Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

Doškář, Martin; Zeman, Jan; Jarušková, Daniela; Novák, Jan

doi:10.1016/j.euromechsol.2017.12.002

Cited by 20 publications

(10 citation statements)

References 72 publications

(159 reference statements)

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“…(a) (b) Nonetheless, to the merit of the proposed modular concept, it is scalable up to the limits of the manufacturing hardware and extensible to three dimensions. New possibilities in terms of the design flexibility are foreseen by incorporating the concept of Wang tiles [24][25][26][27]. As opposed to commonly produced disordered structures, e.g., metal foams, the behavior of the jigsaw puzzle system augmented with Wang tiling would be fully predictable and customizable with respect to specific needs.…”

Section: Discussionmentioning

confidence: 99%

A jigsaw puzzle metamaterial concept

Nežerka

Somr

Janda

et al. 2018

Composite Structures

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A concept of a planar modular mechanical metamaterial inspired by the principle of local adaptivity is proposed. The metamaterial consists of identical pieces similar to jigsaw puzzle tiles. Their rotation within assembly provides a substantial flexibility in terms of structural behavior, whereas mechanical interlocks enable reassembly. The tile design with a diagonal elliptical opening allows us to vary elastic properties from stiff to compliant, with positive, zero, or negative Poisson's ratio. The outcomes of experimental testing on additively manufactured specimens confirm that the assembly properties can be accurately designed using optimization approaches with finite element analysis at heart.

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Section: Discussionmentioning

confidence: 99%

A jigsaw puzzle metamaterial concept

Nežerka

Somr

Janda

et al. 2018

Composite Structures

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“…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].…”

Section: Wang Tilingsmentioning

confidence: 99%

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

Tyburec,

Doškář,

Zeman

et al. 2021

Preprint

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In the optimal design of modular structures and mechanisms, two key questions must be answered: (i) what should the topology of individual modules be like and (ii) how should modules be arranged at the product scale? We address these challenges by proposing a bi-level sequential strategy that combines free material design, clustering techniques, and topology optimization. First, using free material optimization enhanced with post-processing for checkerboard suppression, we determine the distribution of elasticity tensors at the product scale. To extract the sought-after modular arrangement, we partition the obtained elasticity tensors with a novel deterministic clustering algorithm and interpret its outputs within Wang tiling formalism. Finally, we design interiors of individual modules by solving a single-scale topology optimization problem with the design space reduced by modular mapping, conveniently starting from an initial guess provided by free material optimization. We illustrate these developments with three benchmarks first, covering compliance minimization of modular structures, and, for the first time, the design of compliant modular mechanisms. Furthermore, we design a set of modules reusable in an inverter and in gripper mechanisms, which ultimately pave the way towards the rational design of modular architectured (meta)materials.

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“…In the online phase, microstructural realizations are assembled from these domains with a fast, linear algorithm that produces stochastic realizations with suppressed periodicity. The merits of the tiling concept for RVE size analyses were demonstrated in [45,60].…”

Section: Wang Tiling In Microstructure Modellingmentioning

confidence: 99%

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

Doškář

Zeman

Rypl

et al. 2020

Computer-Aided Design

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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.

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Wang tiling aided statistical determination of the Representative Volume Element size of random heterogeneous materials

Cited by 20 publications

References 72 publications

A jigsaw puzzle metamaterial concept

A jigsaw puzzle metamaterial concept

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

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

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