A Smoothed 2-Dimensional Frictionless Mortar Contact

Abstract

Contact mechanics problems are challenging due to both geometric and constitutive nonlinearities as well as the inherent non-smoothness that occurs when contact interfaces transition between separation and contact. This non-smooth behavior limits the applicability of gradient-based optimization and sensitivity analysis, particularly in finite element formulations where the active contact region evolves during deformation. In this work, a smoothed two-dimensional frictionless mortar contact formulation is developed to improve the regularity of the contact response while retaining the accuracy and robustness of mortar-based discretizations. The proposed approach introduces smoothing through a parametric ramping of the contact integration bounds and a modified gap scaling that smoothly deactivates contact as opposing surfaces rotate outof alignment. These modifications are designed to produce a C1-continuous contact energy and corresponding C 0-continuous contact forces. The formulation is presented for both penalty and Lagrange multiplier enforcement, and the resulting first and second variations are derived for use in Newton-based solution procedures. Accuracy and robustness are assessed using a series of contact patch tests, including cases with matching and non-matching meshes. While penalty enforcement yields the expected small approximation error, the Lagrange multiplier formulation recovers the analytical patch test solution to machine precision. A modified smoothing function is shown to eliminate free-edge errors present in the original construction and preserves exact patch test performance across all smoothing parameters considered. These results demonstrate that the proposed smoothing strategy provides a promising foundation for smooth contact analysis and future gradient-based design optimization involving contact