Lineal-path Function Research Articles

Influence of structural and thermophysical parameters of insulating aggregates on the effective thermal conductivity of lightweight concrete

Understanding how structural features influence effective thermal conductivity is an essential step towards the optimization of heterogeneous materials. Generally, thermal conductivity models link the internal structure to the effective thermal conductivity, with the volume fraction as the only parameter that takes into account the internal structure. Yet, the real internal structures of materials are often too complex to be accurately characterized by the volume fraction only. In this study, the influence of three parameters - the particle size, volume fraction, and thermal conductivity ratio of the constituents - on the effective thermal conductivity of concrete is evaluated. Two-point correlation and lineal path functions are used to characterize the internal structure generated by the Random Sequential Addition (RSA) model. A finite element model is developed to evaluate the effective thermal conductivity of concrete. The results demonstrate that the internal structure of concrete is dependent on the volume fraction of the inclusions, and also of the particle size. However, for volume fractions less than 20%, the effective thermal conductivity is dependent only of the volume fraction and thermal conductivity ratio of the constituents.

Accelerating multi-point statistics reconstruction method for porous media via deep learning

Reconstruction of three-dimensional (3D) porous media from a single two-dimensional (2D) image or limited morphological information has been an outstanding problem. Specifically, multi-point statistics (MPS)-based methods have been proved to be effective and promising to address this problem. Besides, recently sampling-based MPS methods that reconstruct 3D microstructure layer by layer have attracted increasing attention, for its relatively lower computational cost and memory demand. However, due to the nature that one has to search and match multi-point information in a data structure like binary tree or list for each simulation, it is very time-consuming, especially for large template used to reconstruct geometrically complex media. In this paper, to overcome this challenge, for the first time to the best of our knowledge, an accelerating reconstruction method based on deep learning is proposed. Specially, conditional generative adversarial networks (CGAN) is leveraged to model and characterize the relation between the sampling image which contains conditioning data and the target void-solid image. Unlike conventional MPS-based method generally performing simulation point by point, our method is able to directly translate the sampling image to the target one instantaneously, thus leading to a significant speedup factor (∼760 for 2D reconstruction and ∼25 for 3D reconstruction on CPU). Meanwhile, to assess the performance of our method, we test it on sandstone samples, and two-point correlation function, lineal-path function and two-point cluster function, as well as local porosity distribution are quantitatively compared with those of the target microstructure. The comparison indicates that our method is of high efficiency while preserving good agreement of statistical functions with that of the target system. We remark that our approach opens the door to a new methodology that incorporating machine learning/deep learning techniques into conventional methods to address CPU- or memory-demanding issues in a wide range of related fields.

3D Numerical Reconstruction of Porous Sandstone Using Improved Simulated Annealing Algorithms

In this paper, a 3D numerical reconstruction method is proposed based on improved simulated annealing algorithms and limited morphological information from experiments. Firstly, X-ray micro-CT experimental tests are applied to obtain the CT images and the mechanical parameters of Xingluokeng sandstone. Then, the voxel block exchanging algorithm is introduced to accelerate the preliminary reconstruction stage of the porous sandstone model. Two-point probability control function, lineal-path control function and fractal control function are employed to characterize complex pore morphologies. The system updating algorithm is improved to avoid wasting time on the unnecessary and unsuccessful position exchange. Finally, to verify the proposed method, the topological and mechanical properties of the reconstructed porous sandstone model are analyzed by a distance-ordered homotopic thinning algorithm as well as the numerical compression. Moreover, the numerical results of the reconstructed model are compared with the experimental data. It is found that the proposed method in this paper has a good ability to predict the properties of the rocks for engineering purposes.

Area of lineal-path function for describing the pore microstructures of cement paste and their relations to the mechanical properties simulated from μ-CT microstructures

The pore distribution of a cement paste strongly affects its mechanical behavior such as its stiffness and strength. Porosity is an influential parameter that can be used to identify the complex pore microstructures of cement paste, but it has limitations as a scalar parameter. In this study, the lineal-path function, a low-order probability function, is investigated as a supplement or an alternative parameter for describing the microstructural characteristics of cement paste microstructures. In particular, the area of the lineal-path function is used as a measure of the pore microstructural characteristics, which can be linked with its properties.A relatively new method for simulating crack propagation, the crack phase field model, is used to evaluate the stiffness and tensile strength of cement paste microstructures and the evaluated properties are linked to the proposed characterization parameters. The evaluation is performed on virtual specimens obtained from micro-level computerized tomography (μ-CT) images of real cement paste specimens. The validity of the microstructure-property relations obtained from the proposed characterization parameters and the crack phase field model are confirmed through the statistical analysis of dozens of specimens.It is concluded that the correlation between the area of the lineal-path function and the mechanical properties is very strong. The parameter could potentially be used as a supplementary or an alternative parameter to describe the pore microstructures of cement paste.

Reconstruction of three-dimensional heterogeneous media from a single two-dimensional section via co-occurrence correlation function

Accurately modeling and reconstructing three-dimensional (3D) microstructure of heterogeneous media from limited morphological information obtained from a single two-dimensional (2D) section remains an outstanding problem. Specially, based on simulated annealing (SA) procedure, two-point correlation function S2 is widely used to reconstruct various 3D microstructures in many different cases. However, two-point correlation function is a lower-order statistical descriptor and is generally not sufficient to accurately characterize a microstructure. From the perspective of texture description, a co-occurrence correlation function (CCF) based on gray-level co-occurrence matrix is presented, and four feature functions are derived from it and used as objective function in SA procedure. Moreover, the CCF can be both manageably measured from a single 2D image and embody non-trivial higher-order morphological information. Furthermore, to avoid completely re-computing the CCF in the reconstruction, a fast updating method for CCF is proposed, and thus accelerating the reconstruction process. The four feature functions are utilized to reconstruct different types of media, including square array of nonoverlapping disks, anode microstructure of solid oxide fuel cell and sandstone core. Meanwhile, the two-point correlation function reconstructions are used for comparison. Unconstrained functions including lineal-path function and two-point cluster function are used to quantitatively ascertain the accuracy of the reconstructions, and the comparison of run time is also demonstrated. The results indicate that the proposed method can accurately reconstruct the microstructures with correlation functions being in good agreement with those of the target.

Analyzing fracture properties of the 3D reconstructed model of porous rocks

The non-negligible fissures, cracks or pore structures, which are contained in rocks, have significant influences on their mechanical properties. In this paper, a novel method is developed to reconstruct a three-dimensional multiple phase rock model from the limited morphological information of Chongqing sandstone. Firstly, X-ray micro-CT experimental tests are applied to Chongqing sandstone to obtain the raw CT images and the mechanical parameters. Then, the two-point correlation function, lineal-path function and fractal function are combined to generate the reconstructed model. Next, the reconstructed model subjected to compressive loads is simulated by Finite Element-Smoothed Particle Hydrodynamics (FE-SPH) coupling method with a brittle model formulation. Finally, the initiation, propagation and coalescence of cracks are investigated during the numerical simulation. The topological and mechanical properties of the reconstructed model are carefully analyzed and compared with that of the reference model. The comparison of the numerical modeling results and the experimental data shows a good consistency. This indicates that the proposed method has the ability to predict macroscopic behaviors of rocks from limited experimental information.

Characterizing apple microstructure via directional statistical correlation functions

Our understanding of food is tightly related to the complex food microstructure. We introduce the use of statistical correlation functions to quantitatively describe the spatial distribution of cell and void phases of ‘Braeburn’ and ‘Kanzi’ apples. The lineal-path distribution function, L(r), and the two-point correlation function, S2(r), were measured from bi-dimensional (2D) microtomographic images. While the void fraction of both apples cultivars was similar, 0.840% and 0.853%, the pores of ‘Braeburn’ apples were bigger in size. Pores with an extension of 25μm were found in percentage of 12.6% and 4.3% for ‘Braeburn’ and ‘Kanzi’ respectively. The cell phase of ‘Braeburn’ apples was described by larger clusters as a result of a greater degree of connectivity among the individual cells. ‘Kanzi’ apple tissue was structured by more separated cell clusters due to the presence of small pores with a greater spatial distribution. The clusters showed a good homogeneity in shape for both varieties while the voids of ‘Kanzi’ apples appeared more inhomogeneous and elongated in one of the dimensions. The obtained structural information were employed to model tissue structures of apples. We found that the cell phase could be modeled by overlapping disks having a mean radius of 58 and 53mm for ‘Braeburn’ and ‘Kanzi’ apples, respectively. In this way macroscopic properties of apple tissue could be estimated precisely.

Packing of nonoverlapping cubic particles: Computational algorithms and microstructural characteristics.

Packing of cubic particles arises in a variety of problems, ranging from biological materials to colloids and the fabrication of new types of porous materials with controlled morphology. The properties of such packings may also be relevant to problems involving suspensions of cubic zeolites, precipitation of salt crystals during CO_{2} sequestration in rock, and intrusion of fresh water in aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We present a detailed simulation and microstructural characterization of packings of nonoverlapping monodisperse cubic particles, following up on our preliminary results [H. Malmir et al., Sci. Rep. 6, 35024 (2016)2045-232210.1038/srep35024]. A modification of the random sequential addition (RSA) algorithm has been developed to generate such packings, and a variety of microstructural descriptors, including the radial distribution function, the face-normal correlation function, two-point probability and cluster functions, the lineal-path function, the pore-size distribution function, and surface-surface and surface-void correlation functions, have been computed, along with the specific surface and mean chord length of the packings. The results indicate the existence of both spatial and orientational long-range order as the the packing density increases. The maximum packing fraction achievable with the RSA method is about 0.57, which represents the limit for a structure similar to liquid crystals.

Efficient computation of the angularly resolved chord length distributions and lineal path functions in large microstructure datasets

Chord length distributions (CLDs) and lineal path functions (LPFs) have been successfully utilized in prior literature as measures of the size and shape distributions of the important microscale constituents in the material system. Typically, these functions are parameterized only by line lengths, and thus calculated and derived independent of the angular orientation of the chord or line segment. We describe in this paper computationally efficient methods for estimating chord length distributions and lineal path functions for 2D (two dimensional) and 3D microstructure images defined on any number of arbitrary chord orientations. These so called fully angularly resolved distributions can be computed for over 1000 orientations on large microstructure images (5003 voxels) in minutes on modest hardware. We present these methods as new tools for characterizing microstructures in a statistically meaningful way.

Compression and reconstruction of random microstructures using accelerated lineal path function

Microstructure reconstruction and compression techniques are designed to identify microstructures with desired properties. While a microstructure reconstruction involves searching for a microstructure with prescribed statistical properties, a microstructure compression focuses on efficient representation of material morphology for the purpose of multiscale modelling. Successful application of these techniques, nevertheless, requires proper understanding of the underlying statistical descriptors quantifying morphology of a material. In this paper, we focus on a lineal path function designed to capture short-range effects and phase connectedness, which can hardly be handled by the commonly used two-point probability function. Usage of the lineal path function is, however, significantly limited because of huge computational requirements. So as to examine the properties of the lineal path function during computationally exhaustive compression and reconstruction processes, we start with an acceleration of the lineal path evaluation, namely by porting part of its code to a graphics processing unit using the CUDA (Compute Unified Device Architecture) programming environment. This allows us to present a unique comparison of the entire lineal path function with the commonly used rough approximation based on the Monte Carlo and/or sampling template. Moreover, this accelerated version of the lineal path function is then compared to the two-point probability function during the compression and reconstruction of two-phase morphologies. Their significant features are discussed and illustrated using a set of artificial periodic as well as real-world random microstructures.

Investigation of characteristics and responses of insulating cement paste specimens with Aer solids using X-ray micro-computed tomography

Insulating concrete is a type of material designed to enhance energy efficiency by reducing thermal conductivity. It contains numerous voids induced by the use of air-entraining admixtures or aggregates, and these voids strongly affect the characteristics and responses of the material. Thus, appropriate methods to examine the void distribution of materials are needed. Here, X-ray micro-computed tomography (micro-CT) is adopted to investigate the spatial distribution of voids in a material without destroying the specimen. By using micro-CT, a series of cross-sectional images with the micrometer range pixel size can be obtained. Then, void density contour and probabilistic description methods, such as two-point correlation and lineal-path functions, are used to describe the void distribution inside the specimen. In this study, Aer solids, small prefabricated hollow air bubbles, are used for insulating cement paste specimens to increase the insulating effect. The cement paste specimens with different void ratios and Aer solids are generated, and their void distribution characteristics are investigated using the methods mentioned above. The thermal and mechanical responses of the insulating specimens are also evaluated by experiments and numerical methods. The effect of Aer solids on the material characteristics and the relationship between void distribution and material responses are confirmed from the obtained results.

On the inverse problem of the reconstruction of food microstructure from limited statistical information. A study on bread

The possibility to reconstruct food microstructure by limited morphological information has fundamental importance for theoretical and practical applications. We implemented the simulated annealing method proposed by Yeong and Torquato (1998) for reconstructing bread structure through the information contained into the lineal-path distribution function, L(r), and two-point correlation function, S2(r). The method enabled the evolution of two-phase random image toward bread structure. When using the information of lineal-path distribution function, the generated images well captured the main morphological features of bread, although several deviations still existed. This was in accordance with the significant differences between the original and reconstructed images as measured by two-point correlation function. By hybrid reconstruction, based on both correlation functions, a better reconstruction in terms of both number and size of pores was obtained. In the future the use of more several statistical correlation functions could enable further improvement in reconstruction of bread microstructure.

Study on Multi-Objective Reconstruction of Random Media

This paper deals with a reconstruction of random media via multi-objective optimization. Two statistical descriptors, namely a two-point probability function and a two-point lineal path function, are repetitively evaluated for the original medium and the reconstructed image to appreciate the improvement in the optimization progress. Because of doubts of the weights setting in the weighted-sum method, purely multi-objective optimization routine Non-dominated Sorting Genetic Algorithm~II is utilized. Three operators are compared for creating new offspring populations that satisfy a prescribed volume fraction constraint. The main contribution is in the testing of the proposed methodology on several benchmark images.

Synthesis, characterization and quantitative analysis of porous metal microstructures: Application to microporous copper produced by solid state foaming

Porous metals can be created through a wide variety of processing techniques, and the pore morphology resulting from these processes is equally diverse. The structural and functional properties of metal foams are directly dependent on the size, shape, interconnectedness and volume fraction of pores, so accurately quantifying the pore characteristics is of great importance. Methods for analyzing porous materials are presented here and applied to a copper-based metallic foam generated through solid state foaming via oxide reduction and expansion. This process results in large voids (10s of microns) between sintered particles and small pores (10 microns to less than 50 nm) within particles. Optical and electron microscopy were used to image the porosity over this wide range, and the pore characteristics were quantified using image segmentation and statistical analysis. Two-dimensional pore analysis was performed using the Chan-Vese method, and two-point correlation and lineal path functions were used to assess three-dimensional reconstructions from FIB tomography. Two-dimensional analysis reveals distinct size and morphological differences in porosity between particles and within them. Three-dimensional analysis adds further information on the high level interconnectedness of the porosity and irregular shape it takes, forming tortuous pathways rather than spherical cells. Mechanical polishing and optical microscopy allow large areas to be created and analyzed quickly, but methods such as focused ion beam (FIB) sectioning can provide additional insight about microstructural features. In particular, after FIB milling is used to create a flat surface, that surface can be analyzed for structural and compositional information.

