# Department of Mechanics: Seminar: Abstract Zeman

#### Jan Zeman, Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague

- (jointly with Jan Novák and Anna Kučerová)

#### Modeling Random Heterogeneous Materials by Wang Tilings

Microstructure generation algorithms have become an irreplaceable tool to study the overall behavior of heterogeneous media. In general,
they can be classified as *reconstruction* and *compression-* based. The goal of the first class of the methods is to reproduce a microstructure corresponding to a given set of statistical descriptors. The objective of the compression algorithms is to replace the original (complex) microstructure with a simpler object, a statistically equivalent unit cell, which approximates the original media with a reasonable
accuracy. Here, the terms 'statistically equivalent' refers to the fact that the error induced by the approximation is quantified by the same statistical descriptors as in the first case.

Compression algorithms are particularly useful in multi-scale homogenization schemes, as they allow capturing the most dominant microstructure features at a feasible computational cost. Their main deficiency, however, remains in the selection of suitable conditions at the cell boundary. The usually adopted assumption of periodicity is known to highly influence the overall response, particularly for the problems characterized by localized fields.

In the present contribution, we discuss as yet another approach to microstructural compression, inspired by successful applications of Wang tiles in computer graphics and game industry. Wang tiles are square cells with distinct codes at their edges, that are not allowed to rotate when tiling is performed.

In this framework, we represent the microstructure by a square tile with edges marked by particular code, and the corresponding tile set created by permutation of the codes in all possible directions. On the basis of this representation, the complete microstructure can be generated by randomly gathering compatible members (compatibility on a contiguous edge codes).

The potential of this methodology is illustrated by compression of an artificial microstructure, corresponding to an equilibrium distribution of approximately 10,000 impenetrable equi-sized discs. The parameters of the compressed representation are the number and positions of disc centers in the reference unit cell and the set tiles, respectively. These are found by minimizing the difference between the one- and the two-point probability functions. We shall demonstrate that the non-periodic set provides substantially better approximation to the original structure, and removes to a great extent artificial regularities of its periodic counterpart.

We refer interested readers to [1] and [2] for additional details.