Level-set based design of Wang tiles for modelling complex microstructures
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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 framework of 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.
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