Computational homogenization with million-way parallelism using domain decomposition methods

Parallel computational homogenization using the well-knwon $FE^2$ approach is described and combined with domain decomposition and algebraic multigrid solvers. It is the purpose of this paper to show that and how the $FE^2$ method can take advantage of the largest supercomputers available and those of the upcoming exascale era for virtual material testing of micro-heterogeneous materials such as advanced steel. The $FE^2$ method is a computational micro-macro homogenization approach where at each Gauss integration point of the macroscopic finite element problem a microscopic finite element problem, defined on a representative volume element (RVE), is attached. Note that the $FE^2$ method is not embarrassingly parallel since the RVE problems are coupled through the macroscopic problem. Numerical results considering different grids on both, the macroscopic and microscopic level as well as weak scaling results for up to a million parallel processes are presented