I'm diving into the fascinating world of finite element analysis and would like to solve a large thermo-mechanical problem (only thermal $\rightarrow$ mechanical, no feedback). For the mechanical problem, I already grasped from Geoff's answer, that I'll need to use iterative solver due to the size of my mesh. I further read in Matt's reply, that the choice of the correct iterative algorithm is a daunting task.I'm asking here if there exist any experience on large 3-d linear-elastic problems that would help me narrow down my search for the best performance? In my case, it's a structure with thin, patterned films, and irregularly placed materials (both high-CTE and low-CTE). There are no large deformations in this thermo-mechanical analysis. I can use my university's HPC [1.314 nodes, with 2 AMD Opteron processors (each 2.2 GHz/ 8 cores)].I think PETSc could contain something interesting, especially the algorithms which do some sort of domain decomposition (FETI, multigrid) but I'm a bit overwhelmed by the options and have no experience. I also like the phrase geometrically informed preconditioners, but am unsure if this helps me. I have not yet found something focussing on linear continuum mechanics.Strong scaling (Amdahl) is very important in my application because my industrial partners can't wait a long time for simulation results. I definitely not only appreciate answers, but also recommendations for further reading in the comments. 
What is a robust, iterative solver for large 3-d linear-elastic problems?
pde;parallel computing;petsc;hpc;iterative method
Assuming that your structures are actually 3D (rather than only thin features, perhaps discretized with shell elements) and that the model is larger than a few hundred thousand dofs, direct solvers become impractical, especially if you only need to solve each problem once. Additionally, unless the structure is always close to a Dirichlet boundary, you will need a multilevel method to be efficient. The community is divided between multigrid and multilevel domain decomposition. For a comparison of the mathematics, see my reply to: What is the advantage of multigrid over domain decomposition preconditioners, and vice versa?The multigrid community has generally been more successful at producing general purpose software. For elasticity, I suggest using smoothed aggregation which requires an approximate near null space. In PETSc, this is done by choosing PCGAMG or PCML (configure with --download-ml) and calling MatSetNearNullSpace() to provide the rigid body modes.Domain decomposition methods offer an opportunity to coarsen faster than smoothed aggregation, thus possibly being more latency tolerant, but the sweet spot in terms of performance tends to be narrower than smoothed aggregation. Unless you want to do research on domain decomposition methods, I suggest just using smoothed aggregation, and perhaps try a domain decomposition method when the software becomes better.