Problems In Heterogeneous Media Research Articles

An Effective Discontinuous Galerkin Method for Solving Acoustic Wave Equations in Heterogeneous Media

Abstract A discontinuous Galerkin (DG) method is introduced for solving 2D and 3D first-order velocity-pressure acoustic wave propagation problems in heterogeneous media. First, acoustic equations are reformulated into a first-order hyperbolic system, which can be integrated into the framework of the DG method. Then, we introduce an effective numerical flux in DG spatial discretization, which preserves physical laws on both sides of interfaces with discontinuous material parameters. It is compared with both the classic Lax-Friedrichs flux and the exact upwind flux, with the latter derived from solving the Riemann problem and satisfying the Rankine-Hugoniot condition. The comparison reveals that while our flux is formally similar to the classic local Lax-Friedrichs flux, its numerical behavior is analogous to the exact upwind flux. The primary advantage of this method lies in its simplicity and straightforward computation yet can maintain physical continuity conditions near interfaces with discontinuous material parameters. Several numerical experiments of acoustic wave propagation in 2D and 3D heterogeneous media are carried out, illustrating the effectiveness of this method.

On the Application of the Nyström Method to Two-Dimensional Neutral Particle Transport Problems in Heterogeneous Media

Over time, several methods were developed to deal with neutral particle transport problems. The interest in these problems is related to their wide range of applications, from neutron transport and heat transfer in nuclear reactors to radiative transfer in atmospheric clouds. Unlike the discrete ordinates or discrete ordinates–like methods, integral methods do not require discretization of angular variables. Instead, angular variables are completely eliminated by an integration procedure over the solid angle, which allows elimination of the ray effect. That said, this paper presents a new approach to estimate the scalar flux in two-dimensional fixed-source neutron transport problems in a heterogeneous medium, considering isotropic scattering and vacuum and reflective boundary conditions. Here, the Nyström method is combined with the singularity-subtraction technique to present an integral formulation for the scalar flux in a mesh grid over all regions of the domain. The iterative method of the Neumann series is used as an alternative to direct methods to solve the resulting system of equations generated from the domain discretization. Numerical results are given to verify the offered method.

Comment on “Numerical study with OpenFOAM on heat conduction problems in heterogeneous media” by K. Zhang, C.-A. Wang, J.-Y. Tan, International Journal of Heat and Mass Transfer, Vol. 124 (2018) 1156–1162

In the paper “Numerical study with OpenFOAM on heat conduction problems in heterogeneous media” by K. Zhang, C.-A. Wang, J.-Y. Tan, International Journal of Heat and Mass Transfer, Vol. 124 (2018) 1156–1162 the authors modify laplacianFoam from open-source CFD package OpenFOAM® to solve heat conduction problems in heterogeneous media by changing the diffusion coefficient from a constant value to a scalar field. Their paper clearly demonstrates a fundamental, overlooked, and common mistake in rearranging the heat conduction equation, which cannot be replaced with the temperature diffusion equation. In this comment, we provide clear evidence through mathematical derivations and numerical examples calculated in OpenFOAM®.

A fast numerical method for the conductivity of heterogeneous media with Dirichlet boundary conditions based on discrete sine–cosine transforms

The aim of this work is to develop a fast numerical method for solving conductivity problems in heterogeneous media subjected to Dirichlet boundary conditions. The method is based on a fixed-point iterative solution to an integral Lippmann–Schwinger type equation that is obtained by a Galerkin discretization of the cell problem using an approximation space spanned by sine series; the solution field is split between a known term verifying the Dirichlet boundary conditions and an unknown term described by sine series, which is null on the boundary by construction. With a suitable numerical integration scheme of the elementary integrals involved in the Galerkin formulation, based on discrete sine–cosine transforms, the method relies on the numerical complexity of fast Fourier transforms. The method is assessed in several problems including composite materials and fibrous networks.

