Level-set Based Design of Wang Tiles for Modelling Complex Microstructures

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.

Modular-topology optimization of structures and mechanisms with free material design and clustering

“…Visually appealing, nonperiodic yet compressed arrangements have impelled the adaptation of Wang tiles in computer graphics to generate naturally-looking textures [41,42] and Poisson disk distributions [41,43]. Sharing objectives, these applications have inspired the use of Wang tiles in modeling materials microstructures, both for geometrical representation [44,45,46] and calculations [47,48,49,50]. Because the cardinality of the Wang tileset balances the computation efficiency of the PUC with precise (micro-)structural description, Wang tilings also appear to be useful in securing automatic connectivity in the topology optimization of modular truss structures [8].…”

Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method

“…In the series of our previous works [33,34,35], we have introduced the framework of Wang tiles as a suitable extension of the (Statistically Equivalent) Periodic Unit Cell methodology for modelling microstructural geometry of random heterogeneous materials. We have shown that replacing the unit cell based representation with a set of domains-Wang tiles-with predefined mutual compatibility enables an efficient generation of stochastic microstructural realizations that feature suppressed periodicity artefacts [33,34,35] Even though the spurious periodicity is significantly reduced in the generated microstructural samples, they are still composed of only a handful of tiles. The Wang tile concept thus provides a finite-size discrete parametrization space of all realizations.…”

“…So far, three design strategies have been proposed: an optimization approach to minimize a discrepancy between spatial statistics of the reference specimen and generated assemblies [33], a sample-based strategy [34], and a levelset based framework for particulate and foam-like microstructures [35]. However, most of the methods developed for generating PUC can in principle be modified to adopt the generalized periodicity constraints of Wang tiles.…”

“…Unlike mathematicians that focus mainly on tile sets for strictly aperiodic tiling, see the overview in [47,Chapter 11], we prefer stochastic tile sets introduced by Cohen et al [39] because these sets perform better in terms of suppressing periodic artefacts [34,35] and are less constrained in their construction. Using the assembly algorithm described next, a sufficient condition for a tile set is that it contains at least one tile for each admissible code combination on the left and top tile edges.…”

“…To enforce continuity of the solution across individual tiles, the remaining boundary DOFs are renumbered and assembled into u s such that the corresponding DOFs at edges with the same code share the same number throughout the tile set. While uniquely enumerating DOFs that belong exclusively to a tile edge is straightforward, enumerating vertex DOFs requires an analysis of the tile set described in [35,Section 2.3] in order to identify how many unique DOFs are needed for the vertices. Depending on the code distribution within the set, it may happen that certain vertices will never coincide during the tile assembly, thus constituting separate vertex group with distinct DOFs.…”