American Institute of Mathematical Sciences

December  2020, 9(4): 961-980. doi: 10.3934/eect.2020059

Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body

 Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

The paper is dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  December 2019 Revised  March 2020 Published  December 2020 Early access  May 2020

We consider a contact process between a body and a foundation. The body is assumed to be viscoelastic and piezoelectric and the contact is dynamic. Unlike many related papers, the body is assumed to be non-clamped. The contact conditions has a form of inclusions involving the Clarke subdifferential of locally Lipschitz functionals and they have nonmonotone character. The problem in its weak formulation has a form of two coupled Clarke subdifferential inclusions, from which the first one is dynamic and the second one is stationary. The main goal of the paper is numerical analysis of the studied problem. The corresponding numerical scheme is based on the spatial and temporal discretization. Furthermore, the spatial discretization is based on the first order finite element method, while the temporal discretization is based on the backward Euler scheme. We show that under suitable regularity conditions the error between the exact solution and the approximate one is estimated in an optimal way, namely it depends linearly upon the parameters of discretization.

Citation: Krzysztof Bartosz. Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body. Evolution Equations and Control Theory, 2020, 9 (4) : 961-980. doi: 10.3934/eect.2020059
References:
 [1] Y. Ayyad, M. Barboteu and J. R. Fernández, A frictionless viscoelastodynamic contact problem with energy consistent properties: Numerical analysis and computational aspects, Comput. Methods Appl. Mech. Engrg., 198 (2009), 669-679.  doi: 10.1016/j.cma.2008.10.004. [2] M. Barboteu, K. Bartosz and D. Danan, Analysis of a dynamic contact problem with nonmonotone friction and non-clamped boundary conditions, Appl. Numer. Math., 126 (2018), 53-77.  doi: 10.1016/j.apnum.2017.12.005. [3] M. Barboteu, K. Bartosz, W. Han and T. Janiczko, Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact, SIAM J. Numer. Anal., 53 (2015), 527-550.  doi: 10.1137/140969737. [4] M. Barboteu, K. Bartosz and P. Kalita, An analytical and numerical approach to a bilateral contact problem with nonmonotone friction, Int. J. Appl. Math. Comp. Sci., 23 (2013), 263-276.  doi: 10.2478/amcs-2013-0020. [5] M. Barboteu, J. R. Fernández and Y. Ouafik, Numerical analysis of a frictionless viscoelastic piezoelectric contact problem, M2AN Math. Model. Numer. Anal., 42 (2008), 667-682.  doi: 10.1051/m2an:2008022. [6] M. Barboteu, J. R. Fernández and R. Tarraf, Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3724-3732.  doi: 10.1016/j.cma.2008.02.023. [7] M. Barboteu and M. Sofonea, Analysis and numerical approach of a piezoelectric contact problem, Ann. Acad. Rom. Sci. Ser. Math. Appl., 1 (2009), 7-30. [8] K. Bartosz, Variable time-step $\theta$-scheme for nonlinear second order evolution inclusion, Int. J. Numer. Anal. Model., 14 (2017), 842-868. [9] K. Bartosz, D. Danan and P. Szafraniec, Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law, Comput. Math. Appl., 73 (2017), 727-746.  doi: 10.1016/j.camwa.2016.12.026. [10] M. Campo, J. R. Fernández, K. L. Kuttler, M. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006. [11] P. G. Ciarlet, The finite element method for elliptic problems, in Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. [13] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, in Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976. [14] W. Han, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Appl. Math., 137 (2001), 377-398.  doi: 10.1016/S0377-0427(00)00707-X. [15] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Sommerville, MA, 2002. [16] I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1993. [17] H. B. Khenous, P. Laborde and Y. Renard, On the discretization of contact problems in elastodynamics, in Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics, 27, Springer, Berlin, Heidelberg, 2006, 31–38. doi: 10.1007/3-540-31761-9_4. [18] H. B. Khenous, J. Pommier and Y. Renard, Hybrid discretization of the Signorini problem with Coulomb friction: Theoretical aspects and comparison of some numerical solvers, Appl. Numer. Math., 56 (2006), 163-192.  doi: 10.1016/j.apnum.2005.03.002. [19] A. Matei and M. Sofonea, A mixed variational formulation for a piezoelectric frictional contact problem, IMA J. Appl. Math., 82 (2017), 334-354.  doi: 10.1093/imamat/hxw052. [20] S. Migórski, A. Ochal and M. Sofonea, A dynamic frictional contact problem for piezoelectric materials, J. Math. Anal. Appl., 361 (2010), 161-176.  doi: 10.1016/j.jmaa.2009.09.004. [21] S. Migórski, A. Ochal and M. Sofonea, Analysis of a quasistatic contact problem for piezoelectric materials, J. Math. Anal. Appl., 382 (2011), 701-713.  doi: 10.1016/j.jmaa.2011.04.082. [22] S. Migórski, A. Ochal and M. Sofonea, Analysis of a piezoelectric contact problem with subdifferential boundary condition, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1007-1025.  doi: 10.1017/S0308210513000607. [23] S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems, in Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [24] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995. [25] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Problems, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1. [26] M. Sofonea and Y. Ouafik, A piezoelectric contact problem with normal compliance, Appl. Math. (Warsaw), 32 (2005), 425-442.  doi: 10.4064/am32-4-5.

