Cohomology in electromagnetic modeling

TL;DR

This paper explores the use of cohomology theory to define and compute cuts in electromagnetic modeling, replacing heuristic methods with a rigorous, general approach that improves the accuracy and efficiency of potential design in complex conductive regions.

Contribution

It introduces a cohomology-based definition of cuts in electromagnetic problems, providing a provably general, automatic, and efficient algorithm for their computation, surpassing heuristic approaches.

Findings

Cohomology theory offers a rigorous framework for defining cuts.

The proposed method automates cut computation in complex geometries.

Heuristic cut definitions are shown to be inadequate through counter-examples.

Abstract

Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational electromagnetics community to develop fast and general techniques for potential design. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problems which involve conductive regions with holes, \textit{cuts} are needed to make the boundary value problem well defined. While an intimate connection with homology theory has been quickly recognized, heuristic definitions of cuts are surprisingly still dominant in the literature. The aim of this paper is first to survey several definitions of cuts together with their shortcomings. Then, cuts are defined as generators of the first cohomology group over integers of a finite CW-complex. This provably general definition has also the virtue of providing an automatic, general and efficient algorithm for the computation of cuts. Some counter-examples show that heuristic definitions of cuts should be abandoned. The use of cohomology theory is not an option but the invaluable tool expressly needed to solve this problem.