Spatial harmonic analysis of FEM results in magnetostatics

Approaches to the spatial harmonic characterization of the field due to a symmetrically shielded permanent bar magnet are presented. At first, exact spherical harmonic coefficients are obtained for a given unshielded magnet and comparisons are made with the coefficients of unshielded numerical models. In the calibration test, the scalar magnetic potential is separately calculated by the integral equation method (IEM) and finite-element method (FEM) with a layer of "infinite" elements on the interface boundary. While the values from the IEM data are nearly exact, the analysis of the FEM data demonstrates greater errors. For the selection of the coefficients from the discrete data from either unshielded or shielded magnets, discrete selective functions are additionally orthonormalized at the boundary surface. For the discrete selective functions generated for the grid of 612 points, which are uniformly distributed over the unit sphere with angular steps of 10° in φ and θ directions, the maximal error of nonorthogonality is below 10^-8.