Stochastic augmented Lagrangian multiplier methods for stochastic contact analysis

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.