerable simplifications of the discontinuous displacement and velocity fields in shell finite elements. The proposed method is implemented for the discrete Kirchhoff triangular (DKT shell element, which is one of the most popular shell elements in engineering analysis. Numerical examples for dynamic failure of shells under impulsive loads including implosion and explosion are presented to demonstrate the effectiveness and robustness of the method.
12. Nonlinear Finite Element Analysis of Reinforced Concrete Shells
Mustafa K. Ahmed
2013-05-01
Full Text Available This investigation is to develop a numerical model suitable for nonlinear analysis of reinforced concrete shells. A nine-node Lagrangian element Figure (1 with enhanced shear interpolation will be used in this study. Table (1 describes shape functions and their derivatives of this element.An assumed transverse shear strain is used in the formulation of this element to overcome shear locking. Degenerated quadratic thick plate elements employing a layered discrelization through the thickness will be adopted. Different numbers of layers for different thickness can be used per element. A number of layers between (6 and 10 have proved to be appropriate to represent the nonlinear material behavior in structures. In this research 8 layers will be adequate. Material nonlinearities due to cracking of concrete, plastic flow or crushing of concrete in compression and yield condition of reinforcing steel are considered. The maximum tensile strength is used as a criterion for crack initiation. Attention is given to the tension stiffening phenomenon and the degrading effect of cracking on the compressive and shear strength of concrete. Perfect bond between concrete and steel is assumed. Attention is given also to geometric nonlinearities. An example have been chosen in order to demonstrate the suitability of the models by comparing the predicted behaviour with the experimental results for shell exhibiting various modes of failure.
13. Unconstrained Finite Element for Geometrical Nonlinear Dynamics of Shells
Humberto Breves Coda
2009-01-01
Full Text Available This paper presents a positional FEM formulation to deal with geometrical nonlinear dynamics of shells. The main objective is to develop a new FEM methodology based on the minimum potential energy theorem written regarding nodal positions and generalized unconstrained vectors not displacements and rotations. These characteristics are the novelty of the present work and avoid the use of large rotation approximations. A nondimensional auxiliary coordinate system is created, and the change of configuration function is written following two independent mappings from which the strain energy function is derived. This methodology is called positional and, as far as the authors' knowledge goes, is a new procedure to approximated geometrical nonlinear structures. In this paper a proof for the linear and angular momentum conservation property of the Newmark algorithm is provided for total Lagrangian description. The proposed shell element is locking free for elastic stress-strain relations due to the presence of linear strain variation along the shell thickness. The curved, high-order element together with an implicit procedure to solve nonlinear equations guarantees precision in calculations. The momentum conserving, the locking free behavior, and the frame invariance of the adopted mapping are numerically confirmed by examples.
14. An isoparametric shell of revolution finite element for harmonic loadings of any order
Johnson, J.J.; Charman, C.M.
1981-01-01
A general isoparametric shell of revolution finite element subjected to any order harmonic loading is presented. Derivation of the element properties, its implementation in a general purpose finite element program, and its application to a sample problem are discussed. The element is isoparametric, that is, the variation of the displacements along the meridian of the shell and the shape of the meridian itself are approximated in an identical manner. The element has been implemented in the computer program MODSAP. A sample problem of a cooling tower subjected to wind loading is presented. (orig./HP)
15. Stress analysis for shells with double curvature by finite element method
Mueller, A.
1981-08-01
A simple triangular finite element for plates and shells, is presented. Since the rotation fields are assumed independent of the displacement fields, simple shape functions of second and third degree were used. An implicit penalty method allows one to solve thin shell problems since the Kirchoff-Love hypothesis are automatically satisfied. (Author) [pt
16. Elastic shells of revolution under nonstationary thermal loading using ring finite elements
Yao Zhenhan
1986-01-01
The report deals with the analysis of elastic shells of revolution under nonstationary thermal loading using ring finite elements. First, a ring element for moderately thick shells is derived which should also be employed for thin shells when either higher Fourier components of the displacements, or deflection patterns with very steep gradients occur. Then, a ring element for the analysis of heat conduction in shells of revolution is derived, and algorithms for the numerical solution of linear stationary, nonlinear stationary, as well as linear nonstationary problems are presented. Finally, a ring element for the coupled thermoelastic analysis of shells of revolution is developed, and an algorithm for the solution of weakly coupled problems is given. (orig.) [de
17. A finite element model for nonlinear shells of revolution
Cook, W.A.
1979-01-01
A shell-of-revolution model was developed to analyze impact problems associated with the safety analysis of nuclear material shipping containers. The nonlinear shell theory presented by Eric Reissner in 1972 was used to develop our model. Reissner's approach includes transverse shear deformation and moments turning about the middle surface normal. With these features, this approach is valid for both thin and thick shells. His theory is formulated in terms of strain and stress resultants that refer to the undeformed geometry. This nonlinear shell model is developed using the virtual work principle associated with Reissner's equilibrium equations. First, the virtual work principle is modified for incremental loading; then it is linearized by assuming that the nonlinear portions of the strains are known. By iteration, equilibrium is then approximated for each increment. A benefit of this approach is that this iteration process makes it possible to use nonlinear material properties. (orig.)
18. Linear and nonlinear symmetrically loaded shells of revolution approximated with the finite element method
Cook, W.A.
1978-10-01
Nuclear Material shipping containers have shells of revolution as a basic structural component. Analytically modeling the response of these containers to severe accident impact conditions requires a nonlinear shell-of-revolution model that accounts for both geometric and material nonlinearities. Present models are limited to large displacements, small rotations, and nonlinear materials. This report discusses a first approach to developing a finite element nonlinear shell of revolution model that accounts for these nonlinear geometric effects. The approach uses incremental loads and a linear shell model with equilibrium iterations. Sixteen linear models are developed, eight using the potential energy variational principle and eight using a mixed variational principle. Four of these are suitable for extension to nonlinear shell theory. A nonlinear shell theory is derived, and a computational technique used in its solution is presented
19. Frequency response analysis of cylindrical shells conveying fluid using finite element method
Seo, Young Soo; Jeong, Weui Bong; Yoo, Wan Suk; Jeong, Ho Kyeong
2005-01-01
A finite element vibration analysis of thin-walled cylindrical shells conveying fluid with uniform velocity is presented. The dynamic behavior of thin-walled shell is based on the Sanders' theory and the fluid in cylindrical shell is considered as inviscid and incompressible so that it satisfies the Laplace's equation. A beam-like shell element is used to reduce the number of degree-of-freedom by restricting to the circumferential modes of cylindrical shell. An estimation of frequency response function of the pipe considering of the coupled effects of the internal fluid is presented. A dynamic coupling condition of the interface between the fluid and the structure is used. The effective thickness of fluid according to circumferential modes is also discussed. The influence of fluid velocity on the frequency response function is illustrated and discussed. The results by this method are compared with published results and those by commercial tools
20. Analysis of thin composite structures using an efficient hex-shell finite element
Shiri, Seddik [Universite Bordeaux, Pessac (France); Naceur, Hakim [Universite de valenciennes, Valenciennes (France)
2013-12-15
In this paper a general methodology for the modeling of material composite multilayered shell structures is proposed using a Hex-shell finite element modeling. The first part of the paper is devoted to the general FE formulation of the present composite 8-node Hex-shell element called SCH8, based only on displacement degrees of freedom. A particular attention is given to alleviate shear, trapezoidal and thickness locking, without resorting to the classical plane-stress assumption. The anisotropic material behavior of layered shells is modeled using a fully three dimensional elastic orthotropic material law in each layer, including the thickness stress component. Applications to laminate thick shell structures are studied to validate the methodology, and good results have been obtained in comparison with ABAQUS commercial code.
1. Nonlinear finite element analysis of reinforced and prestressed concrete shells with edge beams
Srinivasa Rao, P.; Duraiswamy, S.
1994-01-01
The structural design of reinforced and prestressed concrete shells demands the application of nonlinear finite element analysis (NFEM) procedures to ensure safety and serviceability. In this paper the details of a comprehensive NFEM program developed are presented. The application of the program is highlighted by solving two numerical problems and comparing the results with experimental results. (author). 20 refs., 15 figs
2. Shell finite element of reinforced concrete for internal pressure analysis of nuclear containment building
Lee, Hong Pyo, E-mail: hplee@kepri.re.k [Nuclear Power Laboratory, Korea Electric Power Research Institute, 103-16 Munji-Dong, Yuseong-Gu, Daejeon 305-380 (Korea, Republic of)
2011-02-15
Research highlights: Finite element program with 9-node degenerated shell element was developed. The developed program was mainly forced to analyze nuclear containment building. Concrete material model is adapted Niwa and Yamada failure criteria. The performance of program developed is verified through various numerical examples. The numerical analysis results similar to the experimental data. - Abstract: This paper describes a 9-node degenerated shell finite element (FE), an analysis program developed for ultimate pressure capacity evaluation and nonlinear analysis of a nuclear containment building. The shell FE developed adopts the Reissner-Mindlin (RM) assumptions to consider the degenerated shell solidification technique and the degree of transverse shear strain occurring in the structure. The material model of the concrete determines the level of the concrete stress and strain by using the equivalent stress-equivalent strain relationship. When a crack occurs in the concrete, the material behavior is expressed through the tension stiffening model that takes adhesive stress into account and through the shear transfer mechanism and compressive strength reduction model of the crack plane. In addition, the failure envelope proposed by Niwa is adopted as the crack occurrence criteria for the compression-tension region, and the failure envelope proposed by Yamada is used for the tension-tension region. The performance of the program developed is verified through various numerical examples. The analysis based on the application of the shell FE developed from the results of verified examples produced results similar to the experiment or other analysis results.
3. Shell finite element of reinforced concrete for internal pressure analysis of nuclear containment building
Lee, Hong Pyo
2011-01-01
Research highlights: → Finite element program with 9-node degenerated shell element was developed. → The developed program was mainly forced to analyze nuclear containment building. → Concrete material model is adapted Niwa and Yamada failure criteria. → The performance of program developed is verified through various numerical examples. → The numerical analysis results similar to the experimental data. - Abstract: This paper describes a 9-node degenerated shell finite element (FE), an analysis program developed for ultimate pressure capacity evaluation and nonlinear analysis of a nuclear containment building. The shell FE developed adopts the Reissner-Mindlin (RM) assumptions to consider the degenerated shell solidification technique and the degree of transverse shear strain occurring in the structure. The material model of the concrete determines the level of the concrete stress and strain by using the equivalent stress-equivalent strain relationship. When a crack occurs in the concrete, the material behavior is expressed through the tension stiffening model that takes adhesive stress into account and through the shear transfer mechanism and compressive strength reduction model of the crack plane. In addition, the failure envelope proposed by Niwa is adopted as the crack occurrence criteria for the compression-tension region, and the failure envelope proposed by Yamada is used for the tension-tension region. The performance of the program developed is verified through various numerical examples. The analysis based on the application of the shell FE developed from the results of verified examples produced results similar to the experiment or other analysis results.
4. Parameterized Finite Element Modeling and Buckling Analysis of Six Typical Composite Grid Cylindrical Shells
Lai, Changliang; Wang, Junbiao; Liu, Chuang
2014-10-01
Six typical composite grid cylindrical shells are constructed by superimposing three basic types of ribs. Then buckling behavior and structural efficiency of these shells are analyzed under axial compression, pure bending, torsion and transverse bending by finite element (FE) models. The FE models are created by a parametrical FE modeling approach that defines FE models with original natural twisted geometry and orients cross-sections of beam elements exactly. And the approach is parameterized and coded by Patran Command Language (PCL). The demonstrations of FE modeling indicate the program enables efficient generation of FE models and facilitates parametric studies and design of grid shells. Using the program, the effects of helical angles on the buckling behavior of six typical grid cylindrical shells are determined. The results of these studies indicate that the triangle grid and rotated triangle grid cylindrical shell are more efficient than others under axial compression and pure bending, whereas under torsion and transverse bending, the hexagon grid cylindrical shell is most efficient. Additionally, buckling mode shapes are compared and provide an understanding of composite grid cylindrical shells that is useful in preliminary design of such structures.
5. Optimal design of geometrically nonlinear shells of revolution with using the mixed finite element method
Stupishin, L. U.; Nikitin, K. E.; Kolesnikov, A. G.
2018-02-01
The article is concerned with a methodology of optimal design of geometrically nonlinear (flexible) shells of revolution of minimum weight with strength, stability and strain constraints. The problem of optimal design with constraints is reduced to the problem of unconstrained minimization using the penalty functions method. Stress-strain state of shell is determined within the geometrically nonlinear deformation theory. A special feature of the methodology is the use of a mixed finite-element formulation based on the Galerkin method. Test problems for determining the optimal form and thickness distribution of a shell of minimum weight are considered. The validity of the results obtained using the developed methodology is analyzed, and the efficiency of various optimization algorithms is compared to solve the set problem. The developed methodology has demonstrated the possibility and accuracy of finding the optimal solution.
6. Linear dynamic analysis of arbitrary thin shells modal superposition by using finite element method
Goncalves Filho, O.J.A.
1978-11-01
The linear dynamic behaviour of arbitrary thin shells by the Finite Element Method is studied. Plane triangular elements with eighteen degrees of freedom each are used. The general equations of movement are obtained from the Hamilton Principle and solved by the Modal Superposition Method. The presence of a viscous type damping can be considered by means of percentages of the critical damping. An automatic computer program was developed to provide the vibratory properties and the dynamic response to several types of deterministic loadings, including temperature effects. The program was written in FORTRAN IV for the Burroughs B-6700 computer. (author)
7. Vibration isolation design for periodically stiffened shells by the wave finite element method
Hong, Jie; He, Xueqing; Zhang, Dayi; Zhang, Bing; Ma, Yanhong
2018-04-01
Periodically stiffened shell structures are widely used due to their excellent specific strength, in particular for aeronautical and astronautical components. This paper presents an improved Wave Finite Element Method (FEM) that can be employed to predict the band-gap characteristics of stiffened shell structures efficiently. An aero-engine casing, which is a typical periodically stiffened shell structure, was employed to verify the validation and efficiency of the Wave FEM. Good agreement has been found between the Wave FEM and the classical FEM for different boundary conditions. One effective wave selection method based on the Wave FEM has thus been put forward to filter the radial modes of a shell structure. Furthermore, an optimisation strategy by the combination of the Wave FEM and genetic algorithm was presented for periodically stiffened shell structures. The optimal out-of-plane band gap and the mass of the whole structure can be achieved by the optimisation strategy under an aerodynamic load. Results also indicate that geometric parameters of stiffeners can be properly selected that the out-of-plane vibration attenuates significantly in the frequency band of interest. This study can provide valuable references for designing the band gaps of vibration isolation.
8. Dynamic instability analysis of axisymmetric shells by finite element method with convected coordinates
Hsieh, B.J.
1977-01-01
9. Dynamic instability analysis of axisymmetric shells by finite element method with convected coordinates
Hsieh, B.J.
1977-01-01
The instability of axisymmetric shells has been used in engineering fields as a safety device such as the rupture discs used in the LMFBR (Liquid Metal Fast Breeder Reactor) design to relieve the excessive pressure caused by the water and sodium reaction when there is a leak in the piping system. Hence, the analysis of the instability of shells under time varying loading is becoming more and more important. However, notorious discrepancy has been observed between various analytical predications and experimental results for the buckling of shells. Various theories have been proposed to explain these discrepancies. Most of these theories are concerned with two aspects: initial imperfections and asymmetric responses. Both theories do narrow the gap between theoretical and experimental results; however, the remaining discrepancy is still not small. Other possible causes of this discrepancy have to be studied- among them, the boundary conditions. It has been pointed out that the slip at the boundary may have noticeable effect on the transient behavior of a plate. In this paper, the effect of various boundary conditions on the dynamic instability of axisymmetric shells is studied using the numerical discretization technique--convective finite element method
10. Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements
Caty, O.; Maire, E.; Youssef, S.; Bouchet, R.
2008-01-01
Closed-cell cellular materials exhibit several interesting properties. These properties are, however, very difficult to simulate and understand from the knowledge of the cellular microstructure. This problem is mostly due to the highly complex organization of the cells and to their very fine walls. X-ray tomography can produce three-dimensional (3-D) images of the structure, enabling one to visualize locally the damage of the cell walls that would result in the structure collapsing. These data could be used for meshing with continuum elements of the structure for finite element (FE) calculations. But when the density is very low, the walls are fine and the meshes based on continuum elements are not suitable to represent accurately the structure while preserving the representativeness of the model in terms of cell size. This paper presents a shell FE model obtained from tomographic 3-D images that allows bigger volumes of low-density closed-cell cellular materials to be calculated. The model is enriched by direct thickness measurement on the tomographic images. The values measured are ascribed to the shell elements. To validate and use the model, a structure composed of stainless steel hollow spheres is firstly compressed and scanned to observe local deformations. The tomographic data are also meshed with shells for a FE calculation. The convergence of the model is checked and its performance is compared with a continuum model. The global behavior is compared with the measures of the compression test. At the local scale, the model allows the local stress and strain field to be calculated. The calculated deformed shape is compared with the deformed tomographic images
11. A comparison study on the performance of lower order solid finite element for elastic analysis of plate and shell structures
Lee, Young Jung; Lee, Sang Jin; Choun, Young Sun; Seo, Jeong Moon
2003-05-01
The objective of this research is to assess the performance of lower order solid finite elements which will be ultimately applied into the safety analysis of nuclear containment building. For the safety analysis of large structures such as nuclear containment building, efficient lower order finite element is necessarily required to calculate the structural response of containment building with low computational cost. In this study, the state of the art formulations of lower order solid finite element are throughly reviewed and the best possible solid finite element is adopted into the development of nuclear containment analysis system. Three 8-node solid finite elements based on standard strain-displacement relationship, B-bar method and EAS method are implemented as computer modules and completely tested with various plate and shell structures. The present results can be directly applied into the analysis code development for general reinforced concrete structures
12. A layered shell containing patches of piezoelectric fibers and interdigitated electrodes: Finite element modeling and experimental validation
Nielsen, Bo Bjerregaard; Nielsen, Martin S.; Santos, Ilmar
2017-01-01
The work gives a theoretical and experimental contribution to the problem of smart materials connected to double curved flexible shells. In the theoretical part the finite element modeling of a double curved flexible shell with a piezoelectric fiber patch with interdigitated electrodes (IDEs......) is presented. The developed element is based on a purely mechanical eight-node isoparametric layered element for a double curved shell, utilizing first-order shear deformation theory. The electromechanical coupling of piezoelectric material is added to all elements, but can also be excluded by setting...... the piezoelectric material properties to zero. The electrical field applied via the IDEs is aligned with the piezoelectric fibers, and hence the direct d33 piezoelectric constant is utilized for the electromechanical coupling. The dynamic performance of a shell with a microfiber composite (MFC) patch...
13. Modeling deformation and chaining of flexible shells in a nematic solvent with finite elements on an adaptive moving mesh
DeBenedictis, Andrew; Atherton, Timothy J.; Rodarte, Andrea L.; Hirst, Linda S.
2018-03-01
A micrometer-scale elastic shell immersed in a nematic liquid crystal may be deformed by the host if the cost of deformation is comparable to the cost of elastic deformation of the nematic. Moreover, such inclusions interact and form chains due to quadrupolar distortions induced in the host. A continuum theory model using finite elements is developed for this system, using mesh regularization and dynamic refinement to ensure quality of the numerical representation even for large deformations. From this model, we determine the influence of the shell elasticity, nematic elasticity, and anchoring condition on the shape of the shell and hence extract parameter values from an experimental realization. Extending the model to multibody interactions, we predict the alignment angle of the chain with respect to the host nematic as a function of aspect ratio, which is found to be in excellent agreement with experiments.
14. The development and use of a piece-wise continuous finite element for plate and shell analysis
Jobson, D.A.; Knowles, J.A.
1975-01-01
The implementation of general purpose programs for the numerical analysis of plate and shell structures calls for the adoption of finite element stiffness expressions which take into account of both lateral distortion and membrane action. It is important that design-oriented programs of the above kind be perfectly general. In particular the element behaviour must be independent of the choice of base axes, and not prone either to singularities or to doubts over convergence with successive mesh refinement. The basic elements should also be mathematically isotropic and the imposition of rigid body displacements should not cause self-straining. Ideally the program should allow the assembly of a wide variety of elements, oriented in any conceivable way and of a freely chosen shape. The present paper documents a procedure for synthesising the latter from a three node primary element which satisfies the above requirements. (Auth.)
15. Finite element analysis program for shells of revolution: ISTRAN/SR, 4
Chiba, Toshio
1980-01-01
The computational capabilities available in the current version of ISTRAN/SR for stress analysis of shells of revolution have been described in the 1st, 2nd and 3rd reports. This report describes the linear elastic dynamic analysis of shells of revolution under axisymmetric and asymmetric loadings. The shell, idealized as a curved element and cubic function for all displacements, is used. A method for solution of the equations of motion is described with special emphasis on the computational aspect of the solution. Three solution methods, which can be employed for the linear dynamic analysis, are possible - direct integration method, mode superposition method, and spectrum analysis method. Each method involves a numerical method which must be formulated in effective form for computer implementation - solution of linear equations, evaluation of eigenvalues and eigenvectors, and step-by-step numerical integration. In this program, the skyline method is employed for the solution of linear equations, and the subspace method and the determinant search method are employed for eigenproblem. The Newmark-Wilson method is employed for the step-by-step integration. The comparison of the solution of ISTRAN/SR and other numerical solution shows good agreement. (author)
16. A benchmark study of 2D and 3D finite element calculations simulating dynamic pulse buckling tests of cylindrical shells under axial impact
Hoffman, E.L.; Ammerman, D.J.
1993-01-01
A series of tests investigating dynamic pulse buckling of a cylindrical shell under axial impact is compared to several finite element simulations of the event. The purpose of the study is to compare the performance of the various analysis codes and element types with respect to a problem which is applicable to radioactive material transport packages, and ultimately to develop a benchmark problem to qualify finite element analysis codes for the transport package design industry
17. Neotectonics of Asia: Thin-shell finite-element models with faults
Kong, Xianghong; Bird, Peter
1994-01-01
As India pushed into and beneath the south margin of Asia in Cenozoic time, it added a great volume of crust, which may have been (1) emplaced locally beneath Tibet, (2) distributed as regional crustal thickening of Asia, (3) converted to mantle eclogite by high-pressure metamorphism, or (4) extruded eastward to increase the area of Asia. The amount of eastward extrusion is especially controversial: plane-stress computer models of finite strain in a continuum lithosphere show minimal escape, while laboratory and theoretical plane-strain models of finite strain in a faulted lithosphere show escape as the dominant mode. We suggest computing the present (or neo)tectonics by use of the known fault network and available data on fault activity, geodesy, and stress to select the best model. We apply a new thin-shell method which can represent a faulted lithosphere of realistic rheology on a sphere, and provided predictions of present velocities, fault slip rates, and stresses for various trial rheologies and boundary conditions. To minimize artificial boundaries, the models include all of Asia east of 40 deg E and span 100 deg on the globe. The primary unknowns are the friction coefficient of faults within Asia and the amounts of shear traction applied to Asia in the Himalayan and oceanic subduction zones at its margins. Data on Quaternary fault activity prove to be most useful in rating the models. Best results are obtained with a very low fault friction of 0.085. This major heterogeneity shows that unfaulted continum models cannot be expected to give accurate simulations of the orogeny. But, even with such weak faults, only a fraction of the internal deformation is expressed as fault slip; this means that rigid microplate models cannot represent the kinematics either. A universal feature of the better models is that eastern China and southeast Asia flow rapidly eastward with respect to Siberia. The rate of escape is very sensitive to the level of shear traction in the
18. FEATURES APPLICATION CIRCUIT MOMENT FINITE ELEMENT (MSSE) NONLINEAR CALCULATIONS OF PLATES AND SHELLS
Bazhenov V.A.; Sacharov A.S.; Guliar A. I.; Pyskunov S.O.; Maksymiuk Y.V.
2014-01-01
Based MSSE created shell CE general type, which allows you to analyze the stress-strain state of axisymmetrical shells and plates in problems of physical and geometric nonlinearity. The principal nonlinear elasticity theory, algorithms for solving systems of nonlinear equations for determining the temperature and plastic deformation.
19. FEATURES APPLICATION CIRCUIT MOMENT FINITE ELEMENT (MSSE NONLINEAR CALCULATIONS OF PLATES AND SHELLS
Bazhenov V.A.
2014-06-01
Full Text Available Based MSSE created shell CE general type, which allows you to analyze the stress-strain state of axisymmetrical shells and plates in problems of physical and geometric nonlinearity. The principal nonlinear elasticity theory, algorithms for solving systems of nonlinear equations for determining the temperature and plastic deformation.
20. A semi-analytical finite element process for nonlinear elastoplastic analysis of arbitrarily loaded shells of revolution
Rensch, H.J.; Wunderlich, W.
1981-01-01
The governing partial differential equations used are valid for small strains and moderate rotations. Plasticity relations are based on J 2 -flow theory. In order to eliminate the circumferential coordinate, the loading as well as the unkown quantities are expanded in Fourier series in the circumferential direction. The nonlinear terms due to moderate rotations and plastic deformations are treated as pseudo load quantities. In this way, the governing equations can be reduced to uncoupled systems of first-order ordinary differential equations in the meridional direction. They are then integrated over a shell segment via a matrix series expansion. The resulting element transfer matrices are transformed into stiffness matrices, and for the analysis of the total structure the finite element method is employed. Thus, arbitrary branching of the shell geometry is possible. Compared to two-dimensional approximations, the major advantage of the semi-analytical procedure is that the structural stiffness matrix usually has a small handwidth, resulting in shorter computer run times. Moreover, its assemblage and triangularization has to be carried out only once bacause all nonlinear effects are treated as initial loads. (orig./HP)
1. Effects of finite element formulation on optimal plate and shell structural topologies
Long, CS
2009-09-01
Full Text Available , and the other is a 4-node element accounting for in-plane (drilling) rotations. Plate elements selected for evaluation include the discrete Kirchhoff quadrilateral (DKQ) element and two Mindlin–Reissner-based elements, one employing selective reduced integration...
2. FINITE ELEMENT DISPLACEMENT PERTURBATION METHOD FOR GEOMETRIC NONLINEAR BEHAVIORS OF SHELLS OF REVOLUTION OVERALL BENDING IN A MERIDIONAL PLANE AND APPLICATION TO BELLOWS (Ⅰ)
朱卫平; 黄黔
2002-01-01
In order to analyze bellows effectively and practically, the finite-element-displacement-perturbation method (FEDPM) is proposed for the geometric nonlinearbehaviors of shells of revolution subjected to pure bending moments or lateral forces in one of their meridional planes. The formulations are mainly based upon the idea of perturba-tion that the nodal displacement vector and the nodal force vector of each finite elementare expanded by taking root-mean-square value of circumferential strains of the shells as aperturbation parameter. The load steps and the iteration times are not cs arbitrary andunpredictable as in usual nonlinear analysis. Instead, there are certain relations betweenthe load steps and the displacement increments, and no need of iteration for each loadstep. Besides, in the formulations, the shell is idealized into a series of conical frusta for the convenience of practice, Sander' s nonlinear geometric equations of moderate smallrotation are used, and the shell made of more than one material ply is also considered.
3. MEL finite element analysis of water-shell interactions in the context of a PWR-LOCA
Verbiese, S.; Vrije Universiteit Brussels; Goethem, G. van
1979-01-01
In the framework of the computational effort engaged towards and understanding of the transient dynamic fluid-structure phenomena taking place in the very first instants of the PWR loss-of-coolant-accident, before the ebullition crisis and the subsequent two-phase flow, two finite element programs have been selected and coupled to describe this class of events in pressure vessels undergoing moderate plastic deformations. Water is modeled by a compressible inviscid Eulerian (bulk of the fluid) - mixed Eulerian-Lagrangian MEL (boundary elements in contact with the moving structure) program. For the shells a convected coordinates elastic-plastic structural code (EURDYN I) is used. A 1-D discussion on the MEL integration scheme is presented, as well as a flow chart of the combined program. Emphases is placed, during the present calculations limited to very simple axisymmetric configurations, upon the computational aspects in dealing with the interaction of both media at the fluid-structure interface, such as weak code coupling, subcycling and pressure relaxation. (orig.)
4. Dynamic pulse buckling of cylindrical shells under axial impact: A benchmark study of 2D and 3D finite element calculations
Hoffman, E.L.; Ammerman, D.J.
1995-01-01
A series of tests investigating dynamic pulse buckling of a cylindrical shell under axial impact is compared to several 2D and 3D finite element simulations of the event. The purpose of the work is to investigate the performance of various analysis codes and element types on a problem which is applicable to radioactive material transport packages, and ultimately to develop a benchmark problem to qualify finite element analysis codes for the transport package design industry. During the pulse buckling tests, a buckle formed at each end of the cylinder, and one of the two buckles became unstable and collapsed. Numerical simulations of the test were performed using PRONTO, a Sandia developed transient dynamics analysis code, and ABAQUS/Explicit with both shell and continuum elements. The calculations are compared to the tests with respect to deformed shape and impact load history
5. Basic Finite Element Method
Lee, Byeong Hae
1992-02-01
This book gives descriptions of basic finite element method, which includes basic finite element method and data, black box, writing of data, definition of VECTOR, definition of matrix, matrix and multiplication of matrix, addition of matrix, and unit matrix, conception of hardness matrix like spring power and displacement, governed equation of an elastic body, finite element method, Fortran method and programming such as composition of computer, order of programming and data card and Fortran card, finite element program and application of nonelastic problem.
6. Finite element modeling of shell shape in the freshwater turtle Pseudemys concinna reveals a trade-off between mechanical strength and hydrodynamic efficiency.
Rivera, Gabriel; Stayton, C Tristan
2011-10-01
Aquatic species can experience different selective pressures on morphology in different flow regimes. Species inhabiting lotic regimes often adapt to these conditions by evolving low-drag (i.e., streamlined) morphologies that reduce the likelihood of dislodgment or displacement. However, hydrodynamic factors are not the only selective pressures influencing organismal morphology and shapes well suited to flow conditions may compromise performance in other roles. We investigated the possibility of morphological trade-offs in the turtle Pseudemys concinna. Individuals living in lotic environments have flatter, more streamlined shells than those living in lentic environments; however, this flatter shape may also make the shells less capable of resisting predator-induced loads. We tested the idea that "lotic" shell shapes are weaker than "lentic" shell shapes, concomitantly examining effects of sex. Geometric morphometric data were used to transform an existing finite element shell model into a series of models corresponding to the shapes of individual turtles. Models were assigned identical material properties and loaded under identical conditions, and the stresses produced by a series of eight loads were extracted to describe the strength of the shells. "Lotic" shell shapes produced significantly higher stresses than "lentic" shell shapes, indicating that the former is weaker than the latter. Females had significantly stronger shell shapes than males, although these differences were less consistent than differences between flow regimes. We conclude that, despite the potential for many-to-one mapping of shell shape onto strength, P. concinna experiences a trade-off in shell shape between hydrodynamic and mechanical performance. This trade-off may be evident in many other turtle species or any other aquatic species that also depend on a shell for defense. However, evolution of body size may provide an avenue of escape from this trade-off in some cases, as changes in
7. Three dimensional stress analysis of nozzle-to-shell intersections by the finite element method and a auto-mesh generation program
Fujihara, Hirohiko; Ueda, Masahiro
1975-01-01
In the design of chemical reactors or nuclear pressure vessels it is often important to evaluate the stress distribution in nozzle-to-shell intersections. The finite element method is a powerful tool for stress analysis, but it has a defects to require troublesome work in preparing input data. Specially, the mesh data of oblique nozzles and tangential nozzles, in which stress concentration is very high, are very difficult to be prepared. The authors made a mesh generation program which can be used to any nozzle-to-shell intersections, and combining this program with a three dimensional stress analysis program by the finite element method they made the stress analysis of nozzle-to-shell intersections under internal pressure. Consequently, stresses, strains and deformations of nozzles nonsymmetrical to spherical shells and nozzles tangential to cylindrical shells were made clear and it was shown that the curvature of the inner surface of the nozzle corner was a controlling factor in reducing stress concentration. (auth.)
8. Modeling of a fluid-loaded smart shell structure for active noise and vibration control using a coupled finite element–boundary element approach
Ringwelski, S; Gabbert, U
2010-01-01
A recently developed approach for the simulation and design of a fluid-loaded lightweight structure with surface-mounted piezoelectric actuators and sensors capable of actively reducing the sound radiation and the vibration is presented. The objective of this paper is to describe the theoretical background of the approach in which the FEM is applied to model the actively controlled shell structure. The FEM is also employed to model finite fluid domains around the shell structure as well as fluid domains that are partially or totally bounded by the structure. Boundary elements are used to characterize the unbounded acoustic pressure fields. The approach presented is based on the coupling of piezoelectric and acoustic finite elements with boundary elements. A coupled finite element–boundary element model is derived by introducing coupling conditions at the fluid–fluid and fluid–structure interfaces. Because of the possibility of using piezoelectric patches as actuators and sensors, feedback control algorithms can be implemented directly into the multi-coupled structural–acoustic approach to provide a closed-loop model for the design of active noise and vibration control. In order to demonstrate the applicability of the approach developed, a number of test simulations are carried out and the results are compared with experimental data. As a test case, a box-shaped shell structure with surface-mounted piezoelectric actuators and four sensors and an open rearward end is considered. A comparison between the measured values and those predicted by the coupled finite element–boundary element model shows a good agreement
9. A set of pathological tests to validate new finite elements
M. Senthilkumar (Newgen Imaging) 1461 1996 Oct 15 13:05:22
The finite element method entails several approximations. Hence it ... researchers have designed several pathological tests to validate any new finite element. The .... Three dimensional thick shell elements using a hybrid/mixed formu- lation.
10. Finite element bending behaviour of discretely delaminated ...
user
due to their light weight, high specific strength and stiffness properties. ... cylindrical shell roofs respectively using finite element method with centrally located .... where { }ε and { }γ are the direct and shear strains in midplane and { }κ denotes ...
11. Generalized finite elements
Wachspress, E.
2009-01-01
Triangles and rectangles are the ubiquitous elements in finite element studies. Only these elements admit polynomial basis functions. Rational functions provide a basis for elements having any number of straight and curved sides. Numerical complexities initially associated with rational bases precluded extensive use. Recent analysis has reduced these difficulties and programs have been written to illustrate effectiveness. Although incorporation in major finite element software requires considerable effort, there are advantages in some applications which warrant implementation. An outline of the basic theory and of recent innovations is presented here. (authors)
12. Finite elements and approximation
Zienkiewicz, O C
2006-01-01
A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises.Starting with continuum boundary value problems and the need for numerical discretization, the text examines finite difference methods, weighted residual methods in the context of continuous trial functions, and piecewise defined trial functions and the finite element method. Additional topics include higher o
13. Verification of Orthogrid Finite Element Modeling Techniques
Steeve, B. E.
1996-01-01
The stress analysis of orthogrid structures, specifically with I-beam sections, is regularly performed using finite elements. Various modeling techniques are often used to simplify the modeling process but still adequately capture the actual hardware behavior. The accuracy of such 'Oshort cutso' is sometimes in question. This report compares three modeling techniques to actual test results from a loaded orthogrid panel. The finite element models include a beam, shell, and mixed beam and shell element model. Results show that the shell element model performs the best, but that the simpler beam and beam and shell element models provide reasonable to conservative results for a stress analysis. When deflection and stiffness is critical, it is important to capture the effect of the orthogrid nodes in the model.
14. Finite element-implementation of creep of concrete for thin-shell analysis using nonlinear constitutive relations and creep compliance functions
Walter, H.; Mang, H.A.
1991-01-01
A procedure for combining nonlinear short-time behavior of concrete with nonlinear creep compliance functions is presented. It is an important ingredient of a computer code for nonlinear finite element (FE) analysis of prestressed concrete shells, considering creep, shrinkage and ageing of concrete, and relaxation of the prestressing steel. The program was developed at the Institute for Strength of Materials of Technical University of Vienna, Austria. The procedure has resulted from efforts to extend the range of application of a Finite Element program, abbreviated as FESIA, which originally was capable of modeling reinforeced concrete in the context of thin-shell analysis, using nonlinear constitutive relations for both, conrete and steel. The extension encompasses the time-dependent behavior of concrete: Creep, shrinkage and ageing. Creep is modeled with the help of creep compliance functions which may be nonlinear to conform with the short-time constitutive relations. Ageing causes an interdependence between long-time and short-time deformations. The paper contains a description of the physical background of the procedure and hints on the implementation of the algorithm. The focus is on general aspects. Details of the aforementioned computer program are considered only where this is inevitable. (orig.)
15. Inside finite elements
Weiser, Martin
2016-01-01
All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered as far as it gives insight into the construction of algorithms. Throughout the exercises a complete FE-solver for scalar 2D problems will be implemented in Matlab/Octave.
16. Dynamic pulse buckling of cylindrical shells under axial impact: A comparison of 2D and 3D finite element calculations with experimental data
Hoffman, E.L.; Ammerman, D.J.
1995-04-01
A series of tests investigating dynamic pulse buckling of a cylindrical shell under axial impact is compared to several 2D and 3D finite element simulations of the event. The purpose of the work is to investigate the performance of various analysis codes and element types on a problem which is applicable to radioactive material transport packages, and ultimately to develop a benchmark problem to qualify finite element analysis codes for the transport package design industry. Four axial impact tests were performed on 4 in-diameter, 8 in-long, 304 L stainless steel cylinders with a 3/16 in wall thickness. The cylinders were struck by a 597 lb mass with an impact velocity ranging from 42.2 to 45.1 ft/sec. During the impact event, a buckle formed at each end of the cylinder, and one of the two buckles became unstable and collapsed. The instability occurred at the top of the cylinder in three tests and at the bottom in one test. Numerical simulations of the test were performed using the following codes and element types: PRONTO2D with axisymmetric four-node quadrilaterals; PRONTO3D with both four-node shells and eight-node hexahedrons; and ABAQUS/Explicit with axisymmetric two-node shells and four-node quadrilaterals, and 3D four-node shells and eight-node hexahedrons. All of the calculations are compared to the tests with respect to deformed shape and impact load history. As in the tests, the location of the instability is not consistent in all of the calculations. However, the calculations show good agreement with impact load measurements with the exception of an initial load spike which is proven to be the dynamic response of the load cell to the impact. Finally, the PRONIT02D calculation is compared to the tests with respect to strain and acceleration histories. Accelerometer data exhibited good qualitative agreement with the calculations. The strain comparisons show that measurements are very sensitive to gage placement
17. Finite element analysis of tibial fractures
Wong, Christian Nai En; Mikkelsen, Mikkel Peter W; Hansen, Leif Berner
2010-01-01
Project. The data consisted of 21,219 3D elements with a cortical shell and a trabecular core. Three types of load of torsion, a direct lateral load and axial compression were applied. RESULTS: The finite element linear static analysis resulted in relevant fracture localizations and indicated relevant...
18. Using Finite Element Method
M.H.R. Ghoreishy
2008-02-01
Full Text Available This research work is devoted to the footprint analysis of a steel-belted radial tyre (185/65R14 under vertical static load using finite element method. Two models have been developed in which in the first model the tread patterns were replaced by simple ribs while the second model was consisted of details of the tread blocks. Linear elastic and hyper elastic (Arruda-Boyce material models were selected to describe the mechanical behavior of the reinforcing and rubbery parts, respectively. The above two finite element models of the tyre were analyzed under inflation pressure and vertical static loads. The second model (with detailed tread patterns was analyzed with and without friction effect between tread and contact surfaces. In every stage of the analysis, the results were compared with the experimental data to confirm the accuracy and applicability of the model. Results showed that neglecting the tread pattern design not only reduces the computational cost and effort but also the differences between computed deformations do not show significant changes. However, more complicated variables such as shape and area of the footprint zone and contact pressure are affected considerably by the finite element model selected for the tread blocks. In addition, inclusion of friction even in static state changes these variables significantly.
19. Probabilistic finite elements
Belytschko, Ted; Wing, Kam Liu
1987-01-01
In the Probabilistic Finite Element Method (PFEM), finite element methods have been efficiently combined with second-order perturbation techniques to provide an effective method for informing the designer of the range of response which is likely in a given problem. The designer must provide as input the statistical character of the input variables, such as yield strength, load magnitude, and Young's modulus, by specifying their mean values and their variances. The output then consists of the mean response and the variance in the response. Thus the designer is given a much broader picture of the predicted performance than with simply a single response curve. These methods are applicable to a wide class of problems, provided that the scale of randomness is not too large and the probabilistic density functions possess decaying tails. By incorporating the computational techniques we have developed in the past 3 years for efficiency, the probabilistic finite element methods are capable of handling large systems with many sources of uncertainties. Sample results for an elastic-plastic ten-bar structure and an elastic-plastic plane continuum with a circular hole subject to cyclic loadings with the yield stress on the random field are given.
20. Optical Finite Element Processor
Casasent, David; Taylor, Bradley K.
1986-01-01
A new high-accuracy optical linear algebra processor (OLAP) with many advantageous features is described. It achieves floating point accuracy, handles bipolar data by sign-magnitude representation, performs LU decomposition using only one channel, easily partitions and considers data flow. A new application (finite element (FE) structural analysis) for OLAPs is introduced and the results of a case study presented. Error sources in encoded OLAPs are addressed for the first time. Their modeling and simulation are discussed and quantitative data are presented. Dominant error sources and the effects of composite error sources are analyzed.
1. Prediction of Path Deviation in Robot Based Incremental Sheet Metal Forming by Means of a New Solid-Shell Finite Element Technology and a Finite Elastoplastic Model with Combined Hardening
Kiliclar, Yalin; Laurischkat, Roman; Vladimirov, Ivaylo N.; Reese, Stefanie
2011-08-01
The presented project deals with a robot based incremental sheet metal forming process, which is called roboforming and has been developed at the Chair of Production Systems. It is characterized by flexible shaping using a freely programmable path-synchronous movement of two industrial robots. The final shape is produced by the incremental infeed of the forming tool in depth direction and its movement along the part contour in lateral direction. However, the resulting geometries formed in roboforming deviate several millimeters from the reference geometry. This results from the compliance of the involved machine structures and the springback effects of the workpiece. The project aims to predict these deviations caused by resiliences and to carry out a compensative path planning based on this prediction. Therefore a planning tool is implemented which compensates the robots's compliance and the springback effects of the sheet metal. The forming process is simulated by means of a finite element analysis using a material model developed at the Institute of Applied Mechanics (IFAM). It is based on the multiplicative split of the deformation gradient in the context of hyperelasticity and combines nonlinear kinematic and isotropic hardening. Low-order finite elements used to simulate thin sheet structures, such as used for the experiments, have the major problem of locking, a nonphysical stiffening effect. For an efficient finite element analysis a special solid-shell finite element formulation based on reduced integration with hourglass stabilization has been developed. To circumvent different locking effects, the enhanced assumed strain (EAS) and the assumed natural strain (ANS) concepts are included in this formulation. Having such powerful tools available we obtain more accurate geometries.
2. Summary compilation of shell element performance versus formulation.
Heinstein, Martin Wilhelm; Hales, Jason Dean (Idaho National Laboratory, Idaho Falls, ID); Breivik, Nicole L.; Key, Samuel W. (FMA Development, LLC, Great Falls, MT)
2011-07-01
This document compares the finite element shell formulations in the Sierra Solid Mechanics code. These are finite elements either currently in the Sierra simulation codes Presto and Adagio, or expected to be added to them in time. The list of elements are divided into traditional two-dimensional, plane stress shell finite elements, and three-dimensional solid finite elements that contain either modifications or additional terms designed to represent the bending stiffness expected to be found in shell formulations. These particular finite elements are formulated for finite deformation and inelastic material response, and, as such, are not based on some of the elegant formulations that can be found in an elastic, infinitesimal finite element setting. Each shell element is subjected to a series of 12 verification and validation test problems. The underlying purpose of the tests here is to identify the quality of both the spatially discrete finite element gradient operator and the spatially discrete finite element divergence operator. If the derivation of the finite element is proper, the discrete divergence operator is the transpose of the discrete gradient operator. An overall summary is provided from which one can rank, at least in an average sense, how well the individual formulations can be expected to perform in applications encountered year in and year out. A letter grade has been assigned albeit sometimes subjectively for each shell element and each test problem result. The number of A's, B's, C's, et cetera assigned have been totaled, and a grade point average (GPA) has been computed, based on a 4.0-system. These grades, combined with a comparison between the test problems and the application problem, can be used to guide an analyst to select the element with the best shell formulation.
3. Probabilistic fracture finite elements
Liu, W. K.; Belytschko, T.; Lua, Y. J.
1991-05-01
The Probabilistic Fracture Mechanics (PFM) is a promising method for estimating the fatigue life and inspection cycles for mechanical and structural components. The Probability Finite Element Method (PFEM), which is based on second moment analysis, has proved to be a promising, practical approach to handle problems with uncertainties. As the PFEM provides a powerful computational tool to determine first and second moment of random parameters, the second moment reliability method can be easily combined with PFEM to obtain measures of the reliability of the structural system. The method is also being applied to fatigue crack growth. Uncertainties in the material properties of advanced materials such as polycrystalline alloys, ceramics, and composites are commonly observed from experimental tests. This is mainly attributed to intrinsic microcracks, which are randomly distributed as a result of the applied load and the residual stress.
4. Finite element modelling
Tonks, M.R.; Williamson, R.; Masson, R.
2015-01-01
The Finite Element Method (FEM) is a numerical technique for finding approximate solutions to boundary value problems. While FEM is commonly used to solve solid mechanics equations, it can be applied to a large range of BVPs from many different fields. FEM has been used for reactor fuels modelling for many years. It is most often used for fuel performance modelling at the pellet and pin scale, however, it has also been used to investigate properties of the fuel material, such as thermal conductivity and fission gas release. Recently, the United Stated Department Nuclear Energy Advanced Modelling and Simulation Program has begun using FEM as the basis of the MOOSE-BISON-MARMOT Project that is developing a multi-dimensional, multi-physics fuel performance capability that is massively parallel and will use multi-scale material models to provide a truly predictive modelling capability. (authors)
5. Massively Parallel Finite Element Programming
Heister, Timo
2010-01-01
Today\\'s large finite element simulations require parallel algorithms to scale on clusters with thousands or tens of thousands of processor cores. We present data structures and algorithms to take advantage of the power of high performance computers in generic finite element codes. Existing generic finite element libraries often restrict the parallelization to parallel linear algebra routines. This is a limiting factor when solving on more than a few hundreds of cores. We describe routines for distributed storage of all major components coupled with efficient, scalable algorithms. We give an overview of our effort to enable the modern and generic finite element library deal.II to take advantage of the power of large clusters. In particular, we describe the construction of a distributed mesh and develop algorithms to fully parallelize the finite element calculation. Numerical results demonstrate good scalability. © 2010 Springer-Verlag.
6. Massively Parallel Finite Element Programming
Heister, Timo; Kronbichler, Martin; Bangerth, Wolfgang
2010-01-01
Today's large finite element simulations require parallel algorithms to scale on clusters with thousands or tens of thousands of processor cores. We present data structures and algorithms to take advantage of the power of high performance computers in generic finite element codes. Existing generic finite element libraries often restrict the parallelization to parallel linear algebra routines. This is a limiting factor when solving on more than a few hundreds of cores. We describe routines for distributed storage of all major components coupled with efficient, scalable algorithms. We give an overview of our effort to enable the modern and generic finite element library deal.II to take advantage of the power of large clusters. In particular, we describe the construction of a distributed mesh and develop algorithms to fully parallelize the finite element calculation. Numerical results demonstrate good scalability. © 2010 Springer-Verlag.
7. Finite element computational fluid mechanics
Baker, A.J.
1983-01-01
This book analyzes finite element theory as applied to computational fluid mechanics. It includes a chapter on using the heat conduction equation to expose the essence of finite element theory, including higher-order accuracy and convergence in a common knowledge framework. Another chapter generalizes the algorithm to extend application to the nonlinearity of the Navier-Stokes equations. Other chapters are concerned with the analysis of a specific fluids mechanics problem class, including theory and applications. Some of the topics covered include finite element theory for linear mechanics; potential flow; weighted residuals/galerkin finite element theory; inviscid and convection dominated flows; boundary layers; parabolic three-dimensional flows; and viscous and rotational flows
8. Programming the finite element method
Smith, I M; Margetts, L
2013-01-01
Many students, engineers, scientists and researchers have benefited from the practical, programming-oriented style of the previous editions of Programming the Finite Element Method, learning how to develop computer programs to solve specific engineering problems using the finite element method. This new fifth edition offers timely revisions that include programs and subroutine libraries fully updated to Fortran 2003, which are freely available online, and provides updated material on advances in parallel computing, thermal stress analysis, plasticity return algorithms, convection boundary c
9. On symmetric pyramidal finite elements
Liu, L.; Davies, K. B.; Yuan, K.; Křížek, Michal
2004-01-01
Roč. 11, 1-2 (2004), s. 213-227 ISSN 1492-8760 R&D Projects: GA AV ČR IAA1019201 Institutional research plan: CEZ:AV0Z1019905 Keywords : mesh generation * finite element method * composite elements Subject RIV: BA - General Mathematics Impact factor: 0.108, year: 2004
10. FINITE ELEMENT ANALYSIS OF STRUCTURES
PECINGINA OLIMPIA-MIOARA
2015-05-01
Full Text Available The application of finite element method is analytical when solutions can not be applied for deeper study analyzes static, dynamic or other types of requirements in different points of the structures .In practice it is necessary to know the behavior of the structure or certain parts components of the machine under the influence of certain factors static and dynamic . The application of finite element in the optimization of components leads to economic growth , to increase reliability and durability organs studied, thus the machine itself.
11. Finite elements of nonlinear continua
Oden, John Tinsley
1972-01-01
Geared toward undergraduate and graduate students, this text extends applications of the finite element method from linear problems in elastic structures to a broad class of practical, nonlinear problems in continuum mechanics. It treats both theory and applications from a general and unifying point of view.The text reviews the thermomechanical principles of continuous media and the properties of the finite element method, and then brings them together to produce discrete physical models of nonlinear continua. The mathematical properties of these models are analyzed, along with the numerical s
12. Automation of finite element methods
Korelc, Jože
2016-01-01
New finite elements are needed as well in research as in industry environments for the development of virtual prediction techniques. The design and implementation of novel finite elements for specific purposes is a tedious and time consuming task, especially for nonlinear formulations. The automation of this process can help to speed up this process considerably since the generation of the final computer code can be accelerated by order of several magnitudes. This book provides the reader with the required knowledge needed to employ modern automatic tools like AceGen within solid mechanics in a successful way. It covers the range from the theoretical background, algorithmic treatments to many different applications. The book is written for advanced students in the engineering field and for researchers in educational and industrial environments.
13. Finite elements methods in mechanics
Eslami, M Reza
2014-01-01
This book covers all basic areas of mechanical engineering, such as fluid mechanics, heat conduction, beams, and elasticity with detailed derivations for the mass, stiffness, and force matrices. It is especially designed to give physical feeling to the reader for finite element approximation by the introduction of finite elements to the elevation of elastic membrane. A detailed treatment of computer methods with numerical examples are provided. In the fluid mechanics chapter, the conventional and vorticity transport formulations for viscous incompressible fluid flow with discussion on the method of solution are presented. The variational and Galerkin formulations of the heat conduction, beams, and elasticity problems are also discussed in detail. Three computer codes are provided to solve the elastic membrane problem. One of them solves the Poisson’s equation. The second computer program handles the two dimensional elasticity problems, and the third one presents the three dimensional transient heat conducti...
14. Stress analysis in pressure vessels by mixed finite element methods taking into account shear deformation
Franca, L.P.; Toledo, E.M.; Loula, A.F.D.; Garcia, E.L.M.
1988-12-01
A new finite element method is employed to approximate axisymmetric shell problems. This formulation enhances stability and accuracy, from thin to moderately thick shells, compared to the correspondent Galerkin finite element approximations. Numerical results illustrate the good performance of the present method on some typical pressure vessels aplications. (author) [pt
15. Finite element analysis of inclined nozzle-plate junctions
Dixit, K.B.; Seth, V.K.; Krishnan, A.; Ramamurthy, T.S.; Dattaguru, B.; Rao, A.K.
1979-01-01
Estimation of stress concentration at nozzle to plate or shell junctions is a significant problem in the stress analysis of nuclear reactors. The topic is a subject matter of extensive investigations and earlier considerable success has been reported on analysis for the cases when the nozzle is perpendicular to the plate or is radial to the shell. Analytical methods for the estimation of stress concentrations for the practical situations when the intersecting nozzle is inclined to the plate or is non-radial to the shell is rather scanty. Specific complications arise in dealing with the junction region when the nozzle with circular cross-section meets the non-circular cut-out on the plate or shell. In this paper a finite element analysis is developed for inclined nozzles and results are presented for nozzle-plate junctions. A method of analysis is developed with a view to achieving simultaneously accuracy of results and simplicity in the choice of elements and their connectivity. The circular nozzle is treated by axisymmetric conical shell elements. The nozzle portion in the region around the junction and the flat plate is dealt with by triangular flat shell elements. Special transition elements are developed for joining the flat shell elements with the axisymmetric elements under non-axisymmetric loading. A substructure method of analysis is adopted which achieves considerable economy in handling the structure and also conveniently combines the different types of elements in the structure. (orig.)
16. FINITE ELEMENT MODEL FOR PREDICTING RESIDUAL ...
FINITE ELEMENT MODEL FOR PREDICTING RESIDUAL STRESSES IN ... the transverse residual stress in the x-direction (σx) had a maximum value of 375MPa ... the finite element method are in fair agreement with the experimental results.
17. Structural modeling techniques by finite element method
Kang, Yeong Jin; Kim, Geung Hwan; Ju, Gwan Jeong
1991-01-01
This book includes introduction table of contents chapter 1 finite element idealization introduction summary of the finite element method equilibrium and compatibility in the finite element solution degrees of freedom symmetry and anti symmetry modeling guidelines local analysis example references chapter 2 static analysis structural geometry finite element models analysis procedure modeling guidelines references chapter 3 dynamic analysis models for dynamic analysis dynamic analysis procedures modeling guidelines and modeling guidelines.
18. Peridynamic Multiscale Finite Element Methods
Costa, Timothy [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Bond, Stephen D. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Littlewood, David John [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Moore, Stan Gerald [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2015-12-01
The problem of computing quantum-accurate design-scale solutions to mechanics problems is rich with applications and serves as the background to modern multiscale science research. The prob- lem can be broken into component problems comprised of communicating across adjacent scales, which when strung together create a pipeline for information to travel from quantum scales to design scales. Traditionally, this involves connections between a) quantum electronic structure calculations and molecular dynamics and between b) molecular dynamics and local partial differ- ential equation models at the design scale. The second step, b), is particularly challenging since the appropriate scales of molecular dynamic and local partial differential equation models do not overlap. The peridynamic model for continuum mechanics provides an advantage in this endeavor, as the basic equations of peridynamics are valid at a wide range of scales limiting from the classical partial differential equation models valid at the design scale to the scale of molecular dynamics. In this work we focus on the development of multiscale finite element methods for the peridynamic model, in an effort to create a mathematically consistent channel for microscale information to travel from the upper limits of the molecular dynamics scale to the design scale. In particular, we first develop a Nonlocal Multiscale Finite Element Method which solves the peridynamic model at multiple scales to include microscale information at the coarse-scale. We then consider a method that solves a fine-scale peridynamic model to build element-support basis functions for a coarse- scale local partial differential equation model, called the Mixed Locality Multiscale Finite Element Method. Given decades of research and development into finite element codes for the local partial differential equation models of continuum mechanics there is a strong desire to couple local and nonlocal models to leverage the speed and state of the
19. Linear and Nonlinear Finite Elements.
1983-12-01
Metzler. Con/ ugte rapdent solution of a finite element elastic problem with high Poson rato without scaling and once with the global stiffness matrix K...nonzero c, that makes u(0) = 1. According to the linear, small deflection theory of the membrane the central displacement given to the membrane is not... theory is possible based on the approximations (l-y 2 )t = +y’ 2 +y , (1-y)’ 1-y’ 2 - y" (6) that change eq. (5) to V) = , [yŖ(1 + y") - Qy
20. Finite element analysis of the cross-section of wind turbine blades; a comparison between shell and 2D-solid models
Pardo, D.; Branner, K.
2005-01-01
line load. The results are compared with result from similar shell models, which typically are used for practical design. Usually, good agreement between the shell models and the detailed 2D-solid model is found for the deflections, strains and stresses in regions with loads from pure bending. However...
1. Finite element application to global reactor analysis
Schmidt, F.A.R.
1981-01-01
The Finite Element Method is described as a Coarse Mesh Method with general basis and trial functions. Various consequences concerning programming and application of Finite Element Methods in reactor physics are drawn. One of the conclusions is that the Finite Element Method is a valuable tool in solving global reactor analysis problems. However, problems which can be described by rectangular boxes still can be solved with special coarse mesh programs more efficiently. (orig.) [de
2. Domain decomposition methods for mortar finite elements
Widlund, O.
1996-12-31
In the last few years, domain decomposition methods, previously developed and tested for standard finite element methods and elliptic problems, have been extended and modified to work for mortar and other nonconforming finite element methods. A survey will be given of work carried out jointly with Yves Achdou, Mario Casarin, Maksymilian Dryja and Yvon Maday. Results on the p- and h-p-version finite elements will also be discussed.
3. A first course in finite elements
Fish, Jacob
2007-01-01
Developed from the authors, combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems governed by partial differential equations. Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, the book is both introductory and self-contained, as well as being a hands-on experience for any student. This authoritative text on Finite Elements:Adopts
4. About two new efficient nonlinear shell elements
Yin, J.; Suo, X.Z.; Combescure, A.
1989-01-01
The aim of the paper is to present the development of two shell elements for non linear analysis. The first one is an axisymetric curved shell element and it is developed for buckling analysis. The formulation is given, as well as some typical applications. The second one is an extension of the classical DKT element to large strains taking into account all aspects of non linearities. This element is used for the simulation of four point bending of cracked pipes. The whole experiment is simulated by the calculation taking into account very large strains at the crack tip and propagation of the crack
5. Finite element coiled cochlea model
Isailovic, Velibor; Nikolic, Milica; Milosevic, Zarko; Saveljic, Igor; Nikolic, Dalibor; Radovic, Milos; Filipović, Nenad
2015-12-01
Cochlea is important part of the hearing system, and thanks to special structure converts external sound waves into neural impulses which go to the brain. Shape of the cochlea is like snail, so geometry of the cochlea model is complex. The simplified cochlea coiled model was developed using finite element method inside SIFEM FP7 project. Software application is created on the way that user can prescribe set of the parameters for spiral cochlea, as well as material properties and boundary conditions to the model. Several mathematical models were tested. The acoustic wave equation for describing fluid in the cochlea chambers - scala vestibuli and scala timpani, and Newtonian dynamics for describing vibrations of the basilar membrane are used. The mechanical behavior of the coiled cochlea was analyzed and the third chamber, scala media, was not modeled because it does not have a significant impact on the mechanical vibrations of the basilar membrane. The obtained results are in good agreement with experimental measurements. Future work is needed for more realistic geometry model. Coiled model of the cochlea was created and results are compared with initial simplified coiled model of the cochlea.
6. Non-linear finite element modeling
Mikkelsen, Lars Pilgaard
The note is written for courses in "Non-linear finite element method". The note has been used by the author teaching non-linear finite element modeling at Civil Engineering at Aalborg University, Computational Mechanics at Aalborg University Esbjerg, Structural Engineering at the University...
7. Nonlinear finite element modeling of corrugated board
A. C. Gilchrist; J. C. Suhling; T. J. Urbanik
1999-01-01
In this research, an investigation on the mechanical behavior of corrugated board has been performed using finite element analysis. Numerical finite element models for corrugated board geometries have been created and executed. Both geometric (large deformation) and material nonlinearities were included in the models. The analyses were performed using the commercial...
8. Why do probabilistic finite element analysis ?
Thacker, Ben H
2008-01-01
The intention of this book is to provide an introduction to performing probabilistic finite element analysis. As a short guideline, the objective is to inform the reader of the use, benefits and issues associated with performing probabilistic finite element analysis without excessive theory or mathematical detail.
9. Finite-Element Software for Conceptual Design
Lindemann, J.; Sandberg, G.; Damkilde, Lars
2010-01-01
and research. Forcepad is an effort to provide a conceptual design and teaching tool in a finite-element software package. Forcepad is a two-dimensional finite-element application based on the same conceptual model as image editing applications such as Adobe Photoshop or Microsoft Paint. Instead of using...
10. Element-topology-independent preconditioners for parallel finite element computations
Park, K. C.; Alexander, Scott
1992-01-01
A family of preconditioners for the solution of finite element equations are presented, which are element-topology independent and thus can be applicable to element order-free parallel computations. A key feature of the present preconditioners is the repeated use of element connectivity matrices and their left and right inverses. The properties and performance of the present preconditioners are demonstrated via beam and two-dimensional finite element matrices for implicit time integration computations.
