A framework for FFT-based homogenization on anisotropic lattices

Abstract

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. We employ a Fourier transform for these lattices by introducing the corresponding set of sample points, the so called pattern, and its frequency set, the generating set, both representing the anisotropy of both the shape of the unit cell and the chosen preferences in certain sampling directions. In several cases, this Fourier transform is of lower dimension than the space itself. For the so called rank-1-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 give 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.

Approximation of periodic PDE solutions with anisotropic translation invariant spaces

“…With these definitions the discretisation of the PDEs on spaces of translates can be performed as follows: Choosing the function f as the Dirichlet kernel f = D M reduces (18) and (20) to discretised equations which correspond to truncating the Fourier series arising in the continuous equations (11) and (12). The effect of the pattern matrix M in this special case is analysed in [2]. For the Dirichlet kernel on a tensor product grid, which corresponds to a diagonal pattern matrix, Vondřejc et.al.…”

“…The generalized Hashin structure that consists of two confocal ellipsoids (Ω c ) and (Ω e ) embedded in a matrix material (Ω m ) and is depicted in Figure 2 (left). For this structure an analytic expression for the strain ε and for the effective stiffness C eff : ε 0 is known and described in [2,Section 4.2]. The examples were computed using (18) using an iterative scheme based on a Neumann series approach, originally proposed by Moulinec and Suquet [9].…”

“…The following computation was performed with the pattern matrix M = 128 272 0 128 , showing the minimal 2 -error in [2].…”

“…The continuous equation allows for two equivalent formulations. They have the same discrete solution in a space of translates if and only if the space yields the truncated Fourier series approach which corresponds to using spaces of translates generated by a Dirichlet kernel on anisotropic patterns [2].…”

FFT-based homogenization on periodic anisotropic translation invariant spaces

“…[27] show that the method of Moulinec and Suquet can also be understood as a Galerkin projection using truncated Fourier series. This idea is generalized in [3] to anisotropic lattices thus allowing to take directional information on the geometrical structure or the orientation of interfaces between materials into account. Brisard and Dormieux [8,9] use constant finite elements to arrive at the Basic Scheme with a modified linear operator, based on an energy based formulation.…”