Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

TL;DR

This paper introduces a Wang tiling-based method for efficiently determining the size of the Representative Volume Element in heterogeneous materials by generating large microstructure realizations and constructing confidence intervals for their properties.

Contribution

The paper presents a novel approach combining Wang tiling, statistical sampling, and the Partition theorem to adaptively determine RVE size with prescribed accuracy in microstructure homogenization.

Findings

Efficient generation of large, statistically consistent microstructure realizations.

Construction of confidence intervals for apparent properties based on tiling and sampling.

Adaptive microstructure sampling to meet specified tolerance levels.

Abstract

Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative Volume Element size related to the user-defined significance level and the discrepancy between bounds on the apparent properties. First, the tiling concept is employed to efficiently generate arbitrarily large, statistically consistent realizations of investigated microstructures. Second, benefiting from the regular structure inherent to the tiling concept, the Partition theorem, and statistical sampling, we construct confidence intervals of the apparent properties related to the size of a microstructure specimen. Based on the interval width and the upper and lower bounds on the apparent properties, we adaptively generate additional microstructure realizations in order to arrive at an RVE satisfying the prescribed tolerance. The methodology is illustrated with the homogenization of thermo-mechanical properties of three two-dimensional microstructure models: a microstructure with mono-disperse elliptic inclusions, foam, and sandstone.