TY - JOUR

T1 - A stochastic finite element scheme for solving partial differential equations defined on random domains

AU - Zheng, Zhibao

AU - Valdebenito, Marcos

AU - Beer, Michael

AU - Nackenhorst, Udo

N1 - Funding Information: The authors are grateful to the Alexander von Humboldt Foundation, Germany and the International Research Training Group 2657 (IRTG 2657) funded by the German Research Foundation (DFG) (Grant number 433082294 ).

PY - 2023/2/15

Y1 - 2023/2/15

N2 - 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.

AB - 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.

KW - Mesh transformation

KW - Random domains

KW - Random interfaces

KW - Random isoparametric mapping

KW - Stochastic finite element method

UR - http://www.scopus.com/inward/record.url?scp=85145189252&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2022.115860

DO - 10.1016/j.cma.2022.115860

M3 - Article

AN - SCOPUS:85145189252

VL - 405

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 115860

ER -