
-
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
Numerical analysis of a thermal frictional contact problem with long memory
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
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.
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. |
[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. |
[3] |
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-boundary Problems, John Wiley, Chichester, 1984. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |
[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. |
[12] |
W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, RI-International Press, Somerville, 2002.
doi: 10.1090/amsip/030.![]() ![]() |
[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. |
[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. |
[15] |
A. Matei and M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009. |
[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. |
[17] |
Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, M. Dekker, New york, 1995. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-boundary Problems, John Wiley, Chichester, 1984. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. |
[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. |
[12] |
W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, RI-International Press, Somerville, 2002.
doi: 10.1090/amsip/030.![]() ![]() |
[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. |
[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. |
[15] |
A. Matei and M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009. |
[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. |
[17] |
Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, M. Dekker, New york, 1995. |
[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. |
[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. |
[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. |
[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. |
[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. |



[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
Other articles
by authors
[Back to Top]