An extended stochastic finite element method for solving stochastic partial differential equations on random domains

By Anthony Nouy, A. Clément, Franck Schoefs, Nicolas Moës. 2008

116 citations recorded by the source corpus.

Abstract Recently, a new strategy was proposed to solve stochastic partial differential equations on random domains. It is based on the extension to the stochastic framework of the extended finite element method (X-FEM). This method leads by a “direct” calculus to an explicit solution in terms of the variables describing the randomness on the geometry. It relies on two major points: the implicit representation of complex geometries using random level-set functions and the use of a Galerkin approximation at both stochastic and deterministic levels. In this article, we detail the basis of this technique, from theoretical and technical points of view. Several numerical examples illustrate the efficiency of this method and compare it to other approaches.

Subfields: Computational Mechanics, Mechanics of Materials, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a15bcb5529711133890356e118a02084a62e4095

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In Computational Mechanics

In Mechanics of Materials

In Statistics and Probability

Citations in the corpus

Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, background): “6This hypothesis is not verified for classical formulations of PDE defined on random domains [23, 94, 95, 121].”
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison (2011, unknown): “[11, 27, 13, 17, 23]).”
Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics (2012, unknown): “Several deﬁnitions have been proposed for PGD [27, 28, 25].”
A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties (2013, methodology): “They are based on reformulation of the problem on a deterministic domain, by introducing a suitable random mapping to a fixed deterministic domain [32], or by introducing reformulations on a deterministic fictitious domain [28, 29, 30].”
Reengineering Aircraft Structural Life Prediction Using a Digital Twin (2011, methodology): “Other developments of interest include multiscale SFEM [40] and an extended SFEM (X-SFEM) analogous to XFEM for deterministic FEM [41].”
The stochastic finite element method: Past, present and future (2009, methodology): “Finally, the X-SFEM proposed in [124] is an approach relying on two major points: the implicit representation of complex geometries using random level-set functions and the use of a Galerkin approximation at both stochastic and deterministic (FE) levels (Fig.”
Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis (2015, methodology): “An Eulerian approach [11, 12, 40] should be adopted to model the imperfections in case this assumption can not be made.”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, background): “2 For SPDEs de ned on random domains, a suitable reformulation of the problem on a deterministic domain allows to work in a deterministic function space V [54,8,40].”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, methodology): “In order to accurately compute E(AKHαHβ), a suitable integration technique has been proposed in [21], which consists in introducing a Gaussian quadrature de ned on a partition of Ξ which approximates the partition {Ξ1 ,Ξ K 2 ,Ξ K c }.”