FEM Formulation in Magnetostatics

Abstract

The finite element formulation for non-saturated magnetic problems using the magnetic scalar potential ϕ in two dimensions is in every respect identical to its electric scalar potential counterpart. The only difference is that the dielectric permittivity is replaced with the magnetic permeability. For the variational approach, the functional to be minimised is equal to the total magnetic energy W

$$ W = \frac{1}{2}\int\limits_{\Omega } {H\bullet Bd\Omega = \frac{1}{2}} \int\limits_{\Omega } {\mu |\nabla \phi {|^2}} d\Omega $$

(7.1)

where Ω is the total domain. It should be noted that using the above energy expression assumes that the potential is either fixed on the outer boundary or that its normal derivative is zero. For more general boundary conditions, the weak formulation can be used. The boundary conditions on the scalar potential are not trivial to derive. The scalar potential must not span current carrying regions and care should be taken in defining the domain so as to avoid multi-valued boundary conditions. Note that for the axi-symmetric case, where first-order elements are used, corrective action for the nodes on-axis must be taken, just as in the electrostatic case.