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Volume 13, Issue 3
Numerical Analysis of a Dynamic Contact Problem with History-Dependent Operators

Hailing Xuan, Xiaoliang Cheng, Weimin Han & Qichang Xiao

DOI: 10.4208/nmtma.OA-2019-0130

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 569-594.

Published online: 2020-03

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  • Abstract

In this paper, we study a dynamic contact model with long memory which allows both the convex potential and nonconvex superpotentials to depend on history-dependent operators. The deformable body consists of a viscoelastic material with long memory and the process is assumed to be dynamic. The contact involves a nonmonotone Clarke subdifferential boundary condition and the friction is modeled by a version of the Coulomb's law of dry friction with the friction bound depending on the total slip. We introduce and study a fully discrete scheme of the problem, and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method.  This theoretical result is illustrated numerically.


  • Keywords

Variational-hemivariational inequality, history-dependent operators, finite element method, numerical approximation, optimal order error estimate.

  • AMS Subject Headings

65M15, 65N21, 65N22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hailingxuan@zju.edu.cn (Hailing Xuan)

xiaoliangcheng@zju. edu.cn (Xiaoliang Cheng)

weimin-han@uiowa.edu (Weimin Han)

xiao_qc@126.com (Qichang Xiao)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-569, author = {Xuan , HailingCheng , XiaoliangHan , Weimin and Xiao , Qichang}, title = {Numerical Analysis of a Dynamic Contact Problem with History-Dependent Operators}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {13}, number = {3}, pages = {569--594}, abstract = {

In this paper, we study a dynamic contact model with long memory which allows both the convex potential and nonconvex superpotentials to depend on history-dependent operators. The deformable body consists of a viscoelastic material with long memory and the process is assumed to be dynamic. The contact involves a nonmonotone Clarke subdifferential boundary condition and the friction is modeled by a version of the Coulomb's law of dry friction with the friction bound depending on the total slip. We introduce and study a fully discrete scheme of the problem, and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method.  This theoretical result is illustrated numerically.


}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0130}, url = {http://global-sci.org/intro/article_detail/nmtma/15776.html} }
TY - JOUR T1 - Numerical Analysis of a Dynamic Contact Problem with History-Dependent Operators AU - Xuan , Hailing AU - Cheng , Xiaoliang AU - Han , Weimin AU - Xiao , Qichang JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 569 EP - 594 PY - 2020 DA - 2020/03 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0130 UR - https://global-sci.org/intro/article_detail/nmtma/15776.html KW - Variational-hemivariational inequality, history-dependent operators, finite element method, numerical approximation, optimal order error estimate. AB -

In this paper, we study a dynamic contact model with long memory which allows both the convex potential and nonconvex superpotentials to depend on history-dependent operators. The deformable body consists of a viscoelastic material with long memory and the process is assumed to be dynamic. The contact involves a nonmonotone Clarke subdifferential boundary condition and the friction is modeled by a version of the Coulomb's law of dry friction with the friction bound depending on the total slip. We introduce and study a fully discrete scheme of the problem, and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method.  This theoretical result is illustrated numerically.


Xuan , HailingCheng , XiaoliangHan , Weimin and Xiao , Qichang. (2020). Numerical Analysis of a Dynamic Contact Problem with History-Dependent Operators. Numerical Mathematics: Theory, Methods and Applications. 13 (3). 569-594. doi:10.4208/nmtma.OA-2019-0130
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