11. Finite element and finite difference methods in electromagnetic scattering
Morgan, MA
2013-01-01
This second volume in the Progress in Electromagnetic Research series examines recent advances in computational electromagnetics, with emphasis on scattering, as brought about by new formulations and algorithms which use finite element or finite difference techniques. Containing contributions by some of the world's leading experts, the papers thoroughly review and analyze this rapidly evolving area of computational electromagnetics. Covering topics ranging from the new finite-element based formulation for representing time-harmonic vector fields in 3-D inhomogeneous media using two coupled sca
12. Books and monographs on finite element technology
Noor, A. K.
1985-01-01
The present paper proviees a listing of all of the English books and some of the foreign books on finite element technology, taking into account also a list of the conference proceedings devoted solely to finite elements. The references are divided into categories. Attention is given to fundamentals, mathematical foundations, structural and solid mechanics applications, fluid mechanics applications, other applied science and engineering applications, computer implementation and software systems, computational and modeling aspects, special topics, boundary element methods, proceedings of symmposia and conferences on finite element technology, bibliographies, handbooks, and historical accounts.
13. Probabilistic finite elements for fracture mechanics
Besterfield, Glen
1988-01-01
The probabilistic finite element method (PFEM) is developed for probabilistic fracture mechanics (PFM). A finite element which has the near crack-tip singular strain embedded in the element is used. Probabilistic distributions, such as expectation, covariance and correlation stress intensity factors, are calculated for random load, random material and random crack length. The method is computationally quite efficient and can be expected to determine the probability of fracture or reliability.
14. Electrical machine analysis using finite elements
Bianchi, Nicola
2005-01-01
OUTLINE OF ELECTROMAGNETIC FIELDSVector AnalysisElectromagnetic FieldsFundamental Equations SummaryReferencesBASIC PRINCIPLES OF FINITE ELEMENT METHODSIntroductionField Problems with Boundary ConditionsClassical Method for the Field Problem SolutionThe Classical Residual Method (Galerkin's Method)The Classical Variational Method (Rayleigh-Ritz's Method)The Finite Element MethodReferencesAPPLICATIONS OF THE FINITE ELEMENT METHOD TO TWO-DIMENSIONAL FIELDSIntroductionLinear Interpolation of the Function fApplication of the Variational MethodSimple Descriptions of Electromagnetic FieldsAppendix: I
15. Finite element analysis of piezoelectric materials
Lowrie, F.; Stewart, M.; Cain, M.; Gee, M.
1999-01-01
This guide is intended to help people wanting to do finite element analysis of piezoelectric materials by answering some of the questions that are peculiar to piezoelectric materials. The document is not intended as a complete beginners guide for finite element analysis in general as this is better dealt with by the individual software producers. The guide is based around the commercial package ANSYS as this is a popular package amongst piezoelectric material users, however much of the information will still be useful to users of other finite element codes. (author)
16. On higher order pyramidal finite elements
Liu, L.; Davies, K.B.; Křížek, Michal; Guan, L.
2011-01-01
Roč. 3, č. 2 (2011), s. 131-140 ISSN 2070-0733 R&D Projects: GA AV ČR(CZ) IAA100190803 Institutional research plan: CEZ:AV0Z10190503 Keywords : pyramidal polynomial basis functions * finite element method * composite elements * three-dimensional mortar elements Subject RIV: BA - General Mathematics Impact factor: 0.750, year: 2011
17. Finite element methods a practical guide
Whiteley, Jonathan
2017-01-01
This book presents practical applications of the finite element method to general differential equations. The underlying strategy of deriving the finite element solution is introduced using linear ordinary differential equations, thus allowing the basic concepts of the finite element solution to be introduced without being obscured by the additional mathematical detail required when applying this technique to partial differential equations. The author generalizes the presented approach to partial differential equations which include nonlinearities. The book also includes variations of the finite element method such as different classes of meshes and basic functions. Practical application of the theory is emphasised, with development of all concepts leading ultimately to a description of their computational implementation illustrated using Matlab functions. The target audience primarily comprises applied researchers and practitioners in engineering, but the book may also be beneficial for graduate students.
18. Advanced finite element method in structural engineering
Long, Yu-Qiu; Long, Zhi-Fei
2009-01-01
This book systematically introduces the research work on the Finite Element Method completed over the past 25 years. Original theoretical achievements and their applications in the fields of structural engineering and computational mechanics are discussed.
19. ANSYS mechanical APDL for finite element analysis
Thompson, Mary Kathryn
2017-01-01
ANSYS Mechanical APDL for Finite Element Analysis provides a hands-on introduction to engineering analysis using one of the most powerful commercial general purposes finite element programs on the market. Students will find a practical and integrated approach that combines finite element theory with best practices for developing, verifying, validating and interpreting the results of finite element models, while engineering professionals will appreciate the deep insight presented on the program's structure and behavior. Additional topics covered include an introduction to commands, input files, batch processing, and other advanced features in ANSYS. The book is written in a lecture/lab style, and each topic is supported by examples, exercises and suggestions for additional readings in the program documentation. Exercises gradually increase in difficulty and complexity, helping readers quickly gain confidence to independently use the program. This provides a solid foundation on which to build, preparing readers...
20. Review on Finite Element Method * ERHUNMWUN, ID ...
ABSTRACT: In this work, we have discussed what Finite Element Method (FEM) is, its historical development, advantages and ... residual procedures, are examples of the direct approach ... The paper centred on the "stiffness and deflection of ...
1. Bibliography for finite elements. [2200 references
Whiteman, J R [comp.
1975-01-01
This bibliography cites almost all of the significant papers on advances in the mathematical theory of finite elements. Reported are applications in aeronautical, civil, mechanical, nautical and nuclear engineering. Such topics as classical analysis, functional analysis, approximation theory, fluids, and diffusion are covered. Over 2200 references to publications up to the end of 1974 are included. Publications are listed alphabetically by author and also by keywords. In addition, finite element packages are listed.
2. The finite element method in electromagnetics
Jin, Jianming
2014-01-01
A new edition of the leading textbook on the finite element method, incorporating major advancements and further applications in the field of electromagnetics The finite element method (FEM) is a powerful simulation technique used to solve boundary-value problems in a variety of engineering circumstances. It has been widely used for analysis of electromagnetic fields in antennas, radar scattering, RF and microwave engineering, high-speed/high-frequency circuits, wireless communication, electromagnetic compatibility, photonics, remote sensing, biomedical engineering, and space exploration. The
3. Surgery simulation using fast finite elements
Bro-Nielsen, Morten
1996-01-01
This paper describes our recent work on real-time surgery simulation using fast finite element models of linear elasticity. In addition, we discuss various improvements in terms of speed and realism......This paper describes our recent work on real-time surgery simulation using fast finite element models of linear elasticity. In addition, we discuss various improvements in terms of speed and realism...
4. A refined element-based Lagrangian shell element for geometrically nonlinear analysis of shell structures
Woo-Young Jung
2015-04-01
Full Text Available For the solution of geometrically nonlinear analysis of plates and shells, the formulation of a nonlinear nine-node refined first-order shear deformable element-based Lagrangian shell element is presented. Natural co-ordinate-based higher order transverse shear strains are used in present shell element. Using the assumed natural strain method with proper interpolation functions, the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. Furthermore, a refined first-order shear deformation theory for thin and thick shells, which results in parabolic through-thickness distribution of the transverse shear strains from the formulation based on the third-order shear deformation theory, is proposed. This formulation eliminates the need for shear correction factors in the first-order theory. To avoid difficulties resulting from large increments of the rotations, a scheme of attached reference system is used for the expression of rotations of shell normal. Numerical examples demonstrate that the present element behaves reasonably satisfactorily either for the linear or for geometrically nonlinear analysis of thin and thick plates and shells with large displacement but small strain. Especially, the nonlinear results of slit annular plates with various loads provided the benchmark to test the accuracy of related numerical solutions.
5. TAURUS, Post-processor of 3-D Finite Elements Plots
Brown, B.E.; Hallquist, J.O.; Kennedy, T.
2002-01-01
Description of program or function: TAURUS reads the binary plot files generated by the LLNL three-dimensional finite element analysis codes, NIKE3D (NESC 9725), DYNA3D (NESC 9909), TACO3D (NESC 9838), TOPAZ3D (NESC9599) and GEMINI and plots contours, time histories, and deformed shapes. Contours of a large number of quantities may be plotted on meshes consisting of plate, shell, and solid type elements. TAURUS can compute a variety of strain measures, reaction forces along constrained boundaries, and momentum. TAURUS has three phases: initialization, geometry display with contouring, and time history processing
6. Calibration of a finite element composite delamination model by experiments
Gaiotti, M.; Rizzo, C.M.; Branner, Kim
2013-01-01
This paper deals with the mechanical behavior under in plane compressive loading of thick and mostly unidirectional glass fiber composite plates made with an initial embedded delamination. The delamination is rectangular in shape, causing the separation of the central part of the plate into two...... distinct sub-laminates. The work focuses on experimental validation of a finite element model built using the 9-noded MITC9 shell elements, which prevent locking effects and aiming to capture the highly non linear buckling features involved in the problem. The geometry has been numerically defined...
7. Free vibration of finite cylindrical shells by the variational method
Campen, D.H. van; Huetink, J.
1975-01-01
The calculation of the free vibrations of circular cylindrical shells of finite length has been of engineer's interest for a long time. The motive for the present calculations originates from a particular type of construction at the inlet of a sodium heated superheater with helix heating bundle for SNR-Kalkar. The variational analysis is based on a modified energy functional for cylindrical shells, proposed by Koiter and resulting in Morley's equilibrium equations. As usual, the dispacement amplitude is assumed to be distributed harmonically in the circumferential direction of the shell. Following the method of Gontkevich, the dependence between the displacements of the shell middle surface and the axial shell co-ordinate is expressed approximately by a set of eigenfunctions of a free vibrating beam satisfying the desired boundary conditions. Substitution of this displacement expression into the virtual work equation for the complete shell leads to a characteristic equation determining the natural frequencies. The calculations are carried out for a clamped-clamped and a clamped-free cylinder. A comparison is given between the above numerical results and experimental and theoretical results from literature. In addition, the influence of surrounding fluid mass on the above frequencies is analysed for a clamped-clamped shell. The solution for the velocity potential used in this case differs from the solutions used in literature until now in that not only travelling waves in the axial direction are considered. (Auth.)
8. Quadrature representation of finite element variational forms
Ølgaard, Kristian Breum; Wells, Garth N.
2012-01-01
This chapter addresses the conventional run-time quadrature approach for the numerical integration of local element tensors associated with finite element variational forms, and in particular automated optimizations that can be performed to reduce the number of floating point operations...
9. Modelling drawbeads with finite elements and verification
Carleer, B.D.; Carleer, B.D.; Vreede, P.T.; Vreede, P.T.; Louwes, M.F.M.; Louwes, M.F.M.; Huetink, Han
1994-01-01
Drawbeads are commonly used in deep drawing processes to control the flow of the blank during the forming operation. In finite element simulations of deep drawing the drawbead geometries are seldom included because of the small radii; because of these small radii a very large number of elements is
10. Reduction of the radiating sound of a submerged finite cylindrical shell structure by active vibration control.
Kim, Heung Soo; Sohn, Jung Woo; Jeon, Juncheol; Choi, Seung-Bok
2013-02-06
In this work, active vibration control of an underwater cylindrical shell structure was investigated, to suppress structural vibration and structure-borne noise in water. Finite element modeling of the submerged cylindrical shell structure was developed, and experimentally evaluated. Modal reduction was conducted to obtain the reduced system equation for the active feedback control algorithm. Three Macro Fiber Composites (MFCs) were used as actuators and sensors. One MFC was used as an exciter. The optimum control algorithm was designed based on the reduced system equations. The active control performance was then evaluated using the lab scale underwater cylindrical shell structure. Structural vibration and structure-borne noise of the underwater cylindrical shell structure were reduced significantly by activating the optimal controller associated with the MFC actuators. The results provide that active vibration control of the underwater structure is a useful means to reduce structure-borne noise in water.
11. Reduction of the Radiating Sound of a Submerged Finite Cylindrical Shell Structure by Active Vibration Control
Seung-Bok Choi
2013-02-01
Full Text Available In this work, active vibration control of an underwater cylindrical shell structure was investigated, to suppress structural vibration and structure-borne noise in water. Finite element modeling of the submerged cylindrical shell structure was developed, and experimentally evaluated. Modal reduction was conducted to obtain the reduced system equation for the active feedback control algorithm. Three Macro Fiber Composites (MFCs were used as actuators and sensors. One MFC was used as an exciter. The optimum control algorithm was designed based on the reduced system equations. The active control performance was then evaluated using the lab scale underwater cylindrical shell structure. Structural vibration and structure-borne noise of the underwater cylindrical shell structure were reduced significantly by activating the optimal controller associated with the MFC actuators. The results provide that active vibration control of the underwater structure is a useful means to reduce structure-borne noise in water.
12. Finite Element Methods and Their Applications
Chen, Zhangxin
2005-01-01
This book serves as a text for one- or two-semester courses for upper-level undergraduates and beginning graduate students and as a professional reference for people who want to solve partial differential equations (PDEs) using finite element methods. The author has attempted to introduce every concept in the simplest possible setting and maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. Quite a lot of attention is given to discontinuous finite elements, characteristic finite elements, and to the applications in fluid and solid mechanics including applications to porous media flow, and applications to semiconductor modeling. An extensive set of exercises and references in each chapter are provided.
13. The finite element response matrix method
Nakata, H.; Martin, W.R.
1983-02-01
A new technique is developed with an alternative formulation of the response matrix method implemented with the finite element scheme. Two types of response matrices are generated from the Galerkin solution to the weak form of the diffusion equation subject to an arbitrary current and source. The piecewise polynomials are defined in two levels, the first for the local (assembly) calculations and the second for the global (core) response matrix calculations. This finite element response matrix technique was tested in two 2-dimensional test problems, 2D-IAEA benchmark problem and Biblis benchmark problem, with satisfatory results. The computational time, whereas the current code is not extensively optimized, is of the same order of the well estabilished coarse mesh codes. Furthermore, the application of the finite element technique in an alternative formulation of response matrix method permits the method to easily incorporate additional capabilities such as treatment of spatially dependent cross-sections, arbitrary geometrical configurations, and high heterogeneous assemblies. (Author) [pt
14. Finite-element analysis of flawed and unflawed pipe tests
James, R.J.; Nickell, R.E.; Sullaway, M.F.
1989-12-01
Contemporary versions of the general purpose, nonlinear finite element program ABAQUS have been used in structural response verification exercises on flawed and unflawed austenitic stainless steel and ferritic steel piping. Among the topics examined, through comparison between ABAQUS calculations and test results, were: (1) the effect of using variations in the stress-strain relationship from the test article material on the calculated response; (2) the convergence properties of various finite element representations of the pipe geometry, using shell, beam and continuum models; (3) the effect of test system compliance; and (4) the validity of ABAQUS J-integral routines for flawed pipe evaluations. The study was culminated by the development and demonstration of a ''macroelement'' representation for the flawed pipe section. The macroelement can be inserted into an existing piping system model, in order to accurately treat the crack-opening and crack-closing static and dynamic response. 11 refs., 20 figs., 1 tab
15. On the reliability of finite element solutions
1975-01-01
The extent of reliability of the finite element method for analysis of nuclear reactor structures, and that of reactor vessels in particular and the need for the engineer to guard against the pitfalls that may arise out of both physical and mathematical models have been high-lighted. A systematic way of checking the model to obtain reasonably accurate solutions is presented. Quite often sophisticated elements are suggested for specific design and stress concentration problems. The desirability or otherwise of these elements, their scope and utility vis-a-vis the use of large stack of conventional elements are discussed from the view point of stress analysts. The methods of obtaining a check on the reliability of the finite element solutions either through modelling changes or an extrapolation technique are discussed. (author)
16. Finite elements for analysis and design
Akin, J E; Davenport, J H
1994-01-01
The finite element method (FEM) is an analysis tool for problem-solving used throughout applied mathematics, engineering, and scientific computing. Finite Elements for Analysis and Design provides a thoroughlyrevised and up-to-date account of this important tool and its numerous applications, with added emphasis on basic theory. Numerous worked examples are included to illustrate the material.Key Features* Akin clearly explains the FEM, a numerical analysis tool for problem-solving throughout applied mathematics, engineering and scientific computing* Basic theory has bee
17. Finite-element analysis of dynamic fracture
Aberson, J. A.; Anderson, J. M.; King, W. W.
1976-01-01
Applications of the finite element method to the two dimensional elastodynamics of cracked structures are presented. Stress intensity factors are computed for two problems involving stationary cracks. The first serves as a vehicle for discussing lumped-mass and consistent-mass characterizations of inertia. In the second problem, the behavior of a photoelastic dynamic tear test specimen is determined for the time prior to crack propagation. Some results of a finite element simulation of rapid crack propagation in an infinite body are discussed.
18. Crack Propagation by Finite Element Method
Luiz Carlos H. Ricardo
2018-01-01
Full Text Available Crack propagation simulation began with the development of the finite element method; the analyses were conducted to obtain a basic understanding of the crack growth. Today structural and materials engineers develop structures and materials properties using this technique. The aim of this paper is to verify the effect of different crack propagation rates in determination of crack opening and closing stress of an ASTM specimen under a standard suspension spectrum loading from FDandE SAE Keyhole Specimen Test Load Histories by finite element analysis. To understand the crack propagation processes under variable amplitude loading, retardation effects are observed
19. Fluid-structure finite-element vibrational analysis
Feng, G. C.; Kiefling, L.
1974-01-01
A fluid finite element has been developed for a quasi-compressible fluid. Both kinetic and potential energy are expressed as functions of nodal displacements. Thus, the formulation is similar to that used for structural elements, with the only differences being that the fluid can possess gravitational potential, and the constitutive equations for fluid contain no shear coefficients. Using this approach, structural and fluid elements can be used interchangeably in existing efficient sparse-matrix structural computer programs such as SPAR. The theoretical development of the element formulations and the relationships of the local and global coordinates are shown. Solutions of fluid slosh, liquid compressibility, and coupled fluid-shell oscillation problems which were completed using a temporary digital computer program are shown. The frequency correlation of the solutions with classical theory is excellent.
20. Finite element analysis of inelastic structural behavior
Argyris, J.H.; Szimmat, J.; Willam, K.J.
1977-01-01
The paper describes recent achievements in the finite element analysis of inelastic material behavior. The main purpose is to examine the interaction of three disciplines; (i) the finite element formulation of large deformation problems in the light of a systematic linearization, (ii) the constitutive modelling of inelastic processes in the rate-dependent and rate-independent response regime and (iii) the numerical solution of nonlinear rate problems via incremental iteration techniques. In the first part, alternative finite element models are developed for the idealization of large deformation problems. A systematic approach is presented to linearize the field equations locally by an incremental procedure. The finite element formulation is then examined for the description of inelastic material processes. In the second part, nonlinear and inelastic material phenomena are classified and illustrated with representative examples of concrete and metal components. In particular, rate-dependent and rate-independent material behavior is examined and representative constitutive models are assessed for their mathematical characterization. Hypoelastic, elastoplastic and endochronic models are compared for the description rate-independent material phenomena. In the third part, the numerial solution of inelastic structural behavior is discussed. In this context, several incremental techniques are developed and compared for tracing the evolution of the inelastic process. The numerical procedures are examined with regard to stability and accuracy to assess the overall efficiency. The 'optimal' incremental technique is then contrasted with the computer storage requirements to retain the data for the 'memory-characteristics' of the constitutive model
1. Finite element modelling of solidification phenomena
Unknown
Abstract. The process of solidification process is complex in nature and the simulation of such process is required in industry before it is actually undertaken. Finite element method is used to simulate the heat transfer process accompanying the solidification process. The metal and the mould along with the air gap formation ...
2. Image segmentation with a finite element method
Bourdin, Blaise
1999-01-01
regularization results, make possible to imagine a finite element resolution method.In a first time, the Mumford-Shah functional is introduced and some existing results are quoted. Then, a discrete formulation for the Mumford-Shah problem is proposed and its \$\\Gamma\$-convergence is proved. Finally, some...
3. Orthodontic treatment: Introducing finite element analysis
Driel, W.D. van; Leeuwen, E.J. van
1998-01-01
The aim of orthodontic treatment is the displacement of teeth by means ofspecial appliances, like braces and brackets. Through these appliances the orthodontist can apply a set of forces to the teeth which wilt result in its displacement through the jawbone. Finite Element analysis of this process
4. Isogeometric finite element analysis of poroelasticity
Irzal, F.; Remmers, J.J.C.; Verhoosel, C.V.; Borst, de R.
2013-01-01
We present an alternative numerical approach for predicting the behaviour of a deformable fluid-saturated porous medium. The conventional finite element technology is replaced by isogeometric analysis that uses non-uniform rational B-splines. The ability of these functions to provide higher-order
5. Fast finite elements for surgery simulation
Bro-Nielsen, Morten
1997-01-01
This paper discusses volumetric deformable models for modeling human body parts and organs in surgery simulation systems. These models are built using finite element models for linear elastic materials. To achieve real-time response condensation has been applied to the system stiffness matrix...
6. Simplicial Finite Elements in Higher Dimensions
Brandts, J.; Korotov, S.; Křížek, Michal
2007-01-01
Roč. 52, č. 3 (2007), s. 251-265 ISSN 0862-7940 R&D Projects: GA ČR GA201/04/1503 Institutional research plan: CEZ:AV0Z10190503 Keywords : n-simplex * finite element method * superconvergence Subject RIV: BA - General Mathematics
7. Finite element method - theory and applications
Baset, S.
1992-01-01
This paper summarizes the mathematical basis of the finite element method. Attention is drawn to the natural development of the method from an engineering analysis tool into a general numerical analysis tool. A particular application to the stress analysis of rubber materials is presented. Special advantages and issues associated with the method are mentioned. (author). 4 refs., 3 figs
8. Finite element analysis of structures through unified formulation
Carrera, Erasmo; Petrolo, Marco; Zappino, Enrico
2014-01-01
The finite element method (FEM) is a computational tool widely used to design and analyse complex structures. Currently, there are a number of different approaches to analysis using the FEM that vary according to the type of structure being analysed: beams and plates may use 1D or 2D approaches, shells and solids 2D or 3D approaches, and methods that work for one structure are typically not optimized to work for another. Finite Element Analysis of Structures Through Unified Formulation deals with the FEM used for the analysis of the mechanics of structures in the case of linear elasticity. The novelty of this book is that the finite elements (FEs) are formulated on the basis of a class of theories of structures known as the Carrera Unified Formulation (CUF). It formulates 1D, 2D and 3D FEs on the basis of the same ''fundamental nucleus'' that comes from geometrical relations and Hooke''s law, and presents both 1D and 2D refined FEs that only have displacement variables as in 3D elements. It also covers 1D...
9. Finite element evaluation of erosion/corrosion affected reducing elbow
Basavaraju, C.
1996-01-01
Erosion/corrosion is a primary source for wall thinning or degradation of carbon steel piping systems in service. A number of piping failures in the power industry have been attributed to erosion/corrosion. Piping elbow is one of such susceptible components for erosion/corrosion because of increased flow turbulence due to its geometry. In this paper, the acceptability of a 12 in. x 8 in. reducing elbow in RHR service water pump discharge piping, which experienced significant degradation due to wall thinning in localized areas, was evaluated using finite element analysis methodology. Since the simplified methods showed very small margin and recommended replacement of the elbow, a detailed 3-D finite element model was built using shell elements and analyzed for internal pressure and moment loadings. The finite element analysis incorporated the U.T. measured wall thickness data at various spots that experienced wall thinning. The results showed that the elbow is acceptable as-is until the next fuel cycle. FEA, though cumbersome, and time consuming is a valuable analytical tool in making critical decisions with regard to component replacement of border line situation cases, eliminating some conservatism while not compromising the safety
10. Introduction to finite and spectral element methods using Matlab
Pozrikidis, Constantine
2014-01-01
The Finite Element Method in One Dimension. Further Applications in One Dimension. High-Order and Spectral Elements in One Dimension. The Finite Element Method in Two Dimensions. Quadratic and Spectral Elements in Two Dimensions. Applications in Mechanics. Viscous Flow. Finite and Spectral Element Methods in Three Dimensions. Appendices. References. Index.
11. ZONE: a finite element mesh generator
Burger, M.J.
1976-05-01
The ZONE computer program is a finite-element mesh generator which produces the nodes and element description of any two-dimensional geometry. The geometry is subdivided into a mesh of quadrilateral and triangular zones arranged sequentially in an ordered march through the geometry. The order of march can be chosen so that the minimum bandwidth is obtained. The node points are defined in terms of the x and y coordinates in a global rectangular coordinate system. The zones generated are quadrilaterals or triangles defined by four node points in a counterclockwise sequence. Node points defining the outside boundary are generated to describe pressure boundary conditions. The mesh that is generated can be used as input to any two-dimensional as well as any axisymmetrical structure program. The output from ZONE is essentially the input file to NAOS, HONDO, and other axisymmetric finite element programs. 14 figures
12. Multibody dynamic analysis using a rotation-free shell element with corotational frame
Shi, Jiabei; Liu, Zhuyong; Hong, Jiazhen
2018-03-01
Rotation-free shell formulation is a simple and effective method to model a shell with large deformation. Moreover, it can be compatible with the existing theories of finite element method. However, a rotation-free shell is seldom employed in multibody systems. Using a derivative of rigid body motion, an efficient nonlinear shell model is proposed based on the rotation-free shell element and corotational frame. The bending and membrane strains of the shell have been simplified by isolating deformational displacements from the detailed description of rigid body motion. The consistent stiffness matrix can be obtained easily in this form of shell model. To model the multibody system consisting of the presented shells, joint kinematic constraints including translational and rotational constraints are deduced in the context of geometric nonlinear rotation-free element. A simple node-to-surface contact discretization and penalty method are adopted for contacts between shells. A series of analyses for multibody system dynamics are presented to validate the proposed formulation. Furthermore, the deployment of a large scaled solar array is presented to verify the comprehensive performance of the nonlinear shell model.
13. Finite element elastic-plastic analysis of LMFBR components
Levy, A.; Pifko, A.; Armen, H. Jr.
1978-01-01
The present effort involves the development of computationally efficient finite element methods for accurately predicting the isothermal elastic-plastic three-dimensional response of thick and thin shell structures subjected to mechanical and thermal loads. This work will be used as the basis for further development of analytical tools to be used to verify the structural integrity of liquid metal fast breeder reactor (LMFBR) components. The methods presented here have been implemented into the three-dimensional solid element module (HEX) of the Grumman PLANS finite element program. These methods include the use of optimal stress points as well as a variable number of stress points within an element. This allows monitoring the stress history at many points within an element and hence provides an accurate representation of the elastic-plastic boundary using a minimum number of degrees of freedom. Also included is an improved thermal stress analysis capability in which the temperature variation and corresponding thermal strain variation are represented by the same functional form as the displacement variation. Various problems are used to demonstrate these improved capabilities. (Auth.)
14. FINITE ELEMENT ANALYSIS OF ELEMENT ANALYSIS OF A FREE ...
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the stairs and to compare the finite element ana ... tual three dimensional behavior of the stair slab system. ..... due to its close relation of output with the propo .... flights. It is best not to consider any open well when .... thermodynamics of solids.
15. Development of a shell finite element. Application to the thermo-viscoplastic behaviour of a PWR vessel during a severe accident; Developpement d`un element fini coque. Application au comportement thermo-viscoplastique d`une cuve de reacteur nucleaire (REP) en situation d`accident grave
Diaz, V
1998-10-07
The aim of this study is to develop a model for the thermo-viscoplastic behaviour of he power water reactor lower head during a severe accident, so as to implement it in codes representing the whole accident progress (scenario codes). So it has to give a precise solution in a short cpu-time. The main loadings are the internal pressure and the strong longitudinal and transverse thermal gradients. To deal with this problem, the idea is to develop a new shell element with variable mechanical parameters with the temperature. This is possible in taking advantage of the properties of the bending center line, called neutral fiber. Besides, this new shell element has the particularity to be able to melt without modifying the initial dimensions of the structure. Then, we have developed a complete program to study the mechanical resistance of the vessel. The visco-plastic behaviour is considered as a loading (so it is placed in the second member of the system to be solved) and represented by a Norton law whose parameters depend on the temperature, the law is integrated explicitly which necessitates the introduction of criteria limiting the time step. The rupture criterion by creep is defined by a damage law whereas the rupture criterion by plasticity is based on the exceeding of the mean limit stress in the thickness. Then the model was validated by comparing the results with those of a Castem 2000 volume mesh (finite element code). Finally the model was coupled with the scenario codes ICARE2 and MAAP4 and tested on two typical severe accidents. The results are very satisfactory both on accuracy and cpu-time execution. (author) 113 refs.
16. A finite element method for neutron transport
Ackroyd, R.T.
1983-01-01
A completely boundary-free maximum principle for the first-order Boltzmann equation is derived from the completely boundary-free maximum principle for the mixed-parity Boltzmann equation. When continuity is imposed on the trial function for directions crossing interfaces the completely boundary-free principle for the first-order Boltzmann equation reduces to a maximum principle previously established directly from first principles and indirectly by the Euler-Lagrange method. Present finite element methods for the first-order Boltzmann equation are based on a weighted-residual method which permits the use of discontinuous trial functions. The new principle for the first-order equation can be used as a basis for finite-element methods with the same freedom from boundary conditions as those based on the weighted-residual method. The extremum principle as the parent of the variationally-derived weighted-residual equations ensures their good behaviour. (author)
17. Finite element computation of plasma equilibria
Rivier, M.
1977-01-01
The applicability of the finite element method is investigated for the numerical solution of the nonlinear Grad-Shafranov equation with free boundary for the flux function of a plasma at equilibrium. This method is based on the case of variational principles and finite dimensional subspaces whose elements are piecewise polynomial functions obtained by a Lagrange type interpolation procedure over a triangulation of the domain. Two cases of plasma pressure (exponential and quadratic including a vacuum region) were examined. In both cases the nonuniqueness of the solutions was shown in exhibiting a deeper solution in the case of exponential pressure function, and a non-constant solution for a quadratic pressure function. In order to get this ''other'' solution, two linearization methods were tested with two different constraints. Different cross sections are investigated
18. Finite element reliability analysis of fatigue life
Harkness, H.H.; Belytschko, T.; Liu, W.K.
1992-01-01
Fatigue reliability is addressed by the first-order reliability method combined with a finite element method. Two-dimensional finite element models of components with cracks in mode I are considered with crack growth treated by the Paris law. Probability density functions of the variables affecting fatigue are proposed to reflect a setting where nondestructive evaluation is used, and the Rosenblatt transformation is employed to treat non-Gaussian random variables. Comparisons of the first-order reliability results and Monte Carlo simulations suggest that the accuracy of the first-order reliability method is quite good in this setting. Results show that the upper portion of the initial crack length probability density function is crucial to reliability, which suggests that if nondestructive evaluation is used, the probability of detection curve plays a key role in reliability. (orig.)
19. Finite Element Simulation of Fracture Toughness Test
Chu, Seok Jae; Liu, Cong Hao
2013-01-01
Finite element simulations of tensile tests were performed to determine the equivalent stress - equivalent plastic strain curves, critical equivalent stresses, and critical equivalent plastic strains. Then, the curves were used as inputs to finite element simulations of fracture toughness tests to determine the plane strain fracture toughness. The critical COD was taken as the COD when the equivalent plastic strain at the crack tip reached a critical value, and it was used as a crack growth criterion. The relationship between the critical COD and the critical equivalent plastic strain or the reduction of area was found. The relationship between the plane strain fracture toughness and the product of the critical equivalent stress and the critical equivalent plastic strain was also found
20. Introduction to nonlinear finite element analysis
Kim, Nam-Ho
2015-01-01
This book introduces the key concepts of nonlinear finite element analysis procedures. The book explains the fundamental theories of the field and provides instructions on how to apply the concepts to solving practical engineering problems. Instead of covering many nonlinear problems, the book focuses on three representative problems: nonlinear elasticity, elastoplasticity, and contact problems. The book is written independent of any particular software, but tutorials and examples using four commercial programs are included as appendices: ANSYS, NASTRAN, ABAQUS, and MATLAB. In particular, the MATLAB program includes all source codes so that students can develop their own material models, or different algorithms. This book also: · Presents clear explanations of nonlinear finite element analysis for elasticity, elastoplasticity, and contact problems · Includes many informative examples of nonlinear analyses so that students can clearly understand the nonlinear theory · ...
1. Finite element analysis of ARPS structures
Ruhkamp, J.D.; McDougal, J.R.; Kramer, D.P.
1998-01-01
Algor finite element software was used to determine the stresses and deflections in the metallic walls of Advanced Radioisotope Power Systems (ARPS) designs. The preliminary design review of these systems often neglects the structural integrity of the design which can effect fabrication and the end use of the design. Before finite element analysis (FEA) was run on the canister walls of the thermophotovoltaic (TPV) generator, hand calculations were used to approximate the stresses and deflections in a flat plate. These results compared favorably to the FEA results of a similar size flat plate. The AMTEC (Alkali Metal Thermal-to-Electric Conversion) cells were analyzed by FEA and the results compared to two cells that were mechanically tested. The mechanically tested cells buckled in the thin sections, one at the top and one in the lower section. The FEA predicted similar stress and shape results but the critical buckling load was found to be very shape dependent
2. Finite element analysis of human joints
Bossart, P.L.; Hollerbach, K.
1996-09-01
Our work focuses on the development of finite element models (FEMs) that describe the biomechanics of human joints. Finite element modeling is becoming a standard tool in industrial applications. In highly complex problems such as those found in biomechanics research, however, the full potential of FEMs is just beginning to be explored, due to the absence of precise, high resolution medical data and the difficulties encountered in converting these enormous datasets into a form that is usable in FEMs. With increasing computing speed and memory available, it is now feasible to address these challenges. We address the first by acquiring data with a high resolution C-ray CT scanner and the latter by developing semi-automated method for generating the volumetric meshes used in the FEM. Issues related to tomographic reconstruction, volume segmentation, the use of extracted surfaces to generate volumetric hexahedral meshes, and applications of the FEM are described.
3. Finite element analysis of human joints
Bossart, P.L.; Hollerbach, K.
1996-09-01
Our work focuses on the development of finite element models (FEMs) that describe the biomechanics of human joints. Finite element modeling is becoming a standard tool in industrial applications. In highly complex problems such as those found in biomechanics research, however, the full potential of FEMs is just beginning to be explored, due to the absence of precise, high resolution medical data and the difficulties encountered in converting these enormous datasets into a form that is usable in FEMs. With increasing computing speed and memory available, it is now feasible to address these challenges. We address the first by acquiring data with a high resolution C-ray CT scanner and the latter by developing semi-automated method for generating the volumetric meshes used in the FEM. Issues related to tomographic reconstruction, volume segmentation, the use of extracted surfaces to generate volumetric hexahedral meshes, and applications of the FEM are described
4. Finite element simulations with ANSYS workbench 16
Lee , Huei-Huang
2015-01-01
Finite Element Simulations with ANSYS Workbench 16 is a comprehensive and easy to understand workbook. It utilizes step-by-step instructions to help guide readers to learn finite element simulations. Twenty seven real world case studies are used throughout the book. Many of these cases are industrial or research projects the reader builds from scratch. All the files readers may need if they have trouble are available for download on the publishers website. Companion videos that demonstrate exactly how to preform each tutorial are available to readers by redeeming the access code that comes in the book. Relevant background knowledge is reviewed whenever necessary. To be efficient, the review is conceptual rather than mathematical. Key concepts are inserted whenever appropriate and summarized at the end of each chapter. Additional exercises or extension research problems are provided as homework at the end of each chapter. A learning approach emphasizing hands-on experiences spreads through this entire book. A...
5. Finite element based electric motor design optimization
Campbell, C. Warren
1993-01-01
The purpose of this effort was to develop a finite element code for the analysis and design of permanent magnet electric motors. These motors would drive electromechanical actuators in advanced rocket engines. The actuators would control fuel valves and thrust vector control systems. Refurbishing the hydraulic systems of the Space Shuttle after each flight is costly and time consuming. Electromechanical actuators could replace hydraulics, improve system reliability, and reduce down time.
6. Finite element analysis of nonlinear creeping flows
Loula, A.F.D.; Guerreiro, J.N.C.
1988-12-01
Steady-state creep problems with monotone constitutive laws are studied. Finite element approximations are constructed based on mixed Petrov-Galerkin formulations for constrained problems. Stability, convergence and a priori error estimates are proved for equal-order discontinuous stress and continuous velocity interpolations. Numerical results are presented confirming the rates of convergence predicted in the analysis and the good performance of this formulation. (author) [pt
7. Finite element methods for incompressible flow problems
John, Volker
2016-01-01
This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations, and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques.
8. Upstand Finite Element Analysis of Slab Bridges
O'Brien, Eugene J.; Keogh, D.L.
1998-01-01
For slab bridge decks with wide transverse edge cantilevers, the plane grillage analogy is shown to be an inaccurate method of linear elastic analysis due to variations in the vertical position of the neutral axis. The upstand grillage analogy is also shown to give inaccurate results, this time due to inappropriate modelling of in-plane distortions. An alternative method, known as upstand finite element analysis, is proposed which is sufficiently simple to be used on an everyday basis in the ...
9. Crack Propagation by Finite Element Method
H. Ricardo, Luiz Carlos
2017-01-01
Crack propagation simulation began with the development of the finite element method; the analyses were conducted to obtain a basic understanding of the crack growth. Today structural and materials engineers develop structures and materials properties using this technique. The aim of this paper is to verify the effect of different crack propagation rates in determination of crack opening and closing stress of an ASTM specimen under a standard suspension spectrum loading from FD&E SAE Keyh...
10. Finite element simulation of heat transfer
Bergheau, Jean-Michel
2010-01-01
This book introduces the finite element method applied to the resolution of industrial heat transfer problems. Starting from steady conduction, the method is gradually extended to transient regimes, to traditional non-linearities, and to convective phenomena. Coupled problems involving heat transfer are then presented. Three types of couplings are discussed: coupling through boundary conditions (such as radiative heat transfer in cavities), addition of state variables (such as metallurgical phase change), and coupling through partial differential equations (such as electrical phenomena).? A re
11. Development library of finite elements for computer-aided design system of reed sensors
Kozlov, A. S.; Shmakov, N. A.; Tkalich, V. L.; Labkovskaia, R. I.; Kalinkina, M. E.; Pirozhnikova, O. I.
2018-05-01
The article is devoted to the development of a modern highly reliable element base of devices for security and fire alarm systems, in particular, to the improvement of the quality of contact cores (reed and membrane) of reed sensors. Modeling of elastic sensitive elements uses quadrangular elements of plates and shells, considered in the system of curvilinear orthogonal coordinates. The developed mathematical models and the formed finite element library are designed for systems of automated design of reed switch detectors to create competitive devices alarms. The finite element library is used for the automated system production of reed switch detectors both in series production and in the implementation of individual orders.
12. Finite element simulation of impact response of wire mesh screens
Wang Caizheng
2015-01-01
Full Text Available In this paper, the response of wire mesh screens to low velocity impact with blunt objects is investigated using finite element (FE simulation. The woven wire mesh is modelled with homogeneous shell elements with equivalent smeared mechanical properties. The mechanical behaviour of the woven wire mesh was determined experimentally with tensile tests on steel wire mesh coupons to generate the data for the smeared shell material used in the FE. The effects of impacts with a low mass (4 kg and a large mass (40 kg providing the same impact energy are studied. The joint between the wire mesh screen and the aluminium frame surrounding it is modelled using contact elements with friction between the corresponding elements. Damage to the screen of different types compromising its structural integrity, such as mesh separation and pulling out from the surrounding frame is modelled. The FE simulation is validated with results of impact tests conducted on woven steel wire screen meshes.
13. Shell structure and orbit bifurcations in finite fermion systems
Magner, A. G.; Yatsyshyn, I. S.; Arita, K.; Brack, M.
2011-10-01
We first give an overview of the shell-correction method which was developed by V.M. Strutinsky as a practicable and efficient approximation to the general self-consistent theory of finite fermion systems suggested by A.B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M.C. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the "periodic orbit theory". We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called "superdeformed" energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).
14. Variational approach to probabilistic finite elements
Belytschko, T.; Liu, W. K.; Mani, A.; Besterfield, G.
1991-08-01
Probabilistic finite element methods (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.
15. Finite Element Method in Machining Processes
Markopoulos, Angelos P
2013-01-01
Finite Element Method in Machining Processes provides a concise study on the way the Finite Element Method (FEM) is used in the case of manufacturing processes, primarily in machining. The basics of this kind of modeling are detailed to create a reference that will provide guidelines for those who start to study this method now, but also for scientists already involved in FEM and want to expand their research. A discussion on FEM, formulations and techniques currently in use is followed up by machining case studies. Orthogonal cutting, oblique cutting, 3D simulations for turning and milling, grinding, and state-of-the-art topics such as high speed machining and micromachining are explained with relevant examples. This is all supported by a literature review and a reference list for further study. As FEM is a key method for researchers in the manufacturing and especially in the machining sector, Finite Element Method in Machining Processes is a key reference for students studying manufacturing processes but al...
16. Finite element analysis of the Girkmann problem using the modern hp-version and the classical h-version
Niemi, Antti; Babuška, Ivo M.; Pitkä ranta, Juhani; Demkowicz, Leszek F.
2011-01-01
elasticity theory and (2) by using a dimensionally reduced shell-ring model. In the first approach the problem is solved with a fully automatic hp-adaptive finite element solver whereas the classical h-version of the finite element method is used
17. FINELM: a multigroup finite element diffusion code
Higgs, C.E.; Davierwalla, D.M.
1981-06-01
FINELM is a FORTRAN IV program to solve the Neutron Diffusion Equation in X-Y, R-Z, R-theta, X-Y-Z and R-theta-Z geometries using the method of Finite Elements. Lagrangian elements of linear or higher degree to approximate the spacial flux distribution have been provided. The method of dissections, coarse mesh rebalancing and Chebyshev acceleration techniques are available. Simple user defined input is achieved through extensive input subroutines. The input preparation is described followed by a program structure description. Sample test cases are provided. (Auth.)
18. Solid Modeling and Finite Element Analysis of an Overhead Crane Bridge
C. Alkin
2005-01-01
Full Text Available The design of an overhead crane bridge with a double box girder has been investigated and a case study of a crane with 35 ton capacity and 13 m span length has been conducted. In the initial phase of the case study, conventional design calculations proposed by F. E. M. Rules and DIN standards were performed to verify the stress and deflection levels. The crane design was modeled using both solids and surfaces. Finite element meshes with 4-node tetrahedral and 4-node quadrilateral shell elements were generated from the solid and shell models, respectively. After a comparison of the finite element analyses, the conventional calculations and performance of the existing crane, the analysis with quadratic shell elements was found to give the most realistic results. As a result of this study, a design optimization method for an overhead crane is proposed.
19. A flat triangular shell element with Loof nodes
Poulsen, Peter Noe; Damkilde, Lars
1996-01-01
In the formulation of flat shell elements it is difficult to achieve inter-element compatibility between membrane and transverse displacements for non-coplanar elements. Many elements lack proper nodal degrees of freedom to model intersections making the assembly of elements troublesome. A flat...... triangular shell element is established by a combination of a new plate bending element DKTL and the well-known linear membrane strain element LST, and for this element the above-mentioned deficiences are avoided. The plate bending element DKTL is based on Discrete Kirchhoff Theory and Loof nodes. The nodal...
20. A collocation finite element method with prior matrix condensation
Sutcliffe, W.J.
1977-01-01
For thin shells with general loading, sixteen degrees of freedom have been used for a previous finite element solution procedure using a Collocation method instead of the usual variational based procedures. Although the number of elements required was relatively small, nevertheless the final matrix for the simultaneous solution of all unknowns could become large for a complex compound structure. The purpose of the present paper is to demonstrate a method of reducing the final matrix size, so allowing solution for large structures with comparatively small computer storage requirements while retaining the accuracy given by high order displacement functions. Collocation points, a number are equilibrium conditions which must be satisfied independently of the overall compatibility of forces and deflections for a complete structure. (Auth.)
1. Finite-difference analysis of shells impacting rigid barriers
Pirotin, S.D.; Witmer, E.A.
1977-01-01
Nuclear power plants must be protected from the adverse effects of missile impacts. A significant category of missile impact involves deformable structures (pressure vessel components, whipping pipes) striking relatively rigid targets (concrete walls, bumpers) which act as protective devices. The response and interaction of these structures is needed to assess the adequacy of these barriers for protecting vital safety related equipment. The present investigation represents an initial attempt to develop an efficient numerical procedure for predicting the deformations and impact force time-histories of shells which impact upon a rigid target. The general large-deflection equations of motion of the shell are expressed in finite-difference form in space and integrated in time through application of the central-difference temporal operator. The effect of material nonlinearities is treated by a mechanical sublayer material model which handles the strain-hardening, Bauschinger, and strain-rate effects. The general adequacy of this shell treatment has been validated by comparing predictions with the results of various experiments in which structures have been subjected to well-defined transient forcing functions (typically high-explosive impulse loading). The 'new' ingredient addressed in the present study involves an accounting for impact interaction and response of both the target structure and the attacking body. (Auth.)
2. Finite element analysis of an inflatable torus considering air mass structural element
Gajbhiye, S. C.; Upadhyay, S. H.; Harsha, S. P.
2014-01-01
Inflatable structures, also known as gossamer structures, are at high boom in the current space technology due to their low mass and compact size comparing to the traditional spacecraft designing. Internal pressure becomes the major source of strength and rigidity, essentially stiffen the structure. However, inflatable space based membrane structure are at high risk to the vibration disturbance due to their low structural stiffness and material damping. Hence, the vibration modes of the structure should be known to a high degree of accuracy in order to provide better control authority. In the past, most of the studies conducted on the vibration analysis of gossamer structures used inaccurate or approximate theories in modeling the internal pressure. The toroidal shaped structure is one of the important key element in space application, helps to support the reflector in space application. This paper discusses the finite-element analysis of an inflated torus. The eigen-frequencies are obtained via three-dimensional small-strain elasticity theory, based on extremum energy principle. The two finite-element model (model-1 and model-2) have cases have been generated using a commercial finite-element package. The structure model-1 with shell element and model-2 with the combination of the mass of enclosed fluid (air) added to the shell elements have been taken for the study. The model-1 is computed with present analytical approach to understand the convergence rate and the accuracy. The convergence study is made available for the symmetric modes and anti-symmetric modes about the centroidal-axis plane, meeting the eigen-frequencies of an inflatable torus with the circular cross section. The structural model-2 is introduced with air mass element and analyzed its eigen-frequency with different aspect ratio and mode shape response using in-plane and out-plane loading condition are studied.
3. Finite Element Based Design and Optimization for Piezoelectric Accelerometers
Liu, Bin; Kriegbaum, B.; Yao, Q.
1998-01-01
A systematic Finite Element design and optimisation procedure is implemented for the development of piezoelectric accelerometers. Most of the specifications of accelerometers can be obtained using the Finite Element simulations. The deviations between the simulated and calibrated sensitivities...
4. A multiscale mortar multipoint flux mixed finite element method
Wheeler, Mary Fanett; Xue, Guangri; Yotov, Ivan
2012-01-01
In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite
5. Application of finite-element-methods in food processing
Risum, Jørgen
2004-01-01
Presentation of the possible use of finite-element-methods in food processing. Examples from diffusion studies are given.......Presentation of the possible use of finite-element-methods in food processing. Examples from diffusion studies are given....
6. Finite element analysis of plastic recycling machine designed for ...
... design was evaluated using finite element analysis (FEA) tool in Solid Works Computer ... Also, a minimum factor of safety value of 5.3 was obtained for shredder shaft ... Machine; Design; Recycling; Sustainability; Finite Element; Simulation ...
7. Error-controlled adaptive finite elements in solid mechanics
Stein, Erwin; Ramm, E
2003-01-01
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error-controlled Adaptive Finite-element-methods . . . . . . . . . . . . Missing Features and Properties of Today's General Purpose FE Programs for Structural...
8. Modelling bucket excavation by finite element
Pecingina, O. M.
2015-11-01
Changes in geological components of the layers from lignite pits have an impact on the sustainability of the cup path elements and under the action of excavation force appear efforts leading to deformation of the entire assembly. Application of finite element method in the optimization of components leads to economic growth, to increase the reliability and durability of the studied machine parts thus the machine. It is obvious usefulness of knowledge the state of mechanical tensions that the designed piece or the assembly not to break under the action of tensions that must cope during operation. In the course of excavation work on all bucket cutting force components, the first coming into contact with the material being excavated cutting edge. Therefore in the study with finite element analysis is retained only cutting edge. To study the field of stress and strain on the cutting edge will be created geometric patterns for each type of cup this will be subject to static analysis. The geometric design retains the cutting edge shape and on this on the tooth cassette location will apply an areal force on the abutment tooth. The cutting edge real pattern is subjected to finite element study for the worst case of rock cutting by symmetrical and asymmetrical cups whose profile is different. The purpose of this paper is to determine the displacement and tensions field for both profiles considering the maximum force applied on the cutting edge and the depth of the cutting is equal with the width of the cutting edge of the tooth. It will consider the worst case when on the structure will act both the tangential force and radial force on the bucket profile. For determination of stress and strain field on the form design of cutting edge profile will apply maximum force assuming uniform distribution and on the edge surface force will apply a radial force. After geometric patterns discretization on the cutting knives and determining stress field, can be seen that at the
9. The finite element method in engineering, 2nd edition
Rao, S.S.
1986-01-01
This work provides a systematic introduction to the various aspects of the finite element method as applied to engineering problems. Contents include: introduction to finite element method; solution of finite element equations; solid and structural mechanics; static analysis; dynamic analysis; heat transfer; fluid mechanics and additional applications
10. New mixed finite-element methods
Franca, L.P.
1987-01-01
New finite-element methods are proposed for mixed variational formulations. The methods are constructed by adding to the classical Galerkin method various least-squares like terms. The additional terms involve integrals over element interiors, and include mesh-parameter dependent coefficients. The methods are designed to enhance stability. Consistency is achieved in the sense that exact solutions identically satisfy the variational equations.Applied to several problems, simple finite-element interpolations are rendered convergent, including convenient equal-order interpolations generally unstable within the Galerkin approach. The methods are subdivided into two classes according to the manner in which stability is attained: (1) circumventing Babuska-Brezzi condition methods; (2) satisfying Babuska-Brezzi condition methods. Convergence is established for each class of methods. Applications of the first class of methods to Stokes flow and compressible linear elasticity are presented. The second class of methods is applied to the Poisson, Timoshenko beam and incompressible elasticity problems. Numerical results demonstrate the good stability and accuracy of the methods, and confirm the error estimates
11. A finite element method for neutron transport
Ackroyd, R.T.
1978-01-01
A variational treatment of the finite element method for neutron transport is given based on a version of the even-parity Boltzmann equation which does not assume that the differential scattering cross-section has a spherical harmonic expansion. The theory of minimum and maximum principles is based on the Cauchy-Schwartz equality and the properties of a leakage operator G and a removal operator C. For systems with extraneous sources, two maximum and one minimum principles are given in boundary free form, to ease finite element computations. The global error of an approximate variational solution is given, the relationship of one the maximum principles to the method of least squares is shown, and the way in which approximate solutions converge locally to the exact solution is established. A method for constructing local error bounds is given, based on the connection between the variational method and the method of the hypercircle. The source iteration technique and a maximum principle for a system with extraneous sources suggests a functional for a variational principle for a self-sustaining system. The principle gives, as a consequence of the properties of G and C, an upper bound to the lowest eigenvalue. A related functional can be used to determine both upper and lower bounds for the lowest eigenvalue from an inspection of any approximate solution for the lowest eigenfunction. The basis for the finite element is presented in a general form so that two modes of exploitation can be undertaken readily. The model can be in phase space, with positional and directional co-ordinates defining points of the model, or it can be restricted to the positional co-ordinates and an expansion in orthogonal functions used for the directional co-ordinates. Suitable sets of functions are spherical harmonics and Walsh functions. The latter set is appropriate if a discrete direction representation of the angular flux is required. (author)
12. Finite element simulation of piezoelectric transformers.
Tsuchiya, T; Kagawa, Y; Wakatsuki, N; Okamura, H
2001-07-01
Piezoelectric transformers are nothing but ultrasonic resonators with two pairs of electrodes provided on the surface of a piezoelectric substrate in which electrical energy is carried in the mechanical form. The input and output electrodes are arranged to provide the impedance transformation, which results in the voltage transformation. As they are operated at a resonance, the electrical equivalent circuit approach has traditionally been developed in a rather empirical way and has been used for analysis and design. The present paper deals with the analysis of the piezoelectric transformers based on the three-dimensional finite element modelling. The PIEZO3D code that we have developed is modified to include the external loading conditions. The finite element approach is now available for a wide variety of the electrical boundary conditions. The equivalent circuit of lumped parameters can also be derived from the finite element method (FEM) solution if required. The simulation of the present transformers is made for the low intensity operation and compared with the experimental results. Demonstration is made for basic Rosen-type transformers in which the longitudinal mode of a plate plays an important role; in which the equivalent circuit of lumped constants has been used. However, there are many modes of vibration associated with the plate, the effect of which cannot always be ignored. In the experiment, the double resonances are sometimes observed in the vicinity of the operating frequency. The simulation demonstrates that this is due to the coupling of the longitudinal mode with the flexural mode. Thus, the simulation provides an invaluable guideline to the transformer design.
13. On constitutive modelling in finite element analysis
Bathe, K.J.; Snyder, M.D.; Cleary, M.P.
1979-01-01
This compact contains a brief introduction to the problems involved in constitutive modeling as well as an outline of the final paper to be submitted. Attention is focussed on three important areas: (1) the need for using theoretically sound material models and the importance of recognizing the limitations of the models, (2) the problem of developing stable and effective numerical representations of the models, and (3) the necessity for selection of an appropriate finite element mesh that can capture the actual physical response of the complete structure. In the final paper, we will be presenting our recent research results pertaining to each of these problem areas. (orig.)
14. Generalized multiscale finite element methods: Oversampling strategies
Efendiev, Yalchin R.; Galvis, Juan; Li, Guanglian; Presho, Michael
2014-01-01
In this paper, we propose oversampling strategies in the generalized multiscale finite element method (GMsFEM) framework. The GMsFEM, which has been recently introduced in Efendiev et al. (2013b) [Generalized Multiscale Finite Element Methods, J. Comput. Phys., vol. 251, pp. 116-135, 2013], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) construction of the online space (the latter for parameter-dependent problems). In Efendiev et al. (2013b) [Generalized Multiscale Finite Element Methods, J. Comput. Phys., vol. 251, pp. 116-135, 2013], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we develop oversampling techniques to be used in this context (see Hou and Wu (1997) where oversampling is introduced for multiscale finite element methods). It is known (see Hou and Wu (1997)) that the oversampling can improve the accuracy of multiscale methods. In particular, the oversampling technique uses larger regions (larger than the target coarse block) in constructing local basis functions. Our motivation stems from the analysis presented in this paper, which shows that when using oversampling techniques in the construction of the snapshot space and offline space, GMsFEM will converge independent of small scales and high contrast under certain assumptions. We consider the use of a multiple eigenvalue problems to improve the convergence and discuss their relation to single spectral problems that use oversampled regions. The oversampling procedures proposed in this paper differ from those in Hou and Wu (1997). In particular, the oversampling domains are partially used in constructing local
15. TITUS: a general finite element system
Bougrelle, P.
1983-01-01
TITUS is a general finite element structural analysis system which performs linear/non-linear, static/dynamic analyses of heat-transfer/thermo-mechanical problems. One of the major features of TITUS is that it was designed by engineers, to address engineers in an industrial environment. This has resulted in an easy to use system, with a high-level free-formatted problem oriented language, a large selection of pre- and post processors and sophisticated graphic capabilities. TITUS has many references in civil, mechanical and nuclear engineering applications. The TITUS system is available on various types of machines, from large mainframes to mini computers
16. Finite element analysis of permanent magnet motors
Boglietti, A.; Chiampi, M.; Tartaglia, M.; Chiarabaglio, D.
1989-01-01
The analysis of permanent magnet D.C. brushless motors, supplied by current control inverters, is developed employing a finite element package tailored for such devices. The study is devoted to predicting the performance of a set of four poles machines, under different operating conditions (no-load, rated load). The over-load conditions are also considered including the saturation effect. Moreover the influence of such design parameters, as the tooth shape and the number of magnet segments, is investigated. Computed results are found in satisfactory agreement with experimental ones
17. Hybrid finite difference/finite element immersed boundary method.
E Griffith, Boyce; Luo, Xiaoyu
2017-12-01
The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach. © 2017 The Authors International Journal for Numerical Methods in Biomedical Engineering Published by John Wiley & Sons Ltd.
18. Nonlinear finite-element analysis and biomechanical evaluation of the lumbar spine
Wong, Christian; Gehrchen, P Martin; Darvann, Tron
2003-01-01
A finite-element analysis (FEA) model of an intact lumbar disc-body unit was generated. The vertebral body of the FEA model consisted of a solid tetrahedral core of trabecular bone surrounded by a cortical shell. The disc consisted of an incompressible nucleus surrounded by nonlinear annulus fibe...
19. The finite element response Matrix method
Nakata, H.; Martin, W.R.
1983-01-01
A new method for global reactor core calculations is described. This method is based on a unique formulation of the response matrix method, implemented with a higher order finite element method. The unique aspects of this approach are twofold. First, there are two levels to the overall calculational scheme: the local or assembly level and the global or core level. Second, the response matrix scheme, which is formulated at both levels, consists of two separate response matrices rather than one response matrix as is generally the case. These separate response matrices are seen to be quite beneficial for the criticality eigenvalue calculation, because they are independent of k /SUB eff/. The response matrices are generated from a Galerkin finite element solution to the weak form of the diffusion equation, subject to an arbitrary incoming current and an arbitrary distributed source. Calculational results are reported for two test problems, the two-dimensional International Atomic Energy Agency benchmark problem and a two-dimensional pressurized water reactor test problem (Biblis reactor), and they compare well with standard coarse mesh methods with respect to accuracy and efficiency. Moreover, the accuracy (and capability) is comparable to fine mesh for a fraction of the computational cost. Extension of the method to treat heterogeneous assemblies and spatial depletion effects is discussed
20. Finite element analysis of multilayer coextrusion.
Hopkins, Matthew Morgan; Schunk, Peter Randall; Baer, Thomas A. (Proctor & Gamble Company, West Chester, OH); Mrozek, Randy A. (Army Research Laboratory, Adelphi, MD); Lenhart, Joseph Ludlow (Army Research Laboratory, Adelphi, MD); Rao, Rekha Ranjana; Collins, Robert (Oak Ridge National Laboratory); Mondy, Lisa Ann
2011-09-01
Multilayer coextrusion has become a popular commercial process for producing complex polymeric products from soda bottles to reflective coatings. A numerical model of a multilayer coextrusion process is developed based on a finite element discretization and two different free-surface methods, an arbitrary-Lagrangian-Eulerian (ALE) moving mesh implementation and an Eulerian level set method, to understand the moving boundary problem associated with the polymer-polymer interface. The goal of this work is to have a numerical capability suitable for optimizing and troubleshooting the coextrusion process, circumventing flow instabilities such as ribbing and barring, and reducing variability in layer thickness. Though these instabilities can be both viscous and elastic in nature, for this work a generalized Newtonian description of the fluid is used. Models of varying degrees of complexity are investigated including stability analysis and direct three-dimensional finite element free surface approaches. The results of this work show how critical modeling can be to reduce build test cycles, improve material choices, and guide mold design.
1. A multigrid solution method for mixed hybrid finite elements
Schmid, W. [Universitaet Augsburg (Germany)
1996-12-31
We consider the multigrid solution of linear equations arising within the discretization of elliptic second order boundary value problems of the form by mixed hybrid finite elements. Using the equivalence of mixed hybrid finite elements and non-conforming nodal finite elements, we construct a multigrid scheme for the corresponding non-conforming finite elements, and, by this equivalence, for the mixed hybrid finite elements, following guidelines from Arbogast/Chen. For a rectangular triangulation of the computational domain, this non-conforming schemes are the so-called nodal finite elements. We explicitly construct prolongation and restriction operators for this type of non-conforming finite elements. We discuss the use of plain multigrid and the multilevel-preconditioned cg-method and compare their efficiency in numerical tests.
