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.
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Alleche, B.B., Rădulescu, V.D.: The Ekeland variational principle for equilibrium problems revisited and applications. Nonlin. Anal. Real World Appl. 23, 17–25 (2015)
Aubin, J.P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory. Springer, Berlin (1984)
Bartosz, K.: Numerical methods for evolution hemivariational inequalities, Chapter 5. In: Han, W., et al (eds.) Advances in Variational and Hemivariational Inequalities with Applications. Theory, Numerical Analysis, and Applications, Advances in Mechanics and Mathematics, vol. 33, pp 111–144. Springer (2015)
Carstensen, C., Gwinner, J.: A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems. Ann. Mat. Pura Appl. 177, 363–394 (1999)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)
Costea, N., Rădulescu, V. D.: Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term. J. Glob. Optim. 52, 743–756 (2012)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)
Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
Han, J.F., Migórski, S., Zeng, H.: Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response. Nonlin. Anal. Real World Appl. 28, 229–250 (2016)
Han, W., Migórski, S., Sofonea, M.: Analysis of a general dynamic history-dependent variational-hemivariational inequality. Nonlin. Anal. Real World Appl. 36, 69–88 (2017)
Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55, 640–663 (2017)
Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, 2 (2013)
Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity Studies in Advanced Mathematics, vol. 30. Americal Mathematical Society, Providence, International Press, Somerville (2002)
Han, W., Migórski, S., Sofonea, M.: A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)
Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Methods for Hemivariational Inequalities. Kluwer, Dordrecht (1999)
Kac̆ur, J.: Application of Rothe’s method to perturbed linear hyperbolic equations and variational inequalities. Czechoslov. Math. J. 34, 92–106 (1984)
Kacur, J.: Method of Rothe in Evolution Equations Teubner-Texte zur Mathematik, vol. 80. B.G. Teubner, Leipzig (1985)
Liu, Z.H., Migórski, S., Zeng, S.D.: Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces. J. Differ. Equa. 263, 3989–4006 (2017)
Migórski, S.: Existence of solutions for a class of history-dependent evolution hemivariational inequalities. Dyn. Syst. Appl. 21, 319–330 (2012)
Migórski, S., Zeng, S.D.: Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model. Nonlin. Anal. Real World Appl. 43, 121–143 (2018)
Migórski, S., Zeng, S.D.: Penalty and regularization method for variational-hemivariational inequalities with application to frictional contact. ZAMM-Z Angew. Math. Me. 98, 1503–1520 (2018)
Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)
Migórski, S., Ochal, A.: Quasti-static hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)
Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271–290 (2008)
Migórski, S., Ochal, A., Sofonea, M.: History-dependent variational-hemivariational inequalities in contact mechanics. Nonlin. Anal. Real World Appl. 22, 604–618 (2015)
Migórski, S., Zeng, S.D.: A class of differential hemivariational inequalities in Banach spaces. J. Global Optim. 72, 761–779 (2018)
Migórski, S., Ochal, A., Sofonea, M.: Evolutionary inclusions and hemivariational inequalities, Chapter 2 in Advances in Variational and Hemivariational Inequalities. Theory, Numerical Analysis, and Applications. In: Han, W., et al (eds.) Advances in Mechanics and Mathematics Series, vol. 33, pp 39–64. Springer, Heidelberg (2015)
Migórski, S., Ogorzaly, J.: A class of evolution variational inequalities with memory and its application to viscoelastic frictional contact problems. J. Math. Anal. Appl. 442, 685–702 (2016)
Migórski, S., Ogorzaly, J.: Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics. Zeitschrift fü,r angewandte Mathematik und Physik 68, 15 (2017). https://doi.org/10.1007/s00033-016-0758-4
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York (1995)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Basel, Birkhäuser (1985)
Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)
Papageorgiou, N.S., Rădulescu, V.D., Repovs̆, D.D.: Nonhomogeneous hemivariational inequalities with indefinite potential and robin boundary condition. J. Optim. Theory Appl. 175, 293–323 (2017)
Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer, Berlin (2004)
Sofonea, M., Han, W., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman & Hall/CRC, Boca Raton (2005)
Sofonea, M., Han, W., Migórski, S.: Numerical analysis of history-dependent variational-hemivariational inequalities with applications to contact problems. Euro. J. Appl. Math. 26, 427–452 (2015)
Sofonea, M., Migórski, S., Han, W.: A penalty method for history-dependent variational-hemivariational inequalities. Comput. Math. Appl. 75, 2561–2573 (2018)
Sofonea, M., Matei, A.: History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491 (2011)
Sofonea, M., Patrulescu, F.: Penalization of history-dependent variational inequalities. Eur. J. Appl. Math. 25, 155–176 (2014)
Sofonea, M., Xiao, Y.: Fully history-dependent quasivariational inequalities in contact mechanics. Appl. Anal. 95, 2464–2484 (2016)
Zeidler, E.: Nonlinear Functional Analysis and Applications II A/B. Springer, New York (1990)
Zeng, S.D., Liu, Z.H., Migórski, S.: A class of fractional differential hemivariational inequalities with application to contact problem. Z. Angew. Math. Phys. 69, 36,23 (2018)
Zeng, S.D.: A class of time-fractional hemivariational inequalities with application to frictional contact problem. Commun. Nonlinear Sci. 56, 34–48 (2018)
Zeng, S.D., Migórski, S.: Noncoercive hyperbolic variational inequalities with applications to contact mechanics. J. Math. Anal. Appl. 455, 619–637 (2017)
Funding
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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Migórski, S., Zeng, S. Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics. Numer Algor 82, 423–450 (2019). https://doi.org/10.1007/s11075-019-00667-0
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DOI: https://doi.org/10.1007/s11075-019-00667-0
Keywords
- Hemivariational inequality
- Clarke subgradient
- History-dependent operator
- Rothe method
- Finite element method
- Error estimates
- Viscoelastic material
- Frictional contact