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 - 2025/2/15

Y1 - 2025/2/15

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 -