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This is a fine example of an article that represents the worst of all worlds - it fails to accurately present either the mathematical, physical or engineering aspects of FEM. It is overly simplified for applied mathematicians, incomprehensible for lay persons and useless for the practicing engineer/analyst. There are numerous blatantly incorrect / misleading statements and glaring errors of omission, and the level of presentation alternates between sophomoric and overly pedantic. Students and laymen would be well advised not to attempt to learn anything from this article!
I recently edited the article (section 2.1) by removing what was there with a few paragraphs written by Barna Szabo and Mark Ainsworth. It has been reviewed by Ivo Babuska, Ernst Rank and Alexander Duester among others considered the best in the field. Whomever rejected the edit should go back and learn, not only about the finite element method, but also about humility. He dismissed by stating that "a clear technical description works better than theoretical opacity", for a text written by 5 of the best in the field...if it is opaque it may be because your lenses are dirty dear Woodstone... I've seen some of the responses to the criticism be based on how much they are supposed to know instead of recognizing that they are other that may know even more and check their facts. It is truly disappointing to see that people egocentrism is used as a reference. I'll try to set the edit back...if unsuccessful I'll simply create another alternative page and we will let people decide which one contains valuable information. — Preceding unsigned comment added by Snervi ( talk • contribs) 17:05, 27 November 2014 (UTC)
How about John Argyris' contribution? I think there should be a relevant revision in the history section, provided that experts in the field also agree.
One short biography appears here: http://www.cimne.upc.es/webcimne/boletinECCOMAS/20040518.htm in word format. I do not know though if it can be made available to wikipedia...
http://www.mlahanas.de/Greeks/new/Argyris.htm
Dpser 17:57, 16 January 2007 (UTC)
--- Information (3 June 2008):
In my book
Kurrer, Karl-Eugen: The History of the Theory of Structures. From Arch Analysis to Computational Mechanics. Berlin: Ernst & Sohn 2008.
one can find the chapter " 'The computer shapes the theory' (Argyris): the historical roots of the finite element method and the development of computational mechanics" (pp. 619-672).
...And of course brief biographies of the pioneers of fem e.g. Argyris,...
Best regards,
Karl-Eugen Kurrer —Preceding
unsigned comment added by
212.202.96.83 (
talk)
15:13, 3 June 2008 (UTC)
I think that the extremely significant role of
O. C. Zienkiewicz has been overlooked here. He is generally considered a key pioneer, and, as rightly stated on his
Wiki page: "His books on the Finite Element Method were the first to present the subject and to this day remain the standard reference texts."
Kurvenbau ( talk) 08:42, 13 May 2009 (UTC)
I think that the works in matrix analisys of structures from Mohr and Maxwell in the XIX century must be taken in account. This pioneers works can help to the reader to understand the basis of the method because it was basically FEM with 1D elements. -- Cometo22 ( talk) 08:43, 8 March 2011 (UTC)
math should be typed using the math tag
---
I agree.
There is also something not clear about how the integration by parts is actually done. Something was left out there. (BTW does this have anything to do with Lu=g being of the Sturm-Liouville type?)
--
No. When the operator is not Hermitian, there is still a bilinear form, except of course that it is not an inner product. One can obtain existence and uniqueness from the Lax-Milgram theorem, assuming that the bilinear form is coercive. If it is not, it is sometimes still possible to obtain a form of the Fredholm alternative.
When L is of the Sturm-Liouville type, the Dirichlet problem gives rise to a Hermitian operator and so the theory is nicer (and the linear solve is also easier -- conjugate gradient with a preconditionner can often be used.) If the boundary condition is not Dirichlet, the bilinear form is usually not symmetric, even if L is of Sturm-Liouville type. Loisel 11:46, 26 Jul 2004 (UTC)
The benginning of this article states that FEM approximates solutions to PDEs. But isn't the result somtimes exact? For example; p-type elements with shape functions of order greater than two yield exact results of a beam simulation (and neglecting computer rounding errors). Its been many years since I had this class, so my memory might be fuzzy. Pud 00:46, 25 Jul 2004 (UTC)
I don't know what you're talking about, however it is possible in certain cooked-up examples for the numerical scheme to be exact modulo machine precision. However, this is true of almost every numerical method. Numerical differentiation, integration, ode solvers, pde solvers, root-finding, eigenvalue algorithms, singular value decomposition, etc... can all coincidentally be exact under certain circumstances. Loisel 11:42, 26 Jul 2004 (UTC)
Loisel, do you know what p-type elements and shape functions are? Pud 16:15, 26 Jul 2004 (UTC)
As I said, no, I don't. Loisel 11:44, 27 Jul 2004 (UTC)
I know about piecewise polynomial basis functions on a triangular mesh, if that's what you're saying.
When solving an ODE like d^2v/dx^2=c, the solutions are v=cx^2/2+ax+b, for any a,b. Then one can cook up any number of numerical schemes to solve them exactly (that's what I was talking about in my first reply above.) For instance, the two-step method v[k+2]=2*v[k+1]-v[k]+c is exact in this example (if not entirely stable) even though in general it is not -- that is what I was talking about when I said "cooked up example." The FEM in this case can be written as an implicit method and without doubt some such schemes will be exact in this case. However, I'm fairly certain that the similar conclusion is false in the two variable case d^2v/dx^2+d^2v/dy^2=c because those functions are not polynomials. If c=0, one gets the harmonic functions, none of which are polynomials.
Erratum: of course some polynomials are harmonic.
Loisel 14:58, 27 Jul 2004 (UTC)
It's awful! If I had something like that in university I would stop studying physics. Why not at least write the differential equation in its native form first?
I hope the structure of the example, as well of the entire article, will be changed to something more comprehensive. (not signed)
Actually, I am a mathematician and active in the field of finite elements for about 15 years now. Therefore, I would not accept the disqualification above for me. Still, I must confess, I recognize the method only with difficulty from this article. A person trying to find out what it is will be left completely clueless. I can only recommend rewriting most of it, in particular
Guido Kanschat 13:42, 9 December 2005 (UTC)
I was looking for a simple, lay-persons explanation of the difference between a finite element and a finite difference approach to solving a flow simulation equation. I certainly didn't get that. This is supposed to be an encyclopedia for everyone, how you can start an article with "assuming a knowledge of calculus" is beyond me - what percent of the population actually has a working knowledge of calculus? Your approach is arrogant and exclusive, hopefully someone will find a more approachable way to describe these points soon, if Hawkins can describe the big bang then you should be able to describe this.
