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Deep introduction to the finite element method

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Ali baba Sx

Ali baba Sx

Computer
May 8, 2020
12
Hi everyone,

I recently wrote an article about the finite element method. I tried to provide the full picture about the method, while focusing at the details that are the most important in my opinion.

Let me know what you think about this content :)

Ali
 
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Neat article!

I'd recommend modifying the subtitle because I don't think it's a "understand FEA in 30 minutes article." LOL
 
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It is a good introductory article. Well done!

Some minor comments:

Might be good to point out that KU=F is actually never inverted in practice.

Given the simplified form of the elasticity PDE, only the static form of the Hamilton's principle is applied.

Fix a few minor typos here and there over time and, perhaps, in the appendix note that the nodal displacement based FEM is only one form out of many possible FEMs (stress-based, as an example).

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I haven't read the article, so I apologise in advance.

but i worry about it's intent ... to "provide the full picture about the method" seems to point to a high level overview, but "while focusing at the details that are the most important" suggests something much more detailed.

First, what is the target audience ? professionals (not involved with FEA)? non-engineers (who have seen/heard about FEA)? post-grad ? under-grads ? school kids ??

Second, I'm sure the intent is not to teach anyone how to do FEA, but rather to give a flavour of it, hitting the high notes as it were. FEA solves internal loads (stresses) and displacements based on the principle of minimum work. A simple example is a simply supported beam, where you can illustrate mesh convergence very easily. The next example would be a determinate truss, where a very coarse model (one element per bay) captures the solution very well. The next example would be a redundant structure ... either a beam on several supports, or a redundant truss. the end.

another day in paradise, or is paradise one day closer ?
 
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what's your programming language. Derivations of shape functions and elemental stiffness matrix looks good when printed with LaTex, where I find the lack of information on the finite element is the derivations of the shape functions and elemental stiffness matrix through examples using a computer programming language.
 
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Thanks for all your answers!

@271828
I'll try to rework the introduction. It is still a challenge to find a catchy yet precise intro, I probably went too far ^^.

@IceBreakerSours
The matrix inversion is a good point, but in my opinion it is a software implementation topic. Here I just wanted to lay down the math, without entering into the implementation details.

You are right about Hamilton's principle. I'll manage to add this precision.

About your last remark, I am not sure that talking about stress based FEM can bring value to the reader without confusing him. I have to think more about this.

I'll take the time to read your reference. I think it can fit perfectly as a link somewhere. Thanks!

And sorry for the typos. I tried to do my best, but I can't hide the fact that I am not a native english speaker :). I'll reread again the article.

@rb1957
No problem, feedback is always welcome.

The aim of this article is to provide enough detail to make it a good starting reference. But without providing too much to avoid confusing the reader.

From my point of view there should be no target audience. Obviously when you end up reading this article you are not interested about engineering stuff but about math and numerical method. The target audience is whoever wants to read about this subject.

Your suggestion is another approach to introduce the subject. It focuses on the engineering side of FEM, which is not the viewpoint I took.

@rscassar
I plan to write another article about the practical implementation of the method (at least the stiffness computation) using a Python script. I'll let you know when it is ready.

-----
Ali
 
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ok, I'm no writer, but every communication needs to have a target.

You're obviously not targeting FEA professionals (who covered this material years ago).
You could be targeting people entering (or thinking of entering) the field.
People with structural analysis experience (like maybe undergrads thinking of doing a Masters) would (you'd hope) appreciate some of the material.
People with no experience (other than hearing "FEA") would need more background information before starting.

another day in paradise, or is paradise one day closer ?
 
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I agree with the sentiments expressed by rb1957. It seems like the target audience is beginners (students or professionals) who are comfortable enough with mouse clicks and are curious to see what is under the hood.

Ali baba Sx I am surprised how few typos you have made being a non-native English speaker. Typos are one of my many weird quirks; don't waste time fixing them. By the way, COMSOL guys have done a fantastic job with their blog. Their "deep introduction" to FEM and to the weak form here and here is brilliant.

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Hi
I have not read the article in detail but from what I have read it looks good. But I have a few comments [smile].

First the title, what this seems to be is: An introduction to the basic math behind the Finite Element Method. That is a mouthful but I think you need to include "math" in the title since you present it from that point of view.

I also noticed that you include 1D, 2D and 3D elements but no 0D elements (like point masses) or more specialized elements. Why not? Since you don't go into detail anyway why not mention the (perhaps) not so obvious possibilities also?

Finally, the more problematic part of this article, that has already been adressed. Who is the intended reader? I have not written a lot of articles but I have made several presentations on structural analysis for different audiences. The key to be interresting, in my experience, is always to know your audience, at least try. Since you publish this article on a software blog, is the intended reader a software user?

I think the article is overall good. Like I said, I haven't read it in detail. But I can't quite see the intended reader based on where it is published. On the other hand, I assume that you are involved in SesamX so you should know the blog reader better than I do [smile].

Thomas
 
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after many years of learning and use of FEA, I'll admit readily to barely understanding how it works (even though i have attempted to learn as much about it as i reasonably can in my spare time).

The same way a carpenter likely doesnt understand how a screw gun works, but he can still use one to drive in a screw!

thanks for the article though, it was a good brush up.

 
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at least FEA won't cause bodily harm ...

another day in paradise, or is paradise one day closer ?
 
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Given the number of computational biomedical companies that have mushroomed, I am not sure that FEA (its usage, rather) won't cause bodily harm.

Just a hint of cynicism - Way to begin my day!

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@Ali baba Sx

Just a silly questions..

What governs the shape of the elements?? We talk of trias/quad shapes. Why don't we have elements circular in shape?? On the top of my head I can think of using circular shape elements at areas of stress concentration (plate with a hole for example).
 
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@Elphick

The answer lies in the basis functions, which in the most popular commercial codes, are just linear or quadratic polynomials. With those, there is only so much closer to a general curve you can get. With splines as basis functions, you have a much better shot at representing general curves accurately. LS-DYNA is one major general purpose commercial code that has invested in this domain and Coreform is another relative smaller entity.

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Recall the p-elements where the elements were just any shape that conformed to the geometry. See
Somehow it fell out of favor - I think the h-element group found that they could refine the mesh faster than the p-element guys could get solutions to converge and, since h-elements were an established tech, the p-element advantage disappeared.

Apparently p-elements are still used in Creo Simulate; I forget who they bought to get that software.
 
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@IceBreakerSours & 3DDave,

Thx for the information. Your responses made for an intresting read.
 
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circular elements don't pave well. rectangles and triangles are good pavers, as are tets (for solids).

Faceting an arc is generally sufficient. Using higher order elements (with midside nodes) allows for curved edges.

another day in paradise, or is paradise one day closer ?
 
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There is also the issue of mesh compatibility. How do you connect 2 adjacent circular elements? Given an interpolation inside the element, to get a proper compatibility you need to describe similarly, in each element, the frontier between the 2 elements. I haven't thought about this more than that, but I guess this can be an issue with circular elements.

----
Ali
 
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