Electromagnetic Theory and Computation: A Topological Approach

by Paul W. Gross , P. Robert Kotiuga

• ISBN13: 9780521801607
• ISBN10: 0521801605
• Format: Hardcover
• Copyright: 2004-06-14
• Publisher: Cambridge University Press

Summary

Although topology was recognized by Gauss and Maxwell to play apivotal role in the formulation of electromagnetic boundary value problems, it is a largely unexploited tool for field computation. The development of algebraic topology since Maxwell provides a framework for linking data structures, algorithms, and computation to topological aspects of three-dimensional electromagnetic boundary value problems. This book attempts to expose the link between Maxwell and a modern approach to algorithms. The first chapters lay out the relevant facts about homology and cohomology, stressing their interpretations in electromagnetism. These topological structures are subsequently tied to variational formulations in electromagnetics, the finite element method, algorithms, and certain aspects of numerical linear algebra. A recurring theme is the formulation of and algorithms for the problem of making branch cuts for computing magnetic scalar potentials and eddy currents. Appendices bridge the gap between the material presented and standard expositions of differential forms, Hodge decompositions, and tools for realizing representatives of homology classes as embedded manifolds.

Table of Contents

1. Introduction

2. Introduction and prerequisite ideas from homology theory

3. A short course in differential forms

4. A paradigm problem in electromagnetic theory

5. Data structures for geometric and topological aspects of finite element algorithms

6. Finite element-based algorithms to make cuts for magnetic scalar potentials: topological constraints and computational complexity

7. An analysis of a magnetic scalar potential finite element formation of eddy currents confined to sheets.