JohnH
I agree that this assumes a lot of math background. It's great to have a rigorous explanation, but a layman's explanation and an engineering explanation would be good. I propose this article get forked so that this page, Finite element method is a layman's introduction to the history and uses of the technique, then provide links to Mathematical treatment of the Finite Element Method and Engineering treatment of the Finite Element Method much like Tensor has Classical treatment of tensors, Tensor (intrinsic definition), and Intermediate treatment of tensors. — BenFrantzDale 03:40, 22 November 2005 (UTC)
Guys, ... please, hear a humble oppinion from an electrical engineer. I just started looking into FEA recently to simulate electrical field problems and needed good info on FEM. This Wikipedia article was by far best what I could find on the Web after a few days of Google search. But... I could not disagree that the treatment here of FEM is more "mathematically abstracted" than it needs to be for the "popular engineering audience" (although one can argue the FEM/FEA audience is popular in the sense of guys going out on a Friday night... :) Anyway, I applause the effort for more "engineeringly" oriented article. And I hope to be able to contribute to this if I can. Three specific comments:
A good example of how FEM is applied for an engineering FEA will help here as suggested by others, not just to keep non-mathematical audience's attention but simply because FEA/FEM strenght is in fact in the ENGINEERING APPLICATION of the method, really. Do you agree?
Despite all, this, still, was the best info I could find off the Web quickly and understand what FEM is about - big thanks to the aurhor of the article, too bad Wikipedia does not show author's (including contributing authors) info and contact(s)! -- Momchil 22:59, 22 December 2005 (UTC)
---
I'm the mathematician who originally wrote much of this article. I have also read Engineering treatment of the Finite Element Method and I consider it gobbledygook. If further examples are desired, then further examples should be added. If a nontechnical section is desired, then a nontechnical section should be added. However, if you use the word "elastostatics", you can rest assured that you are in fact talking about your own specialized application of the FEM which is simply more complex and less insightful than the current article.
Just to be clear: I vehemently oppose Ben's proposal of rewriting the FEM article in the style of Engineering treatment of the Finite Element Method.
Loisel 23:02, 26 December 2005 (UTC)
Is this better?
Loisel 19:04, 27 December 2005 (UTC)
How to fit Finite element analysis in this article? -- Abdull 16:30, 21 February 2006 (UTC)
I don't think Finite Element should automatically redirect to Finite element analysis. It made me miss this page for a long time. —Preceding unsigned comment added by Knepley ( talk • contribs) 22:11, 30 January 2008 (UTC)
I think there should be a seperate article on the spectral/hp element method, which is NOT the same as the current spectral method article, as this FEM article only mentions SEM in passing and does not give ANY details
there is also no mention of testing functions, galerkin method, etc. also no references to fem textbooks. i will add those in when i get a chance --anon
The definition of the 'spectral element method' given in the article is incorrect. The spectral element method uses a frequency domain formulation of the stiffness matrix ('dynamic stiffness') and was developed for use in dynamics problems. (e.g. wave propagation in structures). Nowhere in the literature is the mere use of higher degree polynomial basis functions considered a 'spectral' method.
How about some links to free libraries (from the polish wikipedia)
Does the subject of finite element meshing deserve its own article? I'm thinking so. It's distinct enough from the computer graphics description. Ojcit 19:21, 2 October 2006 (UTC)
Is the finite element method somehow related to the finite volume method ? —The preceding unsigned comment was added by Domitori ( talk • contribs) 01:11, 17 December 2006 (UTC).
The section on how to choose the basis is in my opinion too much focused on triangles, since the FEM can be applied to general elements, for example also to quads etc.
is referenced in the text, but I can't find it (should be somewhere between (2) and (4) I guess). -- 147.122.2.207 10:51, 26 January 2007 (UTC)
Speaking of references:
I liked the article so far but there is one thing I couldn't help noticing...There are virtually NO references (only one link to cover the history part in general). Not that I doubt the accuracy of what was written but references are still important. Could contributors if they have any free time look into referencing the bits they wrote? I for one would find it useful to direct my wider reading. Cheers,
Pl4t0
02:43, 12 May 2007 (UTC)
Behshour
13:56, 27 June 2007 (UTC)
1. The bilinear form does not define an inner product in since it does not satisfy the "homogeneity" property of the inner product: For any inner product we must have: if and only if
, but this is not the case here since only implies that . Consider the following function: on THE OPEN INTERVAL (0,1) and . Obviously this is an function whose norm equals 3. Its derivative is identically zero with a zero norm. Therefore is in the space . At the same time, , but is not identically zero. Although this bilinear form does not generate an induced norm, but it generates a semi-norm, and it may be considered a "Minkowski functional" instead of an inner product. It's however noteworthy that another bilinear form on the space , namely, does define an inner product and its induced norm is the standard Sobolev space norm. In this particular argument, however, whether the original bilinear form defines an inner product or not is irrelevant. If you are worried about Reisz representation, all it has to be is a bounded linear functional. Therefore I suggest that that remark be omitted.
2. In reference to the space , it is possible that in DIMENSION 1, and due to embedding theorems of Sobolev, along with the reflexivity of the Hilbert space, this space (or its closure) may be associated with the space of functions of bounded variation. But when one is defining a space, the definition must be as fundamental as possible. Associations do not replace definitions and they come next. The Sobolev space is different from the space of functions of "bounded variations" or " absolutely continuous" as it has been changed back and forth; otherwisw, it would be named that way. must be defined as: .
Please note that it's important to verify the above as soon as possible, and to make corrections as needed since otherwise, it would be misinforming.All Best. Behshour 14:17, 27 June 2007 (UTC) Behshour 14:23, 27 June 2007 (UTC)
---
is equivalent to , I refer the reader to "Sobolev Spaces" by Robert A. Adams and John J.F. Fournier for details. In my copy, one find the following on page 183.
6.29 (Domains of Finite Width) Consider the problem of determining for what domains in is the seminorm
actually a norm on equivalent to the standard norm [...]
