Comparison of different FFT-based methods for computing the mechanical response of heteregoneous materials
Résumé
The last decade has witnessed a growing interest for the so-called " FFT-based methods " for computing the overall and local properties of heterogeneous materials submitted to mechanical solicita-tions. Since the original method was introduced by Moulinec and Suquet [1], several authors have proposed different algorithms to better deal with non-linear materials or with materials with highly contrasted mechanical properties between their constituents. The study concerns a linear elastic material-although the methods involved can be extended into the case of non-linear behavior-submitted to a prescribed overall strain E. The stiffness tensor c(x) of the material varies with the position x. The numerical method proposed by Moulinec & Suquet lies on the iterative resolution of the Lippmann-Schwinger equation and can be summarized by the following relation between two successive iterates ε i and ε i+1 of the strain field: ε i+1 (x) = −Γ 0 * (c(x) − c 0) : ε i (x) + E , where c 0 is the stiffness tensor of a reference medium supposed to be linear elastic, where Γ 0 is a Green operator associated to c 0 and where * denotes the convolution operator. Eyre & Milton [2], Michel et al. [3] and Monchiet & Bonnet [4] proposed different schemes to accelerate the convergence of the initial scheme. It has been recently demonstrated in [5] that the two first schemes are particular cases of the last one. On the other hand, Zeman et al. [6] proposed to use a conjugate gradient method for solving the Lippmann-Schwinger equation. The present paper aims to compare these different methods with a special attention paid to their relative efficiency and their rates of convergence.
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