2. Finite element modeling of piezoelectric elements with complex electrode configuration
Paradies, R; Schläpfer, B
2009-01-01
It is well known that the material properties of piezoelectric materials strongly depend on the state of polarization of the individual element. While an unpolarized material exhibits mechanically isotropic material properties in the absence of global piezoelectric capabilities, the piezoelectric material properties become transversally isotropic with respect to the polarization direction after polarization. Therefore, for evaluating piezoelectric elements the material properties, including the coupling between the mechanical and the electromechanical behavior, should be addressed correctly. This is of special importance for the micromechanical description of piezoelectric elements with interdigitated electrodes (IDEs). The best known representatives of this group are active fiber composites (AFCs), macro fiber composites (MFCs) and the radial field diaphragm (RFD), respectively. While the material properties are available for a piezoelectric wafer with a homogeneous polarization perpendicular to its plane as postulated in the so-called uniform field model (UFM), the same information is missing for piezoelectric elements with more complex electrode configurations like the above-mentioned ones with IDEs. This is due to the inhomogeneous field distribution which does not automatically allow for the correct assignment of the material, i.e. orientation and property. A variation of the material orientation as well as the material properties can be accomplished by including the polarization process of the piezoelectric transducer in the finite element (FE) simulation prior to the actual load case to be investigated. A corresponding procedure is presented which automatically assigns the piezoelectric material properties, e.g. elasticity matrix, permittivity, and charge vector, for finite element models (FEMs) describing piezoelectric transducers according to the electric field distribution (field orientation and strength) in the structure. A corresponding code has been
3. Modeling bistable behaviors in morphing structures through finite element simulations.
Guo, Qiaohang; Zheng, Huang; Chen, Wenzhe; Chen, Zi
2014-01-01
Bistable structures, exemplified by the Venus flytrap and slap bracelets, can transit between different configurations upon certain external stimulation. Here we study, through three-dimensional finite element simulations, the bistable behaviors in elastic plates in the absence of terminate loads, but with pre-strains in one (or both) of the two composite layers. Both the scenarios with and without a given geometric mis-orientation angle are investigated, the results of which are consistent with recent theoretical and experimental studies. This work can open ample venues for programmable designs of plant/shell structures with large deformations, with applications in designing bio-inspired robotics for biomedical research and morphing/deployable structures in aerospace engineering.
4. Membrane versus shell type elements in F.E. analysis of box type buildings
Canetta, G.
1979-01-01
Finite element analysis of box-type buildings is discussed under typical loading conditions - gravity, seismic and temperature loads. The computation effort is recognized to be noticeably different, according to whether membrane or shell type elements are used. The relevance of membrane and bending stress components to the total stress distribution is outlined in the table below; the different role of the typical members under the various loading conditions is emphasized. (orig.)
5. Strain-based finite elements for the analysis of cylinders with holes and normally intersecting cylinders
Sabir, A.B.
1983-01-01
A finite element solution to the problems of stress distribution for cylindrical shells with circular and elliptical holes and also for normally intersecting thin elastic cylindrical shells is given. Quadrilateral and triangular curved finite elements are used in the analysis. The elements are of a new class, based on simple independent generalised strain functions insofar as this is allowed by the compatibility equations. The elements also satisfy exactly the requirements of strain-free-rigid body displacements and uses only the external 'geometrical' nodal degrees of freedom to avoid the difficulties associated with unnecessary internal degrees of freedom. We first develop strain based quadrilateral and triangular elements and apply them to the solution of the problem of stress concentrations in the neighbourhood of small and large circular and elliptical holes when the cylinders are subjected to a uniform axial tension. These results are compared with analytical solutions based on shallow shell approximations and show that the use of these strain based elements obviates the need for using an inordinately large number of elements. Normally intersecting cylinders are common configurations in structural components for nuclear reactor systems and design information for such configurations are generally lacking. The opportunity is taken in the present paper to provide a finite element solution to this problem. A method of substructing will be introduced to enable a solution to the large number of non banded set of simultaneous equations encountered. (orig./HP)
6. Friction welding; Magnesium; Finite element; Shear test.
Leonardo Contri Campanelli
2013-06-01
Full Text Available Friction spot welding (FSpW is one of the most recently developed solid state joining technologies. In this work, based on former publications, a computer aided draft and engineering resource is used to model a FSpW joint on AZ31 magnesium alloy sheets and subsequently submit the assembly to a typical shear test loading, using a linear elastic model, in order to conceive mechanical tests results. Finite element analysis shows that the plastic flow is concentrated on the welded zone periphery where yield strength is reached. It is supposed that “through the weld” and “circumferential pull-out” variants should be the main failure behaviors, although mechanical testing may provide other types of fracture due to metallurgical features.
7. Adaptive finite element method for shape optimization
Morin, Pedro; Nochetto, Ricardo H.; Pauletti, Miguel S.; Verani, Marco
2012-01-01
We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution - a new paradigm in adaptivity. © EDP Sciences, SMAI, 2012.
8. Finite element simulation of asphalt fatigue testing
Ullidtz, Per; Kieler, Thomas Lau; Kargo, Anders
1997-01-01
The traditional interpretation of fatigue tests on asphalt mixes has been in terms of a logarithmic linear relationship between the constant stress or strain amplitude and the number of load repetitions to cause failure, often defined as a decrease in modulus to half the initial value...... damage mechanics.The paper describes how continuum damage mechanics may be used with a finite element program to explain the progressive deterioration of asphalt mixes under laboratory fatigue testing. Both constant stress and constant strain testing are simulated, and compared to the actual results from...... three point and four point fatigue test on different mixes. It is shown that the same damage law, based on energy density, may be used to explain the gradual deterioration under constant stress as well as under constant strain testing.Some of the advantages of using this method for interpreting fatigue...
9. Adaptive finite element method for shape optimization
Morin, Pedro
2012-01-16
We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution - a new paradigm in adaptivity. © EDP Sciences, SMAI, 2012.
10. Finite element program Lamcal. (User's manual)
Lamain, L.G.; Blanckenburg, J.F.G.
1982-01-01
The present user's manual gives the input formats, job control and an input example for the finite element part of the Lamcal program. The input data have been organized in a more or less self explaining way, using keywords and standard input formats and is printed at the beginning of every run. To simplify the use of the whole program and to avoid unecessary data handling, all three parts of the Lamcal program, meshgeneration, plotting and, FE, are combined into one load module. This setup allows to do all calculations in one single run. However, preprocessing, postprocessing and restarts can be made in separate runs as well. The same reserved space for the dynamic core storage is used in all three parts, if the available space is not sufficient the FE program will stop
11. Adaptive finite element methods for differential equations
Bangerth, Wolfgang
2003-01-01
These Lecture Notes discuss concepts of `self-adaptivity' in the numerical solution of differential equations, with emphasis on Galerkin finite element methods. The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order ...
12. Finite groups with three conjugacy class sizes of some elements
Conjugacy class sizes; p-nilpotent groups; finite groups. 1. Introduction. All groups ... group G has exactly two conjugacy class sizes of elements of prime power order. .... [5] Huppert B, Character Theory of Finite Groups, de Gruyter Exp. Math.
13. Investigating ASME allowable loads with finite element analyses
Mattar Neto, Miguel; Bezerra, Luciano M.; Miranda, Carlos A. de J.; Cruz, Julio R.B.
1995-01-01
The evaluation of nuclear components using finite element analysis (FEA) does not generally fall into the shell type verification adopted by the ASME Code. Consequently, the demonstration that the modes of failure are avoided sometimes is not straightforward. Allowable limits, developed by limit load theory, require the computation of shell membrane and bending stresses. How to calculate these stresses from FEA is not necessarily self-evident. One approach to be considered is to develop recommendations in a case-by-case basis for the most common pressure vessel geometries and loads based on comparison between the results of elastic and plastic FEA. In this paper, FE analyses of common 2D and complex 3D geometries are examined and discussed. It will be clear that in the cases studied, stress separation and categorization are not self-evident and simple tasks to undertake. Certain unclear recommendations of ASME Code can lead the stress analyst to non conservative designs as will be demonstrated in this paper. At the endo of this paper, taking into account comparison between elastic and elastic-plastic FE results from ANSYS some observations, suggestions and conclusions about the degree of conservatism of the ASME recommendations will be addressed. (author)
14. Finite element analysis theory and application with ANSYS
Moaveni, Saeed
2015-01-01
For courses in Finite Element Analysis, offered in departments of Mechanical or Civil and Environmental Engineering. While many good textbooks cover the theory of finite element modeling, Finite Element Analysis: Theory and Application with ANSYS is the only text available that incorporates ANSYS as an integral part of its content. Moaveni presents the theory of finite element analysis, explores its application as a design/modeling tool, and explains in detail how to use ANSYS intelligently and effectively. Teaching and Learning Experience This program will provide a better teaching and learning experience-for you and your students. It will help: *Present the Theory of Finite Element Analysis: The presentation of theoretical aspects of finite element analysis is carefully designed not to overwhelm students. *Explain How to Use ANSYS Effectively: ANSYS is incorporated as an integral part of the content throughout the book. *Explore How to Use FEA as a Design/Modeling Tool: Open-ended design problems help stude...
15. Impact of new computing systems on finite element computations
Noor, A.K.; Fulton, R.E.; Storaasi, O.O.
1983-01-01
Recent advances in computer technology that are likely to impact finite element computations are reviewed. The characteristics of supersystems, highly parallel systems, and small systems (mini and microcomputers) are summarized. The interrelations of numerical algorithms and software with parallel architectures are discussed. A scenario is presented for future hardware/software environment and finite element systems. A number of research areas which have high potential for improving the effectiveness of finite element analysis in the new environment are identified
16. The finite element method its basis and fundamentals
Zienkiewicz, Olek C; Zhu, JZ
2013-01-01
The Finite Element Method: Its Basis and Fundamentals offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples in the detail required for readers to apply the knowledge to their own engineering problems and understand more advanced applications. This edition sees a significant rearrangement of the book's content to enable clearer development of the finite element method, with major new chapters and sections added to cover: Weak forms Variational forms Multi-dimensional field prob
17. Parallel iterative procedures for approximate solutions of wave propagation by finite element and finite difference methods
Kim, S. [Purdue Univ., West Lafayette, IN (United States)
1994-12-31
Parallel iterative procedures based on domain decomposition techniques are defined and analyzed for the numerical solution of wave propagation by finite element and finite difference methods. For finite element methods, in a Lagrangian framework, an efficient way for choosing the algorithm parameter as well as the algorithm convergence are indicated. Some heuristic arguments for finding the algorithm parameter for finite difference schemes are addressed. Numerical results are presented to indicate the effectiveness of the methods.
18. Introduction to finite element analysis using MATLAB and Abaqus
Khennane, Amar
2013-01-01
There are some books that target the theory of the finite element, while others focus on the programming side of things. Introduction to Finite Element Analysis Using MATLAB(R) and Abaqus accomplishes both. This book teaches the first principles of the finite element method. It presents the theory of the finite element method while maintaining a balance between its mathematical formulation, programming implementation, and application using commercial software. The computer implementation is carried out using MATLAB, while the practical applications are carried out in both MATLAB and Abaqus. MA
19. Adaptive Smoothed Finite Elements (ASFEM) for history dependent material models
Quak, W.; Boogaard, A. H. van den
2011-01-01
A successful simulation of a bulk forming process with finite elements can be difficult due to distortion of the finite elements. Nodal smoothed Finite Elements (NSFEM) are an interesting option for such a process since they show good distortion insensitivity and moreover have locking-free behavior and good computational efficiency. In this paper a method is proposed which takes advantage of the nodally smoothed field. This method, named adaptive smoothed finite elements (ASFEM), revises the mesh for every step of a simulation without mapping the history dependent material parameters. In this paper an updated-Lagrangian implementation is presented. Several examples are given to illustrate the method and to show its properties.
20. Development and applications of a flat triangular element for thin laminated shells
Mohan, P.
Finite element analysis of thin laminated shells using a three-noded flat triangular shell element is presented. The flat shell element is obtained by combining the Discrete Kirchhoff Theory (DKT) plate bending element and a membrane element similar to the Allman element, but derived from the Linear Strain Triangular (LST) element. The major drawback of the DKT plate bending element is that the transverse displacement is not explicitly defined within the interior of the element. In the present research, free vibration analysis is performed both by using a lumped mass matrix and a so called consistent mass matrix, obtained by borrowing shape functions from an existing element, in order to compare the performance of the two methods. Several numerical examples are solved to demonstrate the accuracy of the formulation for both small and large rotation analysis of laminated plates and shells. The results are compared with those available in the existing literature and those obtained using the commercial finite element package ABAQUS and are found to be in good agreement. The element is employed for two main applications involving large flexible structures. The first application is the control of thermal deformations of a spherical mirror segment, which is a segment of a multi-segmented primary mirror used in a space telescope. The feasibility of controlling the surface distortions of the mirror segment due to arbitrary thermal fields, using discrete and distributed actuators, is studied. The second application is the analysis of an inflatable structure, being considered by the US Army for housing vehicles and personnel. The updated Lagrangian formulation of the flat shell element has been developed primarily for the nonlinear analysis of the tent structure, since such a structure is expected to undergo large deformations and rotations under the action of environmental loads like the wind and snow loads. The follower effects of the pressure load have been included in the
1. Finite element analysis of cylindrical pressure vessels having a misalignment in a circumferential joint
Aseer Brabin, T.; Christopher, T.; Nageswara Rao, B.
2010-01-01
Finite element analysis (FEA) has been carried out to obtain the elastic stress distribution at cylinder-to-cylinder junction in pressurized shell structures that have applications in space - vehicle design. To validate the finite element modeling and analysis results, three joint configurations, (viz., unfilleted butt joint with equal thickness, unfilleted butt joint with unequal thickness and filleted butt joint with equal thickness) having test results in open literature were considered. The peak stress values for these configurations obtained from FEA are close to that of test results. The peak stress value is found to reduce due to filleted butt joint as expected and also confirmed through test results.
2. The nonlinear finite element analysis program NUCAS (NUclear Containment Analysis System) for reinforced concrete containment building
Lee, Sang Jin; Lee, Hong Pyo; Seo, Jeong Moon [Korea Atomic Energy Research Institute, Taejeon (Korea)
2002-03-01
The maim goal of this research is to develop a nonlinear finite element analysis program NUCAS to accurately predict global and local failure modes of containment building subjected to internal pressure. In this report, we describe the techniques we developed throught this research. An adequate model to the analysis of containment building such as microscopic material model is adopted and it applied into the development Reissner-Mindlin degenerated shell element. To avoid finite element deficiencies, the substitute strains based on the assumed strain method is used in the shell formulation. Arc-length control method is also adopted to fully trace the peak load-displacement path due to crack formation. In addition, a benchmark test suite is developed to investigate the performance of NUCAS and proposed as the future benchmark tests for nonlinear analysis of reinforced concrete. Finally, the input format of NUCAS and the examples of input/output file are described. 39 refs., 65 figs., 8 tabs. (Author)
3. A New Triangular Flat Shell Element With Drilling Rotations
Damkilde, Lars
2008-01-01
A new flat triangular shell element has been developed based on a newly developed triangular plate bending element by the author and a new triangular membrane element with drilling degrees of freedom. The advantage of the drilling degree of freedom is that no special precautions have to be made...... in connecting with assembly of elements. Due to the drilling rotations all nodal degrees of freedom have stiffness, and therefore no artificial suppression of degrees of freedom are needed for flat or almost flat parts of the shell structure....
4. European column buckling curves and finite element modelling including high strength steels
Jönsson, Jeppe; Stan, Tudor-Cristian
2017-01-01
Eurocode allows for finite element modelling of plated steel structures, however the information in the code on how to perform the analysis or what assumptions to make is quite sparse. The present paper investigates the deterministic modelling of flexural column buckling using plane shell elements...... imperfections may be very conservative if considered by finite element analysis as described in the current Eurocode code. A suggestion is given for a slightly modified imperfection formula within the Ayrton-Perry formulation leading to adequate inclusion of modern high grade steels within the original four...... bucking curves. It is also suggested that finite element or frame analysis may be performed with equivalent column bow imperfections extracted directly from the Ayrton-Perry formulation....
5. Probabilistic finite element modeling of waste rollover
Khaleel, M.A.; Cofer, W.F.; Al-fouqaha, A.A.
1995-09-01
Stratification of the wastes in many Hanford storage tanks has resulted in sludge layers which are capable of retaining gases formed by chemical and/or radiolytic reactions. As the gas is produced, the mechanisms of gas storage evolve until the resulting buoyancy in the sludge leads to instability, at which point the sludge ''rolls over'' and a significant volume of gas is suddenly released. Because the releases may contain flammable gases, these episodes of release are potentially hazardous. Mitigation techniques are desirable for more controlled releases at more frequent intervals. To aid the mitigation efforts, a methodology for predicting of sludge rollover at specific times is desired. This methodology would then provide a rational basis for the development of a schedule for the mitigation procedures. In addition, a knowledge of the sensitivity of the sludge rollovers to various physical and chemical properties within the tanks would provide direction for efforts to reduce the frequency and severity of these events. In this report, the use of probabilistic finite element analyses for computing the probability of rollover and the sensitivity of rollover probability to various parameters is described
6. Finite element modelling of composite castellated beam
Frans Richard
2017-01-01
Full Text Available Nowadays, castellated beam becomes popular in building structural as beam members. This is due to several advantages of castellated beam such as increased depth without any additional mass, passing the underfloor service ducts without changing of story elevation. However, the presence of holes can develop various local effects such as local buckling, lateral torsional buckling caused by compression force at the flange section of the steel beam. Many studies have investigated the failure mechanism of castellated beam and one technique which can prevent the beam fall into local failure is the use of reinforced concrete slab as lateral support on castellated beam, so called composite castellated beam. Besides of preventing the local failure of castellated beam, the concrete slab can increase the plasticity moment of the composite castellated beam section which can deliver into increasing the ultimate load of the beam. The aim of this numerical studies of composite castellated beam on certain loading condition (monotonic quasi-static loading. ABAQUS was used for finite element modelling purpose and compared with the experimental test for checking the reliability of the model. The result shows that the ultimate load of the composite castellated beam reached 6.24 times than the ultimate load of the solid I beam and 1.2 times compared the composite beam.
7. Shakedown analysis by finite element incremental procedures
Borkowski, A.; Kleiber, M.
1979-01-01
8. TACO: a finite element heat transfer code
Mason, W.E. Jr.
1980-02-01
TACO is a two-dimensional implicit finite element code for heat transfer analysis. It can perform both linear and nonlinear analyses and can be used to solve either transient or steady state problems. Either plane or axisymmetric geometries can be analyzed. TACO has the capability to handle time or temperature dependent material properties and materials may be either isotropic or orthotropic. A variety of time and temperature dependent loadings and boundary conditions are available including temperature, flux, convection, and radiation boundary conditions and internal heat generation. Additionally, TACO has some specialized features such as internal surface conditions (e.g., contact resistance), bulk nodes, enclosure radiation with view factor calculations, and chemical reactive kinetics. A user subprogram feature allows for any type of functional representation of any independent variable. A bandwidth and profile minimization option is also available in the code. Graphical representation of data generated by TACO is provided by a companion post-processor named POSTACO. The theory on which TACO is based is outlined, the capabilities of the code are explained, the input data required to perform an analysis with TACO are described. Some simple examples are provided to illustrate the use of the code
9. Nonlinear finite element analysis of concrete structures
Ottosen, N.S.
1980-05-01
This report deals with nonlinear finite element analysis of concrete structures loaded in the short-term up until failure. A profound discussion of constitutive modelling on concrete is performed; a model, applicable for general stress states, is described and its predictions are compared with experimental data. This model is implemented in the AXIPLANE-program applicable for axisymmetrick and plane structures. The theoretical basis for this program is given. Using the AXIPLANE-program various concrete structures are analysed up until failure and compared with experimental evidence. These analyses include panels pressure vessel, beams failing in shear and finally a specific pull-out test, the Lok-Test, is considered. In these analyses, the influence of different failure criteria, aggregate interlock, dowel action, secondary cracking, magnitude of compressive strenght, magnitude of tensile strenght and of different post-failure behaviours of the concrete are evaluated. Moreover, it is shown that a suitable analysis of the theoretical data results in a clear insight into the physical behaviour of the considered structures. Finally, it is demonstrated that the AXISPLANE-program for widely different structures exhibiting very delicate structural aspects gives predictions that are in close agreement with experimental evidence. (author)
10. Finite element simulation for creep crack growth
Miyazaki, Noriyuki; Sasaki, Toru; Nakagaki, Michihiko; Brust, F.W.
1992-01-01
A finite element method was applied to a generation phase simulation of creep crack growth. Experimental data on creep crack growth in a 1Cr-1Mo-1/4V steel compact tension specimen were numerically simulated using a node-release technique and the variations of various fracture mechanics parameters such as CTOA, J, C * and T * during creep crack growth were calculated. The path-dependencies of the integral parameters J, C * and T * were also obtained to examine whether or not they could characterize the stress field near the tip of a crack propagating under creep condition. The following conclusions were obtained from the present analysis. (1) The J integral shows strong path-dependency during creep crack growth, so that it is does not characterize creep crack growth. (2) The C * integral shows path-dependency to some extent during creep crack growth even in the case of Norton type steady state creep law. Strictly speaking, we cannot use it as a fracture mechanics parameter characterizing creep crack growth. It is, however, useful from the practical viewpoint because it correlates well the rate of creep crack growth. (3) The T * integral shows good path-independency during creep crack growth. Therefore, it is a candidate for a fracture mechanics parameter characterizing creep crack growth. (author)
11. An efficient finite element solution for gear dynamics
Cooley, C G; Parker, R G; Vijayakar, S M
2010-01-01
A finite element formulation for the dynamic response of gear pairs is proposed. Following an established approach in lumped parameter gear dynamic models, the static solution is used as the excitation in a frequency domain solution of the finite element vibration model. The nonlinear finite element/contact mechanics formulation provides accurate calculation of the static solution and average mesh stiffness that are used in the dynamic simulation. The frequency domain finite element calculation of dynamic response compares well with numerically integrated (time domain) finite element dynamic results and previously published experimental results. Simulation time with the proposed formulation is two orders of magnitude lower than numerically integrated dynamic results. This formulation admits system level dynamic gearbox response, which may include multiple gear meshes, flexible shafts, rolling element bearings, housing structures, and other deformable components.
12. Finite element analysis of a finite-strain plasticity problem
Crose, J.G.; Fong, H.H.
1984-01-01
A finite-strain plasticity analysis was performed of an engraving process in a plastic rotating band during the firing of a gun projectile. The aim was to verify a nonlinear feature of the NIFDI/RB code: plastic large deformation analysis of nearly incompressible materials using a deformation theory of plasticity approach and a total Lagrangian scheme. (orig.)
13. A Finite Element Analysis of Optimal Variable Thickness Sheets
1996-01-01
A quasimixed Finite Element (FE) method for maximum stiffness of variablethickness sheets is analysed. The displacement is approximated with ninenode Lagrange quadrilateral elements and the thickness is approximated aselementwise constant. One is guaranteed that the FE displacement solutionswill ...
14. Mixed Element Formulation for the Finite Element-Boundary Integral Method
Meese, J; Kempel, L. C; Schneider, S. W
2006-01-01
A mixed element approach using right hexahedral elements and right prism elements for the finite element-boundary integral method is presented and discussed for the study of planar cavity-backed antennas...
15. A parallel finite element procedure for contact-impact problems using edge-based smooth triangular element and GPU
Cai, Yong; Cui, Xiangyang; Li, Guangyao; Liu, Wenyang
2018-04-01
The edge-smooth finite element method (ES-FEM) can improve the computational accuracy of triangular shell elements and the mesh partition efficiency of complex models. In this paper, an approach is developed to perform explicit finite element simulations of contact-impact problems with a graphical processing unit (GPU) using a special edge-smooth triangular shell element based on ES-FEM. Of critical importance for this problem is achieving finer-grained parallelism to enable efficient data loading and to minimize communication between the device and host. Four kinds of parallel strategies are then developed to efficiently solve these ES-FEM based shell element formulas, and various optimization methods are adopted to ensure aligned memory access. Special focus is dedicated to developing an approach for the parallel construction of edge systems. A parallel hierarchy-territory contact-searching algorithm (HITA) and a parallel penalty function calculation method are embedded in this parallel explicit algorithm. Finally, the program flow is well designed, and a GPU-based simulation system is developed, using Nvidia's CUDA. Several numerical examples are presented to illustrate the high quality of the results obtained with the proposed methods. In addition, the GPU-based parallel computation is shown to significantly reduce the computing time.
16. Hydrothermal analysis in engineering using control volume finite element method
Sheikholeslami, Mohsen
2015-01-01
Control volume finite element methods (CVFEM) bridge the gap between finite difference and finite element methods, using the advantages of both methods for simulation of multi-physics problems in complex geometries. In Hydrothermal Analysis in Engineering Using Control Volume Finite Element Method, CVFEM is covered in detail and applied to key areas of thermal engineering. Examples, exercises, and extensive references are used to show the use of the technique to model key engineering problems such as heat transfer in nanofluids (to enhance performance and compactness of energy systems),
17. finite element model for predicting residual stresses in shielded
eobe
This paper investigates the prediction of residual stresses developed ... steel plates through Finite Element Model simulation and experiments. ... The experimental values as measured by the X-Ray diffractometer were of ... Based on this, it can be concluded that Finite Element .... Comparison of Residual Stresses from X.
18. Parallel direct solver for finite element modeling of manufacturing processes
Nielsen, Chris Valentin; Martins, P.A.F.
2017-01-01
The central processing unit (CPU) time is of paramount importance in finite element modeling of manufacturing processes. Because the most significant part of the CPU time is consumed in solving the main system of equations resulting from finite element assemblies, different approaches have been...
19. A geometric toolbox for tetrahedral finite element partitions
Brandts, J.; Korotov, S.; Křížek, M.; Axelsson, O.; Karátson, J.
2011-01-01
In this work we present a survey of some geometric results on tetrahedral partitions and their refinements in a unified manner. They can be used for mesh generation and adaptivity in practical calculations by the finite element method (FEM), and also in theoretical finite element (FE) analysis.
20. An introduction to the UNCLE finite element scheme
Enderby, J.A.
1983-01-01
UNCLE is a completely general finite element scheme which provides common input, output, equation-solving and other facilities for a family of finite element codes for linear and non-linear stress analysis, heat transfer etc. This report describes the concepts on which UNCLE is based and gives a general account of the facilities provided. (author)
1. A simple finite element method for linear hyperbolic problems
Mu, Lin; Ye, Xiu
2017-01-01
Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.
2. Finite Element Modelling of Seismic Liquefaction in Soils
Galavi, V.; Petalas, A.; Brinkgreve, R.B.J.
2013-01-01
Numerical aspects of seismic liquefaction in soils as implemented in the finite element code, PLAXIS, is described in this paper. After description of finite element equations of dynamic problems, three practical dynamic boundary conditions, namely viscous boundary tractions, tied degrees of freedom
3. Analysis of Tube Drawing Process – A Finite Element Approach ...
In this paper the effect of die semi angle on drawing load in cold tube drawing has been investigated numerically using the finite element method. The equation governing the stress distribution was derived and solved using Galerkin finite element method. An isoparametric formulation for the governing equation was utilized ...
4. A finite element thermohydrodynamic analyis of profile bore bearing
Shah Nor bin Basri
1994-01-01
A finite element-based method is presented for analysing the thermohydrodynamic (THD) behaviour of profile bore bearing. A variational statement for the governing equation is derived and used to formulate a non-linear quadrilateral finite element of serendipity family. The predicted behaviour is compared with experimental evidence where possible and favorable correlation is obtained
5. Finite element simulation of laser transmission welding of dissimilar ...
user
materials between polyvinylidene fluoride and titanium ... finite element (FE) thermal model is developed to simulate the laser ... Keywords: Laser transmission welding, Temperature field, Weld dimension, Finite element analysis, Thermal modeling. 1. .... 4) The heating phenomena due to the phase changes are neglected.
6. A Note on Symplectic, Multisymplectic Scheme in Finite Element Method
GUO Han-Ying; JI Xiao-Mei; LI Yu-Qi; WU Ke
2001-01-01
We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimensional case respectively. These results are in fact the intrinsic reason why the numerical experiments show that such finite element algorithms are accurate in practice.``
7. Finite Element Analysis of Pipe T-Joint
P.M.Gedkar; Dr. D.V. Bhope
2012-01-01
This paper reports stress analysis of two pressurized cylindrical intersection using finite element method. The different combinations of dimensions of run pipe and the branch pipe are used to investigate thestresses in pipe at the intersection. In this study the stress analysis is accomplished by finite element package ANSYS.
8. An introduction to the UNCLE finite element scheme
Enderby, J A [UK Atomic Energy Authority, Northern Division, Risley Nuclear Power Development Establishment, Risley, Warrington (United Kingdom)
1983-05-01
UNCLE is a completely general finite element scheme which provides common input, output, equation-solving and other facilities for a family of finite element codes for linear and non-linear stress analysis, heat transfer etc. This report describes the concepts on which UNCLE is based and gives a general account of the facilities provided. (author)
9. THE PRACTICAL ANALYSIS OF FINITE ELEMENTS METHOD ERRORS
Natalia Bakhova
2011-03-01
Full Text Available Abstract. The most important in the practical plan questions of reliable estimations of finite elementsmethod errors are considered. Definition rules of necessary calculations accuracy are developed. Methodsand ways of the calculations allowing receiving at economical expenditures of computing work the best finalresults are offered.Keywords: error, given the accuracy, finite element method, lagrangian and hermitian elements.
10. Finite size effects of a pion matrix element
Guagnelli, M.; Jansen, K.; Palombi, F.; Petronzio, R.; Shindler, A.; Wetzorke, I.
2004-01-01
We investigate finite size effects of the pion matrix element of the non-singlet, twist-2 operator corresponding to the average momentum of non-singlet quark densities. Using the quenched approximation, they come out to be surprisingly large when compared to the finite size effects of the pion mass. As a consequence, simulations of corresponding nucleon matrix elements could be affected by finite size effects even stronger which could lead to serious systematic uncertainties in their evaluation
11. Advances in 3D electromagnetic finite element modeling
Nelson, E.M.
1997-01-01
Numerous advances in electromagnetic finite element analysis (FEA) have been made in recent years. The maturity of frequency domain and eigenmode calculations, and the growth of time domain applications is briefly reviewed. A high accuracy 3D electromagnetic finite element field solver employing quadratic hexahedral elements and quadratic mixed-order one-form basis functions will also be described. The solver is based on an object-oriented C++ class library. Test cases demonstrate that frequency errors less than 10 ppm can be achieved using modest workstations, and that the solutions have no contamination from spurious modes. The role of differential geometry and geometrical physics in finite element analysis is also discussed
12. Free vibration of thin axisymmetric structures by a semi-analytical finite element scheme and isoparametric solid elements
Akeju, T.A.I.; Kelly, D.W.; Zienkiewicz, O.C.; Kanaka Raju, K.
1981-01-01
The eigenvalue equations governing the free vibration of axisymmetric solids are derived by means of a semi-analytical finite element scheme. In particular we investigated the use of an 8-node solid element in structures which exhibit a 'shell-like' behaviour. Bathe-Wilson subspace iteration algorithm is employed for the solution of the equations. The element is shown to give good results for beam and shell vibration problems. It is also utilised to solve a complex solid in the form of an internal component of a modern jet engine. This particular application is of considerable practical importance as the dynamics of such components form a dominant design constraint. (orig./HP)
13. Finite element and boundary element applications in quantum mechanics
Ueta, Tsuyoshi
2003-01-01
Although this book is one of the Oxford Texts in Applied and Engineering Mathematics, we may think of it as a physics book. It explains how to solve the problem of quantum mechanics using the finite element method (FEM) and the boundary element method (BEM). Many examples analysing actual problems are also shown. As for the ratio of the number of pages of FEM and BEM, the former occupies about 80%. This is, however, reasonable reflecting the flexibility of FEM. Although many explanations of FEM and BEM exist, most are written using special mathematical expressions and numerical computation fields. However, this book is written in the 'language of physicists' throughout. I think that it is very readable and easy to understand for physicists. In the derivation of FEM and the argument on calculation accuracy, the action integral and a variation principle are used consistently. In the numerical computation of matrices, such as simultaneous equations and eigen value problems, a description of important points is also fully given. Moreover, the practical problems which become important in the electron device design field and the condensed matter physics field are dealt with as example computations, so that this book is very practical and applicable. It is characteristic and interesting that FEM is applied to solve the Schroedinger and Poisson equations consistently, and to the solution of the Ginzburg--Landau equation in superconductivity. BEM is applied to treat electric field enhancements due to surface plasmon excitations at metallic surfaces. A number of references are cited at the end of all the chapters, and this is very helpful. The description of quantum mechanics is also made appropriately and the actual application of quantum mechanics in condensed matter physics can also be surveyed. In the appendices, the mathematical foundation, such as numerical quadrature formulae and Green's functions, is conveniently described. I recommend this book to those who need to
14. Finite element simulation of the welding process and structural behaviour of welded components
Locci, J.M.; Rouvray, A. de; Barbe, B.; Poirier, J.
1977-01-01
In the field of inelastic analysis of nuclear metal structures, the computation of residual stresses in welds, and their effects on the strength of welded components is of major importance. This paper presents an experimentally checked finite element simulation with the general nonlinear program PAM NEP-D, of the electron beam welding of two thick hemispherical shells, and the behaviour of the welded sphere under various additional thermomechanical sollicitations. (Auth.)
15. Rib fractures under anterior-posterior dynamic loads: experimental and finite-element study.
Li, Zuoping; Kindig, Matthew W; Kerrigan, Jason R; Untaroiu, Costin D; Subit, Damien; Crandall, Jeff R; Kent, Richard W
2010-01-19
The purpose of this study was to investigate whether using a finite-element (FE) mesh composed entirely of hexahedral elements to model cortical and trabecular bone (all-hex model) would provide more accurate simulations than those with variable thickness shell elements for cortical bone and hexahedral elements for trabecular bone (hex-shell model) in the modeling human ribs. First, quasi-static non-injurious and dynamic injurious experiments were performed using the second, fourth, and tenth human thoracic ribs to record the structural behavior and fracture tolerance of individual ribs under anterior-posterior bending loads. Then, all-hex and hex-shell FE models for the three ribs were developed using an octree-based and multi-block hex meshing approach, respectively. Material properties of cortical bone were optimized using dynamic experimental data and the hex-shell model of the fourth rib and trabecular bone properties were taken from the literature. Overall, the reaction force-displacement relationship predicted by both all-hex and hex-shell models with nodes in the offset middle-cortical surfaces compared well with those measured experimentally for all the three ribs. With the exception of fracture locations, the predictions from all-hex and offset hex-shell models of the second and fourth ribs agreed better with experimental data than those from the tenth rib models in terms of reaction force at fracture (difference rib responses and bone fractures for the loading conditions considered, but coarse hex-shell models with constant or variable shell thickness were more computationally efficient and therefore preferred. Copyright 2009 Elsevier Ltd. All rights reserved.
16. High accuracy 3D electromagnetic finite element analysis
Nelson, E.M.
1996-01-01
A high accuracy 3D electromagnetic finite element field solver employing quadratic hexahedral elements and quadratic mixed-order one-form basis functions will be described. The solver is based on an object-oriented C++ class library. Test cases demonstrate that frequency errors less than 10 ppm can be achieved using modest workstations, and that the solutions have no contamination from spurious modes. The role of differential geometry and geometrical physics in finite element analysis will also be discussed
17. Finite element model for heat conduction in jointed rock masses
Gartling, D.K.; Thomas, R.K.
1981-01-01
A computatonal procedure for simulating heat conduction in a fractured rock mass is proposed and illustrated in the present paper. The method makes use of a simple local model for conduction in the vicinity of a single open fracture. The distributions of fractures and fracture properties within the finite element model are based on a statistical representation of geologic field data. Fracture behavior is included in the finite element computation by locating local, discrete fractures at the element integration points
18. High accuracy 3D electromagnetic finite element analysis
Nelson, Eric M.
1997-01-01
A high accuracy 3D electromagnetic finite element field solver employing quadratic hexahedral elements and quadratic mixed-order one-form basis functions will be described. The solver is based on an object-oriented C++ class library. Test cases demonstrate that frequency errors less than 10 ppm can be achieved using modest workstations, and that the solutions have no contamination from spurious modes. The role of differential geometry and geometrical physics in finite element analysis will also be discussed
19. A finite element calculation of flux pumping
Campbell, A. M.
2017-12-01
A flux pump is not only a fascinating example of the power of Faraday’s concept of flux lines, but also an attractive way of powering superconducting magnets without large electronic power supplies. However it is not possible to do this in HTS by driving a part of the superconductor normal, it must be done by exceeding the local critical density. The picture of a magnet pulling flux lines through the material is attractive, but as there is no direct contact between flux lines in the magnet and vortices, unless the gap between them is comparable to the coherence length, the process must be explicable in terms of classical electromagnetism and a nonlinear V-I characteristic. In this paper a simple 2D model of a flux pump is used to determine the pumping behaviour from first principles and the geometry. It is analysed with finite element software using the A formulation and FlexPDE. A thin magnet is passed across one or more superconductors connected to a load, which is a large rectangular loop. This means that the self and mutual inductances can be calculated explicitly. A wide strip, a narrow strip and two conductors are considered. Also an analytic circuit model is analysed. In all cases the critical state model is used, so the flux flow resistivity and dynamic resistivity are not directly involved, although an effective resistivity appears when J c is exceeded. In most of the cases considered here is a large gap between the theory and the experiments. In particular the maximum flux transferred to the load area is always less than the flux of the magnet. Also once the threshold needed for pumping is exceeded the flux in the load saturates within a few cycles. However the analytic circuit model allows a simple modification to allow for the large reduction in I c when the magnet is over a conductor. This not only changes the direction of the pumped flux but leads to much more effective pumping.
20. Finite Element Simulation of Blanking Process
Afzal Ahmed
2012-10-01
daya penembusan sebanyak 42%. Daya tebukan yang diukur melalui eksperimen dan simulasi kekal pada kira-kira 90kN melepasi penembusan punch sebanyak 62%. Apabila ketebalan keputusan kunci ditambah, ketinggian retak dikurangkan dan ini meningkatkan kualiti pengosongan.KEYWORDS: simulation; finite element simulation; blanking; computer aided manufacturing
1. Finite-element solidification modelling of metals and binary alloys
Mathew, P.M.
1986-12-01
In the Canadian Nuclear Fuel Waste Management Program, cast metals and alloys are being evaluated for their ability to support a metallic fuel waste container shell under disposal vault conditions and to determine their performance as an additional barrier to radionuclide release. These materials would be cast to fill residual free space inside the container and allowed to solidify without major voids. To model their solidification characteristics following casting, a finite-element model, FAXMOD-3, was adopted. Input parameters were modified to account for the latent heat of fusion of the metals and alloys considered. This report describes the development of the solidification model and its theoretical verification. To model the solidification of pure metals and alloys that melt at a distinct temperature, the latent heat of fusion was incorporated as a double-ramp function in the specific heat-temperature relationship, within an interval of +- 1 K around the solidification temperature. Comparison of calculated results for lead, tin and lead-tin eutectic melts, unidirectionally cooled with and without superheat, showed good agreement with an alternative technique called the integral profile method. To model the solidification of alloys that melt over a temperature interval, the fraction of solid in the solid-liquid region, as calculated from the Scheil equation, was used to determine the fraction of latent heat to be liberated over a temperature interval within the solid-liquid zone. Comparison of calculated results for unidirectionally cooled aluminum-4 wt.% copper melt, with and without superheat, showed good agreement with alternative finite-difference techniques
2. Mixed finite element - discontinuous finite volume element discretization of a general class of multicontinuum models
Ruiz-Baier, Ricardo; Lunati, Ivan
2016-10-01
We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi-Douglas-Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two- and three-dimensional porous media are presented to demonstrate the accuracy of the method (displaying an optimal convergence rate), the physics-preserving properties of the mixed-primal scheme, as well as the robustness of the method (which is successfully used to simulate diverse physical phenomena such as density fingering, Terzaghi's consolidation
3. Precise magnetostatic field using the finite element method
Nascimento, Francisco Rogerio Teixeira do
2013-01-01
The main objective of this work is to simulate electromagnetic fields using the Finite Element Method. Even in the easiest case of electrostatic and magnetostatic numerical simulation some problems appear when the nodal finite element is used. It is difficult to model vector fields with scalar functions mainly in non-homogeneous materials. With the aim to solve these problems two types of techniques are tried: the adaptive remeshing using nodal elements and the edge finite element that ensure the continuity of tangential components. Some numerical analysis of simple electromagnetic problems with homogeneous and non-homogeneous materials are performed using first, the adaptive remeshing based in various error indicators and second, the numerical solution of waveguides using edge finite element. (author)
4. Finite element formulation for a digital image correlation method
Sun Yaofeng; Pang, John H. L.; Wong, Chee Khuen; Su Fei
2005-01-01
A finite element formulation for a digital image correlation method is presented that will determine directly the complete, two-dimensional displacement field during the image correlation process on digital images. The entire interested image area is discretized into finite elements that are involved in the common image correlation process by use of our algorithms. This image correlation method with finite element formulation has an advantage over subset-based image correlation methods because it satisfies the requirements of displacement continuity and derivative continuity among elements on images. Numerical studies and a real experiment are used to verify the proposed formulation. Results have shown that the image correlation with the finite element formulation is computationally efficient, accurate, and robust
5. Review of Tomographic Imaging using Finite Element Method
2011-12-01
Full Text Available Many types of techniques for process tomography were proposed and developed during the past 20 years. This paper review the techniques and the current state of knowledge and experience on the subject, aimed at highlighting the problems associated with the non finite element methods, such as the ill posed, ill conditioned which relates to the accuracy and sensitivity of measurements. In this paper, considerations for choice of sensors and its applications were outlined and descriptions of non finite element tomography systems were presented. The finite element method tomography system as obtained from recent works, suitable for process control and measurement were also presented.
6. Finite element simulation and testing of ISW CFRP anchorage
Schmidt, Jacob Wittrup; Goltermann, Per; Hertz, Kristian Dahl
2013-01-01
is modelled in the 3D finite Element program ABAQUS, just as digital image correlation (DIC) testing was performed to verify the finite element simulation. Also a new optimized design was produced to ensure that the finite element simulation and anchorage behaviour correlated well. It is seen....... This paper presents a novel mechanical integrated sleeve wedge anchorage which seem very promising when perusing the scope of ultimate utilization of CFRP 8mm rods (with a tension capacity of approximately 140kN). Compression transverse to the CFRP is evaluated to prevent premature failure. The anchorage...
7. Magnetic materials and 3D finite element modeling
Bastos, Joao Pedro A
2014-01-01
Magnetic Materials and 3D Finite Element Modeling explores material characterization and finite element modeling (FEM) applications. This book relates to electromagnetic analysis based on Maxwell’s equations and application of the finite element (FE) method to low frequency devices. A great source for senior undergraduate and graduate students in electromagnetics, it also supports industry professionals working in magnetics, electromagnetics, ferromagnetic materials science and electrical engineering. The authors present current concepts on ferromagnetic material characterizations and losses. They provide introductory material; highlight basic electromagnetics, present experimental and numerical modeling related to losses and focus on FEM applied to 3D applications. They also explain various formulations, and discuss numerical codes.
8. A finite element conjugate gradient FFT method for scattering
Collins, Jeffery D.; Ross, Dan; Jin, J.-M.; Chatterjee, A.; Volakis, John L.
1991-01-01
Validated results are presented for the new 3D body of revolution finite element boundary integral code. A Fourier series expansion of the vector electric and mangnetic fields is employed to reduce the dimensionality of the system, and the exact boundary condition is employed to terminate the finite element mesh. The mesh termination boundary is chosen such that is leads to convolutional boundary operatores of low O(n) memory demand. Improvements of this code are discussed along with the proposed formulation for a full 3D implementation of the finite element boundary integral method in conjunction with a conjugate gradiant fast Fourier transformation (CGFFT) solution.
9. Development of polygon elements based on the scaled boundary finite element method
Chiong, Irene; Song Chongmin
2010-01-01
We aim to extend the scaled boundary finite element method to construct conforming polygon elements. The development of the polygonal finite element is highly anticipated in computational mechanics as greater flexibility and accuracy can be achieved using these elements. The scaled boundary polygonal finite element will enable new developments in mesh generation, better accuracy from a higher order approximation and better transition elements in finite element meshes. Polygon elements of arbitrary number of edges and order have been developed successfully. The edges of an element are discretised with line elements. The displacement solution of the scaled boundary finite element method is used in the development of shape functions. They are shown to be smooth and continuous within the element, and satisfy compatibility and completeness requirements. Furthermore, eigenvalue decomposition has been used to depict element modes and outcomes indicate the ability of the scaled boundary polygonal element to express rigid body and constant strain modes. Numerical tests are presented; the patch test is passed and constant strain modes verified. Accuracy and convergence of the method are also presented and the performance of the scaled boundary polygonal finite element is verified on Cook's swept panel problem. Results show that the scaled boundary polygonal finite element method outperforms a traditional mesh and accuracy and convergence are achieved from fewer nodes. The proposed method is also shown to be truly flexible, and applies to arbitrary n-gons formed of irregular and non-convex polygons.
10. Finite element formulation for fluid-structure interaction in three-dimensional space
Kulak, R.F.
1979-01-01
A development is presented for a three-dimension hexahedral hydrodynamic finite-element. Using trilinear shape functions and assuming a constant pressure field in each element, simple relations were obtained for internal nodal forces. Because the formulation was based upon a rate approach it was applicable to problems involving large displacements. This element was incorporated into an existing plate-shell finite element code. Diagonal mass matrices were used and the resulting discrete equations of motion were solved using explicit temporal integrator. Results for several problems were presented which compare numerical predictions to closed form analytical solutions. In addition, the fluid-structure interaction problem of a fluid-filled, cylindrical vessel containing internal cylinders was studied. The internal cylinders were cantilever supported from the top cover of the vessel and were periodically located circumferentially at a fixed radius. A pressurized cylindrical cavity located at the bottom of the vessel at its centerline provided the loading
11. Modelling optimization involving different types of elements in finite element analysis
Wai, C M; Rivai, Ahmad; Bapokutty, Omar
2013-01-01
Finite elements are used to express the mechanical behaviour of a structure in finite element analysis. Therefore, the selection of the elements determines the quality of the analysis. The aim of this paper is to compare and contrast 1D element, 2D element, and 3D element used in finite element analysis. A simple case study was carried out on a standard W460x74 I-beam. The I-beam was modelled and analyzed statically with 1D elements, 2D elements and 3D elements. The results for the three separate finite element models were compared in terms of stresses, deformation and displacement of the I-beam. All three finite element models yield satisfactory results with acceptable errors. The advantages and limitations of these elements are discussed. 1D elements offer simplicity although lacking in their ability to model complicated geometry. 2D elements and 3D elements provide more detail yet sophisticated results which require more time and computer memory in the modelling process. It is also found that the choice of element in finite element analysis is influence by a few factors such as the geometry of the structure, desired analysis results, and the capability of the computer
12. Complex finite element sensitivity method for creep analysis
Gomez-Farias, Armando; Montoya, Arturo; Millwater, Harry
2015-01-01
The complex finite element method (ZFEM) has been extended to perform sensitivity analysis for mechanical and structural systems undergoing creep deformation. ZFEM uses a complex finite element formulation to provide shape, material, and loading derivatives of the system response, providing an insight into the essential factors which control the behavior of the system as a function of time. A complex variable-based quadrilateral user element (UEL) subroutine implementing the power law creep constitutive formulation was incorporated within the Abaqus commercial finite element software. The results of the complex finite element computations were verified by comparing them to the reference solution for the steady-state creep problem of a thick-walled cylinder in the power law creep range. A practical application of the ZFEM implementation to creep deformation analysis is the calculation of the skeletal point of a notched bar test from a single ZFEM run. In contrast, the standard finite element procedure requires multiple runs. The value of the skeletal point is that it identifies the location where the stress state is accurate, regardless of the certainty of the creep material properties. - Highlights: • A novel finite element sensitivity method (ZFEM) for creep was introduced. • ZFEM has the capability to calculate accurate partial derivatives. • ZFEM can be used for identification of the skeletal point of creep structures. • ZFEM can be easily implemented in a commercial software, e.g. Abaqus. • ZFEM results were shown to be in excellent agreement with analytical solutions
13. Validation of High Displacement Piezoelectric Actuator Finite Element Models
Taleghani, B. K.
2000-01-01
The paper presents the results obtained by using NASTRAN(Registered Trademark) and ANSYS(Regitered Trademark) finite element codes to predict doming of the THUNDER piezoelectric actuators during the manufacturing process and subsequent straining due to an applied input voltage. To effectively use such devices in engineering applications, modeling and characterization are essential. Length, width, dome height, and thickness are important parameters for users of such devices. Therefore, finite element models were used to assess the effects of these parameters. NASTRAN(Registered Trademark) and ANSYS(Registered Trademark) used different methods for modeling piezoelectric effects. In NASTRAN(Registered Trademark), a thermal analogy was used to represent voltage at nodes as equivalent temperatures, while ANSYS(Registered Trademark) processed the voltage directly using piezoelectric finite elements. The results of finite element models were validated by using the experimental results.
14. Finite element model updating using bayesian framework and modal properties
Marwala, T
2005-01-01
Full Text Available Finite element (FE) models are widely used to predict the dynamic characteristics of aerospace structures. These models often give results that differ from measured results and therefore need to be updated to match measured results. Some...
15. Finite element discretization of Darcy's equations with pressure dependent porosity
Girault, Vivette; Murat, Franç ois; Salgado, Abner
2010-01-01
We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and
16. Finite Element Crash Simulations and Impact-Induced Injuries
Jaroslav Mackerle
1999-01-01
Full Text Available This bibliography lists references to papers, conference proceedings and theses/dissertations dealing with finite element simulations of crashes, impact-induced injuries and their protection that were published in 1980–1998. 390 citations are listed.
17. Generalized multiscale finite element method. Symmetric interior penalty coupling
Efendiev, Yalchin R.; Galvis, Juan; Lazarov, Raytcho D.; Moon, M.; Sarkis, Marcus V.
2013-01-01
Motivated by applications to numerical simulations of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose two different finite element spaces on the coarse mesh. The first space is based on a local eigenvalue problem that uses an interior weighted L2-norm and a boundary weighted L2-norm for computing the "mass" matrix. The second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension. The approximation with these multiscale spaces is based on the discontinuous Galerkin finite element method framework. We investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples. © 2013 Elsevier Inc.
18. Finite element analysis of rotating beams physics based interpolation
Ganguli, Ranjan
2017-01-01
This book addresses the solution of rotating beam free-vibration problems using the finite element method. It provides an introduction to the governing equation of a rotating beam, before outlining the solution procedures using Rayleigh-Ritz, Galerkin and finite element methods. The possibility of improving the convergence of finite element methods through a judicious selection of interpolation functions, which are closer to the problem physics, is also addressed. The book offers a valuable guide for students and researchers working on rotating beam problems – important engineering structures used in helicopter rotors, wind turbines, gas turbines, steam turbines and propellers – and their applications. It can also be used as a textbook for specialized graduate and professional courses on advanced applications of finite element analysis.
19. Optical strain measurements and its finite element analysis of cold ...
International Journal of Engineering, Science and Technology ... Online video images of square grid were recorded during the deformation ... Finite element software ANSYS has been applied for the analysis of the upset forming process.
20. Finite element analyses for RF photoinjector gun cavities
Marhauser, F.
2006-01-01
This paper details electromagnetical, thermal and structural 3D Finite Element Analyses (FEA) for normal conducting RF photoinjector gun cavities. The simulation methods are described extensively. Achieved results are presented. (orig.)
1. Generalized multiscale finite element method. Symmetric interior penalty coupling
Efendiev, Yalchin R.
2013-12-01
Motivated by applications to numerical simulations of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose two different finite element spaces on the coarse mesh. The first space is based on a local eigenvalue problem that uses an interior weighted L2-norm and a boundary weighted L2-norm for computing the "mass" matrix. The second choice is based on generation of a snapshot space and subsequent selection of a subspace of a reduced dimension. The approximation with these multiscale spaces is based on the discontinuous Galerkin finite element method framework. We investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples. © 2013 Elsevier Inc.
2. Finite element model to study calcium distribution in oocytes ...
2015-03-20
Mar 20, 2015 ... Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal 462051 ... finite element method has been employed to obtain the solution. ..... Nelson MT, Cheng H, Rubart M. Relaxation of arterial smooth.
3. Finite element concept to derive isostatic residual maps ...
A new space-domain operator based on the shape function concept of finite element analysis has been developed to derive the ... not require explicit assumptions on isostatic models. Besides .... This information is implicit in the Bouguer ...
4. Finite element analyses for RF photoinjector gun cavities
Marhauser, F. [Berliner Elektronenspeicherring-Gesellschaft fuer Synchrotronstrahlung mbH (BESSY), Berlin (Germany)
2006-07-01
This paper details electromagnetical, thermal and structural 3D Finite Element Analyses (FEA) for normal conducting RF photoinjector gun cavities. The simulation methods are described extensively. Achieved results are presented. (orig.)
5. Implementation of a high performance parallel finite element micromagnetics package
Scholz, W.; Suess, D.; Dittrich, R.; Schrefl, T.; Tsiantos, V.; Forster, H.; Fidler, J.
2004-01-01
A new high performance scalable parallel finite element micromagnetics package has been implemented. It includes solvers for static energy minimization, time integration of the Landau-Lifshitz-Gilbert equation, and the nudged elastic band method
6. Finite element analysis of thermal stress distribution in different ...
Nigerian Journal of Clinical Practice • Jan-Feb 2016 • Vol 19 • Issue 1. Abstract ... Key words: Amalgam, finite element method, glass ionomer cement, resin composite, thermal stress ... applications for force analysis and assessment of different.
7. Finite element analysis of thermal stress distribution in different ...
Nigerian Journal of Clinical Practice. Journal Home ... Von Mises and thermal stress distributions were evaluated. Results: In all ... distribution. Key words: Amalgam, finite element method, glass ionomer cement, resin composite, thermal stress ...
8. Comparison of different precondtioners for nonsymmtric finite volume element methods
Mishev, I.D.
1996-12-31
We consider a few different preconditioners for the linear systems arising from the discretization of 3-D convection-diffusion problems with the finite volume element method. Their theoretical and computational convergence rates are compared and discussed.
9. Application of Mass Lumped Higher Order Finite Elements
J. Chen, H.R. Strauss, S.C. Jardin, W. Park, L.E. Sugiyama, G. Fu, J. Breslau
2005-01-01
There are many interesting phenomena in extended-MHD such as anisotropic transport, mhd, 2-fluid effects stellarator and hot particles. Any one of them challenges numerical analysts, and researchers are seeking for higher order methods, such as higher order finite difference, higher order finite elements and hp/spectral elements. It is true that these methods give more accurate solution than their linear counterparts. However, numerically they are prohibitively expensive. Here we give a successful solution of this conflict by applying mass lumped higher order finite elements. This type of elements not only keep second/third order accuracy but also scale closely to linear elements by doing mass lumping. This is especially true for second order lump elements. Full M3D and anisotropic transport models are studied
10. The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion
Moszo, P.; Kristek, J.; Galis, M.; Pazak, P.; Balazovijech, M.
2006-01-01
Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth's structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finite-difference method is the dominant method in the modeling of earthquake motion. Moreover, it is becoming more important in the seismic exploration and structural modeling. At the same time we are convinced that the best time of the finite-difference method in seismology is in the future. This monograph provides tutorial and detailed introduction to the application of the finite-difference, finite-element, and hybrid finite-difference-finite-element methods to the modeling of seismic wave propagation and earthquake motion. The text does not cover all topics and aspects of the methods. We focus on those to which we have contributed. (Author)
11. Implicit three-dimensional finite-element formulation for the nonlinear structural response of reactor components
Kulak, R.F.; Belytschko, T.B.
1975-09-01
The formulation of a finite-element procedure for the implicit transient and static analysis of plate/shell type structures in three-dimensional space is described. The triangular plate/shell element can sustain both membrane and bending stresses. Both geometric and material nonlinearities can be treated, and an elastic-plastic material law has been incorporated. The formulation permits the element to undergo arbitrarily large rotations and translations; but, in its present form it is restricted to small strains. The discretized equations of motion are obtained by a stiffness method. An implicit integration algorithm based on trapezoidal integration formulas is used to integrate the discretized equations of motion in time. To ensure numerical stability, an iterative solution procedure with equilibrium checks is used
12. Mathematical aspects of finite element methods for incompressible viscous flows
Gunzburger, M. D.
1986-01-01
Mathematical aspects of finite element methods are surveyed for incompressible viscous flows, concentrating on the steady primitive variable formulation. The discretization of a weak formulation of the Navier-Stokes equations are addressed, then the stability condition is considered, the satisfaction of which insures the stability of the approximation. Specific choices of finite element spaces for the velocity and pressure are then discussed. Finally, the connection between different weak formulations and a variety of boundary conditions is explored.
13. Finite element modeling of the filament winding process using ABAQUS
Miltenberger, Louis C.
1992-01-01
A comprehensive stress model of the filament winding fabrication process, previously implemented in the finite element program, WACSAFE, was implemented using the ABAQUS finite element software package. This new implementation, referred to as the ABWACSAFE procedure, consists of the ABAQUS software and a pre/postprocessing routine that was developed to prepare necessary ABAQUS input files and process ABAQUS displacement results for stress and strain computation. The ABWACSAF...
14. Thermal stresses in rectangular plates: variational and finite element solutions
Laura, P.A.A.; Gutierrez, R.H.; Sanchez Sarmiento, G.; Basombrio, F.G.
1978-01-01
This paper deals with the development of an approximate method for the analysis of thermal stresses in rectangular plates (plane stress problem) and an evaluation of the relative accuracy of the finite element method. The stress function is expanded in terms of polynomial coordinate functions which identically satisfy the boundary conditions, and a variational approach is used to determine the expansion coefficients. The results are in good agreement with a finite element approach. (Auth.)
15. A finite element primer for beginners the basics
Zohdi, Tarek I
2014-01-01
The purpose of this primer is to provide the basics of the Finite Element Method, primarily illustrated through a classical model problem, linearized elasticity. The topics covered are:(1) Weighted residual methods and Galerkin approximations,(2) A model problem for one-dimensional?linear elastostatics,(3) Weak formulations in one dimension,(4) Minimum principles in one dimension,(5) Error estimation in one dimension,(5) Construction of Finite Element basis functions in one dimension,(6) Gaussian Quadrature,(7) Iterative solvers and element by element data structures,(8) A model problem for th
16. A finite element solution method for quadrics parallel computer
Zucchini, A.
1996-08-01
A distributed preconditioned conjugate gradient method for finite element analysis has been developed and implemented on a parallel SIMD Quadrics computer. The main characteristic of the method is that it does not require any actual assembling of all element equations in a global system. The physical domain of the problem is partitioned in cells of n p finite elements and each cell element is assigned to a different node of an n p -processors machine. Element stiffness matrices are stored in the data memory of the assigned processing node and the solution process is completely executed in parallel at element level. Inter-element and therefore inter-processor communications are required once per iteration to perform local sums of vector quantities between neighbouring elements. A prototype implementation has been tested on an 8-nodes Quadrics machine in a simple 2D benchmark problem
17. Comparison of 3-D finite elements for incompressible fluid flow
Robichaud, M.; Tanguy, P.A.
1985-01-01
In recent years, the finite element method applied to the solution of incompressible fluid flow has been in constant evolution. In the present state-of-the-art, 2-D problems are solved routinely and reliable results are obtained at a reasonable cost. In 3-D the finite element method is still undergoing active research and many methods have been proposed to solve the Navier-Stokes equations at 'low cost'. These methods have in common the choice of the element which has a trilinear velocity and a discontinuous constant pressure (Q1-PO). The prohibitive cost of 3-D finite element method in fluid flow is the reason for this choice: the Q1-PO is the simplest and the cheapest 3-D element. However, as mentioned in (5) and (6), it generates 'spurious' pressure modes phenomenon called checkerboarding. On regular mesh these spurious modes can be filtered but on distorted mesh the pressure solution is meaningless. (author)
18. Finite element analysis of the Girkmann problem using the modern hp-version and the classical h-version
Niemi, Antti
2011-06-03
We perform finite element analysis of the so called Girkmann problem in structural mechanics. The problem involves an axially symmetric spherical shell stiffened with a foot ring and is approached (1) by using the axisymmetric formulation of linear elasticity theory and (2) by using a dimensionally reduced shell-ring model. In the first approach the problem is solved with a fully automatic hp-adaptive finite element solver whereas the classical h-version of the finite element method is used in the second approach. We study the convergence behaviour of the different numerical models and show that accurate stress resultants can be obtained with both models by using effective post-processing formulas. © Springer-Verlag London Limited 2011.
19. A fluid-solid finite element method for the analysis of reactor safety problems
Mitra, Santanu; Kumar, Ashutosh; Sinhamahapatra, K.P.
2006-01-01
The work presented herein can broadly be categorized as a fluid-structure interaction problem. The response of a circular cylindrical structure subjected to cross flow is examined using the finite element method for both the liquid and the structure domains. The cylindrical tube is mounted elastically at the ends and is free to move under the action of the unsteady flow-induced forces. The fluid is considered to be acoustic compressible and viscous. A Galerkin finite element method implemented on a triangular mesh is used to solve the time-dependent Navier-Stokes equations. The cylinder motion is modeled using a five-degrees of freedom generalized shell element structural dynamics model. The numerical simulations of the response of the calandria tubes/pressure tubes, adjustor rod and shut-off rod of a nuclear reactor are presented. A few typical results are presented to assess the accuracy and applicability of the developed modules
20. Numerical experiment on finite element method for matching data
Tokuda, Shinji; Kumakura, Toshimasa; Yoshimura, Koichi.
