American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells
  • CPAA Home
  • This Issue
  • Next Article
    An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well
April  2021, 20(4): 1521-1543. doi: 10.3934/cpaa.2021031

Numerical analysis of a thermal frictional contact problem with long memory

Hailing Xuan , and  Xiaoliang Cheng

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author

Received  October 2020 Revised  January 2021 Published  March 2021

Fund Project: This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grand Agreement No. 823731 CONMECH and Major Scientifc Research Project of Zhejiang Lab No. 2019DB0ZX01

The objective in this paper is to study a thermal frictional contact model. The deformable body consists of a viscoelastic material and the process is assumed to be dynamic. It is assumed that the material behaves in accordance with Kelvin-Voigt constitutive law and the thermal effect is added. The variational formulation of the model leads to a coupled system including a history-dependent hemivariational inequality for the displacement field and an evolution equation for the temperature field. In study of this system, we first consider a fully discrete scheme of it and then focus on deriving error estimates for numerical solutions. Under appropriate assumptions of solution regularity, an optimal order error estimate is obtained. At the end of this manuscript, we report some numerical simulation results for the contact problem so as to verify the theoretical results.

Keywords: hemivariational inequality, history-dependent operators, numerical approximation, optimal order error estimate, thermal.
Mathematics Subject Classification: Primary: 65M15, 74G30, 65N22; Secondary: 74M15, 47J20.
Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1521-1543. doi: 10.3934/cpaa.2021031
References:
[1]

K. T. Andrews, M. Shillor, S. Wright and A. Klarbring, A dynamic thermviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci., 35 (1997), 1291-1309.  doi: 10.1016/S0020-7225(97)87426-5.  Google Scholar

[2]

K. E. Atkinson and W. Han, Theortical Numerical Analysis: A Functional Analysis Framework, 3rd edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar

[3]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-boundary Problems, John Wiley, Chichester, 1984.  Google Scholar

[4]

K. Bartosz, X. Cheng, P. Yu, Y. J. Kalita and C Zheng, Rothe method for parabolic variational–hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862.  doi: 10.1016/j.jmaa.2014.09.078.  Google Scholar

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, 3rd edition, Spring-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[6]

M. Campo, J. R. Fernández and T. V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math., 196 (2006), 180-197.  doi: 10.1016/j.cam.2005.08.025.  Google Scholar

[7]

A. Capatina, Variational Inequalities Frictional Contact Problems. vol. 31, Advances in Mechanics and Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-10163-7.  Google Scholar

[8]

O. Chau, Numerical analysis of a thermal contact problem with adhesion, Comp. Appl. Math., 37 (2018), 5424-5455.  doi: 10.1007/s40314-018-0642-2.  Google Scholar

[9]

X. Cheng, S. Migórski, A. Ochal and M. Sofonea, Anaysis of two quasistatic history-dependent contact models, Discrete Contin. Dyn. Ser. Ser. B, 19 (2014), 2425-2445.  doi: 10.3934/dcdsb.2014.19.2425.  Google Scholar

[10]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.  Google Scholar

[11]

W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar

[12] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, RI-International Press, Somerville, 2002.  doi: 10.1090/amsip/030.  Google Scholar
[13]

W. Han, M. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085.  Google Scholar

[14]

L. Jianu, M. Shillor and M. Sofonea, A viscoelastic frictioness contact problem with adhesion, Appl. Anal., 80 (2001), 233-255.  doi: 10.1080/00036810108840990.  Google Scholar

[15]

A. Matei and M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009.  Google Scholar

[16]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[17]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, M. Dekker, New york, 1995.  Google Scholar

[18]

J. Nedoma and J. Stehlik, Mathematical and computational methods and algorithms in biomechanics, in Human Skeletal Systems, John Wiley and Sons, 2011. Google Scholar

[19]

J. Ogorzaly, Dynamic contact problem with thermal effect, Georgian Math. J, 2016. doi: 10.1515/gmj-2016-0025.  Google Scholar

[20]

M. Rochdi and M. Shillor, Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity, Quart. Appl. Math, 58 (2000), 543-560.  doi: 10.1090/qam/1770654.  Google Scholar

[21]

M. Sofonea, F. Pǎtrulescu and Y. Souleiman, Analysis of contact problem with wear and unilateral constraint, Appl. Anal., 95 (2017), 2590-2607.  doi: 10.1080/00036811.2015.1102892.  Google Scholar

[22]

W. Xu, Z. Huang, W. Han, W. Chen and C. Wang, Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration, Comput. Math. Appl., 77 (2019), 2596-2607.  doi: 10.1016/j.camwa.2018.12.038.  Google Scholar

[23]

H. Xuan, X. Cheng, W. Han and Q. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math. Theory Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.  Google Scholar

show all references

References:
[1]

K. T. Andrews, M. Shillor, S. Wright and A. Klarbring, A dynamic thermviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci., 35 (1997), 1291-1309.  doi: 10.1016/S0020-7225(97)87426-5.  Google Scholar

[2]

