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

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The analytic solution is no longer valid when the resistivity varies as a function of the temperature, but the numerical solutions still apply, and are compared against each other.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

Nous nous proposons d'étudier la diffusion du courant dans un conducteur cylindrique quand il y a une augmentation rapide de zéro jusqu'à un niveau constant, et de trouver de quelle façon la densité transitoire du courant approche l'état d'équilibre. Une caractéristique de ce problème est la condition aux limites pour le courant qui oblige le courant passant au travers d'une section plane à prendre une valeur donnée a chaque instant.

La formulation de ce problème repose sur les équations de Maxwell. Une solution analytique est trouvée dans le cas où la résistance est constante. De plus, des solutions numériques sont obtenues en utilisant la méthode des différences finies et en utilisant la méthode des élements finis dans l'espace et des différences finies dans le temps. En comparant nos résultats, un moyen facile d'évaluer la précision des méthodes considérées est obtenu.

Dans le cas où la résistance est un fonction de la température, notre solution analytique n'est plus valide. Par contre, nos deux solutions numériques sont toujours valide, et nous les comparons l'une à l'autre.

Département de Mathématiques et de Statistique

Thése de M. Sc.

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.

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