On making cuts for magnetic scalar potentials in multiply connected regions

Abstract

The problem of making cuts is of importance to scalar potential formulations of three-dimensional eddy current problems. Its heuristic solution has been known for a century [J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd ed. (Clarendon, Oxford, 1981), Chap. 1, Article 20] and in the last decade, with the use of finite element methods, a restricted combinatorial variant has been proposed and solved [M. L. Brown, Int. J. Numer. Methods Eng. 20, 665 (1984)]. This problem, in its full generality, has never received a rigorous mathematical formulation. This paper presents such a formulation and outlines a rigorous proof of existence. The technique used in the proof expose the incredible intricacy of the general problem and the restrictive assumptions of Brown [Int. J. Numer. Methods Eng. 20, 665 (1984)]. Finally, the results make rigorous Kotiuga’s (Ph. D. Thesis, McGill University, Montreal, 1984) heuristic interpretation of cuts and duality theorems via intersection matrices.

hp-Finite element simulation of three-dimensional eddy current problems on multiply connected domains

“…A common misinterpretation is that the cut is introduced in order to convert a multiply connected domain into one that is simply connected eg [71], it is noteworthy that a statement along these lines led to a heated debate in the literature [18,43]. The correct mathematical definition of a cut has been proposed by Kotigua [41], its purpose being to make every curl free field equal to the gradient of a scalar in Ω minus the cut(s) [19]. For further details of the computational implementation see [19] and [41,42] where an algorithm for the detecting the location of the cutting (or Siefert) surface is described.…”

“…The correct mathematical definition of a cut has been proposed by Kotigua [41], its purpose being to make every curl free field equal to the gradient of a scalar in Ω minus the cut(s) [19]. For further details of the computational implementation see [19] and [41,42] where an algorithm for the detecting the location of the cutting (or Siefert) surface is described.…”

T-Ω Formulation with Higher-Order Hierarchical Basis Functions for Nonsimply Connected Conductors

“…One pioneering attempt to solve this issue for the lowest order -Ω Finite Element formulation was proposed in [5,6]. In these papers Kotiuga proposed various algorithms to produce a set of cuts, whose elements are representatives of generators of a second relative homology group basis [3] of K realized as discrete surfaces in K .…”

Some key developments in computational electromagnetics and their attribution

“…Developments of automatic algorithms to achieve this is essential for reliable codes and has been investigated by several researchers, e.g., Simkin (1985) [34], Kotiuga (1987) [35], Kettunen (1998) [36], and Dular (2004) [37].…”