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On convergence of the penalty method for a static unilateral contact problem with nonlocal friction in electro-elasticity

Published online by Cambridge University Press:  04 June 2015

El-H. BENKHIRA ,
El-H. ESSOUFI  and
R. FAKHAR
El-H. BENKHIRA
Affiliation:
University Moulay Ismaïl, ESTM, Laboratory LEM2A, BP 3103, Toulal-Meknès, Morocco email: benkhirahassan@yahoo.fr
El-H. ESSOUFI
Affiliation:
Univ Hassan 1, Laboratory MISI, 26000 Settat, Morocco email: e.h.essoufi@gmail.com
R. FAKHAR
Affiliation:
Univ Hassan 1, Laboratory LS3M, 25000 Settat, Morocco email: rachidfakhar@yahoo.fr
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Abstract

In this paper, we consider the penalty method to solve the unilateral contact with friction between an electro-elastic body and a conductive foundation. Mathematical properties, such as the existence of a solution to the penalty problem and its convergence to the solution of the original problem, are reported. Then, we present a finite elements approximation for the penalised problem and prove its convergence. Finally, we propose an iterative method to solve the resulting finite element system and establish its convergence.

Keywords

PiezoelectricVariational inequalityStatic friction contactNonlocal frictionSignorini conditionPenalty methodFixed point processFinite element approximation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 1 , February 2016 , pp. 1 - 22
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Barboteu, M., Fernández, J. R. & Ouafik, Y. (2008) Numerical analysis of two frictionless elasto-piezoelectric contact problems. J. Math. Anal. Appl. 329, 905–917.Google Scholar
[2] Bisegna, P., Lebon, F. & Maceri, F. (2002) The unilateral frictional contact of a piezoelectric body with a rigid support. Contact Mechanics, Kluwer, Dordrecht, pp. 347–354.CrossRefGoogle Scholar
[3] Chouly, F. & Hild, P. (2013) On convergence of the penalty method for unilateral contact problems. Appl. Numer. Math. 65, 27–40.CrossRefGoogle Scholar
[4] Ciarlet, P. G. (1978) The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford.Google Scholar
[5] Duvaut, G. & Lions, J.-L. (1972) Les inéquations en mécanique et en physique, Dunod, Paris.Google Scholar
[6] Eck, C. & Jarušek, J. (1998) Existence results for the static contact problem with Coulomb friction. Math. Mod. Methods Appl. Sci. 8, 445–463.Google Scholar
[7] Essoufi, El-H., Benkhira, El-H. & Fakhar, R. (2010) Analysis and numerical approximation of an electroelastic frictional contact problem. Adv. Appl. Math. Mech. 2 (3), 355–378.Google Scholar
[8] Glowinski, R. (1984) Numerical Methods for Nonlinear Variational Problems, Springer, New York.CrossRefGoogle Scholar
[9] Hüeber, S., Matei, A. & Wohlmuth, B. I. (2005) A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48 (96), 209–232.Google Scholar
[10] Kikuchi, N. & Oden, J. T. (1988) Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia.Google Scholar
[11] Lerguet, Z., Shillor, M. & Sofonea, M. (2007) A frictional contact problem for an electro-viscoelastic body. Electron. J. Differ. Equ. 2007 (170), 1–16.Google Scholar
[12] Maceri, F. & Bisegna, P. (1998) The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Modelling 28, 19–28.Google Scholar
[13] Migórski, S. (2006) Hemivariational inequality for a frictional contact problem in elastopiezoelectricity. Discrete Continuous Dyn. Syst. 6, 1339–1356.Google Scholar
[14] Migórski, S. (2008) A class of hemivariational inequality for electroelastic contact problems with slip dependent friction. Discrete Continuous Dyn. Syst. 1, 117–126.CrossRefGoogle Scholar
[15] Migórski, S., Ochal, A. & Sofonea, M. (2009) Weak solvability of a piezoelectric contact problem. Eur. J. Appl. Math. 20, 145–167.Google Scholar
[16] Nečas, J. & Hlaváček, I. (1981) Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction, Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York.Google Scholar
[17] Sofonea, M. & Essoufi, El-H. (2004) A Piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9, 229–242.Google Scholar
[18] Strang, G. & Fix, G. (1973) An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs.Google Scholar
[19] Temam, R. (1983) Problème mathématiques en plasticité, Gauthier-Villars, Paris.Google Scholar
[20] Wriggers, P. (2002) Computational Contact Mechanics, Wiley, Chichester.Google Scholar