Explicit scheme based on integral transforms for estimation of source terms in diffusion problems in heterogeneous media

The estimation of source terms present in differential equations has various applications, ranging from structural assessment, industrial process monitoring, equipment failure detection, environmental pollution source detection to identification applications in medicine. Significant progress has been made in recent years in methodologies capable of estimating this parameter. This work employs a methodology based on an explicit formulation of the integral transformation to characterize the unknown source term, reconstructing it through the expansion in known eigenfunctions of the Sturm-Liouville eigenvalue problem. To achieve this, a linear model is considered in a heterogeneous medium with known and spatially varying physical properties and two heat sources, with both temporal and spatial dependencies, and only spatial dependence. The eigenvalue problem contains information about the heterogeneous properties and is solved using the generalized integral transformation technique. Additionally, an initial interpolation of the sensor data is proposed for each observation time, making the inverse problem computationally lighter. The solutions of the inverse problem exhibit optimal performance, even with noisy input data and sources with abrupt discontinuities. The temperatures recovered by the direct problem considering the recovered source closely match synthetic experimental data, showing errors less than 1%, ensuring the robustness and reliability of the technique for the proposed application.

A hybrid shifted Laplacian multigrid and domain decomposition preconditioner for the elastic Helmholtz equations

In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation. The shifted Laplacian multigrid method is a common preconditioning approach for the discretized acoustic Helmholtz equation. In some cases, like geophysical seismic imaging, one needs to consider the elastic Helmholtz equation, which is harder to solve: it is three times larger and contains a nullity-rich grad-div term. These properties make the solution of the equation more difficult for multigrid solvers. The key idea in this work is combining the shifted Laplacian with approaches for linear elasticity. We provide local Fourier analysis and numerical evidence that the convergence rate of our method is independent of the Poisson's ratio. Moreover, to better handle the problem size, we complement our multigrid method with the domain decomposition approach, which works in synergy with the local nature of the shifted Laplacian, so we enjoy the advantages of both methods without sacrificing performance. We demonstrate the efficiency of our solver on 2D and 3D problems in heterogeneous media.

A meshfree method for the solution of 2D and 3D second order elliptic boundary value problems in heterogeneous media

We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C0 formulations.

Multi-Resolution Localized Orthogonal Decomposition for Helmholtz Problems

We introduce a novel multi-resolution localized orthogonal decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets (gamblets) and proves its applicability for a class of complex-valued, non-hermitian, and indefinite problems. It computes hierarchical bases that block-diagonalize the Helmholtz operator and thereby decouples the discretization scales. Sparsity is preserved by a novel localization strategy that improves stability properties even in the elliptic case. We present a rigorous stability and a priori error analysis of the proposed method for homogeneous media. In addition, we investigate the fast solvability of the blocks by a standard iterative method. A sequence of numerical experiments illustrates the sharpness of the theoretical findings and demonstrates the applicability to scattering problems in heterogeneous media.

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.

Space–time fractional diffusion equations in d-dimensions

We analyze a new space–time fractional diffusion equation encompassing different diffusion processes in d-dimensions. The first-order time derivative is replaced with a time derivative of the Caputo type of arbitrary order β; the spatial-fractional operator of Riesz–Feller or Riesz–Weyl type is replaced with its extension to d-dimensions, defined by means of an extended Fourier transform. The mathematical problem with the spatial-fractional operator proposed here is formulated to tackle anomalous diffusion in heterogeneous media (fractal structures) and incorporating power-law distributions. A formal solution is proposed using the Green function method which, for appropriate initial and boundary conditions, can be expressed in terms of the generalized H-function of Fox—a typical track of anomalous diffusive processes. These mathematical tools provide a new powerful framework to model anomalous diffusion and relaxation problems in heterogeneous media.

