Using the Boundary Element Method to calculate 3-D magnetic fields and potentials

The Boundary Element Method (BEM) is a well-known technique for solving the integral form of potential and flux/field problems. In this method, the solution to a partial differential equation is found on a closed surface, from which the full 3-D solution can be directly calculated anywhere in the interior. Its advantages over directly solving the basic partial differential equation include reducing the dimensionality of the problem from three dimensions to two, the ability to solve so-called external problems - where the variables extend to infinity - without artificial boundaries and boundary conditions, and good scalability of parallel computations because the associated matrices are dense. While scalar potential problems have been solved extensively using the BEM, vector potential problems have not. We have derived a fully 3-D BEM technique to solve the vector Laplace equation for the magnetic vector potential and the vector Laplace equation for the magnetic flux density or field. While the solutions we obtain are strictly valid only for non-conducting media, the technique can be generalized to include magnetic diffusion. In this paper, we describe the 3-D technique, and show how it can be used to calculate magnetic fields on the surfaces of pulsed power system conductors.