Random Heterogeneous Media Research Articles

Improved micromechanical prediction of short fibre reinforced composites using differential Mori-Tanaka homogenization

Due to the spatial distribution of inclusions in random heterogeneous media, individual inclusions are stressed differently. Mean field homogenization (MFH) methods, a popular homogenization method, cannot account for this variability, instead predicting the same mean value of stress for a particular phase. In this paper, a differential scheme is expanded to calculate the stresses in the individual inclusions, including the scatter and variation of local stresses. The accurate prediction of stress variability of stresses within a particular phase has led to more accurate and realistic modelling of inclusion matrix debonding and inclusion breakage.Extensive benchmarking of the proposed models against finite element (FE) results confirms excellent predictive abilities of scatter at three length scales (effective property, individual inclusions, and matrix-inclusion interface). The potential of modelling stress variability post-onset of damage is also demonstrated.Different realisations of the differential Mori Tanaka (MT) lead to varying predictions of scatter in the individual inclusion stresses. However, the effective properties predictions remain consistent.The prediction of individual inclusion stresses, and their scatter constitutes a significant gap in the literature and has been addressed in this paper. This new scheme could lead to the development of several intricate models of damage in various composites without the need for extensive FE modelling.

3D imaging through a highly heterogeneous double-composite random medium by common-path phase-shift digital holography.

A method is proposed for 3D imaging through a highly heterogeneous double-composite random medium made of a thick mildly inhomogeneous medium followed by a thin strongly scattering layer. To realize the immunity to the heterogeneous random medium, a system of common-path phase-shift digital holography is designed in such a manner that the wavefront distortion caused by the first inhomogeneous medium is canceled out by the common-path geometry, and the influence of the random phase introduced by the second scattering layer is removed by the intensity-based recording of the digital hologram on the thin scattering layer. The validity of the method was confirmed by experiments.

Computing the effective crack energy of heterogeneous and anisotropic microstructures via anisotropic minimal surfaces

A variety of materials, such as polycrystalline ceramics or carbon fiber reinforced polymers, show a pronounced anisotropy in their local crack resistance. We introduce an FFT-based method to compute the effective crack energy of heterogeneous, locally anisotropic materials. Recent theoretical works ensure the existence of representative volume elements for fracture mechanics described by the Francfort–Marigo model. Based on these formulae, FFT-based algorithms for computing the effective crack energy of random heterogeneous media were proposed, and subsequently improved in terms of discretization and solution methods. In this work, we propose a maximum-flow solver for computing the effective crack energy of heterogeneous materials with local anisotropy in the material parameters. We apply this method to polycrystalline ceramics with an intergranular weak plane and fiber structures with transversely isotropic crack resistance.

Numerical studies of the statistics of seismic waveform propagation in random heterogeneous media

Incoherent scattered waves are generated when seismic waves propagate through heterogeneous media. These scattered waves contain statistical information, such as random heterogeneity. In order to study the influence of random heterogeneity on seismic waveforms, four series of simulation experiments were performed in the presents study, as follows: (1) Waves were excited by the same source and propagated through four models with different parameters; (2) Waves were excited by different frequencies, but spread in the same model (0.25, 0.50, 1.00, 2 MHz); (3) Waves were excited by the same source and propagated through three different velocity fluctuation models (5%, 10% and 15%); (4) Waves were excited by the same source and propagated through three different propagation distances (30, 60, and 90 mm). In the present study, the correlation between coherent waveforms and observed waveforms, the energy distribution with respect to the elapsed time, and the statistical distribution of the waveform correlation coefficient were calculated. Next, the variation of the waveform parameters with the normalized scale parameter ka was investigated. Stronger scattered waves and more complicated waveforms were observed when the ka value increased, indicating the transition from an equivalent homogeneous medium to a heterogeneous medium.

Quasi-Monte Carlo Finite Element Analysis for Wave Propagation in Heterogeneous Random Media

We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern Numer. Algorithms, 83 (2020), pp. 1441--1487. The price to pay is that the regularity analysis required for QMC methods becomes much more technical. Nevertheless we obtain a complete analysis with error comprising stochastic dimension truncation error, finite element error, and cubature error, with results comparable to those obtained for the diffusion problem.

