Overview of FFT-based homogenization techniques from the Galerkin point of view

Sébastien Brisard (1)

1 navier umr 8205 - Laboratoire Navier

Most engineering materials are highly heterogeneous at various length-scales. As a consequence it is highly impractical to build a model of a structure which would encompass all heterogeneities. Fortunately, when length-scales are separated, homogenization theory can help account in a simplified way for the heterogeneity of the material making-up the structure. A typical multiscale simulation based on homogenization theory is a two-step process. In a first step, the so-called corrector problem formulated over a representative volume element (with adequate boundary conditions) is solved. The macroscopic (homogenized) properties of the heterogeneous material are then retrieved from the solution to this corrector problem. In a second step, a full model of the structure is built, in which the heterogeneous materials are given their homogenized properties as per the previous step. The present paper focuses on the numerical solution to the corrector problem (step 1). For complex microstructures with convoluted interfaces, it might become very difficult to build a conforming mesh, which rules out standard finite element simulations. In such complex situations, numerical methods formulated over regular grids might be preferable. In the method initially proposed by Moulinec and Suquet (1994, 1998), the corrector problem is first reformulated as an integral equation (also known as the Lippmann–Schwinger equation), which is then discretized over a regular grid. The popularity of this method comes from the use of the fast Fourier transform (FFT) to compute efficiently the convolution product appearing in the Lippmann–Schwinger equation with periodic boundary conditions. Since the publication of the original paper, many variants of the so-called basic scheme have been introduced. This method was recently revisited (Brisard and Dormieux, 2012) and a new formulation as a Galerkin discretization of the Lippmann–Schwinger equation was proposed. This approach leads to a unified framework for all variants of the basic scheme. In particular, the concept of discrete Green operator which stem from asymptotically consistent Galerkin discretizations of the Lippmann–Schwinger equation was introduced. This paper briefly reviews FFT-based homogenization methods, with a focus on the Galerkin point of view.

Mots clés: homogenization; elasticity; FFT

Domaines: Matériaux et structures en mécanique [physics.class-ph]

Sébastien Brisard. Overview of FFT-based homogenization techniques from the Galerkin point of view. Conférence Internationale de Géotechnique, des Ouvrages et Structures (CIGOS 2015), ENS Cachan, May 2015, Cachan, France. ⟨hal-01194693⟩