The FFT-based homogenization method of Moulinec-Suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems. In this work, we generalize the method to problems discretized by trilinear hexahedral elements on Cartesian grids and physically nonlinear elasticity problems. We present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation. For benchmark problems in linear elasticity, the solver exhibits mesh- and contrast-independent convergence behavior and enables the computational homogenization of complex structures, for instance, arising from computed tomography computed tomography (CT) imaging techniques. There exist 3D microstructures involving pores and defects, for which the original FFT-based homogenization scheme does not converge

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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.

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The FFT-based homogenization method proposed by Moulinec and Suquet [1] in 1994 to solve the Lippmann-Schwinger equation has recently become more popular due to its computational speed and accuracy. It is based on regular voxel grids and can work on segmented CT images directly. Given an interface description on the sub-voxel scale we show that using interface voxels with an appropriately chosen stiffness significantly enhances the accuracy of the computed effective properties and decreases voxelation effects in the solution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper, we first introduce the reader to the Basic Scheme of Moulinec and Suquet in the setting of quasi-static linear elasticity, which takes advantage of the fast Fourier transform on homogenized microstructures to accelerate otherwise time-consuming computations. By means of an asymptotic expansion, a hierarchy of linear problems is derived, whose solutions are looked at in detail. It is highlighted how these generalized homogenization problems depend on each other. We extend the Basic Scheme to fit this new problem class and give some numerical results for the first two problem orders.

In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann-Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann-Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both the de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.

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