Volumetric locking-free Mixed Virtual Element Methods for Contact Problems

Abstract: We consider the approximation of the 2D frictionless contact problem in elasticity using the Virtual Element Methods (VEMs). To overcome the volumetric locking phenomenon in the nearly incompressible case, we adopt a mixed displacement/pressure ($u/p$) variational formulation, where pressure is introduced as an independent unknown. We present the VEM discretization and develop a general error analysis, keeping explicit track of the constants involved in the error estimates, thus allowing to consider meshes with "small edges". As examples, we consider two possible VEM schemes: a first-order scheme and a second-order scheme. The numerical results confirm the theoretical predictions, specifically both schemes show: 1) robustness with respect to the volumetric parameter $λ$, thus preventing the occurrence of the volumetric locking phenomenon; 2) good behavior even in the presence of "small edges"; 3) achievement of the expected theoretical convergence rates.