(untitled) (REC 2026 - 11th International Workshop on Reliable Engineering Computing: Reliability Computations in a Data and Model-Driven World, 2026)
Stochastic contact problems (SCPs) arise in many engineering applications where uncertainties in material properties, geometry or loading conditions affect contact behavior. Compared with deterministic contact analysis, the coupling of contact mechanics and randomness dramatically increases the computational complexity. In particular, stochastic contact nodes whose probability density functions can be strongly discontinuous or even singular render SCPs highly sensitive to uncertain inputs and computationally demanding. To address these challenges, a unified framework is presented for solving SCPs. It integrates Lagrange multiplier, penalty or augmented Lagrange multiplier formulations with semi-reduced-order approximations. In this framework, stochastic solutions are approximated as sums of products of random variables and deterministic vectors, thereby transforming high-dimensional SCPs into deterministic problems coupled with low-dimensional stochastic algebraic systems. Semi-reduced-order equations further combine smooth reduced-order solutions with full-order nonsmooth components to reduce computational cost without sacrificing accuracy. In addition, the framework is extended to rolling SCPs via the development of a stochastic arbitrary Lagrangian–Eulerian (ALE) formulation, which enables efficient analysis of rolling interfaces with uncertainties. Numerical results on high-dimensional SCP benchmarks and engineering-level cases such as rolling tires demonstrate that the proposed framework is both efficient and scalable. It accurately captures stochastic contact nodes while also significantly reducing computational cost. This framework provides a powerful tool for robust design and reliability assessment of engineering contact systems under uncertainty.