Lineal-path function for random heterogeneous materials

Generating 3D images of material microstructures from a single 2D image: a denoising diffusion approach

Three-dimensional (3D) images provide a comprehensive view of material microstructures, enabling numerical simulations unachievable with two-dimensional (2D) imaging alone. However, obtaining these 3D images can be costly and constrained by resolution limitations. We introduce a novel method capable of generating large-scale 3D images of material microstructures, such as metal or rock, from a single 2D image. Our approach circumvents the need for 3D image data while offering a cost-effective, high-resolution alternative to existing imaging techniques. Our method combines a denoising diffusion probabilistic model with a generative adversarial network framework. To compensate for the lack of 3D training data, we implement chain sampling, a technique that utilizes the 3D intermediate outputs obtained by reversing the diffusion process. During the training phase, these intermediate outputs are guided by a 2D discriminator. This technique facilitates our method's ability to gradually generate 3D images that accurately capture the geometric properties and statistical characteristics of the original 2D input. This study features a comparative analysis of the 3D images generated by our method, SliceGAN (the current state-of-the-art method), and actual 3D micro-CT images, spanning a diverse set of rock and metal types. The results shown an improvement of up to three times in the Frechet inception distance score, a typical metric for evaluating the performance of image generative models, and enhanced accuracy in derived properties compared to SliceGAN. The potential of our method to produce high-resolution and statistically representative 3D images paves the way for new applications in material characterization and analysis domains.

Pair-Variational Autoencoders for Linking and Cross-Reconstruction of Characterization Data from Complementary Structural Characterization Techniques

In materials research, structural characterization often requires multiple complementary techniques to obtain a holistic morphological view of a synthesized material. Depending on the availability and accessibility of the different characterization techniques (e.g., scattering, microscopy, spectroscopy), each research facility or academic research lab may have access to high-throughput capability in one technique but face limitations (sample preparation, resolution, access time) with other technique(s). Furthermore, one type of structural characterization data may be easier to interpret than another (e.g., microscopy images are easier to interpret than small-angle scattering profiles). Thus, it is useful to have machine learning models that can be trained on paired structural characterization data from multiple techniques (easy and difficult to interpret, fast and slow in data collection or sample preparation) so that the model can generate one set of characterization data from the other. In this paper we demonstrate one such machine learning workflow, Pair-Variational Autoencoders (PairVAE), that works with data from small-angle X-ray scattering (SAXS) that present information about bulk morphology and images from scanning electron microscopy (SEM) that present two-dimensional local structural information on the sample. Using paired SAXS and SEM data of newly observed block copolymer assembled morphologies [open access data from Doerk G. S.; et al. Sci. Adv.2023, 9 ( (2), ), eadd3687], we train our PairVAE. After successful training, we demonstrate that the PairVAE can generate SEM images of the block copolymer morphology when it takes as input that sample's corresponding SAXS 2D pattern and vice versa. This method can be extended to other soft material morphologies as well and serves as a valuable tool for easy interpretation of 2D SAXS patterns as well as an engine for generating ensembles of similar microscopy images to create a database for other downstream calculations of structure-property relationships.

Quantifying microstructures of earth materials using higher-order spatial correlations and deep generative adversarial networks

The key to most subsurface processes is to determine how structural and topological features at small length scales, i.e., the microstructure, control the effective and macroscopic properties of earth materials. Recent progress in imaging technology has enabled us to visualise and characterise microstructures at different length scales and dimensions. However, one limitation of these technologies is the trade-off between resolution and sample size (or representativeness). A promising approach to this problem is image reconstruction which aims to generate statistically equivalent microstructures but at a larger scale and/or additional dimension. In this work, a stochastic method and three generative adversarial networks (GANs), namely deep convolutional GAN (DCGAN), Wasserstein GAN with gradient penalty (WGAN-GP), and StyleGAN2 with adaptive discriminator augmentation (ADA), are used to reconstruct two-dimensional images of two hydrothermally rocks with varying degrees of complexity. For the first time, we evaluate and compare the performance of these methods using multi-point spatial correlation functions-known as statistical microstructural descriptors (SMDs)-ultimately used as external tools to the loss functions. Our findings suggest that a well-trained GAN can reconstruct higher-order, spatially-correlated patterns of complex earth materials, capturing underlying structural and morphological properties. Comparing our results with a stochastic reconstruction method based on a two-point correlation function, we show the importance of coupling training/assessment of GANs with higher-order SMDs, especially in the case of complex microstructures. More importantly, by quantifying original and reconstructed microstructures via different GANs, we highlight the interpretability of these SMDs and show how they can provide valuable insights into the spatial patterns in the synthetic images, allowing us to detect common artefacts and failure cases in training GANs.

