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Purpose
– The purpose of this paper is to propose the speed-up of the fixed-point method by updating the reluctivity at each iteration (this is called a modified fixed-point method).
Design/methodology/approach– A modified fixed-point method, which updates the derivative of reluctivity at each iteration, is proposed. It is shown that the formulation of the fixed-point method using the derivative of reluctivity is almost the same as that of the Newton-Raphson method. The convergence characteristic of the newly proposed fixed-point method is compared with those of the Newton-Raphson method.
Findings– The modified fixed-point method has an advantage that the programming is easy and it has a similar convergence property to the Newton-Raphson method for an isotropic nonlinear problem.
Originality/value– This paper presents the formulation and convergence characteristic of the modified fixed-point method are almost the same as those of the Newton-Raphson method.
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 interpolation function of the edge element. The residual G i ( A ) at the kth nonlinear iteration is given by: Equation 6 Equation 7 where, A , ν, etc. in equations (6) and (7) are values at the kth iteration. ∂ν/∂ B 2 is the term which represents nonlinear magnetic properties. The process of calculation is as follows:
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The initial value of ν is determined.
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δA (0) is set to zero.
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A is updated by A (k)= A (k−1)+δA (k) using δA (k) calculated by equation (6).
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ν(k) is calculated using the ν-B2 curve from B obtained by A (k).
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The process from (3) to (5) is repeated.
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It is judged to be converged if δB( A (k)) is less than a specified small value.
(2) Method B (NRM(B))
The static magnetic field equation can be written as follows in the case of the NRM using the B-H curve: Equation 8 The Galerkin equation G i *( A ) of equation (8) is given by: Equation 9 The residual G i ( A ) at the kth nonlinear iteration is given by: Equation 10 ∂ H ( B )/∂ B is the term which represents nonlinear magnetic properties.
The process of calculation is as follows:
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The initial value of ∂ H ( B (0))/∂ B is determined.
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δA (0) is set to zero.
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A is updated by A (k)= A (k−1)+δA (k) using δA (k) calculated by equation (10).
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H (k) is calculated using the B-H curve from B obtained by A (k).
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The process from (3) to (5) is repeated.
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It is judged to be converged if δB( A (k)) is less than a specified small value.
B. Fixed-point method
In the NRM, the reluctivity is updated in each nonlinear iteration as explained above. In the FPM, the reluctivity is fixed at the first step and it is not changed during the nonlinear iterations.
According to the concept of the FPM (Hantila et al., 2000), the magnetic field strength is given by: Equation 11 where, ν FP is the fixed-point reluctivity which is constant during the nonlinear iterations, H FP is an additional magnetic field strength.
The static magnetic field equation can be written as follows in the case of the FPM: Equation 12 where, H FP(k) at the kth nonlinear iteration can be obtained by the following equation: Equation 13 where, H ( B (k−1)) is the magnetic field strength vector on the B-H curve corresponding to the flux density B (k−1) at the (k−1)th nonlinear iteration. H FP(k) converges to some value after iterations. The residual G i ( A ) of equation (12) is given by: Equation 14 By substituting H FP(k) in equation (13) into H FP(k) in equation (14), we obtain: Equation 15 where, δ A (k)= A (k)− A (k−1).
G i *( A (k−1)) is given by: Equation 16 In the actual calculation, equation (14) is used in the FPM.
The process of calculation is as follows:
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The initial value of ν FP is determined.
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H FP(0) is set to zero.
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B (k) is obtained from A ( k) which is calculated by equation (14).
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H FP(k) is obtained by equation (13).
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The right hand side of equation (14) is updated and the process from (3) to (5) is repeated.
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It is judged to be converged if the change of B (k) is less than the specified small value.
Figure 1 shows the concept of the nonlinear magnetic field analysis using the FPM. A white circle on the B -axis is a convergence target. In this method, the reluctivity ν FP shown in Figure 1 is given as an initial value, and ν FP is not changed during the iterations. The flux density B (1) is obtained by the linear magnetic field analysis. Next, the H FP(1) which corresponds to the flux density B (1) on the B-H curve and ν FP B (1) on the line of ν FP shown in Figure 1(a) is obtained. During iterations, H FP(k) becomes the constant value, which means the difference δ H FP(k) becomes almost zero. Then, the converged result can be obtained.
C. Modified fixed-point method
In the MFPM, the derivative of reluctivity is updated at each iteration. In this expression, the H FP(k) at the kth nonlinear iteration in equation (13) can be rewritten by the following equation: Equation 18 The residual G i ( A (k)) is given by: Equation 19 Equation (19) can be written as follows: Equation 20 Equations (10) and (20) show that the formulation of the MFPM is the same as that of the NRM.
In the actual calculation of the MFPM, equation (19) is used.
The process of calculation is as follows:
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The initial value of ∂ H ( B (0))/∂ B is determined.
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H FP(0) is set to zero.
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B (k) is obtained from A (k) which is calculated by equation (19).
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H FP(k) is obtained by equation (18).
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The right hand side of equation (19) is updated and the process from (3) to (5) is repeated.
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It is judged to be converged if the change of B (k) is less than the specified small value.
III. Analyzed model
The MFPM is applied to the analysis of the magnetic field in the billet heater model (Takahashi et al., 2012) shown in Figure 3. The analysis domain of the model is 1/8. The material of the yoke is 35A230 (non-oriented electrical steel). The material of the billet is S45C (carbon steel). The numbers of elements and nodes are 107,632 and 115,101, respectively. The ampere turns of the coil are set to 70,000 AT (60 Hz). The CPU time and number of iteration of the FPM, MFPM, NRM using ν-B2 curve (NRM(B2)), and NRM using B-H curve (NRM(B)) are compared. For simplicity, only the calculation of the first step of the step by step method for the nonlinear eddy current analysis is carried out in order to compare the performance of each method. As the total CPU time is almost equal to the multiple of number of steps, the comparison of only the first step is sufficient for the comparison of each method.
IV. Results and discussion
Figure 4 shows an example of distribution of flux density of NRM(B) and MFPM. The results of NRM(B2) and FPM are also the same as Figure 4. The comparison of the CPU time and the number of iterations are shown in Table I. The convergence property is shown in Figure 5. The convergence criterion is ΔB( A )<2.0×10−3. The convergence criterion ‖G‖(n)/‖G‖(0) of the ICCG method is chosen as less than 10−5. Intel Core2 Duo E8400@ 3.16 GHz, 3 GB RAM is used. These results suggest that the convergence property of MFPM is near to that of NRM. It is also clarified that NRM(B2) is faster than NRM(B) in this analysis model.
V. Conclusions
The obtained results can be summarized as follows:
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The formulation of the MFPM using the derivative of reluctivity is almost the same as that of the NRM.
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The MFPM has an advantage that the CPU time is less than that of the NRM under some condition, or MFPM has almost the same performance as NRM. Moreover, the programming is easy compared with NRM.
Figure 1
Conceptual diagram of FPM
[Figure omitted. See PDF]
Figure 2
Conceptual diagram of MFPM
[Figure omitted. See PDF]
Figure 3
Analyzed model of billet heater
[Figure omitted. See PDF]
Figure 4
Comparison of numerical results of flux distribution using NRM() and MFPM
[Figure omitted. See PDF]
Figure 5
Convergence properties
[Figure omitted. See PDF]
Table I
Comparison of CPU time and iterations
[Figure omitted. See PDF]
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