Compressing random microstructures via stochastic Wang tilings

This Rapid Communication presents a stochastic Wang tiling-based technique to compress or reconstruct disordered microstructures on the basis of given spatial statistics. Unlike the existing approaches based on a single unit cell, it utilizes a finite set of tiles assembled by a stochastic tiling algorithm, thereby allowing to accurately reproduce long-range orientation orders in a computationally efficient manner. Although the basic features of the method are demonstrated for a two-dimensional particulate suspension, the present framework is fully extensible to generic multidimensional media.

Figure 1

(a) A Wang tile set W8/2-2 with edge length $\ell$ and codes $\{\alpha ,\beta ,\gamma ,\delta \}$. (b) Example of an aperiodic valid tiling.

Figure 2

(a) Reference two-phase medium of size $174\rho \times 174\rho$ formed by equilibrium distribution of 1300 equisized disks of volume fraction $26.8\%$ and (b) its two-point probability function $S_2$; $\rho$ is the disk radius.

Figure 3

Optimized microstructures and two point probability functions ${\tilde{S}}_2$ for PUC (a), (b) and set W18/3-3 (c), (d). The arrow in (c) denotes the periodic region due to the local character of tile placement in the CSHD tiling algorithm.

Figure 4

Comparison of ${\tilde{S}}_2$ in the $x_1$-$S_2$ plane for (a) tile set W8/2-2 with respect to the number of disks $n^{\mathrm{d}}$ involved and (b) different sets. (c) Comparison of different sets in terms of objective function (2) and with respect to the number of disks $n^{\mathrm{d}}$ involved. The curves in (b) are plotted for particular values of $n^{\mathrm{d}}$ highlighted in (c).

Figure 5

Building blocks of microstructure compression based on (a) PUC and (b) tile set W18/3-3 with 49 disks assigned to tile edges (gray) and interiors (white).

Figure 6

(a) Dependence of local extremes ${\widehat{S}}_2$ on the particular tile set. (b) Correlation of local peaks obtained from two-point probabilities of optimized microstructures and their predictions given by Eq. (4); $r$ in (b) denotes the Pearson correlation coefficient.