Quantifying microstructural evolution via time-dependent reduced-dimension metrics based on hierarchical n-point polytope functions

We devise reduced-dimension metrics for effectively measuring the distance between two points (i.e., microstructures) in the microstructure space and quantifying the pathway associated with microstructural evolution, based on a recently introduced set of hierarchical n-point polytope functions P_{n}. The P_{n} functions provide the probability of finding particular n-point configurations associated with regular n polytopes in the material system, and are a special subset of the standard n-point correlation functions S_{n} that effectively decompose the structural features in the system into regular polyhedral basis with different symmetries. The nth order metric Ω_{n} is defined as the L_{1} norm associated with the P_{n} functions of two distinct microstructures. By choosing a reference initial state (i.e., a microstructure associated with t_{0}=0), the Ω_{n}(t) metrics quantify the evolution of distinct polyhedral symmetries and can in principle capture emerging polyhedral symmetries that are not apparent in the initial state. To demonstrate their utility, we apply the Ω_{n} metrics to a two-dimensional binary system undergoing spinodal decomposition to extract the phase separation dynamics via the temporal scaling behavior of the corresponding Ω_{n}(t), which reveals mechanisms governing the evolution. Moreover, we employ Ω_{n}(t) to analyze pattern evolution during vapor deposition of phase-separating alloy films with different surface contact angles, which exhibit rich evolution dynamics including both unstable and oscillating patterns. The Ω_{n} metrics have potential applications in establishing quantitative processing-structure-property relationships, as well as real-time processing control and optimization of complex heterogeneous material systems.

Understanding degeneracy of two-point correlation functions via Debye random media

It is well known that the degeneracy of two-phase microstructures with the same volume fraction and two-point correlation function S_{2}(r) is generally infinite. To elucidate the degeneracy problem explicitly, we examine Debye random media, which are entirely defined by a purely exponentially decaying two-point correlation function S_{2}(r). In this work, we consider three different classes of Debye random media. First, we generate the "most probable" class using the Yeong-Torquato construction algorithm [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)1063-651X10.1103/PhysRevE.57.495]. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function. We structurally discriminate these three classes of Debye random media from one another by ascertaining their other statistical descriptors, including the pore-size, surface correlation, chord-length probability density, and lineal-path functions. We also compare and contrast the percolation thresholds as well as the diffusion and fluid transport properties of these degenerate Debye random media. We find that these three classes of Debye random media are generally distinguished by the aforementioned descriptors, and their microstructures are also visually distinct from one another. Our work further confirms the well-known fact that scattering information is insufficient to determine the effective physical properties of two-phase media. Additionally, our findings demonstrate the importance of the other two-point descriptors considered here in the design of materials with a spectrum of physical properties.

Three-dimensional reconstruction of porous media using super-dimension-based adjacent block-matching algorithm

As porous media play an essential role in a variety of industrial applications, it is essential to understand their physical properties. Nowadays, the super-dimensional (SD) reconstruction algorithm is used to stochastically reconstruct a three-dimensional (3D) structure of porous media from a given two-dimensional image. This algorithm exhibits superiority in accuracy compared with classical algorithms because it learns information from the real 3D structure. However, owing to the short development time of the SD algorithm, it also has some limitations, such as inexact porosity characterization, long run time, blocking artifacts, and suboptimal accuracy that may be improved. To mitigate these limitations, this study presents the design of a special template that contains two parts of data (i.e., adjacent blocks and a central block); the proposed method matches adjacent blocks during reconstruction and assigns the matched central block to the area to be reconstructed. Furthermore, we design two important mechanisms during reconstruction: one for block matching and the other for porosity control. To verify the effectiveness of the proposed method compared with an existing SD method, both methods were tested on silica particle material and three homogeneous sandstones with different porosities; meanwhile, we compared the proposed method with a multipoint statistics method and a simulated annealing method. The reconstructed results were then compared with the target both visually and quantitatively. The experimental results indicate that the proposed method can overcome the aforementioned limitations and further improve the accuracy of existing methods. This method achieved 4-6 speedup factor compared with the traditional SD method.

Generation and structural characterization of Debye random media

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semianalytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks) and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here.

Probing information content of hierarchical n-point polytope functions for quantifying and reconstructing disordered systems

Disordered systems are ubiquitous in physical, biological, and material sciences. Examples include liquid and glassy states of condensed matter, colloids, granular materials, porous media, composites, alloys, packings of cells in avian retina, and tumor spheroids, to name but a few. A comprehensive understanding of such disordered systems requires, as the first step, systematic quantification, modeling, and representation of the underlying complex configurations and microstructure, which is generally very challenging to achieve. Recently, we introduced a set of hierarchical statistical microstructural descriptors, i.e., the "n-point polytope functions" P_{n}, which are derived from the standard n-point correlation functions S_{n}, and successively included higher-order n-point statistics of the morphological features of interest in a concise, explainable, and expressive manner. Here we investigate the information content of the P_{n} functions via optimization-based realization rendering. This is achieved by successively incorporating higher-order P_{n} functions up to n=8 and quantitatively assessing the accuracy of the reconstructed systems via unconstrained statistical morphological descriptors (e.g., the lineal-path function). We examine a wide spectrum of representative random systems with distinct geometrical and topological features. We find that, generally, successively incorporating higher-order P_{n} functions and, thus, the higher-order morphological information encoded in these descriptors leads to superior accuracy of the reconstructions. However, incorporating more P_{n} functions into the reconstruction also significantly increases the complexity and roughness of the associated energy landscape for the underlying stochastic optimization, making it difficult to convergence numerically.

