Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

Sébastien Brisard (1) , Luc Dormieux (1)

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

We report on the mathematical analysis of two different, FFT-based, numerical schemes for the homogenization of composite media within the framework of linear elasticity: the basic scheme of Moulinec and Suquet (1994, 1998) [9] and [10], and the energy-based scheme of Brisard and Dormieux (2010) [13]. Casting these two schemes as Galerkin approximations of the same variational problem allows us to assert their well-posedness and convergence. More importantly, we extend in this work their domains of application, by relieving some stringent conditions on the reference material which were previously thought necessary. The origins of the flaws of each scheme are identified, and a third scheme is proposed, which seems to combine the strengths of the basic and energy-based schemes, while leaving out their weaknesses. Finally, a rule is proposed for handling heterogeneous pixels/voxels, a situation frequently met when images of real materials are used as input to these schemes.

Keywords: Galerkin approximation; Heterogeneous media; Linear elasticity; Numerical homogenization; Polarization; Variational problem

Domains: Mechanics of materials [physics.class-ph]; Mechanics of materials [physics.class-ph]

Sébastien Brisard, Luc Dormieux. Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Computer Methods in Applied Mechanics and Engineering, 2012, 217-220, pp.197-212. ⟨10.1016/j.cma.2012.01.003⟩. ⟨hal-00722361⟩