We can easily show the equivalence of the above seminorm and norm for a domain of finite width, that is, a domain in that lies between two parallel planes of dimension (n-1). In particular, this is true for any bounded domain.
6.30 THEOREM (Poincaré's Inequality) If domain has finite width, then there exists a constant such that for all
[...]
6.31 COROLLARY If has finite width, is a norm on equivalent to the standard norm
---
That said, the reason why I skipped this explanation in the text is that it would considerably lengthen the discussion and this is not really about finite elements, but rather, this is a fact about Sobolev spaces or Elliptic boundary value problems.
Loisel 18:22, 27 June 2007 (UTC)
Poincaré's inequality is probably worth adding to Wikipedia. Loisel 18:26, 27 June 2007 (UTC)
Haha! It's already there. Loisel 18:27, 27 June 2007 (UTC)
The actual theorem I have above is also known as Friedrichs' inequality. Loisel 18:43, 27 June 2007 (UTC)
So how many of these codes are we going to add to the page? Is there WP policy regarding this sort of thing, because I feel like this is getting out of hand. - EndingPop ( talk) 16:38, 18 June 2008 (UTC)
Recently finite element analysis was merged with finite element method to create a new finite element methods. I think the merger was a good idea, but I don't understand why the new article was created with plural. Everywhere I've seen the method is written in singular, even if of course there are many variations of the method.
To the author of the merge: can you please explain how you merged things? I tried to figure out what you did from the diff, but all I see is a sea of red. Did you remove significant information?
Thanks. Oleg Alexandrov ( talk) 03:25, 3 July 2008 (UTC)
I suppose you might claim there is only one Finite element method, but in actuality that isn't really true. I would suggest looking at the following books Brenner-Scott, Brezzi-Fortin, Girault-Raviart books or I even see books with the title "The finite element method" and then immediately talk about different finite element methods. Finite element methods are really about putting many pieces together. For example just using Ciarlet's definition gives many possible methods for making an element (which by the way is not just a subregion of the mesh). If you go into the literature a bit more you will find people that assemble the elements in completely different ways, the standard is based on the minimization in an L2 norm but I've seen L1 norm, and least squares minimizations as well. The previous articles basically presented the material as if first order lagrange polynomials were "The Finite Element Method" but in actuality this is a misrepresentation of the field, but perhaps I should bow to convention.
The merge which I proposed here, I wanted to make the material for finite element methods much cleaner. The finite element anaylsis article really was more about computer-aided engineering or scientific simulation and current trends in engineering. The finite element method article presented a mathematical view that, while true for a specific case, did not represent the general framework given, not to mention I found several errors along the way. IMHO the article should give a view of what the methods are, a bit about where they are used and why, and maybe some mathematics. I tried to combine these two pages to do that, unfortunately it might cause some controversy about what is and isn't acceptable. I would propose another page or maybe a series of pages on the different technical parts of the process, perhaps a page about finite element assembly, another on mesh generation and so forth.
- Art187 ( talk) 09:25, 3 July 2008 (UTC)
Also to conform to standard you need to put a link to the old finite element analysis talk page here. Something like:
Article merged from Finite element analysis: See old talk-page here - Art187 ( talk) 09:55, 3 July 2008 (UTC)
Also why is this a physics article? It has as much to do with physics as multiplication. - Art187 ( talk) 09:58, 3 July 2008 (UTC)
Hello, I wrote much of the original article. I think it should be called "Finite Element Method". The Physics tag is probably because a lot of physicists are interested in this. I was the main editor, and I'm a mathematician, but I didn't tag it. Loisel ( talk) 19:23, 3 July 2008 (UTC)
At the end of the day, it is still not correct to say that finite difference method is a type of finite element. The finite difference method has its own derivations, its own analysis, and its own proofs of convergence. At most, you can say that a large classes of problems that can be solved with finite difference can also be solved in a finite element framework. And even this statement would need some good references. Art187 did provide some examples above, but a statement of such generality would need good references to back it up. Oleg Alexandrov ( talk) 16:10, 8 July 2008 (UTC)
In the current version of the article, the section "Outline of general mathematical framework" suggests that finite element problems are always solved in which is not right. Oleg Alexandrov ( talk) 04:38, 8 July 2008 (UTC)
Yes there are many problems still lurking in this rewrite. I kept the notation because it was suggested by other authors in the past. Here are a few other problems:
As you can see this article can get quite long and should probably serve as a jumping off point to FEM not something that is inclusive. There has been talk of making a Mathematical treatment of the finite element method to go along with Engineering treatment of the finite element method. I would suggest doing so and here we can show simple concepts such as coercivity, h-,p-,k-convergence and so on. - Art187 ( talk) 07:28, 8 July 2008 (UTC)
Finite element is a technical subject, that is unavoidable. That being said, the previous version of this article was a gentler explanation of what was going on than the current one. The current version skips over a lot of details, particularly omitting the 1D derivation of the variational formulation via integration by parts, which, I believe, is the key to getting people to understand how you go from a PDE to a variational formulation.