A Coupled Novel Green Element and Embedded Discrete Fracture Model for Simulation of Fluid Flow in Fractured Reservoir

In this study, we present a hybrid model coupled with two-set nodes Green element method (GEM) and embedded discrete fracture model (EDFM) for capturing the effect of transient flow in inhomogeneous fractured porous media. GEM is an excellent advanced algorithm, which can solve nonlinear problems in heterogeneous media. That is also an obvious advantage of GEM against the original boundary element method (BEM). The novel GEM has double nodes of pressure and flux and it is an improvement of classical GEM, which has a second-order precision and fits for triangle structured grids. In the place of adopting the linear flow approximation for original EDFM, the interflows between local triangle matrix grids and fracture elements are derived using the novel GEM, which has higher accuracy than those in previous EDFMs. Consequently, the modified hybrid model can indeed calculate the pressure and flux distribution of transient flow in multifracture porous media. Three numerical cases are presented to show the practicability of our novel model which include (i) multistage fractured horizontal well, (ii) heterogeneous fractured porous media, and (iii) complex fracture networks (CFNs) in an unconventional reservoir.

Distributed learning machines for solving forward and inverse problems in partial differential equations

We conceptualize Distributed Learning Machines (DLMs) – a novel machine learning approach that integrates existing machine learning algorithms with traditional mesh-based numerical methods for solving forward and inverse problems in nonlinear partial differential equations (PDEs). In conventional numerical methods such as finite element method (FEM), the discretization of the computational domain is a standard technique to reduce the representation load of basis functions. Along the same lines, we propose a distributed neural network architecture that facilitates the simultaneous deployment of several localized neural networks to solve PDEs in a unified manner. The most critical requirement of the DLMs is the synchronization of the distributed neural networks. For this, we introduce a new physics-based interface regularization term to the cost function of the existing learning machines like the Physics Informed Neural Network (PINN) and the Physics Informed Extreme Learning Machine (PIELM). To evaluate the efficacy of this approach, we develop three distinct variants of DLM namely, time-marching Distributed PIELM (DPIELM), Distributed PINN (DPINN) and time-marching DPINN. We show that ideas of linearization and time-marching allow DPIELM to be able to solve nonlinear PDEs to some extent. Next, we show that DPINNs have potential advantages over existing PINNs to solve the inverse problems in heterogeneous media. Finally, we propose a rapid, time-marching version of DPINN which leverages the ideas of transfer learning to accelerate the training. Collectively, this framework leads towards the promise of hybrid Neural Network-FVM or Neural Network-FEM schemes in the future.

Transient three-dimensional heat conduction in heterogeneous media: Integral transforms and single domain formulation

A hybrid numerical-analytical methodology based on integral transforms is employed for solving transient three-dimensional heat conduction problems in heterogeneous media comprising arbitrarily space variable thermophysical properties, including configurations with multiple subdomains of different materials and geometries. The Generalized Integral Transform Technique (GITT) is used to solve for both the temperature distribution and the corresponding eigenvalue problem accounting for the spatially variable coefficients. A previously introduced single domain reformulation strategy is adopted when multiple subdomains are involved, so as to rewrite the heat conduction equation in terms of coefficients with abrupt variations in the spatial coordinates. The hybrid numerical-analytical approach is demonstrated for three classes of problems in which the thermophysical properties undergo significant variations throughout the domain, such as: (i) FGMs (Functionally Graded Materials) with three-dimensional space variable thermophysical properties; (ii) composite medium with inclusions of different geometries and thermophysical properties; (iii) a multi-scale/multi-material heat conduction problem in a IGBT (Insulated Gate Bipolar Transistor) module. In each considered application, both the temperature eigenfunction expansions and the associated eigenvalue problem solutions are critically analyzed in terms of convergence rates and co-verified with purely numerical solutions.

Online basis construction for goal-oriented adaptivity in the generalized multiscale finite element method

In this research, we develop an online enrichment framework for goal-oriented adaptivity within the generalized multiscale finite element method for flow problems in heterogeneous media. The method for approximating the quantity of interest involves construction of residual-based primal and dual basis functions used to enrich the multiscale space at each stage of the adaptive algorithm. Three different online enrichment strategies based on the primal-dual online basis construction are proposed: standard, primal-dual combined and primal-dual product based. Numerical experiments are performed to illustrate the efficiency of the proposed methods for high-contrast heterogeneous problems.

