Loop-based Flux Formulation for Three-dimensional Magnetostatic Problems

In this paper, loop basis functions are introduced to expand the magnetic flux density and the magnetostatic subset of Maxwell's equations are solved in a compact and straightforward manner using finite element method. As linear combinations of divconforming Schaubert-Wilton-Glisson basis functions in three-dimensional, loop basis functions are inherently divergence-free and originally constructed to represent solenoidal electric current density in the electric field integral equation. Sharing the same physical property with the solenoidal electric current density, the magnetic flux density can also be represented by the loop basis functions and thus, Gauss' law for magnetism is naturally satisfied; which is out of the capability of general Whitney elements. The relationship between the loop basis functions and Whitney elements, as well as the comparison between the proposed method and traditional method pertinent to magnetic vector potential are investigated.