Unless there are objections in the next few days, I plan to replace the current section "Outline of general mathematical framework" in the article with what was there before, while keeping the rest. That won't be such a big change since what is there now is obtained by cutting things out (things which were important). Oleg Alexandrov ( talk) 06:57, 10 July 2008 (UTC)
this method is most important in engineering especially in CAE with software like ansys and abaqus. it should not be in physics. —Preceding unsigned comment added by Saeed.Veradi ( talk • contribs) 04:44, 20 July 2008 (UTC)
The first point on the FDM was as follows
Both methods are used to approximate the solution of a differential equation: that's what they are for. Although the author of the sentence probably has a good and well-established idea about both methods, this senence does not convey it. To me it sounds like: use FE if you want the solution of a differential equation. Use FDM if you want the differential equation itself. I hope we can agree that that is definitely not true. —Preceding unsigned comment added by 130.89.67.43 ( talk) 16:36, 22 April 2009 (UTC)
The technical section of the article mentions variational form with a link to the calculus of variations. The link is not appropriate because the linked page does not define variational form. Jfgrcar ( talk) 22:29, 7 December 2010 (UTC)
The introduction links to "Variational method" which really is a seemingly unrelated article on Quantum Mechanics. — Preceding unsigned comment added by 2607:EA00:104:3C00:21C9:8DF1:4B1A:451 ( talk) 22:11, 12 March 2013 (UTC)
I have a question about this method, for which I think the answer should be in the article (which it currently not as far as I can see). I wonder how the gradient and the Laplacian of the scalar field is calculated, after the discretization has been carried out? This is highly relevant to the article, since being able to calculate these two is necessary in order to be able to solve most PDE:s at all using the finite element method. -- Kri ( talk) 20:42, 6 April 2011 (UTC)
Why is it "finite element method" instead of "finite-element method"? The first means "finite method of elements", whereas in reality it is "method of finite elements" and thus in the adj. + noun form should be "finite-element" + "method". — Mikhail Ryazanov ( talk) 02:31, 6 February 2014 (UTC)
I agree with Mark Viking on this. I haven't seen this term hyphenated much in the literature. The people who spend their time enforcing some imaginary laws of prescriptive English... should probably find something else to do. Some1Redirects4You ( talk) 16:04, 27 April 2015 (UTC)
As you can see from my user name, I teach PDE's online, in many flavors. The comments on this talk page range from complimentary to outright hostile regarding the complexity level and quality of this article, lots of emotion about math, wow. Since I teach both undergrads and graduate engineers, let me suggest a solution for folks finding this too much or too tough: 1. Schaum's Outline of Finite Element Analysis is a great starting point for those who get lost at a beginning stage. Although FEA is a simple reductionist framework, implementation can involve hundreds of thousands of PDEs, some of which we run on supercomputers, and proofs and derivations are tough. 2. Once you get beyond the basics of proofs and the math, you'll be into algorithms immediately. Dover's 700 page book (The Finite Element Method) for $30 US is a great "next step" -- it is by Thomas J.R. Hughes, and goes from basics to advanced algorithm design (my field). It, however, is grad level and requires facility in PDEs and LP as a background. This is all to help frustrated Wiki editors and visitors, but to stay true to the talk page intent, I'd also like to humbly suggest that a section on algorithms might be warranted. I'd write it, but given the comments here, want to be sure there is at least some consensus that it is worth it. My reasoning is that although the proofs and derivations can get very difficult, the algorithmic designs are simple and elegant in many cases, and just crunch away at those PDEs! Frankly, this is more the reality today in practical solutions with Autocad, matlab, julia and even haskell, and users often don't have to know the full polygon story running beneath. In fact, many current interfaces allow you to drag and drop splines and beziers, OR write your own code in a little drop down command prompt box, OR do both! I would also like to thank the many contributors to this article, because, regardless of your opinion or feelings, it IS a lot of work! Pdecalculus ( talk) 14:57, 30 March 2016 (UTC)
I'm noticing that Finite Element Method and Finite Element Analysis are treated a synonyms in the introduction chapter. Though, "method" and "analysis" don't sound like the same thing to me at all. The way I have learned to understand it, FEM means the way calculations are done to perform FEA.
FEA is the larger context that includes first setting up a model of a practical case, then computing the numbers and finally interpretting the results of into events in the real world (often with recommendations of action).
FEM only means the technical part, essentially chopping the original problem into simpler parts whose behavior is "known" and constructing the bigger picture based on the interaction of all the simple elements. That alone is not yet an analysis.
The reason I wish to point this out is, that even very skilled and experienced people, who use these terms, some times seem unsure of, which to use and why, especially if they are not exactly in the business themselves -- say managing larger development projects, but with their background on a different field of technology. (Saw that happen just today ... again.)
A typical FEA, for example in the case of structural analysis may contain several runs in FEM, with different load cases and alternative structural or material build-ups. The results may contain information on mechanical durability, stability, response to vibration ... and for example a recommendation of a materials set to use.
So, if everybody agrees, that FEM and FEA are not synonyms, I'd be happy if somebody could take the effort of fitting a short explanation in the beginning of the introduction without breaking, what already is in there. :)
Peteihis ( talk) 20:02, 21 May 2018 (UTC)
Sorry, but I am trying to understand what is meant in the section 'Weak formation of p1' when it states that 'if u solves P1...' I mean didn't the problem statement for P1 just declare that u” solves P1? Isn't u related but potentially quite different from u”? So what does that statement mean, 'if u solves P1...'
Maybe I just need to think about what that means. — Preceding unsigned comment added by 134.137.180.129 ( talk) 19:15, 13 July 2018 (UTC)
the article is one of the best introduction.But in weak formulation of p2 the grrens identity is not clear. we are ina plane region. explicit derivation be done. in p1 use of mean value theorem be dmade explicit. Also the approach of distribution via sequential convergence and distributional derivative can be indicated in few lines.
on the contary tooo much space is used for h and the denedence on h. tthe whole can be summarized in h as the diameter of the traingle maximum amongst all traingles thas all very simple notion.
Further use of space H H be made clear. why not take H as space of continuous and differentiable except at finitely many points and make matters simple.
use of riesz representation be explicit write the functional and how to express it as inner product with u by RRT. excellent ouline which can be rigorous proof if we restrict H to be a suitable space.
please avoid lengthy discussions on h and subdivisions can be understood intuitively. But solve an explict one dime problem completely . Also use of Gallerkin is not made explicit . please make that use explicit in the problem. if these changes are done this can be most seductive logical introduction to fem . no good succint explnation exists on NET — Preceding unsigned comment added by Anilped ( talk • contribs) 06:44, 1 June 2020 (UTC)
John Smith Anderson ( talk) 11:59, 14 June 2020 (UTC)Under the heading "The weak form of P1" it is stated that the weak form implies the strong form. i.e. the equation above the line "The proof is easier for twice continuously differentiable u (mean value theorem)"
I cannot find a reference anywhere to a proof of this result. I am interested in a reference to a proof of this result, and I think it will improve the article for future readers who (like me) wonder how this result is proved. John Smith Anderson ( talk) 11:59, 14 June 2020 (UTC)
Franz Roters is not the progenitor of CPFEM it existed long before he even had a PhD. He is, though, involved in the on-going development of DAMASK which is a crystal plasticity software package. 2601:940:C081:4980:E3F1:FD00:EBB0:B69E ( talk) 00:35, 15 December 2022 (UTC)
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This is a fine example of an article that represents the worst of all worlds - it fails to accurately present either the mathematical, physical or engineering aspects of FEM. It is overly simplified for applied mathematicians, incomprehensible for lay persons and useless for the practicing engineer/analyst. There are numerous blatantly incorrect / misleading statements and glaring errors of omission, and the level of presentation alternates between sophomoric and overly pedantic. Students and laymen would be well advised not to attempt to learn anything from this article!
