Low-cost approximate reconstructing of heterogeneous microstructures

Abstract

We propose an approximate reconstruction of random heterogeneous microstructures using the two-exponent power-law (TEPL). This rule originates from the entropic descriptor (ED) that is a multi-scale measure of spatial inhomogeneity for a given microstructure. A digitized target sample is a cube of linear size L in voxels. Then, a number of trial configurations can be generated by a model of overlapping spheres of a fixed radius, which are randomly distributed on a regular lattice. The TEPL describes the averaged maximum of the ED as a function of the phase concentration and the radius. Thus, it can be used to determine the radius. The suggested approach is tested on surrogate samples of ceramic and carbonate. In each of the cases, fifty low-cost trials provided a few good enough candidates to a selection of the optimal reconstruction. When a better accuracy is planned, the final reconstructions can serve as the starting configurations. Then, the resulting reconstructions should be competitive indeed to those starting with random configurations.Comment: In Figure 3, the repeated by mistake Fig. 2b is replaced by the correct Fig. 3b, published versio

Statistical Reconstruction of Microstructures Using Entropic Descriptors

“…Formerly, a specific starting configuration was randomly generated with the overlapping balls of a radius depending indirectly on the structure considered. Recently, we proposed an approximate reconstruction of random heterogeneous microstructures, using the two-exponent power-law of Olchawa et al (2016). This rule originates from the entropic descriptor that is a multi-scale measure of spatial inhomogeneity for a given microstructure.…”

“…If necessary, see the next section, also the quantitative evaluation of the statistical distance between such two curves can be obtained by minimizing a sum over length-scales of the squared proper differences. Exemplary low-cost but approximate reconstructions for ceramics and carbonate samples with the linear size L = 300 were presented by Olchawa et al (2016). At this stage, when a better accuracy is expected, one can use the final reconstructions as the starting configurations to the standard SA technique.…”

“…It describes quantitatively the average degree of spatial inhomogeneity (Piasecki 2000a;2000b). Its q-extension in Tsallis' spirit (Tsallis 1988;2016) can be found in (Piasecki et al 2002). It should be noticed that the usage of the hybrid pair {S  , CS} of EDs, where the CS measures spatial statistical complexity of Piasecki and Plastino (2010), improves the reconstructed binary microstructures.…”

“…The lower value of the cost function is the better statistical similarity between the target microstructure and its approximate quick reconstruction. The method was tested by Olchawa et al (2016) on surrogate microstructures for ceramics and carbonate samples, which were earlier and carefully reconstructed within the SA (Frączek et al arXiv:1508.03857v2). In each of the cases, among no more than fifty low-cost trials, one can select a few candidates, which are "good enough"; this point will be clarified later in Sect.…”

A Two-Stage Reconstruction of Microstructures with Arbitrarily Shaped Inclusions

“…Among the various methods of reconstruction can be mentioned characteristic approaches, e.g., using support vector machines [22], based on the genetic algorithm (GA) compared with the simulated annealing (SA) and with the maximum entropy (MaxEnt) technique [23], statistical entropic descriptors (ED) [24][25][26][27], composition, dispersion, and geometry descriptors [28], two-point correlation functions and cellular automaton [29], multi-point statistics [30,31], texture synthesis [32], watershed transform and cross-correlation function [33], supervised, generative, transfer or deep learning [34][35][36][37][38], a shape library containing morphologies of heterogeneities extracted from micro-computed tomography images [39], a morphological completeness analysis [40], a successive calculation of conditional probability for multi-phase materials with any level of complexity [41], the Laguerre tessellation for ceramic foams preferably with not spherical cells [42] or using a single SEM foam image and a hybrid algorithm to the pore-sphere packing problem [43], to name just a few of them.…”