The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size

Yue, Xingye; Weinan, E

doi:10.1016/j.jcp.2006.07.034

Cited by 95 publications

(104 citation statements)

References 11 publications

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“…This is an improvement for the reconstruction of the fine scale features since ε H in (3.2) is replaced by ε H in (3.5). Yet, as already mentioned and further studied in [59], the estimate (3.4) is not better than (3.1) in terms of convergence rates. In addition, there are examples in [29,59] (using reasonable averaging masks) for which the prefactor in (3.4) is larger than in (3.1).…”

Section: Windowing and Filtering In The Direct Approachmentioning

confidence: 54%

“…Yet, as already mentioned and further studied in [59], the estimate (3.4) is not better than (3.1) in terms of convergence rates. In addition, there are examples in [29,59] (using reasonable averaging masks) for which the prefactor in (3.4) is larger than in (3.1). It is therefore not clear whether the use of a mask yields better results than simply taking the average in general.…”

Section: Windowing and Filtering In The Direct Approachmentioning

confidence: 54%

“…The numerical tests by Yue and E in [59] tend to show that periodic boundary conditions perform usually better than Dirichlet boundary conditions. The method described there reads: for all L > 0, an approximation of A hom is given by, for all ij, ∈ {1, .…”

Section: Comments On the Periodization Methodsmentioning

confidence: 98%

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Numerical homogenization: survey, new results, and perspectives

Gloria

2012

ESAIM: Proc.

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Abstract. These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar's correctors. Hence the convergence analysis of these methods relies on the H-convergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results.Résumé. Ces notes de cours dressent unétat de l'art des méthodes d'homogénéisation numérique pour leséquations elliptiques linéaires. Le fil conducteur choisi est l'analyse. La plupart des méthodes d'homogénéisation numérique s'interprète comme des discrétisations (plus ou moins différentes) d'une même famille de problèmes continus approchés qui H-converge vers le problème homogénéisé. De même, le concept de correcteur numérique s'interprète comme une approximation des correcteurs introduits par Tartar. Ainsi l'analyse de convergence repose essentiellement sur la théorie de la H-convergence. Si on s'intéresse aux estimations quantitatives d'erreur, il faut faire des hypothèses supplémentaires de structure sur les hétérogénéités (périodicité par exemple). Dans ce cas, un outil important est l'interprétation spectrale de l'équation du correcteur introduite par Papanicolaou et Varadhan, qui permet non seulement de démontrer des résultats quantitatifs, mais aussi de développer des méthodes numériques efficaces. Qu'il s'agisse de propriétés qualitatives ou quantitatives, le développement et l'analyse de méthodes d'homogénéisation numérique reposent sur des concepts fondateurs de la théorie de l'homogénéisation. Ces notes contiennent quelques résultats nouveaux. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

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Section: Windowing and Filtering In The Direct Approachmentioning

confidence: 54%

Section: Windowing and Filtering In The Direct Approachmentioning

confidence: 54%

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Numerical homogenization: survey, new results, and perspectives

Gloria

2012

ESAIM: Proc.

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“…The transition between the two scales is obtained via averaging the internal fields within the RVE. Yue and E [211] discussed alternative averaging methods and reported that weighted or truncated averaging introduced in Refs. [212] and [213] can improve the solution in some cases.…”

Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

183

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…Recall that for pure diffusion problems Dirichlet and Neumann coupling conditions are commonly used, e.g., see [56]. Note that Dirichlet coupling…”

Section: Space-discrete Dg-hmmmentioning

confidence: 99%

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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We introduce a numerical homogenization method based on a discontinuous Galerkin finite element heterogeneous multiscale method (DG-HMM) to efficiently approximate the effective solution of parabolic advection-diffusion problems with rapidly varying coefficients, large Péclet number and compressible flows. To estimate the missing data of an effective model, numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion. For compressible flow with periodic data, we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime. Numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non-periodic data.

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The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size

Cited by 95 publications

References 11 publications

Numerical homogenization: survey, new results, and perspectives

Numerical homogenization: survey, new results, and perspectives

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

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