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.

I. Introduction

There are many reasons why designers of electromagnetic equipment would want to know the value of magnetic field or inductive electric field on the surfaces of conductors in the system. For example, in high-current pulsed power devices, magnetic induction exceeding the characteristic induction can cause problems [1]. For example, for induction above approximately 43 T for copper, diffusion becomes significantly nonlinear. In many cases, this process is undesirable, as it can result in enhanced flux loss, surface melting and plasma creation, and magnetohydrodynamic (MHD) motion. For superconductors, one usually wants to keep the surface fields below the critical field to avoid transition out of the superconducting state.