Super-resolution of real-world rock microcomputed tomography images using cycle-consistent generative adversarial networks

Digital rock imaging plays an important role in studying the microstructure and macroscopic properties of rocks, where microcomputed tomography (MCT) is widely used. Due to the inherent limitations of MCT, a balance should be made between the field of view (FOV) and resolution of rock MCT images-a large FOV at low resolution (LR) or a small FOV at high resolution (HR). However, large FOV and HR are both expected for reliable analysis results in practice. Super-resolution (SR) is an effective solution to break through the mutual restriction between the FOV and resolution of rock MCT images, for it can reconstruct an HR image from a LR observation. Most of the existing SR methods cannot produce satisfactory HR results on real-world rock MCT images. One of the main reasons for this is that paired images are usually needed to learn the relationship between LR and HR rock images. However, it is challenging to collect such a dataset in a real scenario. Meanwhile, the simulated datasets may be unable to accurately reflect the model in actual applications. To address these problems, we propose a cycle-consistent generative adversarial network (CycleGAN)-based SR approach for real-world rock MCT images, namely, SRCycleGAN. In the off-line training phase, a set of unpaired rock MCT images is used to train the proposed SRCycleGAN, which can model the mapping between rock MCT images at different resolutions. In the on-line testing phase, the resolution of the LR input is enhanced via the learned mapping by SRCycleGAN. Experimental results show that the proposed SRCycleGAN can greatly improve the quality of simulated and real-world rock MCT images. The HR images reconstructed by SRCycleGAN show good agreement with the targets in terms of both the visual quality and the statistical parameters, including the porosity, the local porosity distribution, the two-point correlation function, the lineal-path function, the two-point cluster function, the chord-length distribution function, and the pore size distribution. Large FOV and HR rock MCT images can be obtained with the help of SRCycleGAN. Hence, this work makes it possible to generate HR rock MCT images that exceed the limitations of imaging systems on FOV and resolution.

Development and Application of the Statistically Similar Representative Volume Element for Numerical Modelling of Multiphase Materials

Modern aerospace, automotive and construction industries rely on materials with non-homogeneous properties like composites or multiphase structures. Such materials offer a lot of advantages, but they also require application of advanced numerical models of exploitation condition, which are of high importance for designers, architects and engineers. However, computational cost is one of the most important problems in this approach, being very high and sometimes unacceptable. In this paper we propose approach based on Statistically Similar Representative Volume Element (SSRVE), which is generated by combination of isogeometric analysis and optimization methods. The proposed solution significantly decreases computational cost of complex multiscale simulations and simultaneously maintains high reliability of solvers. At first, the motivation of the work is described in introduction, which is followed by general idea of the SSRVE as a modelling technique. Afterwards, examples of generated SSRVEs based on two different cases are given and passed further to numerical simulations of exploitation conditions. The results obtained from these calculations are used in the model predicting gradients of material properties, which are crucial results for discussion on uniqueness of the proposed solution. Additionally, some aspects of computational cost reduction are discussed, as well.

Confinement and activity regulate bacterial motion in porous media

Understanding how bacteria move in porous media is critical to applications in healthcare, agriculture, environmental remediation, and chemical sensing. Recent work has demonstrated that E. coli, which moves by run-and-tumble dynamics in a homogeneous medium, exhibits a new form of motility when confined in a disordered porous medium: hopping-and-trapping motility, in which cells perform rapid, directed hops punctuated by intervals of slow, undirected trapping. Here, we use direct visualization to shed light on how these processes depend on pore-scale confinement and cellular activity. We find that hopping is determined by pore-scale confinement, and is independent of cellular activity; by contrast, trapping is determined by the competition between pore-scale confinement and cellular activity, as predicted by an entropic trapping model. These results thus help to elucidate the factors that regulate bacterial motion in porous media, and could help aid the development of new models of motility in heterogeneous environments.