1993-03-01
Numerical experiments are presented on the finite element method by Pletzer-Dewar for matching data of an ordinary differential equation with regular singular points by using model equation. Matching data play an important role in nonideal MHD stability analysis of a magnetically confined plasma. In the Pletzer-Dewar method, the Frobenius series for the 'big solution', the fundamental solution which is not square-integrable at the regular singular point, is prescribed. The experiments include studies of the convergence rate of the matching data obtained by the finite element method and of the effect on the results of computation by truncating the Frobenius series at finite terms. It is shown from the present study that the finite element method is an effective method for obtaining the matching data with high accuracy. (author)
1. MESHREF, Finite Elements Mesh Combination with Renumbering
1973-01-01
1 - Nature of physical problem solved: The program can assemble different meshes stored on tape or cards. Renumbering is performed in order to keep band width low. Voids and/ or local refinement are possible. 2 - Method of solution: Topology and geometry are read according to input specifications. Abundant nodes and elements are eliminated. The new topology and geometry are stored on tape. 3 - Restrictions on the complexity of the problem: Maximum number of nodes = 2000. Maximum number of elements = 1500
2. Applications of the fundamental solution for a thermal shock on a finite orthotropic cylindrical thin shell
Woo, H.K.; Huang, C.L.D.
1979-01-01
The authors investigate the temperature variations in a thin cylindrical shell of graphite materials with finite length, subjected to an instantaneous thermal shock. The solutions for the line source and the area source of thermal shock are obtained. Quasi-linear theory for heat transfer is assumed. Grades ATJ and ZTA graphite are used in the numerical examples. As is expected, the orthotropically thermal properties significantly affect the temperature variations in the shell due to the thermal shocks. (Auth.)
3. Robust mixed finite element methods to deal with incompressibility in finite strain in an industrial framework
Al-Akhrass, Dina
2014-01-01
Simulations in solid mechanics exhibit several difficulties, as dealing with incompressibility, with nonlinearities due to finite strains, contact laws, or constitutive laws. The basic motivation of our work is to propose efficient finite element methods capable of dealing with incompressibility in finite strain context, and using elements of low order. During the three last decades, many approaches have been proposed in the literature to overcome the incompressibility problem. Among them, mixed formulations offer an interesting theoretical framework. In this work, a three-field mixed formulation (displacement, pressure, volumetric strain) is investigated. In some cases, this formulation can be condensed in a two-field (displacement - pressure) mixed formulation. However, it is well-known that the discrete problem given by the Galerkin finite element technique, does not inherit the 'inf-sup' stability condition from the continuous problem. Hence, the interpolation orders in displacement and pressure have to be chosen in a way to satisfy the Brezzi-Babuska stability conditions when using Galerkin approaches. Interpolation orders must be chosen so as to satisfy this condition. Two possibilities are considered: to use stable finite element satisfying this requirement, or to use finite element that does not satisfy this condition, and to add terms stabilizing the FE Galerkin formulation. The latter approach allows the use of equal order interpolation. In this work, stable finite element P2/P1 and P2/P1/P1 are used as reference, and compared to P1/P1 and P1/P1/P1 formulations stabilized with a bubble function or with a VMS method (Variational Multi-Scale) based on a sub-grid-space orthogonal to the FE space. A finite strain model based on logarithmic strain is selected. This approach is extended to three and two field mixed formulations with stable or stabilized elements. These approaches are validated on academic cases and used on industrial cases. (author)
4. A Finite Element Model for convection-dominatel transport problems
Carmo, E.G.D. do; Galeao, A.C.N.R.
1987-08-01
A new Protev-Galerkin Finite Element Model which automatically incorporates the search for the appropriate upwind direction is presented. It is also shown that modifying the Petrov-Galerkin weightin functions associated with elements adjascent to downwing boudaries effectively eliminates numerical oscillations normally obtained near boundary layers. (Author) [pt
5. Stress distributions in finite element analysis of concrete gravity dam ...
Gravity dams are solid structures built of mass concrete material; they maintain their stability against the design loads from the geometric shape, the mass, and the strength of the concrete. The model was meshed with an 8-node biquadratic plane strain quadrilateral (CPE8R) elements, using ABAQUS, a finite element ...
6. Finite element stress analysis of brick-mortar masonry under ...
Stress analysis of a brick-mortar couplet as a substitute for brick wall structure has been performed by finite element method, and algorithm for determining the element stiffness matrix for a plane stress problem using the displacement approach was developed. The nodal displacements were derived for the stress in each ...
7. Behaviour of Lagrangian triangular mixed fluid finite elements
The behaviour of mixed fluid finite elements, formulated based on the Lagrangian frame of reference, is investigated to understand the effects of locking due to incompressibility and irrotational constraints. For this purpose, both linear and quadratic mixed triangular fluid elements are formulated. It is found that there exists a ...
8. Modelling Convergence of Finite Element Analysis of Cantilever Beam
Convergence studies are carried out by investigating the convergence of numerical results as the number of elements is increased. If convergence is not obtained, the engineer using the finite element method has absolutely no indication whether the results are indicative of a meaningful approximation to the correct solution ...
9. Coupling of smooth particle hydrodynamics with the finite element method
Attaway, S.W.; Heinstein, M.W.; Swegle, J.W.
1994-01-01
A gridless technique called smooth particle hydrodynamics (SPH) has been coupled with the transient dynamics finite element code ppercase[pronto]. In this paper, a new weighted residual derivation for the SPH method will be presented, and the methods used to embed SPH within ppercase[pronto] will be outlined. Example SPH ppercase[pronto] calculations will also be presented. One major difficulty associated with the Lagrangian finite element method is modeling materials with no shear strength; for example, gases, fluids and explosive biproducts. Typically, these materials can be modeled for only a short time with a Lagrangian finite element code. Large distortions cause tangling of the mesh, which will eventually lead to numerical difficulties, such as negative element area or ''bow tie'' elements. Remeshing will allow the problem to continue for a short while, but the large distortions can prevent a complete analysis. SPH is a gridless Lagrangian technique. Requiring no mesh, SPH has the potential to model material fracture, large shear flows and penetration. SPH computes the strain rate and the stress divergence based on the nearest neighbors of a particle, which are determined using an efficient particle-sorting technique. Embedding the SPH method within ppercase[pronto] allows part of the problem to be modeled with quadrilateral finite elements, while other parts are modeled with the gridless SPH method. SPH elements are coupled to the quadrilateral elements through a contact-like algorithm. ((orig.))
10. A cohesive finite element formulation for modelling fracture and ...
cohesive elements experience material softening and lose their stress carrying capacity. A few simple ..... In the present work, a Lagrangian finite element procedure is employed. In this formu clation ...... o, is related to 'c o by,. 't o='c o ¼ 1 ہ. 1.
11. Containment penetration design and analysis by finite element methods
Perry, R.F.; Rigamonti, G.; Dainora, J.
1975-01-01
Containment penetration designs which provide complete support to process piping containing high pressure and high temperature fluids and which do not employ cooling coils, require special provisions to sustain loadings associated with normal/abnormal conditions and to limit maximum temperature transmitted to the containment concrete wall. In order to accommodate piping imposed loads and fluid temperatures within code and regulatory limitations, the containment penetration designs require careful analysis of two critical regions: the portion of the penetration sleeve which is exposed to containment ambient conditions and the portion of the penetration which connects the sleeve to process piping (flued head). The length and thickness of the sleeve must be designed to provide maximum heat dissipation to the atmosphere and minimum heat conduction through the sleeve to meet concrete temperature limitations. The sleeve must have the capability to transmit the postulated piping loads to concrete embedments in the containment shell. The penetration flued head design must be strong enough to transfer high mechanical loads and be flexible enough to accommodate the thermal stresses generated by the high temperature fluid. Analytical models using finite element representations of process piping, penetration flued head, and exposed sleeve were employed to investigate the penetration assembly design. By application of flexible multi-step analyses, different penetration configurations were evaluated to determine the effects of key design parameters. Among the parameters studied were flued head profiles, flued head angles with the process piping, sleeve length and wall thickness. Special designs employing fins welded to the sleeve to lower the temperature at the concrete wall interface were investigated and fin geometry effects reported
12. High accuracy 3D electromagnetic finite element analysis
Nelson, E.M.
1997-01-01
A high accuracy 3D electromagnetic finite element field solver employing quadratic hexahedral elements and quadratic mixed-order one-form basis functions will be described. The solver is based on an object-oriented C++ class library. Test cases demonstrate that frequency errors less than 10 ppm can be achieved using modest workstations, and that the solutions have no contamination from spurious modes. The role of differential geometry and geometrical physics in finite element analysis will also be discussed. copyright 1997 American Institute of Physics
13. On the integration of computer aided design and analysis using the finite element absolute nodal coordinate formulation
Sanborn, Graham G.; Shabana, Ahmed A.
2009-01-01
For almost a decade, the finite element absolute nodal coordinate formulation (ANCF) has been used for both geometry and finite element representations. Because of the ANCF isoparametric property in the cases of beams, plates and shells, ANCF finite elements lend themselves easily to the geometric description of curves and surfaces, as demonstrated in the literature. The ANCF finite elements, therefore, are ideal for what is called isogeometric analysis that aims at the integration ofcomputer aided designandanalysis (ICADA), which involves the integration of what is now split into the separate fields of computer aided design (CAD) and computer aided analysis (CAA). The purpose of this investigation is to establish the relationship between the B-spline and NURBS, which are widely used in the geometric modeling, and the ANCF finite elements. It is shown in this study that by using the ANCF finite elements, one can in a straightforward manner obtain the control point representation required for the Bezier, B-spline and NURBS geometry. To this end, a coordinate transformation is used to write the ANCF gradient vectors in terms of control points. Unifying the CAD and CAA will require the use of such coordinate transformations and their inverses in order to transform control points to position vector gradients which are required for the formulation of the element transformations in the case of discontinuities as well as the formulation of the strain measures and the stress forces based on general continuum mechanics theory. In particular, fully parameterized ANCF finite elements can be very powerful in describing curve, surface, and volume geometry, and they can be effectively used to describe discontinuities while maintaining the many ANCF desirable features that include a constant mass matrix, zero Coriolis and centrifugal forces, no restriction on the amount of rotation or deformation within the finite element, ability for straightforward implementation of general
14. Finite element approximation to the even-parity transport equation
Lewis, E.E.
1981-01-01
This paper studies the finite element method, a procedure for reducing partial differential equations to sets of algebraic equations suitable for solution on a digital computer. The differential equation is cast into the form of a variational principle, the resulting domain then subdivided into finite elements. The dependent variable is then approximated by a simple polynomial, and these are linked across inter-element boundaries by continuity conditions. The finite element method is tailored to a variety of transport problems. Angular approximations are formulated, and the extent of ray effect mitigation is examined. Complex trial functions are introduced to enable the inclusion of buckling approximations. The ubiquitous curved interfaces of cell calculations, and coarse mesh methods are also treated. A concluding section discusses limitations of the work to date and suggests possible future directions
15. Linear finite element method for one-dimensional diffusion problems
Brandao, Michele A.; Dominguez, Dany S.; Iglesias, Susana M., E-mail: micheleabrandao@gmail.com, E-mail: dany@labbi.uesc.br, E-mail: smiglesias@uesc.br [Universidade Estadual de Santa Cruz (LCC/DCET/UESC), Ilheus, BA (Brazil). Departamento de Ciencias Exatas e Tecnologicas. Laboratorio de Computacao Cientifica
2011-07-01
We describe in this paper the fundamentals of Linear Finite Element Method (LFEM) applied to one-speed diffusion problems in slab geometry. We present the mathematical formulation to solve eigenvalue and fixed source problems. First, we discretized a calculus domain using a finite set of elements. At this point, we obtain the spatial balance equations for zero order and first order spatial moments inside each element. Then, we introduce the linear auxiliary equations to approximate neutron flux and current inside the element and architect a numerical scheme to obtain the solution. We offer numerical results for fixed source typical model problems to illustrate the method's accuracy for coarse-mesh calculations in homogeneous and heterogeneous domains. Also, we compare the accuracy and computational performance of LFEM formulation with conventional Finite Difference Method (FDM). (author)
16. A preliminary investigation of finite-element modeling for composite rotor blades
Lake, Renee C.; Nixon, Mark W.
1988-01-01
The results from an initial phase of an in-house study aimed at improving the dynamic and aerodynamic characteristics of composite rotor blades through the use of elastic couplings are presented. Large degree of freedom shell finite element models of an extension twist coupled composite tube were developed and analyzed using MSC/NASTRAN. An analysis employing a simplified beam finite element representation of the specimen with the equivalent engineering stiffness was additionally performed. Results from the shell finite element normal modes and frequency analysis were compared to those obtained experimentally, showing an agreement within 13 percent. There was appreciable degradation in the frequency prediction for the torsional mode, which is elastically coupled. This was due to the absence of off-diagonal coupling terms in the formulation of the equivalent engineering stiffness. Parametric studies of frequency variation due to small changes in ply orientation angle and ply thickness were also performed. Results showed linear frequency variations less than 2 percent per 1 degree variation in the ply orientation angle, and 1 percent per 0.0001 inch variation in the ply thickness.
17. On Using Particle Finite Element for Hydrodynamics Problems Solving
E. V. Davidova
2015-01-01
Full Text Available The aim of the present research is to develop software for the Particle Finite Element Method (PFEM and its verification on the model problem of viscous incompressible flow simulation in a square cavity. The Lagrangian description of the medium motion is used: the nodes of the finite element mesh move together with the fluid that allows to consider them as particles of the medium. Mesh cells deform when in time-stepping procedure, so it is necessary to reconstruct the mesh to provide stability of the finite element numerical procedure.Meshing algorithm allows us to obtain the mesh, which satisfies the Delaunay criteria: it is called \\the possible triangles method". This algorithm is based on the well-known Fortune method of Voronoi diagram constructing for a certain set of points in the plane. The graphical representation of the possible triangles method is shown. It is suitable to use generalization of Delaunay triangulation in order to construct meshes with polygonal cells in case of multiple nodes close to be lying on the same circle.The viscous incompressible fluid flow is described by the Navier | Stokes equations and the mass conservation equation with certain initial and boundary conditions. A fractional steps method, which allows us to avoid non-physical oscillations of the pressure, provides the timestepping procedure. Using the finite element discretization and the Bubnov | Galerkin method allows us to carry out spatial discretization.For form functions calculation of finite element mesh with polygonal cells, \
18. Finite Element Analysis of Circular Plate using SolidWorks
Kang, Yeo Jin; Jhung, Myung Jo
2011-01-01
Circular plates are used extensively in mechanical engineering for nuclear reactor internal components. The examples in the reactor vessel internals are upper guide structure support plate, fuel alignment plate, lower support plate etc. To verify the structural integrity of these plates, the finite element analyses are performed, which require the development of the finite element model. Sometimes it is very costly and time consuming to make the model especially for the beginners who start their engineering job for the structural analysis, necessitating a simple method to develop the finite element model for the pursuing structural analysis. Therefore in this study, the input decks are generated for the finite element analysis of a circular plate as shown in Fig. 1, which can be used for the structural analysis such as modal analysis, response spectrum analysis, stress analysis, etc using the commercial program Solid Works. The example problems are solved and the results are included for analysts to perform easily the finite element analysis of the mechanical plate components due to various loadings. The various results presented in this study would be helpful not only for the benchmark calculations and results comparisons but also as a part of the knowledge management for the future generation of young designers, scientists and computer analysts
19. Two-dimensional isostatic meshes in the finite element method
Martínez Marín, Rubén; Samartín, Avelino
2002-01-01
In a Finite Element (FE) analysis of elastic solids several items are usually considered, namely, type and shape of the elements, number of nodes per element, node positions, FE mesh, total number of degrees of freedom (dot) among others. In this paper a method to improve a given FE mesh used for a particular analysis is described. For the improvement criterion different objective functions have been chosen (Total potential energy and Average quadratic error) and the number of nodes and dof's...
20. Validation Assessment of a Glass-to-Metal Seal Finite-Element Model
Jamison, Ryan Dale [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Buchheit, Thomas E. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Emery, John M [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Romero, Vicente J. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Stavig, Mark E. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Newton, Clay S. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Brown, Arthur [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2017-10-01
Sealing glasses are ubiquitous in high pressure and temperature engineering applications, such as hermetic feed-through electrical connectors. A common connector technology are glass-to-metal seals where a metal shell compresses a sealing glass to create a hermetic seal. Though finite-element analysis has been used to understand and design glass-to-metal seals for many years, there has been little validation of these models. An indentation technique was employed to measure the residual stress on the surface of a simple glass-to-metal seal. Recently developed rate- dependent material models of both Schott 8061 and 304L VAR stainless steel have been applied to a finite-element model of the simple glass-to-metal seal. Model predictions of residual stress based on the evolution of material models are shown. These model predictions are compared to measured data. Validity of the finite- element predictions is discussed. It will be shown that the finite-element model of the glass-to-metal seal accurately predicts the mean residual stress in the glass near the glass-to-metal interface and is valid for this quantity of interest.
1. Relativistic quantum chemistry of the superheavy elements. Closed-shell element 114 as a case study
Schwerdtfeger, Peter; Seth, Michael
2002-01-01
The chemistry of superheavy element 114 is reviewed. The ground state of element 114 is closed shell [112]7s 2 7p 1/2 2 and shows a distinct chemical inertness (low reactivity). This inertness makes it rather difficult to study the atom-at-a-time chemistry of 114 in the gas or liquid phase. (author)
2. Hualien forced vibration calculation with a finite element model
Wang, F.; Gantenbein, F.; Nedelec, M.; Duretz, Ch.
1995-01-01
The forced vibration tests of the Hualien mock-up were useful to validate finite element models developed for soil-structure interaction. In this paper the two sets of tests with and without backfill were analysed. the methods used are based on finite element modeling for the soil. Two approaches were considered: calculation of soil impedance followed by the calculation of the transfer functions with a model taking into account the superstructure and the impedance; direct calculation of the soil-structure transfer functions, with the soil and the structure being represented in the same model by finite elements. Blind predictions and post-test calculations are presented and compared with the test results. (author). 4 refs., 8 figs., 2 tabs
3. Engineering computation of structures the finite element method
Neto, Maria Augusta; Roseiro, Luis; Cirne, José; Leal, Rogério
2015-01-01
This book presents theories and the main useful techniques of the Finite Element Method (FEM), with an introduction to FEM and many case studies of its use in engineering practice. It supports engineers and students to solve primarily linear problems in mechanical engineering, with a main focus on static and dynamic structural problems. Readers of this text are encouraged to discover the proper relationship between theory and practice, within the finite element method: Practice without theory is blind, but theory without practice is sterile. Beginning with elasticity basic concepts and the classical theories of stressed materials, the work goes on to apply the relationship between forces, displacements, stresses and strains on the process of modeling, simulating and designing engineered technical systems. Chapters discuss the finite element equations for static, eigenvalue analysis, as well as transient analyses. Students and practitioners using commercial FEM software will find this book very helpful. It us...
4. Finite Element Residual Stress Analysis of Planetary Gear Tooth
Jungang Wang
2013-01-01
Full Text Available A method to simulate residual stress field of planetary gear is proposed. In this method, the finite element model of planetary gear is established and divided to tooth zone and profile zone, whose different temperature field is set. The gear's residual stress simulation is realized by the thermal compression stress generated by the temperature difference. Based on the simulation, the finite element model of planetary gear train is established, the dynamic meshing process is simulated, and influence of residual stress on equivalent stress of addendum, pitch circle, and dedendum of internal and external meshing planetary gear tooth profile is analyzed, according to non-linear contact theory, thermodynamic theory, and finite element theory. The results show that the equivalent stresses of planetary gear at both meshing and nonmeshing surface are significantly and differently reduced by residual stress. The study benefits fatigue cracking analysis and dynamic optimization design of planetary gear train.
5. Probabilistic finite elements for transient analysis in nonlinear continua
Liu, W. K.; Belytschko, T.; Mani, A.
1985-01-01
The probabilistic finite element method (PFEM), which is a combination of finite element methods and second-moment analysis, is formulated for linear and nonlinear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in nonlinear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem. The moments calculated compare favorably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.
6. Analytical and finite element modeling of grounding systems
Luz, Mauricio Valencia Ferreira da [University of Santa Catarina (UFSC), Florianopolis, SC (Brazil)], E-mail: mauricio@grucad.ufsc.br; Dular, Patrick [University of Liege (Belgium). Institut Montefiore], E-mail: Patrick.Dular@ulg.ac.be
2007-07-01
Grounding is the art of making an electrical connection to the earth. This paper deals with the analytical and finite element modeling of grounding systems. An electrokinetic formulation using a scalar potential can benefit from floating potentials to define global quantities such as electric voltages and currents. The application concerns a single vertical grounding with one, two and three-layer soil, where the superior extremity stays in the surface of the soil. This problem has been modeled using a 2D axi-symmetric electrokinetic formulation. The grounding resistance obtained by finite element method is compared with the analytical one for one-layer soil. With the results of this paper it is possible to show that finite element method is a powerful tool in the analysis of the grounding systems in low frequencies. (author)
7. Flow Applications of the Least Squares Finite Element Method
Jiang, Bo-Nan
1998-01-01
The main thrust of the effort has been towards the development, analysis and implementation of the least-squares finite element method (LSFEM) for fluid dynamics and electromagnetics applications. In the past year, there were four major accomplishments: 1) special treatments in computational fluid dynamics and computational electromagnetics, such as upwinding, numerical dissipation, staggered grid, non-equal order elements, operator splitting and preconditioning, edge elements, and vector potential are unnecessary; 2) the analysis of the LSFEM for most partial differential equations can be based on the bounded inverse theorem; 3) the finite difference and finite volume algorithms solve only two Maxwell equations and ignore the divergence equations; and 4) the first numerical simulation of three-dimensional Marangoni-Benard convection was performed using the LSFEM.
8. Finite element simulation of ironing process under warm conditions
2014-01-01
Full Text Available Metal forming is one of the most important steps in manufacturing of a large variety of products. Ironing in deep drawing is done by adjusting the clearance between the punch and the die and allow the material flow over the punch. In the present investigation effect of extent of ironing behavior on the characteristics of the product like thickness distribution with respect to temperature was studied. With the help of finite element simulation using explicit finite element code LS-DYNA the stress in the drawn cup were predicted in the drawn cup. To increase the accuracy in the simulation process, numbers of integration points were increased in the thickness direction and it was found that there is very close prediction of finite element results to that of experimental ones.
9. The Finite Element Numerical Modelling of 3D Magnetotelluric
Ligang Cao
2014-01-01
Full Text Available The ideal numerical simulation of 3D magnetotelluric was restricted by the methodology complexity and the time-consuming calculation. Boundary values, the variation of weighted residual equation, and the hexahedral mesh generation method of finite element are three major causes. A finite element method for 3D magnetotelluric numerical modeling is presented in this paper as a solution for the problem mentioned above. In this algorithm, a hexahedral element coefficient matrix for magnetoelluric finite method is developed, which solves large-scale equations using preconditioned conjugate gradient of the first-type boundary conditions. This algorithm is verified using the homogeneous model, and the positive landform model, as well as the low resistance anomaly model.
10. Adaptive finite-element ballooning analysis of bipolar ionized fields
Al-Hamouz, Z.M.
1995-01-01
This paper presents an adaptive finite-element iterative method for the analysis of the ionized field around high-voltage bipolar direct-current (HVDC) transmission line conductors without resort to Deutsch's assumption. A new iterative finite-element ballooning technique is proposed to solve Poisson's equation wherein the commonly used artificial boundary around the transmission line conductors is simulated at infinity. Unlike all attempts reported in the literature for the solution of ionized field, the constancy of the conductors' surface field at the corona onset value is directly implemented in the finite-element formulation. In order to investigate the effectiveness of the proposed method, a laboratory model was built. It has been found that the calculated V-I characteristics and the ground-plane current density agreed well with those measured experimentally. The simplicity in computer programming in addition to the low number of iterations required to achieve convergence characterize this method of analysis
11. Geometrically Nonlinear Analysis of Shell Structures Using Flat DKT Shell Elements.
1985-11-22
In general 1r is a curved surface and the exact expressions of f1 e I are not simpler than f e 1. In fact they are theorically identical when the...1982. [23] Zienkiewicz, 0. C., The Finite Element Method (3rd Edition), McGraw-Hill, 1977. [24] Bergan, P. G., Holand , I., Soreide, T. H., "Use of
12. Nonlinear finite element formulation for analyzing shape memory alloy cylindrical panels
Mirzaeifar, R; Shakeri, M; Sadighi, M
2009-01-01
In this paper, a general incremental displacement based finite element formulation capable of modeling material nonlinearities based on first-order shear deformation theory (FSDT) is developed for cylindrical shape memory alloy (SMA) shells. The Boyd–Lagoudas phenomenological model with polynomial hardening in conjunction with 3D incremental convex cutting plane explicit algorithm is implemented for preparing the SMA constitutive model in the finite element formulation. Several numerical examples are presented for demonstrating the performance of the proposed formulation in stress, deflection and phase transformation analysis of pseudoelastic behavior of shape memory cylindrical panels with various boundary conditions. Also, it is shown that the presented formulation can be implemented for studying plates and beams with rectangular cross section
13. Matlab and C programming for Trefftz finite element methods
Qin, Qing-Hua
2008-01-01
Although the Trefftz finite element method (FEM) has become a powerful computational tool in the analysis of plane elasticity, thin and thick plate bending, Poisson's equation, heat conduction, and piezoelectric materials, there are few books that offer a comprehensive computer programming treatment of the subject. Collecting results scattered in the literature, MATLAB® and C Programming for Trefftz Finite Element Methods provides the detailed MATLAB® and C programming processes in applications of the Trefftz FEM to potential and elastic problems. The book begins with an introduction to th
14. Stochastic Finite Elements in Reliability-Based Structural Optimization
Sørensen, John Dalsgaard; Engelund, S.
1995-01-01
Application of stochastic finite elements in structural optimization is considered. It is shown how stochastic fields modelling e.g. the modulus of elasticity can be discretized in stochastic variables and how a sensitivity analysis of the reliability of a structural system with respect to optimi......Application of stochastic finite elements in structural optimization is considered. It is shown how stochastic fields modelling e.g. the modulus of elasticity can be discretized in stochastic variables and how a sensitivity analysis of the reliability of a structural system with respect...... to optimization variables can be performed. A computer implementation is described and an illustrative example is given....
15. FINITE ELEMENT MODELING OF THIN CIRCULAR SANDWICH PLATES DEFLECTION
K. S. Kurachka
2014-01-01
Full Text Available A mathematical model of a thin circular sandwich plate being under the vertical load is proposed. The model employs the finite element method and takes advantage of an axisymmetric finite element that leads to the small dimension of the resulting stiffness matrix and sufficient accuracy for practical calculations. The analytical expressions for computing local stiffness matrices are found, which can significantly speed up the process of forming the global stiffness matrix and increase the accuracy of calculations. A software is under development and verification. The discrepancy between the results of the mathematical model and those of analytical formulas for homogeneous thin circularsandwich plates does not exceed 7%.
16. Preconditioning for Mixed Finite Element Formulations of Elliptic Problems
Wildey, Tim; Xue, Guangri
2013-01-01
In this paper, we discuss a preconditioning technique for mixed finite element discretizations of elliptic equations. The technique is based on a block-diagonal approximation of the mass matrix which maintains the sparsity and positive definiteness of the corresponding Schur complement. This preconditioner arises from the multipoint flux mixed finite element method and is robust with respect to mesh size and is better conditioned for full permeability tensors than a preconditioner based on a diagonal approximation of the mass matrix. © Springer-Verlag Berlin Heidelberg 2013.
17. Finite element solution of two dimensional time dependent heat equation
Maaz
1999-01-01
A Microsoft Windows based computer code, named FHEAT, has been developed for solving two dimensional heat problems in Cartesian and Cylindrical geometries. The programming language is Microsoft Visual Basic 3.0. The code makes use of Finite element formulation for spatial domain and Finite difference formulation for time domain. Presently the code is capable of solving two dimensional steady state and transient problems in xy- and rz-geometries. The code is capable excepting both triangular and rectangular elements. Validation and benchmarking was done against hand calculations and published results. (author)
18. Stress analysis of heated concrete using finite elements
Majumdar, P.; Gupta, A.; Marchertas, A.
1994-01-01
Described is a finite element analysis of concrete, which is subjected to rapid heating. Using thermal mass transport calculation, the moisture content, temperature and pore pressure distribution over space and time is obtained first. From these effects, stress at various points of the concrete are computed using the finite element method. Contribution to the stress formulation comes from three components, namely the thermal expansion, pore pressure, and the shrinkage of concrete due to moisture loss (from dehydration). The material properties of concrete are assumed to be homogeneous, elastic, and cracking is not taken into consideration. (orig.)
19. COMPUTER EXPERIMENTS WITH FINITE ELEMENTS OF HIGHER ORDER
Khomchenko A.
2017-12-01
Full Text Available The paper deals with the problem of constructing the basic functions of a quadrilateral finite element of the fifth order by the means of the computer algebra system Maple. The Lagrangian approximation of such a finite element contains 36 nodes: 20 nodes perimeter and 16 internal nodes. Alternative models with reduced number of internal nodes are considered. Graphs of basic functions and cognitive portraits of lines of zero level are presented. The work is aimed at studying the possibilities of using modern information technologies in the teaching of individual mathematical disciplines.
20. Stochastic Finite Elements in Reliability-Based Structural Optimization
Sørensen, John Dalsgaard; Engelund, S.
Application of stochastic finite elements in structural optimization is considered. It is shown how stochastic fields modelling e.g. the modulus of elasticity can be discretized in stochastic variables and how a sensitivity analysis of the reliability of a structural system with respect to optimi......Application of stochastic finite elements in structural optimization is considered. It is shown how stochastic fields modelling e.g. the modulus of elasticity can be discretized in stochastic variables and how a sensitivity analysis of the reliability of a structural system with respect...
1. Fourier analysis of finite element preconditioned collocation schemes
Deville, Michel O.; Mund, Ernest H.
1990-01-01
The spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes is investigated. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the tranverse direction) of the 2-D Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.
2. Finite Element Method Based Modeling of Resistance Spot-Welded Mild Steel
Miloud Zaoui
Full Text Available Abstract This paper deals with Finite Element refined and simplified models of a mild steel spot-welded specimen, developed and validated based on quasi-static cross-tensile experimental tests. The first model was constructed with a fine discretization of the metal sheet and the spot weld was defined as a special geometric zone of the specimen. This model provided, in combination with experimental tests, the input data for the development of the second model, which was constructed with respect to the mesh size used in the complete car finite element model. This simplified model was developed with coarse shell elements and a spring-type beam element was used to model the spot weld behavior. The global accuracy of the two models was checked by comparing simulated and experimental load-displacement curves and by studying the specimen deformed shapes and the plastic deformation growth in the metal sheets. The obtained results show that both fine and coarse finite element models permit a good prediction of the experimental tests.
3. A new periodic imperfect quasi axisymmetric shell element
Combescure, A.; Garuti, G.
1983-08-01
The object of this paper is to give the formulation and the validation of a ''quasi axisymmetric'' shell element: the main idea is to develop the theory of an imperfect quasi axisymmetric shell element. The imperfection is a variation of the circumferential radius of curvature rsub(theta). The equations are obtained by transporting the equilibrium equations from the actual geometry onto the theoretical axisymmetric (rsub(theta)=r 0 geometry. It is shown that the main hypothesis convenient to perform simply this transformation is that the membrane strains associated with that variation of geometry are less than 1% (that is always the case if you suppose that the imperfect structure is obtained from the perfect one by an inextensional displacement field). The formulation of the element is given in the general case. The rigidity matrices, are given in the particular case in which the imperfection has a component on a single Fourier harmonic. The comparison of theoretical and computed, 3D and quasi axisymmetric, solution or a very simple case shows the influence of the number of the Fourier harmonics chosen on the response of the structure. The influence of the initial imperfections on the natural frequency are studied with element and compared with 3D calculations. Comparison of 3D, quasi axisymmetric, and analytical buckling loads are given and explained. This element gives a very efficient tool for the calculation of thin shells of revolution (which are always imperfect) and especially unables easy parametric study of the variation of the buckling load and eigen frequencies with the amplitude and shapes of non axisymmetric imperfections
4. A multiscale mortar multipoint flux mixed finite element method
Wheeler, Mary Fanett
2012-02-03
In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. Some superconvergence results are also derived. The algebraic system is reduced via a non-overlapping domain decomposition to a coarse scale mortar interface problem that is solved using a multiscale flux basis. Numerical experiments are presented to confirm the theory and illustrate the efficiency and flexibility of the method. © EDP Sciences, SMAI, 2012.
5. Evaluation of Concrete Cylinder Tests Using Finite Elements
Saabye Ottosen, Niels
1984-01-01
Nonlinear axisymmetric finite element analyses are performed on the uniaxial compressive test of concrete cylinders. The models include thick steel loading plates, and cylinders with height‐to‐diameter ratios (h/d) ranging from 1‐3 are treated. A simple constitutive model of the concrete is emplo......Nonlinear axisymmetric finite element analyses are performed on the uniaxial compressive test of concrete cylinders. The models include thick steel loading plates, and cylinders with height‐to‐diameter ratios (h/d) ranging from 1‐3 are treated. A simple constitutive model of the concrete...... uniaxial strength the use of geometrically matched loading plates seems to be advantageous. Finally, it is observed that for variations of the element size within limits otherwise required to obtain a realistic analysis, the results are insensitive to the element size....
6. Finite Element Simulation of the Shear Effect of Ultrasonic on Heat Exchanger Descaling
Lu, Shaolv; Wang, Zhihua; Wang, Hehui
2018-03-01
The shear effect on the interface of metal plate and its attached scale is an important mechanism of ultrasonic descaling, which is caused by the different propagation speed of ultrasonic wave in two different mediums. The propagating of ultrasonic wave on the shell is simulated based on the ANSYS/LS-DYNA explicit dynamic analysis. The distribution of shear stress in different paths under ultrasonic vibration is obtained through the finite element analysis and it reveals the main descaling mechanism of shear effect. The simulation result is helpful and enlightening to the reasonable design and the application of the ultrasonic scaling technology on heat exchanger.
7. A finite element code for electric motor design
Campbell, C. Warren
1994-01-01
FEMOT is a finite element program for solving the nonlinear magnetostatic problem. This version uses nonlinear, Newton first order elements. The code can be used for electric motor design and analysis. FEMOT can be embedded within an optimization code that will vary nodal coordinates to optimize the motor design. The output from FEMOT can be used to determine motor back EMF, torque, cogging, and magnet saturation. It will run on a PC and will be available to anyone who wants to use it.
8. A finite element field solver for dipole modes
Nelson, E.M.
1992-01-01
A finite element field solver for dipole modes in axisymmetric structures has been written. The second-order elements used in this formulation yield accurate mode frequencies with no spurious modes. Quasi-periodic boundaries are included to allow travelling waves in periodic structures. The solver is useful in applications requiring precise frequency calculations such as detuned accelerator structures for linear colliders. Comparisons are made with measurements and with the popular but less accurate field solver URMEL. (author). 7 refs., 4 figs
9. Finite elements for the thermomechanical calculation of massive structures
Argyris, J.H.; Szimmat, J.; Willam, K.J.
1978-01-01
The paper examines the fine element analysis of thermal stress and deformation problems in massive structures. To this end compatible idealizations are utilized for heat conduction and static analysis in order to minimize the data transfer. For transient behaviour due to unsteady heat flow and/or inelastics material processes the two computational parts are interwoven in form of an integrated software package for finite element analysis of thermomechanical problems in space and time. (orig.) [de
10. Nonlinear Finite Element Analysis of Pull-Out Test
Saabye Ottesen, N
1981-01-01
A specific pull-out test used to determine in-situ concrete compressive strength is analyzed. This test consists of a steel disc that is extracted from the structure. The finite element analysis considers cracking as well as strain hardening and softening in the pre- and post-failure region...
11. Piezoelectric Accelerometers Modification Based on the Finite Element Method
Liu, Bin; Kriegbaum, B.
2000-01-01
The paper describes the modification of piezoelectric accelerometers using a Finite Element (FE) method. Brüel & Kjær Accelerometer Type 8325 is chosen as an example to illustrate the advanced accelerometer development procedure. The deviation between the measurement and FE simulation results...
12. Optimization of forging processes using finite element simulations
Bonte, M.H.A.; Fourment, Lionel; Do, Tien-tho; van den Boogaard, Antonius H.; Huetink, Han
2010-01-01
During the last decades, simulation software based on the Finite Element Method (FEM) has significantly contributed to the design of feasible forming processes. Coupling FEM to mathematical optimization algorithms offers a promising opportunity to design optimal metal forming processes rather than
13. Finite element method for solving neutron transport problems
Ferguson, J.M.; Greenbaum, A.
1984-01-01
A finite element method is introduced for solving the neutron transport equations. Our method falls into the category of Petrov-Galerkin solution, since the trial space differs from the test space. The close relationship between this method and the discrete ordinate method is discussed, and the methods are compared for simple test problems
14. Reliability-Based Shape Optimization using Stochastic Finite Element Methods
Enevoldsen, Ib; Sørensen, John Dalsgaard; Sigurdsson, G.
1991-01-01
stochastic fields (e.g. loads and material parameters such as Young's modulus and the Poisson ratio). In this case stochastic finite element techniques combined with FORM analysis can be used to obtain measures of the reliability of the structural systems, see Der Kiureghian & Ke (6) and Liu & Der Kiureghian...
15. Finite element concept to derive isostatic residual maps
A new space-domain operator based on the shape function concept of finite element analysis has been developed to derive the residual maps of the Gorda Plate of western United States. The technique does not require explicit assumptions on isostatic models. Besides delineating the Gorda Plate boundary, the residual ...
16. Total hip reconstruction in acetabular dysplasia : a finite element study
Schüller, H.M.; Dalstra, M.; Huiskes, H.W.J.; Marti, R.K.
1993-01-01
In acetabular dysplasia, fixation of the acetabular component of a cemented total hip prosthesis may be insecure and superolateral bone grafts are often used to augment the acetabular roof. We used finite element analysis to study the mechanical importance of the lateral acetabular roof and found
17. A mixed finite element method for particle simulation in lasertron
Le Meur, G.
1987-03-01
A particle simulation code is being developed with the aim to treat the motion of charged particles in electromagnetic devices, such as Lasertron. The paper describes the use of mixed finite element methods in computing the field components, without derivating them from scalar or vector potentials. Graphical results are shown
18. Steam generator tube rupture simulation using extended finite element method
Mohanty, Subhasish, E-mail: smohanty@anl.gov; Majumdar, Saurin; Natesan, Ken
2016-08-15
Highlights: • Extended finite element method used for modeling the steam generator tube rupture. • Crack propagation is modeled in an arbitrary solution dependent path. • The FE model is used for estimating the rupture pressure of steam generator tubes. • Crack coalescence modeling is also demonstrated. • The method can be used for crack modeling of tubes under severe accident condition. - Abstract: A steam generator (SG) is an important component of any pressurized water reactor. Steam generator tubes represent a primary pressure boundary whose integrity is vital to the safe operation of the reactor. SG tubes may rupture due to propagation of a crack created by mechanisms such as stress corrosion cracking, fatigue, etc. It is thus important to estimate the rupture pressures of cracked tubes for structural integrity evaluation of SGs. The objective of the present paper is to demonstrate the use of extended finite element method capability of commercially available ABAQUS software, to model SG tubes with preexisting flaws and to estimate their rupture pressures. For the purpose, elastic–plastic finite element models were developed for different SG tubes made from Alloy 600 material. The simulation results were compared with experimental results available from the steam generator tube integrity program (SGTIP) sponsored by the United States Nuclear Regulatory Commission (NRC) and conducted at Argonne National Laboratory (ANL). A reasonable correlation was found between extended finite element model results and experimental results.
19. FINELM: a multigroup finite element diffusion code. Part II
Davierwalla, D.M.
1981-05-01
The author presents the axisymmetric case in cylindrical coordinates for the finite element multigroup neutron diffusion code, FINELM. The numerical acceleration schemes incorporated viz. the Lebedev extrapolations and the coarse mesh rebalancing, space collapsing, are discussed. A few benchmark computations are presented as validation of the code. (Auth.)
20. Nonlinear nonstationary analysis with the finite element method
Vaz, L.E.
1981-01-01
In this paper, after some introductory remarks on numerical methods for the integration of initial value problems, the applicability of the finite element method for transient diffusion analysis as well as dynamic and inelastic analysis is discussed, and some examples are presented. (RW) [de
1. A particle finite element method for machining simulations
Sabel, Matthias; Sator, Christian; Müller, Ralf
2014-07-01
The particle finite element method (PFEM) appears to be a convenient technique for machining simulations, since the geometry and topology of the problem can undergo severe changes. In this work, a short outline of the PFEM-algorithm is given, which is followed by a detailed description of the involved operations. The -shape method, which is used to track the topology, is explained and tested by a simple example. Also the kinematics and a suitable finite element formulation are introduced. To validate the method simple settings without topological changes are considered and compared to the standard finite element method for large deformations. To examine the performance of the method, when dealing with separating material, a tensile loading is applied to a notched plate. This investigation includes a numerical analysis of the different meshing parameters, and the numerical convergence is studied. With regard to the cutting simulation it is found that only a sufficiently large number of particles (and thus a rather fine finite element discretisation) leads to converged results of process parameters, such as the cutting force.
2. Possibilities of Particle Finite Element Methods in Industrial Forming Processes
Oliver, J.; Cante, J. C.; Weyler, R.; Hernandez, J.
2007-04-01
The work investigates the possibilities offered by the particle finite element method (PFEM) in the simulation of forming problems involving large deformations, multiple contacts, and new boundaries generation. The description of the most distinguishing aspects of the PFEM, and its application to simulation of representative forming processes, illustrate the proposed methodology.
3. The future of the finite element method in geotechnics
Brinkgreve, R.B.J.
2012-01-01
In this presentation a vision is given on tlie fiiture of the finite element method (FEM) for geotechnical engineering and design. In the past 20 years the FEM has proven to be a powerful method for estimating deformation, stability and groundwater flow in geoteclmical stmctures. Much has been
4. Design, development and use of the finite element machine
Adams, L. M.; Voigt, R. C.
1983-01-01
Some of the considerations that went into the design of the Finite Element Machine, a research asynchronous parallel computer are described. The present status of the system is also discussed along with some indication of the type of results that were obtained.
5. Aranha: a 2D mesh generator for triangular finite elements
Fancello, E.A.; Salgado, A.C.; Feijoo, R.A.
1990-01-01
A method for generating unstructured meshes for linear and quadratic triangular finite elements is described in this paper. Some topics on the C language data structure used in the development of the program Aranha are also presented. The applicability for adaptive remeshing is shown and finally several examples are included to illustrate the performance of the method in irregular connected planar domains. (author)
6. 3D finite element simulation of optical modes in VCSELs
Rozova, M.; Pomplun, J.; Zschiedrich, L.; Schmidt, F.; Burger, S.
2011-01-01
We present a finite element method (FEM) solver for computation of optical resonance modes in VCSELs. We perform a convergence study and demonstrate that high accuracies for 3D setups can be attained on standard computers. We also demonstrate simulations of thermo-optical effects in VCSELs.
7. Finite element analysis of tubular joints in offshore structures ...
... representing a 2-D model of the joint between the brace and the chord walls. This was subsequently followed but finite element analysis of six tubular joints. A global analysis was initially undertaken, then the submodel analysis carried in the areas of stress concentration. Journal of Civil Engineering, JKUAT (2001) Vol 6, ...
8. A mixed finite element method for particle simulation in Lasertron
Le Meur, G.
1987-01-01
A particle simulation code is being developed with the aim to treat the motion of charged particles in electromagnetic devices, such as Lasertron. The paper describes the use of mixed finite element methods in computing the field components, without derivating them from scalar or vector potentials. Graphical results are shown
9. Steam generator tube rupture simulation using extended finite element method
Mohanty, Subhasish; Majumdar, Saurin; Natesan, Ken
2016-01-01
Highlights: • Extended finite element method used for modeling the steam generator tube rupture. • Crack propagation is modeled in an arbitrary solution dependent path. • The FE model is used for estimating the rupture pressure of steam generator tubes. • Crack coalescence modeling is also demonstrated. • The method can be used for crack modeling of tubes under severe accident condition. - Abstract: A steam generator (SG) is an important component of any pressurized water reactor. Steam generator tubes represent a primary pressure boundary whose integrity is vital to the safe operation of the reactor. SG tubes may rupture due to propagation of a crack created by mechanisms such as stress corrosion cracking, fatigue, etc. It is thus important to estimate the rupture pressures of cracked tubes for structural integrity evaluation of SGs. The objective of the present paper is to demonstrate the use of extended finite element method capability of commercially available ABAQUS software, to model SG tubes with preexisting flaws and to estimate their rupture pressures. For the purpose, elastic–plastic finite element models were developed for different SG tubes made from Alloy 600 material. The simulation results were compared with experimental results available from the steam generator tube integrity program (SGTIP) sponsored by the United States Nuclear Regulatory Commission (NRC) and conducted at Argonne National Laboratory (ANL). A reasonable correlation was found between extended finite element model results and experimental results.
10. Discontinuous Galerkin finite element methods for hyperbolic differential equations
van der Vegt, Jacobus J.W.; van der Ven, H.; Boelens, O.J.; Boelens, O.J.; Toro, E.F.
2002-01-01
In this paper a suryey is given of the important steps in the development of discontinuous Galerkin finite element methods for hyperbolic partial differential equations. Special attention is paid to the application of the discontinuous Galerkin method to the solution of the Euler equations of gas
11. Can finite element models detect clinically inferior cemented hip implants?
Stolk, J.; Maher, S.A.; Verdonschot, N.J.J.; Prendergast, P.J.; Huiskes, R.
2003-01-01
Rigorous preclinical testing of cemented hip prostheses against the damage accumulation failure scenario will reduce the incidence of aseptic loosening. For that purpose, a finite element simulation is proposed that predicts damage accumulation in the cement mantle and prosthetic migration. If the
12. a finite element model for the analysis of bridge decks
Dr Obe
A FINITE ELEMENT MODEL FOR THE ANALYSIS OF BRIDGE DECKS. NIGERIAN JOURNAL OF TECHNOLOGY, VOL. 27 NO.1, MARCH 2008. 59. (a) Beam-plate system. (b) T-beam structural model. Fig. 1 Beam-plate structure idealisations. The matrix displacement method of analysis is used. The continuum structure is.
13. Deflation in preconditioned conjugate gradient methods for Finite Element Problems
Vermolen, F.J.; Vuik, C.; Segal, A.
2002-01-01
We investigate the influence of the value of deflation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods applied to a Finite Element discretization for an elliptic equation. Our set-up is a Poisson problem in two dimensions with continuous or discontinuous
14. Finite element modelling of fibre-reinforced brittle materials
Kullaa, J.
1997-01-01
The tensile constitutive behaviour of fibre-reinforced brittle materials can be extended to two or three dimensions by using the finite element method with crack models. The three approaches in this study include the smeared and discrete crack concepts and a multi-surface plasticity model. The
15. Finite element simulations of two rock mechanics tests
Dahlke, H.J.; Lott, S.A.
1986-04-01
Rock mechanics tests are performed to determine in situ stress conditions and material properties of an underground rock mass. To design stable underground facilities for the permanent storage of high-level nuclear waste, determination of these properties and conditions is a necessary first step. However, before a test and its associated equipment can be designed, the engineer needs to know the range of expected values to be measured by the instruments. Sensitivity studies by means of finite element simulations are employed in this preliminary design phase to evaluate the pertinent parameters and their effects on the proposed measurements. The simulations, of two typical rock mechanics tests, the plate bearing test and the flat-jack test, by means of the finite element analysis, are described. The plate bearing test is used to determine the rock mass deformation modulus. The flat-jack test is used to determine the in situ stress conditions of the host rock. For the plate bearing test, two finite element models are used to simulate the classic problem of a load on an elastic half space and the actual problem of a plate bearing test in an underground tunnel of circular cross section. For the flat-jack simulation, a single finite element model is used to simulate both horizontal and vertical slots. Results will be compared to closed-form solutions available in the literature
16. Finite element investigation of the prestressed jointed concrete ...
Precast prestressed concrete pavement (PCP) technology is of recent origin, and the information on PCP performance is not available in literature. This research presents a finite-element analysis of the potential benefits of prestressing on the jointed concrete pavements (JCP). With using a 3-dimensional (3D) ...
17. Appendix F : finite element analysis of end region.
2013-03-01
FE (finite element) modeling was conducted to 1) provide a better understanding of the : elastic behavior of the end region prior to cracking and 2) to evaluate the effects of bearing pad : stiffness and width on end region elastic stresses. The FEA ...
18. GRIZ: Visualization of finite element analysis results on unstructured grids
Dovey, D.; Loomis, M.D.
1994-01-01
GRIZ is a general-purpose post-processing application that supports interactive visualization of finite element analysis results on three-dimensional unstructured grids. GRIZ includes direct-to-videodisc animation capabilities and is being used as a production tool for creating engineering animations
19. Integral finite element analysis of turntable bearing with flexible rings
Deng, Biao; Liu, Yunfei; Guo, Yuan; Tang, Shengjin; Su, Wenbin; Lei, Zhufeng; Wang, Pengcheng
2018-03-01
This paper suggests a method to calculate the internal load distribution and contact stress of the thrust angular contact ball turntable bearing by FEA. The influence of the stiffness of the bearing structure and the plastic deformation of contact area on the internal load distribution and contact stress of the bearing is considered. In this method, the load-deformation relationship of the rolling elements is determined by the finite element contact analysis of a single rolling element and the raceway. Based on this, the nonlinear contact between the rolling elements and the inner and outer ring raceways is same as a nonlinear compression spring and bearing integral finite element analysis model including support structure was established. The effects of structural deformation and plastic deformation on the built-in stress distribution of slewing bearing are investigated on basis of comparing the consequences of load distribution, inner and outer ring stress, contact stress and other finite element analysis results with the traditional bearing theory, which has guiding function for improving the design of slewing bearing.
20. Polarization effects on spectra of spherical core/shell nanostructures: Perturbation theory against finite difference approach
Ibral, Asmaa; Zouitine, Asmaa; Assaid, El Mahdi
2015-01-01
Poisson equation is solved analytically in the case of a point charge placed anywhere in a spherical core/shell nanostructure, immersed in aqueous or organic solution or embedded in semiconducting or insulating matrix. Conduction and valence band-edge alignments between core and shell are described by finite height barriers. Influence of polarization charges induced at the surfaces where two adjacent materials meet is taken into account. Original expressions of electrostatic potential created everywhere in the space by a source point charge are derived. Expressions of self-polarization potential describing the interaction of a point charge with its own image–charge are deduced. Contributions of double dielectric constant mismatch to electron and hole ground state energies as well as nanostructure effective gap are calculated via first order perturbation theory and also by finite difference approach. Dependencies of electron, hole and gap energies against core to shell radii ratio are determined in the case of ZnS/CdSe core/shell nanostructure immersed in water or in toluene. It appears that finite difference approach is more efficient than first order perturbation method and that the effect of polarization charge may in no case be neglected as its contribution can reach a significant proportion of the value of nanostructure gap
1. Polarization effects on spectra of spherical core/shell nanostructures: Perturbation theory against finite difference approach
Ibral, Asmaa [Equipe d' Optique et Electronique du Solide, Département de Physique, Faculté des Sciences, Université Chouaïb Doukkali, B. P. 20 El Jadida principale, El Jadida, Royaume du Maroc (Morocco); Laboratoire d' Instrumentation, Mesure et Contrôle, Département de Physique, Faculté des Sciences, Université Chouaïb Doukkali, B. P. 20 El Jadida principale, El Jadida, Royaume du Maroc (Morocco); Zouitine, Asmaa [Département de Physique, Ecole Nationale Supérieure d' Enseignement Technique, Université Mohammed V Souissi, B. P. 6207 Rabat-Instituts, Rabat, Royaume du Maroc (Morocco); Assaid, El Mahdi, E-mail: eassaid@yahoo.fr [Equipe d' Optique et Electronique du Solide, Département de Physique, Faculté des Sciences, Université Chouaïb Doukkali, B. P. 20 El Jadida principale, El Jadida, Royaume du Maroc (Morocco); Laboratoire d' Instrumentation, Mesure et Contrôle, Département de Physique, Faculté des Sciences, Université Chouaïb Doukkali, B. P. 20 El Jadida principale, El Jadida, Royaume du Maroc (Morocco); and others
2015-02-01
Poisson equation is solved analytically in the case of a point charge placed anywhere in a spherical core/shell nanostructure, immersed in aqueous or organic solution or embedded in semiconducting or insulating matrix. Conduction and valence band-edge alignments between core and shell are described by finite height barriers. Influence of polarization charges induced at the surfaces where two adjacent materials meet is taken into account. Original expressions of electrostatic potential created everywhere in the space by a source point charge are derived. Expressions of self-polarization potential describing the interaction of a point charge with its own image–charge are deduced. Contributions of double dielectric constant mismatch to electron and hole ground state energies as well as nanostructure effective gap are calculated via first order perturbation theory and also by finite difference approach. Dependencies of electron, hole and gap energies against core to shell radii ratio are determined in the case of ZnS/CdSe core/shell nanostructure immersed in water or in toluene. It appears that finite difference approach is more efficient than first order perturbation method and that the effect of polarization charge may in no case be neglected as its contribution can reach a significant proportion of the value of nanostructure gap.
2. Development of a partitioned finite volume-finite element fluid-structure interaction scheme for strongly-coupled problems
Suliman, Ridhwaan
2012-07-01
Full Text Available -linear deformations are accounted for. As will be demonstrated, the finite volume approach exhibits similar disad- vantages to the linear Q4 finite element formulation when undergoing bending. An enhanced finite volume approach is discussed and compared with finite...
3. Finite element analysis of degraded concrete structures - Workshop proceedings
1999-09-01
This workshop is related to the finite element analysis of degraded concrete structures. It is composed of three sessions. The first session (which title is: the use of finite element analysis in safety assessments) comprises six papers which titles are: Historical Development of Concrete Finite Element Modeling for Safety Evaluation of Accident-Challenged and Aging Concrete Structures; Experience with Finite Element Methods for Safety Assessments in Switzerland; Stress State Analysis of the Ignalina NPP Confinement System; Prestressed Containment: Behaviour when Concrete Cracking is Modelled; Application of FEA for Design and Support of NPP Containment in Russia; Verification Problems of Nuclear Installations Safety Software of Strength Analysis (NISS SA). The second session (title: concrete containment structures under accident loads) comprises seven papers which titles are: Two Application Examples of Concrete Containment Structures under Accident Load Conditions Using Finite Element Analysis; What Kind of Prediction for Leak rates for Nuclear Power Plant Containments in Accidental Conditions; Influence of Different Hypotheses Used in Numerical Models for Concrete At Elevated Temperatures on the Predicted Behaviour of NPP Core Catchers Under Severe Accident Conditions; Observations on the Constitutive Modeling of Concrete Under Multi-Axial States at Elevated Temperatures; Analyses of a Reinforced Concrete Containment with Liner Corrosion Damage; Program of Containment Concrete Control During Operation for the Temelin Nuclear Power Plant; Static Limit Load of a Deteriorated Hyperbolic Cooling Tower. The third session (concrete structures under extreme environmental load) comprised five papers which titles are: Shear Transfer Mechanism of RC Plates After Cracking; Seismic Back Calculation of an Auxiliary Building of the Nuclear Power Plant Muehleberg, Switzerland; Seismic Behaviour of Slightly Reinforced Shear Wall Structures; FE Analysis of Degraded Concrete
4. A Novel Polygonal Finite Element Method: Virtual Node Method
Tang, X. H.; Zheng, C.; Zhang, J. H.
2010-05-01
Polygonal finite element method (PFEM), which can construct shape functions on polygonal elements, provides greater flexibility in mesh generation. However, the non-polynomial form of traditional PFEM, such as Wachspress method and Mean Value method, leads to inexact numerical integration. Since the integration technique for non-polynomial functions is immature. To overcome this shortcoming, a great number of integration points have to be used to obtain sufficiently exact results, which increases computational cost. In this paper, a novel polygonal finite element method is proposed and called as virtual node method (VNM). The features of present method can be list as: (1) It is a PFEM with polynomial form. Thereby, Hammer integral and Gauss integral can be naturally used to obtain exact numerical integration; (2) Shape functions of VNM satisfy all the requirements of finite element method. To test the performance of VNM, intensive numerical tests are carried out. It found that, in standard patch test, VNM can achieve significantly better results than Wachspress method and Mean Value method. Moreover, it is observed that VNM can achieve better results than triangular 3-node elements in the accuracy test.
5. Investigations on Actuator Dynamics through Theoretical and Finite Element Approach
Somashekhar S. Hiremath
2010-01-01
Full Text Available This paper gives a new approach for modeling the fluid-structure interaction of servovalve component-actuator. The analyzed valve is a precision flow control valve-jet pipe electrohydraulic servovalve. The positioning of an actuator depends upon the flow rate from control ports, in turn depends on the spool position. Theoretical investigation is made for No-load condition and Load condition for an actuator. These are used in finite element modeling of an actuator. The fluid-structure-interaction (FSI is established between the piston and the fluid cavities at the piston end. The fluid cavities were modeled with special purpose hydrostatic fluid elements while the piston is modeled with brick elements. The finite element method is used to simulate the variation of cavity pressure, cavity volume, mass flow rate, and the actuator velocity. The finite element analysis is extended to study the system's linearized response to harmonic excitation using direct solution steady-state dynamics. It was observed from the analysis that the natural frequency of the actuator depends upon the position of the piston in the cylinder. This is a close match with theoretical and simulation results. The effect of bulk modulus is also presented in the paper.
6. An efficient structural finite element for inextensible flexible risers
Papathanasiou, T. K.; Markolefas, S.; Khazaeinejad, P.; Bahai, H.
2017-12-01
A core part of all numerical models used for flexible riser analysis is the structural component representing the main body of the riser as a slender beam. Loads acting on this structural element are self-weight, buoyant and hydrodynamic forces, internal pressure and others. A structural finite element for an inextensible riser with a point-wise enforcement of the inextensibility constrain is presented. In particular, the inextensibility constraint is applied only at the nodes of the meshed arc length parameter. Among the virtues of the proposed approach is the flexibility in the application of boundary conditions and the easy incorporation of dissipative forces. Several attributes of the proposed finite element scheme are analysed and computation times for the solution of some simplified examples are discussed. Future developments aim at the appropriate implementation of material and geometric parameters for the beam model, i.e. flexural and torsional rigidity.
7. Finite element and discontinuous Galerkin methods for transient wave equations
Cohen, Gary
2017-01-01
This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem ...
8. Nonlinear finite element analysis of nuclear reinforced prestressed concrete containments up to ultimate load capacity
Gupta, A.; Singh, R.K.; Kushwaha, H.S.; Mahajan, S.C.; Kakodkar, A.
1996-01-01
For safety evaluation of nuclear structures a finite element code ULCA (Ultimate Load Capacity Assessment) has been developed. Eight/nine noded isoparametric quadrilateral plate/shell element with reinforcement as a through thickness discrete but integral smeared layer of the element is presented to analyze reinforced and prestressed concrete structures. Various constitutive models such as crushing, cracking in tension, tension stiffening and rebar yielding are studied and effect of these parameters on the reserve strength of structures is brought out through a number of benchmark tests. A global model is used to analyze the prestressed concrete containment wall of a typical 220 MWe Pressurized Heavy Water Reactor (PHWR) up to its ultimate capacity. This demonstrates the adequacy of Indian PHWR containment design to withstand severe accident loads
9. Nonlinear Finite Element Analysis of a General Composite Shell
1988-12-01
for (t) in Equation (B.15) (Appendix B) and writes it as a function of displacements for I the nonlinear problem one obtains [8] 3 29 (*(a)) - [K(a...linked to the main program before execution. Isubroutine upress(t,pa,pb,iunit, ielt ,x,y,z,live,press) c c Pressure distribution subroutine for c...then compiled and linked to the main program before execution. I SUBROUTINE UPRESS(T,PA,PB,IUNIT, IELT ,X,Y,Z,LIVE,PRESS) C c Pressure distribution
10. Analysis of axisymmetric shells subjected to asymmetric loads using field consistent shear flexible curved element
Balakrishna, C; Sarma, B S [Defence Research and Development Laboratory, Hyderabad (India)
1989-02-01
A formulation for axisymmetric shell analysis under asymmetric load based on Fourier series representation and using field consistent 3 noded curved axisymmetric shell element is presented. Different field inconsistent/consistent interpolations for an element based on shear flexible theory have been studied for thick and thin shells under asymmetric loads. Various examples covering axisymmetric as well as asymmetric loading cases have been analyzed and numerical results show a good agreement with the available results in the case of thin shells. 12 refs.