On the multiscale characterization of effective hydraulic conductivity in random heterogeneous media: A historical survey and some new perspectives

In large scale heterogeneous aquifer simulations, determining the appropriate coarsening scale λ to define an effective hydraulic conductivity Keff is a challenging task, that involves a trade-off between accuracy and cost. Efficiently adjusting the scale λ is then key, in particular for uncertainty quantification. In this paper, we obtain improved analytical results for the variance of Keff, valid at any scale, in the context of energy dissipation formulation. Using this formulation, we then derive an efficient Keff numerical estimator, and compare it with those of the potential-flow average and permeameter formulations in 2D, for lognormal and binary media, over a wide range of λ and of heterogeneity. We analyze the probability density function (pdf), mean, and variance, of these estimators, comparing them with the analytical results. In the lognormal case, the pdf’s are rather similar for the three estimators, and remain lognormal at all scales. In the binary case, slow convergence to an asymptotic regime is observed close to the percolation threshold.

Stochastic multiscale modeling of crack propagation in random heterogeneous media

SummaryA stochastic approach to model crack propagation in random heterogeneous media, using mesoscopic representations of elastic and fracture properties, is presented. In order to obtain reference results, Monte‐Carlo simulations are first conducted on microstructural samples in which a pre‐existing crack is propagated by means of a phase‐field approach. These computations are used to estimate the subscale‐induced randomness on the macroscopic response of the domain. Mesoscopic descriptors are then introduced to investigate scale transition. Elasticity tensor random fields are specifically defined, at that stage, through a moving‐window upscaling approach. The mesoscopic fracture toughness, which is assumed homogeneous and deterministic, is identified by solving an inverse problem involving the macroscopic peak force. A stochastic model is subsequently constructed in which the mesoscopic elasticity is described as a non‐Gaussian random field. This model allows the multiscale‐informed elastic counterpart in the phase‐field formulation to be sampled without resorting to computational homogenization. The results obtained with the sample‐based and model‐based mesoscopic descriptions are finally compared with those corresponding to the full‐scale microscopic model. It is shown, in particular, that the mesoscopic elasticity‐phase‐field formulation associated with statically uniform boundary conditions enables the accurate predictions of the mean elastic response and mean peak force.

Uncertainty Quantification for Flow and Transport in Highly Heterogeneous Porous Media Based on Simultaneous Stochastic Model Dimensionality Reduction

Groundwater flow models are usually subject to uncertainty as a consequence of the random representation of the conductivity field. In this paper, we use a Gaussian process model based on the simultaneous dimension reduction in the conductivity input and flow field output spaces in order quantify the uncertainty in a model describing the flow of an incompressible liquid in a random heterogeneous porous medium. We show how to significantly reduce the dimensionality of the high-dimensional input and output spaces while retaining the qualitative features of the original model, and secondly how to build a surrogate model for solving the reduced-order stochastic model. A Monte Carlo uncertainty analysis on the full-order model is used for validation of the surrogate model.

Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

In a previous paper (Graham et al. in J Comput Phys 230:3668–3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

Heterogeneous random medium plasticity and fracture model of additively-manufactured Ti-6Al-4V

The large deformation response of Ti-6Al-4V structures made through shaped metal deposition is heavily affected by the prior-beta grain structure induced by the manufacturing process. Based on more than 25 tension and shear experiments on individual prior-beta grains, the statistical distribution of the isotropic hardening response of prior-beta grains is identified. In close analogy with Crystal Plasticity Finite Element (CPFE) analysis, Voronoi tesselation is used to generate finite element models of additively-manufactured structures containing multiple prior-beta grains. For each prior-beta grain, a different hardening curve is assigned using a random pull from the truncated Gaussian distribution of Hockett-Sherby type of hardening curves. Similarly, the parameters of the Hosford-Coulomb fracture initiation model are assigned. The comparison of the CPFE simulation results with the experimental results on tension specimens containing multiple prior-beta grains, shows good qualitative and quantitative agreement with regards to the force-displacement response, the surface strain fields, and fracture occurrences. The results demonstrate that the failure of additively-manufactured Ti-6Al-4V components through ductile fracture requires a heterogeneous random medium modeling approach. While the present study focuses on strength and hardening variations, it is noted that further improvements are expected when considering the known texture mismatch between prior-beta grains along with detailed models of the prior-beta grain boundaries.

