7 ms·

Eighty Years of the Finite Element Method

179 points by leephillips 4y ago

sd2022al 4y ago
Very nice summary paper on FEM! FEM has now become a key part of any physical product development process. I used to work in FEM area and few years back switched to unrelated software domains - so felt a bit of nostalgia reading the paper. Thanks for posting !
- kwkelly 4y ago
  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.
  - mbustamanter 4y ago
    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.
    - stergios 4y ago
      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 https://github.com/fit087/fem_hughes
MichaelZuo 4y ago
Thanks for posting! Reminds me of long night spent fussing over ANSYS models.
mkoubaa 4y ago
I still work on FEA and it's as technically interesting as ever
revskill 4y ago
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.
- mjhay 4y ago
  The algorithm for linear PDEs usually boils down to some kind of meshing/discretization (often the hard part) to make a (usually sparse) linear system to solve using standard numerical methods. In the basic 1d, first-order case, it winds up being exactly the same thing as equivalent finite difference method.
- mazokum 4y ago
  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-AJ_CC_FS-50_Lines_of_Matlab.pdf https://www.math.hu-berlin.de/~cc/cc_homepage/download/1999-...
  - shoo 4y ago
    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.
    - kqr 4y ago
icks 4y ago
My first programming role was on the back of work by an engineer in this article. The core of the solver was a FORTRAN implementation of a paper on p-convergence. It was really amazing seeing our software predict how a small crack in a part of an aircraft would propagate. The 3D model produced matched the photograph shared later. The lead developer (at the time) once said that the biggest software failure we can have is not incorrect results, but incorrect results without the user knowing. This is probably why I am so bothered by silent failures in my big company role now.
- wheelinsupial 4y ago
  I know it's not a simple answer, but how would you embed checks to flag or highlight potentially incorrect results to the user?
  - kqr 4y ago
    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.
    - meheleventyone 4y ago
      A lot of computer models are deliberately wrong but useful as well which is another ball game in terms of analysing results.
    - badpun 4y ago
      > 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...
      - thrown_22 4y ago
        All models are wrong, some are useful.
rpep 4y ago
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.html http://hplgit.github.io/num-methods-for-PDEs/doc/web/index.h... http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.html http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.h...
- laingc 4y ago
  Used it in its infancy for my PhD work - absolutely loved it, despite its teething problems. Also +1 for Langtangen!
- magicalhippo 4y ago
  > FEniCs made FEM so easy https://fenicsproject.org/ 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.
netman21 4y ago
This is great. I studied FEM in school and got my first job in industry as a structural analyst. I started my own firm when I was 25 and employed 22 analysts that I contracted to Pontiac Motors.
- Scramblejams 4y ago
  Former structural analyst here. :waves: If you worked on consumer products there (aka cars and their components), do you happen to remember what the design-to fatigue life was? I talked with another analyst several years ago who had worked at Chrysler, and he told me they had used 125,000 miles. (I was in aerospace so not sure how many miles comprised a fatigue cycle, etc.)
  - auxym 4y ago
    Not exactly what you asked, but: I worked as a structural analyst on passenger train cars (metros, LRVs, etc) for a while, as my first job after grad school actually. Depending on the project (client requirements), we designed for 25-35 year lifetimes, with 12-24h operation typical. That usually amounted to millions of kilometers. We had load cases with varying numbers of cycles. Eg curves with light loading might have been millions of cycles, but max (or even over max) loading might have been 10s or 100s of thousands of cycles. All load cases were determined based on on usage stats from the operator and testing conducted to measure accelerations on the operator's infrastructure.
TheRealPomax 4y ago
A Springer publication that's free instead of a hyper-inflated price to extrort academia for just past what it can afford? What year is it?
pclmulqdq 4y ago
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.
- hydrogen7800 4y ago
  I once read or heard a similar anecdote about Ludwig Prandtl who set up computation method using a room full of people each doing one step of a calculation. I recall it describing a finite element method, but that would predate the origin of FEM in this article by a few decades. Maybe it was just some numerical method.
  - pclmulqdq 4y ago
    I assume it would be a finite differences calculation, which is also very easy to parallelize/vectorize. They solve the same class of problems, but calculations with finite differences don't always converge the way that finite elements do.
    - marmetio 4y ago
      That's correct, it was finite difference methods. There are many variations with different properties such as whether they'll definitely converge or not.
  - perilunar 4y ago
    Huh? Theres's a long history of doing this — 'computers' were people long before they are machines.
deleted 4y ago
[deleted]
rqtwteye 4y ago
I learned FEM at school. It really drove home the power of computers. The basics of FEM are pretty simple. You should need to do many of these simple calculations.

