Synthesis of microstructure-informed enrichment functions by means of Wang tiles

The Extended Finite Element Method (XFEM) enhances the approximation space of the standard Finite Element Method (FEM) with functions reflecting local features in order to yield more accurate results with less degrees of freedom. XFEM performance is, thus, closely related to the quality of enrichment functions. Analogously to our previous works, in which we have employed the concept of Wang tiles to assembly microstructure geometries, in this contribution we use Wang tiles to assemble microstructure-informed enrichment functions. We compare two ways of generating the enrichments: (i) inspired by the first-order numerical homogenization and (ii) based on spectral analysis of the global stiffness matrix for the whole set. The methodology and performance of both approaches are illustrated through a linear diffusion problem in two dimensions

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An alternative for Wang tiles

ACM Transactions on Graphics ◽

10.1145/1183287.1183296 ◽

2006 ◽

Vol 25 (4) ◽

pp. 1442-1459 ◽

Cited By ~ 58

Author(s):

Ares Lagae ◽

Philip Dutré

Keyword(s):

Wang Tiles

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Wang Tiles for image and texture generation

ACM Transactions on Graphics ◽

10.1145/882262.882265 ◽

2003 ◽

Vol 22 (3) ◽

pp. 287-294 ◽

Cited By ~ 167

Author(s):

Michael F. Cohen ◽

Jonathan Shade ◽

Stefan Hiller ◽

Oliver Deussen

Keyword(s):

Wang Tiles ◽

Texture Generation

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A small aperiodic set of Wang tiles

Discrete Mathematics ◽

10.1016/0012-365x(95)00120-l ◽

1996 ◽

Vol 160 (1-3) ◽

pp. 259-264 ◽

Cited By ~ 65

Author(s):

Jarkko Kari

Keyword(s):

Wang Tiles

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WANG TILES LOCAL TILINGS CREATED BY MERGING EDGE-COMPATIBLE FINITE ELEMENT MESHES

Acta Polytechnica CTU Proceedings ◽

10.14311/app.2017.13.0162 ◽

2017 ◽

Vol 13 ◽

pp. 161

Author(s):

Lukáš Zrůbek ◽

Anna Kučerová ◽

Martin Doškář

Keyword(s):

Finite Element ◽

Mechanical Response ◽

Unit Cell ◽

Cell Method ◽

Mechanical Field ◽

Wang Tiles ◽

Small Set ◽

Finite Element Meshes ◽

Heterogeneous Microstructures ◽

Periodic Unit Cell

In this contribution, we present the concept of Wang Tiles as a surrogate of the periodic unit cell method (PUC) for modelling of materials with heterogeneous microstructures and for synthesis of micro-mechanical fields. The concept is based on a set of specifically designed cells that compresses the stochastic microstructure into a small set of statistical volume elements – tiles. Tiles are placed side by side according to matching edges like in a game of domino. Opposite to the repeating pattern of PUC the Wang Tiles method with the stochastic tiling algorithm preserves the randomness for reconstructed microstructures. The same process is applied to obtain the micro-mechanical response of domains where the evaluation as one piece would be time consuming. Therefore the micro-mechanical quantities are evaluated only on tiles (with surrounding layers of tiles of each addressed tile included into the evaluation) and then synthesized to the micro-mechanical field of whole domain.

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Compression of Heterogeneous Material Systems Based on Wang Tilings

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.592-593.149 ◽

2013 ◽

Vol 592-593 ◽

pp. 149-152 ◽

Cited By ~ 1

Author(s):

Martin Doškář ◽

Jan Novák

Keyword(s):

Spatial Orientation ◽

Random Media ◽

Heterogeneous Materials ◽

Heterogeneous Material ◽

Central Idea ◽

Synthesis Algorithm ◽

Planar Domains ◽

Wang Tiles ◽

Material Systems ◽

Finite Set

The present study is on the concept of modeling of heterogeneous materials by means of Wang tilings. The central idea is to store a microstructural information within a finite set of Wang Tiles, which allow for reconstructing heterogeneity patterns of random media in planar domains of arbitrary sizes. The particular objective of presented work is our automatic tile set designer in conjunction with stochastic tiling synthesis algorithm. The proposed methodology is demonstrated on different examples. The proximity of synthesized microstructures to reference media is explored by statistical descriptors and discussed in terms of parasitic spatial orientation orders that may occur.

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