11. Hermitian Mindlin Plate Wavelet Finite Element Method for Load Identification
Xiaofeng Xue
2016-01-01
Full Text Available A new Hermitian Mindlin plate wavelet element is proposed. The two-dimensional Hermitian cubic spline interpolation wavelet is substituted into finite element functions to construct frequency response function (FRF. It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform. By simulating different excitation cases, Hermitian cubic spline wavelets on the interval (HCSWI finite elements are used to reverse load identification in the Mindlin plate. The singular value decomposition (SVD method is adopted to solve the ill-posed inverse problem. Compared with ANSYS results, HCSWI Mindlin plate element can accurately identify the applied load. Numerical results show that the algorithm of HCSWI Mindlin plate element is effective. The accuracy of HCSWI can be verified by comparing the FRF of HCSWI and ANSYS elements with the experiment data. The experiment proves that the load identification of HCSWI Mindlin plate is effective and precise by using the FRF and response spectrums to calculate the loads.
12. Use of a finite range nucleon-nucleon interaction in the continuum shell model
Faes, Jean-Baptiste
2007-01-01
The unification of nuclear structure and nuclear reactions was always a great challenge of nuclear physics. The extreme complexity of finite quantum systems lead in the past to a separate development of the nuclear structure and the nuclear reactions. A unified description of structure and reactions is possible within the continuum shell model. All previous applications of this model used the zero-range residual interaction and the finite depth local potential to generate the single-particle basis. In the thesis, we have presented an extension of the continuum shell model for finite-range nucleon-nucleon interaction and an arbitrary number of nucleons in the scattering continuum. The great advantage of the present formulation is the same two-body interaction used both to generate the single-particle basis and to describe couplings to the continuum states. This formulation opens a possibility for an ab initio continuum shell model studies with the same nucleon-nucleon interaction generating the nuclear mean field, the configuration mixing and the coupling to the scattering continuum. First realistic applications of the above model has been shown for spectra of "1"7F and "1"7O, and elastic phase-shifts in the reaction "1"6O(p, p)"1"6O. (author)
13. Study of characterization of trace elements in marine shells of Sambaqui: correlation between recent and old shells
Gomez, Mauro Roger Batista Pousada; Rocha, Flavio Roberto; Silva, Paulo Sergio Cardoso da
2013-01-01
Calcium carbonate of recent and ancient C. rhizophorae oyster shells was analyzed for the determination of trace elements by instrumental neutron activation analysis. The ancient shells belong to a Sambaqui located in Cananeia region, South of Sao Paulo state and the recent ones are from an oyster production farm in the same region Studies related to the element concentrations in molluscs shell has been done as a tentative of establishing the element concentrations with palio-environmental factor. In this study it was aimed to verify differences in the elemental constitution of recent and ancient oyster shells that present potential for being used as indicator of marine changes. Results indicated that the elements Br, Ce, La, Na, Sm and An are higher in recent shells and the elements Cr, Fe Sc and Th are higher in ancient shells. Statistical analyses performed indicated that the enrichment of the light rare earth elements related to Ca are possibly good candidates for these palio-environmental studies. (author)
14. The finite element method and applications in engineering using ANSYS
2015-01-01
This textbook offers theoretical and practical knowledge of the finite element method. The book equips readers with the skills required to analyze engineering problems using ANSYS®, a commercially available FEA program. Revised and updated, this new edition presents the most current ANSYS® commands and ANSYS® screen shots, as well as modeling steps for each example problem. This self-contained, introductory text minimizes the need for additional reference material by covering both the fundamental topics in finite element methods and advanced topics concerning modeling and analysis. It focuses on the use of ANSYS® through both the Graphics User Interface (GUI) and the ANSYS® Parametric Design Language (APDL). Extensive examples from a range of engineering disciplines are presented in a straightforward, step-by-step fashion. Key topics include: • An introduction to FEM • Fundamentals and analysis capabilities of ANSYS® • Fundamentals of discretization and approximation functions • Modeling techniq...
15. Finite element design procedure for correcting the coining die profiles
Alexandrino, Paulo; Leitão, Paulo J.; Alves, Luis M.; Martins, Paulo A. F.
2018-05-01
This paper presents a new finite element based design procedure for correcting the coining die profiles in order to optimize the distribution of pressure and the alignment of the resultant vertical force at the end of the die stroke. The procedure avoids time consuming and costly try-outs, does not interfere with the creative process of the sculptors and extends the service life of the coining dies by significantly decreasing the applied pressure and bending moments. The numerical simulations were carried out in a computer program based on the finite element flow formulation that is currently being developed by the authors in collaboration with the Portuguese Mint. A new experimental procedure based on the stack compression test is also proposed for determining the stress-strain curve of the materials directly from the coin blanks.
16. Domain Decomposition Solvers for Frequency-Domain Finite Element Equations
Copeland, Dylan; Kolmbauer, Michael; Langer, Ulrich
2010-01-01
The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine- and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method. © 2011 Springer-Verlag Berlin Heidelberg.
17. Introduction to assembly of finite element methods on graphics processors
Cecka, Cristopher; Lew, Adrian; Darve, Eric
2010-01-01
Recently, graphics processing units (GPUs) have had great success in accelerating numerical computations. We present their application to computations on unstructured meshes such as those in finite element methods. Multiple approaches in assembling and solving sparse linear systems with NVIDIA GPUs and the Compute Unified Device Architecture (CUDA) are presented and discussed. Multiple strategies for efficient use of global, shared, and local memory, methods to achieve memory coalescing, and optimal choice of parameters are introduced. We find that with appropriate preprocessing and arrangement of support data, the GPU coprocessor achieves speedups of 30x or more in comparison to a well optimized serial implementation on the CPU. We also find that the optimal assembly strategy depends on the order of polynomials used in the finite-element discretization.
18. Finite cover method with mortar elements for elastoplasticity problems
Kurumatani, M.; Terada, K.
2005-06-01
Finite cover method (FCM) is extended to elastoplasticity problems. The FCM, which was originally developed under the name of manifold method, has recently been recognized as one of the generalized versions of finite element methods (FEM). Since the mesh for the FCM can be regular and squared regardless of the geometry of structures to be analyzed, structural analysts are released from a burdensome task of generating meshes conforming to physical boundaries. Numerical experiments are carried out to assess the performance of the FCM with such discretization in elastoplasticity problems. Particularly to achieve this accurately, the so-called mortar elements are introduced to impose displacement boundary conditions on the essential boundaries, and displacement compatibility conditions on material interfaces of two-phase materials or on joint surfaces between mutually incompatible meshes. The validity of the mortar approximation is also demonstrated in the elastic-plastic FCM.
19. A finite element model of ferroelectric/ferroelastic polycrystals
HWANG,STEPHEN C.; MCMEEKING,ROBERT M.
2000-02-17
A finite element model of polarization switching in a polycrystalline ferroelectric/ferroelastic ceramic is developed. It is assumed that a crystallite switches if the reduction in potential energy of the polycrystal exceeds a critical energy barrier per unit volume of switching material. Each crystallite is represented by a finite element with the possible dipole directions assigned randomly subject to crystallographic constraints. The model accounts for both electric field induced (i.e. ferroelectric) switching and stress induced (i.e. ferroelastic) switching with piezoelectric interactions. Experimentally measured elastic, dielectric, and piezoelectric constants are used consistently, but different effective critical energy barriers are selected phenomenologically. Electric displacement versus electric field, strain versus electric field, stress versus strain, and stress versus electric displacement loops of a ceramic lead lanthanum zirconate titanate (PLZT) are modeled well below the Curie temperature.
20. Finite element modeling of trolling-mode AFM.
Sajjadi, Mohammadreza; Pishkenari, Hossein Nejat; Vossoughi, Gholamreza
2018-06-01
Trolling mode atomic force microscopy (TR-AFM) has overcome many imaging problems in liquid environments by considerably reducing the liquid-resonator interaction forces. The finite element model of the TR-AFM resonator considering the effects of fluid and nanoneedle flexibility is presented in this research, for the first time. The model is verified by ABAQUS software. The effect of installation angle of the microbeam relative to the horizon and the effect of fluid on the system behavior are investigated. Using the finite element model, frequency response curve of the system is obtained and validated around the frequency of the operating mode by the available experimental results, in air and liquid. The changes in the natural frequencies in the presence of liquid are studied. The effects of tip-sample interaction on the excitation of higher order modes of the system are also investigated in air and liquid environments. Copyright © 2018 Elsevier B.V. All rights reserved.
1. Domain Decomposition Solvers for Frequency-Domain Finite Element Equations
Copeland, Dylan
2010-10-05
The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine- and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method. © 2011 Springer-Verlag Berlin Heidelberg.
2. Assembly of finite element methods on graphics processors
Cecka, Cris
2010-08-23
Recently, graphics processing units (GPUs) have had great success in accelerating many numerical computations. We present their application to computations on unstructured meshes such as those in finite element methods. Multiple approaches in assembling and solving sparse linear systems with NVIDIA GPUs and the Compute Unified Device Architecture (CUDA) are created and analyzed. Multiple strategies for efficient use of global, shared, and local memory, methods to achieve memory coalescing, and optimal choice of parameters are introduced. We find that with appropriate preprocessing and arrangement of support data, the GPU coprocessor using single-precision arithmetic achieves speedups of 30 or more in comparison to a well optimized double-precision single core implementation. We also find that the optimal assembly strategy depends on the order of polynomials used in the finite element discretization. © 2010 John Wiley & Sons, Ltd.
3. Finite Element Method for Analysis of Material Properties
Rauhe, Jens Christian
and the finite element method. The material microstructure of the heterogeneous material is non-destructively determined using X-ray microtomography. A software program has been generated which uses the X-ray tomographic data as an input for the mesh generation of the material microstructure. To obtain a proper...... which are used for the determination of the effective properties of the heterogeneous material. Generally, the properties determined using the finite element method coupled with X-ray microtomography are in good agreement with both experimentally determined properties and properties determined using......The use of cellular and composite materials have in recent years become more and more common in all kinds of structural components and accurate knowledge of the effective properties is therefore essential. In this wok the effective properties are determined using the real material microstructure...
4. Finite element modeling of micromachined MEMS photon devices
Evans, Boyd M., III; Schonberger, D. W.; Datskos, Panos G.
1999-09-01
The technology of microelectronics that has evolved over the past half century is one of great power and sophistication and can now be extended to many applications (MEMS and MOEMS) other than electronics. An interesting application of MEMS quantum devices is the detection of electromagnetic radiation. The operation principle of MEMS quantum devices is based on the photoinduced stress in semiconductors, and the photon detection results from the measurement of the photoinduced bending. These devices can be described as micromechanical photon detectors. In this work, we have developed a technique for simulating electronic stresses using finite element analysis. We have used our technique to model the response of micromechanical photon devices to external stimuli and compared these results with experimental data. Material properties, geometry, and bimaterial design play an important role in the performance of micromechanical photon detectors. We have modeled these effects using finite element analysis and included the effects of bimaterial thickness coating, effective length of the device, width, and thickness.
5. Finite Element Modeling of Micromachined MEMS Photon Devices
Datskos, P.G.; Evans, B.M.; Schonberger, D.
1999-01-01
The technology of microelectronics that has evolved over the past half century is one of great power and sophistication and can now be extended to many applications (MEMS and MOEMS) other than electronics. An interesting application of MEMS quantum devices is the detection of electromagnetic radiation. The operation principle of MEMS quantum devices is based on the photoinduced stress in semiconductors, and the photon detection results from the measurement of the photoinduced bending. These devices can be described as micromechanical photon detectors. In this work, we have developed a technique for simulating electronic stresses using finite element analysis. We have used our technique to model the response of micromechanical photon devices to external stimuli and compared these results with experimental data. Material properties, geometry, and bimaterial design play an important role in the performance of micromechanical photon detectors. We have modeled these effects using finite element analysis and included the effects of bimaterial thickness coating, effective length of the device, width, and thickness
6. Finite element predictions of active buckling control of stiffened panels
Thompson, Danniella M.; Griffin, O. H., Jr.
1993-04-01
Materials systems and structures that can respond 'intelligently' to their environment are currently being proposed and investigated. A series of finite element analyses was performed to investigate the potential for active buckling control of two different stiffened panels by embedded shape memory alloy (SMA) rods. Changes in the predicted buckling load increased with the magnitude of the actuation level for a given structural concept. Increasing the number of actuators for a given concept yielded greater predicted increases in buckling load. Considerable control authority was generated with a small number of actuators, with greater authority demonstrated for those structural concepts where the activated SMA rods could develop greater forces and moments on the structure. Relatively simple and inexpensive analyses were performed with standard finite elements to determine such information, indicating the viability of these types of models for design purposes.
7. An adaptive finite element method for steady and transient problems
Benner, R.E. Jr.; Davis, H.T.; Scriven, L.E.
1987-01-01
Distributing integral error uniformly over variable subdomains, or finite elements, is an attractive criterion by which to subdivide a domain for the Galerkin/finite element method when localized steep gradients and high curvatures are to be resolved. Examples are fluid interfaces, shock fronts and other internal layers, as well as fluid mechanical and other boundary layers, e.g. thin-film states at solid walls. The uniform distribution criterion is developed into an adaptive technique for one-dimensional problems. Nodal positions can be updated simultaneously with nodal values during Newton iteration, but it is usually better to adopt nearly optimal nodal positions during Newton iteration upon nodal values. Three illustrative problems are solved: steady convection with diffusion, gradient theory of fluid wetting on a solid surface and Buckley-Leverett theory of two phase Darcy flow in porous media
8. Finite Element Analysis and Design of Experiments in Engineering Design
Eriksson, Martin
1999-01-01
Projects with the objective of introducing Finite Element Analysis (FEA) into the early phases of the design process have previously been carried out at the Department of Machine Design, see e.g. the Doctoral thesis by Burman [13]. These works clearly highlight the usefulness of introducing design analysis early in the design process. According to Bjärnemo and Burman [10] the most significant advantage of applying design analysis early in the design process was the shift from verification to ...
9. Three-dimensional modeling with finite element codes
Druce, R.L.
1986-01-17
This paper describes work done to model magnetostatic field problems in three dimensions. Finite element codes, available at LLNL, and pre- and post-processors were used in the solution of the mathematical model, the output from which agreed well with the experimentally obtained data. The geometry used in this work was a cylinder with ports in the periphery and no current sources in the space modeled. 6 refs., 8 figs.
10. Finite element computation of natural convection in enclosures
Kushwaha, H.S.
1982-01-01
Compared to U-V-P-T formulation and stream-vorticity temperature formulation, penalty function formulation is simple and computationally competitive. Incremental New-Raphons method employed in this study is effective and efficient. From this study it is established that very fine mesh is not required for a low Rayleigh number considered in this study. The upwinding finite element may be necessary to avoid oscillations for higher Rayleigh numbers. (author)
11. The Development of Piezoelectric Accelerometers Using Finite Element Analysis
Liu, Bin
1999-01-01
This paper describes the application of Finite Element (FE) approach for the development of piezoelectric accelerometers. An accelerometer is simulated using the FE approach as an example. Good agreement is achieved between simulated results and calibrated results. It is proved that the FE modeling...... can be effectively used to predict the specifications of the accelerometer, especially when modification of the accelerometer is required. The FE developing technology forms the bases of fast responsiveness and flexible customized design of piezoelectric accelerometers....
12. A finite element method for SSI time history calculation
Ni, X.; Gantenbein, F.; Petit, M.
1989-01-01
The method which is proposed is based on a finite element modelization for the soil and the structure and a time history calculation. It has been developed for plane and axisymmetric geometries. The principle of this method is presented, then applications are given, first to a linear calculation for which results will be compared to those obtained by standard methods. Then results for a non linear behavior are described
13. Convergence of a residual based artificial viscosity finite element method
Nazarov, Murtazo
2013-02-01
We present a residual based artificial viscosity finite element method to solve conservation laws. The Galerkin approximation is stabilized by only residual based artificial viscosity, without any least-squares, SUPG, or streamline diffusion terms. We prove convergence of the method, applied to a scalar conservation law in two space dimensions, toward an unique entropy solution for implicit time stepping schemes. © 2012 Elsevier B.V. All rights reserved.
14. Imposing orthogonality to hierarchic higher-order finite elements
Šolín, P.; Vejchodský, Tomáš; Zítka, M.; Ávila, F.
2007-01-01
Roč. 76, 1-3 (2007), s. 211-217 ISSN 0378-4754 R&D Projects: GA ČR GP201/04/P021 Institutional research plan: CEZ:AV0Z10190503 Keywords : optimal shape functions * energetic inner product * Laplace equation * symmetric linear elliptic problems * numerical experiments * hp-finite element method Subject RIV: BA - General Mathematics Impact factor: 0.738, year: 2007
15. Finite elements for partial differential equations: An introductory survey
Succi, S.
1988-03-01
After presentation of the basic ideas behind the theory of the Finite Element Method, the application of the method to three equations of particular interest in Physics and Engineering is discussed in some detail, namely, a one-dimensional Sturm-Liouville problem, a two-dimensional linear Fokker-Planck equation and a two-dimensional nonlinear Navier-Stokes equation. 6 refs, 8 figs
16. [Application of Finite Element Method in Thoracolumbar Spine Traumatology].
Zhang, Min; Qiu, Yong-gui; Shao, Yu; Gu, Xiao-feng; Zeng, Ming-wei
2015-04-01
The finite element method (FEM) is a mathematical technique using modern computer technology for stress analysis, and has been gradually used in simulating human body structures in the biomechanical field, especially more widely used in the research of thoracolumbar spine traumatology. This paper reviews the establishment of the thoracolumbar spine FEM, the verification of the FEM, and the thoracolumbar spine FEM research status in different fields, and discusses its prospects and values in forensic thoracolumbar traumatology.
17. A finite element method for flow problems in blast loading
Forestier, A.; Lepareux, M.
1984-06-01
This paper presents a numerical method which describes fast dynamic problems in flow transient situations as in nuclear plants. A finite element formulation has been chosen; it is described by a preprocessor in CASTEM system: GIBI code. For these typical flow problems, an A.L.E. formulation for physical equations is used. So, some applications are presented: the well known problem of shock tube, the same one in 2D case and a last application to hydrogen detonation
18. A code for obtaining temperature distribution by finite element method
Bloch, M.
1984-01-01
The ELEFIB Fortran language computer code using finite element method for calculating temperature distribution of linear and two dimensional problems, in permanent region or in the transient phase of heat transfer, is presented. The formulation of equations uses the Galerkin method. Some examples are shown and the results are compared with other papers. The comparative evaluation shows that the elaborated code gives good values. (M.C.K.) [pt
19. On angle conditions in the finite element method
Brandts, J.; Hannukainen, A.; Korotov, S.; Křížek, Michal
2011-01-01
Roč. 56, - (2011), s. 81-95 ISSN 1575-9822 R&D Projects: GA AV ČR(CZ) IAA100190803 Institutional research plan: CEZ:AV0Z10190503 Keywords : simplicial finite elements * minimum and maximum angle condition * ball conditions Subject RIV: BA - General Mathematics http://www.sema.org.es/ojs/index.php?journal=journal&page=article&op=viewArticle&path%5B%5D=612
20. Three dimensional mathematical model of tooth for finite element analysis
Puškar Tatjana
2010-01-01
Full Text Available Introduction. The mathematical model of the abutment tooth is the starting point of the finite element analysis of stress and deformation of dental structures. The simplest and easiest way is to form a model according to the literature data of dimensions and morphological characteristics of teeth. Our method is based on forming 3D models using standard geometrical forms (objects in programmes for solid modeling. Objective. Forming the mathematical model of abutment of the second upper premolar for finite element analysis of stress and deformation of dental structures. Methods. The abutment tooth has a form of a complex geometric object. It is suitable for modeling in programs for solid modeling SolidWorks. After analyzing the literature data about the morphological characteristics of teeth, we started the modeling dividing the tooth (complex geometric body into simple geometric bodies (cylinder, cone, pyramid,.... Connecting simple geometric bodies together or substricting bodies from the basic body, we formed complex geometric body, tooth. The model is then transferred into Abaqus, a computational programme for finite element analysis. Transferring the data was done by standard file format for transferring 3D models ACIS SAT. Results. Using the programme for solid modeling SolidWorks, we developed three models of abutment of the second maxillary premolar: the model of the intact abutment, the model of the endodontically treated tooth with two remaining cavity walls and the model of the endodontically treated tooth with two remaining walls and inserted post. Conclusion Mathematical models of the abutment made according to the literature data are very similar with the real abutment and the simplifications are minimal. These models enable calculations of stress and deformation of the dental structures. The finite element analysis provides useful information in understanding biomechanical problems and gives guidance for clinical research.
1. Thermohydraulic analysis in pipelines using the finite element method
Costa, L.E.; Idelsohn, S.R.
1984-01-01
The Finite Element Method (FEM) is employed for the numerical solution of fluid flow problems with combined heat transfer mechanisms. Boussinesq approximations are used for the solution of the governing equations. The application of the FEM leads to a set of simultaneous nonlinear equations. The development of the method, for the solution of bidimensional and axisymmetric problems, is presented. Examples of fluid flow in pipes, including natural and forced convection, are solved with the proposed method and discussed in the paper. (Author) [pt
2. A finite element method for SSI time history calculations
Ni, X.M.; Gantenbein, F.; Petit, M.
1989-01-01
The method which is proposed is based on a finite element modelisation for the soil and the structure and a time history calculation. It has been developed for plane and axisymmetric geometries. The principle of this method will be presented, then applications will be given, first to a linear calculation for which results will be compared to those obtained by standard methods. Then results for a non linear behavior will be described
3. Piezoelectric theory for finite element analysis of ultrasonic motors
Emery, J.D.; Mentesana, C.P.
1997-06-01
The authors present the fundamental equations of piezoelectricity and references. They show how a second form of the equations and a second set of coefficients can be found, through inversions involving the elasticity tensor. They show how to compute the clamped permittivity matrix from the unclamped matrix. The authors list the program pzansys.ftn and present examples of its use. This program does the conversions and calculations needed by the finite element program ANSYS.
4. Finite element approximation to a model problem of transonic flow
Tangmanee, S.
1986-12-01
A model problem of transonic flow ''the Tricomi equation'' in Ω is contained in IR 2 bounded by the rectangular-curve boundary is posed in the form of symmetric positive differential equations. The finite element method is then applied. When the triangulation of Ω-bar is made of quadrilaterals and the approximation space is the Lagrange polynomial, we get the error estimates. 14 refs, 1 fig
5. Eigenvalue solutions in finite element thermal transient problems
Stoker, J.R.
1975-01-01
The eigenvalue economiser concept can be useful in solving large finite element transient heat flow problems in which the boundary heat transfer coefficients are constant. The usual economiser theory is equivalent to applying a unit thermal 'force' to each of a small sub-set of nodes on the finite element mesh, and then calculating sets of resulting steady state temperatures. Subsequently it is assumed that the required transient temperature distributions can be approximated by a linear combination of this comparatively small set of master temperatures. The accuracy of a reduced eigenvalue calculation depends upon a good choice of master nodes, which presupposes at least a little knowledge about what sort of shape is expected in the unknown temperature distributions. There are some instances, however, where a reasonably good idea exists of the required shapes, permitting a modification to the economiser process which leads to greater economy in the number of master temperatures. The suggested new approach is to use manually prescribed temperature distributions as the master distributions, rather than using temperatures resulting from unit thermal forces. Thus, with a little pre-knowledge one may write down a set of master distributions which, as a linear combination, can represent the required solution over the range of interest to a reasonable engineering accuracy, and using the minimum number of variables. The proposed modified eigenvalue economiser technique then uses the master distributions in an automatic way to arrive at the required solution. The technique is illustrated by some simple finite element examples
6. Finite-element pre-analysis for pressurized thermoshock tests
Keinaenen, H.; Talja, H.; Lehtonen, M.; Rintamaa, R.; Bljumin, A.; Timofeev, B.
1992-05-01
The behaviour of a model pressure vessel is studied in a pressurized thermal shock loading. The tests were performed at the Prometey Institute in St. Petersburg. The calculations were performed at the Technical Research Centre of Finland. The report describes the preliminary finite-element analyses for the fourth, fifth and sixth thermoshock tests with the first model pressure vessel. Seven pressurized thermoshock tests were made with the same model using five different flaw geometries. In the first three tests the flaw was actually a blunt notch. In the two following tests (tests 4 and 5) a sharp pre-crack was produced before the test. In the last two test (tests 6 and 7) the old crack was used. According to the measurements and post-test ultrasonic examination of the crack front, the sixth test led to significant crack extension. Both temperatures and stresses were calculated using the finite-element method. The calculations were made using the idealized initial flaw geometry and preliminary material data. Both two-and three-dimensional models were used in the calculations. J-integral values were calculated from the elastic-plastic finite-element results. The stress intensity factor values were evaluated on the basis of the calculated J-integrals and compared with the preliminary material fracture toughness data obtained from the Prometey Institute
7. Thermal buckling comparative analysis using Different FE (Finite Element) tools
Banasiak, Waldemar; Labouriau, Pedro [INTECSEA do Brasil, Rio de Janeiro, RJ (Brazil); Burnett, Christopher [INTECSEA UK, Surrey (United Kingdom); Falepin, Hendrik [Fugro Engineers SA/NV, Brussels (Belgium)
2009-12-19
High operational temperature and pressure in offshore pipelines may lead to unexpected lateral movements, sometimes call lateral buckling, which can have serious consequences for the integrity of the pipeline. The phenomenon of lateral buckling in offshore pipelines needs to be analysed in the design phase using FEM. The analysis should take into account many parameters, including operational temperature and pressure, fluid characteristic, seabed profile, soil parameters, coatings of the pipe, free spans etc. The buckling initiation force is sensitive to small changes of any initial geometric out-of-straightness, thus the modeling of the as-laid state of the pipeline is an important part of the design process. Recently some dedicated finite elements programs have been created making modeling of the offshore environment more convenient that has been the case with the use of general purpose finite element software. The present paper aims to compare thermal buckling analysis of sub sea pipeline performed using different finite elements tools, i.e. general purpose programs (ANSYS, ABAQUS) and dedicated software (SAGE Profile 3D) for a single pipeline resting on an the seabed. The analyses considered the pipeline resting on a flat seabed with a small levels of out-of straightness initiating the lateral buckling. The results show the quite good agreement of results of buckling in elastic range and in the conclusions next comparative analyses with sensitivity cases are recommended. (author)
8. Coupling nonlinear Stokes and Darcy flow using mortar finite elements
Ervin, Vincent J.
2011-11-01
We study a system composed of a nonlinear Stokes flow in one subdomain coupled with a nonlinear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes-Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the algorithm and to compare two treatments of the defective boundary conditions. © 2010 Published by Elsevier B.V. on behalf of IMACS.
9. Advances in dynamic relaxation techniques for nonlinear finite element analysis
Sauve, R.G.; Metzger, D.R.
1995-01-01
Traditionally, the finite element technique has been applied to static and steady-state problems using implicit methods. When nonlinearities exist, equilibrium iterations must be performed using Newton-Raphson or quasi-Newton techniques at each load level. In the presence of complex geometry, nonlinear material behavior, and large relative sliding of material interfaces, solutions using implicit methods often become intractable. A dynamic relaxation algorithm is developed for inclusion in finite element codes. The explicit nature of the method avoids large computer memory requirements and makes possible the solution of large-scale problems. The method described approaches the steady-state solution with no overshoot, a problem which has plagued researchers in the past. The method is included in a general nonlinear finite element code. A description of the method along with a number of new applications involving geometric and material nonlinearities are presented. They include: (1) nonlinear geometric cantilever plate; (2) moment-loaded nonlinear beam; and (3) creep of nuclear fuel channel assemblies
10. Discontinuous finite element treatment of duct problems in transport calculations
Mirza, A. M.; Qamar, S.
1998-01-01
A discontinuous finite element approach is presented to solve the even-parity Boltzmann transport equation for duct problems. Presence of ducts in a system results in the streaming of particles and hence requires the employment of higher order angular approximations to model the angular flux. Conventional schemes based on the use of continuous trial functions require the same order of angular approximations to be used everywhere in the system, resulting in wastage of computational resources. Numerical investigations for the test problems presented in this paper indicate that the discontinuous finite elements eliminate the above problems and leads to computationally efficient and economical methods. They are also found to be more suitable for treating the sharp changes in the angular flux at duct-observer interfaces. The new approach provides a single-pass alternate to extrapolation and interactive schemes which need multiple passes of the solution strategy to acquire convergence. The method has been tested with the help of two case studies, namely straight and dog-leg duct problems. All results have been verified against those obtained from Monte Carlo simulations and K/sup +/ continuous finite element method. (author)
11. Finite element analysis of the cyclic indentation of bilayer enamel
Jia, Yunfei; Xuan, Fu-zhen; Chen, Xiaoping; Yang, Fuqian
2014-01-01
Tooth enamel is often subjected to repeated contact and often experiences contact deformation in daily life. The mechanical strength of the enamel determines the biofunctionality of the tooth. Considering the variation of the rod arrangement in outer and inner enamel, we approximate enamel as a bilayer structure and perform finite element analysis of the cyclic indentation of the bilayer structure, to mimic the repeated contact of enamel during mastication. The dynamic deformation behaviour of both the inner enamel and the bilayer enamel is examined. The material parameters of the inner and outer enamel used in the analysis are obtained by fitting the finite element results with the experimental nanoindentation results. The penetration depth per cycle at the quasi-steady state is used to describe the depth propagation speed, which exhibits a two-stage power-law dependence on the maximum indentation load and the amplitude of the cyclic load, respectively. The continuous penetration of the indenter reflects the propagation of the plastic zone during cyclic indentation, which is related to the energy dissipation. The outer enamel serves as a protective layer due to its great resistance to contact deformation in comparison to the inner enamel. The larger equivalent plastic strain and lower stresses in the inner enamel during cyclic indentation, as calculated from the finite element analysis, indicate better crack/fracture resistance of the inner enamel. (paper)
12. Finite element analysis of the cyclic indentation of bilayer enamel
Jia, Yunfei; Xuan, Fu-zhen; Chen, Xiaoping; Yang, Fuqian
2014-04-01
Tooth enamel is often subjected to repeated contact and often experiences contact deformation in daily life. The mechanical strength of the enamel determines the biofunctionality of the tooth. Considering the variation of the rod arrangement in outer and inner enamel, we approximate enamel as a bilayer structure and perform finite element analysis of the cyclic indentation of the bilayer structure, to mimic the repeated contact of enamel during mastication. The dynamic deformation behaviour of both the inner enamel and the bilayer enamel is examined. The material parameters of the inner and outer enamel used in the analysis are obtained by fitting the finite element results with the experimental nanoindentation results. The penetration depth per cycle at the quasi-steady state is used to describe the depth propagation speed, which exhibits a two-stage power-law dependence on the maximum indentation load and the amplitude of the cyclic load, respectively. The continuous penetration of the indenter reflects the propagation of the plastic zone during cyclic indentation, which is related to the energy dissipation. The outer enamel serves as a protective layer due to its great resistance to contact deformation in comparison to the inner enamel. The larger equivalent plastic strain and lower stresses in the inner enamel during cyclic indentation, as calculated from the finite element analysis, indicate better crack/fracture resistance of the inner enamel.
13. Finite element modeling of TFTR poloidal field coils
Baumgartner, J.A.; O'Toole, J.A.
1986-01-01
The Tokamak Fusion Test Reactor (TFTR) Poloidal Field (PF) coils were originally analyzed to TFTR design conditions. The coils have been reanalyzed by PPPL and Grumman to determine operating limits under as-built conditions. Critical stress levels, based upon data obtained from the reanalysis of each PF coil, are needed for input to the TFTR simulation code algorithms. The primary objective regarding structural integrity has been to ascertain the magnitude and location of critical internal stresses in each PF coil due to various combinations of electromagnetic and thermally induced loads. For each PF coil, a global finite element model (FEM) of a coil sector is being analyzed to obtain the basic coil internal loads and displacements. Subsequent fine mesh local models of the coil lead stem and lead spur regions produce the magnitudes and locations of peak stresses. Each copper turn and its surrounding insulation are modeled using solid finite elements. The corresponding electromagnetic and thermal analyses are similarly modeled. A series of test beams were developed to determine the best combination of MSC/NASTRAN-type finite elements for use in PF coil analysis. The results of this analysis compare favorably with those obtained by the earlier analysis which was limited in scope
14. Periodic Boundary Conditions in the ALEGRA Finite Element Code
Aidun, John B.; Robinson, Allen C.; Weatherby, Joe R.
1999-01-01
This document describes the implementation of periodic boundary conditions in the ALEGRA finite element code. ALEGRA is an arbitrary Lagrangian-Eulerian multi-physics code with both explicit and implicit numerical algorithms. The periodic boundary implementation requires a consistent set of boundary input sets which are used to describe virtual periodic regions. The implementation is noninvasive to the majority of the ALEGRA coding and is based on the distributed memory parallel framework in ALEGRA. The technique involves extending the ghost element concept for interprocessor boundary communications in ALEGRA to additionally support on- and off-processor periodic boundary communications. The user interface, algorithmic details and sample computations are given
15. Finite elements for non-linear analysis of pipelines
Benjamim, A.C.; Ebecken, N.F.F.
1982-01-01
The application of a three-dimensional lagrangian formulation for the great dislocations analysis and great rotation of pipelines systems is studied. This formulation is derived from the soil mechanics and take into account the shear stress effects. Two finite element models are implemented. The first, of right axis, uses as interpolation functions the conventional gantry functions, defined in relation to mobile coordinates. The second, of curve axis and variable cross sections, is obtained from the degeneration of the three-dimensional isoparametric element, and uses as interpolation functions third degree polynomials. (E.G.) [pt
16. Analysis of Piezoelectric Solids using Finite Element Method
Aslam, Mohammed; Nagarajan, Praveen; Remanan, Mini
2018-03-01
Piezoelectric materials are extensively used in smart structures as sensors and actuators. In this paper, static analysis of three piezoelectric solids is done using general-purpose finite element software, Abaqus. The simulation results from Abaqus are compared with the results obtained using numerical methods like Boundary Element Method (BEM) and meshless point collocation method (PCM). The BEM and PCM are cumbersome for complex shape and complicated boundary conditions. This paper shows that the software Abaqus can be used to solve the governing equations of piezoelectric solids in a much simpler and faster way than the BEM and PCM.
17. OPTIM, Minimization of Band-Width of Finite Elements Problems
Huart, M.
1977-01-01
1 - Nature of the physical problem solved: To minimize the band-width of finite element problems. 2 - Method of solution: A surface is constructed from the x-y-coordinates of each node using its node number as z-value. This surface consists of triangles. Nodes are renumbered in such a way as to minimize the surface area. 3 - Restrictions on the complexity of the problem: This program is applicable to 2-D problems. It is dimensioned for a maximum of 1000 elements
18. Navier-Stokes equations by the finite element method
Portella, P.E.
1984-01-01
A computer program to solve the Navier-Stokes equations by using the Finite Element Method is implemented. The solutions variables investigated are stream-function/vorticity in the steady case and velocity/pressure in the steady state and transient cases. For steady state flow the equations are solved simultaneously by the Newton-Raphson method. For the time dependent formulation, a fractional step method is employed to discretize in time and artificial viscosity is used to preclude spurious oscilations in the solution. The element used is the three node triangle. Some numerical examples are presented and comparisons are made with applications already existent. (Author) [pt
19. Model Reduction in Dynamic Finite Element Analysis of Lightweight Structures
Flodén, Ola; Persson, Kent; Sjöström, Anders
2012-01-01
models may be created by assembling models of floor and wall structures into large models of complete buildings. When assembling the floor and wall models, the number of degrees of freedom quickly increases to exceed the limits of computer capacity, at least in a reasonable amount of computational time...... Hz. Three different methods of model reduction were investigated; Guyan reduction, component mode synthesis and a third approach where a new finite element model was created with structural elements. Eigenvalue and steady-state analyses were performed in order to compare the errors...
20. Bolted Ribs Analysis for the ITER Vacuum Vessel using Finite Element Submodelling Techniques
Zarzalejos, José María, E-mail: jose.zarzalejos@ext.f4e.europa.eu [External at F4E, c/Josep Pla, n.2, Torres Diagonal Litoral, Edificio B3, E-08019, Barcelona (Spain); Fernández, Elena; Caixas, Joan; Bayón, Angel [F4E, c/Josep Pla, n.2, Torres Diagonal Litoral, Edificio B3, E-08019, Barcelona (Spain); Polo, Joaquín [Iberdrola Ingeniería y Construcción, Avenida de Manoteras 20, 28050 Madrid (Spain); Guirao, Julio [Numerical Analysis Technologies, S L., Marqués de San Esteban 52, Entlo, 33209 Gijon (Spain); García Cid, Javier [Iberdrola Ingeniería y Construcción, Avenida de Manoteras 20, 28050 Madrid (Spain); Rodríguez, Eduardo [Mechanical Engineering Department EPSIG, University of Oviedo, Gijon (Spain)
2014-10-15
Highlights: • The ITER Vacuum Vessel Bolted Ribs assemblies are modelled using Finite Elements. • Finite Element submodelling techniques are used. • Stress results are obtained for all the assemblies and a post-processing is performed. • All the elements of the assemblies are compliant with the regulatory provisions. • Submodelling is a time-efficient solution to verify the structural integrity of this type of structures. - Abstract: The ITER Vacuum Vessel (VV) primary function is to enclose the plasmas produced by the ITER Tokamak. Since it acts as the first radiological barrier of the plasma, it is classified as a class 2 welded box structure, according to RCC-MR 2007. The VV is made of an inner and an outer D-shape, 60 mm-thick double shell connected through thick massive bars (housings) and toroidal and poloidal structural stiffening ribs. In order to provide neutronic shielding to the ex-vessel components, the space between shells is filled with borated steel plates, called In-Wall Shielding (IWS) blocks, and water. In general, these blocks are connected to the IWS ribs which are connected to adjacent housings. The development of a Finite Element model of the ITER VV including all its components in detail is unaffordable from the computational point of view due to the large number of degrees of freedom it would require. This limitation can be overcome by using submodelling techniques to simulate the behaviour of the bolted ribs assemblies. Submodelling is a Finite Element technique which allows getting more accurate results in a given region of a coarse model by generating an independent, finer model of the region under study. In this paper, the methodology and several simulations of the VV bolted ribs assemblies using submodelling techniques are presented. A stress assessment has been performed for the elements involved in the assembly considering possible types of failure and including stress classification and categorization techniques to analyse
1. Finite element analysis of FRP-strengthened RC beams
Teeraphot Supaviriyakit
2004-05-01
Full Text Available This paper presents a non-linear finite element analysis of reinforced concrete beam strengthened with externally bonded FRP plates. The finite element modeling of FRP-strengthened beams is demonstrated. Concrete and reinforcing bars are modeled together as 8-node isoparametric 2D RC element. The FRP plate is modeled as 8-node isoparametric 2D elastic element. The glue is modeled as perfect compatibility by directly connecting the nodes of FRP with those of concrete since there is no failure at the glue layer. The key to the analysis is the correct material models of concrete, steel and FRP. Cracks and steel bars are modeled as smeared over the entire element. Stress-strain properties of cracked concrete consist of tensile stress model normal to crack, compressive stress model parallel to crack and shear stress model tangential to crack. Stressstrain property of reinforcement is assumed to be elastic-hardening to account for the bond between concrete and steel bars. FRP is modeled as elastic-brittle material. From the analysis, it is found that FEM can predict the load-displacement relation, ultimate load and failure mode of the beam correctly. It can also capture the cracking process for both shear-flexural peeling and end peeling modes similar to the experiment.
2. Three dimensional finite element linear analysis of reinforced concrete structures
Inbasakaran, M.; Pandarinathan, V.G.; Krishnamoorthy, C.S.
1979-01-01
A twenty noded isoparametric reinforced concrete solid element for the three dimensional linear elastic stress analysis of reinforced concrete structures is presented. The reinforcement is directly included as an integral part of the element thus facilitating discretization of the structure independent of the orientation of reinforcement. Concrete stiffness is evaluated by taking 3 x 3 x 3 Gauss integration rule and steel stiffness is evaluated numerically by considering three Gaussian points along the length of reinforcement. The numerical integration for steel stiffness necessiates the conversion of global coordiantes of the Gaussian points to nondimensional local coordinates and this is done by Newton Raphson iterative method. Subroutines for the above formulation have been developed and added to SAP and STAP routines for solving the examples. The validity of the reinforced concrete element is verified by comparison of results from finite element analysis and analytical results. It is concluded that this finite element model provides a valuable analytical tool for the three dimensional elastic stress analysis of concrete structures like beams curved in plan and nuclear containment vessels. (orig.)
3. Application of finite element numerical technique to nuclear reactor geometries
Rouai, N M [Nuclear engineering department faculty of engineering Al-fateh universty, Tripoli (Libyan Arab Jamahiriya)
1995-10-01
Determination of the temperature distribution in nuclear elements is of utmost importance to ensure that the temperature stays within safe limits during reactor operation. This paper discusses the use of Finite element numerical technique (FE) for the solution of the two dimensional heat conduction equation in geometries related to nuclear reactor cores. The FE solution stats with variational calculus which considers transforming the heat conduction equation into an integral equation I(O) and seeks a function that minimizes this integral and hence gives the solution to the heat conduction equation. In this paper FE theory as applied to heat conduction is briefly outlined and a 2-D program is used to apply the theory to simple shapes and to two gas cooled reactor fuel elements. Good results are obtained for both cases with reasonable number of elements. 7 figs.
4. Application of finite element numerical technique to nuclear reactor geometries
Rouai, N. M.
1995-01-01
Determination of the temperature distribution in nuclear elements is of utmost importance to ensure that the temperature stays within safe limits during reactor operation. This paper discusses the use of Finite element numerical technique (FE) for the solution of the two dimensional heat conduction equation in geometries related to nuclear reactor cores. The FE solution stats with variational calculus which considers transforming the heat conduction equation into an integral equation I(O) and seeks a function that minimizes this integral and hence gives the solution to the heat conduction equation. In this paper FE theory as applied to heat conduction is briefly outlined and a 2-D program is used to apply the theory to simple shapes and to two gas cooled reactor fuel elements. Good results are obtained for both cases with reasonable number of elements. 7 figs
5. Higher Order Lagrange Finite Elements In M3D
Chen, J.; Strauss, H.R.; Jardin, S.C.; Park, W.; Sugiyama, L.E.; Fu, G.; Breslau, J.
2004-01-01
The M3D code has been using linear finite elements to represent multilevel MHD on 2-D poloidal planes. Triangular higher order elements, up to third order, are constructed here in order to provide M3D the capability to solve highly anisotropic transport problems. It is found that higher order elements are essential to resolve the thin transition layer characteristic of the anisotropic transport equation, particularly when the strong anisotropic direction is not aligned with one of the Cartesian coordinates. The transition layer is measured by the profile width, which is zero for infinite anisotropy. It is shown that only higher order schemes have the ability to make this layer converge towards zero when the anisotropy gets stronger and stronger. Two cases are considered. One has the strong transport direction partially aligned with one of the element edges, the other doesn't have any alignment. Both cases have the strong transport direction misaligned with the grid line by some angles
6. Finite-element time evolution operator for the anharmonic oscillator
Milton, Kimball A.
1995-01-01
The finite-element approach to lattice field theory is both highly accurate (relative errors approximately 1/N(exp 2), where N is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is H = p(exp 2)/2 + lambda q(exp 4)/4. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a results for the anharmonic oscillator ground state energy accurate to better than 1 percent, while a two-state approximation reduces the error to less than 0.1 percent.
7. Experimental validation of finite element analysis of human vertebral collapse under large compressive strains.
Hosseini, Hadi S; Clouthier, Allison L; Zysset, Philippe K
2014-04-01
Osteoporosis-related vertebral fractures represent a major health problem in elderly populations. Such fractures can often only be diagnosed after a substantial deformation history of the vertebral body. Therefore, it remains a challenge for clinicians to distinguish between stable and progressive potentially harmful fractures. Accordingly, novel criteria for selection of the appropriate conservative or surgical treatment are urgently needed. Computer tomography-based finite element analysis is an increasingly accepted method to predict the quasi-static vertebral strength and to follow up this small strain property longitudinally in time. A recent development in constitutive modeling allows us to simulate strain localization and densification in trabecular bone under large compressive strains without mesh dependence. The aim of this work was to validate this recently developed constitutive model of trabecular bone for the prediction of strain localization and densification in the human vertebral body subjected to large compressive deformation. A custom-made stepwise loading device mounted in a high resolution peripheral computer tomography system was used to describe the progressive collapse of 13 human vertebrae under axial compression. Continuum finite element analyses of the 13 compression tests were realized and the zones of high volumetric strain were compared with the experiments. A fair qualitative correspondence of the strain localization zone between the experiment and finite element analysis was achieved in 9 out of 13 tests and significant correlations of the volumetric strains were obtained throughout the range of applied axial compression. Interestingly, the stepwise propagating localization zones in trabecular bone converged to the buckling locations in the cortical shell. While the adopted continuum finite element approach still suffers from several limitations, these encouraging preliminary results towards the prediction of extended vertebral
8. A least squares principle unifying finite element, finite difference and nodal methods for diffusion theory
Ackroyd, R.T.
1987-01-01
A least squares principle is described which uses a penalty function treatment of boundary and interface conditions. Appropriate choices of the trial functions and vectors employed in a dual representation of an approximate solution established complementary principles for the diffusion equation. A geometrical interpretation of the principles provides weighted residual methods for diffusion theory, thus establishing a unification of least squares, variational and weighted residual methods. The complementary principles are used with either a trial function for the flux or a trial vector for the current to establish for regular meshes a connection between finite element, finite difference and nodal methods, which can be exact if the mesh pitches are chosen appropriately. Whereas the coefficients in the usual nodal equations have to be determined iteratively, those derived via the complementary principles are given explicitly in terms of the data. For the further development of the connection between finite element, finite difference and nodal methods, some hybrid variational methods are described which employ both a trial function and a trial vector. (author)
9. Generalization of mixed multiscale finite element methods with applications
Lee, C S [Texas A & M Univ., College Station, TX (United States)
2016-08-01
Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms. For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces. Two enrichment approaches are proposed, one is based on generalized multiscale finte element metthods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (rAMGe). The former one, which is called mixed GMsFEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor. The latter enrichment approach is based on rAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces. Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. iii
10. Unified Formulation Applied to Free Vibrations Finite Element Analysis of Beams with Arbitrary Section
E. Carrera
2011-01-01
Full Text Available This paper presents hierarchical finite elements on the basis of the Carrera Unified Formulation for free vibrations analysis of beam with arbitrary section geometries. The displacement components are expanded in terms of the section coordinates, (x, y, using a set of 1-D generalized displacement variables. N-order Taylor type expansions are employed. N is a free parameter of the formulation, it is supposed to be as high as 4. Linear (2 nodes, quadratic (3 nodes and cubic (4 nodes approximations along the beam axis, (z, are introduced to develop finite element matrices. These are obtained in terms of a few fundamental nuclei whose form is independent of both N and the number of element nodes. Natural frequencies and vibration modes are computed. Convergence and assessment with available results is first made considering different type of beam elements and expansion orders. Additional analyses consider different beam sections (square, annular and airfoil shaped as well as boundary conditions (simply supported and cantilever beams. It has mainly been concluded that the proposed model is capable of detecting 3-D effects on the vibration modes as well as predicting shell-type vibration modes in case of thin walled beam sections.
11. Numerical modeling of the dynamic behavior of structures under impact with a discrete elements / finite elements coupling
Rousseau, J.
2009-07-01
That study focuses on concrete structures submitted to impact loading and is aimed at predicting local damage in the vicinity of an impact zone as well as the global response of the structure. The Discrete Element Method (DEM) seems particularly well suited in this context for modeling fractures. An identification process of DEM material parameters from macroscopic data (Young's modulus, compressive and tensile strength, fracture energy, etc.) will first be presented for the purpose of enhancing reproducibility and reliability of the simulation results with DE samples of various sizes. Then, a particular interaction, between concrete and steel elements, was developed for the simulation of reinforced concrete. The discrete elements method was validated on quasi-static and dynamic tests carried out on small samples of concrete and reinforced concrete. Finally, discrete elements were used to simulate impacts on reinforced concrete slabs in order to confront the results with experimental tests. The modeling of a large structure by means of DEM may lead to prohibitive computation times. A refined discretization becomes required in the vicinity of the impact, while the structure may be modeled using a coarse FE mesh further from the impact area, where the material behaves elastically. A coupled discrete-finite element approach is thus proposed: the impact zone is modeled by means of DE and elastic FE are used on the rest of the structure. An existing method for 3D finite elements was extended to shells. This new method was then validated on many quasi-static and dynamic tests. The proposed approach is then applied to an impact on a concrete structure in order to validate the coupled method and compare computation times. (author)
12. Study of Finite Element Number Influence over the Obtained Results in Finite Element Analyses of a Mechanical Structure
Ana-Maria Budai
2013-05-01
Full Text Available This paper present the results of a study that was made to establish the influence of finite element number used to determined the real load of a structure. Actually, the study represent a linear static analyze for a link gear control mechanism of a Kaplan turbine. The all analyze was made for the normal condition of functioning having like final scope to determine de life time duration of mentioned mechanism.
13. Finite element analysis of car hood for impact test by using ...
Finite element analysis of car hood for impact test by using solidworks software ... high safety and at the same time can be built according to market demands. ... Keywords: finite element analysis; impact test; Solidworks; automation, car hood.
14. Expanded Mixed Multiscale Finite Element Methods and Their Applications for Flows in Porous Media
Jiang, L.; Copeland, D.; Moulton, J. D.
2012-01-01
We develop a family of expanded mixed multiscale finite element methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed multiscale finite element formulation in the sense that four
15. Multiphase poroelastic finite element models for soft tissue structures
Simon, B.R.
1992-01-01
During the last two decades, biological structures with soft tissue components have been modeled using poroelastic or mixture-based constitutive laws, i.e., the material is viewed as a deformable (porous) solid matrix that is saturated by mobile tissue fluid. These structures exhibit a highly nonlinear, history-dependent material behavior; undergo finite strains; and may swell or shrink when tissue ionic concentrations are altered. Give the geometric and material complexity of soft tissue structures and that they are subjected to complicated initial and boundary conditions, finite element models (FEMs) have been very useful for quantitative structural analyses. This paper surveys recent applications of poroelastic and mixture-based theories and the associated FEMs for the study of the biomechanics of soft tissues, and indicates future directions for research in this area. Equivalent finite-strain poroelastic and mixture continuum biomechanical models are presented. Special attention is given to the identification of material properties using a porohyperelastic constitutive law ans a total Lagrangian view for the formulation. The associated FEMs are then formulated to include this porohyperelastic material response and finite strains. Extensions of the theory are suggested in order to include inherent viscoelasticity, transport phenomena, and swelling in soft tissue structures. A number of biomechanical research areas are identified, and possible applications of the porohyperelastic and mixture-based FEMs are suggested. 62 refs., 11 figs., 3 tabs
16. Nonlinear magnetohydrodynamics simulation using high-order finite elements
Plimpton, Steven James; Schnack, D.D.; Tarditi, A.; Chu, M.S.; Gianakon, T.A.; Kruger, S.E.; Nebel, R.A.; Barnes, D.C.; Sovinec, C.R.; Glasser, A.H.
2005-01-01
A conforming representation composed of 2D finite elements and finite Fourier series is applied to 3D nonlinear non-ideal magnetohydrodynamics using a semi-implicit time-advance. The self-adjoint semi-implicit operator and variational approach to spatial discretization are synergistic and enable simulation in the extremely stiff conditions found in high temperature plasmas without sacrificing the geometric flexibility needed for modeling laboratory experiments. Growth rates for resistive tearing modes with experimentally relevant Lundquist number are computed accurately with time-steps that are large with respect to the global Alfven time and moderate spatial resolution when the finite elements have basis functions of polynomial degree (p) two or larger. An error diffusion method controls the generation of magnetic divergence error. Convergence studies show that this approach is effective for continuous basis functions with p (ge) 2, where the number of test functions for the divergence control terms is less than the number of degrees of freedom in the expansion for vector fields. Anisotropic thermal conduction at realistic ratios of parallel to perpendicular conductivity (x(parallel)/x(perpendicular)) is computed accurately with p (ge) 3 without mesh alignment. A simulation of tearing-mode evolution for a shaped toroidal tokamak equilibrium demonstrates the effectiveness of the algorithm in nonlinear conditions, and its results are used to verify the accuracy of the numerical anisotropic thermal conduction in 3D magnetic topologies.
17. Accounting for straight parts effects on elbow's flexibilities in a beam type finite element program
Millard, A.
1983-01-01
An extension of Von Karman's theory is applied to the calculations of the flexibility factor of a pipe bend terminated by a straight part or a flange. This analysis is restricted to the linear elastic deformation behaviour under in plane bending. Analytical solutions are given for the propagation of ovalization in the elbow and in the straight part. Considering the response of the piping structures, we note that the ovalization of the piping systems are reduced significantly when the straight parts or flanges effects are included. This results are presented in terms of global as well local flexibility factors. They have been compared to numerical results obtained by shell type finite elements method. A complete piping system is analyzed, for economical reasons, with a beam type approach. Also, we show how it is possible to take into account an elbow's flexibilities the straight parts effects by means of flexibilities factors introduced in a beam type elements. We have implemented this method in the computer program TEDEL. In some specific geometrical features, we compare solutions using shell type elements and our formulation. (orig.)
18. Accounting for straight parts effects on elbow's flexibilities in a beam type finite element program
Millard, A.; Vaghi, H.; Ricard, A.
1983-08-01
An extension of Von Karman's theory is applied to the calculations of the flexibility factor of a pipe bend terminated by a straight part or a flange. This analysis is restricted to the linear elastic deformation behaviour under in plane bending. Analytical solutions are given for the propagation of ovalization in the elbow and in the straight part. Considering the response of the piping structures, we note that the ovalization of the piping systems are reduced significantly when the straight parts or flanges effects are included. The results are presented in terms of global as well local flexibility factors. They have been compared to numerical results obtained by shell type finite element method. A complete piping system is analyzed, for economical reasons, with a beam type approach. Also, we show how it is possible to take into account on elbow's flexibilities the straight parts effects by means of flexibilities factors introduced in a beam type element. We have implemented this method in the computer program TEDEL. In some specific geometrical features, we compare solutions using shell type elements and our formulation
19. Quasi-Static Viscoelastic Finite Element Model of an Aircraft Tire
Johnson, Arthur R.; Tanner, John A.; Mason, Angela J.
1999-01-01
An elastic large displacement thick-shell mixed finite element is modified to allow for the calculation of viscoelastic stresses. Internal strain variables are introduced at the element's stress nodes and are employed to construct a viscous material model. First order ordinary differential equations relate the internal strain variables to the corresponding elastic strains at the stress nodes. The viscous stresses are computed from the internal strain variables using viscous moduli which are a fraction of the elastic moduli. The energy dissipated by the action of the viscous stresses is included in the mixed variational functional. The nonlinear quasi-static viscous equilibrium equations are then obtained. Previously developed Taylor expansions of the nonlinear elastic equilibrium equations are modified to include the viscous terms. A predictor-corrector time marching solution algorithm is employed to solve the algebraic-differential equations. The viscous shell element is employed to computationally simulate a stair-step loading and unloading of an aircraft tire in contact with a frictionless surface.
20. SPLAI: Computational Finite Element Model for Sensor Networks
Ruzana Ishak
2006-01-01
Full Text Available Wireless sensor network refers to a group of sensors, linked by a wireless medium to perform distributed sensing task. The primary interest is their capability in monitoring the physical environment through the deployment of numerous tiny, intelligent, wireless networked sensor nodes. Our interest consists of a sensor network, which includes a few specialized nodes called processing elements that can perform some limited computational capabilities. In this paper, we propose a model called SPLAI that allows the network to compute a finite element problem where the processing elements are modeled as the nodes in the linear triangular approximation problem. Our model also considers the case of some failures of the sensors. A simulation model to visualize this network has been developed using C++ on the Windows environment.
1. Probabilistic finite elements for fracture and fatigue analysis
Liu, W. K.; Belytschko, T.; Lawrence, M.; Besterfield, G. H.
1989-01-01
The fusion of the probabilistic finite element method (PFEM) and reliability analysis for probabilistic fracture mechanics (PFM) is presented. A comprehensive method for determining the probability of fatigue failure for curved crack growth was developed. The criterion for failure or performance function is stated as: the fatigue life of a component must exceed the service life of the component; otherwise failure will occur. An enriched element that has the near-crack-tip singular strain field embedded in the element is used to formulate the equilibrium equation and solve for the stress intensity factors at the crack-tip. Performance and accuracy of the method is demonstrated on a classical mode 1 fatigue problem.
2. Role of shell structure in the 2νββ nuclear matrix elements
1998-01-01
Significance of the nuclear shell structure in the ββ nuclear matrix elements is pointed out. The 2νββ processes are mainly mediated by the low-lying 1 + states. The shell structure also gives rise to concentration or fragmentation of the 2νββ components over intermediate states, depending on nuclide. These roles of the shell structure are numerically confirmed by realistic shell model calculations. Some shell structure effects are suggested for 0νββ matrix elements; dominance of low-lying intermediate states and nucleus-dependence of their spin-parities. (orig.)
3. Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches
Chróścielewski, Jacek; Schmidt, Rüdiger; Eremeyev, Victor A.
2018-05-01
This paper addresses modeling and finite element analysis of the transient large-amplitude vibration response of thin rod-type structures (e.g., plane curved beams, arches, ring shells) and its control by integrated piezoelectric layers. A geometrically nonlinear finite beam element for the analysis of piezolaminated structures is developed that is based on the Bernoulli hypothesis and the assumptions of small strains and finite rotations of the normal. The finite element model can be applied to static, stability, and transient analysis of smart structures consisting of a master structure and integrated piezoelectric actuator layers or patches attached to the upper and lower surfaces. Two problems are studied extensively: (i) FE analyses of a clamped semicircular ring shell that has been used as a benchmark problem for linear vibration control in several recent papers are critically reviewed and extended to account for the effects of structural nonlinearity and (ii) a smart circular arch subjected to a hydrostatic pressure load is investigated statically and dynamically in order to study the shift of bifurcation and limit points, eigenfrequencies, and eigenvectors, as well as vibration control for loading conditions which may lead to dynamic loss of stability.
4. Finite-element model of ultrasonic NDE [nondestructive evaluation
Lord, W.
1989-07-01
An understanding of the way in which ultrasound interacts with defects in materials is essential to the development of improved nondestructive testing procedures for the inspection of critical power plant components. Traditionally, the modeling of such phenomena has been approached from an analytical standpoint in which appropriate assumptions are made concerning material properties, geometrical constraints and defect boundaries in order to arrive at closed form solutions. Such assumptions, by their very nature, tend to inhibit the development of complete input/output NDT system models suitable for predicting realistic piezoelectric transducer signals from the interaction of pulsed, finite-aperture ultrasound with arbitrarily shaped defects in the kinds of materials of interest to the utilities. The major thrust of EPRI Project RP 2687-2 is to determine the feasibility of applying finite element analysis techniques to overcome these problems. 85 refs., 64 figs., 3 tabs
5. FEHM, Finite Element Heat and Mass Transfer Code
Zyvoloski, G.A.
2002-01-01
1 - Description of program or function: FEHM is a numerical simulation code for subsurface transport processes. It models 3-D, time-dependent, multiphase, multicomponent, non-isothermal, reactive flow through porous and fractured media. It can accurately represent complex 3-D geologic media and structures and their effects on subsurface flow and transport. Its capabilities include flow of gas, water, and heat; flow of air, water, and heat; multiple chemically reactive and sorbing tracers; finite element/finite volume formulation; coupled stress module; saturated and unsaturated media; and double porosity and double porosity/double permeability capabilities. 2 - Methods: FEHM uses a preconditioned conjugate gradient solution of coupled linear equations and a fully implicit, fully coupled Newton Raphson solution of nonlinear equations. It has the capability of simulating transport using either a advection/diffusion solution or a particle tracking method. 3 - Restriction on the complexity of the problem: Disk space and machine memory are the only limitations
6. Finite-element-analysis of fields radiated from ICRF antenna
Yamanaka, Kaoru; Sugihara, Ryo.
1984-04-01
In several simple geometries, electromagnetic fields radiated from a loop antenna, on which a current oscillately flows across the static magnetic field B-vector 0 , are calculated by the finite element method (FEM) as well as by analytic methods in a cross section of a plasma cylinder. A finite wave number along B-vector 0 is assumed. Good agreement between FEM and the analytic solutions is obtained, which indicates the accuracy of FEM solutions. The method is applied to calculations of fields from a half-turn antenna and reasonable results are obtained. It is found that a straightforward application of FEM to problems in an anisotropic medium may bring about erroneous results and that an appropriate coordinate transformation is needed for FEM to become applicable. (author)
7. Finite element method for neutron diffusion problems in hexagonal geometry
Wei, T.Y.C.; Hansen, K.F.
