Electromagnetic Theory and Computation

Abstract

Although topology was recognized by Gauss and Maxwell to play a pivotal 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, first published in 2004, 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.

Differential form approach to the analysis of electromagnetic cloaking and masking

“…The basic feature underlying this remarkable possibility is the invariance of Maxwell equations under diffeomorphisms of the metric (metric invariance) [1], [2], [3], [4], [5], i.e., the fact that a change on the metric of space can be mimicked by a proper change of the constitutive tensors. In this work, we discuss how this feature of Maxwell equations is obviated using differential forms and the exterior calculus framework [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. We then illustrate how this feature also allows for generic masking of objects (again, under idealized conditions) via appropriate metamaterial coatings.…”

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

“…The lattice translation of the discrete exterior derivative d is based on the Generalized Stokes' Theorem (GST) [5,15]. As its name suggests, the GST encompasses the Fundamental Theorem of Calculus in 1-d, Stokes' Theorem in 2-d, and Gauss' Theorem in 3-d, and is valid for any number of dimensions.…”

A geometric integral formulation for eddy‐currents

“…where C is the f ×l face-edge incidence matrix for the grid G. Otherwise for a non-trivial domain, we resort to the classical technique based on the cuts computation, for the details refer to [20][21][22][23]. According to this technique an additional unknown T c is associated with each cut c, with c = 1, .…”

“…We will concentrate mainly on the geometric aspects at the base of the integral formulation and on the discretization process which yields to a discrete differential system of equations from the original integral equation governing an eddy-current problem in linear media. The resulting integral geometric formulation can also treat non-topologically trivial domains by resorting to the techniques already described originally in [16,19] for multiply connected regions or more generally in [20][21][22] and we will not give the details here.…”