A New Vector Potential BEM for Magnetic Fields Bounded by Perfect Conductors

A novel formulation of the boundary integral equation for the magnetic vector potential is presented, where its normal component is imposed to be zero while, instead of enforcing its tangential component, only the circulations of the potential are imposed along any closed paths on the boundary. When the boundary is modelled to be a perfect conductor, these circulations are equal to 0. It is proposed to represent the tangential component of the vector potential as a linear combination of specialized vector functions obtained from the gradients of the nodal element functions. The tangential component of the magnetic induction can be represented in the same way if the boundary is not crossed by electric currents. The integral equation is projected on these vector functions and on an orthogonal set of vector functions simply obtained from the same nodal element functions. This yields an improved conditioning of the system matrix. The number of unknowns is only twice the number of nodes, thus making this method more efficient than existing methods employing the perfect conductor model. The proposed procedure can also straightforwardly be applied to the case when the region outside the perfect conductors is multiply connected, by introducing scalar unknowns associated with the “cuts” employed. It is of practical importance for efficient engineering computations of three-dimensional magnetic fields and inductances in complex conductor systems. Computation examples are given to illustrate the efficiency of the method presented for simply and for multiply connected regions.