179 points by leephillips 4y ago

sd2022al 4y ago

Very nice summary paper on FEM! FEM has now become a key part of any physical product development process. I used to work in FEM area and few years back switched to unrelated software domains - so felt a bit of nostalgia reading the paper. Thanks for posting !

kwkelly 4y ago
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.
- mbustamanter 4y ago
  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.
  - stergios 4y ago
    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 https://github.com/fit087/fem_hughes

MichaelZuo 4y ago

Thanks for posting! Reminds me of long night spent fussing over ANSYS models.

mkoubaa 4y ago

I still work on FEA and it's as technically interesting as ever

revskill 4y ago

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.

mjhay 4y ago
The algorithm for linear PDEs usually boils down to some kind of meshing/discretization (often the hard part) to make a (usually sparse) linear system to solve using standard numerical methods. In the basic 1d, first-order case, it winds up being exactly the same thing as equivalent finite difference method.
mazokum 4y ago
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-AJ_CC_FS-50_Lines_of_Matlab.pdf https://www.math.hu-berlin.de/~cc/cc_homepage/download/1999-...
- shoo 4y ago
  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.
  - kqr 4y ago

icks 4y ago

My first programming role was on the back of work by an engineer in this article. The core of the solver was a FORTRAN implementation of a paper on p-convergence. It was really amazing seeing our software predict how a small crack in a part of an aircraft would propagate. The 3D model produced matched the photograph shared later. The lead developer (at the time) once said that the biggest software failure we can have is not incorrect results, but incorrect results without the user knowing. This is probably why I am so bothered by silent failures in my big company role now.

wheelinsupial 4y ago
I know it's not a simple answer, but how would you embed checks to flag or highlight potentially incorrect results to the user?
- kqr 4y ago
  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.
  - meheleventyone 4y ago
    A lot of computer models are deliberately wrong but useful as well which is another ball game in terms of analysing results.
  - badpun 4y ago
    > 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...
    - thrown_22 4y ago
      All models are wrong, some are useful.

rpep 4y ago

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.html http://hplgit.github.io/num-methods-for-PDEs/doc/web/index.h... http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.html http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.h...

laingc 4y ago
Used it in its infancy for my PhD work - absolutely loved it, despite its teething problems. Also +1 for Langtangen!
magicalhippo 4y ago
> FEniCs made FEM so easy https://fenicsproject.org/ 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.

netman21 4y ago

This is great. I studied FEM in school and got my first job in industry as a structural analyst. I started my own firm when I was 25 and employed 22 analysts that I contracted to Pontiac Motors.

Scramblejams 4y ago
Former structural analyst here. :waves: If you worked on consumer products there (aka cars and their components), do you happen to remember what the design-to fatigue life was? I talked with another analyst several years ago who had worked at Chrysler, and he told me they had used 125,000 miles. (I was in aerospace so not sure how many miles comprised a fatigue cycle, etc.)
- auxym 4y ago
  Not exactly what you asked, but: I worked as a structural analyst on passenger train cars (metros, LRVs, etc) for a while, as my first job after grad school actually. Depending on the project (client requirements), we designed for 25-35 year lifetimes, with 12-24h operation typical. That usually amounted to millions of kilometers. We had load cases with varying numbers of cycles. Eg curves with light loading might have been millions of cycles, but max (or even over max) loading might have been 10s or 100s of thousands of cycles. All load cases were determined based on on usage stats from the operator and testing conducted to measure accelerations on the operator's infrastructure.

TheRealPomax 4y ago

A Springer publication that's free instead of a hyper-inflated price to extrort academia for just past what it can afford? What year is it?

pclmulqdq 4y ago

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.

hydrogen7800 4y ago
I once read or heard a similar anecdote about Ludwig Prandtl who set up computation method using a room full of people each doing one step of a calculation. I recall it describing a finite element method, but that would predate the origin of FEM in this article by a few decades. Maybe it was just some numerical method.
- pclmulqdq 4y ago
  I assume it would be a finite differences calculation, which is also very easy to parallelize/vectorize. They solve the same class of problems, but calculations with finite differences don't always converge the way that finite elements do.
  - marmetio 4y ago
    That's correct, it was finite difference methods. There are many variations with different properties such as whether they'll definitely converge or not.
- perilunar 4y ago
  Huh? Theres's a long history of doing this — 'computers' were people long before they are machines.

deleted 4y ago

[deleted]

rqtwteye 4y ago

I learned FEM at school. It really drove home the power of computers. The basics of FEM are pretty simple. You should need to do many of these simple calculations.