I recently edited the article (section 2.1) by removing what was there with a few paragraphs written by Barna Szabo and Mark Ainsworth. It has been reviewed by Ivo Babuska, Ernst Rank and Alexander Duester among others considered the best in the field. Whomever rejected the edit should go back and learn, not only about the finite element method, but also about humility. He dismissed by stating that "a clear technical description works better than theoretical opacity", for a text written by 5 of the best in the field...if it is opaque it may be because your lenses are dirty dear Woodstone... I've seen some of the responses to the criticism be based on how much they are supposed to know instead of recognizing that they are other that may know even more and check their facts. It is truly disappointing to see that people egocentrism is used as a reference. I'll try to set the edit back...if unsuccessful I'll simply create another alternative page and we will let people decide which one contains valuable information. — Preceding unsigned comment added by Snervi ( talk • contribs) 17:05, 27 November 2014 (UTC)
How about John Argyris' contribution? I think there should be a relevant revision in the history section, provided that experts in the field also agree.
One short biography appears here: http://www.cimne.upc.es/webcimne/boletinECCOMAS/20040518.htm in word format. I do not know though if it can be made available to wikipedia...
http://www.mlahanas.de/Greeks/new/Argyris.htm
Dpser 17:57, 16 January 2007 (UTC)
--- Information (3 June 2008):
In my book
Kurrer, Karl-Eugen: The History of the Theory of Structures. From Arch Analysis to Computational Mechanics. Berlin: Ernst & Sohn 2008.
one can find the chapter " 'The computer shapes the theory' (Argyris): the historical roots of the finite element method and the development of computational mechanics" (pp. 619-672).
...And of course brief biographies of the pioneers of fem e.g. Argyris,...
Best regards,
Karl-Eugen Kurrer —Preceding
unsigned comment added by
212.202.96.83 (
talk)
15:13, 3 June 2008 (UTC)
I think that the extremely significant role of
O. C. Zienkiewicz has been overlooked here. He is generally considered a key pioneer, and, as rightly stated on his
Wiki page: "His books on the Finite Element Method were the first to present the subject and to this day remain the standard reference texts."
Kurvenbau ( talk) 08:42, 13 May 2009 (UTC)
I think that the works in matrix analisys of structures from Mohr and Maxwell in the XIX century must be taken in account. This pioneers works can help to the reader to understand the basis of the method because it was basically FEM with 1D elements. -- Cometo22 ( talk) 08:43, 8 March 2011 (UTC)
math should be typed using the math tag
---
I agree.
There is also something not clear about how the integration by parts is actually done. Something was left out there. (BTW does this have anything to do with Lu=g being of the Sturm-Liouville type?)
--
No. When the operator is not Hermitian, there is still a bilinear form, except of course that it is not an inner product. One can obtain existence and uniqueness from the Lax-Milgram theorem, assuming that the bilinear form is coercive. If it is not, it is sometimes still possible to obtain a form of the Fredholm alternative.
When L is of the Sturm-Liouville type, the Dirichlet problem gives rise to a Hermitian operator and so the theory is nicer (and the linear solve is also easier -- conjugate gradient with a preconditionner can often be used.) If the boundary condition is not Dirichlet, the bilinear form is usually not symmetric, even if L is of Sturm-Liouville type. Loisel 11:46, 26 Jul 2004 (UTC)
The benginning of this article states that FEM approximates solutions to PDEs. But isn't the result somtimes exact? For example; p-type elements with shape functions of order greater than two yield exact results of a beam simulation (and neglecting computer rounding errors). Its been many years since I had this class, so my memory might be fuzzy. Pud 00:46, 25 Jul 2004 (UTC)
I don't know what you're talking about, however it is possible in certain cooked-up examples for the numerical scheme to be exact modulo machine precision. However, this is true of almost every numerical method. Numerical differentiation, integration, ode solvers, pde solvers, root-finding, eigenvalue algorithms, singular value decomposition, etc... can all coincidentally be exact under certain circumstances. Loisel 11:42, 26 Jul 2004 (UTC)
Loisel, do you know what p-type elements and shape functions are? Pud 16:15, 26 Jul 2004 (UTC)
As I said, no, I don't. Loisel 11:44, 27 Jul 2004 (UTC)
I know about piecewise polynomial basis functions on a triangular mesh, if that's what you're saying.
When solving an ODE like d^2v/dx^2=c, the solutions are v=cx^2/2+ax+b, for any a,b. Then one can cook up any number of numerical schemes to solve them exactly (that's what I was talking about in my first reply above.) For instance, the two-step method v[k+2]=2*v[k+1]-v[k]+c is exact in this example (if not entirely stable) even though in general it is not -- that is what I was talking about when I said "cooked up example." The FEM in this case can be written as an implicit method and without doubt some such schemes will be exact in this case. However, I'm fairly certain that the similar conclusion is false in the two variable case d^2v/dx^2+d^2v/dy^2=c because those functions are not polynomials. If c=0, one gets the harmonic functions, none of which are polynomials.
Erratum: of course some polynomials are harmonic.
Loisel 14:58, 27 Jul 2004 (UTC)
It's awful! If I had something like that in university I would stop studying physics. Why not at least write the differential equation in its native form first?
I hope the structure of the example, as well of the entire article, will be changed to something more comprehensive. (not signed)
Actually, I am a mathematician and active in the field of finite elements for about 15 years now. Therefore, I would not accept the disqualification above for me. Still, I must confess, I recognize the method only with difficulty from this article. A person trying to find out what it is will be left completely clueless. I can only recommend rewriting most of it, in particular
Guido Kanschat 13:42, 9 December 2005 (UTC)
I was looking for a simple, lay-persons explanation of the difference between a finite element and a finite difference approach to solving a flow simulation equation. I certainly didn't get that. This is supposed to be an encyclopedia for everyone, how you can start an article with "assuming a knowledge of calculus" is beyond me - what percent of the population actually has a working knowledge of calculus? Your approach is arrogant and exclusive, hopefully someone will find a more approachable way to describe these points soon, if Hawkins can describe the big bang then you should be able to describe this.