1975-06-01
The use of the finite element method for solving two-dimensional static neutron diffusion problems in hexagonal reactor configurations is considered. It is investigated as a possible alternative to the low-order finite difference method. Various piecewise polynomial spaces are examined for their use in hexagonal problems. The central questions which arise in the design of these spaces are the degree of incompleteness permissible and the advantages of using a low-order space fine-mesh approach over that of a high-order space coarse-mesh one. There is also the question of the degree of smoothness required. Two schemes for the construction of spaces are described and a number of specific spaces, constructed with the questions outlined above in mind, are presented. They range from a complete non-Lagrangian, non-Hermite quadratic space to an incomplete ninth order space. Results are presented for two-dimensional problems typical of a small high temperature gas-cooled reactor. From the results it is concluded that the space used should at least include the complete linear one. Complete spaces are to be preferred to totally incomplete ones. Once function continuity is imposed any additional degree of smoothness is of secondary importance. For flux shapes typical of the small high temperature gas-cooled reactor the linear space fine-mesh alternative is to be preferred to the perturbation quadratic space coarse-mesh one and the low-order finite difference method is to be preferred over both finite element schemes
8. A study on the improvement of shape optimization associated with the modification of a finite element
Sung, Jin Il; Yoo, Jeong Hoon
2002-01-01
In this paper, we investigate the effect and the importance of the accuracy of finite element analysis in the shape optimization based on the finite element method and improve the existing finite element which has inaccuracy in some cases. And then, the shape optimization is performed by using the improved finite element. One of the main stream to improve finite element is the prevention of locking phenomenon. In case of bending dominant problems, finite element solutions cannot be reliable because of shear locking phenomenon. In the process of shape optimization, the mesh distortion is large due to the change of the structure outline. So, we have to raise the accuracy of finite element analysis for the large mesh distortion. We cannot guarantee the accurate result unless the finite element itself is accurate or the finite elements are remeshed. So, we approach to more accurate shape optimization to diminish these inaccuracies by improving the existing finite element. The shape optimization using the modified finite element is applied to a two and three dimensional simple beam. Results show that the modified finite element has improved the optimization results
9. Thermal analysis of cracked bodies using finite element techniques
Hellen, T.K.; Price, R.H.; Harrison, R.P.
1975-01-01
The paper develops the potential energy equation in terms of finite element theory including thermal loads. Following this, the energy release rate and consequently the stress intensity factors are derived. Considerations of the classical near crack tip equations are made and deficiencies with the popular substitution methods are highlighted. A method of removing these deficiencies is described. Various energy methods are reconsidered in terms of the role of the thermal energy contribution to the potential energy. These methods include work of crack closure, energy compliance and virtual crack extensions with no other change in nodal geometry, and therefore only requires the recalculation of the stiffness matrices of the crack tip elements. An example of a quadratic temperature gradient parallel to the crack plane in an edge cracked plate is described. Comparisons of the various finite element methods are made and generally show good agreement. A second application compares the virtual crack extension method with an approximate analytical solution in determining stress intensity factors for a thick hollow cylinder with an axial crack for various depths through the wall thickness and for different times. Initially the cylinder is at a uniform high temperature and is then subjected to a sustained cooling shock. Analytical solutions are available for temperature and stress distributions in the uncracked pipe. The stress intensity for a shallow crack in the early stages of the transient has been determined using a superposition procedure. Comparison of the analytical and computed results shows good agreement between the methods
10. Finite Elements Based on Strong and Weak Formulations for Structural Mechanics: Stability, Accuracy and Reliability
Francesco Tornabene
2017-07-01
Full Text Available The authors are presenting a novel formulation based on the Differential Quadrature (DQ method which is used to approximate derivatives and integrals. The resulting scheme has been termed strong and weak form finite elements (SFEM or WFEM, according to the numerical scheme employed in the computation. Such numerical methods are applied to solve some structural problems related to the mechanical behavior of plates and shells, made of isotropic or composite materials. The main differences between these two approaches rely on the initial formulation – which is strong or weak (variational – and the implementation of the boundary conditions, that for the former include the continuity of stresses and displacements, whereas in the latter can consider the continuity of the displacements or both. The two methodologies consider also a mapping technique to transform an element of general shape described in Cartesian coordinates into the same element in the computational space. Such technique can be implemented by employing the classic Lagrangian-shaped elements with a fixed number of nodes along the element edges or blending functions which allow an “exact mapping” of the element. In particular, the authors are employing NURBS (Not-Uniform Rational B-Splines for such nonlinear mapping in order to use the “exact” shape of CAD designs.
11. Nonlinear finite element modeling of concrete deep beams with openings strengthened with externally-bonded composites
Hawileh, Rami A.; El-Maaddawy, Tamer A.; Naser, Mohannad Z.
2012-01-01
Highlights: ► A 3D nonlinear FE model is developed of RC deep beams with web openings. ► We used cohesion elements to simulate bond. ► The developed FE model is suitable for analysis of such complex structures. -- Abstract: This paper aims to develop 3D nonlinear finite element (FE) models for reinforced concrete (RC) deep beams containing web openings and strengthened in shear with carbon fiber reinforced polymer (CFRP) composite sheets. The web openings interrupted the natural load path either fully or partially. The FE models adopted realistic materials constitutive laws that account for the nonlinear behavior of materials. In the FE models, solid elements for concrete, multi-layer shell elements for CFRP and link elements for steel reinforcement were used to simulate the physical models. Special interface elements were implemented in the FE models to simulate the interfacial bond behavior between the concrete and CFRP composites. A comparison between the FE results and experimental data published in the literature demonstrated the validity of the computational models in capturing the structural response for both unstrengthened and CFRP-strengthened deep beams with openings. The developed FE models can serve as a numerical platform for performance prediction of RC deep beams with openings strengthened in shear with CFRP composites.
12. C1 finite elements on non-tensor-product 2d and 3d manifolds
Nguyen, Thien; Karčiauskas, Kęstutis; Peters, Jörg
2015-01-01
Geometrically continuous (Gk) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that are Ck also for non-tensor-product layout. This paper describes and analyzes one such concrete C1 geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregularities. The new gIGA element is based on a recently-developed G1 surface construction that recommends itself by its a B-spline-like control net, low (least) polynomial degree, good shape properties and reproduction of quadratics at irregular (extraordinary) points. Remarkably, for Poisson’s equation on the disk using interior vertices of valence 3 and symmetric layout, we observe O(h3) convergence in the L∞ norm for this family of elements. Numerical experiments confirm the elements to be effective for solving the trivariate Poisson equation on the solid cylinder, deformations thereof (a turbine blade), modeling and computing geodesics on smooth free-form surfaces via the heat equation, for solving the biharmonic equation on the disk and for Koiter-type thin-shell analysis. PMID:26594070
13. Energy Finite Element Analysis Developments for Vibration Analysis of Composite Aircraft Structures
Vlahopoulos, Nickolas; Schiller, Noah H.
2011-01-01
The Energy Finite Element Analysis (EFEA) has been utilized successfully for modeling complex structural-acoustic systems with isotropic structural material properties. In this paper, a formulation for modeling structures made out of composite materials is presented. An approach based on spectral finite element analysis is utilized first for developing the equivalent material properties for the composite material. These equivalent properties are employed in the EFEA governing differential equations for representing the composite materials and deriving the element level matrices. The power transmission characteristics at connections between members made out of non-isotropic composite material are considered for deriving suitable power transmission coefficients at junctions of interconnected members. These coefficients are utilized for computing the joint matrix that is needed to assemble the global system of EFEA equations. The global system of EFEA equations is solved numerically and the vibration levels within the entire system can be computed. The new EFEA formulation for modeling composite laminate structures is validated through comparison to test data collected from a representative composite aircraft fuselage that is made out of a composite outer shell and composite frames and stiffeners. NASA Langley constructed the composite cylinder and conducted the test measurements utilized in this work.
14. On the finite element modeling of the asymmetric cracked rotor
2013-05-01
The advanced phase of the breathing crack in the heavy duty horizontal rotor system is expected to be dominated by the open crack state rather than the breathing state after a short period of operation. The reason for this scenario is the expected plastic deformation in crack location due to a large compression stress field appears during the continuous shaft rotation. Based on that, the finite element modeling of a cracked rotor system with a transverse open crack is addressed here. The cracked rotor with the open crack model behaves as an asymmetric shaft due to the presence of the transverse edge crack. Hence, the time-varying area moments of inertia of the cracked section are employed in formulating the periodic finite element stiffness matrix which yields a linear time-periodic system. The harmonic balance method (HB) is used for solving the finite element (FE) equations of motion for studying the dynamic behavior of the system. The behavior of the whirl orbits during the passage through the subcritical rotational speeds of the open crack model is compared to that for the breathing crack model. The presence of the open crack with the unbalance force was found only to excite the 1/2 and 1/3 of the backward critical whirling speed. The whirl orbits in the neighborhood of these subcritical speeds were found to have nearly similar behavior for both open and breathing crack models. While unlike the breathing crack model, the subcritical forward whirling speeds have not been observed for the open crack model in the response to the unbalance force. As a result, the behavior of the whirl orbits during the passage through the forward subcritical rotational speeds is found to be enough to distinguish the breathing crack from the open crack model. These whirl orbits with inner loops that appear in the neighborhood of the forward subcritical speeds are then a unique property for the breathing crack model.
15. Stress categorization in nozzle to pressure vessel connections finite elements models
Albuquerque, Levi Barcelos de
1999-01-01
The ASME Boiler and Pressure Vessel Code, Section III , is the most important code for nuclear pressure vessels design. Its design criteria were developed to preclude the various pressure vessel failure modes throughout the so-called 'Design by Analysis', some of them by imposing stress limits. Thus, failure modes such as plastic collapse, excessive plastic deformation and incremental plastic deformation under cyclic loading (ratchetting) may be avoided by limiting the so-called primary and secondary stresses. At the time 'Design by Analysis' was developed (early 60's) the main tool for pressure vessel design was the shell discontinuity analysis, in which the results were given in membrane and bending stress distributions along shell sections. From that time, the Finite Element Method (FEM) has had a growing use in pressure vessels design. In this case, the stress results are neither normally separated in membrane and bending stress nor classified in primary and secondary stresses. This process of stress separation and classification in Finite Element (FE) results is what is called stress categorization. In order to perform the stress categorization to check results from FE models against the ASME Code stress limits, mainly from 3D solid FE models, several research works have been conducted. This work is included in this effort. First, a description of the ASME Code design criteria is presented. After that, a brief description of how the FEM can be used in pressure vessel design is showed. Several studies found in the literature on stress categorization for pressure vessel FE models are reviewed and commented. Then, the analyses done in this work are presented in which some typical nozzle to pressure vessel connections subjected to internal pressure and concentrated loads were modeled with solid finite elements. The results from linear elastic and limit load analyses are compared to each other and also with the results obtained by formulae for simple shell
16. 2D Finite Element Model of a CIGS Module
Janssen, G.J.M.; Slooff, L.H.; Bende, E.E. [ECN Solar Energy, P.O.Box 1, NL-1755 ZG Petten (Netherlands)
2012-06-15
The performance of thin-film CIGS (Copper indium gallium selenide) modules is often limited due to inhomogeneities in CIGS layers. A 2-dimensional Finite Element Model for CIGS modules is presented that predicts the impact of such inhomogeneities on the module performance. Results are presented of a module with a region of poor diode characteristics. It is concluded that according to this model the effects of poor diodes depend strongly on their location in the module and on their dispersion over the module surface. Due to its generic character the model can also be applied to other series connections of photovoltaic cells.
17. Finite element modeling of ultrasonic inspection of weldments
Dewey, B.R.; Adler, L.; Oliver, B.F.; Pickard, C.A.
1983-01-01
High performance weldments for critical service applications require 100% inspection. Balanced against the adaptability of the ultrasonic method for automated inspection are the difficulties encountered with nonhomogeneous and anisotropic materials. This research utilizes crystals and bicrystals of nickel to model austenitic weld metal, where the anisotropy produces scattering and mode conversion, making detection and measurement of actual defects difficult. Well characterized samples of Ni are produced in a levitation zone melting facility. Crystals in excess of 25 mm diameter and length are large enough to permit ultrasonic measurements of attenuation, wave speed, and spectral content. At the same time, the experiments are duplicated as finite element models for comparison purposes
18. Finite element calculation of stress induced heating of superconductors
Akin, J.E.; Moazed, A.
1976-01-01
This research is concerned with the calculation of the amount of heat generated due to the development of mechanical stresses in superconducting composites. An emperical equation is used to define the amount of stress-induced heat generation per unit volume. The equation relates the maximum applied stress and the experimental measured hysteresis loop of the composite stress-strain diagram. It is utilized in a finite element program to calculate the total stress-induced heat generation for the superconductor. An example analysis of a solenoid indicates that the stress-induced heating can be of the same order of magnitude as eddy current effects
19. Finite Element Simulation of Diametral Strength Test of Hydroxyapatite
Ozturk, Fahrettin; Toros, Serkan; Evis, Zafer
2011-01-01
In this study, the diametral strength test of sintered hydroxyapatite was simulated by the finite element software, ABAQUS/Standard. Stress distributions on diametral test sample were determined. The effect of sintering temperature on stress distribution of hydroxyapatite was studied. It was concluded that high sintering temperatures did not reduce the stress on hydroxyapatite. It had a negative effect on stress distribution of hydroxyapatite after 1300 deg. C. In addition to the porosity, other factors (sintering temperature, presence of phases and the degree of crystallinity) affect the diametral strength of the hydroxyapatite.
20. Assessing performance and validating finite element simulations using probabilistic knowledge
Dolin, Ronald M.; Rodriguez, E. A. (Edward A.)
2002-01-01
Two probabilistic approaches for assessing performance are presented. The first approach assesses probability of failure by simultaneously modeling all likely events. The probability each event causes failure along with the event's likelihood of occurrence contribute to the overall probability of failure. The second assessment method is based on stochastic sampling using an influence diagram. Latin-hypercube sampling is used to stochastically assess events. The overall probability of failure is taken as the maximum probability of failure of all the events. The Likelihood of Occurrence simulation suggests failure does not occur while the Stochastic Sampling approach predicts failure. The Likelihood of Occurrence results are used to validate finite element predictions.
1. Finite-element modeling and micromagnetic modeling of perpendicular writers
Heinonen, Olle; Bozeman, Steven P.
2006-04-01
We compare finite-element modeling (FEM) and fully micromagnetic modeling results of four prototypical writers for perpendicular recording. In general, the agreement between the two models is quite good in the vicinity of saturated or near-saturated magnetic material, such as the pole tip, for quantities such as the magnetic field, the gradient of the magnetic field and the write width. However, in the vicinity of magnetic material far from saturation, e.g., return pole or trailing edge write shield, there can be large qualitative and quantitative differences.
2. 2D - Finite element model of a CIGS module
Janssen, G.J.M.; Slooff, L.H.; Bende, E.E. [ECN Solar Energy, Petten (Netherlands)
2012-09-15
The performance of thin-film CIGS modules is often limited due to inhomogeneities in CIGS layers. A 2-dimensional Finite Element Model for CIGS modules is demonstrated that predicts the impact of such inhomogeneities on the module performance. Results are presented of a module with a region of poor diode characteristics. It is concluded that according to this model the effects of poor diodes depend strongly on their location in the module and on their dispersion over the module surface. Due to its generic character the model can also be applied to other series connections of photovoltaic cells.
3. Seakeeping with the semi-Lagrangian particle finite element method
Nadukandi, Prashanth; Servan-Camas, Borja; Becker, Pablo Agustín; Garcia-Espinosa, Julio
2017-07-01
The application of the semi-Lagrangian particle finite element method (SL-PFEM) for the seakeeping simulation of the wave adaptive modular vehicle under spray generating conditions is presented. The time integration of the Lagrangian advection is done using the explicit integration of the velocity and acceleration along the streamlines (X-IVAS). Despite the suitability of the SL-PFEM for the considered seakeeping application, small time steps were needed in the X-IVAS scheme to control the solution accuracy. A preliminary proposal to overcome this limitation of the X-IVAS scheme for seakeeping simulations is presented.
4. Finite element analysis of stemming loads on pipes
Maiden, D.E.
1979-08-01
A computational model has been developed for calculating the loads and displacements on a pipe placed in a hole which is subsequently filled with soil. A composite soil-pipe finite element model which employs fundamental material constants in its formalism is derived. The shear modulus of the soil, and the coefficient of friction at the pipe are the important constants to be specified. The calculated loads on the pipe are in agreement with experimental data for layered and unlayered stemming designs. As a result more economical designs of the pipe string can be realized
5. An introduction to the mathematical theory of finite elements
Oden, J T
2011-01-01
This introduction to the theory of Sobolev spaces and Hilbert space methods in partial differential equations is geared toward readers of modest mathematical backgrounds. It offers coherent, accessible demonstrations of the use of these techniques in developing the foundations of the theory of finite element approximations.J. T. Oden is Director of the Institute for Computational Engineering & Sciences (ICES) at the University of Texas at Austin, and J. N. Reddy is a Professor of Engineering at Texas A&M University. They developed this essentially self-contained text from their seminars and co
6. Finite element modeling and experimentation of bone drilling forces
Lughmani, W A; Bouazza-Marouf, K; Ashcroft, I
2013-01-01
Bone drilling is an essential part of many orthopaedic surgery procedures, including those for internal fixation and for attaching prosthetics. Estimation and control of bone drilling forces are critical to prevent drill breakthrough, excessive heat generation, and mechanical damage to the bone. This paper presents a 3D finite element (FE) model for prediction of thrust forces experienced during bone drilling. The model incorporates the dynamic characteristics involved in the process along with the accurate geometrical considerations. The average critical thrust forces and torques obtained using FE analysis, for set of machining parameters are found to be in good agreement with the experimental results
7. Applications of finite-element scaling analysis in primatology.
Richtsmeier, J T
1989-01-01
The study of biological shape in three dimensions using landmark data can now be accomplished using several alternative methods. This report focuses on the use of finite-element scaling analysis in primate craniofacial morphology. The method is particularly useful in its ability to localize the differences between forms, thereby indicating those loci that differ most between specimens. Several examples of this feature are provided from primatological research. Particulars of the methods are also discussed in an attempt to provide the reader with cautionary knowledge for prudent application of the method in future research.
8. Finite element method for time-space-fractional Schrodinger equation
Xiaogang Zhu
2017-07-01
Full Text Available In this article, we develop a fully discrete finite element method for the nonlinear Schrodinger equation (NLS with time- and space-fractional derivatives. The time-fractional derivative is described in Caputo's sense and the space-fractional derivative in Riesz's sense. Its stability is well derived; the convergent estimate is discussed by an orthogonal operator. We also extend the method to the two-dimensional time-space-fractional NLS and to avoid the iterative solvers at each time step, a linearized scheme is further conducted. Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods.
9. Finite element analysis of reticulated ceramics under compression
D’Angelo, Claudio; Ortona, Alberto; Colombo, Paolo
2012-01-01
Graphical abstract: - Abstract: This paper shows how finite element analysis can be used to study the effect of the morphological features of reticulated ceramics on their mechanical properties under compression. Quantitative morphological data, obtained by X-ray computed tomography (XCT) for a commercially available Si–SiC foam produced by the replica method, have been linked to a set of computer generated cells in which one morphological parameter was varied at a time. The findings indicate how the modification of some morphological features, which depend on the careful selection of appropriate and specific processing parameters, would enable the production of ceramic foams possessing higher strength for a given total porosity value.
10. Piezoelectric Analysis of Saw Sensor Using Finite Element Method
2013-06-01
Full Text Available In this contribution modeling and simulation of surface acoustic waves (SAW sensor using finite element method will be presented. SAW sensor is made from piezoelectric GaN layer and SiC substrate. Two different analysis types are investigated - modal and transient. Both analyses are only 2D. The goal of modal analysis, is to determine the eigenfrequency of SAW, which is used in following transient analysis. In transient analysis, wave propagation in SAW sensor is investigated. Both analyses were performed using FEM code ANSYS.
11. Eddy current analysis by the finite element circuit method
Kameari, A.; Suzuki, Y.
1977-01-01
The analysis of the transient eddy current in the conductors by ''Finite Element Circuit Method'' is developed. This method can be easily applied to various geometrical shapes of thin conductors. The eddy currents on the vacuum vessel and the upper and lower support plates of JT-60 machine (which is now being constructed by Japan Atomic Energy Research Institute) are calculated by this method. The magnetic field induced by the eddy current is estimated in the domain occupied by the plasma. And the force exerted to the vacuum vessel is also estimated
12. Finite element investigation of explosively formed projectiles (EFP)
1999-01-01
This thesis report represents the numerical simulation of explosively formed projectiles (EFP), a type of linear self-forging fragment device. The simulation is performed using a finite element code DYNA2D. It also explicates that how the shape, velocity and kinetic energy of an explosively formed projectile is effected by various parameters. Different parameters investigated are mesh density, material, thickness, contour and types of liner. Effect of shape of casing and material model is also analyzed. The shapes of projectiles at different times after detonation are shown. The maximum velocity and kinetic energy of the projectile have been used to ascertain the effect of above mentioned parameters. (author)
13. Finite Element Approximation of the FENE-P Model
Barrett , John ,; Boyaval , Sébastien
2017-01-01
We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain \$D \$\\subset\$ R d , d = 2 or 3\$, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by c...
14. Accurate evaluation of exchange fields in finite element micromagnetic solvers
Chang, R.; Escobar, M. A.; Li, S.; Lubarda, M. V.; Lomakin, V.
2012-04-01
Quadratic basis functions (QBFs) are implemented for solving the Landau-Lifshitz-Gilbert equation via the finite element method. This involves the introduction of a set of special testing functions compatible with the QBFs for evaluating the Laplacian operator. The results by using QBFs are significantly more accurate than those via linear basis functions. QBF approach leads to significantly more accurate results than conventionally used approaches based on linear basis functions. Importantly QBFs allow reducing the error of computing the exchange field by increasing the mesh density for structured and unstructured meshes. Numerical examples demonstrate the feasibility of the method.
15. A finite element model for the quench front evolution problem
Folescu, J.; Galeao, A.C.N.R.; Carmo, E.G.D. do.
1985-01-01
A model for the rewetting problem associated with the loss of coolant accident in a PWR reactor is proposed. A variational formulation for the time-dependent heat conduction problem on fuel rod cladding is used, and appropriate boundary conditions are assumed in order to simulate the thermal interaction between the fuel rod cladding and the fluid. A numerical procedure which uses the finite element method for the spatial discretization and a Crank-Nicolson-like method for the step-by-step integration is developed. Some numerical results are presented showing the quench front evolution and its stationary profile. (Author) [pt
16. Finite element method for simulation of the semiconductor devices
Zikatanov, L.T.; Kaschiev, M.S.
1991-01-01
An iterative method for solving the system of nonlinear equations of the drift-diffusion representation for the simulation of the semiconductor devices is worked out. The Petrov-Galerkin method is taken for the discretization of these equations using the bilinear finite elements. It is shown that the numerical scheme is a monotonous one and there are no oscillations of the solutions in the region of p-n transition. The numerical calculations of the simulation of one semiconductor device are presented. 13 refs.; 3 figs
17. Finite element analysis of advanced neutron source fuel plates
Luttrell, C.R.
1995-08-01
The proposed design for the Advanced Neutron Source reactor core consists of closely spaced involute fuel plates. Coolant flows between the plates at high velocities. It is vital that adjacent plates do not come in contact and that the coolant channels between the plates remain open. Several scenarios that could result in problems with the fuel plates are studied. Finite element analyses are performed on fuel plates under pressure from the coolant flowing between the plates at a high velocity, under pressure because of a partial flow blockage in one of the channels, and with different temperature profiles
18. 3D-finite element impact simulation on concrete structures
Heider, N.
1989-12-15
The analysis of impact processes is an interesting application of full 3D Finite Element calculations. This work presents a simulation of the penetration process of a Kinetic Energy projectile into a concrete target. Such a calculation requires an adequate FE model, especially a proper description of the crack opening process in front of the projectile. The aim is the prediction of the structural survival of the penetrator case with the help of an appropriate failure criterion. Also, the computer simulation allows a detailed analysis of the physical phenomena during impact. (orig.) With 4 refs., 14 figs.
19. Computation of stress intensity factors for nozzle corner cracks by various finite element procedures
Broekhoven, M.J.G.
1975-01-01
The present study aims at deriving accurate K-factors for a series of 5 elliptical nozzle corner cracks of increasing size by various finite element procedures, using a three-level recursive substructuring scheme to perform the computations in an economic way on an intermediate size computer (IBM 360/65 system). A nozzle on a flat plate has been selected for subsequent experimental verification, this configuration being considered an adequate simulation of a nozzle on a shallow shell. The computations have been performed with the ASKA finite element system using mainly HEXEC-27 (incomplete quartic) elements. The geometry has been subdivided into 5 subnets with a total of 3515 nodal points and 6250 unknowns, two main nets and one hyper net. Each crack front is described by 11 nodal points and all crack front nodes are inserted in the hyper net, which allows for the realization of the successive crack geometries by changing only a relatively small hyper net (615 to 725 unknowns). Output data have been interpreted in terms of K-factors by the global energy method, the displacement method and the stress method. Besides, a stiffness derivative procedure, recently developed at Brown University, which takes full advantage of the finite element formulation to calculate local K-factors, has been applied. Finally it has been investigated whether sufficiently accurate results can be obtained by analyzing a considerably smaller part than one half of the geometry (as strictly required by symmetry considerations), using fixed boundary conditions derived from a far cheaper analysis of the uncracked structure
20. A fully coupled finite element framework for thermal fracturing simulation in subsurface cold CO2 injection
Shunde Yin
2018-03-01
Simulation of thermal fracturing during cold CO2 injection involves the coupled processes of heat transfer, mass transport, rock deforming as well as fracture propagation. To model such a complex coupled system, a fully coupled finite element framework for thermal fracturing simulation is presented. This framework is based on the theory of non-isothermal multiphase flow in fracturing porous media. It takes advantage of recent advances in stabilized finite element and extended finite element methods. The stabilized finite element method overcomes the numerical instability encountered when the traditional finite element method is used to solve the convection dominated heat transfer equation, while the extended finite element method overcomes the limitation with traditional finite element method that a model has to be remeshed when a fracture is initiated or propagating and fracturing paths have to be aligned with element boundaries.
1. Finite element simulation of thermal, elastic and plastic phenomena in fuel elements
Soba, Alejandro; Denis, Alicia C.
1999-01-01
Taking as starting point an irradiation experiment of the first Argentine MOX fuel prototype, performed at the HFR reactor of Petten, Holland, the deformation suffered by the fuel element materials during burning has been numerically studied. Analysis of the pellet-cladding interaction is made by the finite element method. The code determines the temperature distribution and analyzes elastic and creep deformations, taking into account the dependency of the physical parameters of the problem on temperature. (author)
2. The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion
Moczo, P.; Kristek, J.; Pazak, P.; Balazovjech, M.; Moczo, P.; Kristek, J.; Galis, M.
2007-01-01
Numerical modeling of seismic wave propagation and earthquake motion is an irreplaceable tool in investigation of the Earth's structure, processes in the Earth, and particularly earthquake phenomena. Among various numerical methods, the finite-difference method is the dominant method in the modeling of earthquake motion. Moreover, it is becoming more important in the seismic exploration and structural modeling. At the same time we are convinced that the best time of the finite-difference method in seismology is in the future. This monograph provides tutorial and detailed introduction to the application of the finite difference (FD), finite-element (FE), and hybrid FD-FE methods to the modeling of seismic wave propagation and earthquake motion. The text does not cover all topics and aspects of the methods. We focus on those to which we have contributed. We present alternative formulations of equation of motion for a smooth elastic continuum. We then develop alternative formulations for a canonical problem with a welded material interface and free surface. We continue with a model of an earthquake source. We complete the general theoretical introduction by a chapter on the constitutive laws for elastic and viscoelastic media, and brief review of strong formulations of the equation of motion. What follows is a block of chapters on the finite-difference and finite-element methods. We develop FD targets for the free surface and welded material interface. We then present various FD schemes for a smooth continuum, free surface, and welded interface. We focus on the staggered-grid and mainly optimally-accurate FD schemes. We also present alternative formulations of the FE method. We include the FD and FE implementations of the traction-at-split-nodes method for simulation of dynamic rupture propagation. The FD modeling is applied to the model of the deep sedimentary Grenoble basin, France. The FD and FE methods are combined in the hybrid FD-FE method. The hybrid
3. Ship vibration analysis by finite element technique. Pt. II: Vibration analysis / Analyse van scheepstrillingen door middel van de elementenmethode. Dl. II: Trillingsanalyse
Hylarides, S.
1971-01-01
In the calculation of the natural frequencies of ships more accurate values are expected when the shell-like structure of ships is taken into account by the finite element technique, especially in the higher-node vibration modes. To avoid large matrix systems an elimination process has been
4. Hybrid Fundamental Solution Based Finite Element Method: Theory and Applications
Changyong Cao
2015-01-01
Full Text Available An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM and its application in engineering problems is presented in this paper. The framework and formulations of HFS-FEM for potential problem, plane elasticity, three-dimensional elasticity, thermoelasticity, anisotropic elasticity, and plane piezoelectricity are presented. In this method, two independent assumed fields (intraelement filed and auxiliary frame field are employed. The formulations for all cases are derived from the modified variational functionals and the fundamental solutions to a given problem. Generation of elemental stiffness equations from the modified variational principle is also described. Typical numerical examples are given to demonstrate the validity and performance of the HFS-FEM. Finally, a brief summary of the approach is provided and future trends in this field are identified.
5. Structural optimisation of cage induction motors using finite element analysis
Palko, S.
The current trend in motor design is to have highly efficient, low noise, low cost, and modular motors with a high power factor. High torque motors are useful in applications like servo motors, lifts, cranes, and rolling mills. This report contains a detailed review of different optimization methods applicable in various design problems. Special attention is given to the performance of different methods, when they are used with finite element analysis (FEA) as an objective function, and accuracy problems arising from the numerical simulations. Also an effective method for designing high starting torque and high efficiency motors is presented. The method described in this work utilizes FEA combined with algorithms for the optimization of the slot geometry. The optimization algorithm modifies the position of the nodal points in the element mesh. The number of independent variables ranges from 14 to 140 in this work.
6. OXYGEN PRESSURE REGULATOR DESIGN AND ANALYSIS THROUGH FINITE ELEMENT MODELING
Asterios KOSMARAS
2017-05-01
Full Text Available Oxygen production centers produce oxygen in high pressure that needs to be defused. A regulator is designed and analyzed in the current paper for medical use in oxygen production centers. This study aims to design a new oxygen pressure regulator and perform an analysis using Finite Element Modeling in order to evaluate its working principle. In the design procedure,the main elements and the operating principles of a pressure regulator are taking into account. The regulator is designed and simulations take place in order to assessthe proposed design. Stress analysis results are presented for the main body of the regulator, as well as, flow analysis to determine some important flow characteristics in the inlet and outlet of the regulator.
7. Probabilistic finite elements for fatigue and fracture analysis
Belytschko, Ted; Liu, Wing Kam
1993-04-01
An overview of the probabilistic finite element method (PFEM) developed by the authors and their colleagues in recent years is presented. The primary focus is placed on the development of PFEM for both structural mechanics problems and fracture mechanics problems. The perturbation techniques are used as major tools for the analytical derivation. The following topics are covered: (1) representation and discretization of random fields; (2) development of PFEM for the general linear transient problem and nonlinear elasticity using Hu-Washizu variational principle; (3) computational aspects; (4) discussions of the application of PFEM to the reliability analysis of both brittle fracture and fatigue; and (5) a stochastic computational tool based on stochastic boundary element (SBEM). Results are obtained for the reliability index and corresponding probability of failure for: (1) fatigue crack growth; (2) defect geometry; (3) fatigue parameters; and (4) applied loads. These results show that initial defect is a critical parameter.
8. GOMESH, Finite Elements Structure Plot with Triangular Mesh
Draper, J.
1977-01-01
1 - Nature of the physical problem solved: Graphical representation of calculations on structures with finite subdivision. 2 - Method of solution: GOMESH treats meshes with triangular basic elements. The program uses the same punched cards as those required for the input to the 'STAG' series of stress analysis codes and can prepare three basic mesh diagrams which differ in their mode of numbering. One objective of using these diagrams is to show up errors in the card deck by making them visually recognisable. Furthermore, digital tests are made within the program to check that certain requirements have been observed in the production of the lattice. The program 'GOMESH', can provide, superimposed in the graphical representation, stress and temperature values in numerical form, can represent the displacement of the mesh before and after a specified irradiation time, and give the directions and sense of the principal stresses occurring in the individual elements, in the form of arrows of varying length
9. 3D unstructured mesh discontinuous finite element hydro
Prasad, M.K.; Kershaw, D.S.; Shaw, M.J.
1995-01-01
The authors present detailed features of the ICF3D hydrodynamics code used for inertial fusion simulations. This code is intended to be a state-of-the-art upgrade of the well-known fluid code, LASNEX. ICF3D employs discontinuous finite elements on a discrete unstructured mesh consisting of a variety of 3D polyhedra including tetrahedra, prisms, and hexahedra. The authors discussed details of how the ROE-averaged second-order convection was applied on the discrete elements, and how the C++ coding interface has helped to simplify implementing the many physics and numerics modules within the code package. The author emphasized the virtues of object-oriented design in large scale projects such as ICF3D
10. QUADRATIC SERENDIPITY FINITE ELEMENTS ON POLYGONS USING GENERALIZED BARYCENTRIC COORDINATES.
Rand, Alexander; Gillette, Andrew; Bajaj, Chandrajit
2014-01-01
We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n -gon, our construction produces 2 n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n ( n + 1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.
11. Elasto-viscoplastic finite element model for prestressed concrete structures
Prates Junior, N.P.; Silva, C.S.B.; Campos Filho, A.; Gastal, F.P.S.L.
1995-01-01
This paper presents a computational model, based on the finite element method, for the study of reinforced and prestressed concrete structures under plane stress states. It comprehends short and long-term loading situations, where creep and shrinkage in concrete and steel relaxation are considered. Elasto-viscoplastic constitutive models are used to describe the behavior of the materials. The model includes prestressing and no prestressing reinforcement, on situation with pre- and post-tension with and without bond. A set of prestressed concrete slab elements were tested under instantaneous and long-term loading. The experimental data for deflections, deformations and ultimate strength are used to compare and validate the results obtained through the proposed model. (author). 11 refs., 5 figs
12. Fracture and Fragmentation of Simplicial Finite Elements Meshes using Graphs
Mota, A; Knap, J; Ortiz, M
2006-10-18
An approach for the topological representation of simplicial finite element meshes as graphs is presented. It is shown that by using a graph, the topological changes induced by fracture reduce to a few, local kernel operations. The performance of the graph representation is demonstrated and analyzed, using as reference the 3D fracture algorithm by Pandolfi and Ortiz [22]. It is shown that the graph representation initializes in O(N{sub E}{sup 1.1}) time and fractures in O(N{sub I}{sup 1.0}) time, while the reference implementation requires O(N{sub E}{sup 2.1}) time to initialize and O(N{sub I}{sup 1.9}) time to fracture, where NE is the number of elements in the mesh and N{sub I} is the number of interfaces to fracture.
13. Stress and Deformation Analysis in Base Isolation Elements Using the Finite Element Method
Claudiu Iavornic
2011-01-01
Full Text Available In Modern tools as Finite Element Method can be used to study the behavior of elastomeric isolation systems. The simulation results obtained in this way provide a large series of data about the behavior of elastomeric isolation bearings under different types of loads and help in taking right decisions regarding geometrical optimizations needed for improve such kind of devices.
14. Automating the generation of finite element dynamical cores with Firedrake
Ham, David; Mitchell, Lawrence; Homolya, Miklós; Luporini, Fabio; Gibson, Thomas; Kelly, Paul; Cotter, Colin; Lange, Michael; Kramer, Stephan; Shipton, Jemma; Yamazaki, Hiroe; Paganini, Alberto; Kärnä, Tuomas
2017-04-01
The development of a dynamical core is an increasingly complex software engineering undertaking. As the equations become more complete, the discretisations more sophisticated and the hardware acquires ever more fine-grained parallelism and deeper memory hierarchies, the problem of building, testing and modifying dynamical cores becomes increasingly complex. Here we present Firedrake, a code generation system for the finite element method with specialist features designed to support the creation of geoscientific models. Using Firedrake, the dynamical core developer writes the partial differential equations in weak form in a high level mathematical notation. Appropriate function spaces are chosen and time stepping loops written at the same high level. When the programme is run, Firedrake generates high performance C code for the resulting numerics which are executed in parallel. Models in Firedrake typically take a tiny fraction of the lines of code required by traditional hand-coding techniques. They support more sophisticated numerics than are easily achieved by hand, and the resulting code is frequently higher performance. Critically, debugging, modifying and extending a model written in Firedrake is vastly easier than by traditional methods due to the small, highly mathematical code base. Firedrake supports a wide range of key features for dynamical core creation: A vast range of discretisations, including both continuous and discontinuous spaces and mimetic (C-grid-like) elements which optimally represent force balances in geophysical flows. High aspect ratio layered meshes suitable for ocean and atmosphere domains. Curved elements for high accuracy representations of the sphere. Support for non-finite element operators, such as parametrisations. Access to PETSc, a world-leading library of programmable linear and nonlinear solvers. High performance adjoint models generated automatically by symbolically reasoning about the forward model. This poster will present
15. Finite element and analytical models for twisted and coiled actuator
Tang, Xintian; Liu, Yingxiang; Li, Kai; Chen, Weishan; Zhao, Jianguo
2018-01-01
Twisted and coiled actuator (TCA) is a class of recently discovered artificial muscle, which is usually made by twisting and coiling polymer fibers into spring-like structures. It has been widely studied since discovery due to its impressive output characteristics and bright prospects. However, its mathematical models describing the actuation in response to the temperature are still not fully developed. It is known that the large tensile stroke is resulted from the untwisting of the twisted fiber when heated. Thus, the recovered torque during untwisting is a key parameter in the mathematical model. This paper presents a simplified model for the recovered torque of TCA. Finite element method is used for evaluating the thermal stress of the twisted fiber. Based on the results of the finite element analyses, the constitutive equations of twisted fibers are simplified to develop an analytic model of the recovered torque. Finally, the model of the recovered torque is used to predict the deformation of TCA under varying temperatures and validated against experimental results. This work will enhance our understanding of the deformation mechanism of TCAs, which will pave the way for the closed-loop position control.
16. A Finite Element Method for Simulation of Compressible Cavitating Flows
Shams, Ehsan; Yang, Fan; Zhang, Yu; Sahni, Onkar; Shephard, Mark; Oberai, Assad
2016-11-01
This work focuses on a novel approach for finite element simulations of multi-phase flows which involve evolving interface with phase change. Modeling problems, such as cavitation, requires addressing multiple challenges, including compressibility of the vapor phase, interface physics caused by mass, momentum and energy fluxes. We have developed a mathematically consistent and robust computational approach to address these problems. We use stabilized finite element methods on unstructured meshes to solve for the compressible Navier-Stokes equations. Arbitrary Lagrangian-Eulerian formulation is used to handle the interface motions. Our method uses a mesh adaptation strategy to preserve the quality of the volumetric mesh, while the interface mesh moves along with the interface. The interface jump conditions are accurately represented using a discontinuous Galerkin method on the conservation laws. Condensation and evaporation rates at the interface are thermodynamically modeled to determine the interface velocity. We will present initial results on bubble cavitation the behavior of an attached cavitation zone in a separated boundary layer. We acknowledge the support from Army Research Office (ARO) under ARO Grant W911NF-14-1-0301.
17. Finite element based composite solution for neutron transport problems
Mirza, A.N.; Mirza, N.M.
1995-01-01
A finite element treatment for solving neutron transport problems is presented. The employs region-wise discontinuous finite elements for the spatial representation of the neutron angular flux, while spherical harmonics are used for directional dependence. Composite solutions has been obtained by using different orders of angular approximations in different parts of a system. The method has been successfully implemented for one dimensional slab and two dimensional rectangular geometry problems. An overall reduction in the number of nodal coefficients (more than 60% in some cases as compared to conventional schemes) has been achieved without loss of accuracy with better utilization of computational resources. The method also provides an efficient way of handling physically difficult situations such as treatment of voids in duct problems and sharply changing angular flux. It is observed that a great wealth of information about the spatial and directional dependence of the angular flux is obtained much more quickly as compared to Monte Carlo method, where most of the information in restricted to the locality of immediate interest. (author)
18. Nonlinear finite element analyses: advances and challenges in dental applications.
Wakabayashi, N; Ona, M; Suzuki, T; Igarashi, Y
2008-07-01
To discuss the development and current status of application of nonlinear finite element method (FEM) in dentistry. The literature was searched for original research articles with keywords such as nonlinear, finite element analysis, and tooth/dental/implant. References were selected manually or searched from the PUBMED and MEDLINE databases through November 2007. The nonlinear problems analyzed in FEM studies were reviewed and categorized into: (A) nonlinear simulations of the periodontal ligament (PDL), (B) plastic and viscoelastic behaviors of dental materials, (C) contact phenomena in tooth-to-tooth contact, (D) contact phenomena within prosthodontic structures, and (E) interfacial mechanics between the tooth and the restoration. The FEM in dentistry recently focused on simulation of realistic intra-oral conditions such as the nonlinear stress-strain relationship in the periodontal tissues and the contact phenomena in teeth, which could hardly be solved by the linear static model. The definition of contact area critically affects the reliability of the contact analyses, especially for implant-abutment complexes. To predict the failure risk of a bonded tooth-restoration interface, it is essential to assess the normal and shear stresses relative to the interface. The inclusion of viscoelasticity and plastic deformation to the program to account for the time-dependent, thermal sensitive, and largely deformable nature of dental materials would enhance its application. Further improvement of the nonlinear FEM solutions should be encouraged to widen the range of applications in dental and oral health science.
19. Finite element analysis for temperature distributions in a cold forging
Kim, Dong Bum; Lee, In Hwan; Cho, Hae Yong; Kim, Sung Wook; Song, In Chul; Jeon, Byung Cheol
2013-01-01
In this research, the finite element method is utilized to predict the temperature distributions in a cold-forging process for a cambolt. The cambolt is mainly used as a part of a suspension system of a vehicle. The cambolt has an off-centered lobe that manipulates the vertical position of the knuckle and wheel to a slight degree. The cambolt requires certain mechanical properties, such as strength and endurance limits. Moreover, temperature is also an important factor to realize mass production and improve efficiency. However, direct measurement of temperature in a forging process is infeasible with existing technology; therefore, there is a critical need for a new technique. Accordingly, in this study, a thermo-coupled finite element method is developed for predicting the temperature distribution. The rate of energy conversion to heat for the workpiece material is determined, and the temperature distribution is analyzed throughout the forging process for a cambolt. The temperatures associated with different punch speeds are also studied, as well as the relationships between load, temperature, and punch speed. Experimental verification of the technique is presented.
20. Mixed Generalized Multiscale Finite Element Methods and Applications
Chung, Eric T.
2015-03-03
In this paper, we present a mixed generalized multiscale finite element method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries, and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of oversampling techniques is to introduce a small dimensional snapshot space. We present numerical results for two-phase flow and transport, without updating basis functions in time. Our numerical results show that one can achieve good accuracy with a few basis functions per coarse edge if one selects appropriate offline spaces. © 2015 Society for Industrial and Applied Mathematics. | 95,285 | 482,539 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2020-34 | latest | en | 0.854283 |
https://www.jiskha.com/questions/102949/write-the-standard-form-of-the-equation-of-the-circle-that-passes-through-the-points-at | 1,624,561,849,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488556482.89/warc/CC-MAIN-20210624171713-20210624201713-00151.warc.gz | 753,797,831 | 5,894 | # math-equation of circle
Write the standard form of the equation of the circle that passes through the points at (4,5) (-2, 3), and (-4,-3).
a. (x-5)^2+(y+4)^2=49
b. (x-3)^2+(y+2)^2=50
c. (x+4)^2+(y-2)^2=36
d. (x-2)^2+(y+2)^2=25
I'm having trouble with solving for the variables...after I make the system of equations, I use elimination and wind up with odd numbers (1/9 for E, 307/9 for F).
If somebody who is good at this could show me their steps it would really help me understand what I'm doing wrong.
Thanks so much in advance, I really appreciate the help! MUCH needed!
1. 👍
2. 👎
3. 👁
1. Since this is multiple choice, the quickest way to determine the correct solution would be to plug in the given coordinates into each of the solutions.
1. 👍
2. 👎
2. Well, call center at (xc,yc) and radius r
then
(x-xc)^2 + (y-yc)^2 = r^2
distance from center to those points is the same = r
(4-xc)^2 + (5-yc)^2 = r^2
(-2-xc)^2 + (3-yc^2) = r^2
(-4-xc)^2 + (-3-yc^2) = r^2
16-8xc+xc^2+25-10yc+yc^2 = r^2
4+4xc+xc^2+9-6yc+yc^2 = r^2
16+8xc+xc^2+9+6yc+yc^2 = r^2
-8xc+xc^2-10yc+yc^2 = r^2-41
+4xc+xc^2-6yc+yc^2 = r^2-13
+8xc+xc^2+6yc+yc^2 = r^2-25
12 xc +4 yc = 28 changing sign of first and adding to second
4xc +12 yc = -12 changing sign of second and adding to third
12 xc +4 yc = 28
12 xc +36 yc = -36
-32yc = 64
yc = -2
ok, take it from there
1. 👍
2. 👎
3. of course it has to be b or d, but solve for xc to tell
4 xc + 12(-2) = -12
4 xc = 12
xc = 3
so center at
(3,-2)
and of form
(x-3)^2 +(y+2)^2 = r^2
1. 👍
2. 👎
4. Thanks Damon!!
This helped a lot...I was totally backwards on some things.
1. 👍
2. 👎
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https://bitbucket.org/alfonse/gltut/commits/ed68323e25f64a1910c34e111d9d87471822f362?at=0.3.1 | 1,440,967,896,000,000,000 | text/html | crawl-data/CC-MAIN-2015-35/segments/1440644065341.3/warc/CC-MAIN-20150827025425-00173-ip-10-171-96-226.ec2.internal.warc.gz | 857,543,786 | 24,996 | # Commits
committed ed68323
Corrections to tutorial docs.
• Participants
• Parent commits 750ee01
# File Documents/Illumination/Tutorial 12.xml
` </section>`
` <section>`
` <?dbhtml filename="Tut12 Monitors and Gamma.html" ?>`
`- <title>Monitors and Gamma</title>`
`+ <title>Linearity and Gamma</title>`
` <para>There is one major issue left, and it is one that has been glossed over since the`
` beginning of our look at lighting: your screen.</para>`
`- <para>The fundamental assumption underlying all of our lighting equations is the idea that`
`- the surface colors and light intensities are all in a linear`
`+ <para>The fundamental assumption underlying all of our lighting equations since the very`
`+ beginning is the idea that the surface colors and light intensities are all in a linear`
` <glossterm>colorspace</glossterm>. A colorspace defines how we translate from a set`
` of numerical values to actual, real colors that you can see. A colorspace is a`
` <glossterm>linear colorspace</glossterm> if doubling any value in that colorspace`
` we multiplied the sun and ambient intensities by 3 in the last section, we were`
` increasing the brightness by 3x. Multiplying the maximum intensity by 3 had the effect`
` of reducing the overall brightness by 3x.</para>`
`- <para>There's just one problem. Your screen doesn't work that way. Time for a short history`
`+ <para>There's just one problem: tour screen doesn't work that way. Time for a short history`
` of television/monitors.</para>`
` <para>The original televisions used an electron gun fired at a phosphor surface to generate`
` light and images; this is called a <acronym>CRT</acronym> display (cathode ray tube).`
` image.</para>`
` <para>The easiest way to deal with that in the earliest days of TV was to simply modify the`
` incoming image at the source. TV broadcasts sent image data that was non-linear in the`
`- opposite direction of the CRT's normal non-linearity. This resulted in a color`
`- reproduction in a linear colorspace.</para>`
`+ opposite direction of the CRT's normal non-linearity. Thus, the final output was`
`+ displayed linearly, as it was originally captured by the camera.</para>`
` <para>The term for this process, de-linearizing an image to compensate for a non-linear`
`- display, is <glossterm>gamma correction</glossterm>.</para>`
`+ display, is called <glossterm>gamma correction</glossterm>.</para>`
` <para>You may be wondering why this matters. After all, odds are, you don't have a CRT-based`
` monitor; you probably have some form of LCD, plasma, LED, or similar technology. So what`
` does the vagaries of CRT monitors matter to you?</para>`
` monitor's gamma-correction circuitry has been mangling.</para>`
` <section>`
` <title>Gamma Functions</title>`
`- <para>A <glossterm>gamma function</glossterm> is the function mapping linear RGB space`
`+ <para>A <glossterm>gamma function</glossterm> is the function that maps linear RGB space`
` to non-linear RGB space. The gamma function for CRT displays was fairly standard,`
` and all non-CRT displays mimic this standard. It is ultimately based on a math`
` function of CRT displays. The strength of the electron beam is controlled by the`
` };</programlisting>`
` </example>`
` <para>For the sake of clarity in this tutorial, we send the actual gamma value. For`
`- performance's sake, we should send 1/gamma, so that we don't have to do it in every`
`- fragment.</para>`
`+ performance's sake, we ought send 1/gamma, so that we don't have to needlessly do a`
`+ division in every fragment.</para>`
` <para>The gamma is applied in the fragment shader as follows:</para>`
` <example>`
` <title>Fragment Gamma Correction</title>`
` outputColor = pow(accumLighting, gamma);</programlisting>`
` </example>`
` <para>Otherwise, the code is mostly unchanged from the HDR tutorial. Speaking of which,`
`- gamma correction does not require HDR per se, but both are necessary for quality`
`- lighting results.</para>`
`+ gamma correction does not require HDR per se, nor does HDR require gamma correction.`
`+ However, the combination of the two has the power to create substantial improvements`
`+ in overall visual quality.</para>`
`+ <para>One final change in the code is for light values that are written directly,`
`+ without any lighting computations. The background color is simply the clear color`
`+ for the framebuffer. Even so, it needs gamma correction too; this is done on the CPU`
`+ by gamma correcting the color before drawing it. If you have any other colors that`
`+ are drawn directly, <emphasis>do not</emphasis> forget to do this.</para>`
` </section>`
` <section>`
` <title>Gamma Correct Lighting</title>`
`- <para>This is what happens when you apply HDR lighting to a scene who's light properties`
`- were defined <emphasis>without</emphasis> gamma correction. Look at the scene at`
`- night; the point lights are extremely bright, and their lighting effects seem to go`
`- farther than before. This last point bears investigating.</para>`
`+ <para>What we have seen is what happens when you apply HDR lighting to a scene who's`
`+ light properties were defined <emphasis>without</emphasis> gamma correction. Look at`
`+ the scene at night; the point lights are extremely bright, and their lighting`
`+ effects seem to go much farther than before. This last point bears`
`+ investigating.</para>`
` <para>When we first talked about light attenuation, we said that the correct attenuation`
` function for a point light was an inverse-square relationship with respect to the`
` distance to the light. We also said that this usually looked wrong, so people often`
` <para>Gamma is the reason for this. Or rather, lack of gamma correction is the reason.`
` Without correcting for the display's gamma function, the attenuation of <inlineequation>`
` <mathphrase>1/r<superscript>2</superscript></mathphrase>`
`- </inlineequation> becomes <inlineequation>`
`+ </inlineequation> effectively becomes <inlineequation>`
` <mathphrase>(1/r<superscript>2</superscript>)<superscript>2.2</superscript></mathphrase>`
` </inlineequation>, which is <inlineequation>`
` <mathphrase>1/r<superscript>4.4</superscript></mathphrase>`
` attenuation of lights. A simple <inlineequation>`
` <mathphrase>1/r</mathphrase>`
` </inlineequation> relationship looks better without gamma correction because the`
`- display's gamma function turns it into something that is much closer to being`
`- physically correct: <inlineequation>`
`+ display's gamma function turns it into something that is much closer to being in a`
`+ linear colorspace: <inlineequation>`
` <mathphrase>1/r<superscript>2.2</superscript></mathphrase>`
` </inlineequation>.</para>`
`- <para>Since this lighting was not designed while looking at gamma correct results, let's`
`- look at some scene lighting that was developed that way. Turn on gamma correction`
`- and set the gamma value to 2.2 (the default if you did not change it). The press <keycombo>`
`+ <para>Since this lighting environment was not designed while looking at gamma correct`
`+ results, let's look at some scene lighting that was developed that way. Turn on`
`+ gamma correction and set the gamma value to 2.2 (the default if you did not change`
`+ it). The press <keycombo>`
` <keycap>Shift</keycap>`
` <keycap>L</keycap>`
` </keycombo>:</para>`
` <para>If there is one point you should learn from this exercise, it is this: make sure`
` that you implement gamma correction and HDR <emphasis>before</emphasis> trying to`
` light your scenes. If you don't, then you may have to adjust all of the lighting`
`- parameters again. In this case, it wasn't even possible to use simple math to make`
`- the lighting work right. This lighting environment was developed from`
`+ parameters again, and you may need to change materials as well. In this case, it`
`+ wasn't even possible to use simple corrective math on the lighting environment to`
`+ make it work right. This lighting environment was developed essentially from`
` scratch.</para>`
` <para>One thing we can notice when looking at the gamma correct lighting is that proper`
` gamma correction improves shadow details substantially:</para>`
` </imageobject>`
` </mediaobject>`
` </figure>`
`- <para>These two images use HDR lighting; the one on the left doesn't have gamma`
`+ <para>These two images use the HDR lighting; the one on the left doesn't have gamma`
` correction, and the one on the right does. Notice how easy it is to make out the`
` details in the hills near the triangle on the right.</para>`
` <para>Looking at the gamma function, this makes sense. Without proper gamma correction,`
` fully half of the linearRGB range is shoved into the bottom one-fifth of the`
`- available light intensity. That doesn't leave much room for areas that are dark, but`
`+ available light intensity. That doesn't leave much room for areas that are dark but`
` not too dark to see anything.</para>`
` <para>As such, gamma correction is a key process for producing color-accurate rendered`
` images.</para>`
# File Documents/Illumination/Tutorial 13.xml
` <para>Computing the position is also easy. The position of a point on the surface of a`
` sphere is the normal at that position scaled by the radius and offset by the center`
` point of the sphere.</para>`
`+ <para>One final thing. Notice the square-root at the end, being applied to our`
`+ accumulated lighting. This effectively simulates a gamma of 2.0, but without the`
`+ expensive <function>pow</function> function call. A <function>sqrt</function> call`
`+ is much less expensive and far more likely to be directly built into the shader`
`+ hardware. Yes, this is not entirely accurate, since most displays simulate the 2.2`
`+ gamma of CRT displays. But it's a lot less inaccurate than applying no correction at`
`+ all. We'll discuss a much cheaper way to apply proper gamma correction in future`
`+ tutorials.</para>`
` </section>`
` </section>`
` <section>`
` <?dbhtml filename="Tut13 Correct Chicanery.html" ?>`
` <title>Correct Chicanery</title>`
`- <para>Our perfect sphere looks pretty nice. However, it is unfortunately very wrong.</para>`
`+ <para>Our perfect sphere looks pretty nice. It has no polygonal outlines and you can zoom in`
`+ on it forever. However, it is unfortunately very wrong.</para>`
` <para>To see how, toggle back to rendering the mesh on sphere <keycap>1</keycap> (the`
` central blue one). Then move the camera so that the sphere is at the left edge of the`
` screen. Then toggle back to impostor rendering.</para>`
` </imageobject>`
` </mediaobject>`
` </figure>`
`- <para>The line through the circle represents the square we drew. When viewing the sphere off`
`- to the side like this, we shouldn't be able to see the left-edge of the sphere facing`
`- perpendicular to the camera. And we should see some of the sphere on the right that is`
`- behind the plane.</para>`
`+ <para>The dark line through the circle represents the square we drew. When viewing the`
`+ sphere off to the side like this, we shouldn't be able to see the left-edge of the`
`+ sphere facing perpendicular to the camera. And we should see some of the sphere on the`
`+ right that is behind the plane.</para>`
` <para>So how do we solve this?</para>`
` <para>Use better math. Our last algorithm is a decent approximation if the spheres are`
` somewhat small. But if the spheres are reasonably large (which also can mean close to` | 2,965 | 13,583 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2015-35 | latest | en | 0.845825 |
http://mathcs.chapman.edu/~jipsen/structures/doku.php/reflexive_relations?rev=1280443597 | 1,576,505,290,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540565544.86/warc/CC-MAIN-20191216121204-20191216145204-00459.warc.gz | 94,287,654 | 5,092 | This is an old revision of the document!
## Reflexive relations
Abbreviation: RefRel
### Definition
A reflexive relation is a structure $\mathbf{X}=\langle X,R\rangle$ such that $R$ is a binary relation on $X$ (i.e. $R\subseteq X\times X$) that is
reflexive: $xRx$
Remark: This is a template. If you know something about this class, click on the Edit text of this page'' link at the bottom and fill out this page.
It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
##### Morphisms
Let $\mathbf{X}$ and $\mathbf{Y}$ be reflexive relations. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:A\rightarrow B$ that is a homomorphism: $xR^{\mathbf X} y\Longrightarrow h(x)R^{\mathbf Y}h(y)$
Example 1:
### Properties
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
Classtype quasivariety yes no no
### Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
### Superclasses
[[Directed graphs]] supervariety | 386 | 1,279 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2019-51 | latest | en | 0.761002 |
https://stage.geogebra.org/m/x5crrvt5 | 1,653,209,153,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662545090.44/warc/CC-MAIN-20220522063657-20220522093657-00426.warc.gz | 618,959,867 | 10,237 | # Rotation Station
Topic:
Rotation
Step 1 - Choose where you want your point of rotation to be by clicking the circle at the origin and moving it around. Click and move the X to rotate triangle ABC. A) Describe what changes as you move the reflection line around. Describe what stays the same. Step 2 - Reflect triangle ABC to a new location on the coordinate plane and write down the original coordinates and the new coordinates (triangle A'B'C'). Be sure to use the correct format for coordinate pairs (x,y). B) How could you be precise with your description of the transformation (how the shape moved from its original position to its new position)? | 137 | 654 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2022-21 | latest | en | 0.855673 |
https://ru.scribd.com/presentation/7290252/Hill-Climbing-1st-in-Class | 1,579,362,836,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250592636.25/warc/CC-MAIN-20200118135205-20200118163205-00526.warc.gz | 647,101,419 | 85,419 | Вы находитесь на странице: 1из 32
# When A* doesn’t work
## A few slides adapted from CS 471, Fall 2004, UBMC
(which were adapted from notes by Charles R.
Outline
• Local Search: Hill Climbing
• Escaping Local Maxima: Simulated Annealing
• Genetic Algorithms (if time allows)
## CIS 391 - Intro to AI
2
Local search and optimization
• Local search:
• Use single current state and move to neighboring states.
incrementally improve it until it is one
• Use very little memory
• Find often reasonable solutions in large or infinite state
spaces.
• Also useful for pure optimization problems.
• Find best state according to some objective function.
• e.g. survival of the fittest as a metaphor for optimization.
## CIS 391 - Intro to AI
3
Hill Climbing
Hill climbing on a surface of states
Height Defined by
Evaluation
Function
## CIS 391 - Intro to AI
5
Hill-climbing search: Take I & II
• While (∃ uphill points):
• Move in the direction of increasing value, lessening distance to goal
• If (∃ a successor si for the current state n such that
— h(si) < h(n)
— h(si) ≤ h(sj) for all successors sj of n, j≠ i,):
• then move from n to si.
• Otherwise, halt at n.
• Properties:
## • Terminates when a peak is reached.
• Does not look ahead of the immediate neighbors of the current state.
• Chooses randomly among the set of best successors, if there is more than
one.
• Doesn’t backtrack, since it doesn’t remember where it’s been
• a.k.a. greedy local search
## CIS 391 - Intro "Like
to AI climbing Everest in thick fog with amnesia"
6
Hill-climbing search: Take III
function HILL-CLIMBING( problem) return a state that is a local maximum
input: problem, a problem
local variables: current, a node.
neighbor, a node.
current ← MAKE-NODE(INITIAL-STATE[problem])
loop do
neighbor ← a highest valued successor of current
if VALUE [neighbor] ≤ VALUE[current] then return STATE[current]
current ← neighbor
## CIS 391 - Intro to AI
7
Hill climbing Example I
3 1 2 1 2
start 4 5 8 h=5 goal 3 4 5
6 7 6 7 8
5 h=4 5 2 h=0
3 1 2 3 1 2
4 5 8 h(n) = (number of 4 5
6 7 tiles out of place) 6 7 8
5 h=3 4 h=1
3 1 2 3 1 2
4 5 h=2 4 5
6 7 8 6 7 8
4
CIS 391 - Intro to AI
8
Hill-climbing Example: n-queens
• Put n queens on an n × n board with no two
queens on the same row, column, or diagonal
## CIS 391 - Intro to AI
9
Hill-climbing example: 8-queens
a) b)
## h = number of pairs of queens that are attacking each other
a) A state with h=17 and the h-value for each possible successor.
b) A local minimum of h in the 8-queens state space (h=1).
