Fast numerical upscaling of heat equation for fibrous materials

Iliev, Oleg; Lazarov, Raytcho; Willems, J.

doi:10.1007/s00791-010-0144-2

Cited by 13 publications

(5 citation statements)

References 10 publications

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“…, m, as the following derivation shows. Consider the non-homogeneous displacement problem (11). The correction is attained from…”

Section: Non-zero Fixed Boundary Conditionsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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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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“…, m, as the following derivation shows. Consider the non-homogeneous displacement problem (11). The correction is attained from…”

Section: Non-zero Fixed Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical upscaling of discrete network models

et al. 2019

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“…Because of the above considerations, some model reduction methods are necessary. In the past several decades, many researchers proposed various model reduction techniques, for example, upscaling methods [6,29,22] and homogenization [12]. These methods aim to obtain a reduced model and one can therefore solve the problem in a coarse grid.…”

Section: Introductionmentioning

confidence: 99%

A conservative multiscale method for stochastic highly heterogeneous flow

Chung¹,

Fu²

2022

Preprint

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In this paper, we propose a local model reduction approach for subsurface flow problems in stochastic and highly heterogeneous media. To guarantee the mass conservation, we consider the mixed formulation of the flow problem and aim to solve the problem in a coarse grid to reduce the complexity of a large-scale system. We decompose the entire problem into a training and a testing stage, namely the offline coarse-grid multiscale basis generation stage and online simulation stage with different parameters. In the training stage, a parameter-independent and small-dimensional multiscale basis function space is constructed, which includes the media, source and boundary information. The key part of the basis generation stage is to solve some local problems defined specially. With the parameter-independent basis space, one can efficiently solve the concerned problems corresponding to different samples of permeability field in a coarse grid without repeatedly constructing a multiscale space for each new sample. A rigorous analysis on convergence of the proposed method is proposed. In particular, we consider a generalization error, where bases constructed with one source will be used to a different source. In the numerical experiments, we apply the proposed method for both single-phase and twophase flow problems. Simulation results for both 2D and 3D representative models demonstrate the high accuracy and impressive performance of the proposed model reduction techniques.

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“…Model reduction techniques usually depend on a coarse grid approximation, which can be obtained by discretizing the problem on a coarse grid and choosing a suitable coarse-grid formulation of the problem. In the literature, several approaches have been developed to obtain the coarse-grid formulation, including upscaling methods [29,22,23,20,25,3,2,27,6,8] and multiscale methods [19,24,17,16,4,9,5,1]. Among these approaches, GMsFEM (Generalized Multiscale Finite Element Methods) [18,12,11,10,15,7] provides a systematic way of adding degrees of freedom for problems with high contrast and multiple scales.…”

Section: Introductionmentioning

confidence: 99%

Space-time multiscale model reduction for transport equations

Chung¹,

Efendiev²,

Li³

2018

Preprint

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In this paper, we propose a space-time GMsFEM for transport equations. Multiscale transport equations occur in many geoscientific applications, which include subsurface transport, atmospheric pollution transport, and so on. Most of existing multiscale approaches use spatial multiscale basis functions or upscaling, and there are very few works that design space-time multiscale functions to solve the transport equation on a coarse grid. For the time dependent problems, the use of space-time multiscale basis functions offers several advantages as the spatial and temporal scales are intrinsically coupled. By using the GMsFEM idea with a space-time framework, one obtains a better dimension reduction taking into account features of the solutions in both space and time. In addition, the time-stepping can be performed using much coarser time step sizes compared to the case when spatial multiscale basis are used. Our scheme is based on space-time snapshot spaces and model reduction using space-time spectral problems derived from the analysis. We give the analysis for the well-posedness and the spectral convergence of our method. We also present some numerical examples to demonstrate the performance of the method. In all examples, we observe a good accuracy with a few basis functions.

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Fast numerical upscaling of heat equation for fibrous materials

Cited by 13 publications

References 10 publications

Numerical upscaling of discrete network models

Numerical upscaling of discrete network models

A conservative multiscale method for stochastic highly heterogeneous flow

Space-time multiscale model reduction for transport equations

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