Cuts for 3-D magnetic scalar potentials: Visualizing unintuitive surfaces arising from trivial knots

Abstract

A wealth of literature exists on computing and visualizing cuts for the magnetic scalar potential of a current carrying conductor via Finite Element Methods (FEM) and harmonic maps to the circle. By a cut we refer to an orientable surface bounded by a given current carrying path (such that the flux through it may be computed) that restricts contour integrals on a curl-zero vector field to those that do not link the current-carrying path, analogous to branch cuts of complex analysis. This work is concerned with a study of a peculiar contour that illustrates topologically unintuitive aspects of cuts obtained from a trivial loop and raises questions about the notion of an optimal cut. Specifically, an unknotted curve that bounds only high genus surfaces in its convex hull is analyzed. The current work considers the geometric realization as a current-carrying wire in order to construct a magnetic scalar potential. Moreover, we consider the problem of choosing an energy functional on the space of maps, suggesting an algorithm for computing cuts via minimizing a conformally invariant functional utilizing Newton iteration.

COMSOL implementation of the H-$ϕ$-formulation with thin cuts for modeling superconductors with transport currents

“…Finally, although using the magnetic scalar potential as opposed to the full vector field greatly reduces the number of DOFs in non-conducting domains, the need for imposing cuts in complex topologies can be very tricky. Solutions proposed include algorithms for automatically generating the proper cuts and using thick cuts [12], [13], [28], [32]- [36]. However, these methods are not currently implementable in COMSOL Multiphysics.…”