Symmetries in stochastic homogenization and adjustments for the RVE method
Binh Huy Nguyen, Matti Schneider
TL;DR
This work analyzes how ensemble symmetries shape the effective conductivity tensor $\bm{A}^{\mathrm{eff}}$ and its fluctuations, within stochastic homogenization and the RVE framework. It develops symmetry-informed orthogonal projections to enforce $\mathcal{G}$-invariance in postprocessing, proving that such projections yield unbiased variance reduction while preserving eigenvalue bounds. The authors derive invariance properties for isotropic, transversely isotropic, and orthotropic ensembles and demonstrate, via large-scale FFT-based simulations, that symmetry projections significantly reduce random errors with minimal impact on systematic errors across isotropic and TI cases, while offering modest gains for orthotropic configurations. The proposed approach provides a practical, low-cost tool to enforce physical symmetries in homogenization, with potential extensions to elasticity via analogous projection operators.
Abstract
We investigate the implications of a given symmetry of a random microstructure on the obtained effective tensor and its fluctuation in the context of thermal conductivity, and study strategies for enforcing these symmetries in postprocessing via orthogonal projectors. Within the framework of the representative volume element (RVE) method, we establish the invariance conditions for the effective tensor and its fluctuation under different symmetry groups of the microstructure. Interestingly, the symmetry of the considered cell type in the RVE method may break the ensemble symmetry and compromise the approximation of the effective properties. To rectify this issue, we introduce dedicated techniques which permit to enforce the expected symmetries in postprocessing and study the implications on the bounds for the effective properties as well as the total, the random and the systematic errors. We provide theoretical arguments that suitable projections lead to unbiased variance-reduction strategies which furthermore enforce the expected symmetries exactly. Through large-scale FFT-based homogenization simulations, we study the symmetry structure of the estimated effective conductivities and their fluctuations. Moreover, we demonstrate the power of the symmetry-projection techniques for fiber-reinforced composite microstructures of industrial scale.