# 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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [8] K. Bartosz, Variable time-step $\theta$-scheme for nonlinear second order evolution inclusion, Int. J. Numer. Anal. Model., 14 (2017), 842-868.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [13] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, in Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [8] K. Bartosz, Variable time-step $\theta$-scheme for nonlinear second order evolution inclusion, Int. J. Numer. Anal. Model., 14 (2017), 842-868.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.  Google Scholar [13] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, in Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [2] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [3] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [4] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [5] Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 [6] Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001 [7] Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021001 [8] Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 [9] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351 [10] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 [11] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [12] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [13] Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 [14] Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 [15] Tengfei Yan, Qunying Liu, Bowen Dou, Qing Li, Bowen Li. An adaptive dynamic programming method for torque ripple minimization of PMSM. Journal of Industrial & Management Optimization, 2021, 17 (2) : 827-839. doi: 10.3934/jimo.2019136 [16] Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143 [17] Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4019-4037. doi: 10.3934/dcds.2020056 [18] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [19] Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322 [20] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

2019 Impact Factor: 0.953

Article outline