Multiple scattering of an ultrasonic shock wave in bubbly media.

This experimental study deals with the propagation of an ultrasonic shock wave in a random heterogeneous medium, constituted of identical 75μm radius bubbles, trapped in a yield-stress fluid. The fundamental frequency of the incident wave (in the MHz range) was much larger than the resonance frequency of bubbles (38kHz). A well-expanded coda, resulting from the multiple scattering of the incident shock wave through the heterogeneous medium, was experimentally measured in transmission. Despite the significant amplitude of the shock wave (90kPa), no sign of nonlinear response of the bubbles was detected. Both the coherent and incoherent fields were successfully described by linear theories. Using a shock wave presents the advantage of characterizing the medium over a large frequency range (1.5-15MHz).

Treatment of Double Heterogeneity in the Resonance and Thermal Energy Regions in High-Temperature Reactors

The fuel in high-temperature reactors (HTRs) consists of a large number of tiny (a few hundred microns in size) TRISO-coated particles dispersed randomly in a graphite matrix. At the resonances of the major nuclide (238U or 232Th) of the fuel, the neutron mean free path in the fuel is often comparable to the kernel dimensions, and hence the dispersion of particles in the graphite matrix must be treated as a heterogeneous medium for obtaining self-shielded cross sections in the resonance (epithermal) groups. HTRs containing only plutonium as fuel (as may be the case in reactors designed for burning plutonium) have high concentrations of the isotopes 239Pu and 240Pu in the kernels. This fact together with their rather large resonance cross section in the thermal groups results in a very short neutron mean free path in the fuel that is comparable to the kernel dimensions. In such cases the fuel zone must be treated as a heterogeneous medium in the thermal energy region as well. However, the resonance treatment method in libraries such as the WIMS library does not cover the two large resonances of plutonium lying in the thermal region. Instead, a large number of groups are used to cover the details of cross-section variation in this region.The heterogeneity of the fuel region together with the heterogeneous distribution of fuel region, graphite moderator, and coolant is referred to as the double heterogeneity of HTRs. The paper describes work we carried out for addressing these problems. A new method for generating a random distribution of TRISO particles in the fuel zone of a pebble or fuel compact and Monte Carlo calculation of the Dancoff factor in the heterogeneous random medium, required for calculating the self-shielded resonance group cross sections, is presented. Dancoff factors obtained by our method are compared with values available in published literature on the subject. The paper also discusses a new methodology developed to solve the double-heterogeneity problem at the stage of the multigroup transport theory solution, which is particularly important in the thermal region occurring in high-content Pu-fueled HTRs. These features have been incorporated in the WIMS library–based lattice code BOXER3. An option for handling the spherical geometry of the lattice cell of a pebble bed reactor has been added in the code. Results of analysis of a number of HTR lattice cell benchmark problems are presented.

Bayesian and variational Bayesian approaches for flows in heterogeneous random media

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

Numerical Simulations of the Lunar Penetrating Radar and Investigations of the Geological Structures of the Lunar Regolith Layer at the Chang’E 3 Landing Site

In the process of lunar exploration, and specifically when studying lunar surface structure and thickness, the established lunar regolith model is usually a uniform and ideal structural model, which is not well-suited to describe the real structure of the lunar regolith layer. The present study aims to explain the geological structural information contained in the channel 2 LPR (lunar penetrating radar) data. In this paper, the random medium theory and Apollo drilling core data are used to construct a modeling method based on discrete heterogeneous random media, and the simulation data are processed and collected by the electromagnetic numerical method FDTD (finite-difference time domain). When comparing the LPR data with the simulated data, the heterogeneous random medium model is more consistent with the actual distribution of the media in the lunar regolith layer. It is indicated that the interior structure of the lunar regolith layer at the landing site is not a pure lunar regolith medium but rather a regolith-rock mixture, with rocks of different sizes and shapes. Finally, several reasons are given to explain the formation of the geological structures of the lunar regolith layer at the Chang’E 3 landing site, as well as the possible geological stratification structure.