JohnH
I agree that this assumes a lot of math background. It's great to have a rigorous explanation, but a layman's explanation and an engineering explanation would be good. I propose this article get forked so that this page, Finite element method is a layman's introduction to the history and uses of the technique, then provide links to Mathematical treatment of the Finite Element Method and Engineering treatment of the Finite Element Method much like Tensor has Classical treatment of tensors, Tensor (intrinsic definition), and Intermediate treatment of tensors. — BenFrantzDale 03:40, 22 November 2005 (UTC)
Guys, ... please, hear a humble oppinion from an electrical engineer. I just started looking into FEA recently to simulate electrical field problems and needed good info on FEM. This Wikipedia article was by far best what I could find on the Web after a few days of Google search. But... I could not disagree that the treatment here of FEM is more "mathematically abstracted" than it needs to be for the "popular engineering audience" (although one can argue the FEM/FEA audience is popular in the sense of guys going out on a Friday night... :) Anyway, I applause the effort for more "engineeringly" oriented article. And I hope to be able to contribute to this if I can. Three specific comments:
A good example of how FEM is applied for an engineering FEA will help here as suggested by others, not just to keep non-mathematical audience's attention but simply because FEA/FEM strenght is in fact in the ENGINEERING APPLICATION of the method, really. Do you agree?
Despite all, this, still, was the best info I could find off the Web quickly and understand what FEM is about - big thanks to the aurhor of the article, too bad Wikipedia does not show author's (including contributing authors) info and contact(s)! -- Momchil 22:59, 22 December 2005 (UTC)
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I'm the mathematician who originally wrote much of this article. I have also read Engineering treatment of the Finite Element Method and I consider it gobbledygook. If further examples are desired, then further examples should be added. If a nontechnical section is desired, then a nontechnical section should be added. However, if you use the word "elastostatics", you can rest assured that you are in fact talking about your own specialized application of the FEM which is simply more complex and less insightful than the current article.
Just to be clear: I vehemently oppose Ben's proposal of rewriting the FEM article in the style of Engineering treatment of the Finite Element Method.
Loisel 23:02, 26 December 2005 (UTC)
Is this better?
Loisel 19:04, 27 December 2005 (UTC)
How to fit Finite element analysis in this article? -- Abdull 16:30, 21 February 2006 (UTC)
I don't think Finite Element should automatically redirect to Finite element analysis. It made me miss this page for a long time. —Preceding unsigned comment added by Knepley ( talk • contribs) 22:11, 30 January 2008 (UTC)
I think there should be a seperate article on the spectral/hp element method, which is NOT the same as the current spectral method article, as this FEM article only mentions SEM in passing and does not give ANY details
there is also no mention of testing functions, galerkin method, etc. also no references to fem textbooks. i will add those in when i get a chance --anon
The definition of the 'spectral element method' given in the article is incorrect. The spectral element method uses a frequency domain formulation of the stiffness matrix ('dynamic stiffness') and was developed for use in dynamics problems. (e.g. wave propagation in structures). Nowhere in the literature is the mere use of higher degree polynomial basis functions considered a 'spectral' method.
How about some links to free libraries (from the polish wikipedia)
Does the subject of finite element meshing deserve its own article? I'm thinking so. It's distinct enough from the computer graphics description. Ojcit 19:21, 2 October 2006 (UTC)
Is the finite element method somehow related to the finite volume method ? —The preceding unsigned comment was added by Domitori ( talk • contribs) 01:11, 17 December 2006 (UTC).
The section on how to choose the basis is in my opinion too much focused on triangles, since the FEM can be applied to general elements, for example also to quads etc.
is referenced in the text, but I can't find it (should be somewhere between (2) and (4) I guess). -- 147.122.2.207 10:51, 26 January 2007 (UTC)
Speaking of references:
I liked the article so far but there is one thing I couldn't help noticing...There are virtually NO references (only one link to cover the history part in general). Not that I doubt the accuracy of what was written but references are still important. Could contributors if they have any free time look into referencing the bits they wrote? I for one would find it useful to direct my wider reading. Cheers,
Pl4t0
02:43, 12 May 2007 (UTC)
Behshour
13:56, 27 June 2007 (UTC)
1. The bilinear form does not define an inner product in since it does not satisfy the "homogeneity" property of the inner product: For any inner product we must have: if and only if
, but this is not the case here since only implies that . Consider the following function: on THE OPEN INTERVAL (0,1) and . Obviously this is an function whose norm equals 3. Its derivative is identically zero with a zero norm. Therefore is in the space . At the same time, , but is not identically zero. Although this bilinear form does not generate an induced norm, but it generates a semi-norm, and it may be considered a "Minkowski functional" instead of an inner product. It's however noteworthy that another bilinear form on the space , namely, does define an inner product and its induced norm is the standard Sobolev space norm. In this particular argument, however, whether the original bilinear form defines an inner product or not is irrelevant. If you are worried about Reisz representation, all it has to be is a bounded linear functional. Therefore I suggest that that remark be omitted.
2. In reference to the space , it is possible that in DIMENSION 1, and due to embedding theorems of Sobolev, along with the reflexivity of the Hilbert space, this space (or its closure) may be associated with the space of functions of bounded variation. But when one is defining a space, the definition must be as fundamental as possible. Associations do not replace definitions and they come next. The Sobolev space is different from the space of functions of "bounded variations" or " absolutely continuous" as it has been changed back and forth; otherwisw, it would be named that way. must be defined as: .
Please note that it's important to verify the above as soon as possible, and to make corrections as needed since otherwise, it would be misinforming.All Best. Behshour 14:17, 27 June 2007 (UTC) Behshour 14:23, 27 June 2007 (UTC)
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is equivalent to , I refer the reader to "Sobolev Spaces" by Robert A. Adams and John J.F. Fournier for details. In my copy, one find the following on page 183.
6.29 (Domains of Finite Width) Consider the problem of determining for what domains in is the seminorm
actually a norm on equivalent to the standard norm [...]
