Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-11T09:43:38.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysis of the quasistatic thermoviscoelastic thermistor problem

Published online by Cambridge University Press:  21 June 2006

José R. Fernández
José R. Fernández*
Affiliation:
Departamento de Matemática Aplicada, Facultade de Matemáticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. jramon@usc.es
Get access

Abstract

In this work, the quasistatic thermoviscoelastic thermistor problem is considered. The thermistor model describes the combination of the effects due to the heat, electrical current conduction and Joule's heat generation. The variational formulation leads to a coupled system of nonlinear variational equations for which the existence of a weak solution is recalled. Then, a fully discrete algorithm is introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the scheme is deduced. Finally, some numerical simulations are performed in order to show the behaviour of the algorithm.

Keywords

Thermoviscoelastic thermistorerror estimatesfinite elementsnumerical simulations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 2 , March 2006 , pp. 353 - 366
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allegretto, W. and Xie, H., A non-local thermistor problem. Eur. J. Appl. Math. 6 (1995) 83–94. CrossRef
Allegreto, W., Lin, Y. and Zhou, A., A box scheme for coupled systems resulting from microsensor thermistor problems. Dynam. Contin. Discret. S. 5 (1999) 209–223.
Allegreto, W., Lin, Y. and Existence, S. Ma and long time behaviour of solutions to obstacle thermistor equations. Discrete Contin. Dyn. S. 8 (2002) 757–780.
Antontsev, S.N. and Chipot, M., The thermistor problem: existence, smoothness, uniqueness, blowup. SIAM J. Math. Anal. 25 (1994) 1128–1156. CrossRef
Bahadir, A.R., Application of cubic B-spline finite element technique to the thermistor problem. Appl. Math. Comput. 149 (2004) 379–387.
Bermúdez, A., Muñiz, M.C. and Quintela, P., Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminum electrolytic cell. Comput. Method Appl. M. 106 (1993) 129–142. CrossRef
Chau, O., Fernández, J.R., Han, W. and Sofonea, M., A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. Comput. Method Appl. M. 191 (2002) 5007–5026. CrossRef
Chen, X., Existence and regularity of solutions of a nonlinear degenerate elliptic system arising from a thermistor problem. J. Partial Differential Equations 7 (1994) 19–34.
P.G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, Vol. II, Part 1, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991) 17–352.
Cimatti, G., Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. J. Mech. Appl. Math. 47 (1989) 117–121.
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Springer, New-York (1976).
J.R. Fernández, K.L. Kuttler, M.C. Muñiz and M. Shillor, A model and simulations of the thermoviscoelastic thermistor. Eur. J. Appl. Math. (submitted).
W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, Americal Mathematical Society–International Press (2002).
Howison, S.D., A note on the thermistor problem in two space dimension. Quart. J. Mech. Appl. Math. 47 (1989) 509–512.
Howison, S.D., Rodrigues, J. and Shillor, M., Stationary solutions to the thermistor problem. J. Math. Anal. Appl. 174 (1993) 573–588. CrossRef
Kutluay, S., Bahadir, A.R. and Ozdeć, A., A variety of finite difference methods to the thermistor with a new modified electrical conductivity. Appl. Math. Comput. 106 (1999) 205–213.
Kutluay, S., Bahadir, A.R. and Ozdeć, A., Various methods to the thermistor problem with a bulk electrical conductivity. Int. J. Numer. Method. Engrg. 45 (1999) 1–12. 3.0.CO;2-0>CrossRef
Kutluay, S. and Esen, E., B-spline, A finite element method for the thermistor problem with the modified electrical conductivity. Appl. Math. Comput. 156 (2004) 621–632.
Kutluay, S. and Wood, A.S., Numerical solutions of the thermistor problem with a ramp electrical conductivity. Appl. Math. Comput. 148 (2004) 145–162.
K.L. Kuttler, M. Shillor and J.R. Fernández, Existence for the thermoviscoelastic thermistor problem. Differential Equations Dynam. Systems (to appear).
Xie, H. and Allegretto, W., $C^{\alpha}(\bar \Omega)$ solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. Anal. 22 (1991) 1491–1499. CrossRef
The, X. Xu thermistor problem with conductivity vanishing for large temperature. P. Roy. Soc. Edinb. A 124 (1994) 1–21.
On, X. Xu the existence of bounded temperature in the thermistor problem with degeneracy. Nonlinear Anal. 42 (2000) 199–213.
X. Xu, On the effects of thermal degeneracy in the thermistor problem. SIAM J. Math. Anal. 35 (4) (2003) 1081–1098.
Local, X. Xu regularity theorems for the stationary thermistor problem. P. Roy. Soc. Edinb. A 134 (2004) 773–782.
Zhou, S. and Westbrook, D.R., Numerical solutions of the thermistor equations. J. Comput. Appl. Math. 79 (1997) 101–118. CrossRef