Numerical analysis of a quasistatic elasto-piezoelectric contact problem with damage

Abstract

In this work, a contact problem between an elasto-piezoelectric body and a deformable obstacle is numerically studied. The damage of the material, caused by internal tension or compression, is also included into the model. The variational formulation leads to a coupled system composed of a nonlinear variational equation for the displacement field, a linear variational equation for the electric potential, and a nonlinear parabolic variational equation for the damage field. The existence of a unique weak solution is recalled. Then, a fully discrete scheme is introduced by using a finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, and the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, a two-dimensional example is presented to demonstrate the behaviour of the solution.

Keywords:

• elasto-piezoelectricity
• damage
• normal compliance
• finite elements
• error estimates
• numerical simulations

2000 AMS Subject Classifications :

• 74B20
• 74R05
• 74M15
• 65M15
• 65M60
• 74S05

Acknowledgements

This work was partially supported by the Ministerio de Educación y Ciencia (Project MTM2006-13981).