We can easily show the equivalence of the above seminorm and norm for a domain of finite width, that is, a domain in that lies between two parallel planes of dimension (n-1). In particular, this is true for any bounded domain.
6.30 THEOREM (Poincaré's Inequality) If domain has finite width, then there exists a constant such that for all
[...]
6.31 COROLLARY If has finite width, is a norm on equivalent to the standard norm
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That said, the reason why I skipped this explanation in the text is that it would considerably lengthen the discussion and this is not really about finite elements, but rather, this is a fact about Sobolev spaces or Elliptic boundary value problems.
Loisel 18:22, 27 June 2007 (UTC)
Poincaré's inequality is probably worth adding to Wikipedia. Loisel 18:26, 27 June 2007 (UTC)
Haha! It's already there. Loisel 18:27, 27 June 2007 (UTC)
The actual theorem I have above is also known as Friedrichs' inequality. Loisel 18:43, 27 June 2007 (UTC)
So how many of these codes are we going to add to the page? Is there WP policy regarding this sort of thing, because I feel like this is getting out of hand. - EndingPop ( talk) 16:38, 18 June 2008 (UTC)
Recently finite element analysis was merged with finite element method to create a new finite element methods. I think the merger was a good idea, but I don't understand why the new article was created with plural. Everywhere I've seen the method is written in singular, even if of course there are many variations of the method.
To the author of the merge: can you please explain how you merged things? I tried to figure out what you did from the diff, but all I see is a sea of red. Did you remove significant information?
Thanks. Oleg Alexandrov ( talk) 03:25, 3 July 2008 (UTC)
I suppose you might claim there is only one Finite element method, but in actuality that isn't really true. I would suggest looking at the following books Brenner-Scott, Brezzi-Fortin, Girault-Raviart books or I even see books with the title "The finite element method" and then immediately talk about different finite element methods. Finite element methods are really about putting many pieces together. For example just using Ciarlet's definition gives many possible methods for making an element (which by the way is not just a subregion of the mesh). If you go into the literature a bit more you will find people that assemble the elements in completely different ways, the standard is based on the minimization in an L2 norm but I've seen L1 norm, and least squares minimizations as well. The previous articles basically presented the material as if first order lagrange polynomials were "The Finite Element Method" but in actuality this is a misrepresentation of the field, but perhaps I should bow to convention.
The merge which I proposed here, I wanted to make the material for finite element methods much cleaner. The finite element anaylsis article really was more about computer-aided engineering or scientific simulation and current trends in engineering. The finite element method article presented a mathematical view that, while true for a specific case, did not represent the general framework given, not to mention I found several errors along the way. IMHO the article should give a view of what the methods are, a bit about where they are used and why, and maybe some mathematics. I tried to combine these two pages to do that, unfortunately it might cause some controversy about what is and isn't acceptable. I would propose another page or maybe a series of pages on the different technical parts of the process, perhaps a page about finite element assembly, another on mesh generation and so forth.
- Art187 ( talk) 09:25, 3 July 2008 (UTC)
Also to conform to standard you need to put a link to the old finite element analysis talk page here. Something like:
Article merged from Finite element analysis: See old talk-page here - Art187 ( talk) 09:55, 3 July 2008 (UTC)
Also why is this a physics article? It has as much to do with physics as multiplication. - Art187 ( talk) 09:58, 3 July 2008 (UTC)
Hello, I wrote much of the original article. I think it should be called "Finite Element Method". The Physics tag is probably because a lot of physicists are interested in this. I was the main editor, and I'm a mathematician, but I didn't tag it. Loisel ( talk) 19:23, 3 July 2008 (UTC)
At the end of the day, it is still not correct to say that finite difference method is a type of finite element. The finite difference method has its own derivations, its own analysis, and its own proofs of convergence. At most, you can say that a large classes of problems that can be solved with finite difference can also be solved in a finite element framework. And even this statement would need some good references. Art187 did provide some examples above, but a statement of such generality would need good references to back it up. Oleg Alexandrov ( talk) 16:10, 8 July 2008 (UTC)
In the current version of the article, the section "Outline of general mathematical framework" suggests that finite element problems are always solved in which is not right. Oleg Alexandrov ( talk) 04:38, 8 July 2008 (UTC)
Yes there are many problems still lurking in this rewrite. I kept the notation because it was suggested by other authors in the past. Here are a few other problems:
As you can see this article can get quite long and should probably serve as a jumping off point to FEM not something that is inclusive. There has been talk of making a Mathematical treatment of the finite element method to go along with Engineering treatment of the finite element method. I would suggest doing so and here we can show simple concepts such as coercivity, h-,p-,k-convergence and so on. - Art187 ( talk) 07:28, 8 July 2008 (UTC)
Finite element is a technical subject, that is unavoidable. That being said, the previous version of this article was a gentler explanation of what was going on than the current one. The current version skips over a lot of details, particularly omitting the 1D derivation of the variational formulation via integration by parts, which, I believe, is the key to getting people to understand how you go from a PDE to a variational formulation.
