  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.
