An algorithm to make cuts for magnetic scalar potentials in tetrahedral meshes based on the finite element method

A general approach to making cuts for magnetic scalar potentials that utilizes the formalism of algebraic topology and points to the external role played by mappings into circles was presented by D. Rodger and J.F. Eastham (1987). The author extends these previous results to practical algorithms by considering the variational formulation of harmonic maps into the circle and their numerical discretization by the finite element method. The harmonic map functional is nonquadratic and nonconvex. It is shown that a finite element discretization can make the problem reduce to that of harmonic functions subject to peculiar interelement constraints. Analyzing the continuity requirements of the harmonic map and the degrees of freedom in the interelement constraints which leave the discretized harmonic map functional invariant, the effective degrees of freedom in the element assembly are identified with topological constraints. The interelement constraints are then rephrased to make them amenable to well-known techniques of electric circuit theory. Finally, the entire development is summarized in the form of an algorithm. The results are important for the magnetic scalar potential analysis of eddy currents and nondestructive testing methodology in three dimensions