A Hölder estimate for non-uniform elliptic equations in a random medium

Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Hölder estimate in ω,ϵ,λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ∈(0,1] is the length scale of the interfaces, and λ2∈(0,1] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Hölder estimate in ω,ϵ,λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.

Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium

We consider the very challenging problem of efficient uncertainty quantification for acoustic wave propagation in a highly heterogeneous, possibly layered, random medium, characterized by possibly anisotropic, piecewise log-exponentially distributed Gaussian random fields. A multi-level Monte Carlo finite volume method is proposed, along with a novel, bias-free upscaling technique that allows to represent the input random fields, generated using spectral FFT methods, efficiently. Combined together with a recently developed dynamic load balancing algorithm that scales to massively parallel computing architectures, the proposed method is able to robustly compute uncertainty for highly realistic random subsurface formations that can contain a very high number (millions) of sources of uncertainty. Numerical experiments, in both two and three space dimensions, illustrating the efficiency of the method are presented.

Homogenization of randomly deformed conductivity resistant membranes

We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris and Lions \cite{BLBL06}. Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that on an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole domain with infinitely many interfaces, in a setting that is neither periodic nor stationary ergodic in the usual sense.

An efficient and automated 3D FE approach to evaluate effective elastic properties of overlapping random fibre composites

The paper deals with an automated and double-scale finite element (FE) model to evaluate the effective elastic properties of 3D overlapping random fibre composites. An efficient and perfectly reliable approach to generate meshes in the context of random heterogeneous media for which some difficulties are encountered, is proposed. The basic idea is to attain an approximation of the geometry relative to a grid of regular hexahedral elements. The concept called voxelisation is tremendously sped up using an adaptive mesh refinement (AMR) which enables one to locally and efficiently smooth each heterogeneity. Thus, a fast, reliable and fully automated generation of representative volume elements (RVE) is possible whatever the complexity of the geometry is. In the present work, algorithms to generate RVE according to the concept of voxelisation are provided. In addition, numerical tests are performed to evaluate both the efficiency of the model and the reliability of the results in the framework of linear elasticity and random fibre composites.

Multi-Level Monte Carlo Finite Volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium

We consider the very challenging problem of efficient uncertainty quantification for acoustic wave propagation in a highly heterogeneous, possibly layered, random medium, characterized by possibly anisotropic, piecewise log-exponentially distributed Gaussian random fields. A multi-level Monte Carlo finite volume method is proposed, along with a novel, bias-free upscaling technique that allows to represent the input random fields, generated using spectral FFT methods, efficiently. Combined together with a recently developed dynamic load balancing algorithm that scales to massively parallel computing architectures, the proposed method is able to robustly compute uncertainty for highly realistic random subsurface formations that can contain a very high number (millions) of sources of uncertainty. Numerical experiments, in both two and three space dimensions, illustrating the efficiency of the method are presented.

Simulation of heterogeneous two-phase media using random fields and level sets

The accurate and efficient simulation of random heterogeneous media is important in the framework of modeling and design of complex materials across multiple length scales. It is usually assumed that the morphology of a random microstructure can be described as a non-Gaussian random field that is completely defined by its multivariate distribution. A particular kind of non-Gaussian random fields with great practical importance is that of translation fields resulting from a simple memory-less transformation of an underlying Gaussian field with known second-order statistics. This paper provides a critical examination of existing random field models of heterogeneous two-phase media with emphasis on level-cut random fields which are a special case of translation fields. The case of random level sets, often used to represent the geometry of physical systems, is also examined. Two numerical examples are provided to illustrate the basic features of the different approaches.