Reconstruction of porous media from extremely limited information using conditional generative adversarial networks

Porous media are ubiquitous in both nature and engineering applications. Therefore, their modeling and understanding is of vital importance. In contrast to direct acquisition of three-dimensional (3D) images of this type of medium, obtaining its subregion (s) such as 2D images or several small areas can be feasible. Therefore, reconstructing whole images from limited information is a primary technique in these types of cases. Given data in practice cannot generally be determined by users and may be incomplete or only partially informed, thus making existing reconstruction methods inaccurate or even ineffective. To overcome this shortcoming, in this study we propose a deep-learning-based framework for reconstructing full images from their much smaller subareas. In particular, conditional generative adversarial network is utilized to learn the mapping between the input (a partial image) and output (a full image). To ensure the reconstruction accuracy, two simple but effective objective functions are proposed and then coupled with the other two functions to jointly constrain the training procedure. Because of the inherent essence of this ill-posed problem, a Gaussian noise is introduced for producing reconstruction diversity, thus enabling the network to provide multiple candidate outputs. Our method is extensively tested on a variety of porous materials and validated by both visual inspection and quantitative comparison. It is shown to be accurate, stable, and even fast (∼0.08 s for a 128×128 image reconstruction). The proposed approach can be readily extended by, for example, incorporating user-defined conditional data and an arbitrary number of object functions into reconstruction, while being coupled with other reconstruction methods.

Dictionary optimization and constraint neighbor embedding-based dictionary mapping for superdimension reconstruction of porous media

The three-dimensional (3D) structure of a digital core can be reconstructed from a single two-dimensional (2D) image via mathematical modeling. In classical mathematical modeling algorithms, such as multipoint geostatistics algorithms, the optimization of pattern sets (dictionaries) and the mapping problems are important issues. However, they have rarely been discussed thus far. Pattern set (dictionary)-related problems include the pattern set (dictionary) size problem and the one-to-many mapping problem in a pattern set (dictionary). The former directly affects the completeness of the dictionary, while the latter is manifested such that a single to-be-matched 2D patch has multiple matching patterns in the library and it is hence necessary to select these modes to establish an optimal mapping relationship. Whether the two above-mentioned problems can be properly resolved is directly related to the accuracy of the reconstruction results. Super-dimension reconstruction is a new 3D reconstruction method proposed by introducing the concepts of training dictionary, prior model, and mapping into the reconstruction of the digital core from the field of super-resolution reconstruction. In addition, mapping relationship extraction and dictionary building are also key issues in super-dimension reconstruction. Therefore, this paper discusses these common dictionary-related problems from the perspective of super-dimension dictionaries. We propose dictionary optimization using augmentation dictionaries and clustering based on the boundary features of the dictionary elements to improve the completeness and expand the expression ability of the dictionary. Furthermore, we propose constraint neighbor embedding-based dictionary mapping to establish a more reasonable dictionary mapping relationship for super-dimension reconstruction, and we solve the one-to-many mapping problem in the dictionary. Our experimental results show that the performance of the super-dimension dictionary can be improved by the above-mentioned algorithm. Thus, through the optimized dictionary structure and mapping relationship determined by the above-mentioned methods, the 2D patch to be reconstructed can match a more accurate 3D block in the dictionary. Consequently, the reconstruction precision is improved.

Correlation-driven direct sampling method for geostatistical simulation and training image evaluation

Multiple-point geostatistics (MPS) is a competitive algorithm that produces a set of geologically realistic models. Viewing a training image (TI) as a prior model, MPS extracts patterns from the TI and reproduces patterns which are compatible with the hard data (HD). However, two challenges within the MPS framework are the geologically complex simulation and the TI evaluation. With the objective to achieve a high-quality simulation, we explore a way to address these two issues. First, correlation-driven direct sampling (CDS) is proposed to realize geostatistical simulation. We introduce the correlation-driven distance as a measure of similarity between two patterns. The weights in our distance measurement are derived by the concepts of the ellipse, correlation coefficient, Gaussian function, and affine transformation. Second, we fulfill TI evaluation on the basis of the consistency between TI and HD. Inspired by CDS, the minimum correlation-driven distance (MCD) is proposed to improve the evaluation accuracy. We suggest a conditioning pattern extraction history strategy to speed up the evaluation program. Third, the local consistency is presented to address nonstationarity. The program automatically divides the simulation domain into several subareas. A two-dimensional (2D) channelized reservoir image and a three-dimensional (3D) rock image are used to validate our proposed method. In comparison with previous methods, CDS yields better simulation quality. The further applications include a set of 2D TI evaluations and a 3D simulation domain segmentation. MCD exhibits sensible evaluation accuracy, excellent computational efficiency, and the ability to deal with nonstationarity.

Microstructural damage sensitivity prediction using spatial statistics

The vast compositional space of metallic materials provides ample opportunity to design stronger, more ductile and cheaper alloys. However, the substantial complexity of deformation micro-mechanisms makes simulation-based prediction of microstructural performance exceedingly difficult. In absence of predictive tools, tedious experiments have to be conducted to screen properties. Here, we develop a purely empirical model to forecast microstructural performance in advance, bypassing these challenges. This is achieved by combining in situ deformation experiments with a novel methodology that utilizes n-point statistics and principle component analysis to extract key microstructural features. We demonstrate this approach by predicting crack nucleation in a complex dual-phase steel, achieving substantial predictive ability (84.8% of microstructures predicted to crack, actually crack), a substantial improvement upon the alternate simulation-based approaches. This significant accuracy illustrates the utility of this alternate approach and opens the door to a wide range of alloy design tools.

