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.

a b s t r a c tIn this short note, we present a new technique to accelerate the convergence of a FFT-based solver for numerical homogenization of complex periodic media proposed by Moulinec and Suquet [1]. The approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to Vainikko [2], to give a linear system which can be efficiently solved by conjugate gradient methods. Computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high-contrast coefficients at a low overhead per iteration.

Fourier solvers have become efficient tools to establish structure-property relations in heterogeneous materials. Introduced as an alternative to the Finite Element (FE) method, they are based on fixed-point solutions of the Lippmann-Schwinger type integral equation. Their computational efficiency results from handling the kernel of this equation by the Fast Fourier Transform (FFT). However, the kernel is derived from an auxiliary homogeneous linear problem, which renders the extension of FFT-based schemes to non-linear problems conceptually difficult. This paper aims to establish a link between FE-and FFT-based methods, in order to develop a solver applicable to general history-and time-dependent material models. For this purpose, we follow the standard steps of the FE method, starting from the weak form, proceeding to the Galerkin discretization and the numerical quadrature, up to the solution of non-linear equilibrium equations by an iterative Newton-Krylov solver. No auxiliary linear problem is thus needed. By analyzing a two-phase laminate with non-linear elastic, elasto-plastic, and visco-plastic phases, and by elasto-plastic simulations of a dual-phase steel microstructure, we demonstrate that the solver exhibits robust convergence. These results are achieved by re-using the non-linear FE technology, with the potential of further extensions beyond small-strain inelasticity considered in this paper.

Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model