Nonlinear composite voxels and FFT-based homogenization

The FFT-based homogenization method of Moulinec-Suquet [14] has reached a degree of sophistication and maturity, where it can be applied to microstructures of industrial size and realistic scope. However, for non-linear or load-path-dependent problems the method reaches its limits, in particular if variations of the geometry are considered or the determination of the full material law on the macro-scale is required. Time and memory considerations are primarily responsible for these limitations. This work focuses on the composite voxel technique, where sub-voxels are merged into bigger voxels to which an effective material law based on laminates is assigned. Due to the down-sampled grid, both the memory requirements and the computational effort are severely reduced, while retaining the original accuracy. We discuss the extensions of linear elastic ideas [6, 9] to incremental problems at small strains. In contrast to conventional model order reduction methods, our approach does neither rely upon a "offline phase" nor on preselected "modes". We demonstrate our ideas with several numerical experiments, comparing to full-resolution computations heavily relying upon our MPI-parallel implementation FeelMath [1].