An accurate investigation on numerical methods for nonlinear magnetic field problems

Abstract

This paper investigates different iterative techniques (direct iteration, Newton-Raphson and fixed point methods) for the solution of nonlinear magnetic problems neglecting hysteresis. The investigations are carried out on a one-dimensional model problem whose exact nonlinear solution is computed when the magnetic flux is assigned. In spite of its simplicity, this model can present the same computational difficulties as most actual problems. Since the exact solution is known, it is possible to analyze the convergence of the different methods by comparing exact and computed solutions both in terms of magnetic potential and of magnetic flux density. The analysis shows that the direct iteration method is not always convergent and that the Newton-Raphson technique is usually fast, but some difficulties can arise in the case of magnetic characteristics presenting inflections. In contrast, the fixed point method is always convergent but its convergence rate can be slow. The paper investigates the capability of different norms between two successive iterations to indicate the convergence of the method, and the advantages of coupling the fixed point and the Newton-Raphson methods are presented.