A Transfer Learning Approach for Microstructure Reconstruction and Structure-property Predictions

Stochastic microstructure reconstruction has become an indispensable part of computational materials science, but ongoing developments are specific to particular material systems. In this paper, we address this generality problem by presenting a transfer learning-based approach for microstructure reconstruction and structure-property predictions that is applicable to a wide range of material systems. The proposed approach incorporates an encoder-decoder process and feature-matching optimization using a deep convolutional network. For microstructure reconstruction, model pruning is implemented in order to study the correlation between the microstructural features and hierarchical layers within the deep convolutional network. Knowledge obtained in model pruning is then leveraged in the development of a structure-property predictive model to determine the network architecture and initialization conditions. The generality of the approach is demonstrated numerically for a wide range of material microstructures with geometrical characteristics of varying complexity. Unlike previous approaches that only apply to specific material systems or require a significant amount of prior knowledge in model selection and hyper-parameter tuning, the present approach provides an off-the-shelf solution to handle complex microstructures, and has the potential of expediting the discovery of new materials.

Improved multipoint statistics method for reconstructing three-dimensional porous media from a two-dimensional image via porosity matching

Reconstructing a three-dimensional (3D) structure from a single two-dimensional training image (TI) is a challenging issue. Multiple-point statistics (MPS) is an effective method to solve this problem. However, in the traditional MPS method, errors occur while statistical features of reconstruction, such as porosity, connectivity, and structural properties, deviate from those of TI. Due to the MPS reconstruction mechanism that the voxel being reconstructed is dependent on the reconstructed voxel, it may cause error accumulation during simulations, which can easily lead to a significant difference between the real 3D structure and the reconstructed result. To reduce error accumulation and improve morphological similarity, an improved MPS method based on porosity matching is proposed. In the reconstruction, we search the matching pattern in the TI directly. Meanwhile, a multigrid approach is also applied to capture the large-scale structures of the TI. To demonstrate its superiority over the traditional MPS method, our method is tested on different sandstone samples from many aspects, including accuracy, stability, generalization, and flow characteristics. Experimental results show that the reconstruction results by the improved MPS method effectively match the CT sandstone samples in correlation functions, local porosity distribution, morphological parameters, and permeability.

Accurate modeling and evaluation of microstructures in complex materials

Accurate characterization of heterogeneous materials is of great importance for different fields of science and engineering. Such a goal can be achieved through imaging. Acquiring three- or two-dimensional images under different conditions is not, however, always plausible. On the other hand, accurate characterization of complex and multiphase materials requires various digital images (I) under different conditions. An ensemble method is presented that can take one single (or a set of) I(s) and stochastically produce several similar models of the given disordered material. The method is based on a successive calculating of a conditional probability by which the initial stochastic models are produced. Then, a graph formulation is utilized for removing unrealistic structures. A distance transform function for the Is with highly connected microstructure and long-range features is considered which results in a new I that is more informative. Reproduction of the I is also considered through a histogram matching approach in an iterative framework. Such an iterative algorithm avoids reproduction of unrealistic structures. Furthermore, a multiscale approach, based on pyramid representation of the large Is, is presented that can produce materials with millions of pixels in a matter of seconds. Finally, the nonstationary systems-those for which the distribution of data varies spatially-are studied using two different methods. The method is tested on several complex and large examples of microstructures. The produced results are all in excellent agreement with the utilized Is and the similarities are quantified using various correlation functions.

Process-Structure Linkages Using a Data Science Approach: Application to Simulated Additive Manufacturing Data

A novel data science workflow is developed and demonstrated to extract process-structure linkages (i.e., reduced-order model) for microstructure evolution problems when the final microstructure depends on (simulation or experimental) processing parameters. This workflow consists of four main steps: data pre-processing, microstructure quantification, dimensionality reduction, and extraction/validation of process-structure linkages. Methods that can be employed within each step vary based on the type and amount of available data. In this paper, this data-driven workflow is applied to a set of synthetic additive manufacturing microstructures obtained using the Potts-kinetic Monte Carlo (kMC) approach. Additive manufacturing techniques inherently produce complex microstructures that can vary significantly with processing conditions. Using the developed workflow, a low-dimensional data-driven model was established to correlate process parameters with the predicted final microstructure. Additionally, the modular workflows developed and presented in this work facilitate easy dissemination and curation by the broader community.