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References:
 [1] Y. Ayyad, M. Barboteu and J. R. Fernández, A frictionless viscoelastodynamic contact problem with energy consistent properties: Numerical analysis and computational aspects, Comput. Methods Appl. Mech. Engrg., 198 (2009), 669-679.  doi: 10.1016/j.cma.2008.10.004. [2] M. Barboteu, K. Bartosz and D. Danan, Analysis of a dynamic contact problem with nonmonotone friction and non-clamped boundary conditions, Appl. Numer. Math., 126 (2018), 53-77.  doi: 10.1016/j.apnum.2017.12.005. [3] M. Barboteu, K. Bartosz, W. Han and T. Janiczko, Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact, SIAM J. Numer. Anal., 53 (2015), 527-550.  doi: 10.1137/140969737. [4] M. Barboteu, K. Bartosz and P. Kalita, An analytical and numerical approach to a bilateral contact problem with nonmonotone friction, Int. J. Appl. Math. Comp. Sci., 23 (2013), 263-276.  doi: 10.2478/amcs-2013-0020. [5] M. Barboteu, J. R. Fernández and Y. Ouafik, Numerical analysis of a frictionless viscoelastic piezoelectric contact problem, M2AN Math. Model. Numer. Anal., 42 (2008), 667-682.  doi: 10.1051/m2an:2008022. [6] M. Barboteu, J. R. Fernández and R. Tarraf, Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3724-3732.  doi: 10.1016/j.cma.2008.02.023. [7] M. Barboteu and M. Sofonea, Analysis and numerical approach of a piezoelectric contact problem, Ann. Acad. Rom. Sci. Ser. Math. Appl., 1 (2009), 7-30. [8] K. Bartosz, Variable time-step $\theta$-scheme for nonlinear second order evolution inclusion, Int. J. Numer. Anal. Model., 14 (2017), 842-868. [9] K. Bartosz, D. Danan and P. Szafraniec, Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law, Comput. Math. Appl., 73 (2017), 727-746.  doi: 10.1016/j.camwa.2016.12.026. [10] M. Campo, J. R. Fernández, K. L. Kuttler, M. Shillor and J. M. Viaño, Numerical analysis and simulations of a dynamic frictionless contact problem with damage, Comput. Methods Appl. Mech. Engrg., 196 (2006), 476-488.  doi: 10.1016/j.cma.2006.05.006. [11] P. G. Ciarlet, The finite element method for elliptic problems, in Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. [13] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, in Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976. [14] W. Han, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Appl. Math., 137 (2001), 377-398.  doi: 10.1016/S0377-0427(00)00707-X. [15] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Sommerville, MA, 2002. [16] I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1993. [17] H. B. Khenous, P. Laborde and Y. Renard, On the discretization of contact problems in elastodynamics, in Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics, 27, Springer, Berlin, Heidelberg, 2006, 31–38. doi: 10.1007/3-540-31761-9_4. [18] H. B. Khenous, J. Pommier and Y. Renard, Hybrid discretization of the Signorini problem with Coulomb friction: Theoretical aspects and comparison of some numerical solvers, Appl. Numer. Math., 56 (2006), 163-192.  doi: 10.1016/j.apnum.2005.03.002. [19] A. Matei and M. Sofonea, A mixed variational formulation for a piezoelectric frictional contact problem, IMA J. Appl. Math., 82 (2017), 334-354.  doi: 10.1093/imamat/hxw052. [20] S. Migórski, A. Ochal and M. Sofonea, A dynamic frictional contact problem for piezoelectric materials, J. Math. Anal. Appl., 361 (2010), 161-176.  doi: 10.1016/j.jmaa.2009.09.004. [21] S. Migórski, A. Ochal and M. Sofonea, Analysis of a quasistatic contact problem for piezoelectric materials, J. Math. Anal. Appl., 382 (2011), 701-713.  doi: 10.1016/j.jmaa.2011.04.082. [22] S. Migórski, A. Ochal and M. Sofonea, Analysis of a piezoelectric contact problem with subdifferential boundary condition, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1007-1025.  doi: 10.1017/S0308210513000607. [23] S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems, in Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [24] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995. [25] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Problems, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1. [26] M. Sofonea and Y. Ouafik, A piezoelectric contact problem with normal compliance, Appl. Math. (Warsaw), 32 (2005), 425-442.  doi: 10.4064/am32-4-5.
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