Geometric formulation of edge and nodal finite element equations in electromagnetics

Abstract

-Finite element equations for electromagnetic fields are examined, in particular nodal elements using scalar potential formulation and edge elements for vector potential formulation. It is shown how the equations usually obtained via variational approach may be more conveniently derived using integral methods employing a geometrical description of the interpolating functions of edge and facet elements. Moreover, the resultant equations describe the equivalent multi-branch circuit models.

A Magnetic Flux–Electric Current Volume Integral Formulation Based on Facet Elements for Solving Electromagnetic Problems

“…On the other hand, Whitney facet interpolation can be used for representing current density J and flux density B. With this approach, an electromagnetic problem can be represented by an equivalent circuit using of magnetic fluxes and electric currents through the facets as proposed in [11].…”

A comparative analysis between classical and modified approach of description of the electrical machine windings by means ofT0method

“…where: w fq , w fq are interpolation functions of facet element [10]. For determining the edge values (i 0τ ) of potential T 0 on the boundary surface Γ T0 , the relation between edge quantities i 0τ and facet currents of EFN network (facet quantities i f 0 of current density vector J 0 , where J 0 curlT 0 ) has been employed [10].…”

“…To solve (3) the 3D Edge Element Method has been applied [9,10], for which equations (3) can be expressed in the following form:…”

“…For determining the edge values (i 0τ ) of potential T 0 on the boundary surface Γ T0 , the relation between edge quantities i 0τ and facet currents of EFN network (facet quantities i f 0 of current density vector J 0 , where J 0 curlT 0 ) has been employed [10]. To determine the i 0τ , it is necessary to include also condition that current density in normal direction to surface Γ T0 is equal to zero [6]:…”

“…where: i 0c and i 0n are the vectors describing the edge values of the potential T 0 inside region (Ωc) of the coil and beyond this region (Ωn) respectively [6]; ks is the transposed loop matrix of EFN [10], and the vector i 0τ describes given source of edge values of T 0 on surface Γ T0 . The matrices R Ωc and R Ωn are the coefficient matrices of the EE equations for considered regions Ωc and Ωn in the coil calculated from the following expressions:…”