Computational Algorithms for Contact Mechanics

Contact mechanics is a fundamental branch of computational solid mechanics that deals with the interaction between deformable or rigid bodies at their interfaces. This article provides a comprehensive overview of key computational algorithms used to solve contact problems, spanning classical analytical solutions, finite element formulations, constraint enforcement methods (such as penalty, Lagrange multipliers, and augmented Lagrangian), and modern approaches including mortar methods and discrete element modeling. We discuss mathematical formulations, including the Signorini problem for frictionless unilateral contact, frictional models like Coulomb's law, and numerical challenges such as non-smoothness, detection, and convergence. Applications in engineering—from Hertzian contacts to large-scale simulations in automotive and biomechanical systems—are highlighted. The review bridges classical continuum mechanics with contemporary numerical techniques, emphasizing accuracy, robustness, and efficiency. Future directions, including machine learning integration, are briefly noted.