A Framework for FFT-based Homogenization on Anisotropic Lattices

TL;DR

This paper extends the FFT-based homogenization method to arbitrary anisotropic lattices, enabling more efficient and accurate analysis of composite materials with anisotropic features and complex unit cell geometries.

Contribution

It generalizes the Basic Scheme for homogenization to arbitrary sampling lattices using Fourier transforms, reducing computational cost and error in anisotropic material analysis.

Findings

Reduced computation time compared to classical grids

Lower numerical error with anisotropic subsampling

Analytical solutions derived for specific structures

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. 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.