Numerical analysis of a frictionless viscoelastic piezoelectric contact problem

¹ Laboratoire de Mathématiques et Physique pour les Systèmes (MEPS), Bâtiment B3, case courrier 12, 52 Avenue Paul Alduy, 66860 Perpignan, France. barboteu@univ-perp.fr; youssef.ouafik@univ-perp.fr
² Departamento de Matemática Aplicada, Facultade de Matemáticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. jramon@usc.es

Received: 23 July 2007
Revised: 18 January 2008

Abstract

In this work, we consider the quasistatic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelastic constitutive law is employed to model the piezoelectric material and the normal compliance condition is used to model the contact. The variational formulation is derived in a form of a coupled system for the displacement and electric potential fields. 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 an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.