## CIS 391 - Intro to AI
10
Search Space features
## CIS 391 - Intro to AI
11
Drawbacks of hill climbing
• Problems:
• Local Maxima (foothills): peaks that aren’t the
highest point in the space
• Plateaus: the space has a broad flat region that
gives the search algorithm no direction (random
walk)
• Ridges: flat like a plateau, but with dropoffs to
the sides; steps to the North, East, South and
West may go down, but a step to the NW may
go up.
## CIS 391 - Intro to AI
12
Example of a local maximum
4 2
2
3 1 5
start 6 7 8 goal
4 1 2 4 1 2 1 2
3 5 3 7 5 2 3 4 5 0
6 7 8 6 8 6 7 8
1
4 1 2
3 5 2
6 7 8
CIS 391 - Intro to AI
13
The Shape of an Easy Problem
## CIS 391 - Intro to AI
14
The Shape of a Harder Problem
## CIS 391 - Intro to AI
15
The Shape of a Yet Harder Problem
## CIS 391 - Intro to AI
16
Remedies to drawbacks of hill climbing
• Random restart
• Problem reformulation
## • In the end: Some problem spaces are great for
hill climbing and others are terrible.
## CIS 391 - Intro to AI
17
Simulated Annealing
Simulated annealing (SA)
• Annealing: the process by which a metal cools and freezes
into a minimum-energy crystalline structure (the annealing
process)
• SA exploits an analogy between annealing and the search
for a minimum [or maximum] in a more general system.
## CIS 391 - Intro to AI
19
Simulated annealing
• Idea:
• Escape local maxima by allowing “bad” moves.
• But gradually decrease their size and frequency.
• Bouncing ball analogy:
• Shaking hard (= high temperature).
• Shaking less (= lower the temperature).
• Control parameter T
• By analogy with the original application is known as the system
“temperature.”
• T starts out high and gradually decreases toward 0.
• If T decreases slowly enough, then finds a global optimum with
probability approaching 1.
• Applied for VLSI layout, airline scheduling, etc.
## CIS 391 - Intro to AI
20
The Simulated Annealing Algorithm
function SIMULATED-ANNEALING( problem, schedule) return a solution state
input: problem, a problem
schedule, a mapping from time to temperature
local variables: current, a node.
next, a node.
T, a “temperature” controlling the probability of downward steps
current ← MAKE-NODE(INITIAL-STATE[problem])
for t ← 1 to ∞ do
T ← schedule[t]
if T = 0 then return current
next ← a randomly selected successor of current
∆E ← VALUE[next] - VALUE[current]
if ∆E > 0 then current ← next
else current ← next only with probability e∆E /T
## CIS 391 - Intro to AI
21
Local beam search
• Keep track of k states instead of one
• Initially: k random states
• Next: determine all successors of k states
• If any of successors is goal → finished
• Else select k best from successors and repeat.
## • Major difference with random-restart search
• Information is shared among k search threads.
• Can suffer from lack of diversity.
• Stochastic variant: choose k successors at proportionally to
state success.
## CIS 391 - Intro to AI
22
Genetic Algorithms
(only if time allows)
Genetic algorithms
• New states are generated by either
• “Mutation” of a single state or
• “Sexual Reproduction” (combining) of two parent states
(selected according to their fitness)
• Encoding used for the “genome” of an individual
strongly affects the behavior of the search
• Similar (in some ways) to stochastic beam
search
## CIS 391 - Intro to AI
24
Representation: Strings of genes
• Each chromosome
• represents a possible solution
• made up of a string of genes
• Each gene encodes some property of the solution
• There is a fitness metric on phenotypes of
chromosomes
• Evaluation of how well a solution with that set of properties
solves the problem.
• New generations are formed by
• Crossover: sexual reproduction
• Mutation: asexual reproduction
## CIS 391 - Intro to AI
25
Encoding of a Chromosome
• The chromosome encodes characteristics of the
solution which it represents, often as a string of
binary digits.
Chromosome 1 1101100100110110
Chromosome 2 1101111000011110
## • Each bit or set of bits in this string represents
some aspect of the solution.
## CIS 391 - Intro to AI
26
Example: Genetic Algorithm for Drive Train
Genes for:
• Number of Cylinders
• RPM: 1st -> 2nd
• RPM 2nd -> 3rd
• RPM 3rd -> Drive
• Rear end gear ratio
• Size of wheels
## CIS 391 - Intro to AI
27
Reproduction
• Reproduction by crossover selects genes from two parent
chromosomes and creates two new offspring.
• To do this, randomly choose some crossover point
(perhaps none).
• For the first child, everything before this point comes from
the first parent and everything after a from the second
parent.
• Crossover can then look like this ( | is the crossover point):
## Chromosome 1 11001 | 00100110110
Chromosome 2 10011 | 11000011110
## Offspring 1 11001 | 11000011110
Offspring 2 10011 | 00100110110
28
Mutation
## • Mutation randomly changes genes in the new
offspring.
• For binary encoding we can switch a few
randomly chosen bits from 1 to 0 or from 0 to 1.
## Original offspring 1101111000011110
Mutated offspring 1100111000001110
## CIS 391 - Intro to AI
29
The Basic Genetic Algorithm
1. Generate random population of chromosomes
2. Until the end condition is met, create a new
population by repeating following steps
• Evaluate the fitness of each chromosome
• Select two parent chromosomes from a population,
weighed by their fitness
• With probability pc cross over the parents to form a new
offspring.
• With probability pm mutate new offspring at each position on
the chromosome.
• Place new offspring in the new population
• Return the best solution in current population
## CIS 391 - Intro to AI
30
Genetic algorithm
function GENETIC_ALGORITHM( population, FITNESS-FN) return an individual
input: population, a set of individuals
FITNESS-FN, a function which determines the quality of the individual
repeat
new_population ← empty set
loop for i from 1 to SIZE(population) do
x ← RANDOM_SELECTION(population, FITNESS_FN)
y ← RANDOM_SELECTION(population, FITNESS_FN)
child ← REPRODUCE(x,y)
if (small random probability) then child ← MUTATE(child )
population ← new_population
until some individual is fit enough or enough time has elapsed
return the best individual
## CIS 391 - Intro to AI
31
Genetic algorithms:8-queens
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Industrial Relations And Labour Laws
Arung73
Service
+1 Other
Dear All,
There is some confusion regarding the over time payment. As per law it should be double payment of wages and wages includes the many components. My question is which component should be taken for OT calculation. Can we take all component or only Basic & DA. Our wages components are basic, DA, HRA and Conveyance. What the law says?
Regards
Arun Gupta
7th February 2011 From India, Delhi
As per law wages means basic wage and dearness allowance and as such you can calculate OT on basic and DA only. However, there can be a different argument from the workers that their basic wages is put at a lower level purposefully to reduce so many burdens like bonus, EPF contribution by the employer etc and their salary have been bifurcated in to small components just to benefit the employer and not the employee. Accepting their stand, the total salary for which they are entitled had they work for a day will have to be taken as base for calculation of overtime wage. Invariably, the same wage will be deducted if they absent themselves from work or they take leave without pay.
Regards,
7th February 2011 From India, Kannur
Thanks Madhu, We have not bifurcated the minimum wages. Apart from minimum wages we are giving other allowances. If case like this then what? Regds Arun Gupta
8th February 2011 From India, Delhi
Hello,
I have a doubt after calculating OT.
To calculate PT, as per law do we need to add OT to gross salary to calculate PT?
Salary Rs 8,980
Basic - 3592
HRA - 1437
Conv - 800
Med - 1200
Other Allow - 1901
OT - 1356
__________________
Gross Salary - 10,336
PT - 150
PF - 431
In this case emp salary is Rs 8,980. If i add OT within Gross salary then gross salary will increase more than Rs 10,000 then emp need to pay P.T.
Thanks
Sree
Edit/Delete Message
20th July 2011 From India, Bangalore
Profession Tax is not actually on 'salary' but it has a wider interpretation that it should be calculated on incomes received by an employee or profession. A Doctor or Lawyer is also expected to pay Profession Tax though they are not in receipt of any salary. Therefore, strictly speaking, all components of salary or remuneration irrespective of whether it is basis salary or OT wages or even bonus are to be taken into account for calculation of profession tax. However, the tax on profession being a state subject, there may be different interpretations in this regard. Therefore, you are requested to go through the definition of income received/ salary etc as gven in your state's Act.
Regards,
21st July 2011 From India, Kannur
i am working in a factory. here every one saying that OT should not be considered to calculate PT.
now i am just trying to find any written document on that. please let me know if you have any.
Thanks
Sree
22nd July 2011 From India, Bangalore
Cite.Co - is a repository of information created by your industry peers and experienced seniors. Register Here and help by adding your inputs to this topic/query page.
Prime Sponsor: TALENTEDGE - Certification Courses for career growth from top institutes like IIM / XLRI direct to device (online digital learning) | 813 | 3,386 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2019-35 | longest | en | 0.924155 |
https://rohanurich.wordpress.com/tag/chapter-thoughts/ | 1,540,160,044,000,000,000 | text/html | crawl-data/CC-MAIN-2018-43/segments/1539583514355.90/warc/CC-MAIN-20181021203102-20181021224602-00408.warc.gz | 760,199,950 | 31,015 | # The Dragon Curve (and a bit on Fractals)
If you have ever read Michael Crichton’s “Jurassic Park” (Yes, the 1993 movie was adapted from a book as most tend to be) then you would have noticed the unorthodox chapter headings.
Each chapter (or iteration as the book called it) was found to have a seemingly meaningless illustration of some lines and squares adjunct to a quotation from one the book’s characters.
The First Chapter/First Iteration contained the above image with the quote: “At the earliest drawings of the fractal curve, few clues to the underlying mathematical structure will be seen.” Ian Malcolm.
With each subsequent chapter, the quotations begin to resemble the events in the story – the idea of unpredictability; chaos theory – and the illustrations become more elaborate – and eerily reptilian.
# What we have is called a Dragon Curve.
It is a fractal with a very simple iterative process:
1. Draw a Line
2. Rotate a copy of the Line from Step 1 90 degrees (clockwise or anti-clockwise is arbitrary) and attach it to the end of the First Line. (You will now have two perpendicular lines – an L shape).
3. Rotate a copy of the entire L shape from Step 2 90 degrees (continue rotating it the same direction as you did in Step 2) and attach it to the end of the First L shape. (You will now have a saucepan shape).
4. Rotate a copy of the entire saucepan shape from Step 3 90 degrees (continue rotating it the same direction) and attach it to the end of the First saucepan shape.
5. Rotate a copy of the entire image and attach it to the end of the original.
Each step corresponds to the panel in the image below (read from left to right, top to bottom)
*Some things to be aware of is the overlapping that occurs from Step 5 onwards. If you’re attempting to draw the curve by hand, it would be helpful to use different colours to help yourself differentiate between the original and the rotated copy.
As you can see it is an extremely simple fractal design. You can see that the number of lines double with each iteration; this is because we are copying the previous iteration. Another thing to notice is that each section of the Dragon Curve (more noticeable in higher iterations) is reduced by a factor of and rotated by 45°.
Dragon Curves, like many other fractals, posses a property called Self-Similarity. For a fractal to be self-similar means that if you zoom at any specific region of the fractal, no matter you far you go, it will remain the same image. An example of this can be seen in the Sierpinski Triangle or Koch Snowflake.
Unlike some fractals, Dragon Curves can be produced with different iterative processes, though they are more complicated to pursue compared to the one mentioned previously.
The first method requires you to:
1. Take a square
2. Divide it in half horizontally.
3. Translate both halves in opposite directions horizontally such that their edges touch the midpoints of the longer edge.
4. Divide the shape in vertically in sixths
5. Translate these divisions vertically similar to Step 3.
6. Divide the shape in horizontally in tenths
7. Translate these divisions horizontally similar to Step 3.
8. and so on…
The second method requires you to:
1. Essentially draw a lot of triangles…I have been unable to understand the integral mechanics of this particular iterative process. But mathtuber TheMathGuy’s video on the Dragon Fractal (another name for the Dragon Curve; also called the Jurassic Park Dragon, Heighway Dragon, etc…) provides a detailed demonstration on another way to draw the Dragon Curve.
While the Dragon Curve is just another good-looking fractal that can be a fairly decent (yet cliché)
desktop wallpaper, fractals have numerous applications in fields such as astronomy, graphics design, meteorology, geology, economics and many more.
The iterative nature of fractals allow computer designers to map landscapes in video games and 3-D maps. Fractals can be used to understand the nature of different crystal lattice structures and correlate the iterative process to their strength. Price trends are found to possess self-similarity, just like fractals, as patterns in price fluctuations follow similar patterns in different time periods, regardless of magnitude. You can calculate the length of coastlines using fractal mathematics.
Fractals ominously occur in real life: the branches of a tree, the arrangement of flower petals, the design of a seashell, tributaries in a delta, crystalline structures of compounds and rocks (ice, diamond, halite), lightning, DNA, animal horns, and many many more.
The dragon is symbolic for its mystery and revered power, and the Dragon Curve is likewise the perfect testimonial to illustrate – and confirm – the power of fractals.
Works Cited:
123 4 5 6 7 8
# the Pledge
Imagine you’re in a game-show. You have managed to blaze through the quarter-finals and semi-finals and end up as one of the two finalist contestants at the threshold of receiving the Grand Prize of 1 Million Dollars!
Your final challenge is a simple guessing game. There are three doors: one of which contains the Grand Prize, and the remaining two contain a beautifully groomed and domesticated goat each. You must guess which door contains the Grand Prize. If you guess correctly, you receive the Grand Prize and are declared official winner of the game show. If you guess incorrectly, you will receive the goats, leaving the Grand Prize to your opponent.
You have no way of knowing which door conceals which item. Whichever door you pick, you will receive the prize behind it.
The game-show host asks you to pick a door. Let us say you had a good feeling about Door A and picked it. Here’s the twist. Before the game-show host reveals the contents behind Door A, he reveals to you the content of one of the remaining two doors. The game-show host will always reveal the location of one of the goats. Let’s say he opened Door B, which contained a goat.
Then game-show host offers you a second chance: do you want to stick with your initial choice, or would like to switch doors?
# the Turn
If your answer is that it doesn’t really make a difference whether you switch doors or not, then you are unfortunately, very very wrong. Surprising isn’t it? It in fact does make a difference, a very large one in fact, as you actually double your chances by switching from your initial choice. Don’t believe me?
Never fear, Mathematics is here!
# the Prestige
Assuming you are among the many who believed it didn’t make a difference (if you’re a smarty-pants and chose correctly to swap doors, then go be smart some place else…), then you probably made the false assumption that there was a 50% chance that the Grand Prize would be in Door A as well as Door C.
Here’s an example.
We will use the same arrangement, where Door A contains a Goat, Door B contains a Goat, and Door C contains the Grand Prize.
Let us set some cases:
1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat]
2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat]
3. You pick Door C [it contains the Grand Prize], the Host shows you Door A/B [it contains a Goat]
### Let’s count the number of times you win if you do not switch doors and remain with your initial choice.
1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat], you remain with Door A.
YOU LOSE
2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat], you remain with Door B.
YOU LOSE
3. You pick Door C [it contains the Grand Prize], the Host shows you Door A/B [it contains a Goat], you remain with Door C.
YOU WIN
Therefore:
• P(Lose if you do not switch) = 2/3
• P(Win if you do not switch) = 1/3
### Let’s count the number of times you win if you switch doors rather than remaining with your initial choice.
1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat], you switch to Door C.
YOU WIN
2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat], you switch to Door C.
YOU WIN
3. You pick Door C [it contains the Grand Prize], the Host shows you Door B [it contains a Goat], you switch to Door A.
YOU LOSE
Therefore:
• P(Lose if you switch) = 1/3
• P(Win if you switch) = 2/3
You can also come up with the same answer by drawing a tree diagram,
or using Conditional Probability calculations,
As you can see, the Probability of Winning doubles when you decide to switch doors and the Probability of Losing halves.
This problem is known as the Monty Hall Problem or Monty Hall Paradox, and is named after the host of the TV Game Show “Let’s Make a Deal”. This problem is recognized for its counter-intuitive result – a Veridical Paradox.
For many years, it has baffled people, especially renowned mathematicians, for its paradoxical solution. Numerous simulations have been created to test whether you are more likely to win if you switch doors. The New York Times created a Flash Simulation on their website that allows you to test for yourself the Monty Hall Problem.
The Mythbusters and many other science shows have all tried their hand at this problem, and switching has always proven to be the most profitable choice.
The most important lesson to take from the Monty Hall Problem is to realize that our gut/instinctive feeling is not always correct, and that we must always be skeptical of everything.
Works Cited
1 – 2 – 3 – 4 – 5 – 6
# Contact – Probable Impossibility or Improbable Possibility?
Introduction
It was a dry and blistering August 15, 1977 at the Ohioan Big Ear Telescope. SETI (Search for Extra-Terrestrial Intelligence) Radio Astronomer Jerry Ehman spent this afternoon laboriously examining the shelves of printed perforated paper that detailed radio signals emanating from the cosmos. Hours spent scrutinizing over every seemingly inconsequential signal led to him to come across an anomaly curiously inconsistent with the library of signal data. The 6EQUJ5 code circled with his red pen indicated a radio signal thirty times stronger than any other normal signal received and a frequency ominously identical to the hydrogen atom’s resonance frequency. Ehmer’s comment of the code – Wow! – modestly sums up the discovery. Was it first contact? Was it just an emission from celestial bodies like quasars, known for their violent ejections of electromagnetic energy? Alternatively, was it an alien races’ misguided attempt at invading our planet by kidnapping the Earth’s then “King”, Elvis Presley, who coincidentally “passed away” the following day?
This 72 second supposed alien message has since chosen not to reveal itself after our scouring the skies for a signal with the slightest resemblance. Whatever the origin and cause, the “Wow! Signal” had become an iconic symbol for the SETI’s goal, and by extension, humanity’s evolution as a technological and cosmological species. The implications of such a discovery were subject to our imagination. As Carl Sagan, astronomer once said, “Imagination will often carry us to worlds that never were. But without it we go nowhere”.
# The Drake Equation
An example of this use of imagination is the Drake Equation.
The Drake Equation, a mathematical equation created by astronomer Frank Drake in 1961, is used to approximate the number of detectable intelligent extraterrestrial civilizations in the Milky Way, and by extension our chances of encountering them. As you can see, the equation involves a variety of variables:
• N = the number of civilizations in our galaxy with which communication might be possible
• R* = number of new stars born each year
• fp = percentage/fraction of stars with planets
• ne = average number of habitable planets per solar system
• fℓ = percentage chance a habitable planet develops life
• fi = percentage chance that life develops intelligence
• fc = percentage chance life can communicate across space
• L = the length of time for which such civilizations release detectable signals into space
If you notice, the type of information for each following variable becomes more and more specific; it calculates the approximation from a macro-perspective and narrows the specifications of the variables. Essentially, the equation is structured using the Multiplicative/Fundamental Counting Principle.
The chances of finding intelligent extraterrestrial life in our galaxy can either be extremely high or extremely low. We must understand that the Drake Equation presently exists to provide perspective, rather than calculate a specific number. With the level of technology and knowledge we have so far, we can accurately come up with values for the first three variables by observing the billions of galaxies and stars in the night sky. But the values for the remaining four variables is up to our own pessimism or optimism and are the ones scientists are searching for. The problem with trying to predict the chance that a planet develops life that becomes intelligent that achieves interstellar communication is that we have only one data point, and that is us on our pale blue dot.
However, the Drake Equation is not necessarily perfect, as there have been numerous refinements and abridgements to it. For example, the nr variable describes the percentage chance of new civilizations evolving from previously extinct civilizations. This is an optional variable for the equation as it takes into consideration the billion year lifespans most terrestrial planets possess.
Each variable is laden with assumptions; for example, the fp variable factors only the number of terrestrial planets rather than gas giants Saturn and Jupiter. This makes the assumption that gas giants are unable to host intelligent life, as far as we know.
If we look at the types of values that would be used in the estimate, the values R*, fp, and ne are closer to powers of 10 rather than 1. The fl, fi, fc factors would all be numbers less than 1 (
10^-1 or even smaller). If we multiply these values together we can approximately get a value of 1. Therefore, based on these crude estimates, the number of civilization in our galaxy with which communication might be possible, N, is approximately equal to L, the longevity of a civilization.
Probabilities of there being intelligent aliens, when it comes to something as infinite as outer space, has the potential of being extremely large. Because space is so large and the formation of planets and other phenomenon is seemingly random, there is undoubtedly at least one (apart from our own) planet with intelligent life. This is analogous to finding one’s phone number in the digits of pi or any other transcendental number like e or phi. Because of pi’s random digits, there exists a chance of finding any conceivable number permutation, based on the notion of infinity and randomness. Based on these assumptions, that the universe is in fact infinite (or just terribly large) and its processes are random in nature, intelligent life should exist elsewhere. The issue is whether we can contact them or not.
# “Where are they?”
– was the big question asked by Italian Physicist Enrico Fermi. If Earth is considered an atypical planet orbiting an ordinary young yellow star like billions of other planets, there should be a multitude of civilizations in space – yet where are they? So far, we have not one strong piece of evidence supporting this assumption, just evidence of the contrary.
This idea is called the Fermi Paradox – “the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilization and humanity’s lack of contact with, or evidence for, such civilizations”.
There are numerous reasons for this being the case:
• The Universe is unimaginably large – the diameter of the observable universe (notice the word “observable”) is 93 billion light years. Intelligent life can possibly exist anywhere.
• The Speed of Light is finite – the speed of light in a vacuum is approximately 299 792 458 m / s. It would take a photon of light 93 000 000 000 years to travel from one edge of the observable universe to the opposite end.
• The Universe is still expanding – as discovered by astronomer Edwin Hubble from observing Doppler shifts from neighbouring celestial objects, the Universe is still expanding after 13.77 billion years. Like ants on an inflatable balloon, they get farther and farther apart even if they stay still. Similarly, everything is essentially flying away from us, and shouting “come back” into the night sky won’t help.
• Radio Transmissions never get to reach us – radio waves are just low frequency electromagnetic radiation or light. Light, in the presence of a large mass like a star or black hole, will have its path bent due to gravitational influence. Because of this, not only do the chances of alien radio transmissions reaching us slim, but our ability to accurately locate its source will reduce.
There are other explanations for this paradox:
• We are in fact alone – perhaps it takes extremely precise conditions for intelligent life to arise. This is known as the Rare Earth Hypothesis. This goes against the Mediocrity Principle – that Earth is like any other planet in the universe and thus the development of intelligent life is common. Proof against the Rare Earth Principle is the thousands of exo-planets that we have discovered that have Earth-like conditions.
• We’re not looking properly – perhaps aliens are transmitting data on frequencies unknown to us, or have encrypted transmissions that we might pass for insignificant.
• Intelligent Life tends to destroy itself – perhaps developing as an intelligent species inadvertently killed them in the process such as by destroying the ecosystem or mutual assured destruction through war.
• Natural disasters eradicate Intelligent Species – perhaps earthquakes, volcanic eruptions, emerging viruses, or meteor impacts may have caused intelligent life to die out.
• We have not searched long enough to detect and comprehend interstellar transmissions
There are also other explanations that can be slightly 1984-esque:
• We are purposely not contacted – perhaps extraterrestrial intelligence has developed and has decided that we never contact them. This is known as the Zoo Hypothesis, where we would be the animals in the cage being observed. Their reasons for doing so? They might be conducting experiments on us. They might be waiting for us to reach a higher technological level before contacting us.
• The Alien Civilization does not agree with itself. The question of who speaks for Earth/Humanity might be similar to what other civilizations might undergo.
• It would be dangerous to communicate with other alien civilizations – perhaps alien civilizations have found out that contacting other civilizations might be disastrous, and so have decided to avoid us.
• They are here on Earth unobserved – We might underestimate their size or form. They could be microscopic or they have taken our own human forms. Perhaps they are here among us. Maybe its the bus driver, your neighbour, your teacher…
# Conclusion
Calculating the probability of intelligent extraterrestrial life in the Universe is subject to numerous variables. Our chances of encountering them is largely determined by our physical constraints and perhaps extraterrestrials’ decisions
As Arthur C. Clarke once said: “Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.”
Works Cited
1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10
# Deriving the Power Rule using Binomial Theorem
This derivation goes to show once again the pervasiveness of mathematical concepts in different areas of the field. While Euler’s Formula does a better job of doing that, this simple derivation will suffice.
The Power Rule is often the first Derivative Rule many Calculus students will learn. It is commonly used to calculate the derivatives of polynomial functions, as well as reciprocal functions and nth roots of functions.
There are numerous and varied derivations and proofs on the internet that show the derivation of the Power Rule, but the one I will focus on involves using the Binomial Theorem through the First Principle or Difference Quotient.
When trying to come up with this derivation, I tried to avoid expanding the Summation. But evaluating the limit alongside manipulating the summation made it harder to see a clear set of steps to derive the Power Rule.
This proof makes you wonder (at least me) how Isaac Newton (or Gottfried Liebniz, but nobody really cares about him…) derived this Derivative Rule. Was it through observing the patterns in different polynomials? Was it using this exact derivation? Or was it…
Interestingly, the Power Rule was actually created by Liebniz rather than Newton, as commonly believed. Newton’s method of calculating polynomial derivatives was more complicated, thus being the reason why we continue using Liebniz’ notation. Newton and Liebniz were both contemporaries in the 17th century, with Newton being English and Liebniz being German. Newton had made his discoveries in 1666 and had them published in 1693, while Liebniz had made his in 1676 and published in 1684, sparking much controversy. This controversy was taken to court as it had become a matter of national pride for both countries. Being credited as the country for first inventing Calculus – then considered the forefront of mathematical innovation – would be immensely advantageous. As a result, it was taken to court under the mediation of the Royal Society (a society for the sciences – still present today). Coincidentally, Isaac Newton was the society’s president, thus leaving poor Gottfried Liebniz with charges of plagiarism. This story reveals the false perceptions we have of those in history.
Who would have considered Newton as someone who abused their own authority? It is similar to wondering if Leonardo Da Vinci was a bigot or Ludwig van Beethoven a drunkard.
If it is any consolation, England was far behind Europe for most of the 18th century in terms of mathematical innovation due to Newton’s complex notation.
In conclusion (a terrible way to end, but it gets the job done nonetheless), this specific derivation of the Power Rule reveals the interconnectedness of mathematical fields. The Binomial Theorem is useful in numerous situations and contexts, and this derivation proves so.
Works Cited
1 – 2 – 3
# Proving Binomial Theorem using Mathematical Induction
The Binomial Theorem is the perfect example to show how different streams in mathematics are connected to one another: its coefficients have combinatorial roots and can be traced to terms in Pascal’s Triangle, and expansion of binomials to different orders of power can describe probability and combination distributions. The combinatorial proof as under requires no need for proving again, but after learning a method called Mathematical Induction from incessant internet browsing on a late Saturday night, I thought, why not give it a shot?
Mathematical Induction is a method of mathematical proof used to prove an expression true for all natural numbers. The steps are as under:
1. State the proposition P(n) that needs proving.
2. The Basis: Show P(n) is true, when n=1.
3. The Inductive Step:
1. Assume n=k
2. If P(k) is true, show that P(k+1) is true
4. If P(k+1) is true, therefore P(n) is true.
(Side-note: It’s not everyday you get to use Q.E.D.)
# The Game of Permutations
What do mathematicians do? Simply, they try to find a mathematical model that describes the patterns that emerge in life, the universe, and everything. Mathematicians are employed in numerous industries in everything from manufacturing to finance. Mathematicians also, however, research in other topics that appear to have very little mathematical foundation. Take for example, board games. Sudoku are some examples that have been under the scrutiny of mathematicians all over the world.
The quest to accurately find all the possible Sudoku structures has been going on for many decades, and can often require the use of super computers to deal with the complex calculations involved. While it may appear, at first, to involve simple permutation calculations, the rules of the game restrict certain permutations from being included. A variety of research has been conducted to solve these problems, and I will try to do my best to explain them as easily as possible.
The research paper credited to have accurately calculated all the possible Sudoku arrangements goes Professor Bertram Felgenhauer (Dresden University of Technology, Germany) and Professor Frazer Jarvis’ (University of Sheffield, UK) paper Enumerating Possible Sudoku Grids (2005).
If we were to look at all the possible permutations for a 9 x 9 Sudoku grid exempt of its rules, it would total up to 9!^9 = 1.0911069e+50 (9! for each of the nine 3 x 3 grids). Their final calculation amounted to 6670903752021072936960 ≈ 6.671 × 1021.
The process in which they have calculated the answer is simplified as follows:
We shall create a classification system to identify parts of the grid.
• The 9 x 9 Sudoku grid is called a grid
• each 3 x 3 grid will be called a block. Therefore there are 9 blocks in total, creating the 9 x 9 grid.
• the three rows of blocks will be called bands of the grid.
• the three columns of blocks will be called the stacks.
• A cell in the ith row and jth column is said to be in (i,j) position.
• The number of distinct Sudoku grids will be denoted N.
• The digits 1-9 that fill up each block will be referred to as symbols or digits.
Fig. 1: Each block is denoted a name as B#.
How many ways are there to fill B1 in a valid way? (Remember that each block must contain nine symbols of 1 to 9 with no repetitions)
Looking solely at B1, we know that the first cell (1,1) has nine possible options (i.e., 1, 2, 3, 4, 5, 6, 7, 8, 9). The second cell (1,2) now has eight possible options, since the digit used in the (1,1) cannot be repeated. The third cell (1,3) now has seven remaining possible options and so onWhat we are essentially doing is counting the number of valid of permutations of nine symbols for B1: how we could arrange 9 symbols into 9 cells/order 9 symbols. Therefore the total permutations for B1 are 9P9 = 9!. However, we need to calculate the number of valid arrangements, since B1 will depend on the arrangements of other blocks.
Fig. 2: One possible permutation for B1.
If N represents the number of distinct grids (abiding by game rules) and Nthe number of distinct grids with the B1 permutation shown in Fig. 2, the total number of valid grids will be N1×9!, so N1=N/9!.
The authors decide to permute B2 and B3 in terms of one permutation of B1 as shown in Fig. 2.
Since 1, 2, and 3 occur in the first row in B1, these numbers cannot occur in the rest of the row. So only the numbers 4, 5, 6, 7, 8, and 9 from the second and third rows of B1 can be used in the first row in B2 and B3.
They classify these two possibilities as pure top rows – when the numbers {4,5,6} as in the second row of B1 are kept together in B2 and the numbers {7,8,9} as in the third row of B1 are kept together in B3, and the swapped version of this – and the rest of the possibilities as mixed top rows – where the sets {4,5,6} and {7,8,9} are mixed up when using them to fill in the first rows of B2 and B3.
With the use of pure and mixed top rows, twenty possibilities occur. With this, they try and complete the first band.
(Note: the permutation of B1 remains fixed)
There are (3!)6 ways to complete the first band in terms of the the pure top row 1,2,3;{4,5,6};{7,8,9}.
The cases of the mixed top rows are more complicated. Let’s consider the top row 1,2,3;{4,6,8};{5,7,9}. This can be completed to the first band as in Fig. 3 , where a, b, and c are the numbers 1, 2, 3 in any order.
Once a is picked, b and c are the remaining two numbers in any order, since they are in the same row. Since there are three choices for a, and the three digits in each of the six sets in B2 and B3 can be permuted to result in various valid first bands, the total number of configurations in this case is 3×(3!)6. You can similarly work out each of the remaining seventeen cases of first rows to obtain the same number.
Now we have the number of possible first bands given the standard B1: it is 2×(3!)6+18×3×(3!)6=2612736, where the first part of the sum is the number of first band completions from the pure top rows and the second is the number of first band completions from the mixed top rows.
Instead of calculating how many full grid completions each of these 2612736 possibilities has, Felgenhauer and Jarvis next determined which first bands share the same number of full grid completions. Such an analysis reduces the number of first bands that need to be considered when counting.
Here are some operations which leave the number of grid completions of the first band unchanged: relabeling the numbers, permuting any of the blocks in the first band, permuting the columns within any block, and permuting the three rows of the band. When any of these changes B1, we can relabel the digits to recover its standard form.
Permuting B1, B2, and B3 preserves the number of grid completions because if we start with any valid Sudoku grid, the only way to keep it valid would be to permute B4, B5, B6 and B7, B8, B9 correspondingly so that the stacks remain the same. In other words, every valid grid completion for the first band gives exactly one valid grid completion for the first band being any permutation of B1, B2, B3.
Such considerations allow us to reduce the number of specific first bands we need to consider when counting. Following Felgenhauer and Jarvis, we permute the columns of B2 and B3 so that the top row entries of each are in increasing order, and then swap B2 and B3 if necessary to make the first entry of B2 smaller than that of B3. This is called lexicographical reduction. Since there are 6 permutations of the columns in each of the two blocks and two ways to permute the blocks, lexicographical reduction tells us that, given a first band, there are 62×2=72 other first bands with the same number of grid completions. So now we only need to consider 2612736/72=36288 first bands.
For each of these possibilities, we consider permutations of all three top blocks: there are 6 of them. For each of these, there are 63 permutations of the columns within each block. After performing these operations, we relabel to get B1 back into standard form. We can similarly permute the top three rows of the band, and relabel to get B1 back into standard form. These operations preserve the number of grid completions of a first band. Felgenhauer and Jarvis used a computer program to determine that these operations reduce the number of first bands to be considered from 36288 to just 416.
The main point is that the band you started with and the different band you ended with have the same number of completions to a full Sudoku grid. So instead of calculating the number of grid completions for each of these bands, we can count it for just one of them.
There are more steps that reduce the number of grids to be considered. If we have a pair of digits {a,b} in one column with a in the ith row and b in the jth row, and the same pair in a different column with b in the ith row and a in the jth row, then swapping the places of a and b in each pair will result in a band that has the same number of grid completions as the original. This is because each pair lies in the same column, and swapping both at once keeps the One Rule satisfied for the rows involved as well. As an example, look at the numbers 8 and 9 in the sixth and ninth columns of the above example. Considering all possible cases of this left Felgenhauer and Jarvis with 174 out of the 416 first bands with which to proceed.
They also considered other configurations of the same set of digits lying in two different columns or rows, which can be permuted within their columns or rows leaving the number of grid completions invariant. This reduced the number of first bands to consider to 71, and searching through each of these 71 cases let them know that there are actually only 44 first bands whose number of grid completions need to be found. Each of these 44 bands has the same number of completions to a full grid.
Let C stand for one of these 44 bands. Then the number of ways that C can be completed to a full Sudoku grid can be calculated: call it nc. We also need the number mC of first bands that share this number nC of grid completions. Then the total number of Sudoku grids will just be N=ΣCmCnC, or the sum of mCnC over all of the 44 bands.
Felgenhauer and Jarvis wrote a computer program to carry out the final calculations. They computed the number N1 of valid completions with B1 in standard form, starting with the 44 bands. Then they multiplied this number by 9! to get the answer. They discovered that the number of possible 9 by 9 Sudoku grids is N=6670903752021072936960 which is approximately 6.671×1021.
Works Cited
1 | 7,400 | 33,242 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2018-43 | latest | en | 0.939904 |
https://essayflix.com/tsunamis-ocean-and-wave-shape-changes/ | 1,611,556,850,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703565376.63/warc/CC-MAIN-20210125061144-20210125091144-00770.warc.gz | 329,342,874 | 10,823 | # Tsunamis: Ocean and Wave Shape Changes
6 05 Tsunami Nicolai Kortendick 1. The web site presents extensive information regarding tsunamis. Survey the site. A. Select the five facts about tsunamis that were the most interesting or surprising to you. Make a list of your facts. 1. A tsunami is made up of a series of traveling ocean waves of extremely long wavelength. 2. They are triggered by earthquakes and undersea volcanic eruptions and deep sea landslides. 3. The wave shape changes and the height increases as it approaches the coastline. . Far field tsunamis have a long travel time so it is easier to predict their effects. 5. Near field tsunami have a travel time of one or two hours, making it harder to evacuate people to safe, high areas before the tsunami reaches the coast. B. Now look over your list. In your opinion, what is the most intriguing item on your list? Explain. The most intriguing item on the list to me is that tsunamis resemble waves that I see a lot every day and they have extremely long wavelengths. 2.
If you were on a ship at sea, and a tsunami passed under your ship, what would probably be your reaction? Explain. I would be pretty scared if I knew it was a tsunami, and I would be worried for the people on the coast it was heading for. It probably wouldn’t be a very big wave if I was far out in the ocean so it wouldn’t scare me as much. 3. The site offers a tsunami quiz. Take the quiz. What was your score? I got 7 out of the 10 questions correct. 4. When you viewed the “Introduction to Waves” video, you learned several terms that apply to all waves.
How do the following terms apply to tsunamis and what are typical values for a tsunami’s wavelength and amplitude? Use the following sites to look for answers: http://www. enchantedlearning. com/subjects/tsunami http://hyperphysics. phy-astr. gsu. edu/HBASE/Waves/tsunami. html C. Wavelength Tsunamis have an extremely long wavelength (which is the distance between the crest of one wave and the crest of the next wave) – up to several hundred miles long. D. amplitude
The amplitude of a wave is the height of a wave from the still water level to the top of the wave crest. As a tsunami reaches a coastline its amplitude increase greatly. E. crest The crest is the top of a wave. The wavelength of a wave is measured from the crest of one wave to the crest of another. F. trough The trough is the bottom of a wave. As a tsunami approaches the coast (where the sea becomes shallow), the trough of a wave hits the beach floor, causing the wave to slow down, to increase in height and to decrease in wavelength.
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STAY HOME, SAVE LIVES. Order your paper today and save 15% with the discount code FLIX | 682 | 2,899 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2021-04 | latest | en | 0.945959 |
https://www.physicsforums.com/threads/relativistic-velocity-scenario.912358/ | 1,600,439,609,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400187899.11/warc/CC-MAIN-20200918124116-20200918154116-00755.warc.gz | 1,047,428,297 | 18,359 | # Relativistic velocity scenario
• I
Lets suppose there is an observer at a certain distance from a planet that is moving at 0.9c as seen by the observer. We take as a reference, a frame where the observer is at rest. A spaceship leaves the planet and begins accelerating relative to the planet in the oposite direction. The observer should see the space ship decelerating and at some point having zero velocity, then the ship will apear to acceletate further in the opposite direction. Relative to the observer I suppose it is possible in principle can reach 0.9c. But when the observer checks the relative velocity between the planet and the spaceship it will conclude the spaceship managed to change its velocity bu 1.8c. Is this correct ?
Related Special and General Relativity News on Phys.org
PeterDonis
Mentor
2019 Award
Moderator's note: moved to relativity forum.
calinvass
PeterDonis
Mentor
2019 Award
when the observer checks the relative velocity between the planet and the spaceship it will conclude the spaceship managed to change its velocity bu 1.8c.
You have described this incorrectly. A correct description is: when the observer checks the difference in velocity, in his reference frame, between the planet and the spaceship, he will find it to be 1.8c. This is because the ship and the planet are moving in opposite directions, each at 0.9c, in his frame.
However, this 1.8c is not "the relative velocity between the planet and the spaceship". The latter term means either the velocity of the planet in the spaceship's frame, or the velocity of the spaceship in the planet's frame. (Numerically these are the same, they only differ in sign.) This can be found by the relativistic velocity addition formula: ##(0.9 + 0.9) / (1 + 0.9^2)##, which gives about 0.995c.
calinvass
Thank you for the correction. In other words the observer sees the spaceship and the planet are moving away from each other faster than the speed of light. But that is acceptable since the observer basically sees two objects traveling at 0.9c that happen to be in opposite directions.
But it is still funny because the ship will see the planet going away at 0.995c and the observer can talk to the crew and tell them he sees them separating at 1.9c.
Another scenario can be with two spaceship. They both are initially in a frame at rest then continuously accelerate (constant proper acceleration) in opposite directions. The relative velocity between them can in principle will approach c, but there must exist a reference frame where the spacecraft are separating away from each other at more than 1.9 c after a certain amount of time has passed.
Nugatory
Mentor
there must exist a reference frame where the spacecraft are separating away from each other at more than 1.9 c after a certain amount of time has passed.
Yes. The frame in which they were originally rest in one of these.
calinvass
Mister T
Gold Member
In other words the observer sees the spaceship and the planet are moving away from each other faster than the speed of light.
No. He sees the distance between them increasing at a rate that's faster than the speed of light. He knows, as a result of a calculation, how fast they move relative to each other. Do the calculation incorrectly and you get 1.8 c. Do it correctly and you get 0.95 c.
The important takeaway here is that it is still the case that nothing with mass can reach or exceed the speed of light relative to you, but you can observe two different things with mass approach or move apart from each other at a rate faster than c (but always slower than 2c).
From your perspective, light emitted from the ship will barely outpace the ship at 0.1c, and slowly catch up to earth at 0.1c, so nothing is actually moving faster than light relative to you.
calinvass
Yes. The frame in which they were originally rest in one of these.
Supposing only one ship starts to accelerate continuously , and the other remains still in the observer's frame. Again at some point I suppose there should be a frame where the difference between them is 1.9c (or whatever value greater than c we choose, depending on how much we wait), although in the observer's frame, this time the difference will never reach c.
Nugatory
Mentor
Supposing only one ship starts to accelerate continuously , and the other remains still in the observer's frame. Again at some point I suppose there should be a frame where the difference between them is 1.9c (or whatever value greater than c we choose, depending on how much we wait), although in the observer's frame, this time the difference will never reach c.
If by "whatever value greater than c we choose" you mean "whatever value greater than c and less than 2c we choose", and by "the observer's frame" you mean "the frame in which both ships are initially at rest", then yes.
calinvass
Thank you. Of course, less than 2c. But in this case, I think, if the observer remains in the frame where both spacecraft were at rest, it will only see a maximum just below c. (In this second example one of the spacecraft remains in the initial position) In order to see a difference of just below 2c it needs to follow the first spacecraft that leaves and to accelerate until it sees the spacecraft that remained at the initial position, just below c, then to stop accelerating and wait until the first one reaches near c.
Last edited:
FactChecker
Gold Member
Thank you. Of course, less than 2c. But in this case, I think, if the observer remains in the frame where both spacecraft were at rest, it will only see a maximum just below c. (In this second example one of the spacecraft remains in the initial position) In order to see a difference of just below 2c it needs to follow the first spacecraft that leaves and to accelerate until it sees the spacecraft that remained at the initial position, just below c, then to stop accelerating and wait until the first one reaches near c.
Not sure exactly what this all means, but the situation is fairly simple:
1) Nothing can move in any inertial reference frame faster than c.
2) Two objects can move apart in a "stationary" reference frame faster than c.
a) Suppose two objects are moving apart faster than c in a "stationary" reference frame. In either of their moving reference frames, the other is not moving faster than c.
b) A person in the "stationary" frame can use SR to understand why a person in one moving reference frame would say that the other moving object was not moving away faster than c.
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calinvass
Not sure exactly what this all means, but the situation is fairly simple:
1) Nothing can move in any inertial reference frame faster than c.
2) Two objects can move apart in a "stationary" reference frame faster than c.
a) Suppose two objects are moving apart faster than c in a "stationary" reference frame. In either of their moving reference frames, the other is not moving faster than c.
b) A person in the "stationary" frame can use SR to understand why a person in one moving reference frame would say that the other moving object was not moving away faster than c.
Thank you, I agree with the above statements. My example uses them but if what I said is correct (to me it looks clear) it may not be so obvious to say how it the thought experiment works. | 1,629 | 7,289 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2020-40 | latest | en | 0.929836 |
https://studydaddy.com/question/what-is-9-2-into-a-mixed-number | 1,638,740,303,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363216.90/warc/CC-MAIN-20211205191620-20211205221620-00475.warc.gz | 616,758,294 | 7,393 | Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
QUESTION
# What is 9/2 into a mixed number?
##9/2 = 4 1/2##
An improper fraction means that all the whole numbers are given as fractions.
'Counting in halves' would be:
##1/2" "2/2" "3/2" "4/2" "5/2" "6/2" "7/2" "8/2" "9/2##
Or in mixed numbers:
##1/2" "1" "1 1/2" "2" "2 1/2" "3" " 3 1/2" "4" "4 1/2##
To convert an improper fraction into a mixed number, divide the numerator by the denominator to find out how many whole numbers there are. The remainder is given as a fraction with the same denominator.
##9/2 = 9 div2##
##9div 2 = 4 " remainder " 1##
##9/2 = 4 1/2##
Another example: ##23/5 = 23div5##
##23 div 5 = 4 " remainder "3##
##23/5 = 4 3/5## | 280 | 781 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2021-49 | latest | en | 0.893353 |
https://nrich.maths.org/public/leg.php?code=27&cl=3&cldcmpid=5776 | 1,508,641,102,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825057.91/warc/CC-MAIN-20171022022540-20171022042540-00761.warc.gz | 798,268,663 | 9,060 | # Search by Topic
#### Resources tagged with Ratio similar to Twisting and Turning:
Filter by: Content type:
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### There are 40 results
Broad Topics > Fractions, Decimals, Percentages, Ratio and Proportion > Ratio
### Sitting Pretty
##### Stage: 4 Challenge Level:
A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
### How Big?
##### Stage: 3 Challenge Level:
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
### Do Unto Caesar
##### Stage: 3 Challenge Level:
At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won \$1 200. What were the. . . .
### Semi-square
##### Stage: 4 Challenge Level:
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
### Mixing More Paints
##### Stage: 3 Challenge Level:
Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?
### Racing Odds
##### Stage: 3 Challenge Level:
In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?
### Ratio or Proportion?
##### Stage: 2 and 3
An article for teachers which discusses the differences between ratio and proportion, and invites readers to contribute their own thoughts.
### Pythagoras’ Comma
##### Stage: 4 Challenge Level:
Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.
### Mixing Paints
##### Stage: 3 Challenge Level:
A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.
### Cereal Mix
##### Stage: 3 Challenge Level:
A farmer is supplying a mix of seeds, nuts and dried apricots to a manufacturer of crunchy cereal bars. What combination of ingredients costing £5 per kg could he supply?
### One and Three
##### Stage: 4 Challenge Level:
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
### Orbiting Billiard Balls
##### Stage: 4 Challenge Level:
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
### Golden Thoughts
##### Stage: 4 Challenge Level:
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
### Ratio Sudoku 3
##### Stage: 3 and 4 Challenge Level:
A Sudoku with clues as ratios or fractions.
##### Stage: 3 Challenge Level:
Can you work out which drink has the stronger flavour?
### Six Notes All Nice Ratios
##### Stage: 4 Challenge Level:
The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.
### Around and Back
##### Stage: 4 Challenge Level:
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
### Equal Temperament
##### Stage: 4 Challenge Level:
The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.
### Speeding Boats
##### Stage: 4 Challenge Level:
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
### Star Gazing
##### Stage: 4 Challenge Level:
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
### Oh for the Mathematics of Yesteryear
##### Stage: 3 Challenge Level:
A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45. . . .
### Ratio Sudoku 2
##### Stage: 3 and 4 Challenge Level:
A Sudoku with clues as ratios.
### Rhombus in Rectangle
##### Stage: 4 Challenge Level:
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
### Circuit Training
##### Stage: 4 Challenge Level:
Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .
### Ratio Sudoku 1
##### Stage: 3 and 4 Challenge Level:
A Sudoku with clues as ratios.
### Ratios and Dilutions
##### Stage: 4 Challenge Level:
Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions
### Rati-o
##### Stage: 3 Challenge Level:
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
### Trapezium Four
##### Stage: 4 Challenge Level:
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
### Ratio Pairs 3
##### Stage: 3 and 4 Challenge Level:
Match pairs of cards so that they have equivalent ratios.
### Pent
##### Stage: 4 and 5 Challenge Level:
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
### Triangle in a Triangle
##### Stage: 4 Challenge Level:
Can you work out the fraction of the original triangle that is covered by the inner triangle?
### Exact Dilutions
##### Stage: 4 Challenge Level:
Which exact dilution ratios can you make using only 2 dilutions?
### From All Corners
##### Stage: 4 Challenge Level:
Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
### Bus Stop
##### Stage: 4 Challenge Level:
Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .
### A Scale for the Solar System
##### Stage: 4 Challenge Level:
The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?
### The Rescaled Map
##### Stage: 4 Challenge Level:
We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.
### Slippage
##### Stage: 4 Challenge Level:
A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .
### Points in Pairs
##### Stage: 4 Challenge Level:
In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?
### Tin Tight
##### Stage: 4 Challenge Level:
What's the most efficient proportion for a 1 litre tin of paint?
### Pi, a Very Special Number
##### Stage: 2 and 3
Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible. | 1,915 | 8,013 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2017-43 | latest | en | 0.916879 |
https://www.nayuki.io/page/gauss-jordan-elimination-over-any-field-java | 1,480,860,844,000,000,000 | application/xhtml+xml | crawl-data/CC-MAIN-2016-50/segments/1480698541322.19/warc/CC-MAIN-20161202170901-00417-ip-10-31-129-80.ec2.internal.warc.gz | 1,004,360,293 | 4,184 | # Gauss-Jordan elimination over any field (Java)
While it’s typical to solve a system of linear equations in real numbers, it’s also possible to solve a linear system over any mathematical field. Here is some Java code that defines various fields and provides a version of Gauss-Jordan elimination that works on any field. The basic Gauss-Jordan elimination algorithm can be adapted to solve systems of linear equations, invert matrices, calculate determinants, calculate ranks, and more.
## Source code
Core classes:
• Field.java: Defines the constants and operations that every field must support.
• Matrix.java: Represents a matrix and implements a collection of operations:
• Basic information: Dimensions, get/set cell
• Row operations: Swap rows, multiply row, add two rows
• Advanced operations: Gauss-Jordan elimination, determinant, inverse
Runnable classes:
• Main.java: A short demonstration program, which implements the “Rational numbers” example below.
• MatrixTest.java: A JUnit test suite for Matrix.reducedRowEchelonForm(), Matrix.determinantAndRef(), and Matrix.invert() – using PrimeField(11).
Included fields:
Ideas for more fields you can implement:
• Complex rationals (a special case of quadratic surds)
• Quadratic surds (useful for numbers that contain a square root)
• Finite fields based on irreducible polynomials (very useful for error correction codes and cryptography)
## Examples
Rational numbers
Suppose we want to solve this system of linear equations in rational numbers:
$$\begin{cases} 2x + 5y + 3z &=& 7 \\ x + z &=& 1 \\ -4x + 2y - 9z &=& 6 \end{cases}$$
First we convert the system into an augmented matrix:
$$\begin{bmatrix} 2 & 5 & 3 & 7 \\ 1 & 0 & 1 & 1 \\ -4 & 2 & -9 & 6 \end{bmatrix}$$
Then we perform Gauss-Jordan elimination on the matrix:
$$\begin{bmatrix} 1 & 0 & 0 & 67/27 \\ 0 & 1 & 0 & 35/27 \\ 0 & 0 & 1 & -40/27 \end{bmatrix}$$
And as long as the left side is the identity matrix, we can read the answer off the rightmost column:
$$\begin{cases} x &=& 67/27 \\ y &=& 35/27 \\ z &=& -40/27 \end{cases}$$
Here is the code to perform this example:
// Nice-looking matrix
int[][] input = {
{2, 5, 3, 7},
{1, 0, 1, 1},
{-4, 2, -9, 6},
};
// The actual matrix object
Matrix<Fraction> mat = new Matrix<Fraction>(
input.length, input[0].length, RationalField.FIELD);
for (int i = 0; i < mat.rowCount(); i++) {
for (int j = 0; j < mat.columnCount(); j++)
mat.set(i, j, new Fraction(input[i][j], 1));
}
// Gauss-Jordan elimination
mat.reducedRowEchelonForm();
Integers modulo a prime
Suppose we want to solve this system of linear equations modulo 7:
$$\begin{cases} 3x + 1y + 4z &\equiv& 1 \mod 7 \\ 5x + 2y + 6z &\equiv& 5 \mod 7 \\ 0x + 5y + 2z &\equiv& 1 \mod 7 \end{cases}$$
First we convert the system into an augmented matrix:
$$\begin{bmatrix} 3 & 1 & 4 & 1 \\ 5 & 2 & 6 & 5 \\ 0 & 5 & 2 & 1 \end{bmatrix}$$
Then we perform Gauss-Jordan elimination on the matrix:
$$\begin{bmatrix} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$
And as long as the left side is the identity matrix, we can read the answer off the rightmost column:
$$\begin{cases} x &=& 4 \\ y &=& 3 \\ z &=& 0 \end{cases}$$
Here is the code to perform this example:
// Nice-looking matrix
int[][] input = {
{3, 1, 4, 1},
{5, 2, 6, 5},
{0, 5, 2, 1},
};
// The actual matrix object
Matrix<Integer> mat = new Matrix<Integer>(
input.length, input[0].length, new PrimeField(7));
for (int i = 0; i < mat.rowCount(); i++) {
for (int j = 0; j < mat.columnCount(); j++)
mat.set(i, j, input[i][j]);
}
// Gauss-Jordan elimination
mat.reducedRowEchelonForm(); | 1,120 | 3,619 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2016-50 | latest | en | 0.698635 |
https://www.speedsolving.com/wiki/index.php/Old_Pochmann | 1,556,292,966,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578841544.98/warc/CC-MAIN-20190426153423-20190426175423-00354.warc.gz | 830,565,479 | 6,549 | # Classic Pochmann
(Redirected from Old Pochmann)
Classic Pochmann, earlier named Old Pochmann is a 2-cycle blindfold method invented by Stefan Pochmann. The general idea is that you solve one piece at a time, using PLL algorithms T and J for the edges as well as Y for the corners, and appropriate setup moves for each possible target. It is considered one of the simplest and easiest to learn blindfold methods.
The drawback is the move count, because it only solves one piece at the time and you need to do setup moves and a PLL alg each time a full solve can be well over 200 turns, advanced methods like freestyle BLD uses some 70-80.
## Classic Pochmann on 2x2x2
One of the easiest ways to solve the 2x2x2 cube blindfolded is via a PBL corner-swap algorithm. Conceptually, it is similar to corners phase of Classic Pochmann on 3x3x3. Unlike CP on the 3x3x3 there are no parity issues because single corner swaps can be done without affecting any other pieces.
### Method
To solve a corner in the UFL buffer position do the following:
1. Look at the upward facing sticker of the piece in the UFL position. Locate the position this sticker would normally occupy when solved. Move this sticker position into the U-face of the URF position, without disturbing the UFL corner. This is the setup move.
2. Apply the adjacent corner swap algorithm (R' F R' B2 R F' R' B2 R2 U') to move the corner in UFL into the URF position.
3. Undo the setup move. | 360 | 1,456 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2019-18 | latest | en | 0.895602 |
https://blinkx.in/en/knowledge-base/intraday-trading/intraday-momentum-index | 1,721,714,155,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763518014.29/warc/CC-MAIN-20240723041947-20240723071947-00384.warc.gz | 116,650,268 | 67,899 | Existing Customer
• 13 Jun 2024
The Intraday Momentum Index (IMI) is a technical indicator designed to measure the momentum of price fluctuations within a single trading day. By combining both price and volume data, it offers a holistic assessment of market strengths or weaknesses. Day traders primarily use this indicator to identify possible entry and exit points, especially in the volatile market.
The Intraday Momentum Index is widely known in the world of trading and investing. It proves to be valuable for recognising short-term trends, capturing instances of price reversals, and generating signals to buy or sell. Its effectiveness is particularly notable in fast-moving markets, where rapid shifts in momentum can signify potential trading opportunities. In this article, we will discuss in detail about Intraday Momentum Index and its practical application.
## What is the Intraday Momentum Index?
The Intraday Momentum Index (IMI) is designed to capture intraday price momentum by analyzing the relationship between price changes and intraday price ranges.
The calculation of the IMI involves comparing the ratio of positive price changes to negative price changes with the ratio of positive intraday price ranges to negative intraday price ranges over a specific period. This calculation results in a value between 0 and 100, representing the strength of intraday momentum. A reading above 50 suggests bullish momentum, while a reading below 50 indicates bearish momentum.
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### Table of Content
1. What is the Intraday Momentum Index?
2. Formula For IMI Calculations
3. Practical Application and Strategies of IMI Indicator
## Formula For IMI Calculations
The IMI doesn't focus on how much the open and close prices differ from one day to the next. Instead, it measures these differences within a single day.
### The IMI calculation formula is as follows:
IMI = {[∑(d=1)(n)Gains)] / [(∑(d=1)(n)Gains) + (∑(d=1)(n)Losses)]} × 100
Gains represent the profit made when the closing price is higher than the opening price on Up Days. On these days, calculate gains by subtracting the opening price (OP) from the closing price (CP). Conversely, losses occur on Down Days when the closing price is lower than the opening price. Here, losses are calculated by subtracting the closing price from the opening price.
The variables are: "d" stands for days, and "n" represents the total number of days, typically set at 14. To gauge Intraday Momentum, add up all the gains on Up Days and divide that by the total gains on Up Days plus the sum of losses on Down Days. The ratio is subsequently multiplied by 100.
## Practical Application and Strategies of IMI Indicator
The IMI indicator provides valuable insights into intraday price movements. Here are some practical applications and strategies for using the IMI in your trading.
### 1. Overbought and Oversold Conditions
One of the primary applications of the IMI is to identify overbought and oversold conditions in the market. When the IMI reading reaches or exceeds the 70 level, it indicates that the market is overbought, suggesting a potential reversal or correction in prices. Conversely, when the IMI falls to or below the 30 level, it signals an oversold condition, which may present buying opportunities. Traders can use these levels to make decisions on when to enter or exit positions.
### 2. Divergence Analysis
Divergence analysis involves comparing the IMI with the price action of the underlying asset. A bullish divergence occurs when the IMI forms higher highs while the price forms lower highs, indicating potential upward momentum. Conversely, bearish divergence occurs when the Intraday momentum index forms lower lows while the price forms higher lows, suggesting potential downward momentum. Traders can use these divergences as signals for potential trend reversals and adjust their trading strategies accordingly.
### 3. Confirmation with Other Indicators
To enhance the accuracy of trading signals, traders often combine the IMI with other technical indicators. For example, using the intraday momentum strategy in conjunction with oscillators like the Relative Strength Index (RSI) or the Moving Average Convergence Divergence (MACD) can provide additional confirmation. When multiple indicators align and generate similar signals, it strengthens the conviction in the trade setup.
### 4. Breakout and Trend-Following Strategies
The IMI can be effectively used in breakout strategies, where traders look for significant price movements and breakouts of key levels. By monitoring the intraday momentum indicator alongside price action, traders can identify periods of increasing momentum and potential breakouts. When the Intraday momentum index confirms a breakout with a strong reading above 50, It can provide confidence to enter trades in the direction of the breakout.
### 5. Risk Management
Intraday trading requires strict risk management. Traders can use the IMI to help manage risk by setting stop-loss orders and profit targets based on the identified momentum. For example, if the IMI suggests an overbought condition, a trader may set a tight stop-loss order to protect against potential reversals. Similarly, when the IMI indicates an oversold condition, a trader may set profit targets at key resistance levels.
### 6. Timeframe Selection
The Intraday momentum strategy can be applied to various timeframes, ranging from minutes to hours, depending on the trader's preferred trading style. Shorter timeframes may provide more frequent signals but may also be more prone to noise and false signals. Longer timeframes, on the other hand, may generate fewer signals but can offer more reliable indications of momentum.
Conclusion
The Intraday Momentum Index (IMI) is a valuable tool for intraday traders in the stock market. By calculating and analysing the IMI, traders can gain insights into intraday momentum and make informed trading decisions. However, it's important to remember that the IMI should not be used in isolation. It is most effective when combined with other technical analysis tools, risk management strategies, and market knowledge.
Traders should consider using the IMI in conjunction with indicators like moving averages, oscillators, or support and resistance levels to enhance the accuracy of their trading signals. To optimise your investments seamlessly, you can consider downloading the stock trading app. These apps are to empower users with a lot of innovative tools and real-time insights for a dynamic and informed investment experience.
You may also be interested to know:
The Intraday Momentum Index (IMI) is a technical indicator used to measure intraday momentum in the stock market. It helps traders assess the strength and direction of short-term price movements.
The time frame for calculating the IMI depends on the trader's preference and trading style. It can vary from minutes to hours. Shorter time frames provide more frequent signals but may be more prone to noise, while longer time frames generate fewer signals but offer more reliable indications of momentum.
The Intraday Momentum Index is primarily designed for short-term trading and capturing intraday price movements. It may not be as effective for longer-term trading strategies, which require different indicators and methodologies.
While the IMI can provide valuable insights into intraday momentum, it is generally recommended to use it as part of a comprehensive trading strategy. Combining the IMI with other technical analysis tools, risk management techniques, and market knowledge can enhance the overall trading approach.
The Intraday Momentum Index is available on many popular charting platforms and trading software. Traders can usually find it in the list of technical indicators or create a custom indicator by using the necessary formulas and parameters.
