A comparative study on low-memory iterative solvers for FFT-based homogenization of periodic media

Abstract

In this paper, we assess the performance of four iterative algorithms for solving non-symmetric rank-deficient linear systems arising in the FFT-based homogenization of heterogeneous materials defined by digital images. Our framework is based on the Fourier–Galerkin method with exact and approximate integrations that has recently been shown to generalize the Lippmann–Schwinger setting of the original work by Moulinec and Suquet from 1994. It follows from this variational format that the ensuing system of linear equations can be solved by general-purpose iterative algorithms for symmetric positive-definite systems, such as the Richardson, the Conjugate gradient, and the Chebyshev algorithms, that are compared here to the Eyre–Milton scheme — the most efficient specialized method currently available. Our numerical experiments, carried out for two-dimensional elliptic problems, reveal that the Conjugate gradient algorithm is the most efficient option, while the Eyre–Milton method performs comparably to the Chebyshev semi-iteration. The Richardson algorithm, equivalent to the still widely used original Moulinec–Suquet solver, exhibits the slowest convergence. Besides this, we hope that our study highlights the potential of the well-established techniques of numerical linear algebra to further increase the efficiency of FFT-based homogenization methods.