Limitations and merits of wavelet discretizations in FFT-based computational micromechanics

FFT-based computational micromechanics enables to efficiently analyze the solid mechanical behavior of industrial-scale heterogeneous microstructures. This approach relies on the discretization of the microstructure on a regular grid, which makes it particularly convenient for working with 2D or 3D images. However, efficient local refinement of the grid to capture fine-scale phenomena is not possible within the original framework. Among the various strategies addressing this limitation, Kaiser et al. recently introduced a multiresolution wavelet collocation approach for FFT-based computational micromechanics. They showed the benefits of local adaptivity for examples in one and two dimensions. The present contribution focuses on the properties of the wavelet discretizations in three spatial dimensions. More precisely, we analyze the Deslauriers–Dubuc (DD) wavelets of degrees one to four, connecting and comparing them to established FFT-based homogenization frameworks. Numerical studies on regular grids serve to analyze the convergence behavior of the associated Lippmann-Schwinger solvers, the emerging macroscopic properties, and the resulting local fields obtained with these discretizations on industrial-type microstructures. Also, we study the performance for inelastic constituents. Of particular interest is the result that only first-order DD wavelets ensure stable convergence for porous microstructures.