An FFT-based Galerkin Method for Homogenization of Periodic Media
This provides theoretical foundations and a practical improvement (conjugate gradient solver) for FFT-based homogenization methods, which are widely used in materials science and engineering.
The authors prove that the Moulinec-Suquet FFT-based method for homogenization of periodic media is equivalent to a Galerkin discretization with trigonometric polynomials, establishing convergence and a priori error estimates. They also show that the resulting linear system can be solved via conjugate gradient, improving solver performance.
In 1994, Moulinec and Suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media. The scheme is based on a fixed-point iterative solution to an integral equation of the Lippmann-Schwinger type, with action of its kernel efficiently evaluated by the Fast Fourier Transform techniques. The aim of this work is to demonstrate that the Moulinec-Suquet setting is actually equivalent to a Galerkin discretization of the cell problem, based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme. For the latter framework and scalar elliptic setting, we prove convergence of the approximate solution to the weak solution, including a-priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration. Moreover, we also show that the variational structure implies that the resulting non-symmetric system of linear equations can be solved by the conjugate gradient method. Apart from providing a theoretical support to Fast Fourier Transform-based methods for numerical homogenization, these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature.