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

Three-dimensional construction of hyperuniform, nonhyperuniform, and antihyperuniform disordered heterogeneous materials and their transport properties via spectral density functions

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ[over ̃]_{_{V}}(k), which can be obtained experimentally from scattering data, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant ε_{e} for electromagnetic wave propagation. Moreover, χ[over ̃]_{_{V}}(k) determines rigorous upper bounds on the fluid permeability K. Given the importance of χ[over ̃]_{_{V}}(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical functional forms for χ[over ̃]_{_{V}}(k) that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by autocovariance function χ_{_{V}}(r) with a power-law tail, resulting in microstructures that contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability K associated with all of the constructed materials directly from the corresponding spectral densities. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ[over ̃]_{_{V}}(k), with the stealthy hyperuniform and antihyperuniform media, respectively, achieving the most and least efficient transport, we find that varying the length-scale parameter within each class of χ[over ̃]_{_{V}}(k) functions can also lead to orders of magnitude variation of S(t) at intermediate and long time scales. Moreover, we find that increasing the solid volume fraction ϕ_{1} and correlation length a in the constructed media generally leads to a decrease in the dimensionless fluid permeability K/a^{2}, while the antihyperuniform media possess the largest K/a^{2} among the four classes of materials with the same ϕ_{1} and a. These results indicate the feasibility of employing parameterized spectral densities for designing composites with targeted transport properties.

Evaluation of three-point correlation functions from structural images on CPU and GPU architectures: Accounting for anisotropy effects

Structures, or spatial arrangements of matter and energy, including some fields (e.g., velocity or pressure) are ubiquitous in research applications and frequently require description for subsequent analysis, or stochastic reconstruction from limited data. The classical descriptors are two-point correlation functions (CFs), but the computation of three-point statistics is known to be advantageous in some cases as they can probe non-Gaussian signatures, not captured by their two-point counterparts. Moreover, n-point CFs with n≥3 are believed to possess larger information content and provide more information about studied structures. In this paper, we have developed algorithms and code to compute S_{3},C_{3},F_{sss},F_{ssv}, and F_{svv} with a right-angle and arbitrary triangle pattern. The former was believed to be faster to compute, but with the help of precomputed regular positions we achieved the same speed for arbitrary pattern. In this work we also implement and demonstrate computations of directional three-point CFs-for this purpose right-triangular pattern seems to be superior due to explicit orientation and high coverage. Moreover, we assess the errors in CFs' evaluation due to image or pattern rotations and show that they have minor effect on accuracy of computations. The execution times of our algorithms for the same number of samples are orders of magnitude lower than in existing published counterparts. We show that the volume of data produced gets unwieldy very easily, especially if computations are performed in frequency domain. For these reasons until information content of different sets of correlation functions with different "n-pointness" is known, advantages of CFs with n>3 are not clear. Nonetheless, developed algorithms and code are universal enough to be easily extendable to any n with increasing computational and random access memory (RAM) burden. All results are available as part of open-source package correlationfunctions.jl [V. Postnicov et al., Comput. Phys. Commun. 299, 109134 (2024)10.1016/j.cpc.2024.109134.]. As described in this paper, three-point CFs computations can be immediately applied in a great number of research applications, for example: (1) flow and transport velocity fields analysis or any data with non-Gaussian signatures, (2) deep learning for structural and physical properties, and (3) structure taxonomy and categorization. In all these and numerous other potential cases the ability to compute directional three-point functions may be crucial. Notably, the organization of the code functions allows computation of cross correlation, i.e., one can compute three-point CFs for multiphase images (while binary structures were used in this paper for simplicity of explanations).

Robust surface-correlation-function evaluation from experimental discrete digital images

Correlation functions (CFs) are universal structural descriptors; surface-surface F_{ss} and surface-void F_{sv} CFs are a subset containing additional information about the interface between the phases. The description of the interface between pores and solids in porous media is of particular importance and recently Ma and Torquato [Phys. Rev. E 98, 013307 (2018)2470-004510.1103/PhysRevE.98.013307] proposed an elegant way to compute these functions for a wide variety of cases. However, their "continuous" approach is not always applicable to digital experimental 2D and 3D images of porous media as obtained using x-ray tomography or scanning electron microscopy due to nonsingularities in chemical composition or local solid material's density and partial volume effects. In this paper we propose to use edge-detecting filters to compute surface CFs in the "digital" fashion directly in the images. Computed this way, surface correlation functions are the same as analytically known for Poisson disks in case the resolution of the image is adequate. Based on the multiscale image analysis we developed a C_{0.5} criterion that can predict if the imaging resolution is enough to make an accurate evaluation of the surface CFs. We also showed that in cases when the input image contains all major features, but do not pass the C_{0.5} criterion, it is possible with the help of image magnification to sample CFs almost similar to those obtained for high-resolution image of the same structure with high C_{0.5}. The computational framework as developed here is open source and available within the CorrelationFunctions.jl package developed by our group. Our "digital" approach was applied to a wide variety of real porous media images of different quality. We discuss critical aspects of surface correlation functions computations as related to different applications. The developed methodology allows applying surface CFs to describe the structure of porous materials based on their experimental images and enhance stochastic reconstructions or super-resolution procedures, or serve as an efficient metrics in machine learning applications due to computationally effective GPU implementation.