Numerical approximation of an electro-elastic frictional contact problem modeled by hemivariational inequality
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Numerical approximation of an electro-elastic frictional contact problem modeled by hemivariational inequality Wei Xu1,2 · Ziping Huang1,2 · Weimin Han3 · Wenbin Chen4 · Cheng Wang1 Received: 23 July 2019 / Revised: 14 July 2020 / Accepted: 14 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, an electro-elastic frictional contact problem is studied numerically as a hemivariational inequality. Convergence of the Galerkin approximation for the hemivariational inequality is proved, and Céa’s type inequalities are derived for error estimation. The results are applied to the electro-elastic contact problem, and an optimal order error estimate is deduced for linear element approximation. Finally, two numerical examples are reported, providing numerical evidence of the optimal convergence order theoretically predicted. Keywords Hemivariational inequality · Galerkin approximation · Optimal order error estimate · Electro-elastic material · Frictional contact Mathematics Subject Classification 65N30 · 65N15 · 74M10 · 74M15
1 Introduction This paper is devoted to numerical analysis of a static problem for contact between an electro-elastic body and an electrically conductive foundation. Such contact phenomena arise in engineering devices, e.g., switches in radiotronics and measuring equipment. The main feature of the electro-elastic contact problem is the coupling between the mechanical and electrical properties of the materials. In the coupled system, the forces acting on the body affect the appearance of electric charge, and in turn, the action of the electric field generates stress or strain in the body. Recently, various contact problems for piezoelectric materials have attracted much interest. For existence and uniqueness of a weak solution of the contact problems, the reader is referred to Lerguet et al. (2007), Migórski (2006), Migórski et al. (2010), Migórski et al. (2011). On numerical approximation of piezoelectric contact problems, only
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Cheng Wang [email protected]
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School of Mathematical Sciences, Tongji University, Shanghai 200092, China
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Tongji Zhejiang College, Jiaxing 314051, China
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Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
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School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China 0123456789().: V,-vol
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a few references can be found in the literature. In Barboteu et al. (2008a), two frictionless contact models for electro-elastic materials are studied in the framework of a coupled system consisting of a variational inequality and an elliptic equation. Linear finite element is used to solve the problem numerically, and some error estimates are derived. In Barboteu and Sofonea (2009), an algorithm based on the finite element method and the backward Euler scheme is applied to solve a quasistatic contact model for electro-viscoelastic materials. An augmented Lagrangian method is des
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