A Comprehensive Overview of Computational Algorithms for Contact Mechanics: From Classical Formulations to Modern Machine Learning Approaches

The accurate and efficient simulation of contact phenomena is a cornerstone of computational solid mechanics, underpinning a vast array of engineering applications from automotive braking systems to biomedical implants. This paper provides a comprehensive overview of the fundamental algorithms developed for solving contact mechanics problems. We systematically review the core mathematical formulations, beginning with the strong form of the contact problem and its weak variational formulation. The discussion then progresses through the primary solution strategies, including the Lagrange multiplier method, the penalty method, and the augmented Lagrangian method, highlighting their respective advantages and numerical challenges. We detail the computational geometry aspects, focusing on robust contact detection algorithms for both node-to-surface and surface-to-surface discretizations. The paper further explores advanced topics such as finite deformation contact, friction modeling, and the incorporation of adaptive mesh refinement. A critical analysis of algorithm performance is presented through benchmark numerical results. Finally, we conclude with a forward-looking perspective on emerging trends, including isogeometric analysis for smooth contact surfaces and the integration of machine learning techniques to accelerate contact computations, outlining the future trajectory of this critical field.