Analysis of the Convergence of the Fixed-Point Method Used for Solving Nonlinear Rotational Magnetic Field Problems

The fixed-point method has not been widely used for solving nonlinear electromagnetic field problems, except for the hysteretic problem, for which it is the prevailing method. The method converges stably with a slow rate, exactly opposite to the Newton-Raphson method, which can easily suffer from instability, but which, if it converges, does so remarkably fast. In this paper, we analyze the convergence of the fixed-point method, examine the barriers behind the slow convergence, and show how to overcome them. The analysis has proved fruitful and provided sound techniques for speeding up the convergence of the fixed-point method. We have given special attention to the 2D and 3D problems and devised a general formula for the fastest convergence. We used a time-stepping finite-element formulation to test the convergence of the fixed-point method and compare it to the Newton-Raphson method. We investigated certain factors including the time-step size in magnetic field simulations of two rotating electrical machines.