Efficient parametric model order reduction in contact mechanics

Abstract

Contact problems are inherently non-linear and present significant computational challenges in simulations. Traditional proper orthogonal decomposition-based non-linear system reduction often proves inefficient due to the complexity of handling full-scale models. This article presents a generalized parametric model order reduction framework tailored for dynamic contact problems involving arbitrarily-shaped inclusions, incorporating various specialized approaches. The proposed framework features a two-tier reduction process: the first tier applies a proper orthogonal decomposition-based model order reduction approach, while the second tier employs either the discrete empirical interpolation method or energy conserving sampling and weighting to address the reduction of non-linear terms. Discrete empirical interpolation method identifies nodes in contact as indices for the reduction of non-linear terms, and energy conserving sampling and weighting highlight contact elements as the most contributing elements to virtual work associated with non-linear forces. These results align with physics of the problem, as in this article, everything is linear except for the contact non-linearity. Two construction strategies are evaluated: one utilizing a single global basis and another using interpolated local bases. The framework leveraging local bases interpolation on the tangent space to the Grassmannian manifold demonstrates better accuracy compared to the global basis approach. The effectiveness of the different approaches within the generalized parametric model order reduction framework is evaluated through standard 2D dynamic contact problems involving perfectly bonded inclusions.