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Eighty Years of the Finite Element Method

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Re: Eighty Years of the Finite Element Method

#21

Post by pclmulqdq »

My numerical analysis professor once told the class a story about how he used the finite element method to solve stresses on pieces of metal for the soviet space program. Computers were too expensive, so they did it by hand.

They cleared a university classroom of all the chairs and desks, and rolled out pieces of paper to cover the floor. The team took their shoes off proceeded from one corner of the room to the other. If you found that someone had made a mistake, you had a record available to find it, and you could simply rip up the paper at the point where the mistake was made and roll out fresh paper to take its place.

Apparently a triangle took a few days.

SolidWorks does the same math today.

Re: Eighty Years of the Finite Element Method

#22

Post by shoo »

mazokum wrote: post #11
revskill wrote: post #7

I'm not sure if there's a simple implementation of FEM (might be < 50 lines of code) to understand the virtual of the algorithm then.

You can check this paper. Is the one my advisor recommended when I started my PhD. Is a FEM implementation in 50 lines of MATLAB. https://www.math.hu-berlin.de/~cc/cc_homepage/download/1999-...

What I found amazing about FEM is not the detail of how to implement it in code, but all the PDE theory & approximation theory -- how you can express the original continuous-domain infinite dimensional problem in a weak form using an infinite dimensional space of test functions, then approximating the weak form of the original problem with a finite dimensional Galerkin approximation, using a finite dimensional space of test functions, and use that to define a finite dimensional system of equations to solve. Then the theory for under what conditions you can guarantee that approximate solutions obtained from your finite dimensional approximation converge toward the true solution, as you increase the mesh resolution, and how fast the convergence rate will be.

Some of this is summarised in this paper in 2. model problem & 3. Galerkin discretisation of the problem, but not in a way that will communicate the mathematical ideas to anyone who hasn't already taken a course on the theory -- probably need a couple of courses on real analysis & a course on PDE as pre-reqs.

Re: Eighty Years of the Finite Element Method

#25

Post by magicalhippo »

rpep wrote: post #13

My favourite resources for FEM were the book and courses that Hans Petter Langtangen (RIP) wrote at Simula. FEniCs made FEM so easy, it’s truly an excellent software project that is unfortunately not as well known as it should be within the community. http://hplgit.github.io/num-methods-for-PDEs/doc/web/index.h... http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.h...

> FEniCs made FEM so easy

https://fenicsproject.org/

Indeed, was blown away when I saw it for the first time over a decade ago, compared to the convoluted C++ FEM libraries I had seen before that.

Re: Eighty Years of the Finite Element Method

#27

Post by thrown_22 »

badpun wrote: post #19
kqr wrote: post #12

Earlier quoted context omitted.

For physical processes, you can sometimes lean on conservation laws. Financial, no-arbitrage "laws". Sometimes it can help to run two different models, or different numerical methods, and compare their results. More generally, it's very, very hard. I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong.

> I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong. I wonder what that means for the accuraccy of the climate models...

All models are wrong, some are useful.

Re: Eighty Years of the Finite Element Method

#28

Post by geysersam »

badpun wrote: post #19
kqr wrote: post #12

Earlier quoted context omitted.

For physical processes, you can sometimes lean on conservation laws. Financial, no-arbitrage "laws". Sometimes it can help to run two different models, or different numerical methods, and compare their results. More generally, it's very, very hard. I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong.

> I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong. I wonder what that means for the accuraccy of the climate models...

It's probably impossible to accurately simulate the climate of the planet decades into the future.

That does not meaningfully detract from the evidence for human caused global warming however.

Re: Eighty Years of the Finite Element Method

#29

Post by halffaday »

geysersam wrote: post #28
badpun wrote: post #19

Earlier quoted context omitted.

> I am fairly sure there are multiple published results in computational fluid dynamics that are subtly wrong. I wonder what that means for the accuraccy of the climate models...

It's probably impossible to accurately simulate the climate of the planet decades into the future. That does not meaningfully detract from the evidence for human caused global warming however.

I have to applaud the subtle elegance of the sarcasm here

Re: Eighty Years of the Finite Element Method

#30

Post by stergios »

mbustamanter wrote: post #5
kwkelly wrote: post #3

Earlier quoted context omitted.

I just skimmed this and I never worked in FEM but this really brought me back. I had the pleasure of meeting several of the people in the article (Hughes, Oden, Babuska, Demkowicz) and couldn’t help feeling some nostalgia as well.

Great! Hughes, Oden and Babuška are three of my heroes. My work is in FEM, specifically, the theory of Mixed Finite Elements and Eigenalue problems, to which both Babuška made important contributions. Oden I know from elasticity for the most part.

I was lucky enough to have taken two FEA classes from Hughes in the 80s. I worked with his DLEARN [1] code many times back then. He was a great teacher.

I never had the pleasure of taking classes from Juan Carlos Simo, but he was known to have outstanding classes. His was a very brilliant light & life cut too short by cancer at the young age of 42.

[1]. DLEARN is a linear static and dynamic finite element code written in Fortran. https://github.com/fit087/fem_hughes

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