K. E. Atkinson and W. Han, Theortical Numerical Analysis: A Functional Analysis Framework, 3rd edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar

[3]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-boundary Problems, John Wiley, Chichester, 1984.  Google Scholar

[4]

K. Bartosz, X. Cheng, P. Yu, Y. J. Kalita and C Zheng, Rothe method for parabolic variational–hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862.  doi: 10.1016/j.jmaa.2014.09.078.  Google Scholar

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, 3rd edition, Spring-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[6]

M. Campo, J. R. Fernández and T. V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math., 196 (2006), 180-197.  doi: 10.1016/j.cam.2005.08.025.  Google Scholar

[7]

A. Capatina, Variational Inequalities Frictional Contact Problems. vol. 31, Advances in Mechanics and Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-10163-7.  Google Scholar

[8]

O. Chau, Numerical analysis of a thermal contact problem with adhesion, Comp. Appl. Math., 37 (2018), 5424-5455.  doi: 10.1007/s40314-018-0642-2.  Google Scholar

[9]

X. Cheng, S. Migórski, A. Ochal and M. Sofonea, Anaysis of two quasistatic history-dependent contact models, Discrete Contin. Dyn. Ser. Ser. B, 19 (2014), 2425-2445.  doi: 10.3934/dcdsb.2014.19.2425.  Google Scholar

[10]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.  Google Scholar

[11]

W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar

[12] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, RI-International Press, Somerville, 2002.  doi: 10.1090/amsip/030.  Google Scholar
[13]

W. Han, M. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085.  Google Scholar

[14]

L. Jianu, M. Shillor and M. Sofonea, A viscoelastic frictioness contact problem with adhesion, Appl. Anal., 80 (2001), 233-255.  doi: 10.1080/00036810108840990.  Google Scholar

[15]

A. Matei and M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009.  Google Scholar

[16]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[17]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, M. Dekker, New york, 1995.  Google Scholar

[18]

J. Nedoma and J. Stehlik, Mathematical and computational methods and algorithms in biomechanics, in Human Skeletal Systems, John Wiley and Sons, 2011. Google Scholar

[19]

J. Ogorzaly, Dynamic contact problem with thermal effect, Georgian Math. J, 2016. doi: 10.1515/gmj-2016-0025.  Google Scholar

[20]

M. Rochdi and M. Shillor, Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity, Quart. Appl. Math, 58 (2000), 543-560.  doi: 10.1090/qam/1770654.  Google Scholar

[21]

M. Sofonea, F. Pǎtrulescu and Y. Souleiman, Analysis of contact problem with wear and unilateral constraint, Appl. Anal., 95 (2017), 2590-2607.  doi: 10.1080/00036811.2015.1102892.  Google Scholar

[22]

W. Xu, Z. Huang, W. Han, W. Chen and C. Wang, Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration, Comput. Math. Appl., 77 (2019), 2596-2607.  doi: 10.1016/j.camwa.2018.12.038.  Google Scholar

[23]

H. Xuan, X. Cheng, W. Han and Q. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math. Theory Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.  Google Scholar

Figure 1.  Reference configuration of the two-dimensional example
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  the deformed configuration at several times
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  the deformed configuration at $ c_1 = c_2 = a = 0.1 $ and $ c_1 = c_2 = a = 1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  the temperature field at $ \theta_R = 0 $ and $ \theta_R = 8 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  the deformed configuration at t = 0.5s and t = 1s
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  the deformed configuration at $ \theta_R = 0 $ and $ \theta_R = 10 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  the deformed configuration at $ \theta_R = 0 $ and $ \theta_R = 8 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations & Control Theory, 2020, 9 (4) : 1089-1114. doi: 10.3934/eect.2020047
[2]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044
[3]

Xiaoliang Cheng, Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of two quasistatic history-dependent contact models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2425-2445. doi: 10.3934/dcdsb.2014.19.2425
[4]

Mircea Sofonea, Meir Shillor. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Communications on Pure & Applied Analysis, 2014, 13 (1) : 371-387. doi: 10.3934/cpaa.2014.13.371
[5]

Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325
[6]

Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545
[7]

Changjie Fang, Weimin Han. Stability analysis and optimal control of a stationary Stokes hemivariational inequality. Evolution Equations & Control Theory, 2020, 9 (4) : 995-1008. doi: 10.3934/eect.2020046
[8]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[9]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056
[10]

Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036
[11]

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
[12]

Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659
[13]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations & Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058
[14]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[15]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[16]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[17]

Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339
[18]

Lijing Xi, Yuying Zhou, Yisheng Huang. A class of quasilinear elliptic hemivariational inequality problems on unbounded domains. Journal of Industrial & Management Optimization, 2014, 10 (3) : 827-837. doi: 10.3934/jimo.2014.10.827
[19]

Zhenhai Liu, Stanislaw Migórski. Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 129-143. doi: 10.3934/dcdsb.2008.9.129
[20]

Rua Murray. Approximation error for invariant density calculations. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 535-557. doi: 10.3934/dcds.1998.4.535

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (44)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences