Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 321, Issue -, Pages 151-168Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2016.05.041
Keywords
Numerical homogenization; Fourier-Galerkin method; Fast Fourier Transform; Guaranteed bounds; Richardson iteration; Conjugate gradient algorithm; Chebyshev semi-iterative method; Eyre-Milton scheme
Funding
- European Social Fund
- state budget of the Czech Republic [CZ.1.07/2.3.00/30.0034]
- Czech Science Foundation [13-22230S]
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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. (C) 2016 Elsevier Inc. All rights reserved.
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