A stochastic finite element scheme for solving partial differential equations defined on random domains

Zheng, Zhibao, Valdebenito, Marcos, Beer, Michael ORCID: 0000-0002-0611-0345 and Nackenhorst, Udo (2023) A stochastic finite element scheme for solving partial differential equations defined on random domains. Computer Methods in Applied Mechanics and Engineering, 405. p. 115860.

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Official URL: http://dx.doi.org/10.1016/j.cma.2022.115860

Abstract

This paper proposes a novel stochastic finite element scheme to solve partial differential equations defined on random domains. A geometric mapping algorithm first transforms the random domain into a reference domain. By combining the mesh topology (i.e. the node numbering and the element numbering) of the reference domain and random nodal coordinates of the random domain, random meshes of the original problem are obtained by only one mesh of the reference domain. In this way, the original problem is still discretized and solved on the random domain instead of the reference domain. A random isoparametric mapping of random meshes is then proposed to generate the stochastic finite element equation of the original problem. We adopt a weak-intrusive method to solve the obtained stochastic finite element equation. In this method, the unknown stochastic solution is decoupled into a sum of the products of random variables and deterministic vectors. Deterministic vectors are computed by solving deterministic finite element equations, and corresponding random variables are solved by a proposed sampling method. The computational effort of the proposed method does not increase dramatically as the stochastic dimension increases and it can solve high-dimensional stochastic problems with low computational effort, thus the proposed method avoids the curse of dimensionality successfully. Four numerical examples are given to demonstrate the good performance of the proposed method.

Item Type:	Article
Uncontrolled Keywords:	Random domains, Mesh transformation, Random isoparametric mapping, Stochastic finite element method, Random interfaces
Divisions:	Faculty of Science and Engineering > School of Engineering
Depositing User:	Symplectic Admin
Date Deposited:	10 Jan 2023 09:33
Last Modified:	28 Dec 2023 02:30
DOI:	10.1016/j.cma.2022.115860
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3166944