Numerical Studies of a Hemivariational Inequality for a Viscoelastic Contact Problem with Damage

TL;DR

This paper develops and analyzes a numerical scheme for a hemivariational inequality modeling a viscoelastic contact problem with damage, proving existence, uniqueness, and convergence of the method with supporting simulations.

Contribution

It introduces a fully discrete finite element scheme for a complex hemivariational inequality with damage, providing error estimates and numerical validation.

Findings

The scheme achieves optimal order error estimates.

Numerical examples confirm theoretical convergence rates.

The model effectively captures damage in viscoelastic contact problems.

Abstract

This paper is devoted to the study of a hemivariational inequality modeling the quasistatic bilateral frictional contact between a viscoelastic body and a rigid foundation. The damage effect is built into the model through a parabolic differential inclusion for the damage function. A solution existence and uniqueness result is presented. A fully discrete scheme is introduced with the time derivative of the damage function approximated by the backward finite different and the spatial derivatives approximated by finite elements. An optimal order error estimate is derived for the fully discrete scheme when linear elements are used for the velocity and displacement variables, and piecewise constants are used for the damage function. Simulation results on numerical examples are reported illustrating the performance of the fully discrete scheme and the theoretically predicted convergence orders.