Efficient three-phase reconstruction of heterogeneous material from 2D cross-sections via phase-recovery algorithm

Digital reconstruction of a complex heterogeneous media from the limited statistical information, mostly provided by different imaging techniques, is the key to the successful computational analysis of this important class of materials. In this study, a novel approach is presented for three-dimensional (3D) reconstruction of a three-phase microstructure from its statistical information provided by two-dimensional (2D) cross-sections. In this three-step method, first two-point correlation functions (TPCFs) are extracted from the cross-section(s) using a spectral method suitable for the three-phase media. In the next step, 3D TPCFs are approximated for all vectors in a representative volume element (RVE). Finally, the 3D microstructure is realized from the full-set TPCFs obtained in the previous step, using a modified phase-recovery algorithm. The method is generally applicable to any complex three-phase media, here illustrated for an SOFC anode microstructure. The capabilities and shortcomings of the method are then investigated by performing a qualitative comparison between example cross-sections obtained computationally and their experimental equivalents. Finally, it is shown that the method almost conserves key microstructural properties of the media including tortuosity, percolation and three-phase boundary length (TPBL).

Dynamic reconstruction of heterogeneous materials and microstructure evolution

Reconstructing heterogeneous materials from limited structural information has been a topic that attracts extensive research efforts and still poses many challenges. The Yeong-Torquato procedure is one of the most popular reconstruction techniques, in which the material reconstruction problem based on a set of spatial correlation functions is formulated as a constrained energy minimization (optimization) problem and solved using simulated annealing [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)]. The standard two-point correlation function S2 has been widely used in reconstructions, but can also lead to large structural degeneracy for certain nearly percolating systems. To improve reconstruction accuracy and reduce structural degeneracy, one can successively incorporate additional morphological information (e.g., nonconventional or higher-order correlation functions), which amounts to reshaping the energy landscape to create a deep (local) energy minimum. In this paper, we present a dynamic reconstruction procedure that allows one to use a series of auxiliary S2 to achieve the same level of accuracy as those incorporating additional nonconventional correlation functions. In particular, instead of randomly sampling the microstructure space as in the simulated annealing scheme, our procedure utilizes a series of auxiliary microstructures that mimic a physical structural evolution process (e.g., grain growth). This amounts to constructing a series auxiliary energy landscapes that bias the convergence of the reconstruction to a favored (local) energy minimum. Moreover, our dynamic procedure can be naturally applied to reconstruct an actual microstructure evolution process. In contrast to commonly used evolution reconstruction approaches that separately generate individual static configurations, our procedure continuously evolves a single microstructure according to a time-dependent correlation function. The utility of our procedure is illustrated by successfully reconstructing nearly percolating hard-sphere packings and particle-reinforced composites as well as the coarsening process in a binary metallic alloy.

Stochastic reconstruction using multiple correlation functions with different-phase-neighbor-based pixel selection

A reconstruction methodology based on threshold energy based energy minimization (TA) and different-phase-neighbor (DPN)-based pixel swapping is presented. The TA method uses an energy threshold rather than probabilities as an acceptance criteria for annealing steps. The DPN-based pixel selection method gives priority to pixels which are segregated from clusters instead of random selection. An in-house solver has been developed to obtain two-dimensional reconstructions of heterogeneous two-phase mediums. Compared to conventional simulated annealing with random pixel swapping, the proposed method was found to achieve an optimal structure with up to an order of magnitude reduction in energy. When selecting a threshold tolerance value, the proposed method showed a 50% improvement in convergence time compared to conventional simulated annealing with random pixel swapping. The improved algorithm is used to study the effect of multiple correlation functions during the reconstruction. It was found that a combination of two-point correlation function and lineal path function for both phases results in most accurate reconstructions.

Reconstruction of three-dimensional porous media using a single thin section

The purpose of any reconstruction method is to generate realizations of two- or multiphase disordered media that honor limited data for them, with the hope that the realizations provide accurate predictions for those properties of the media for which there are no data available, or their measurement is difficult. An important example of such stochastic systems is porous media for which the reconstruction technique must accurately represent their morphology--the connectivity and geometry--as well as their flow and transport properties. Many of the current reconstruction methods are based on low-order statistical descriptors that fail to provide accurate information on the properties of heterogeneous porous media. On the other hand, due to the availability of high resolution two-dimensional (2D) images of thin sections of a porous medium, and at the same time, the high cost, computational difficulties, and even unavailability of complete 3D images, the problem of reconstructing porous media from 2D thin sections remains an outstanding unsolved problem. We present a method based on multiple-point statistics in which a single 2D thin section of a porous medium, represented by a digitized image, is used to reconstruct the 3D porous medium to which the thin section belongs. The method utilizes a 1D raster path for inspecting the digitized image, and combines it with a cross-correlation function, a grid splitting technique for deciding the resolution of the computational grid used in the reconstruction, and the Shannon entropy as a measure of the heterogeneity of the porous sample, in order to reconstruct the 3D medium. It also utilizes an adaptive technique for identifying the locations and optimal number of hard (quantitative) data points that one can use in the reconstruction process. The method is tested on high resolution images for Berea sandstone and a carbonate rock sample, and the results are compared with the data. To make the comparison quantitative, two sets of statistical tests consisting of the autocorrelation function, histogram matching of the local coordination numbers, the pore and throat size distributions, multiple-points connectivity, and single- and two-phase flow permeabilities are used. The comparison indicates that the proposed method reproduces the long-range connectivity of the porous media, with the computed properties being in good agreement with the data for both porous samples. The computational efficiency of the method is also demonstrated.