Stochastic microstructure characterization and reconstruction via supervised learning

Microstructure characterization and reconstruction have become indispensable parts of computational materials science. The main contribution of this paper is to introduce a general methodology for practical and efficient characterization and reconstruction of stochastic microstructures based on supervised learning. The methodology is general in that it can be applied to a broad range of microstructures (clustered, porous, and anisotropic). By treating the digitized microstructure image as a set of training data, we generically learn the stochastic nature of the microstructure via fitting a supervised learning model to it (we focus on classification trees). The fitted supervised learning model provides an implicit characterization of the joint distribution of the collection of pixel phases in the image. Based on this characterization, we propose two different approaches to efficiently reconstruct any number of statistically equivalent microstructure samples. We test the approach on five examples and show that the spatial dependencies within the microstructures are well preserved, as evaluated via correlation and lineal-path functions. The main advantages of our approach stem from having a compact empirically-learned model that characterizes the stochastic nature of the microstructure, which not only makes reconstruction more computationally efficient than existing methods, but also provides insight into morphological complexity.

Evaluation of effect of glass beads on thermal conductivity of insulating concrete using micro CT images and probability functions

Insulating concrete is a type of concrete that is designed to reduce thermal conductivity. Insulating concrete contains numerous voids that play an important role in reducing heat conduction. Therefore, appropriate nondestructive methods are required to examine the spatial distribution of voids and constituents in a concrete specimen. In this study, an insulating concrete specimen containing hollow glass beads to increase the insulating effect is adopted. Then, micro computed tomography (CT) is used to investigate the spatial distribution of the voids in this specimen. By using a micro CT device, a series of cross-sectional images of the specimen at micrometer-order pixel size are generated by X-rays. To quantitatively describe the spatial distribution of voids in the specimen, probability functions such as two-point correlation, lineal-path, and two-point cluster functions are adopted. In addition, the thermal conductivity of the specimen is evaluated using finite element simulation. The results clarify the insulating effect of glass beads on the concrete specimen and reveal a strong relationship between the probabilistic characteristics of the void distribution and the material responses of insulating concrete.

Prediction and Evaluation of Time-Dependent Effective Self-diffusivity of Water and Other Effective Transport Properties Associated with Reconstructed Porous Solids

We reconstructed pore structures of three porous solids that differ from each other in morphology and topology of pore space. To achieve this, we used a stochastic method based on simulated annealing and X-ray computed microtomography. Simulated annealing was constrained by the following microstructural descriptors sampled along the principal and diagonal directions: the two-point probability function for the void phase and the lineal-path functions for both void and solid phases. The stochastic method also assumed the isotropic pore structures in accordance with a recent paper (Capek et al. in Transp Porous Media 88(1): 87–106 (2011)). With the exception of the solid with the widest pores, we made tomographic volume images in high and low resolution, which enabled us to study the effect of resolution on microstructural descriptors and effective transport properties. A comparison of the two-point probability function and the lineal-path function sampled in the principal directions revealed that the pore structures derived from the tomographic volume images were slightly anisotropic, in opposition to the assumption of the stochastic method. Besides the anisotropy, other microstructural descriptors including the pore-size function and the total fraction of percolating cells indicated that the morphological and topological characteristics of the pore structures depended on the reconstruction method and its parameters. Particularly, the pore structures reproduced using the stochastic method contained wider pores than those obtained using X-ray tomography. Deviations between the pore structures derived from low- and high-resolution tomographic volume images were also observed and imputed to partial volume artefacts. Then, viscous flow of incompressible liquid, ordinary diffusion, Knudsen flow and self-diffusion of water in the reconstructed pore spaces were simulated. As counterparts, experimental data were measured by means of permeation and Wicke–Kallenbach cells and pulsed field gradient NMR. Deviations between the simulated quantities on the one hand and experimental data on the other hand were generally acceptable, which corroborated the pore-space models. As expected, the predictions based on the tomographic models of pore space were more successful than those derived from the stochastic models. The stationary effective transport properties, i.e. the effective permeability, the effective pore size and the geometric factor, were sensitive to a bias in long-range pore connectivity. Furthermore, the time-dependent effective diffusivity was found to be especially sensitive to relatively small morphological deviations between the real and reconstructed pore structures. It is concluded that the combined predictions of the effective permeability, the effective pore size, the geometric factor and time-dependent effective self-diffusivity of water are needed for the reliable evaluation of pore-space reconstruction.

