Elegant analytical solutions to solid and fluid mechanics problems on simplified domains can strengthen engineering understanding. Solving the Laplace equation on the unit square is also good for complexion and digestion, which is part of why I went and got a mechanical engineering MSc after working for several years. Love me some analytical solutions.
But the real world is frighteningly complicated. Sometimes simplified domains and straightforward boundary conditions aren't sufficient to really understand a system, much less accurately predict failure. Enter numerical methods, and, in particular, the finite element method. And 50+ years of elevating computing power and refining algorithms. Now you can solve lots of cool things. But...it helps to know what you're doing. Commercial packages (like ANSYS) and others make straightforward analyses...more straightforward - but there's quite a bit of theoretical baggage that one has lumpf around if you're going to be robust about interpreting results.
In the ever-further-receding-past, I had an inkling of what I was doing and started exploring deal.ii. It's an open-source finite element package that has insane amounts of functionality. Adaptive methods? Discontinuous Galerkin methods? Time-stepping algorithms? Got 'em all. And it's free.
I literally only scratched the surface of deal.ii and can only applaud Wolfgang Bangerth and the rest of the deal.ii authors for this freely available software. Highly recommend you check it out.
Also just made some of my own basic write-ups for a simplified case of a beam in plane strain (pdf) and of solving the heat equation on a cylinder with some partial boundary conditions and a heat source (pdf).
Obviously I had more time on my hands when I wrote these. Code is at my github.
But if you're looking for great open-source finite element packages and are handy with C++ and willing to go through some of the tutorials, take a gander at deal.ii.