A Galerkin approach to FFT-based homogenization methods

Sébastien Brisard (1) , Luc Dormieux (1)

1 multi-échelle - Modélisation et expérimentation multi-échelle pour les solides hétérogènes

Since their introduction by Moulinec and Suquet [1, 2], FFT-based full-field simulations of the mechanical properties of composites have become increasingly popular, with applications ranging from the linear elastic behaviour of cementitious materials [3], to the plasticity of polycrystals [4]. Recently, the authors have proposed [5] a new formulation of these numerical schemes, based on the energy principle of Hashin and Shtrikman [6]. While similar in principle to the original scheme of Moulinec and Suquet, the new scheme was shown to be much better-behaved. Indeed, convergence of the scheme is guaranteed for any contrast, without having to resort to augmented Lagrangian approaches [7]. Besides, convergence of the new scheme is generally much faster. However, the new scheme has two drawbacks. First, the reference material must be stiffer (or softer) than all constituants of the composite; this is not always possible, for example when the composite contains both pores and rigid inclusions. Second, the scheme requires the preliminary computation of the so-called consistent Green operator, which turned out to be a difficult task in three dimensions. In order to relax these requirements, an in-depth mathematical analysis of these schemes was carried out by the authors [8]. In this paper, the Lippmann-Schwinger equation and its variational form will briefly be recalled. The Galerkin approach will then be adopted for the discretization of this equation, and it will be shown that the basic scheme of [1] as well as the energy scheme proposed in [5] can both be viewed as well-posed Galerkin approximations of the Lippmann-Schwinger equation. Contrary to what was previously believed [7, 5] these approximations are convergent, regardless of the reference material (provided that its stiffness is positive definite). Comparison of these two approximations leads to the derivation of the so-called filtered, non-consistent approach, which combines the assets of the two former methods. Finally, some applications will be shown. In particular, the important problem of heterogeneous voxels will be addressed.

Domaines: Analyse numérique [math.NA]; Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]; Matériaux

Sébastien Brisard, Luc Dormieux. A Galerkin approach to FFT-based homogenization methods. 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012, Sep 2012, Vienna, Austria. pp.1709-1718. ⟨hal-00738040⟩