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

## TL;DR

This paper extends a microstructural generation framework to non-periodic Wang tiles, enabling the creation of complex, large, stochastic 3D microstructures with improved compatibility and reduced artificial periodicity.

## Contribution

It introduces a robust procedure for designing complex 3D Wang tile morphologies and broadens the tiling concept's applicability for microstructure modeling.

## Key findings

- Vertex-defined tile sets outperform in 2D microstructure generation.
- The method effectively reduces artificial periodicity in reconstructed samples.
- Demonstrated capabilities in 2D and 3D microstructure synthesis.

## Abstract

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.