Details
| Original language | English |
|---|---|
| Article number | 117661 |
| Journal | Computer Methods in Applied Mechanics and Engineering |
| Volume | 435 |
| Early online date | 18 Dec 2024 |
| Publication status | E-pub ahead of print - 18 Dec 2024 |
Abstract
This article presents stochastic augmented Lagrangian multiplier methods to solve contact problems with uncertainties, in which stochastic contact constraints are imposed by weak penalties and stochastic Lagrangian multipliers. The stochastic displacements of original stochastic contact problems are first decomposed into two parts, including contact and non-contact stochastic solutions. Each part is approximated by a summation of a set of products of random variables and deterministic vectors. Two alternating iterative algorithms are then proposed to solve each pair of random variable and deterministic vector in a greedy way, named stochastic Uzawa algorithm and generalized stochastic Uzawa algorithm. The stochastic Uzawa algorithm is considered as a stochastic extension of the classical Uzawa algorithm, which involves a global alternating iteration between stochastic Lagrangian multipliers and each pair of random variable and deterministic vector, and a local alternating iteration between the random variable and the deterministic vector. The generalized stochastic Uzawa algorithm does not require the local iteration and only relies on a three-component alternating iteration between the random variable, the deterministic vector and the stochastic Lagrangian multipliers. To further improve computational accuracy, the stochastic solution is recalculated by an equivalent stochastic contact interface system that is constructed using the obtained deterministic vectors. It only involves the contact stochastic solution and therefore has good convergence. Furthermore, since the proposed solution approximation and iterative algorithms are not sensitive to stochastic dimensions, the proposed methods can be applied to high-dimensional stochastic contact problems without modifications. Three benchmarks demonstrate the promising performance of the proposed methods.
Keywords
- Curse of dimensionality, Stochastic augmented Lagrangian multiplier method, Stochastic contact interface system, Stochastic contact problems, Stochastic Uzawa iteration
ASJC Scopus subject areas
- Engineering(all)
- Computational Mechanics
- Engineering(all)
- Mechanics of Materials
- Engineering(all)
- Mechanical Engineering
- Physics and Astronomy(all)
- General Physics and Astronomy
- Computer Science(all)
- Computer Science Applications
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In: Computer Methods in Applied Mechanics and Engineering, Vol. 435, 117661, 15.02.2025.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Stochastic augmented Lagrangian multiplier methods for stochastic contact analysis
AU - Zheng, Zhibao
AU - Nackenhorst, Udo
N1 - Publisher Copyright: © 2024 The Authors
PY - 2024/12/18
Y1 - 2024/12/18
N2 - This article presents stochastic augmented Lagrangian multiplier methods to solve contact problems with uncertainties, in which stochastic contact constraints are imposed by weak penalties and stochastic Lagrangian multipliers. The stochastic displacements of original stochastic contact problems are first decomposed into two parts, including contact and non-contact stochastic solutions. Each part is approximated by a summation of a set of products of random variables and deterministic vectors. Two alternating iterative algorithms are then proposed to solve each pair of random variable and deterministic vector in a greedy way, named stochastic Uzawa algorithm and generalized stochastic Uzawa algorithm. The stochastic Uzawa algorithm is considered as a stochastic extension of the classical Uzawa algorithm, which involves a global alternating iteration between stochastic Lagrangian multipliers and each pair of random variable and deterministic vector, and a local alternating iteration between the random variable and the deterministic vector. The generalized stochastic Uzawa algorithm does not require the local iteration and only relies on a three-component alternating iteration between the random variable, the deterministic vector and the stochastic Lagrangian multipliers. To further improve computational accuracy, the stochastic solution is recalculated by an equivalent stochastic contact interface system that is constructed using the obtained deterministic vectors. It only involves the contact stochastic solution and therefore has good convergence. Furthermore, since the proposed solution approximation and iterative algorithms are not sensitive to stochastic dimensions, the proposed methods can be applied to high-dimensional stochastic contact problems without modifications. Three benchmarks demonstrate the promising performance of the proposed methods.
AB - This article presents stochastic augmented Lagrangian multiplier methods to solve contact problems with uncertainties, in which stochastic contact constraints are imposed by weak penalties and stochastic Lagrangian multipliers. The stochastic displacements of original stochastic contact problems are first decomposed into two parts, including contact and non-contact stochastic solutions. Each part is approximated by a summation of a set of products of random variables and deterministic vectors. Two alternating iterative algorithms are then proposed to solve each pair of random variable and deterministic vector in a greedy way, named stochastic Uzawa algorithm and generalized stochastic Uzawa algorithm. The stochastic Uzawa algorithm is considered as a stochastic extension of the classical Uzawa algorithm, which involves a global alternating iteration between stochastic Lagrangian multipliers and each pair of random variable and deterministic vector, and a local alternating iteration between the random variable and the deterministic vector. The generalized stochastic Uzawa algorithm does not require the local iteration and only relies on a three-component alternating iteration between the random variable, the deterministic vector and the stochastic Lagrangian multipliers. To further improve computational accuracy, the stochastic solution is recalculated by an equivalent stochastic contact interface system that is constructed using the obtained deterministic vectors. It only involves the contact stochastic solution and therefore has good convergence. Furthermore, since the proposed solution approximation and iterative algorithms are not sensitive to stochastic dimensions, the proposed methods can be applied to high-dimensional stochastic contact problems without modifications. Three benchmarks demonstrate the promising performance of the proposed methods.
KW - Curse of dimensionality
KW - Stochastic augmented Lagrangian multiplier method
KW - Stochastic contact interface system
KW - Stochastic contact problems
KW - Stochastic Uzawa iteration
UR - http://www.scopus.com/inward/record.url?scp=85212349897&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2024.117661
DO - 10.1016/j.cma.2024.117661
M3 - Article
AN - SCOPUS:85212349897
VL - 435
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
M1 - 117661
ER -