Choosing the relaxation parameter for the solution of nonlinear magnetic field problems by the Newton-Raphson method

The use of the modified Newton method for the solution of nonlinear magnetostatic problems arising from the vector potential Finite Element analysis is investigated. In particular the optimum choice of the relaxation factor α is investigated. A new method is developed for determining the relaxation factor which minimizes the energy functional in the direction along the solution update at each nonlinear iteration. This method is based on approximating the functional with a fourth order polynomial. In this way the optimum relaxation factor can be quickly determined with the minimum number of extra function evaluations. This choice of α is compared to choosing the relaxation factor which minimizes the residual norm at each iteration. The modified Newton methods are compared to the standard Newton-Raphson method for the solution of 2D and 3D problems. For problems involving saturated iron parts convergence rates are greatly improved by use of the modified Newton methods. Of the two methods for choosing the relaxation factor the one which minimizes the functional is shown to be the better. Using the new algorithm to determine this relaxation factor results in substantial reductions in solution times