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Open House: Can you define FEM in one line?

Can you define FEM in one line?

If yes, what would it be? And, in that case, permit me a second question: How?

 

...Really interested in knowing what the members of this community think (of this matter), if they do...

 

--Ajit

[E&OE]

 

Comments

I guess mathematically all Galerkin methods are in the form:

Find a(uh,vh)=<f,vh> for any vh in Vh

To specify FEM, we need add in above line that uh and vh are with compact/"small" support.

 

Ritz method + interpolation function with compact suppor.

0. Thanks, Xu, Yi,  for answering the question.

1. As to my own attempt at answering the question, here we go:

FEM in one line: Expanding the sought, usually continuous (or field) solution, typically of a differential equation problem, by recasting the original strong-form problem into its corresponding weak-form in reference to a finite set of ansatz (or trial) basis-functions having compact but finite supports over their respective finite regions (which usually decompose the domain in a mutually exclusive and collectively exhaustive manner), the weak-form itself being obtained by appeal to a suitable procedure arising from the principle of stationarity of energy or from the more general mathematical method of weighted residulals whose physical meaning has never been identified explicitly.

2. I am not too happy with my answer, because though one would have liked to give some reference to the matrix-nature of the end-formulation, to the method of handling the time variable (esp. as in spectral analysis), and some indication of handling nonlinearities essentially through iterations, these aspects are left outside of the definition. I am not sure if all indeed are easily implied by whatever stuff there is that has made it into the definition.


* * * * *

3. I was asked the title question during an interview for a professor' post, but without the attendent question: "how," which I myself added (and haven't answered here).

At the time of interview (after my PhD), I didn't have an answer ready. However, even back then, I had explicitly thought that the question was very stupid in view of its bad epistemological standing (and the interviewers asking the question very stupid, malicious, or both, in view of the sort of decisions they eventually did take).

As is evident above, I do now seem to have something by way of an answer. However, my opinion of the question, and of the interviewers asking it, has not undergone a sea-change. Even if there might be some change, it certainly has not been for the better. May be, only for the worse. But, I really cannot be sure about this very last part.


* * * * * 


4. The technical part of my answer remains open to criticism, and suggestions for improvement. And, of course, the original question itself remains open for further discussion, if any one of you wish to do so. 


[However, I am not ready for subjecting this write up to delete the "middle part" during moderation, if any. Not at all. I, instead, am willing to permanently leave this forum if necessary.]

 

--Ajit

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[E&OE]

ramdas chennamsetti's picture

Hi Sir,

I just thought on the question 'FEM in one line'.....I could make it as follows.

'A numerical technique that makes use of piecewise interpolation'.

Best regards,

- Ramadas

Hi Ramdas,

Thanks. You do have a good start, but your definition, as given, is incomplete. Is FEM just about numerical interpolations (even if done piecewise)? In what way is FEM different from curve-fitting? ... In your definition, you have to add something to indicate the meaning of the terms *finite* *elements*, and how the technique involves the weak form (variational principles, weighted residuals, integral form, whatever etc.). In a way, your definition, as given, is too broad and fails to distinguish between FEM and other interpolations.

BTW, in my answer at the time of interview, I had began by saying something like: you take a 3D region and divide or mesh it into a finite number of elements, and then over each such element, you prescribe a trial function.... At this point of time, I was abruptly and forcefully cut short (using all the collective, sly, and interceptive powers wrested in the committee, sitting physically all around one). And, I was reminded, in a loud, openly deridingly laughing way, that what I was doing was explaining, not defining. One senior guy even ventured: Why can't you stick to the questions asked? Why do you have to run here and there? Why do you talk so much? etc.

Which experience tells me that though you have a good start in defining over there, and though you could have improved the definition on the fly, still, you, too, wouldn't have done any better than I did---you, too, would have been rejected. After all, they would not allow you to begin at some concrete place and then tie-in further more abstract elements. They wouldn't allow you this bottom-to-top strategy. You would have to have this (the above sort of) "definition" ready-made in your mind, and you would be expected to rattle it off, like a parrot, in that metaphorically "Punjab mail" speed. Else, your knowledge would be deemed to be loose. And, going by whatever interaction I had earlier with them (we had teaching presentation sessions before interviews), if you wanted to get selected, you would better answer anything on the nature of FEM, not in the way the engineers originally invented the technique. You would better make direct reference only to abstract mathematics in your answer. You better stick to abstract mathematical terms alone, lest someone suspect that the method actually might have engineering or physical origins or applications. This means you had better rattle off (vomit out) the monster of a "definition" such as what I have tried to indicate above.

Today, I think that any definition that makes reference to abstract vector spaces (basis-sets, etc.) would have been found satisfying to them---assuming that they were going to be satisfied by anything a certain candidate they had already decided their mind they were going to reject, was going to say.

But assuming that the above definition would have been satisfactory to them, here is what I find so idiotic/weak about such definitions:

Why do we need to couch FEM in terms of abstract spaces, expansion, etc.? I mean, it's obvious that when it comes to FEM, the basis-set is always going to be only finite, never infinite---else, it won't be implentable on a computer. So, there is never an occasion to treat the finite and the infinte types of basis-sets together. Thus, we are never going to have an occasion when we will have the need to distinguish FEM from these other kinds of abstract spaces. If so, isn't couching FEM in such abstract spaces terms a kind of cognitive dead-end? (In Objectivist epistemology terms: there is no CCD from FEM to these other ideas inasmuch as many of those other ideas aren't in principle capable of being a numerical technique.)

If so, why insist on defining FEM in such unnecessarily highly abstract terms? Just to let mechanical engineers feel good that they can talk the language of their colleagues from the physics and mathematics departments, and thereby to let them derive some second-hander's sort of glory? As far as that particular interviewing panel goes, most members had precisely such pathalogical sort of working epistemology. Their outlook and convictions was shaped by a pathological variety of epistemology.

Enough about them. I wrote the answer primarily to show that FEM is essentially a loose cluster of various elements, and if you are going to be IIT Bombay-proof, you had better rope in as many elements as I did, even if it means that none can hold the entirety of the "definition" in one's mind as a single unit---the crow-epistemology principle is violated. (More on the latter, in Ayn Rand's book on epistemology.)

You all are welcome to discuss further, but I realize that I need to take a break from this unpleasant matter, and so, I am not going to respond on this thread for a week or so. But, it doesn't mean anything. Feel free to discuss it among yourselves.

--Ajit

(Written on the fly. Will clarify points, if necessary, sometime later. Sorry for poor compositional skills.)

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[E&OE]

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