A Simplified Method for Upscaling Composite Materials with High Contrast of the Conductivity

Ewing, Richard E.; Iliev, Oleg; Lazarov, Raytcho; Rybak, Iryna; Willems, J.

doi:10.1137/080731906

Cited by 25 publications

(19 citation statements)

References 16 publications

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“…First, the basis for the multiscale space V ms is constructed by calculating each φ i from (6). Secondly, the attained modified basis is used to solve the global multiscale problem (5). In this section, elementwise assembling and localization, described in Sec.…”

Section: Algebraic Formulationmentioning

confidence: 99%

Numerical upscaling of discrete network models

et al. 2019

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In this paper a numerical multiscale method for discrete networks is presented. The method gives an accurate coarse scale representation of the full network by solving sub-network problems. The method is used to solve problems with highly varying connectivity or random network structure, showing optimal order convergence rates with respect to the mesh size of the coarse representation. Moreover, a network model for paper-based materials is presented. The numerical multiscale method is applied to solve problems governed by the presented network model. IntroductionNetwork structures are used to model a wide variety of phenomena, such as flow in porous media, traffic flows, elasticity of materials, body deformation in computer graphics, molecular dynamics, and fiber materials. In these applications, the microscale behaviour determines the macroscale properties of the system. Often a full microscale model is difficult or impossible to work with because of the vast computational complexity. Therefore, there is an interest in constructing coarser, but still accurate, representations of the entire system. Such a procedure is sometimes referred to as upscaling or homogenization. In this work a numerical upscaling method for discrete networks is presented.There exist several numerical upscaling methods for partial differential equations (PDE) based on the idea of homogenization, such as the Heterogeneous Multiscale Method (HMM) [20], the Multiscale Finite Element Method (MsFEM) [8], and the more recent works [3,17]. The upscaling approach presented in this paper is based on the Localized Orthogonal Decomposition Method (LOD) [15,4], which in turn is inspired by the Variational Multiscale Method (VMM) [9]. Multiscale methods applied to network problems are for instance investigated by Ewing, Ilev et al. [5,11] who study the heat conductivity of network materials and develop an upscaling method by solving the heat equation locally over small sub-domains. These local solutions are used to compute an effective global thermal conductivity tensor. Della Rossa et al. investigate network models of traffic flows [2] and derive a governing PDE for the macroscale by formulating traffic flow equations for single network nodes and interpreting the relations as finite difference approximations. The macroscale parameters are resolved using a two-scale averaging technique. Chu et al. develop a multiscale method for networks representing flows in a porous medium [1]. The medium is modelled as a network where nodes represent pores and edges represent throats. The conductance of each throat is assumed to be given by Hagen-Poiseuille equation, and using mass conservation equations for the flow through the network, a model for the microscale is attained.The numerical upscaling method proposed in this work is developed for general unstructured networks. The network is supposed to represent the microscale, and the macroscale is represented by a finite element mesh which is coarse in comparison to the fine scale network. The coarse grid does not...

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Section: Algebraic Formulationmentioning

confidence: 99%

Numerical upscaling of discrete network models

et al. 2019

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“…Additionally, we mention [11,17,38] among others, which focus on multilevel methods targeting problems with high-contrast and discontinuous coefficients. Numerical methods based on different asymptotic analysis are detailed in [4,9,14,21,27].…”

Section: Introductionmentioning

confidence: 99%

“…Fast numerical upscaling techniques can also be constructed with the first order terms of (2) or (3). See [21] where the authors develop fast numerical upscaling methods based on some asymptotic analysis. We also mention [14,27] where numerical approximations are designed using asymptotic analysis.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Expansions for High-Contrast Elliptic Equations

Calo

Efendiev

Galvis

2013

Math. Models Methods Appl. Sci.

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In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The computation of the expansion does not depend on the contrast which is important for simulations. The latter allows avoiding increased mesh resolution around high conductivity features. This work is partly motivated by our earlier work in [Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model Simul. 8 (2010) 1461-1483] where we design efficient numerical procedures for solving high-contrast problems. These multiscale approaches require local solutions and our proposed high-order expansion can be used to approximate these local solutions inexpensively. In the case of a large-number of inclusions, the proposed analysis can help to design localization techniques for computing the terms in the expansion. In the * Also at Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia. † Corresponding author 465 Math. Models Methods Appl. Sci. 2014.24:465-494. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 02/03/15. For personal use only. 466 V. M. Calo, Y. Efendiev & J. Galvispaper, we present a rigorous analysis of the proposed high-order expansion and estimate the remainder of it. We consider both high-and low-conductivity inclusions.

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“…for v, w ∈ V i snap , and n i is the fixed unit normal vector for E i . After solving the spectral problem (12) in ω i , we arrange the eigenvalues in ascending order…”

Section: Generalized Multiscale Spacementioning

confidence: 99%

A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media

Yang

Chung

2019

Journal of Computational Physics

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In this paper, we consider flow simulation in highly heterogeneous media that has many practical applications in industry. To enhance mass conservation, we write the elliptic problem in a mixed formulation and introduce a robust two-grid preconditioner to seek the solution. We first need to transform the indefinite saddle problem to a positive definite problem by preprocessing steps. The preconditioner consists of a local smoother and a coarse preconditioner. For the coarse preconditioner, we design an adaptive spectral coarse space motivated by the GMsFEM (Generalized Multiscale Finite Element Method). We test our preconditioner for both Darcy flow and two phase flow and transport simulation in highly heterogeneous porous media. Numerical results show that the proposed preconditioner is highly robust and efficient.

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A Simplified Method for Upscaling Composite Materials with High Contrast of the Conductivity

Cited by 25 publications

References 16 publications

Numerical upscaling of discrete network models

Numerical upscaling of discrete network models

Asymptotic Expansions for High-Contrast Elliptic Equations

A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media

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