Discussions about Finite Element Analysis (PDFs)

[maistral]

Hi. Can someone point me to a good (and really basic) PDF file or text regarding Finite Element analysis? I would prefer it having an example where it would solve the 1D heat equation or the Laplace equation so I can extend what I know from finite difference methods to it.

Sorry if this seems too much, but I seriously cannot understand the text that I keep on finding. I am trying to study these methods on my own as I wanted to know the methods well (and at the bare minimum, execute manual calculations regarding these), as I believe knowing the method's concept is important before using programs that would execute them easily (I don't want to use programs blindly).

I am asking for help because what happens is that every single paper I find follows a single format: It would introduce the big three PDEs, then it would show the advantage of using numerical solutions, then it would then introduce (apparently) weak and strong formulations and there would be integrations out of nowhere. Basically everything went ballistic lol and I am so inebriated now

Thank you very much for your help.

 

[fresh_42]

You might have a look whether here:
https://www.ams.org/open-math-notes
is something you're looking for or perhaps help you otherwise.

 

[Coelum]

Hi maistral,
you might have a look at this: http://www.maths.manchester.ac.uk/~djs/primer.pdf . Depending on the area of application, there may be other texts worth your time. If you can be more specific, maybe I can point you to more documents.
Hope it helps...

Francesco

 

[maistral]

Thanks for the help guys but it made my head ache more

As of this moment I am still looking for that, Laplace equation solution with constant boundary conditions using finite elements. Sadly I'm still ending up empty-handed... but I'm not stopping yet as I don't want to give up on this (cliche, lol).

 

[maistral]

You might have a look whether here:
https://www.ams.org/open-math-notes
is something you're looking for or perhaps help you otherwise.

I seem to be unable to find the Finite Elements here. I mean, I opened the Numerical Analysis PDF file here, and it seems to end on numerical ODE.

 

[maistral]

Hi maistral,
you might have a look at this: http://www.maths.manchester.ac.uk/~djs/primer.pdf . Depending on the area of application, there may be other texts worth your time. If you can be more specific, maybe I can point you to more documents.
Hope it helps...

Francesco

It went ballistic as well

 

[S.G. Janssens]

I wanted to know the methods well (and at the bare minimum, execute manual calculations regarding these), as I believe knowing the method's concept is important before using programs that would execute them easily (I don't want to use programs blindly).

Very good, more people should do that.

I am asking for help because what happens is that every single paper I find follows a single format: It would introduce the big three PDEs, then it would show the advantage of using numerical solutions, then it would then introduce (apparently) weak and strong formulations and there would be integrations out of nowhere. Basically everything went ballistic lol and I am so inebriated now

What do you mean by "everything went ballistic"?

I get the impression the concept of weak solution is not clear to you, yet? Before going to FEM, I think you should look at a basic PDE text to clarify that. Essentially, one shows that strong solutions satisfy a certain identity when integrated against suitable test functions. Then one takes that identity as the defining property of a weak solution.

Maybe follow up on what Francesco offered and give some more information on your background knowledge.

 

[maistral]

I guess I get terrified when I see the 'weak' and 'strong' solution then I get stormed by Rⁿ. I instantly get demoralized and confused after that.

 

[S.G. Janssens]

I guess I get terrified when I see the 'weak' and 'strong' solution then I get stormed by Rn. I instantly get demoralized and confused after that.

Did you take a multivariable calculus course?

 

[maistral]

Did you take a multivariable calculus course?

Sadly, no.

Actually I studied how to solve PDEs analytically and numerically via FDM all on my own. I feel so deprived actually. I wanted someone to teach me, but since there isn't anyone available and I just don't want to give up, I ended up being able to study them.

That's why my foundations are somewhat 'imbalanced', I lack knowledge on some things, but I ended up knowing a lot on other things even if I don't want to.

May I also ask, Is this concept of weak and strong solutions an important prerequisite in FEM solutions? Not that I don't want to study them (I wanted to actually), it's just that I don't have much time and I don't have much foundation on them. I mean, in FDM solutions I can just use the concept of numerical derivatives right?