Grain reconstruction of porous media: application to a Bentheim sandstone

The two-point correlation measured on a thin section can be used to derive the probability density of the radii of a population of penetrable spheres. The geometrical, transport, and deformation properties of samples derived by this method compare well with the properties of the digitized real sample and of the samples generated by the standard grain reconstruction method.

A simple model for characterizing non-uniform fibre-based composites and networks

A mean-field model is presented that describes non-uniformities in the spatial distribution of fibres in networks and composites in terms of fluctuations in the local composition. The mean pore radius, specific surface area, lineal path function, and chord length probability density are expressed as functions of the fibre volume fraction within a heuristic formalism. The impact of statistical heterogeneities in the fibre distribution upon the elastic moduli is assessed within the semi-empirical Reuss-Voigt-Hill averaging scheme. Results from illustrative calculations suggest that such macroscopically averaged material properties are particularly sensitive to variations in the mean pore radius.

Statistically reconstructing continuous isotropic and anisotropic two-phase media while preserving macroscopic material properties

We propose a method to generate statistically similar reconstructions of two-phase media. As with previous work, we initially characterize the microstructure of the material using two-point correlation functions (a subset of spatial correlation functions) and then generate numerical reconstructions using a simulated annealing method that preserves the geometric relationships of the material's phase of interest. However, in contrast to earlier contributions that consider reconstructions composed of discrete arrays of pixels or voxels alone, we generate reconstructions based on assemblies of continuous, three-dimensional, interpenetrating objects. The result is a continuum description of the material microstructure (as opposed to a discretized or pixelated description), capable of efficiently representing large disparities in scale. Different reconstruction methods are considered based on distinct combinations of two-point correlation functions of varying degrees of complexity. The quality of the reconstruction methods are evaluated by comparing the total pore fraction, specific surface area of the percolating cluster, pore fraction of the percolating cluster, tortuosity, and permeability of the reconstructions to those of a set of reference assemblies. Elsewhere it has been proposed that two-phase media could be statistically reproduced with only two spatial correlation functions: the two-point probability function (the probability that two points lie within the same phase) and the lineal path function (the probability that a line between two points lies entirely within the same phase). We find that methods employing the two-point probability function and lineal path function are improved if the percolating cluster volume is also considered in the reconstruction. However, to reproduce more complicated geometric assemblies, we find it necessary to employ the two-point probability, two-point cluster, and lineal path function in addition to the percolating cluster volume to produce a generally accurate statistical reconstruction.

Microstructure reconstruction using entropic descriptors

A multi-scale approach to the inverse reconstruction of a pattern’s microstructure is reported. Instead of a correlation function, a pair of entropic descriptors (EDs) is proposed for a stochastic optimization method. The first of them measures a spatial inhomogeneity, for a binary pattern, or compositional one, for a greyscale image. The second one quantifies a spatial or compositional statistical complexity. The EDs reveal structural information that is dissimilar, at least in part, to that given by correlation functions at almost all discrete length scales. The method is tested on a few digitized binary and greyscale images. In each of the cases, the persuasive reconstruction of the microstructure is found.

Geometrical ambiguity of pair statistics. II. Heterogeneous media

In the first part of this series of two papers [Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 011105 (2010)], we considered the geometrical ambiguity of pair statistics associated with point configurations. Here we focus on the analogous problem for heterogeneous media (materials). Heterogeneous media are ubiquitous in a host of contexts, including composites and granular media, biological tissues, ecological patterns, and astrophysical structures. The complex structures of heterogeneous media are usually characterized via statistical descriptors, such as the n -point correlation function Sn. An intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two-point correlation function S2 of a target medium. Recently, general claims of the uniqueness of reconstructions using S2 have been made based on numerical studies, which implies that S2 suffices to uniquely determine the structure of a medium within certain numerical accuracy. In this paper, we provide a systematic approach to characterize the geometrical ambiguity of S2 for both continuous two-phase heterogeneous media and their digitized representations in a mathematically precise way. In particular, we derive the exact conditions for the case where two distinct media possess identical S2 , i.e., they form a degenerate pair. The degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations, respectively. By examining these equations and constructing their rigorous solutions for specific examples, we conclusively show that in general S2 is indeed not sufficient information to uniquely determine the structure of the medium, which is consistent with the results of our recent study on heterogeneous-media reconstruction [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. U.S.A. 106, 17634 (2009)]. The analytical examples include complex patterns composed of building blocks bearing the letter "T" and the word "WATER" as well as degenerate stacking variants of the densest sphere packing in three dimensions (Barlow films). Several numerical examples of degeneracy (e.g., reconstructions of polycrystal microstructures, laser-speckle patterns and sphere packings) are also given, which are virtually exact solutions of the degeneracy equations. The uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed, including two-point quantities that contain topological connectedness information about the phases.