A double distribution lattice Boltzmann scheme for unsteady Conjugate Heat Transfer: The DD-CHT LB method

In this paper, we propose an innovative formulation of the “well-known” thermal lattice Boltzmann method for solving transient conduction heat transfer problems in heterogeneous media: the DD-CHT LB method for Double Distribution – Conjugate Heat Transfer. The main idea of this new formulation is to bring out a second distribution function that represent the effect of thermal inertia and its variation in a heterogeneous media. By adopting this original formulation we achieved several goals, mainly conservation of local properties of the studied media and ensuring the stability of the proposed formulation by avoiding any alteration of the “well-known” TLBM collision step. Validation of the formulation is fulfilled through three different benchmark conduction heat transfer problems in heterogeneous media. All results are compared with those obtained by the Control Volume Method (CVM) and excellent agreements were found.

Numerical homogenization for poroelasticity problem in heterogeneous media

We consider the poroelasticity problems in heterogeneous porous media. Mathematical model contains coupled system of the equations for pressure and displacements. We construct coarse grid solver using numerical homogenization technique, where we calculate effective permeability and elastic properties of the media by solution of the local problems. We present numerical results for the model problem in two and three-dimensional formulations. The results show that method can provide good accuracy and computationally effective.

A Variable Wave Number Plane Wave Enriched Partition of Unity Isogeometric Analysis for Acoustic Problems in Heterogeneous Media

A Variable Wave Number Plane Wave Enriched Partition of Unity Isogeometric Analysis for Acoustic Problems in Heterogeneous Media

An enriched meshless finite volume method for the modeling of material discontinuity problems in 2D elasticity

Abstract A 2D formulation for incorporating material discontinuities into the meshless finite volume method is proposed. In the proposed formulation, the moving least squares approximation space is enriched by local continuous functions that contain discontinuity in the first derivative at the location of the material interfaces. The formulation utilizes space-filling Voronoi-shaped finite volumes in order to more intelligently model irregular geometries. Numerical experiments for elastostatic problems in heterogeneous media are presented. The results are compared with the corresponding solutions obtained using the standard meshless finite volume method and element free Galerkin method in order to highlight the improvements achieved by the proposed formulation. It is demonstrated that the enriched meshless finite volume method could alleviate the expecting oscillations in derivative fields around the material discontinuities. The results have revealed the potential of the proposed method in studying the mechanics of heterogeneous media with complex micro-structures.

Numerical study with OpenFOAM on heat conduction problems in heterogeneous media

OpenFOAM was used to numerically investigate heat conduction in heterogeneous media. The main advantage of this platform is that it offers an open flexible interface with great convenience for mesh generation and solver modification. Cases of 1D and 2D heat conduction problems in media with different levels of heterogeneity were considered to check the performance of OpenFOAM. The results were compared with references and other commercial software and showed good accuracy. Note that all of the processes, including the mesh generation, numerical simulation, and post-processing, relied on free open-source codes.

Low-Rank Approximation to Heterogeneous Elliptic Problems

In this work, we investigate the low-rank approximation of elliptic problems in heterogeneous media by means of Kolmogrov $n$-width and asymptotic expansion. This class of problems arises in many practical applications involving high-contrast media, and their efficient numerical approximation often relies crucially on certain low-rank structure of the solutions. We provide conditions on the permeability coefficient $\kappa$ that ensure a favorable low-rank approximation. These conditions are expressed in terms of the distribution of the inclusions in the coefficient $\kappa$, e.g., the values, locations, and sizes of the heterogeneous regions. Further, we provide a new asymptotic analysis for high-contrast elliptic problems based on the perfect conductivity problem and layer potential techniques, which allows deriving new estimates on the spectral gap for such high-contrast problems. These results provide theoretical underpinnings for several multiscale model reduction algorithms.