Math vs. FE Method: Comparing Mechanical Engineering Focus

[Trying2Learn]

TL;DR Summary

How would you describe the FE method

I have written FE codes.
I have seen FE classes from mechanical engineering
I have seen FE classes from the math departments

When offered from a math department, the focus is the theory (with a lot of attention paid to general differential equations)
They are good courses and they suggest, I think AN ALGORITHM

However in the Finite Element Method, to ME the operative word is METHOD

And I am hoping someone can say what I am about to say, more precisely:

The FE METHOD was developed by engineers (civil: trusses, frames, etc.; and mechanical: solids, plane stress, etc.) As such, it is a METHOD of applying the ALGORITHM in pre-set ways for mechanical engineering: to read in connectivity, set up Jacobians, set up Gauss Q. integration, set up the B matrix, B-transpose * B * B, etc. I don't think the math departments teach this properly for mechanical engineers. I have seen students take FE classes from math departments and come out without any idea that there really is an established approach to a METHOD.

Can someone help me say this better? (Or, maybe I am wrong?) I am not sure I have the right to abuse the word METHOD and ALGORITHM as I do.

 

[pasmith]

Is it fair to say that the math course still leaves you with some decisions to make based on your specific application, whereas the ME course is already specialized to a particular application?

It would also not surprise me if the math course goes into more detail about how and why the method works.

 

[FEAnalyst]

I don't think the math departments teach this properly for mechanical engineers. I have seen students take FE classes from math departments and come out without any idea that there really is an established approach to a METHOD.

Well, when I was studying at the mechanical faculty, the FEM course was strictly about the math behind this method - vector spaces, weak forms, Galerkin method and basis/shape functions. Not a single word about applications of this mathematical method in mechanics. Now they teach the opposite - pure practice (how to use a particular FEA software). Neither of these approaches is good and the first one is literally terrible for mechanical engineers. It should be theory (not math but its application in mechanical problems) plus some practice. Like many other FEA engineers, I had to spend a lot of time on self-learning. Fortunately, apart from usually very academically focused books about FEA, there are also several blogs with a good introduction to FEM theory and after understanding the basics one may proceed to more advanced content included in classic books (I wouldn't recommend the trilogy by Zienkiewicz though).

I think that in order to understand how FEA software actually works, it's best to take a closer look at examples involving hand calculations of simple structures. Starting from springs, proceeding to bars/beams and finally flat plates. The latter are particularly important, among others, because they show how the stiffness matrix is obtained when the direct approach cannot be used and thus how it's actually implemented in FEA codes.