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.

中文翻译：

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非夹紧压电粘弹性体非单调动态接触问题的数值分析

我们考虑身体与基础之间的接触过程。假定该主体是粘弹性和压电的，并且接触是动态的。与许多相关论文不同，该身体被假定为不夹紧的。接触条件具有包含Lipschitz泛函的Clarke次微分的包含物形式，并且具有非单调性。弱公式中的问题具有两个耦合的Clarke次微分夹杂物的形式，第一个是动态的，第二个是平稳的。本文的主要目的是对所研究问题进行数值分析。相应的数值方案基于空间和时间离散化。此外，空间离散化基于一阶有限元方法，而时间离散化则基于后向Euler方案。我们表明，在适当的规则性条件下，精确解和近似解之间的误差是以最佳方式估算的，即它线性依赖于离散化的参数。