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

verfasst von

Zhibao Zheng, Marcos Valdebenito, Michael Beer, Udo Nackenhorst

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.

Organisationseinheit(en)

Institut für Risiko und Zuverlässigkeit
Institut für Baumechanik und Numerische Mechanik
Internationales GRK 2657: Methoden der Numerischen Mechanik in höheren Dimensionen

Externe Organisation(en)

Technische Universität Dortmund
The University of Liverpool
Tongji University

Typ

Artikel

Journal

Computer Methods in Applied Mechanics and Engineering

Band

405

ISSN

0045-7825

Publikationsdatum

15.02.2023

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Numerische Mechanik, Werkstoffmechanik, Maschinenbau, Physik und Astronomie (insg.), Angewandte Informatik

Elektronische Version(en)

https://doi.org/10.1016/j.cma.2022.115860 (Zugang: Geschlossen)