Design of 3D statistically similar Representative Volume Elements based on Minkowski functionals

In this paper an extended optimization procedure is proposed for the construction of statistically similar RVEs (SSRVEs) which are defined as artificial microstructures showing a lower complexity than the associated real microstructures. This enables a computationally efficient discretization required for numerical calculations of microscopic boundary value problems and leads therefore to more efficient computational two-scale schemes. The optimization procedure is staggered and consists of an outer and an inner optimization problem. The outer problem treats different types of morphology parameterizations, different sets of statistical measures and different sets of weighting factors needed in the inner problem to minimize differences of mechanical errors that compare the response of the SSRVE with a target (real) microstructure. The inner problem minimizes differences of statistical measures describing the microstructure morphology for fixed parameterization type, statistical measures and weighting factors. The main contribution here is the analysis of new microstructure descriptors based on tensor-valued Minkowski functionals, whose numerical calculation requires less time compared to e.g. lineal-path functions. Thereby, a more efficient inner optimization problem can be realized and thus, an automated solution of the outer optimization problem becomes more practicable. Representative examples demonstrate the performance of the proposed method. It turns out that the evaluation of objective functions formulated in terms of the Minkowski functionals is almost 2000 times faster than functions taking into account lineal-path functions.

Reconstruction of three-dimensional porous media from a single two-dimensional image using three-step sampling.

A random three-dimensional (3D) porous medium can be reconstructed from a two-dimensional (2D) image by reconstructing an image from the original 2D image, and then repeatedly using the result to reconstruct the next 2D image. The reconstructed images are then stacked together to generate the entire reconstructed 3D porous medium. To perform this successfully, a very important issue must be addressed, i.e., controlling the continuity and variabilityamong adjacent layers. Continuity and variability, which areconsistentwith the statistics characteristic of the training image (TI), ensure that the reconstructed result matches the TI. By selecting the number and location of the sampling points in the sampling process, the continuity and variability can be controlled directly, and thus the characteristics of the reconstructed image can be controlled indirectly. In this paper, we propose and develop an original sampling method called three-step sampling. In our sampling method, sampling points are extracted successively from the center of 5×5 and 3×3 sampling templates and the edge area based on a two-point correlation function. The continuity and variability of adjacent layers were considered during the three steps of the sampling process. Our method was tested on a Berea sandstone sample, and the reconstructed result was compared with the original sample, using tests involving porosity distribution, the lineal path function, the autocorrelation function, the pore and throat size distributions, and two-phase flow relative permeabilities. The comparison indicates that many statistical characteristics of the reconstructed result match with the TI and the reference 3D medium perfectly.

Spatial Distribution of Voids in Insulating Concrete Analyzed by Micro-CT Images and Probability Functions

Insulating concrete is a multiphase material designed for reduced thermal conductivity, and the void distribution in concrete strongly affects its physical properties such as mechanical response and heat conduction. Therefore, it is essential to develop a method for identifying the spatial distribution of voids. To examine the voids of insulating concrete specimens, micro-CT (computed tomography) images can be effectively used. The micro-CT images are binarized to visualize the void distribution and stacked to generate 3D specimen images. From the obtained images, the spatial distribution of the voids and the microscopic constituents inside the insulating concrete specimens can be identified. The void distribution in the material can be characterized using low-order probability functions such as two-point correlation, lineal-path, and two-point cluster functions. It is confirmed that micro-CT images and low-order probability functions are effective in describing the relative degree of void clustering and void connectivity in insulating concrete.