Commun. Comput. Phys., 14 (2013), pp. 48-76.


Cohomology in 3D Magneto-Quasistatics Modeling

Pawel Dlotko 1, Ruben Specogna 2*

1 Institute of Computer Science, Jagiellonian University, ul. St. Lojasiewicza 6, 30-348 Krakow, Poland.
2 Universita di Udine, Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Via delle Scienze 208, 33100 Udine, Italy.

Received 15 November 2011; Accepted (in revised version) 18 July 2012
Available online 18 October 2012
doi:10.4208/cicp.151111.180712a

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 defining potentials. When magneto-quasi-static discrete formulations based on magnetic scalar potential are employed in problems which involve conductive regions with holes, 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.

AMS subject classifications: 55M05, 55N33, 55N99, 78M10, 78M25

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Key words: Algebraic topology, (co)homology, computational electromagnetics, cuts.

*Corresponding author.
Email: pawel.dlotko@uj.edu.pl (P. Dlotko), ruben.specogna@uniud.it (R. Specogna)
 

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