Unless there are objections in the next few days, I plan to replace the current section "Outline of general mathematical framework" in the article with what was there before, while keeping the rest. That won't be such a big change since what is there now is obtained by cutting things out (things which were important). Oleg Alexandrov ( talk) 06:57, 10 July 2008 (UTC)
this method is most important in engineering especially in CAE with software like ansys and abaqus. it should not be in physics. —Preceding unsigned comment added by Saeed.Veradi ( talk • contribs) 04:44, 20 July 2008 (UTC)
The first point on the FDM was as follows
Both methods are used to approximate the solution of a differential equation: that's what they are for. Although the author of the sentence probably has a good and well-established idea about both methods, this senence does not convey it. To me it sounds like: use FE if you want the solution of a differential equation. Use FDM if you want the differential equation itself. I hope we can agree that that is definitely not true. —Preceding unsigned comment added by 130.89.67.43 ( talk) 16:36, 22 April 2009 (UTC)
The technical section of the article mentions variational form with a link to the calculus of variations. The link is not appropriate because the linked page does not define variational form. Jfgrcar ( talk) 22:29, 7 December 2010 (UTC)
The introduction links to "Variational method" which really is a seemingly unrelated article on Quantum Mechanics. — Preceding unsigned comment added by 2607:EA00:104:3C00:21C9:8DF1:4B1A:451 ( talk) 22:11, 12 March 2013 (UTC)
I have a question about this method, for which I think the answer should be in the article (which it currently not as far as I can see). I wonder how the gradient and the Laplacian of the scalar field is calculated, after the discretization has been carried out? This is highly relevant to the article, since being able to calculate these two is necessary in order to be able to solve most PDE:s at all using the finite element method. -- Kri ( talk) 20:42, 6 April 2011 (UTC)
Why is it "finite element method" instead of "finite-element method"? The first means "finite method of elements", whereas in reality it is "method of finite elements" and thus in the adj. + noun form should be "finite-element" + "method". — Mikhail Ryazanov ( talk) 02:31, 6 February 2014 (UTC)
I agree with Mark Viking on this. I haven't seen this term hyphenated much in the literature. The people who spend their time enforcing some imaginary laws of prescriptive English... should probably find something else to do. Some1Redirects4You ( talk) 16:04, 27 April 2015 (UTC)
As you can see from my user name, I teach PDE's online, in many flavors. The comments on this talk page range from complimentary to outright hostile regarding the complexity level and quality of this article, lots of emotion about math, wow. Since I teach both undergrads and graduate engineers, let me suggest a solution for folks finding this too much or too tough: 1. Schaum's Outline of Finite Element Analysis is a great starting point for those who get lost at a beginning stage. Although FEA is a simple reductionist framework, implementation can involve hundreds of thousands of PDEs, some of which we run on supercomputers, and proofs and derivations are tough. 2. Once you get beyond the basics of proofs and the math, you'll be into algorithms immediately. Dover's 700 page book (The Finite Element Method) for $30 US is a great "next step" -- it is by Thomas J.R. Hughes, and goes from basics to advanced algorithm design (my field). It, however, is grad level and requires facility in PDEs and LP as a background. This is all to help frustrated Wiki editors and visitors, but to stay true to the talk page intent, I'd also like to humbly suggest that a section on algorithms might be warranted. I'd write it, but given the comments here, want to be sure there is at least some consensus that it is worth it. My reasoning is that although the proofs and derivations can get very difficult, the algorithmic designs are simple and elegant in many cases, and just crunch away at those PDEs! Frankly, this is more the reality today in practical solutions with Autocad, matlab, julia and even haskell, and users often don't have to know the full polygon story running beneath. In fact, many current interfaces allow you to drag and drop splines and beziers, OR write your own code in a little drop down command prompt box, OR do both! I would also like to thank the many contributors to this article, because, regardless of your opinion or feelings, it IS a lot of work! Pdecalculus ( talk) 14:57, 30 March 2016 (UTC)
I'm noticing that Finite Element Method and Finite Element Analysis are treated a synonyms in the introduction chapter. Though, "method" and "analysis" don't sound like the same thing to me at all. The way I have learned to understand it, FEM means the way calculations are done to perform FEA.
FEA is the larger context that includes first setting up a model of a practical case, then computing the numbers and finally interpretting the results of into events in the real world (often with recommendations of action).
FEM only means the technical part, essentially chopping the original problem into simpler parts whose behavior is "known" and constructing the bigger picture based on the interaction of all the simple elements. That alone is not yet an analysis.
The reason I wish to point this out is, that even very skilled and experienced people, who use these terms, some times seem unsure of, which to use and why, especially if they are not exactly in the business themselves -- say managing larger development projects, but with their background on a different field of technology. (Saw that happen just today ... again.)
A typical FEA, for example in the case of structural analysis may contain several runs in FEM, with different load cases and alternative structural or material build-ups. The results may contain information on mechanical durability, stability, response to vibration ... and for example a recommendation of a materials set to use.
So, if everybody agrees, that FEM and FEA are not synonyms, I'd be happy if somebody could take the effort of fitting a short explanation in the beginning of the introduction without breaking, what already is in there. :)
Peteihis ( talk) 20:02, 21 May 2018 (UTC)
Sorry, but I am trying to understand what is meant in the section 'Weak formation of p1' when it states that 'if u solves P1...' I mean didn't the problem statement for P1 just declare that u” solves P1? Isn't u related but potentially quite different from u”? So what does that statement mean, 'if u solves P1...'
Maybe I just need to think about what that means. — Preceding unsigned comment added by 134.137.180.129 ( talk) 19:15, 13 July 2018 (UTC)
the article is one of the best introduction.But in weak formulation of p2 the grrens identity is not clear. we are ina plane region. explicit derivation be done. in p1 use of mean value theorem be dmade explicit. Also the approach of distribution via sequential convergence and distributional derivative can be indicated in few lines.
on the contary tooo much space is used for h and the denedence on h. tthe whole can be summarized in h as the diameter of the traingle maximum amongst all traingles thas all very simple notion.
Further use of space H H be made clear. why not take H as space of continuous and differentiable except at finitely many points and make matters simple.
use of riesz representation be explicit write the functional and how to express it as inner product with u by RRT. excellent ouline which can be rigorous proof if we restrict H to be a suitable space.
please avoid lengthy discussions on h and subdivisions can be understood intuitively. But solve an explict one dime problem completely . Also use of Gallerkin is not made explicit . please make that use explicit in the problem. if these changes are done this can be most seductive logical introduction to fem . no good succint explnation exists on NET — Preceding unsigned comment added by Anilped ( talk • contribs) 06:44, 1 June 2020 (UTC)
John Smith Anderson ( talk) 11:59, 14 June 2020 (UTC)Under the heading "The weak form of P1" it is stated that the weak form implies the strong form. i.e. the equation above the line "The proof is easier for twice continuously differentiable u (mean value theorem)"
I cannot find a reference anywhere to a proof of this result. I am interested in a reference to a proof of this result, and I think it will improve the article for future readers who (like me) wonder how this result is proved. John Smith Anderson ( talk) 11:59, 14 June 2020 (UTC)
Franz Roters is not the progenitor of CPFEM it existed long before he even had a PhD. He is, though, involved in the on-going development of DAMASK which is a crystal plasticity software package. 2601:940:C081:4980:E3F1:FD00:EBB0:B69E ( talk) 00:35, 15 December 2022 (UTC)