zbMATH — the first resource for mathematics

A B-spline finite element method for the thermistor problem with the modified electrical conductivity. (English) Zbl 1108.78018
Summary: Approximate steady-state solutions of a one-dimensional positive temperature coefficient thermistor problem with a modified step function electrical conductivity are obtained by using the Galerkin cubic B-spline finite element method. It is shown that the computational results obtained by the method display the correct physical characteristics of the problem, and they are found to be in very good agreement with the exact solution. It is also shown that the numerical solution exhibits the expected convergence to the exact one as the mesh size is refined. Further a Fourier stability analysis of the method is investigated.

MSC:
 78M20 Finite difference methods applied to problems in optics and electromagnetic theory 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text:
References:
 [1] Macklen, E.D., Thermistors, (1979), Electrochemical Publications Scotland [2] Cimatti, G., Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Q. appl. math., 47, 117-121, (1989) · Zbl 0694.35137 [3] Cimatti, G.; Prodi, G., Existence results for a non-linear elliptic system modelling a temperature dependent electrical resistor, Ann. mat. pura appl., 162, 33-42, (1992) [4] Fowler, A.; Frigaard, I.; Howison, S.D., Temperature surges in thermistors, SIAM J. appl. math., 52, 998-1011, (1992) · Zbl 0800.80001 [5] Howison, S.D.; Rodrigues, J.F.; Shillor, M., Stationary solutions to the thermistor problem, J. math. anal. appl., 174, 573-588, (1993) · Zbl 0787.35033 [6] Wiedmann, J., The thermistor problem, Nonlinear differ. equ. appl., 4, 133-148, (1997) · Zbl 0992.35102 [7] Zhou, S.; Westbrook, D.R., Numerical solutions of the thermistor equations, J. comp. appl. math., 5, 259-273, (1997) · Zbl 0885.65147 [8] Westbrook, D.R., The thermistor: a problem in heat and current flow, Numer. meth. pdes, 79, 101-118, (1997) · Zbl 0885.65147 [9] Preis, K.; Biro, O.; Dyczij-Edlinger, R.; Richter, K.R.; Badics, Zs.; Riedler, H.; StĂ¶nger, H., Application of FEM to coupled electric, thermal and mechanical problems, IEEE trans. magn., 30, 5, 3316-3319, (1994) [10] Kutluay, S.; Bahadir, A.R.; Ozdes, A., Various methods to the thermistor problem with a bulk electrical conductivity, Int. J. numer. meth. engng., 45, 1-12, (1999) · Zbl 0941.78011 [11] Kutluay, S.; Bahadir, A.R.; Ozdes, A., A variety of finite difference methods to the thermistor with a new modified electrical conductivity, Appl. math. comput., 106, 205-213, (1999) · Zbl 1049.80501 [12] S. Kutluay, A.S. Wood, Numerical solutions of the thermistor problem with a ramp electrical conductivity, Appl. Math. Comput., in press · Zbl 1072.78523 [13] Wood, A.S., Modelling the thermistor, (), 397-400 [14] Wood, A.S.; Kutluay, S., A heat balance integral model of the thermistor, Int. J. heat mass transfer., 38, 10, 1831-1840, (1995) · Zbl 0924.73029 [15] Prcnter, P.M., Splines and variational methods, (1975), John-Wiley New York [16] Smith, G.D., Numerical solution of partial differential equations: finite difference methods, (1987), Clarendon Oxford
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.