Microstructural geometry plays a critical role in a response of heterogeneous materials. Consequently, methods for generating microstructural samples are becoming an integral part of advanced numerical analyses. Here, we extend the unified frameworkof Sonon and co-workers , developed originally for generating particulate and foam-like microstructural geometries of Periodic Unit Cells, to non-periodic microstructural representations based on the formalism of Wang tiles. The formalism has been recently proposed as a generalization of the Periodic Unit Cell approach, enabling a fast synthesis of arbitrarily large, stochastic microstructural samples from a handful of domains with predefined microstructural compatibility constraints. However, a robust procedure capable of designing complex, three-dimensional, foam-like and cellular morphologies of Wang tiles has been missing . This contribution thus significantly broadens the applicability of the tiling concept. Since the original  framework builds on a random sequential addition of particles enhanced with a level-set description , we first devise an analysis based on a connectivity graph of a tile set, resolving the question where should a particle be copied when it intersects a tile boundary. Next, we introduce several modifications to the original algorithm that are necessary to ensure microstructural compatibility in the generalized periodicity setting of Wang tiles. Having  a universal procedure for generating tile morphologies at hand , we compare strictly aperiodic and stochastic sets of the same cardinality in terms of reducing the artificial periodicity in reconstructed microstructural samples , and demonstrate the superiority of the vertex-defined tile sets for two-dimensional problems . Finally, we illustrate the capabilities of the algorithm with two- and three-dimensional examples.
Microstructural geometry plays a critical role in the response of heterogeneous materials. Consequently, methods for generating microstructural samples are increasingly crucial to advanced numerical analyses. We extend Sonon et al.'s unified framework , developed originally for generating particulate and foam-like microstructural geometries of Periodic Unit Cells, to non-periodic microstructural representations based on the formalism of Wang tiles. This formalism has been recently proposed in order to generalize the Periodic Unit Cell approach, enabling a fast synthesis of arbitrarily large, stochastic microstructural samples from a handful of domains with predefined compatibility constraints. However, a robust procedure capable of designing complex, three-dimensional, foam-like and cellular morphologies of Wang tiles has not yet been proposed . This contribution fills the gap by significantly broadening the applicability of the tiling concept. Since the original Sonon et al.'s framework builds on a random sequential addition of particles enhanced with an implicit representation of particle boundaries by the level-set field , we first devise an analysis based on a connectivity graph of a tile set, resolving the question where a particle should be copied when it intersects a tile boundary. Next, we introduce several modifications to the original algorithm that are necessary to ensure microstructural compatibility in the generalized periodicity setting of Wang tiles. Having established a universal procedure for generating tile morphologies , we compare strictly aperiodic and stochastic sets with the same cardinality in terms of reducing the artificial periodicity in reconstructed microstructural samples . We demonstrate the superiority of the vertex-defined tile sets for two-dimensional problems and illustrate the capabilities of the algorithm with two- and three-dimensional examples.
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