Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity

A dynamic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle is numerically studied in this paper. The contact is modelled using the normal compliance contact condition and the linear electro-viscoelastic constitutive law is employed to simulate the piezoelectric effects. The variational formulation is a coupled system composed of a parabolic nonlinear variational equation for the velocity field and a linear variational equation for the electric potential. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the accuracy and the performance of the algorithm.