No single indicator can guarantee profitable trades. The IMI is a tool that provides insights into intraday momentum, but it should be used in conjunction with other analysis techniques, risk management strategies, and trader's judgement.
Stock directory: | 1,643 | 8,217 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2024-30 | latest | en | 0.878987 |
https://support.nag.com/numeric/nl/nagdoc_28.7/examples/source/e04rffe.f90.html | 1,696,338,507,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233511106.1/warc/CC-MAIN-20231003124522-20231003154522-00085.warc.gz | 583,094,884 | 3,831 | NAG Library Manual, Mark 28.7
``` Program e04rffe
! E04RFF Example Program Text
! Compute the nearest correlation matrix in Frobenius norm
! using nonlinear semidefinite programming. By default,
! preserve the nonzero structure of the input matrix
! (preserve_structure = .True.).
! Mark 28.7 Release. NAG Copyright 2022.
! .. Use Statements ..
Use, Intrinsic :: iso_c_binding, Only: c_null_ptr, &
c_ptr
Use nag_library, Only: e04raf, e04rff, e04rnf, e04rzf, e04svf, e04zmf, &
nag_wp, x04caf
! .. Implicit None Statement ..
Implicit None
! .. Parameters ..
Integer, Parameter :: nin = 5, nout = 6
Logical, Parameter :: preserve_structure = .True.
! .. Local Scalars ..
Type (c_ptr) :: h
Integer :: dima, i, idblk, idx, ifail, inform, &
j, n, nblk, nnzasum, nnzc, nnzh, &
nnzu, nnzua, nnzuc, nvar
! .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: a(:), g(:,:), hmat(:), x(:)
Real (Kind=nag_wp) :: rdummy(1), rinfo(32), stats(32)
Integer, Allocatable :: blksizea(:), icola(:), icolh(:), &
irowa(:), irowh(:), nnza(:)
Integer :: idummy(1)
! .. Executable Statements ..
Write (nout,*) 'E04RFF Example Program Results'
Write (nout,*)
Flush (nout)
! Skip heading in data file.
! Read in the problem size.
Allocate (g(n,n))
! Read in the matrix G.
! Symmetrize G: G = (G + G')/2
Do j = 2, n
Do i = 1, j - 1
g(i,j) = (g(i,j)+g(j,i))/2.0_nag_wp
g(j,i) = g(i,j)
End Do
End Do
! Initialize handle.
h = c_null_ptr
! There are as many variables as nonzeros above the main diagonal in
! the input matrix. The variables are corrections of these elements.
nvar = 0
Do j = 2, n
Do i = 1, j - 1
If (.Not. preserve_structure .Or. g(i,j)/=0.0_nag_wp) Then
nvar = nvar + 1
End If
End Do
End Do
Allocate (x(nvar))
! Initialize an empty problem handle with NVAR variables.
ifail = 0
Call e04raf(h,nvar,ifail)
! Set up the objective - minimize Frobenius norm of the corrections.
! Our variables are stored as a vector thus, just minimize
! sum of squares of the corrections --> H is identity matrix, c = 0.
nnzc = 0
nnzh = nvar
Allocate (irowh(nnzh),icolh(nnzh),hmat(nnzh))
Do i = 1, nvar
irowh(i) = i
icolh(i) = i
hmat(i) = 1.0_nag_wp
End Do
ifail = 0
Call e04rff(h,nnzc,idummy,rdummy,nnzh,irowh,icolh,hmat,ifail)
! Construct linear matrix inequality to request that
! matrix G with corrections X is positive semidefinite.
! (Don't forget the sign at A_0!)
! How many nonzeros do we need? Full triangle for A_0 and
! one nonzero element for each A_i.
nnzasum = n*(n+1)/2 + nvar
Allocate (nnza(nvar+1),irowa(nnzasum),icola(nnzasum),a(nnzasum))
nnza(1) = n*(n+1)/2
nnza(2:nvar+1) = 1
! Copy G to A_0, only upper triangle with different sign (because -A_0)
! and set the diagonal to 1.0 as that's what we want independently
! of what was in G.
idx = 1
Do j = 1, n
Do i = 1, j - 1
irowa(idx) = i
icola(idx) = j
a(idx) = -g(i,j)
idx = idx + 1
End Do
! Unit diagonal.
irowa(idx) = j
icola(idx) = j
a(idx) = -1.0_nag_wp
idx = idx + 1
End Do
! A_i has just one nonzero - it binds x_i with its position as
! a correction.
Do j = 2, n
Do i = 1, j - 1
If (.Not. preserve_structure .Or. g(i,j)/=0.0_nag_wp) Then
irowa(idx) = i
icola(idx) = j
a(idx) = 1.0_nag_wp
idx = idx + 1
End If
End Do
End Do
! Just one matrix inequality of the dimension of the original matrix.
nblk = 1
Allocate (blksizea(nblk))
dima = n
blksizea(:) = (/dima/)
! Add the constraint to the problem formulation.
idblk = 0
ifail = 0
Call e04rnf(h,nvar,dima,nnza,nnzasum,irowa,icola,a,nblk,blksizea,idblk, &
ifail)
! Set optional arguments of the solver.
ifail = 0
Call e04zmf(h,'Print Options = No',ifail)
ifail = 0
Call e04zmf(h,'Initial X = Automatic',ifail)
! Pass the handle to the solver, we are not interested in
! Lagrangian multipliers.
nnzu = 0
nnzuc = 0
nnzua = 0
ifail = 0
Call e04svf(h,nvar,x,nnzu,rdummy,nnzuc,rdummy,nnzua,rdummy,rinfo,stats, &
inform,ifail)
! Destroy the handle.
ifail = 0
Call e04rzf(h,ifail)
! Form the new nearest correlation matrix as the sum
! of G and the correction X.
idx = 1
Do j = 1, n
Do i = 1, j - 1
If (.Not. preserve_structure .Or. g(i,j)/=0.0_nag_wp) Then
g(i,j) = g(i,j) + x(idx)
idx = idx + 1
End If
End Do
g(j,j) = 1.0_nag_wp
End Do
! Print the matrix.
ifail = 0
Call x04caf('Upper','N',n,n,g,n,'Nearest Correlation Matrix',ifail)
End Program e04rffe
``` | 1,610 | 4,609 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2023-40 | latest | en | 0.540461 |
https://www.reference.com/health/blood-pressure-high-a6dc540f84e7d796 | 1,480,974,429,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541839.36/warc/CC-MAIN-20161202170901-00280-ip-10-31-129-80.ec2.internal.warc.gz | 1,008,314,881 | 21,164 | Q:
# How do you know if your blood pressure is too high?
A:
A person can know if his blood pressure is too high by using a device that checks the diastolic and systolic pressure, says the Centers for Disease Control. Then, the readings are compared to a chart that tells the patient what is a normal or elevated blood pressure. The device that reads blood pressure is called a sphygmomanometer, says Dictionary.com.
## Keep Learning
Credit: Fuse Fuse Getty Images
A doctor attaches a cuff to the patient's arm, inflates it, and lets the air out slowly, says the CDC. All the while, he uses a stethoscope to hear the patient's pulse as he monitors the gauge attached to the sphygmomanometer. The systolic pressure is the measurement of the patient's pressure when his heart beats, while the diastolic pressure is the measurement of the pressure between beats. The result is written as a fraction with the systolic pressure the dividend and the diastolic pressure the divisor.
If a person has high blood pressure, his systolic pressure is 140 millimeters of mercury, or mmHg, or higher, while his diastolic pressure is 90 mmHG or higher, claims the CDC. A normal pressure is 120 mmHg or lower, while the diastolic pressure is lower than 80 mmHg. If the blood pressure numbers are 120 to 139 mmHg over 80 to 89 mmHg, the patient is at risk for high blood pressure.
Sources:
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PEOPLE SEARCH FOR | 507 | 2,183 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2016-50 | longest | en | 0.904095 |
http://mathhelpforum.com/advanced-algebra/98874-algebraic-elements-evaluation-homomorphism.html | 1,527,172,265,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794866326.60/warc/CC-MAIN-20180524131721-20180524151721-00155.warc.gz | 186,946,110 | 10,196 | # Thread: algebraic elements and evaluation homomorphism
1. ## algebraic elements and evaluation homomorphism
Let F be a subfield of E, $\displaystyle \alpha\in E$and $\displaystyle \alpha$is an algebraic over $\displaystyle F$. Let $\displaystyle \phi_\alpha$be the evaluation homomorphism. $\displaystyle F(\alpha)$is the minimal field containing $\displaystyle F,\alpha$
True or false: $\displaystyle \phi_\alpha(F[x])\cong F(\alpha)$?
One main problem is:$\displaystyle \phi_\alpha(F[x])$is a field????
2. Originally Posted by ynj
Let F be a subfield of E, $\displaystyle \alpha\in E$and $\displaystyle \alpha$is an algebraic over $\displaystyle F$. Let $\displaystyle \phi_\alpha$be the evaluation homomorphism. $\displaystyle F(\alpha)$is the minimal field containing $\displaystyle F,\alpha$
True or false: $\displaystyle \phi_\alpha(F[x])\cong F(\alpha)$?
One main problem is:$\displaystyle \phi_\alpha(F[x])$is a field????
yes! let $\displaystyle f(x) \in F[x]$ be the minimal polynomial of $\displaystyle \alpha,$ i.e. $\displaystyle f$ is monic and it has the smallest degree among all $\displaystyle g(x) \in F[x]$ with this property that $\displaystyle g(\alpha)=0.$ then $\displaystyle \ker \phi_{\alpha} = <f(x)>$ and, since
$\displaystyle \phi_{\alpha}:F[x] \longrightarrow F[\alpha]$ is surjective, we have $\displaystyle \frac{F[x]}{<f(x)>} \cong \phi_{\alpha}(F[x]) = F[\alpha].$ finally, since $\displaystyle f(x)$ is irreducible and $\displaystyle F[x]$ is a PID, the ideal $\displaystyle <f(x)>$ is maximal and thus $\displaystyle F[\alpha]$ is a field. as a result
$\displaystyle F[\alpha]=F(\alpha).$ the converse is also true: if $\displaystyle F[\alpha]$ is a field, then $\displaystyle \alpha$ is algebraic over $\displaystyle F.$ the reason is that if $\displaystyle \alpha$ is transcendental over $\displaystyle F,$ then $\displaystyle F[\alpha] \cong F[x]$ and obviously $\displaystyle F[x]$ is never a field.
3. emm...I ignored the fact that $\displaystyle f(x)$is irreducible.. | 569 | 2,015 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2018-22 | latest | en | 0.650737 |
https://ucmp.berkeley.edu/diapsids/buzz/locomotion.html | 1,701,800,373,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100555.27/warc/CC-MAIN-20231205172745-20231205202745-00499.warc.gz | 688,773,011 | 3,896 | # Inferring the PossibleSpeeds of Dinosaurs?
We know that dinosaurs moved; they were vertebrates, and we have their trackways (sequences of footprints) preserved in certain sediments as fossils. One popular question is how fast did they move? Did some of them run as fast as the modern cheetah? Most scientists involved in the investigation of dinosaur locomotion think not. We have several lines of evidence that can help us estimate how dinosaurs could move.
One line of evidence is the information given to us by those trackways. Two samples of the UCMP's large collection of theropod dinosaur footprints are shown at left; note the amazing similarity to the tracks of birds on a beach or muddy ground. We don't know of any giant birds that were around in the Mesozoic era, and no other vertebrates had feet quite like those of theropod dinosaurs, so we can be confident that these footprints were made by the carnivorous dinosaurs.
A good sequence of preserved footprints (called a trackway) can be extrapolated to give a rough estimate of how fast that particular animal was traveling at that moment. This method uses simple equations based on the distance between footfalls and the size of the feet. The fastest speeds evident from dinosaur tracks (a medium-sized theropod in this case) are about 12 meters per second (about 27 mph); a little faster than the best Olympic sprinters. Problem: It's hard to tell who made those tracks! We usually can narrow down our identification to large vs. small theropods, sauropods, or whatever. Then, knowing what sorts of those dinosaurs lived around there at the time the tracks were made, we can get ideas of who might have made those tracks. Then we have a shaky guess as to what dinosaur was moving at what speed at that instant. The obvious problem: "At that instant" is the key phrase; most animals are not running at top speed all of the time, especially when on soft ground, where tracks are most likely to be made. So we just have a glimpse into a brief moment in time; not a thorough analysis of dinosaur behavior. Good trackways are quite rare, too. But trackways are what we have to work with, so paleontologists must make do with that evidence.
The morphology (shape and structure, or anatomy) of dinosaurs may be a more useful tool, but it is much more difficult to use properly. We can reconstruct dinosaur skeletons to figure out how the bones were connected, and make predictions about their functional morphology (how their bodies moved and worked) from muscle scars and other anatomical features. We have done that, and learned long ago that dinosaurs stood erect (like birds and most mammals); they did not keep their legs sprawling out to the side of their body like most lizards and salamanders do. Also, from the trackways of dinosaurs, we know that they rarely dragged their tail on the ground — normally, their vertebral column was oriented roughly horizontally with respect to the ground. So, mammals and birds are probably better models for understanding dinosaur locomotion than lizards are (but all are useful to some degree, and all are limited in their usefulness). Now it gets tricky!
There are at least two ways we can go from here: one is to simply compare dinosaurs with extant (living) animals whose motion we understand better, and make assumptions based on the similarities and differences between the two. This can be called the morphological paradigm. The hadrosaurs and theropods had many members whose skeletal structure was similar to that of some modern cursors (animals that are good runners, like horses and ostriches): long legs, digitigrade stance (walking on one's toes), and so on. So we might think that some of those dinosaurs were “cursorial,” or specialized for locomotion, but because their locomotory features are not as specialized as those of many of the faster extant runners, we think that it is unlikely that any non-avian dinosaurs ran incredibly fast. Similarly, many sauropods, thyreophorans (armored dinosaurs), and ceratopsians were similar to modern graviportal (non-cursorial, heavily-built) animals like elephants, so paleontologists think that such large dinosaurs were less speedy. In general, big land animals use less strenous activities than their smaller relatives.
A second direction we can take is more conclusive, but much harder. We can use the laws of physics and apply them to our dinosaurs; this is called biomechanics. We can "reconstruct" a dinosaur's muscles (using the musculature of the dinosaur's closest living relatives — the crocodilians and birds — as guides), estimate its weight, and apply established engineering principles to figure out how fast that particular dinosaur could move if it wanted to. Or so we think. The problem is that it is very hard to do this with any living animal! The difficulties are staggering when we try to do this with 65 million year old fossils (which are often incomplete). When you hear quotes about T.rex moving 40-60 mph, ask for the evidence and judge for yourself.
Back to DinoBuzz | 1,066 | 5,066 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.125 | 3 | CC-MAIN-2023-50 | latest | en | 0.976313 |
https://colleenyoung.org/category/for-students/ | 1,702,238,476,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679102637.84/warc/CC-MAIN-20231210190744-20231210220744-00001.warc.gz | 202,987,089 | 31,007 | # Resources for Students
One of my personal resolutions for the coming year is to carry on with my practice of using resources that students can then refer to or use at home if they wish. Mathematics notes and calculators are a good example of such resources.
To consider an example, early this term with the Further Mathematicians I will be studying matrices and I will let them know the sources of any resources I use in lessons. I use a blog to provide the details of my students’ homework so I can simply add the links to their homework page. Sometimes where there are several useful resources I think maybe of interest to a wider audience I also add a post to Mathematics for Students, see for example, Polar Coordinates. In fact I think I will do that more this year.
On the AQA website the Teaching and learning resources page for A Level Further Maths includes three online textbooks under the Resources for students heading. For example if I want a worked example of finding the inverse of a 3×3 matrix then we can look at Chapter 5 of AQA’s Further Pure 4 text. This also has an exercise with the answers at the back if they want additional examples.
The Math Centre
More sources of notes and examples include Chapter 9 on Matrices and Transformations from the CIMT Further Pure Mathematics A Level material, Just the Mathsthe Math Centre and The HELM Project. If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include clear worked examples. You can see on the list that a very small number of titles (that you are unlikely to want A Level) are ‘not ready yet’; for the sake of completeness I discovered the complete set hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.
Obviously we need to keep an eye on the specification when looking at alternative sources of examples but surely that can only be a good thing, particularly for our students who will be off to university in the near future.
Matrices is an example of a topic where it can be very useful to check work with WolframAlpha; I have created a new slideshow of Matrix Examples to add to the WolframAlpha slideshow series so we can easily check any work.
The series is on Mathematics for Students also and a post including the matrices resources discussed here has been added also.
# Transformations with the Desmos Graphing Calculator
This week Year 10 (UK age 14-15) have been exploring different graph types and also transformations and graphs.
For homework I asked them to draw just a small number of graphs by hand but wanted them to check their work and explore further graphs using the Desmos graphing calculator. Early in the week I made sure they could all use Desmos including the use of tables so in an IT room they used the slideshow here and created several graphs of their own.
Once all the students were confident to use Desmos to create various lines and curves I asked them to explore a series of graphs so that this coming week we can discuss transformations and graphs. Using Desmos allowed them to explore many graphs in a short space of time and several students chose to take screenshots and make notes for themselves.
Desmos – simple transformations example
Having used sliders they were able to create
this type of graph page.
I have created the slideshow below to use in class to summarise our work and act as a revision aid for them.
These slideshows are both available here for students.
# Websites for Students…and more!
One of the most popular posts on this blog is Top >10 Mathematics Websites. It struck me that it might be useful to think about my top recommendations for students; once again using some categories as well as individual sites gives me the excuse to mention more than 10! So for your students:
Top >10 Mathematics Websites for Students
Back to the teachers!
I have mentioned TED-Ed before with its collection of Mathematics videos, note the feature now offered by TED-Ed to ‘find and flip’ which allows you to use a video and turn it into a lesson; see ‘Flip This Video’.
Looking at some videos, it struck me that something like Gaurav Tekriwal’s The magic of Vedic Math would be ideal to tinker with! (These ‘tricks’ can make ideal starters, I have linked to some further videos on this page on Number on Mathematics Starters.)
On the subject of videos and TED, have you heard Ken Robinson’s latest talk,
How to escape education’s death valley?
# Some resources used in my classes….
Below I have given details of some resources I am currently using with my classes or have recommend to my students so they can explore examples further themselves.
Year 12 (age 16-17)
I want to talk about quadratic inequalities this week so I thought I’d use the Desmos graphing calculator to draw some pictures! Click on the image for the Desmos page and select ‘projector ‘ mode for display on the interactive whiteboard. Last week a student in this class asked where she could find some additional resources on polynomial division.
Note: I use the Desmos calculator so much I have decided it deserves a page of its own here (under Resources).
Year 13 (age 17-18)
Some students in a Further Maths class asked for some Polar coordinates resources to support their studies – so a post for them on my blog for students – these resources would also work well on the interactive whiteboard for use in class. As regular readers know I am a great fan of WolframAlpha and use it with all my classes (WolframAlpha now have a paid for service but it is still completely free to use to check answers for an unlimited number of queries, the free use limits step by step solutions to 3 a day). One of this class showed me that he has the WolframAlpha app on his phone.
Year 11 (age 15-16)
My Year 11 group are studying the AQA Level 2 Certificate in Further Mathematics (a course I am very much enjoying teaching) as well as completing their GCSE course this year. We have been studying calculus and I have found the Desmos graphing calculator very useful to illustrate problems we have been solving. This class have mock examinations coming up and I wanted to recommend some additional resources for them (we have various texts and the AQA support is excellent but the more the better and there is currently no textbook for the course); one site with some very useful resources for some parts of the course such as introductory Calculus is David Smith’s ‘The Maths Teacher.’
Year 8 (age 12-13)
I have a Year 8 class this year, none of whom I have taught before, we have been looking at surface area and volume. Math Open Ref has a rather nice animation which helps when looking at the surface area of a cylinder. (More on John Page’s Math Open Ref). I will also use this site when we look at constructions soon. Most had not seen WolframAlpha before so were quite impressed at how easy it is to check working! There are slideshows available for students showing the syntax for a selection of examples on my blog for students.
# Hello to the Students!
It struck me recently that so many blogs are addressed to the teachers, so perhaps it’s time to provide one for the students!
Hence the new Mathematics for Students a companion blog to this one but addressed to students. | 1,567 | 7,483 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2023-50 | latest | en | 0.944082 |
http://mathforum.org/kb/thread.jspa?threadID=2328360 | 1,524,507,487,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125946120.31/warc/CC-MAIN-20180423164921-20180423184921-00470.warc.gz | 200,436,102 | 4,658 | Search All of the Math Forum:
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Topic: Claimed approximation for Kullback-Leibler distance D(p||q) when p
and q are close, apparently based on Taylor series
Replies: 0
jdm Posts: 16 Registered: 11/22/10
Claimed approximation for Kullback-Leibler distance D(p||q) when p
and q are close, apparently based on Taylor series
Posted: Jan 1, 2012 6:36 PM
The following claim featured in a research paper I've been studying -
however, no proof accompanied it beyond a statement that the
approximation could be obtained using Taylor series at order 2 - and
it wasn't clear what the variable was supposed to be or around which
point.
Let p and q be discrete probability distributions of random variables
taking on values from a set with M+1 elements;
p=(p_0, ..., p_M)
(p_i = P(random variable with distribution p is equal to i))
Likewise, q=(q_0, ..., q_M)
Where p and q are close - defined as |p_{i} - q_{i}| << q_{i} \forall
i - let e_i denote the value (p_i - q_i).
Then, according to the paper, D(p||q) \approx D(q||p) \approx sum_{i=0}
^{M}(e_{i}^{2}/q_{i})/2.
I haven't been able to verify this approximation for myself, as I
stated, and if anyone reading this can help (even by arguing that the
approximation isn't in fact valid) it would be much appreciated!
Many thanks,
James McLaughlin. | 401 | 1,478 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2018-17 | longest | en | 0.927803 |
https://www.physicsforums.com/threads/magnetic-compression.315896/ | 1,581,962,547,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875143079.30/warc/CC-MAIN-20200217175826-20200217205826-00448.warc.gz | 869,096,206 | 15,369 | # Magnetic compression
## Main Question or Discussion Point
Hi,
I want to study magnetic compression theory.
I want to able to calculate and simulate how magnetic compression can store or give back energy from a circuit
Any differential equation are available for this ?
Thanks
Related Electrical Engineering News on Phys.org
vk6kro
A dense plasma focus (DPF) is a plasma machine that produces, by electromagnetic acceleration and compression, short-lived plasma that is so hot and dense that it becomes a copious multi-radiation source. It was invented in the early 1960s by J.W. Mather and also independently by N.V. Filippov. The electromagnetic compression of a plasma is called a "pinch". The plasma focus is similar to the high-intensity plasma gun device (HIPGD) (or just plasma gun), which ejects plasma in the form of a plasmoid, without pinching it.
If it is the latter, there is an article in Wikipedia under "dense plasma focus". Not something most of us get to play with.
Would storing energy in a magnetic coil, such as a solenoid or a torus be OK? How much energy do you want to store? For how long? There are superconducting magnets that can store a lot of magnetic energy. One Tesla stored in 1 cubic meter is
E = (1/2u0) Integral[B2 dV] = 398 kilojoules | 298 | 1,278 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2020-10 | longest | en | 0.932006 |
https://www.educalc.net/page/428086/ | 1,686,330,589,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224656788.77/warc/CC-MAIN-20230609164851-20230609194851-00535.warc.gz | 850,613,416 | 2,480 | # HP10BII Lump Sums
To begin, we consider TVM calculations with single (lump) sums. In this situation, we do not use the PMT key, so be sure to either clear all, which sets the payment (PMT) equal to 0, or enter 0 as the PMT when entering the input data. If you know any 3 variables, you can find the value of the 4th.
Example 1:
What is the FV (future value) after 3 years if the interest rate is 26%? First, clear with ‘gold shift’ C ALL. If you observe other than 1 P_Yr, change by entering 1 ‘gold shift’ P/YR. Check by ‘gold shift’ C ALL.
Next, enter the data.
3 N
26 I/YR
100 PV
To determine the FV simply press FV key and the FV of -\$200.04 is displayed.
The HP is programmed so that if the PV is + then the FV is displayed as - and vice versa, because the HP assumes that one is an inflow and the other is an outflow.
Example 2:
What is the PV of \$500 due in 5 years if the interest rate is 10%? Clear first and then enter the following data.
5 N
10 I/YR
500 FV
Pressing the PV key reveals that \$310.46 will grow to \$500 in 5 years at a 10% rate.
Example 3:
Assume a bond can be purchased today for \$200. It will return \$1,000 after 14 years. The bond pays no interest during its life. What rate of return would you earn if you bought the bond?
14 N
-200 PV (key in 200 and then use the +/- key to change sign)
1000 FV
Simple press the I/YR key and the HP calculates the rate of return to be 12.18%.
Remember that the HP is programmed so that if the PV is + then the FV is displayed as - and vice versa because the HP assumes that one is an inflow and the other is an outflow.
Now suppose you learn that the bond will actually cost \$300. What rate of return will you earn?
Override the -200 by entering 300 =/- PV , then press I/YR to get 8.98%. If you pay more for the bond, you earn less on it. The important thing, though, is that you can do ‘what if’ analyses with the calculator.
Now do nothing except ‘gold shift’ OFF to turn off the calculator. Then turn on the calculator ON. The display shows 0.00. Is the memory erased? Not completely. What was on the screen is gone, but press RCL N to get N = 14.00. The other memory registers also retain info unless you press ‘gold shift’ C ALL. | 613 | 2,232 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.25 | 4 | CC-MAIN-2023-23 | latest | en | 0.896557 |
https://facileessays.com/engineering/ | 1,670,635,687,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446711637.64/warc/CC-MAIN-20221210005738-20221210035738-00424.warc.gz | 285,387,570 | 16,855 | # Engineering
Consider an abrupt Si pn + junction that has 1015 acceptors cm-3 on the p-side and 1019 donors on the inside.
The minority carrier recombination times are τe = 490 ns for electrons in the p-side and τh = 2.5 ns
for holes in the n -side. The cross-sectional area is 1 mm2. Assuming a long diode, calculate the
current, I, through the diode at room temperature when the voltage, V, across it is 0.6 V. What are V/I
and the incremental resistance (rd) of the diode and why are they different? | 138 | 508 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2022-49 | longest | en | 0.894553 |
https://logosconcarne.com/tag/quantum-computing/ | 1,656,419,765,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103516990.28/warc/CC-MAIN-20220628111602-20220628141602-00701.warc.gz | 440,801,287 | 24,320 | Tag Archives: quantum computing
Matter Waves
A single line from a blog post I read got me wondering if maybe (just maybe) the answer to a key quantum question has been figuratively lurking under our noses all along.
Put as simply as possible, the question is this: Why is the realm of the very tiny so different from the larger world? (There’s a cosmological question on the other end involving gravity and the realm of the very vast, but that’s another post.)
Here, the answer just might involve the wavelength of matter.
The Power of Qubits
I’ve been working my way through The Principles of Quantum Mechanics (1930), by Paul Dirac. (It’s available as a Kindle eBook for only 6.49 USD.) It’s perhaps best known for being where he defines and describes his 〈bra|ket〉 notation (which I posted about in QM 101: Bra-Ket Notation). More significantly, Dirac shows how to build a mathematical quantum theory from the ground up.
This is not a pop-science book. Common wisdom is that including even a single equation in a science book greatly reduces reader interest. Dirac’s book, in its 82 chapters, has 785 equations! (And no diagrams, which is a pity. I like diagrams.)
What I wanted to post about is something he mentioned about qubits.
QM 101: Bloch Sphere
One small hill I had to climb involved the object I’ve been using as the header image in these posts. It’s called the Bloch sphere, and it depicts a two-level quantum system. It’s heavily used in quantum computing because qubits typically are two-level systems.
So is quantum spin, which I wrote about last time. The sphere idea dates back to 1892 when Henri Poincaré defined the Poincaré sphere to describe light polarization (which is the quantum spin of photons).
All in all, it’s a handy device for visualizing these quantum states.
QM 101: Introduction
The word “always” always finds itself in phrases such as “I’ve always loved Star Trek!” I’ve always wondered about that — it’s rarely literally true. (I suppose it could be “literally” true, though. Language is odd, not even.) The implied sense, obviously, is “as long as I could have.”
The last years or so I’ve always been trying to instead say, “I’ve long loved Star Trek!” (although, bad example, I don’t anymore; 50 years was enough). Still, it remains true I loved Star Trek for a long (long) time.
On the other hand, it is literally true that I’ve always loved science.
Whither 2020
I think we all agree 2020 has been, as the curse puts it, an “interesting” year. Going into it, I had intentions about making changes. Most fell by the wayside due to COVID-19; I still haven’t taken the bus to watch the St. Paul Saints play. Or the bus-light rail combo to Target Field.
As a life long hard-core introvert, “social isolation” mostly meant I shopped for groceries less often but stocked up more when I did. The pain was fewer occasions of meeting a friend for tasty food, drink, and chat. I’m really looking forward to dining out again.
All-in-all, the last four years, this year… It’s been exhausting.
Square Root of NOT
Since I retired, I’ve been learning and exploring the mathematics and details of quantum mechanics. There is a point with quantum theory where language and intuition fail, and only the math expresses our understanding. The irony of quantum theory is that no one understands what the math means (but it works really well).
Recently I’ve felt comfortable enough with the math to start exploring a more challenging aspect of the mechanics: quantum computing. As with quantum anything, part of the challenge involves “impossible” ideas.
Like the square root of NOT. | 825 | 3,627 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2022-27 | latest | en | 0.942038 |
https://www.aqua-calc.com/one-to-all/acceleration/preset/foot-per-hour-squared/8 | 1,716,418,539,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058575.96/warc/CC-MAIN-20240522224707-20240523014707-00617.warc.gz | 562,693,003 | 8,581 | Convert feet per hour squared [ft/h²] to other units of acceleration
feet/hour² [ft/h²] acceleration conversions
8 ft/h² = 1 881.48 angstroms per second squared ft/h² to Å/s² 8 ft/h² = 1.26 × 10-18 astronomical unit per second squared ft/h² to au/s² 8 ft/h² = 188 148 148 152 attometers per second squared ft/h² to am/s² 8 ft/h² = 1.88 × 10-5 centimeter per second squared ft/h² to cm/s² 8 ft/h² = 9.35 × 10-9 chain per second squared ft/h² to ch/s² 8 ft/h² = 1.88 × 10-6 decimeter per second squared ft/h² to dm/s² 8 ft/h² = 1.88 × 10-8 dekameter per second squared ft/h² to dam/s² 8 ft/h² = 1.88 × 10-25 Exameter per second squared ft/h² to Em/s² 8 ft/h² = 1.03 × 10-7 fathom per second squared ft/h² to ftm/s² 8 ft/h² = 188 148 148 femtometers per second squared ft/h² to fm/s² 8 ft/h² = 188 148 148 fermis per second squared ft/h² to fermi/s² 8 ft/h² = 6.17 × 10-7 foot per second squared ft/h² to ft/s² 8 ft/h² = 9.35 × 10-10 furlong per second squared ft/h² to fur/s² 8 ft/h² = 1.88 × 10-16 Gigameter per second squared ft/h² to Gm/s² 8 ft/h² = 1.88 × 10-9 hectometer per second squared ft/h² to hm/s² 8 ft/h² = 7.41 × 10-6 inch per second squared ft/h² to in/s² 8 ft/h² = 1.88 × 10-10 kilometer per second squared ft/h² to km/s² 8 ft/h² = 1.99 × 10-23 light year per second squared ft/h² to ly/s² 8 ft/h² = 1.88 × 10-13 Megameter per second squared ft/h² to Mm/s² 8 ft/h² = 1.88 × 10-7 meter per second squared ft/h² to m/s² 8 ft/h² = 7.41 microinches per second squared ft/h² to µin/s² 8 ft/h² = 0.19 micrometer per second squared ft/h² to µm/s² 8 ft/h² = 0.19 micron per second squared ft/h² to µ/s² 8 ft/h² = 0.01 mil per second squared ft/h² to mil/s² 8 ft/h² = 1.17 × 10-10 mile per second squared ft/h² to mi/s² 8 ft/h² = 0.0002 millimeter per second squared ft/h² to mm/s² 8 ft/h² = 188.15 nanometers per second squared ft/h² to nm/s² 8 ft/h² = 1.02 × 10-10 nautical mile per second squared ft/h² to nmi/s² 8 ft/h² = 6.1 × 10-24 parsec per second squared ft/h² to pc/s² 8 ft/h² = 1.88 × 10-22 Petameter per second squared ft/h² to Pm/s² 8 ft/h² = 188 148.15 picometers per second squared ft/h² to pm/s² 8 ft/h² = 1.88 × 10-19 Terameter per second squared ft/h² to Tm/s² 8 ft/h² = 0.01 thou per second squared ft/h² to thou/s² 8 ft/h² = 2.06 × 10-7 yard per second squared ft/h² to yd/s² 8 ft/h² = 1.88 × 10+17 yoctometers per second squared ft/h² to ym/s² 8 ft/h² = 1.88 × 10-31 Yottameter per second squared ft/h² to Ym/s² 8 ft/h² = 1.88 × 10+14 zeptometers per second squared ft/h² to zm/s² 8 ft/h² = 1.88 × 10-28 Zettameter per second squared ft/h² to Zm/s² 8 ft/h² = 6 773 333.34 angstroms per minute squared ft/h² to Å/min² 8 ft/h² = 4.53 × 10-15 astronomical unit per minute squared ft/h² to au/min² 8 ft/h² = 6.77 × 10+14 attometers per minute squared ft/h² to am/min² 8 ft/h² = 0.07 centimeter per minute squared ft/h² to cm/min² 8 ft/h² = 3.37 × 10-5 chain per minute squared ft/h² to ch/min² 8 ft/h² = 0.01 decimeter per minute squared ft/h² to dm/min² 8 ft/h² = 6.77 × 10-5 dekameter per minute squared ft/h² to dam/min² 8 ft/h² = 6.77 × 10-22 Exameter per minute squared ft/h² to Em/min² 8 ft/h² = 0.0004 fathom per minute squared ft/h² to ftm/min² 8 ft/h² = 677 333 333 336 femtometers per minute squared ft/h² to fm/min² 8 ft/h² = 677 333 333 336 fermis per minute squared ft/h² to fermi/min² 8 ft/h² = 0.002 foot per minute squared ft/h² to ft/min² 8 ft/h² = 3.37 × 10-6 furlong per minute squared ft/h² to fur/min² 8 ft/h² = 6.77 × 10-13 Gigameter per minute squared ft/h² to Gm/min² 8 ft/h² = 6.77 × 10-6 hectometer per minute squared ft/h² to hm/min² 8 ft/h² = 0.03 inch per minute squared ft/h² to in/min² 8 ft/h² = 6.77 × 10-7 kilometer per minute squared ft/h² to km/min² 8 ft/h² = 7.16 × 10-20 light year per minute squared ft/h² to ly/min² 8 ft/h² = 6.77 × 10-10 Megameter per minute squared ft/h² to Mm/min² 8 ft/h² = 0.001 meter per minute squared ft/h² to m/min² 8 ft/h² = 26 666.67 microinches per minute squared ft/h² to µin/min² 8 ft/h² = 677.33 micrometers per minute squared ft/h² to µm/min² 8 ft/h² = 677.33 microns per minute squared ft/h² to µ/min² 8 ft/h² = 26.67 mils per minute squared ft/h² to mil/min² 8 ft/h² = 4.21 × 10-7 mile per minute squared ft/h² to mi/min² 8 ft/h² = 0.68 millimeter per minute squared ft/h² to mm/min² 8 ft/h² = 677 333.33 nanometers per minute squared ft/h² to nm/min² 8 ft/h² = 3.66 × 10-7 nautical mile per minute squared ft/h² to nmi/min² 8 ft/h² = 2.2 × 10-20 parsec per minute squared ft/h² to pc/min² 8 ft/h² = 6.77 × 10-19 Petameter per minute squared ft/h² to Pm/min² 8 ft/h² = 677 333 333.6 picometers per minute squared ft/h² to pm/min² 8 ft/h² = 6.77 × 10-16 Terameter per minute squared ft/h² to Tm/min² 8 ft/h² = 26.67 thous per minute squared ft/h² to thou/min² 8 ft/h² = 0.001 yard per minute squared ft/h² to yd/min² 8 ft/h² = 6.77 × 10+20 yoctometers per minute squared ft/h² to ym/min² 8 ft/h² = 6.77 × 10-28 Yottameter per minute squared ft/h² to Ym/min² 8 ft/h² = 6.77 × 10+17 zeptometers per minute squared ft/h² to zm/min² 8 ft/h² = 6.77 × 10-25 Zettameter per minute squared ft/h² to Zm/min² 8 ft/h² = 24 384 000 000 angstroms per hour squared ft/h² to Å/h² 8 ft/h² = 1.63 × 10-11 astronomical unit per hour squared ft/h² to au/h² 8 ft/h² = 2.44 × 10+18 attometers per hour squared ft/h² to am/h² 8 ft/h² = 243.84 centimeters per hour squared ft/h² to cm/h² 8 ft/h² = 0.12 chain per hour squared ft/h² to ch/h² 8 ft/h² = 24.38 decimeters per hour squared ft/h² to dm/h² 8 ft/h² = 0.24 dekameter per hour squared ft/h² to dam/h² 8 ft/h² = 2.44 × 10-18 Exameter per hour squared ft/h² to Em/h² 8 ft/h² = 1.33 fathoms per hour squared ft/h² to ftm/h² 8 ft/h² = 2.44 × 10+15 femtometers per hour squared ft/h² to fm/h² 8 ft/h² = 2.44 × 10+15 fermis per hour squared ft/h² to fermi/h² 8 ft/h² = 0.01 furlong per hour squared ft/h² to fur/h² 8 ft/h² = 2.44 × 10-9 Gigameter per hour squared ft/h² to Gm/h² 8 ft/h² = 0.02 hectometer per hour squared ft/h² to hm/h² 8 ft/h² = 96 inches per hour squared ft/h² to in/h² 8 ft/h² = 0.002 kilometer per hour squared ft/h² to km/h² 8 ft/h² = 2.58 × 10-16 light year per hour squared ft/h² to ly/h² 8 ft/h² = 2.44 × 10-6 Megameter per hour squared ft/h² to Mm/h² 8 ft/h² = 2.44 meters per hour squared ft/h² to m/h² 8 ft/h² = 96 000 000 microinches per hour squared ft/h² to µin/h² 8 ft/h² = 2 438 400 micrometers per hour squared ft/h² to µm/h² 8 ft/h² = 2 438 400 microns per hour squared ft/h² to µ/h² 8 ft/h² = 96 000 mils per hour squared ft/h² to mil/h² 8 ft/h² = 0.002 mile per hour squared ft/h² to mi/h² 8 ft/h² = 2 438.4 millimeters per hour squared ft/h² to mm/h² 8 ft/h² = 2 438 400 000 nanometers per hour squared ft/h² to nm/h² 8 ft/h² = 0.001 nautical mile per hour squared ft/h² to nmi/h² 8 ft/h² = 7.9 × 10-17 parsec per hour squared ft/h² to pc/h² 8 ft/h² = 2.44 × 10-15 Petameter per hour squared ft/h² to Pm/h² 8 ft/h² = 2 438 400 000 096 picometers per hour squared ft/h² to pm/h² 8 ft/h² = 2.44 × 10-12 Terameter per hour squared ft/h² to Tm/h² 8 ft/h² = 96 000 thous per hour squared ft/h² to thou/h² 8 ft/h² = 2.67 yards per hour squared ft/h² to yd/h² 8 ft/h² = 2.44 × 10+24 yoctometers per hour squared ft/h² to ym/h² 8 ft/h² = 2.44 × 10-24 Yottameter per hour squared ft/h² to Ym/h² 8 ft/h² = 2.44 × 10+21 zeptometers per hour squared ft/h² to zm/h² 8 ft/h² = 2.44 × 10-21 Zettameter per hour squared ft/h² to Zm/h²
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LYCHEE IN SYRUP, UPC: 083737532013 contain(s) 79 calories per 100 grams (≈3.53 ounces) [ price ]
11 foods that contain Inositol. List of these foods starting with the highest contents of Inositol and the lowest contents of Inositol
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CaribSea, Marine, Arag-Alive, Hawaiian Black weighs 1 281.5 kg/m³ (80.00143 lb/ft³) with specific gravity of 1.2815 relative to pure water. Calculate how much of this gravel is required to attain a specific depth in a cylindricalquarter cylindrical or in a rectangular shaped aquarium or pond [ weight to volume | volume to weight | price ]
Calcium phosphate, dibasic dihydrate [CaHPO4 ⋅ 2H2O] weighs 2 310 kg/m³ (144.20859 lb/ft³) [ weight to volume | volume to weight | price | mole to volume and weight | mass and molar concentration | density ]
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Volume to Weight conversions for common substances and materials | 3,385 | 8,863 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2024-22 | latest | en | 0.60061 |
https://thespectrumofriemannium.wordpress.com/2012/11/27/log055-will-hunting-problems/ | 1,501,123,029,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549426951.85/warc/CC-MAIN-20170727022134-20170727042134-00593.warc.gz | 727,562,756 | 52,201 | # LOG#055. Will Hunting problems.
The off-topic post today (I am trying to edit a complete long thread about “fundamental constants” and “units” in Physics for the next posts, so be patient, please) is graph theory through a wonderful film. I have always loved the movie Good Will Hunting. It has some remarkable problems I solved long ago…Here you are my solutions ( and a final challenge at the end of this blog post to test your mathematical skills).
Problem A. Given the graph G below:
1) Find the adjacency matrix L of G.
2) Find the matrix giving the number of 3 step walks in G.
3) Find the generating function of the graph G for walks from node (i) to node (j).
4) Find the generating function for walks from point/node 1 to point/node 3. Explain the final result.
The adjacency matrix L or $L=(L_{ij})$ encodes some beautiful features of the graph. The entry $L_{ij}$ is equal to k if there are k connections/links/edges between node i and j. Otherwise, the entry is zero. Problem A1 is to find L and A2 asks to find the matrix which encodes all possible paths of length 3. $L^2_{ij}$ is by definition of the matrix product the sum
$L_{i1}L_{1j}+L_{i2}L_{2j}+\ldots+L_{in}L_{nj}$
Each term in this sum, like $L_{i1}L_{1j}$ is not 0 if and only if there is at least one path of length 2 going from i to j passing through k. Therefore, the matrix $(L^2)_{ij}$ is the number of paths of length 2 leading from node i to j.
Similarly, $L^n_{ij}$ will be the number of paths of length n going from i to j. A walk/path of length k is an ordered sucession of vertices and edges $(v_1,e_1,\ldots,v_n,e_n)$ and such as $v_1=v_n$ if the walk is “closed” after k-steps/jumps from node to node.
Generating function/Generating function matrix. To a graph one can assign for pair of nodes (i,j) a series
$\displaystyle{f(z)=\sum_{n=0}^\infty a_n^{(ij)}z^n}$
and where $a_n^{(ij)}$ is the number of walks from i to j with n steps. Problem A3 asks for a formula for f(z) and problem A4 asks an explicit expression in the case i=1,j=3. In the movie, the generating function is written in a different notation
$\displaystyle{\Gamma^\omega \left(P_i\rightarrow P_j; z\right)=\sum_{n=0}^\infty \omega_n \left( i\rightarrow j\right)z^n}$
There, $a_n^{(ij)}=\omega_n \left( i\rightarrow j\right)=\omega_n^{(ij)}$
The formula $\displaystyle{\sum_n x^n=(1-x)^{-1}}$ for the summation of a geometric series holds also for matrices, since
$\displaystyle{f(z)_{ij}=\sum_{ n=0}^\infty L^n_{ij}z^n=\left[\sum_{n=0}^\infty L^nz^n\right]_{ij}=\left[\left( 1-Lz\right)^{-1}\right]_{ij}}$
Solution A1.
$\boxed{A=L_{ij}=\begin{pmatrix} 0 & 1 & 0 & 1\\ 1 & 0 & 2 & 1\\ 0 & 2 & 0 & 0\\ 1 & 1& 0 & 0\end{pmatrix}}$
Solution A2.
$A^2=L^2_{ij}=\begin{pmatrix}2 & 1 & 2 & 1\\ 1 & 6 & 0 & 1\\ 2 & 0 & 4 & 2\\ 1 & 1 & 2 & 2\end{pmatrix}$
$\boxed{A^3=L^3_{ij}=\begin{pmatrix}2 & 7 & 2 & 3\\ 7 & 2 & 12 & 7\\ 2 & 12 & 0 & 2\\ 3 & 7 & 2 & 2\end{pmatrix}}$
Solution A3.
The Cramer’s rule for the inverse of a matrix is
$A^{-1}=\dfrac{\mbox{Adj}(A)^T}{\det A}$
and it leads to the equivalent formulae
$A^{-1}_{ij}=\dfrac{\mbox{Adj}(A^T)_{ij}}{\det A_{ij}}$
and with $A=1-zL$
$(1-zL)^{-1}_{ij}=\dfrac{\mbox{Adj}((1-zL)^T)_{ij}}{\det (1-zL)_{ij}}$
$\vert A^{-1}_{ij}\vert =\dfrac{\vert \mbox{Adj}(A^T)_{ij}\vert }{\vert A_{ij}\vert }=\dfrac{\vert \mbox{Adj}(A)_{ij}\vert }{\vert A_{ij}\vert }$
as $\det A=\det A^T$. For $A=\mathbb{I}-z\mathbb{L}$, we get
$\det A^{-1}_{ij}=\dfrac{\det \left(\mathbb{I}-z\mathbb{L}\right)}{\det (1-zL)}=\dfrac{\det (\mathbb{I}_{ij}-zL_{ij})}{\det (1-zL)}$
It can also be written as
$\det A^{-1}_{ij}=\dfrac{\det \left(\mbox{Adj}(L^{-1}-z)_{ij}\right)}{\det (L^{-1}-z)}$
Solution A4.
Especially, when i=1 and j=3, we get
$\det (\mbox{Adj}(L)_{13})=2z^2+2z^3$ and $\det (1-zL)=1-7z^2-2z^3+4z^4$. Then, the generating function can be written as the fraction
$f(z)=\dfrac{2z^2+2z^3}{1-7z^2-2z^3+4z^4}=\dfrac{2z^2(1+z)}{1-7z^2-2z^3+4z^4}=\dfrac{2z^2}{1-z-6z^2+4z^3}$
Indeed, the computation can be done for every pair of nodes. The generating function matrix comes from
$A=L_{ij}=\begin{pmatrix}0 & 1 & 0 & 1\\ 1 & 0 & 2 & 1\\ 0 & 2 & 0 & 0\\ 1 & 1& 0 & 0\end{pmatrix}$
and then
$A^{-1}=L_{ij}^{-1}=f(z)_{ij}=W(z)_{ij}=(I-zL)^{-1}=\begin{pmatrix}1 & -z & 0 & -z\\ -z & 1 & -2z & -z\\ 0 & -2z & 1 & 0\\ -z & -z& 0 & 1\end{pmatrix}^{-1}$
A long and tedious calculation provides
$W(z)_{ij}=\dfrac{1}{1-z-6z^2+4z^3}\begin{pmatrix}\frac{1-5z^2}{1+z} & z & 2z^2 & \frac{z+z^2-4z^3}{1+z}\\ z & 1-z & 2(z-z^2) & z \\ 2z^2 & 2(z-z^2) & 1-z-2z^2 & 2z^2\\ \frac{z+z^2-4z^3}{1+z} & z & 2z^2 & \frac{1-5z^2}{1+z}\end{pmatrix}$
Finally, let us remark that if we expand the generating function into an infinite series, we can obtain the number of walks of k-length as the numerical coefficient in the kth-power of z. In our example and problem from the Will Hunting’s movie, we deduce that
$f(z)=\dfrac{2z^2}{1-z-6z^2+4z^3}$
and then
$f(z)\approx 2z^2+2z^3+14z^4+18z^5+96z^6+146z^7+638z^8+1138z^9+\mathcal{O}(z^{10})$
or up to 12th degree in z
$f(z)\approx 2z^2+2z^3+14z^4+18z^5+96z^6+146z^7+638z^8+1138z^9+4382z^{10}+8568z^{11}+30398z^{12}+\mathcal{O}(z^{13})$
Then, from the node 1 to the node 3 there is 2 walks of length 2, 2 walks of length 3, 14 walks of length 4, 18 paths of length 5, 96 paths of length 6, 146 walks of length 7, 638 paths of length 8, 1138 walks of length 9, and so on.
The movie provides this cool picture of the problem and the solutions I have already deduced
Problem B.
1) State the Cayley’s theorem/formula and explain its meaning.
2) Find every homeomorphically irreducible tree of degree 10 ( i.e., irreducible trees of 10 nodes non-isomorphic to each other, and without vertices of degree 2).
Answer B1. Cayley’s formula is a result in graph theory named after the mathematician Arthur Cayley. It states that for any positive integer $n$ the number of trees with labeled vertices is $n^{n-2}$, i.e, it says that the number of trees on n-labeled vertices is $n^{n-2}$.
Remark: The formula equivalently counts the number of “spanning trees” of a complete graph $K_n$ with n-labeled vertices.
Answer B2. The solution is given by the following trees
Only 8 of these graphs are in the movie, since when Will Hunting is writing them on the blackboard, Prof. Lambeau appeared and Will didn’t continue with the solution to the problem.
Remark: The counting function for homeomorphically (inequivalent) irreducible trees is
$h(x)=x+x^2+x^4+x^5+2x^6+2x^7+4x^8+5x^9+10x^{10}+14x^{11}+26x^{12}+\mathcal{O}(x^{13})$
Homeomorphically irreducible trees are also called series-reduced trees or topological trees. Indeed, the generating function expansion is a series formed by the so-called number of reduced trees with n-nodes. It is the Sloane series A000014, and it is given by the following numbers:
$a_n=0, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, 249, 445, 842, 1561, 2988, 5671, 10981,$
$21209, 41472, 81181, 160176, 316749, 629933, 1256070, 2515169, 5049816, 10172638,$
$20543579, 41602425, 84440886, 171794492, 350238175, 715497037, 146440711,\ldots$
Here you are the complete list of homeomorphically irreducible trees up to n=12 ( n=0 is absent since it is the empty set) :
PS: A final poster of the Good Will Hunting movie and some of the problem B solutions, with solution in b) being wrong in one case AND incomplete ( be aware!)…
PS(II): A challenge for eager readers is…Are you able to guess AND prove the complicated formulae providing the last series and numbers? I mean, are you able to find the generating function producing the series above?
$a_n=0, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, 249, 445, 842, 1561, 2988, 5671, 10981,$
$21209, 41472, 81181, 160176, 316749, 629933, 1256070, 2515169, 5049816, 10172638,$
$20543579, 41602425, 84440886, 171794492, 350238175, 715497037, 146440711,\ldots$
It is not easy but it is fun and enlightening!
STAY TUNED!!!
### 6 Comments on “LOG#055. Will Hunting problems.”
1. Hi . Could you provide the solution for problem PS(II)? . I do not have the proper mathematical knowledge to arrive at the answer yet but I am very interested in knowing the generating function for this series.
• amarashiki says:
Well, the solution is tricky and very hard, because there is no analytical generating function per se. You have to guess a numerical solution for an implicit equation of 3 specific graph types and then you solve the implicit equation with a computer or by hand by brute force(I did the latter long ago). As far as I know, there is no analytical closed formula to that series, but an implicit solution in terms of 3 generating functions makes the job. You can also search for the solution in a mathematical page devoted to the origin and uses of series if you are clever enough. However, it is a nice excercise to face with the problem, as I did myself some years ago, and compute analitically the first numbers and/or use your computer (with the aid of Mathematica, Maple,…) to solve the 3 equations providing the solution implicitly. I will not provide solutions of this class of excercises in my blog (explicitly) since I believe that “young people” go too fast for the solution without thinking in general the procedure to get them! It is important the process to arrive at solutions! Otherwise, and more now in the internet era, you google it and you have the solution. That is not how the world works. Scientists like me face problems without knowing most of the time how to solve it…Hehehe…Engineers and other people are interested only in “the solution”… It also happens with problems like those I wrote long ago relative to neutrinos in some Cherenkov detectors. I really believe that what matters for people interested in science should also be how to get the solutions. Of course, some of these ideas I have could be preposterous in the far future, when every question has been solved, it is likely that people were only interested in the solutions, and not how to get them (with computers only?)…A pity! Try harder! You will learn a lot if you face the problem yourself… I urge you to post the solution in the future, when you find it. I will be honored and glad to know some people do/solve the problems I proposed here :).
2. srandy6977 says:
Two of the entries in your matrix L^2 are incorrect. The final two entries of Row 2 should be 01. In your matrix they are transposed as 10.
3. just wondering if n=10 is the only case witch the number of nodes is equal to the number of solutions. | 3,542 | 10,562 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 47, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.96875 | 4 | CC-MAIN-2017-30 | longest | en | 0.862628 |
https://mail.python.org/pipermail/python-list/2004-February/247980.html | 1,398,299,140,000,000,000 | text/html | crawl-data/CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00313-ip-10-147-4-33.ec2.internal.warc.gz | 977,472,843 | 1,776 | # finding all sublists of list2 that are identical to list1
Klaus Neuner klaus_neuner82 at yahoo.de
Mon Feb 9 15:59:35 CET 2004
```Hello,
the function given below returns all indexes of list2 where a sublist
of list2 that is identical to list1 begins.
As I will need this function quite often, I would like to know if more
experienced programmers would agree with the way I defined the
function:
- Is there a more efficient way to do it? (Apart from building
automata.)
- Is there a more elegant/Pythonic way to write the function?
Klaus
def sublist(list1, list2):
if list1 == list2:
return [0]
elif len(list1) > len(list2):
return []
else:
result = []
shift = 0
i = 0
while shift <= len(list2)-len(list1):
if list1[i] == list2[shift+i]:
if i == len(list1)-1:
result.append(shift)
i = 0
shift += 1
else:
i += 1
else:
i = 0
shift += 1
return result
``` | 255 | 859 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2014-15 | latest | en | 0.638054 |
https://www.answers.com/natural-sciences/How_many_inches_in_6_mm | 1,675,455,860,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00575.warc.gz | 648,191,748 | 50,204 | 0
How many inches in 6 mm?
Wiki User
2013-06-14 06:13:17
There are 152.4 millimeters in 6 inches. 6 inches x 25.4 millimeters/1 inch = 152.4 millimeters 1 inch = 25.4 millimeters
Wiki User
2009-04-25 18:05:44
Study guides
20 cards
How many miles equals a km
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118 Reviews
Wiki User
2013-06-14 06:13:17
0.23622"
Direct Conversion Formula 6 mm*
1 in
25.4 mm
=
0.2362204724 in | 159 | 409 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2023-06 | latest | en | 0.659321 |
https://brainsanswers.com/physics/question9451997 | 1,600,844,324,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400209999.57/warc/CC-MAIN-20200923050545-20200923080545-00702.warc.gz | 292,389,074 | 31,428 | 480 kilograms horse runs across the field at a rate of 40 km/hr what is the magnitude of the force
, 26.07.2019 08:00, lizzyboo32
# 480 kilograms horse runs across the field at a rate of 40 km/hr what is the magnitude of the force
### Other questions on the subject: Physics
Physics, 22.06.2019 05:30, mlopezmanny5722
The a992 steel rod bc has a diameter of 50 mm and is used as a strut to support the beam. determine the maximum intensity w of the uniform distributed load that can be applied to the beam without risk of causing the strut to buckle. take f. s. = 2 against bucklin
Physics, 22.06.2019 05:30, xaviiaquino3378
Choose the most likely outcome of this scenario: jen decided to go bike riding without a helmet. while no one is around during her ride, she is thrown from her bike when her wheel goes into a pothole. she is not injured, but she is terrified to get back on her bike. what happens next? a. her physical health is affected even though she wasn't hurt. b. her mental and emotional health are affected because she is afraid to get back on her bike. c. her social health is affected because she is worried her friends saw the fall. d. her overall health is not affected at all by her fall. | 312 | 1,214 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2020-40 | latest | en | 0.961212 |
https://stats.stackexchange.com/questions/57807/confusion-with-augmented-dickey-fuller-test?noredirect=1 | 1,596,941,003,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738380.22/warc/CC-MAIN-20200809013812-20200809043812-00592.warc.gz | 520,736,492 | 35,133 | # Confusion with Augmented Dickey Fuller test
I am working on the data set electricity available in R package TSA. My aim is to find out if an arima model will be appropriate for this data and eventually fit it. So I proceeded as follows:
1st: Plot the time series which resulted if the following graph:
2nd: I wanted to take log of electricity to stabilize variance and afterward differenced the series as appropriate, but just before doing so, I tested for stationarity on the original data set using the adf (Augmented Dickey Fuller) test and surprisingly, it resulted as follows:
### Code and Results:
adf.test(electricity)
Augmented Dickey-Fuller Test
data: electricity
Dickey-Fuller = -9.6336, Lag order = 7, p-value = 0.01
alternative hypothesis: stationary
Warning message: In adf.test(electricity) : p-value smaller than printed p-value
Well, as per my beginner's notion of time series, I suppose it means that the data is stationary (small p-value, reject null hypothesis of non-stationarity). But looking at the ts plot, I find no way that this can be stationary. Does anyone has a valid explanation for this?
• ADF only tests for unit root stationary, this could be trend stationary. So you should use the KPSS test, see stats.stackexchange.com/questions/30569/… In general, there is a difference, between DS (difference-stationary) and TS (trend stationary) models. KPSS is the better test to distinguish between those models, see the link for more details. – Stat Tistician May 1 '13 at 13:28
• Looks like the series has seasonals and trend. Integrate in the ADF-test a deterministic trend + seasonal dummies and run the test. Check also for autocorrelated residuals. – Pantera May 1 '13 at 23:07
Since you take the default value of k in adf.test, which in this case is 7, you're basically testing if the information set of the past 7 months helps explain $x_t - x_{t-1}$. Electricity usage has strong seasonality, as your plot shows, and is likely to be cyclical beyond a 7-month period. If you set k=12 and retest, the null of unit root cannot be rejected,
> adf.test(electricity, k=12)
Augmented Dickey-Fuller Test
data: electricity
Dickey-Fuller = -1.9414, Lag order = 12, p-value = 0.602
alternative hypothesis: stationary
Assuming that "adf.test" really comes from the "tseries" package (directly or indirectly), the reason would be that it automatically includes a linear time trend. From the tseries doc (version 0.10-35): "The general regression equation which incorporates a constant and a linear trend is used [...]" So the test result indeed indicates trend stationarity (which despite the name is not stationary).
I also agree with Pantera that the seasonal effects could distort the result. The series could in reality be a time trend + deterministic seasonals + stochastic unit root process, but the ADF test might mis-interpret the seasonal fluctuations as stochastic reversions to the deterministic trend, which would imply roots smaller than unity. (On the other hand, given that you have included enough lags, this should rather show up as (spurious) unit roots at seasonal frequencies, not the zero/long-run frequency that the ADF test looks at. In any case, given the seasonal pattern it's better to include the seasonals.) | 773 | 3,275 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2020-34 | latest | en | 0.918607 |
http://www.chegg.com/homework-help/questions-and-answers/water-92-degrees-celsius-pumped-storage-tank-rate-300-l-min-motor-pump-supplies-work-rate--q1639468 | 1,475,273,969,000,000,000 | text/html | crawl-data/CC-MAIN-2016-40/segments/1474738662400.75/warc/CC-MAIN-20160924173742-00245-ip-10-143-35-109.ec2.internal.warc.gz | 393,728,753 | 13,673 | Water at 92 degrees Celsius is pumped from a storage tank at a rate of 300 L/min.
The motor of the pump supplies work at the rate of 3 hp. The water is passed
through a heat exchanger, where it gives up heat at a rate of 800 kW, and it is delivered to a second storage tank at an elevation of 20 m above the first tank.
What is the mass flow rate of the hot water? [Hint: you may not simply
assume that the water has a density of 1.00 g/cm^3] | 118 | 443 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2016-40 | latest | en | 0.94749 |
http://quizlet.com/12336654/math-geometry-equations-flash-cards/ | 1,427,594,008,000,000,000 | text/html | crawl-data/CC-MAIN-2015-14/segments/1427131298020.57/warc/CC-MAIN-20150323172138-00213-ip-10-168-14-71.ec2.internal.warc.gz | 221,279,365 | 14,764 | # Math Geometry equations
### 7 terms by idontknowanymore
#### Study only
Flashcards Flashcards
Scatter Scatter
Scatter Scatter
## Create a new folder
### Surface Area of a cone
SA=πr² + πrl (l = Slant of cone)
SA=6s²
### Surface area of a rectangular prism
SA=2[(l×w)+(l×h)+(w×h)]
### Surface area of a Triangular prism
SA=2×triangle sides+3 rectangles
2×end face
V=A×h
V=1/3×A×h
Example: | 138 | 408 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.125 | 3 | CC-MAIN-2015-14 | latest | en | 0.537411 |
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