Details
| Original language | English |
|---|---|
| Article number | 103455 |
| Journal | Probabilistic Engineering Mechanics |
| Volume | 73 |
| Early online date | 25 Apr 2023 |
| Publication status | Published - Jul 2023 |
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.
Keywords
- Domain transformation, Random domains, Random fields, Stochastic eigenvalue equations, Stochastic Karhunen–Loève expansion
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Engineering(all)
- Civil and Structural Engineering
- Energy(all)
- Nuclear Energy and Engineering
- Physics and Astronomy(all)
- Condensed Matter Physics
- Engineering(all)
- Aerospace Engineering
- Engineering(all)
- Ocean Engineering
- Engineering(all)
- Mechanical Engineering
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In: Probabilistic Engineering Mechanics, Vol. 73, 103455, 07.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Simulation of random fields 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 and the International Research Training Group 2657 (IRTG 2657) funded by the German Research Foundation (DFG) (Grant number 433082294 ).
PY - 2023/7
Y1 - 2023/7
N2 - 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.
AB - 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.
KW - Domain transformation
KW - Random domains
KW - Random fields
KW - Stochastic eigenvalue equations
KW - Stochastic Karhunen–Loève expansion
UR - http://www.scopus.com/inward/record.url?scp=85154064754&partnerID=8YFLogxK
U2 - 10.1016/j.probengmech.2023.103455
DO - 10.1016/j.probengmech.2023.103455
M3 - Article
AN - SCOPUS:85154064754
VL - 73
JO - Probabilistic Engineering Mechanics
JF - Probabilistic Engineering Mechanics
SN - 0266-8920
M1 - 103455
ER -