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.

中文翻译：

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用于随机接触分析的随机增广拉格朗日乘子方法


本文提出了随机增广拉格朗日乘子方法来解决具有不确定性的接触问题，其中随机接触约束由弱罚和随机拉格朗日乘子施加。首先将原始随机接触问题的随机位移分解为接触和非接触式随机解两部分。每个部分都是由一组随机变量和确定性向量的乘积之和近似的。然后，提出了两种交替迭代算法，以贪婪的方式求解每对随机变量和确定性向量，分别称为随机 Uzawa 算法和广义随机 Uzawa 算法。随机 Uzawa 算法被认为是经典 Uzawa 算法的随机扩展，它涉及随机拉格朗日乘子与每对随机变量和确定性向量之间的全局交替迭代，以及随机变量和确定性向量之间的局部交替迭代。广义随机 Uzawa 算法不需要局部迭代，仅依赖于随机变量、确定性向量和随机拉格朗日乘子之间的三分量交替迭代。为了进一步提高计算精度，随机解由等效随机接触界面系统重新计算，该系统使用获得的确定性向量构建。它只涉及接触随机解，因此具有良好的收敛性。 此外，由于所提出的解近似和迭代算法对随机维度不敏感，因此所提出的方法无需修改即可应用于高维随机接触问题。三个基准证明了所提出的方法的有希望的性能。