Three-dimensional nonlinear magnetic field boundary value problem and its numerical solution

An accurate solution to the nonlinear magnetic field boundary value problem which results when ferromagnetic materials are present is generally very difficult to obtain. Several computer codes based on a magnetic vector potential have been published in the literature. While those codes can be used to solve 2-dimensional problems, difficulties arise in their application to 3-dimensional problems. One difficulty is the complexity of the resulting differential equations which must be solved. This paper treats the more complex 3-dimensional solutions as derived from the concept of a scalar magnetic potential. The most desirable feature of this approach is that the magnetic material characteristics are incorporated directly into one second order nonlinear partial differential equation which can be solved on a digital computer using standard finite difference schemes. In order to add credence to the validity of the technique presented, a simple 3-dimensional magnetic field boundary value problem is posed and solved.