A Framework for FFT-based Homogenization on Anisotropic Lattices

In order to take structural anisotropies of a given composite and different shapes of its unit cell into account, we generalize the Basic Scheme in Homogenization by Moulinec and Suquet to arbitrary sampling lattices and tilings of the d-dimensional Euclidean space. Employing a Fourier transform on arbitrary lattices, which generate sampling patterns in the unit cell of interest, we derive a generalization of this scheme. In several cases, this Fourier transform is of lower dimension than the space itself; for many lattices it even reduces to a one-dimensional Fourier transform having the same leading coefficient as the fastest Fourier transform implementation available. We illustrate the generalized Basic Scheme on an anisotropic laminate and a generalized ellipsoidal Hashin structure. For both we derive an analytical solution to the elasticity problem, in two- and three dimensions, respectively. We then illustrate the possibilities of choosing a pattern. Compared to classical grids this introduces both a reduction of computation time and a reduced error of the numerical method. It also allows for anisotropic subsampling, i.e. choosing a sub lattice of a pixel or voxel grid based on anisotropy information of the material at hand.