Stabilization-free virtual element method for 2D third medium contact

The third medium contact has been proven to be an effective approach for simulating contact problems involving large deformations. Unlike traditional contact algorithms, the third medium contact introduces a third medium between two contacting bodies, thereby avoiding the complex treatment of the contact constraints. The approach has been successfully applied in different applications in the framework of the finite element method (FEM). As a generalization of the finite element method, the virtual element method (VEM) can handle arbitrary polygonal elements, providing greater flexibility for modeling third medium contact. However, due to the introduction of the projection operator, VEM requires additional stabilization terms to control the rank of the stiffness matrix. Moreover, the regularization term in the third medium contact formulation requires a second-order numerical scheme, which further complicates the application of VEM to such problems. In this work, the stabilization-free virtual element method (SFVEM) is developed and applied to solve the third medium contact problems. Different from the traditional VEM, SFVEM does not require additional stabilization terms, which simplifies the construction of the regularization term in third medium contact. Building upon the traditional second-order FEM framework, we present the specific format of SFVEM for solving third medium contact, including the construction of high-order projection operator and the tangent stiffness matrix. Numerical examples are provided to demonstrate the effectiveness and applicability of SFVEM in solving complex 2D third medium contact problems.