Constitutive relation error for FFT-based methods

Sébastien Brisard (1) , L Chamoin (2)

1 navier umr 8205 - Laboratoire Navier
2 LMT - Laboratoire de Mécanique et Technologie

Since their introduction by Moulinec and Suquet (1994, 1998), FFT-based methods have become popular alternatives to finite element methods (FEM) for the numerical homogenization of heterogeneous materials. They have been successfully applied to a variety of constitutive laws. Even geometric non-linearities can now be accounted for (Kabel et al., 2014). Theoretical analysis of this numerical technique has also greatly improved. Convergence of the iterative solvers is now well understood (Michel et al., 2001; Brisard & Dormieux, 2010; Zeman et al., 2010; Monchiet & Bonnet, 2012; Moulinec & Silva, 2014), as well as convergence with respect to the grid-size (Brisard & Dormieux, 2012; Schneider 2014). Alternative discretizations aiming at improving the quality of the solution (at fixed grid-size) have also been proposed (Brisard & Dormieux, 2012; Willot et al., 2014; Vondřec et al., 2015; Willot, 2015). However, this quality improvement is usually assessed in a qualitative way. Some progress has recently been made by Vondřec et al. (2015) towards quantitative analysis of the quality of the solution, but their approach is restricted to discretizations by trigonometric polynomials. In this talk, we present a strategy for the aposteriori error analysis of the numerical solution to the Lippmann–Schwinger equation. We use the framework of the error in constitutive relation, initially introduced by Ladevèze and Leguillon (1983) for the FEM. This requires to produce a kinematically admissible displacement field and a statically admissible stress field. We will show that a kinematically admissible displacement field can be efficiently reconstructed. Our approximation does not use trigonometric polynomials; rather it is local (Q1) in space (thus free of spurious Gibbs-like oscillations). Besides, the reconstructed displacement field has some properties that allow us to reuse standard techniques from the FEM world to produce a statically admissible stress field. A posteriori error estimates are then readily computed. Our strategy applies to any discretization/solver, which allows a quantitative comparison of the various FFT-based strategies proposed in the literature. This will be illustrated on a few examples, both in two and three dimensions.

Domaines: Mécanique des matériaux [physics.class-ph]

Sébastien Brisard, L Chamoin. Constitutive relation error for FFT-based methods. VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016), Jun 2016, Crete Island, Greece. ⟨hal-01330790v2⟩