Simulation of random fields on random domains

verfasst von

Zhibao Zheng, Marcos Valdebenito, Michael Beer, Udo Nackenhorst

Abstract

This paper focuses on the simulation of random fields on random domains. This is an important class of problems in fields such as topology optimization and multiphase material analysis. However, there is still a lack of effective methods to simulate this kind of random fields. To this end, we extend the classical Karhunen–Loève expansion (KLE) to this class of problems, and we denote this extension as stochastic Karhunen–Loève expansion (SKLE). We present three numerical algorithms for solving the stochastic integral equations arising in the SKLE. The first algorithm is an extension of the classical Monte Carlo simulation (MCS), which is used to solve the stochastic integral equation on each sampled domain. However, such approach demands remeshing each sampled domain and solving the corresponding integral equation, which can become computationally very demanding. In the second algorithm, a domain transformation is used to map the random domain into a reference domain, and only one mesh for the reference domain is required. In this way, remeshing different sample realizations of the random domain is avoided and much computational effort is thus saved. MCS is then adopted to solve the corresponding stochastic integral equation. Further, to avoid the computational effort of MCS, the third algorithm proposed in this contribution involves a reduced-order method to solve the stochastic integral equation efficiently. In this third algorithm, stochastic eigenvectors are represented as a sum of products of unknown random variables and deterministic vectors, where the deterministic vectors are efficiently computed by solving deterministic eigenvalue problems. The random variables and stochastic eigenvalues that appear in this third algorithm are calculated by a reduced-order stochastic eigenvalue problem constructed by the obtained deterministic vectors. Based on the obtained stochastic eigenvectors, the target random field is then simulated and reformulated as a classical KLE-like representation. Finally, three numerical examples are presented to demonstrate the performance of the proposed methods.

Organisationseinheit(en)

Institut für Risiko und Zuverlässigkeit
Institut für Baumechanik und Numerische Mechanik

Externe Organisation(en)

Technische Universität Dortmund
The University of Liverpool
Tongji University

Typ

Artikel

Journal

Probabilistic Engineering Mechanics

Band

73

ISSN

0266-8920

Publikationsdatum

07.2023

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Statistische und nichtlineare Physik, Tief- und Ingenieurbau, Kernenergie und Kernkraftwerkstechnik, Physik der kondensierten Materie, Luft- und Raumfahrttechnik, Meerestechnik, Maschinenbau

Elektronische Version(en)

https://doi.org/10.1016/j.probengmech.2023.103455 (Zugang: Geschlossen)