Statistical Reconstruction of Microstructures Using Entropic Descriptors

Abstract

We report a multiscale approach of broad applicability to stochastic reconstruction of multiphase materials, including porous ones. The approach devised uses an optimization method, such as the simulated annealing (SA) and the so-called entropic descriptors (EDs). For a binary pattern, they quantify spatial inhomogeneity or statistical complexity at discrete length scales. The EDs extract dissimilar structural information to that given by two-point correlation functions (CFs). Within the SA, we use an appropriate cost function consisting of EDs or comprised of EDs and CFs. It was found that the stochastic reconstruction is computationally efficient when we begin with a preliminary synthetic configuration having in part desirable features. Another option is low-cost approximate reconstructing of the entire multiphase medium beyond the SA technique. The information included in the target ED-curve was utilized for this purpose. For a given volume fraction, the low-cost trial microstructures can be generated in two ways. In the first one, applied to ceramics and carbonate samples, the interpenetrating spheres generate a number of trial configurations. In the second one, with phase-EDs, here used to the sandstone sample, the overlapping superspheres do it. Both methods use a radius determined from the EDs-linked two-exponent power-law. However, the supersphere deformation parameter allows controlling of the spatial inhomogeneity of prototypical microstructures. At last, even for a hypothetical ED-curve (under reasonable assumptions), the specific microstructure can be found, if it is realizable for a given volume fraction. In general, the EDs-based methods offer a compromise between computational efficiency and the accuracy of reconstructions.

Appendix

Omitting in formulas dimension d for simplicity and using Eq. (1), the definition of the spatial statistical complexity $C_{S}$ given by Eq. (5) can be rewritten as

$$\begin{aligned} C_S (k) = \frac{1}{\lambda (k)} \frac{[S_{\max } (k) - S(k)][S(k) - S_{\min } (k)]}{[S_{\max } (k) - S_{\min } (k)]} \equiv S_{\Delta } (k) \gamma (k) \end{aligned}$$

(A1)

where

$$\begin{aligned} \gamma (k) \equiv \frac{S(k) - S_{\min } (k)}{S_{\max } (k) - S_{\min } (k)}. \end{aligned}$$

(A2)

Thus, the $C_{S}$ can be treated as the $S_{\Delta }$ corrected by a factor $\gamma$ linear in S. However, the whole descriptor $C_{S}$ is a nonlinear function of S in contrast to the $S_{\Delta }$ alone that is linear in S. This has some meaning for improving the quality of the stochastic reconstructions when the hybrid cost function is employed.

On the other hand, for $0<\gamma (k)<1$, the power expansion of the function $[1 - \gamma (k)]^{-1}$ includes $\gamma$-terms of any order. Thus, taking into account all the components in the hypothetical series the simple relation is fulfilled at every scale k

$$\begin{aligned} S_{\Delta } (k) [1 + \gamma (k) + \gamma ^{2}(k) +\cdots ] = \frac{S_{\Delta } (k)}{1 - \gamma (k) } \equiv \frac{S_{\max } (k) - S_{\min } (k)}{\lambda (k)}. \end{aligned}$$

(A3)

The hybrid EDs-pair, $\{S_{\Delta }(k),~C_{S}(k)\}~\equiv ~\{S_{\Delta }(k),~S_{\Delta }(k) \gamma ~(k)\}$, has been frequently employed by the entropic method of multiscale statistical reconstruction. One can suppose that the next term of the third order in S, i.e. the $S_{\Delta }(k) \gamma ^{2}(k)$, may provide an additional information useful for the SA approach. At last, the slightly better structural accuracy could be potentially obtained by using the triplet of the hybrid EDs, $\{S_{\Delta }(k),S_{\Delta }(k) \gamma (k),\, S_{\Delta }(k)\gamma ^{2}(k)\}$.