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Cohomology in 3D Magneto-Quasistatics Modeling

Published online by Cambridge University Press:  03 June 2015

Paweł Dłotko and
Ruben Specogna
Paweł Dłotko*
Affiliation:
Institute of Computer Science, Jagiellonian University, ul.. St. Łojasiewicza 6, 30-348 Kraków, Poland
Ruben Specogna*
Affiliation:
Università di Udine, Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Via delle Scienze 208, 33100 Udine, Italy
*
Email:pawel.dlotko@uj.edu.pl
Corresponding author.Email:ruben.specogna@uniud.it
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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 problemswhich involve conductive regionswith 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.

Keywords

55M0555N3355N9978M1078M25Algebraic topology(co)homologycomputational electromagneticscuts

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Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013, pp. 48 - 76
Copyright
Copyright © Global Science Press Limited 2013

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