Keywords: microstructure, Wang tilings, microscale fluctuation fields, stress field synthesis, Schur complement method, parallel computing.

This paper is devoted to the high performance analysis of the micro-scale quantities such as stresses, strains and displacements at strongly heterogeneous materials. We recall a recently reported technique based on Wang tilings [1] that instead of an abrupt evaluation of the micro-scopic fields in entire macro-scopic domains uses a small set of statistical volume elements - tiles, from which the fields are synthesized using stochastic tiling algorithms [2].

The proposed methodology is demostrated on tilings synthesized from the set that contains only eight different tiles. As the sought fields are nonlocal, at least the nearest neighbours of each tile in the tiling must be taken into account. Therefore, thousands of micro-scale problems, square tilings of nine tiles in the simplest setting, have to be solved. At this time, structured discretizations arising from a raster image representation of microstructures are used and solved efficiently using the Moulinec-Suquet fast Fourier transform (FFT) algorithm [3].

Keeping up with the fact that there are only eight different tiles, each may be discretized by a finite element mesh with a regular distribution of nodes on the edges. It means, if any of two tiles are placed side by side in either spatial direction, their finite element meshes are conforming. Hence, the solution to all admissible micro-scale problems can be obtained effectively using the Schur complement method, which is the domain decomposition method of a non-overlapping type.

Regarding the independent solution of all admissible tilings, the problem is ideal for processor farming, since the number of processors is usually significantly smaller. All available processors will share the computational effort when tackling the basic Schur complements with very fine meshes.

We are fully aware of the fact that these outcomes are rather provisional. So far, no numerical experiments have been executed. However, on the basis of the theoretical analysis above we conjecture that the method of Wang tilings applied to the synthesis of microstructural fluctuation fields brings a significant compression of the data, still preserving the important features of underlying microstructures, and the analysis of the task complexity by means of the conventional Schur method proves its feasibility.

1

H. Wang, "Proving theorems by pattern recognition-II", Bell Systems Technical Journal, 1-41, ISSN 0005-8580, 1961.

2

M.F. Cohen, J. Shade, S. Hiller, O. Deussen, "Wang tiles for image and texture generation", ACM Transactions on Graphics, Citeseer, 287-294, ISSN 0730-0301, 2003.

3

H. Moulinec, P. Suquet, "A fast numerical method for computing the linear and nonlinear mechanical properties of composites", Comptes rendus de l'Académie des sciences. Série II, Mécanique, physique, chimie, astronomie, 1417-1423, 1994.

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