Wang Tilings in Numerical Homogenization

Stochastic Wang tiling has been shown to bring unexpected insights to microstructure representation efforts as it generalizes the conventional unit-cell approach. It allows to reconstruct stochastic realizations of the compressed medium without prior periodic assumptions on microstructural patterns. Moreover, once the microstructure is compressed, its realizations of various sizes can be generated at almost negligible cost. In this paper, we follow the standard numerical homogenization procedure and make use of the realizations as domains over which local quantities are computed and averaged subsequently. In order to alleviate computational cost, a domain decomposition method is adopted such that it benefits from the fact that the computational domains are composed of limited number of repetitive patterns -- tiles.