Three Dimensional Finite Element Vector Potential Formulation of Magnetic Fields in Electrical Apparatus

The three dimensional magnetic vector potential (m.v.p.) variational formulation for magnetostatic field problems, with ideal current carrying conductors is given. An appropriate functional was chosen for this class of problems. Variational techniques were used to prove that the chosen functional is stationary whenever the partial differential equations, the Neumann, and the specified Dirichlet boundary conditions are satisfied, within and on the boundary of the volume under investigation, respectively. The finite element method is used to obtain a numerical approximation of the stationary point of the functional. A three dimensional discretization scheme involving first order tetrahedral elements was used to implement the method. The m. v. p. at each of the four tetrahedral vertices (nodes) has three degrees of freedom (or components) in the x, y and z directions due to the three dimensional nature of the field.