In this paper, we present the concept of the Wang tiles method that compresses the stochastic microstructure into a small set of statistical volume elements – tiles. These tiles are then used for modeling of materials with heterogeneous stochastic microstructures and to streamline the calculations on the micro-scale level. In the following text, we focus on the fluctuation fields that are obtained as the main tiling reconstructed from the micro-mechanical quantities evaluated on individual tiles. But because of the non-local character of mechanical quantities the synthesized fluctuation field contain jumps between adjacent tiles. To prevent this phenomenon, the nearest surrounding tiles are included into the evaluation for each tile from the main tiling. Then the mechanical response is solved on these small so-called local tilings and results for middle tiles are saved. The main tiling is then synthesized using these results and further can be utilized as an enrichment functions for the finite element method.

<p>In this paper we present our recent work focused on the analysis of the abilities of Wang Tiles method and Automatic tile design method to synthesize the micro-structure of cellular materials, especially particular type of metal foam.</p><p>Wang Tiles method stores and compress the micro-structure in a set of Wang Tiles and by the means of stochastic tiling algorithms the planar domain is reconstructed. The used tiles are created by the Automatic tile design method from respective number of small specimens extracted from the original micro-structure image. As an additional step the central areas of automatically designed tiles are patched to suppress the influence of repeating tile edges (and relevant tile quarters) on inducing artifacts. In the presented analysis the performance of raw and patched tiles of different sizes in conjunction of various tile sets is investigated.</p>

Abstract. 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.

In this paper we present our approach to estimate mechanical fields (strains, stresses or displacements) inside isotropic infinite body with isotropic inclusions. Solution can be obtained easily for inclusions with ellipsoidal-like shapes by means of J. D. Eshelby's analytical solution given in 1957. Unfortunately for other distinct shapes of inclusions there is no analytic solution and finite element analysis is quite time consuming option. In our work, we focus on prediction of mechanical response for inclusions in form of short cylinders (e.g. steel fibers in steel-fiber-reinforced concrete) by means of artificial neural network. Which if trained on sufficiently large set of reference examples can predict desired mechanical fields and achieve considerable speed-up at the cost of approximate solution.

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