An optimal preconditioned FFT-accelerated finite element solver for homogenization

Abstract

We generalize and provide a linear algebra-based perspective on a finite element (FE) homogenization scheme, pioneered by Schneider et al. (2017)[1] and Leuschner and Fritzen (2018)[2]. The efficiency of the scheme is based on a preconditioned, well-scaled reformulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner—a discretized Green’s function of a periodic homogeneous reference problem—has a block-diagonal structure in the Fourier space which permits its efficient inversion using fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as O(nlog(n)), like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to classical spectral solvers, the proposed scheme works with FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon. We show that the scheme achieves a number of iterations that are almost independent of spatial discretization. The scheme also scales mildly with phase contrast. We also discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection.