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Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

  • Stanisław MigórskiEmail author
  • Shengda Zeng
Open Access
Original Paper
  • 41 Downloads

Abstract

In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate.

Keywords

Hemivariational inequality Clarke subgradient History-dependent operator Rothe method Finite element method Error estimates Viscoelastic material Frictional contact 

Mathematics Subject Classification (2010)

35L15 35L86 35L87 74Hxx 74M10 

Notes

Funding information

This project was supported by the H2020-MSCA-RISE-2018 Research and Innovation Staff Exchange Scheme Fellowship within the Project No. 823731 CONMECH, the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0, Natural Sciences Foundation of Guangxi Grant No. 2018JJA110006, Beibu Gulf University Project No. 2018KYQD06, National Natural Science Foundation of China (Grant Nos. 11561007).

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Copyright information

© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Applied MathematicsChengdu University of Information TechnologyChengduPeople’s Republic of China
  2. 2.Chair of Optimization and ControlJagiellonian University in KrakowKrakowPoland
  3. 3.Faculty of Mathematics and Computer ScienceJagiellonian University in KrakowKrakowPoland

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