A multiscale stochastic finite element method on elliptic problems involving uncertainties

Xu, X. Frank

doi:10.1016/j.cma.2007.02.002

Cited by 88 publications

(60 citation statements)

References 43 publications

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“…Recently proposed methods. The most recent developments in spectral-Galerkin-based SFEM include the stochastic reduced basis methods (SRBMs) introduced in [118,154,116], the non-intrusive approaches proposed in [13][14][15], the use of the method in a multi-scale setting [191] and the extension to the stochastic framework of the eXtended finite element method (X-FEM) [124]. The SRBMs constitute an efficient alternative which is also limited to the analysis of random linear systems (at least in its present formulation).…”

Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

“…This is why these methods can take advantage of powerful deterministic FE codes and using them as a black-box. The multi-scale SFEM developed in [191] combines a stochastic variational approach and scale-bridging multi-scale shape functions in order to solve stochastic elliptic problems. The applicability and efficiency of the method are demonstrated with the analysis of a simplified benchmark multi-scale model of groundwater flow in porous media.…”

Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

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The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

869

407

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a b s t r a c tA powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.

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Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

Section: The Spectral Stochastic Finite Element Methods -Ssfemmentioning

confidence: 99%

The stochastic finite element method: Past, present and future

Stefanou

2009

Computer Methods in Applied Mechanics and Engineering

869

407

View full text Add to dashboard Cite

show abstract

“…Under suitable assumptions on the underlying random field, a rigorous homogenization theory available in [3,19] was used to construct efficient stochastic homogenization solvers, based on spectral collocation methods [18] or Fourier-Galerkin approaches [46,47]. Recently, these methods were extended by Xu [45] to treat heterogeneous media with small but finite lengthscale contrast.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Modeling of Chaotic Masonry via Mesostructural Characterization

Lombardo

Zeman

Šejnoha

et al. 2009

Int J Mult Comp Eng

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Abstract. The purpose of this study is to explore three numerical approaches to the elastic homogenization of disordered masonry structures with moderate meso/macrolengthscale ratio. The methods investigated include a representative of perturbation methods, the Karhunen-Loève expansion technique coupled with Monte-Carlo simulations and a solver based on the Hashin-Shtrikman variational principles. In all cases, parameters of the underlying random field of material properties are directly derived from image analysis of a real-world structure. Added value as well as limitations of individual schemes are illustrated by a case study of an irregular masonry panel.

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“…A number of authors have recently considered domain decomposition methods [50,25] and multiscale methods [4,29,18,46,16] for stochastic partial differential equations. These papers focus on the combination of a deterministic algorithm with a stochastic approximation method for efficient uncertainty quantification for porous media flow.…”

Section: Introductionmentioning

confidence: 99%

A multiscale preconditioner for stochastic mortar mixed finite elements

Wheeler

Wildey

Yotov

2011

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThe aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy's law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media.

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A multiscale stochastic finite element method on elliptic problems involving uncertainties

Cited by 88 publications

References 43 publications

The stochastic finite element method: Past, present and future

The stochastic finite element method: Past, present and future

Stochastic Modeling of Chaotic Masonry via Mesostructural Characterization

A multiscale preconditioner for stochastic mortar mixed finite elements

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