Could I not do the same thing on FEM? Thanks a lot for answering my questions.

 

[maistral]

To add. To be honest I don't even know what I lack in terms of mathematics. I just try and solve problems, then if I don't know how to solve a certain thing, I try to research and look for solutions on how to kill off the problem. I think I got the grip of FDM solution fairly well (I think I know the concepts and can execute Crank Nicolson, derive computational stencils, use tridiagonal matrix methods, Gauss-SOR among other things - I don't know, I'm blindly guessing here) but when I get here at FEM I ended up with a bloody nose. The amount of Rⁿ is terrifying, and the sudden appearance of integration terrified me even more (I didn't see these things in FDM!)

I'm an engineering graduate student (if that matters).

 

[S.G. Janssens]

Actually I studied how to solve PDEs all on my own. I feel so deprived actually. I wanted someone to teach me, but since there isn't anyone available and I just don't want to give up, I ended up being able to study them.

It is nice that you went ahead anyway, it means you have an intrinsic interest in the topic.

May I also ask, Is this concept of weak and strong solutions an important prerequisite in FEM solutions?

Yes, I think it is not an overstatement to say that understanding weak and strong solutions and their relationship is very important.

One can argue about how much analysis is necessary to understand FEM and PDE in general, but I think everyone would agree that a good understanding of multivariable calculus (including the integral theorems) cannot be missed. If I were you, I would take a step back, and learn this first. It will pay you rich dividends for the rest of your life. So, roughly, the order would be:

1. multivariable / vector calculus (including the integral theorems of Stokes and Gauss),
2. applied functional analysis, so you understand at least the concept of a function space,
3. the basic theory of the classical PDE,
4. FEM.

You can probably interchange 2 and 3 if you like. Also, there is no need (at least initially) to overdo the functional analysis, but I believe you should understand its basics before moving to 4.

As far as required background knowledge goes, FDM is considerably less demanding than FEM, but FEM allows for much more physical domain flexibility, which probably matters to you as an engineering graduate student.

EDIT: $\mathbb{R}^n$ is just $n$-dimensional space. You will learn all about it when you study 1.

 

[S.G. Janssens]

the sudden appearance of integration terrified me even more (I didn't see these things in FDM!)

This is because in FDM you discretize the differential operators directly. It is, in a sense, the most natural and straightforward approach to solving a (linear) PDE, but it is also not a very flexible one.

 

[maistral]

Just a curious question. I had this problem about diffusion from a point source. While I can solve this extremely easily using FDM, I wanted to try the analytical method then everything (and everyone) is pointing me to 'Green's Functions'.

Is this also covered by those? I'm getting excited actually. Thanks!

 

[S.G. Janssens]

Just a curious question. I had this problem about diffusion from a point source. While I can solve this extremely easily using FDM, I wanted to try the analytical method then everything (and everyone) is pointing me to 'Green's Functions'.

Is this also covered by those? I'm getting excited actually. Thanks!

Yes, this is covered in 2. and/or 3. and preferably in both. Green's functions should be discussed in any book about the classical PDEs and they often appear in applied functional analysis books as well. If you want to be sure, study the TOC and perhaps also the index.

 

[Coelum]

To add. To be honest I don't even know what I lack in terms of mathematics. I just try and solve problems, then if I don't know how to solve a certain thing, I try to research and look for solutions on how to kill off the problem. I think I got the grip of FDM solution fairly well (I think I know the concepts and can execute Crank Nicolson, derive computational stencils, use tridiagonal matrix methods, Gauss-SOR among other things - I don't know, I'm blindly guessing here) but when I get here at FEM I ended up with a bloody nose. The amount of Rⁿ is terrifying, and the sudden appearance of integration terrified me even more (I didn't see these things in FDM!)

I'm an engineering graduate student (if that matters).

I just found a document with Matlab code for a 1D FEM problem: http://www4.ncsu.edu/~zhilin/TEACHING/MA587/chap6.pdf . At first glance, it seems very clear. It still contains a significant amount of math, but you can check your understanding by looking at the code, running it and modifying it. Good luck!

Francesco