Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body

Krzysztof Bartosz

doi:10.3934/eect.2020059

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2020, Volume 9, Issue 4: 961-980. Doi: 10.3934/eect.2020059

This issue Previous Article Measurable solutions to general evolution inclusions Next Article A dynamic viscoelastic problem with friction and rate-depending contact interactions

Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body

Krzysztof Bartosz

Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

The paper is dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received: December 2019

Revised: March 2020

Early access: May 2020

Published: December 2020

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Abstract

We consider a contact process between a body and a foundation. The body is assumed to be viscoelastic and piezoelectric and the contact is dynamic. Unlike many related papers, the body is assumed to be non-clamped. The contact conditions has a form of inclusions involving the Clarke subdifferential of locally Lipschitz functionals and they have nonmonotone character. The problem in its weak formulation has a form of two coupled Clarke subdifferential inclusions, from which the first one is dynamic and the second one is stationary. The main goal of the paper is numerical analysis of the studied problem. The corresponding numerical scheme is based on the spatial and temporal discretization. Furthermore, the spatial discretization is based on the first order finite element method, while the temporal discretization is based on the backward Euler scheme. We show that under suitable regularity conditions the error between the exact solution and the approximate one is estimated in an optimal way, namely it depends linearly upon the parameters of discretization.

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Mathematics Subject Classification: 65M60, 65M15, 74M10, 74M15, 74S05, 74S20.

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