Level-set Based Design of Wang Tiles
for Modelling Complex Microstructures
Martin Doškář
Department of Mechanics
Faculty of Civil Engineering
Czech Technical University in Prague
Thákurova 7, 166 29 Prague, Czech Republic
Experimental centre
Faculty of Civil Engineering
Czech Technical University in Prague
Thákurova 7, 166 29 Prague, Czech Republic
Jan Zeman
Department of Mechanics, Experimental centre
Faculty of Civil Engineering
Czech Technical University in Prague
Thákurova 7, 166 29 Prague, Czech Republic
The Institute of Information Theory and Automation
Academy of Sciences of the Czech Republic
Pod Vodárenskou věží 4, 182 08 Prague, Czech Republic
Daniel Rypl
Department of Mechanics
Faculty of Civil Engineering
Czech Technical University in Prague
Thákurova 7, 166 29 Prague, Czech Republic
Jan Novák
Experimental centre
Faculty of Civil Engineering
Czech Technical University in Prague
Thákurova 7, 166 29 Prague, Czech Republic
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
microstructural 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.