A superior descriptor of random textures and its predictive capacity

Two-phase random textures abound in a host of contexts, including porous and composite media, ecological structures, biological media, and astrophysical structures. Questions surrounding the spatial structure of such textures continue to pose many theoretical challenges. For example, can two-point correlation functions be identified that can be manageably measured and yet reflect nontrivial higher-order structural information about the textures? We present a solution to this question by probing the information content of the widest class of different types of two-point functions examined to date using inverse "reconstruction" techniques. This enables us to show that a superior descriptor is the two-point cluster function C(2)(r), which is sensitive to topological connectedness information. We demonstrate the utility of C(2)(r) by accurately reconstructing textures drawn from materials science, cosmology, and granular media, among other examples. Our work suggests a theoretical pathway to predict the bulk physical properties of random textures and that also has important ramifications for atomic and molecular systems.

The evolution of robust development and homeostasis in artificial organisms

During embryogenesis, multicellular animals are shaped via cell proliferation, cell rearrangement, and apoptosis. At the end of development, tissue architecture is then maintained through balanced rates of cell proliferation and loss. Here, we take an in silico approach to look for generic systems features of morphogenesis in multicellular animals that arise as a consequence of the evolution of development. Using artificial evolution, we evolved cellular automata-based digital organisms that have distinct embryonic and homeostatic phases of development. Although these evolved organisms use a variety of strategies to maintain their form over time, organisms of different types were all found to rapidly recover from environmental damage in the form of wounds. This regenerative response was most robust in an organism with a stratified tissue-like architecture. An evolutionary analysis revealed that evolution itself contributed to the ability of this organism to maintain its form in the face of genetic and environmental perturbation, confirming the results of previous studies. In addition, the exceptional robustness of this organism to surface injury was found to result from an upward flux of cells, driven in part by cell divisions with a stable niche at the tissue base. Given the general nature of the model, our results lead us to suggest that many of the robust systems properties observed in real organisms, including scar-free wound-healing in well-protected embryos and the layered tissue architecture of regenerating epithelial tissues, may be by-products of the evolution of morphogenesis, rather than the direct result of selection.

During development, multicellular animals are shaped by cell proliferation, cell rearrangement, and cell death to generate an adult whose form is maintained over time. Disruption of this finely balanced state can have devastating consequences, including aging, psoriasis, and cancer. Typically, however, development is robust, so that animals achieve the same final form even when challenged by environmental damage such as wounding. To see how morphogenetic robustness arises, we have taken an in silico approach to evolve digital organisms that exhibit distinct phases of growth and homeostasis. During the homeostasis period, organisms were found to use a variety of strategies to maintain their form. Remarkably, however, all recovered from severe wounds, despite having evolved in the absence of selection pressure to do so. This ability to regenerate was most striking in an organism with a tissue-like architecture, where it was enhanced by a directional flux of cells that drives tissue turnover. This identifies a stratified architecture, like that seen in human skin and gut, as an evolutionarily accessible and robust form of tissue organisation, and suggests that wound-healing may be a general feature of evolved morphogenetic systems. Both may therefore contribute to homeostasis, wound-healing, and regeneration in real animals.

Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications

In the first part of this series of two papers, we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials (media) via two-point correlation functions S(2) and introduced an efficient heterogeneous-medium (re)construction algorithm called the "lattice-point" algorithm. Here we discuss the algorithmic details of the lattice-point procedure and an algorithm modification using surface optimization to further speed up the (re)construction process. The importance of the error tolerance, which indicates to what accuracy the media are (re)constructed, is also emphasized and discussed. We apply the algorithm to generate three-dimensional digitized realizations of a Fontainebleau sandstone and a boron-carbide/aluminum composite from the two-dimensional tomographic images of their slices through the materials. To ascertain whether the information contained in S(2) is sufficient to capture the salient structural features, we compute the two-point cluster functions of the media, which are superior signatures of the microstructure because they incorporate topological connectedness information. We also study the reconstruction of a binary laser-speckle pattern in two dimensions, in which the algorithm fails to reproduce the pattern accurately. We conclude that in general reconstructions using S(2) only work well for heterogeneous materials with single-scale structures. However, two-point information via S(2) is not sufficient to accurately model multiscale random media. Moreover, we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two-point correlation functions.

Random-walk approach to the d-dimensional disordered Lorentz gas

A correlated random walk approach to diffusion is applied to the disordered nonoverlapping Lorentz gas. By invoking the Lu-Torquato theory for chord-length distributions in random media [J. Chem. Phys. 98, 6472 (1993)], an analytic expression for the diffusion constant in arbitrary number of dimensions d is obtained. The result corresponds to an Enskog-like correction to the Boltzmann prediction, being exact in the dilute limit, and better or nearly exact in comparison to renormalized kinetic theory predictions for all allowed densities in d=2,3 . Extensive numerical simulations were also performed to elucidate the role of the approximations involved.