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There're commercial FEA software for most applications. Actually coding one is pretty time consuming and not the best use of your time.
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Coding one is how you learn what the commercial FEA apps are doing.
FEA software is very prone to garbage-in, garbage-out. FEA experience is precisely the reason why I've gotten offers in aerospace and sub turbine design, and why when I was casually talking to a NASA-Ames engineer (at a wedding shower, no less) a month or so ago, I was offered a job on the spot. If you don't know exactly what you're doing with boundary conditions, you're going to screw it up and to someone who doesn't know material behaviors extremely well, it'll look right. That's why FE analysts are in such high demand.
FEA is definitely graduate work. Don't worry, nobody's going to ask you to do finite element theory without some pretty heavy theoretical mechanics courses first, which you'll start in grad school.
I think it's fun and exciting. It's very challenging. It requires a lot of really hard-core analysis, and if you get a job doing that kind of stuff, your brain's gonna feel gelatinous at the end of the day... That's the exhausting part, and it's why a lot of people give it up.
Finite element is extremely applicable, and Payne's buddy could've done anything from designing missile nosecones to designing engine blades... My non-academic experience was in blast-resistant enclosures (basically explosion-proof guardhouses) for a government contractor, using ABAQUS and ADINA.
FEA is just breaking things into itty-bitty, highly analyzable and very regular pieces and making the computer do all the heavy lifting for you. The theory behind it is very complex, though, and like pretty much all of engineering computing, you can do things very badly if you don't actually know your jazz. It just *looks* deceptively easy to use...