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I. Introduction
The fixed-point method (FPM) (Hantila et al., 2000; Chiampi et al., 1995) has an advantage that the convergence can be obtained even for a complicated nonlinear problems (Miyagi et al., 2012) such as the analysis considering vector magnetic properties treating an anisotropic material (Urata et al., 2006; Fujiwara et al., 2002), in which the convergence is sometimes difficult. In addition, it has an advantage that the software for nonlinear analysis can be easily obtained by adding a small change to that for linear analysis. But, the FPM requires a number of iterations and long CPU time compared with those of the Newton-Raphson method (NRM) (Nakata et al., 1992). It is reported that the CPU time can be reduced by using a constant reluctivity in the beginning of nonlinear iterations (Dlala et al., 2007, 2008). However, nearly ten times longer CPU time is still necessary compared with the NRM.
In this paper, a modified fixed-point method (MFPM), which updates the derivative of reluctivity at each iteration, is proposed. Furthermore, it is pointed out that the formulation of the FPM using the derivative of reluctivity is the same as the NRM. The convergence characteristic of the newly proposed FPM is compared with those of the NRM.
II. Formulation of NRM and FPM
A. Newton-Raphson method
There are two kinds of methods which deal with the nonlinearity in the NRM. One is the method A (NRM(B2)) which uses the ν-B2 curve. In this method, the magnetic field strength H is given by: Equation 1 B is the flux density. The reluctivity ν is given by: Equation 2 The other is the method B (NRM(B)) which uses the B-H curve directly. In this method, the magnetic field strength H is given by: Equation 3
(1) Method A (NRM(B2)
The static magnetic field equation can be written as follows in the case of the NRM using the ν-B2 curve: Equation 4 where, A is the magnetic vector potential. J 0 is the forced current density. The Galerkin equation G i *( A (k)) of equation (4) is given by: Equation 5 where, N i is the...