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

TL;DR

This study compares four iterative algorithms for solving linear systems in FFT-based homogenization of heterogeneous materials, finding the Conjugate gradient method most efficient among them.

Contribution

It evaluates and compares the performance of general-purpose and specialized iterative solvers within a Fourier-Galerkin framework for homogenization problems.

Findings

Conjugate gradient is the most efficient solver tested.

Eyre-Milton scheme performs comparably to Chebyshev semi-iteration.

Richardson algorithm shows the slowest convergence.

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.