FEA, at its base, is solving for [x] in [F] = [k][x] where F is your input forces on a structure, k is the stiffness matrix and x is the resultant displacement,

FEA, at its base, is solving for [x] in [F] = [k][x] where F is your input forces on a structure, k is the stiffness matrix and x is the resultant displacement, which when combined with material properties produces a strain matrix and therefore body stresses of the structure. It basically has to invert a many-dimensional matrix to do that, which is where most of the 'core' computations of a linear-static FEA solution sit. Of course, desire for more detailed simulation has always exceeded computational capacity, so slicing up the matrix operation and writing it to permanent storage (tape, disk, etc.) is a routine operation for all solvers like this. Expanding on from that, working out eigenvalues and eigenvectors for a matrix allows you to solve for harmonics, as you're well aware. Harmonics in a structural context are used to identify modal vibrations (e.g.: at what resonant frequency a drum skin vibrates for a given surface area, material and tension). It's essentially analogous to solving harmonics in electrical systems, like how a mass-spring-damper system is analogous to an RLC circuit. NASTRAN is the granddaddy of FEA packages and was produced for NASA (hence its name) way back in the early days of computational simulation for their space programme. Well before the days of desktop workstations, companies (including the one I work for) would bundle up their model, ship it off to a company that had spare mainframe capacity and a solver licence, have them run it during their downtime and then receive their results back some time later. Basically, as a kid of the 80s all of this sits in 'computing prehistory' for me so it was a fascinating (to me) exercise to place your work in the context of big software packages with which I'm familiar from a similar era. My last comment about beating NASA at writing a numerical solver for matrix inversion was just a bit of a silly joke, but it's similarly cool to realise that it's in fact true for other big numerical packages like LINPACK! :)