Volumetric locking-free Mixed Virtual Element Methods for Contact Problems

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 ($u\mathrm{/}p$) 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 $\lambda$, 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.