January  2018, 23(1): 387-400. doi: 10.3934/dcdsb.2018027

Numerical treatment of contact problems with thermal effect

Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Lojasiewicza 6, 30-348 Krakow, Poland

* Corresponding author: Anna Ochal

Received  December 2016 Revised  May 2017 Published  January 2018

Fund Project: The research was supported by the National Center Science of Poland under Maestro Advanced Project no DEC-2012/06/A/ST1/00262.

The paper deals with the formulation and the finite element approximation of a quasi-static thermoviscoelastic problem which describes frictional contact between a deformable body and a rigid foundation. The contact is modeled by normal damped response condition whereas the friction is described by the Coulomb law of dry friction. The weak formulation of the model consists of a coupled system of the variational inequality for the displacement and the parabolic equation for the temperature. The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.

Citation: Anna Ochal, Michal Jureczka. Numerical treatment of contact problems with thermal effect. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 387-400. doi: 10.3934/dcdsb.2018027
References:
[1]

K. BartoszD. Danan and P. Szafraniec, Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law, Computers & Mathematics with Applications, 73 (2017), 727-746. doi: 10.1016/j.camwa.2016.12.026. Google Scholar

[2]

O. Chau and R. Oujja, Numerical treatment of a class of thermal contact problems, Mathematics and Computers in Simulation, 118 (2015), 163-176. doi: 10.1016/j.matcom.2014.12.007. Google Scholar

[3]

L. GasinkiA. Ochal and M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Journal of Analysis and its Applications (ZAA), 34 (2015), 251-275. doi: 10.4171/ZAA/1538. Google Scholar

[4] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society and International Press, 2002.
[5]

S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems Advances in Mechanics and Mathematics, vol. 26, Springer, 2013. doi: 10.1007/978-1-4614-4232-5. Google Scholar

[6] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, 1993. doi: 10.1007/978-3-642-51677-1.
[7] M. ShillorM. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Springer-Verlag, 2004. doi: 10.1007/b99799.
[8]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics Lecture Note Series, vol. 398, Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. Google Scholar

[9] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

K. BartoszD. Danan and P. Szafraniec, Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law, Computers & Mathematics with Applications, 73 (2017), 727-746. doi: 10.1016/j.camwa.2016.12.026. Google Scholar

[2]

O. Chau and R. Oujja, Numerical treatment of a class of thermal contact problems, Mathematics and Computers in Simulation, 118 (2015), 163-176. doi: 10.1016/j.matcom.2014.12.007. Google Scholar

[3]

L. GasinkiA. Ochal and M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Journal of Analysis and its Applications (ZAA), 34 (2015), 251-275. doi: 10.4171/ZAA/1538. Google Scholar

[4] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society and International Press, 2002.
[5]

S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems Advances in Mechanics and Mathematics, vol. 26, Springer, 2013. doi: 10.1007/978-1-4614-4232-5. Google Scholar

[6] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, 1993. doi: 10.1007/978-3-642-51677-1.
[7] M. ShillorM. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Springer-Verlag, 2004. doi: 10.1007/b99799.
[8]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics Lecture Note Series, vol. 398, Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. Google Scholar

[9] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
Figure 1.  Body's shape and temperature at t = 0:25
Figure 2.  Body's shape and temperature at t = 0:5
Figure 3.  Body's shape and temperature at t = 0:75
Figure 4.  Body's shape and temperature at t = 1
[1]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[2]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663

[3]

Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems & Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947

[4]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[5]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75

[6]

Daniel Peterseim. Robustness of finite element simulations in densely packed random particle composites. Networks & Heterogeneous Media, 2012, 7 (1) : 113-126. doi: 10.3934/nhm.2012.7.113

[7]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

[8]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

[9]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[10]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

[11]

Géry de Saxcé. Modelling contact with isotropic and anisotropic friction by the bipotential approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 409-425. doi: 10.3934/dcdss.2016004

[12]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[13]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[14]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387

[15]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[16]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[17]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[18]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[19]

Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389

[20]

Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (25)
  • HTML views (54)
  • Cited by (0)

Other articles
by authors

[Back to Top]