Analogies Between Finite-Difference and Finite-Element Methods for Scalar and Vector Potential Formulations in Magnetic Field Calculations

Numerical 3-D formulations using scalar Ω and vector A potentials are examined for magnetic fields with an emphasis on the finite-difference method (FDM) and finite-element method (FEM) using nodal and facet elements. It is shown that for hexahedral elements, the FDM equations may be presented in a form similar to the FEM equations; to accomplish this, the coefficients defining volume integrals in the FEM need to be expressed in an approximate manner, while the nodes in the FDM require supplementary association with middle points of edges, facets, and volumes. The analogy between a description of magnetic field sources arising from the classical MMF distribution approach, and when expressed in terms of edge values of vector potential, T
₀
is emphasized. Comparisons are made between the results obtained using the FDM and the FEM for both the scalar and vector potential formulations. Forces in systems containing permanent magnets and torques in permanent magnet machines are calculated and compared using both the approaches for scalar and vector formulations. A unified form of the stress tensor has been applied to the FDM and FEM.