×

zbMATH — the first resource for mathematics

Numerical analysis of the quasistatic thermoviscoelastic thermistor problem. (English) Zbl 1108.74013
Summary: We consider the quasistatic thermoviscoelastic thermistor model which describes the combination of 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.

MSC:
74F05 Thermal effects in solid mechanics
74F15 Electromagnetic effects in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] W. Allegretto and H. Xie , A non-local thermistor problem . Eur. J. Appl. Math. 6 ( 1995 ) 83 - 94 . Zbl 0826.35120 · Zbl 0826.35120 · doi:10.1017/S0956792500001686
[2] W. Allegreto , Y. Lin and A. Zhou , A box scheme for coupled systems resulting from microsensor thermistor problems . Dynam. Contin. Discret. S. 5 ( 1999 ) 209 - 223 . Zbl 0979.78023 · Zbl 0979.78023
[3] W. Allegreto , Y. Lin and S. Ma , Existence and long time behaviour of solutions to obstacle thermistor equations . Discrete Contin. Dyn. S. 8 ( 2002 ) 757 - 780 . Zbl 1062.35187 · Zbl 1062.35187 · doi:10.3934/dcds.2002.8.757
[4] S.N. Antontsev and M. Chipot , The thermistor problem: existence, smoothness, uniqueness, blowup . SIAM J. Math. Anal. 25 ( 1994 ) 1128 - 1156 . Zbl 0808.35059 · Zbl 0808.35059 · doi:10.1137/S0036141092233482
[5] A.R. Bahadir , Application of cubic B-spline finite element technique to the thermistor problem . Appl. Math. Comput. 149 ( 2004 ) 379 - 387 . Zbl 1038.65057 · Zbl 1038.65057 · doi:10.1016/S0096-3003(03)00146-2
[6] A. Bermúdez , M.C. Muñiz and P. Quintela , Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminum electrolytic cell . Comput. Method Appl. M. 106 ( 1993 ) 129 - 142 . Zbl 0800.76240 · Zbl 0800.76240 · doi:10.1016/0045-7825(93)90188-4
[7] O. Chau , J.R. Fernández , W. Han and M. Sofonea , A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage . Comput. Method Appl. M. 191 ( 2002 ) 5007 - 5026 . Zbl 1042.74039 · Zbl 1042.74039 · doi:10.1016/S0045-7825(02)00423-1
[8] X. Chen , Existence and regularity of solutions of a nonlinear degenerate elliptic system arising from a thermistor problem . J. Partial Differential Equations 7 ( 1994 ) 19 - 34 . Zbl 0830.35045 · Zbl 0830.35045
[9] 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 . Zbl 0875.65086 · Zbl 0875.65086
[10] G. Cimatti , Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions . Quart. J. Mech. Appl. Math. 47 ( 1989 ) 117 - 121 . Zbl 0694.35137 · Zbl 0694.35137
[11] G. Duvaut and J.L. Lions , Inequalities in Mechanics and Physics , Springer, New-York ( 1976 ). MR 521262 | Zbl 0331.35002 · Zbl 0331.35002
[12] 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).
[13] W. Han and M. Sofonea , Quasistatic contact problems in viscoelasticity and viscoplasticity , Americal Mathematical Society-International Press ( 2002 ). Zbl 1013.74001 · Zbl 1013.74001
[14] S.D. Howison , A note on the thermistor problem in two space dimension . Quart. J. Mech. Appl. Math. 47 ( 1989 ) 509 - 512 . Zbl 0692.35094 · Zbl 0692.35094
[15] S.D. Howison , J. Rodrigues and M. Shillor , Stationary solutions to the thermistor problem . J. Math. Anal. Appl. 174 ( 1993 ) 573 - 588 . Zbl 0787.35033 · Zbl 0787.35033 · doi:10.1006/jmaa.1993.1142
[16] S. Kutluay , A.R. Bahadir and A. Ozdeć , A variety of finite difference methods to the thermistor with a new modified electrical conductivity . Appl. Math. Comput. 106 ( 1999 ) 205 - 213 . Zbl 1049.80501 · Zbl 1049.80501 · doi:10.1016/S0096-3003(98)10123-6
[17] S. Kutluay , A.R. Bahadir and A. Ozdeć , Various methods to the thermistor problem with a bulk electrical conductivity . Int. J. Numer. Method. Engrg. 45 ( 1999 ) 1 - 12 . Zbl 0941.78011 · Zbl 0941.78011 · doi:10.1002/(SICI)1097-0207(19990510)45:1<1::AID-NME565>3.0.CO;2-0
[18] S. Kutluay and E. Esen , A B-spline finite element method for the thermistor problem with the modified electrical conductivity . Appl. Math. Comput. 156 ( 2004 ) 621 - 632 . Zbl 1108.78018 · Zbl 1108.78018 · doi:10.1016/j.amc.2003.08.014
[19] S. Kutluay and A.S. Wood , Numerical solutions of the thermistor problem with a ramp electrical conductivity . Appl. Math. Comput. 148 ( 2004 ) 145 - 162 . Zbl 1072.78523 · Zbl 1072.78523 · doi:10.1016/S0096-3003(02)00834-2
[20] K.L. Kuttler , M. Shillor and J.R. Fernández , Existence for the thermoviscoelastic thermistor problem . Differential Equations Dynam. Systems (to appear). · Zbl 1178.74033
[21] H. Xie and W. Allegretto , \(C^{\alpha }(\bar{\Omega })\) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem . SIAM J. Math. Anal. 22 ( 1991 ) 1491 - 1499 . Zbl 0744.35016 · Zbl 0744.35016 · doi:10.1137/0522096
[22] X. Xu , The thermistor problem with conductivity vanishing for large temperature . P. Roy. Soc. Edinb. A 124 ( 1994 ) 1 - 21 . Zbl 0807.35143 · Zbl 0807.35143 · doi:10.1017/S0308210500029164
[23] X. Xu , On the existence of bounded temperature in the thermistor problem with degeneracy . Nonlinear Anal. 42 ( 2000 ) 199 - 213 . Zbl 0964.35005 · Zbl 0964.35005 · doi:10.1016/S0362-546X(98)00340-X
[24] X. Xu , On the effects of thermal degeneracy in the thermistor problem . SIAM J. Math. Anal. 35 (4) ( 2003 ) 1081 - 1098 . Zbl 1055.35069 · Zbl 1055.35069 · doi:10.1137/S0036141002417401
[25] X. Xu , Local regularity theorems for the stationary thermistor problem . P. Roy. Soc. Edinb. A 134 ( 2004 ) 773 - 782 . Zbl 1081.35038 · Zbl 1081.35038 · doi:10.1017/S0308210500003474
[26] S. Zhou and D.R. Westbrook , Numerical solutions of the thermistor equations . J. Comput. Appl. Math. 79 ( 1997 ) 101 - 118 . Zbl 0885.65147 · Zbl 0885.65147 · doi:10.1016/S0377-0427(96)00166-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.