7 ! t , , , , , : t
x !n ! ! * ! , USING THE FINITE DIFFERENCE AND THE FINITE ELEMENT METHOD
TO SOLVE AN ELECTRIC CURRENT DIFFUSION PROBLEM
by
Walter Heger
A thesis submitted to the Faculty of Graduate Studies and Research of McGill University, in partial fulfillment of the requirements for the degree of Master of Science.
Department of Mathematics and Statistics
McGill University, Montreal
March 1987
Walter Heger, 1987
USING THE FINITE DIFFERENCE AND THE FINITE ELEMENT METHOD
TO SOLVE AN ELECTRIC CURRENT DIFFUSION PROBLEM
by
Walter Heger
Abstract
The problem considered here is how an initial current which increases rapidly from zero to a maintained constant value diffuses through a cylinderical conductor, and how the transient current density approaches the steady state value. One unusual feature of the problem is the current constraint boundary condition, which requires that the current passing through a plane cross-section must have a specified value.
The problem is formulated using Maxwell's equations, and an analytic solution is found when the resistivity is constant. The solution is also found numerically using finite difference methods and also using finite elements in space and finite differences in time. These results are compared to get a practical evaluation of the accuracy of the methods.
The analytic solution is no longer valid when the resistivity varies as a function of the temperature, but the numerical solutions still apply, and are compared against each other.
Department of Mathematics and Statistics
M. Sc. Thesis
L'UTLISATION DE LA METHODE DE DIFFERENCE FINIES ET
D'ELEMENT FINIS POUR RESOUDRE UN PROBLEM
DE DIFFUSION DE COURANT DANS UN CONDUCTEUR
par
Walter Heger
Rsum
Nous nous proposons d'tudier la diffusion du courant dans un conducteur cylindrique quand il y a une augmentation rapide de zro jusqu' un niveau constant, et de trouver de quelle faon la densit transitoire du courant approche l'tat d'quilibre. Une caractristique de ce problme est la condition aux limites pour le courant qui oblige le courant passant au travers d'une section plane prendre une valeur donne a chaque instant.
La formulation de ce problme repose sur les quations de Maxwell. Une solution analytique est trouve dans le cas o la rsistance est constante. De plus, des solutions numriques sont obtenues en utilisant la mthode des diffrences finies et en utilisant la mthode des lements finis dans l'espace et des diffrences finies dans le temps. En comparant nos rsultats, un moyen facile d'valuer la prcision des mthodes considres est obtenu.
Dans le cas o la rsistance est un fonction de la temprature, notre solution analytique n'est plus valide. Par contre, nos deux solutions numriques sont toujours valide, et nous les comparons l'une l'autre.
Dpartement de Mathmatiques et de Statistique
Thse de M. Sc.
Dedication
I dedicate this thesis to my father, Dr. Frank Heger. He gave me his utmost support and waited the last twenty years to see this thesis.
Acknowledgements
It is my pleasure to thank Professor A. Evans and Professor J. Webb for their time and assistance during this thesis. I have benefited from many hours of enlightening discussions with them. I also thank Dr. M. Perry and Dr. W. Condit for helping me getting started and discussing the physical significance of the problem. I thank the General Electric Company for the use of their computer and facilities and Miss J. Hewitt for her help in obtaining literature at the General Electric Library. Finally, I thank Mr. B. Dionne for his help in the French translation of the abstract.
Using the Finite Difference and the Finite Element Method to Solve an Electric Current Diffusion Problem
Table of Contents
1. Introduction. 1
2. A Formulation of the Problem. 10
3. The Analytic Method for a Circular Cylinder. 33
4. The Finite Difference Method for a Circular Cylinder. 50
5. Finite Element Method of the General Problem. 77
6. Discussion and Results. 106
7. Appendix A: The finite difference programs FD and FDT. 134
Appendix B: Calculation of the conductivity and heat
capacity as a function of the temperature.
Appendix C: Symbol Table.
HEGER M.Sc. Thesis Copy I
HEGER M.Sc. Thesis Copy II
HEGER M.Sc. Thesis Copy III
Using the Finite Difference and the Finite Element Method
to Solve an Electric Current Diffusion Problem
by
Walter Heger
Department of Mathematics and Statistics
M. Sc. Thesis, March 1987
Copy I
Using the Finite Difference and the Finite Element Method
to Solve an Electric Current Diffusion Problem
by
Walter Heger
Department of Mathematics and Statistics
M. Sc. Thesis, March 1987
Copy II
Using the Finite Difference and the Finite Element Method
to Solve an Electric Current Diffusion Problem
by
Walter Heger
Department of Mathematics and Statistics
M. Sc. Thesis, March 1987
Copy III
qI@e @
$J@Q@`
@@ >0800>0000 ><0F00> 0000`0~ >0000x> 0 0 00x > 0 0,0L>00 0 0 >~0`0`0|>000F0< > 000`0|>f0f0f0f0< >~000> 0000 ><0f0f0f><0f0f0f0< N O Z G H t y z @ ; j m z { | } ~ 2 3 \ x ' ּug
h '
! h '
! h '
! h '
! h ' ! h ' ' ! ' ! '
! ' ! ' ( J s p 6
= M N O Z F G H 7 u G s㹫p____________ ! `"`*/ ! `"`*/
! '
'
! '
! '
h )
! h )
! h )
! h ) ! h ) % s
M | I L Y Z % & O k t v y zｽ
! @# x `*/ ! x `*/ ! `"`*/ `"`*/ ! `"`*/ < z } ! x `*/ ! `p # Regular
O H
s z
H P ( H P P ' =/p J @ P
-:ImageWriter
( K (