Speed-up of nonlinear magnetic field analysis using a modified fixed-point method

Abstract

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.

Recent advances in computational electromagnetics

Edited by Prof. Oszkár Bíró, Prof. David A. Lowther and Prof. Piergiorgio Alotto

I. Introduction

The fixed-point method (FPM) ([1] Hantila et al. , 2000; [2] Chiampi et al. , 1995) has an advantage that the convergence can be obtained even for a complicated nonlinear problems ([3] Miyagi et al. , 2012) such as the analysis considering vector magnetic properties treating an anisotropic material ([4] Urata et al. , 2006; [5] 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) ([6] 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 ([7], [8] 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 [Figure omitted. See Article Image.] B is the flux density. The reluctivity ν is given by: Equation 2 [Figure omitted. See Article Image.] 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 [Figure omitted. See Article Image.]

(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 [Figure omitted. See Article Image.] where, A is the magnetic vector potential. J₀ is the forced current density. The Galerkin equation G_i^* (A^{(k )} ) of equation (4) is given by: Equation 5 [Figure omitted. See Article Image.] where, N_i is the interpolation function of the edge element. The residual G_i (A ) at the k th nonlinear iteration is given by: Equation 6 [Figure omitted. See Article Image.] Equation 7 [Figure omitted. See Article Image.] where, A , ν , etc. in equations (6) and (7) are values at the k th iteration. ∂ν /∂B² is the term which represents nonlinear magnetic properties. The process of calculation is as follows:

The initial value of ν is determined.

δA⁽⁰⁾ is set to zero.

A is updated by A^{(k )} =A^{(k -1)} +δA^{(k )} using δA^{(k )} calculated by equation (6).

ν^{(k )} is calculated using the ν -B² curve from B obtained by A^{(k )} .

The process from (3) to (5) is repeated.

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 [Figure omitted. See Article Image.] The Galerkin equation G_i^* (A ) of equation (8) is given by: Equation 9 [Figure omitted. See Article Image.] The residual G_i (A ) at the k th nonlinear iteration is given by: Equation 10 [Figure omitted. See Article Image.] ∂H (B )/∂B is the term which represents nonlinear magnetic properties.

The process of calculation is as follows:

The initial value of ∂H (B⁽⁰⁾ )/∂B is determined.

δA⁽⁰⁾ is set to zero.

A is updated by A^{(k )} =A^{(k -1)} +δA^{(k )} using δA^{(k )} calculated by equation (10).

H^{(k )} is calculated using the B -H curve from B obtained by A^{(k )} .

The process from (3) to (5) is repeated.

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 ([1] Hantila et al. , 2000), the magnetic field strength is given by: Equation 11 [Figure omitted. See Article Image.] 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 [Figure omitted. See Article Image.] where, H_FP^{(k )} at the k th nonlinear iteration can be obtained by the following equation: Equation 13 [Figure omitted. See Article Image.] 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 [Figure omitted. See Article Image.] By substituting H_FP^{(k )} in equation (13) into H_FP^{(k )} in equation (14), we obtain: Equation 15 [Figure omitted. See Article Image.] where, δA^{(k )} =A^{(k )} -A^{(k -1)} .

G_i^* (A^{(k -1)} ) is given by: Equation 16 [Figure omitted. See Article Image.] In the actual calculation, equation (14) is used in the FPM.

The process of calculation is as follows:

The initial value of ν_FP is determined.

H_FP⁽⁰⁾ is set to zero.

B^{(k )} is obtained from A^{(k )} which is calculated by equation (14).

H_FP^{(k )} is obtained by equation (13).

The right hand side of equation (14) is updated and the process from (3) to (5) is repeated.

It is judged to be converged if the change of B^{(k )} is less than the specified small value.

According to equation (15), we found that δH_FP^{(k )} is given by: Equation 17 [Figure omitted. See Article Image.] Equation (17) means that the difference δH_FP^{(k )} is the same as δH in equation (9) of the NRM and it can be used as the judgment of the convergence.

Figure 1 [Figure omitted. See Article Image.] 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 [Figure omitted. See Article Image.] 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_FPB⁽¹⁾ on the line of ν_FP shown in Figure 1(a) [Figure omitted. See Article Image.] is obtained. During iterations, H_FP^{(k )} becomes the constant value, which means the difference δH_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 k th nonlinear iteration in equation (13) can be rewritten by the following equation: Equation 18 [Figure omitted. See Article Image.] The residual G_i (A^{(k )} ) is given by: Equation 19 [Figure omitted. See Article Image.] Equation (19) can be written as follows: Equation 20 [Figure omitted. See Article Image.] 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:

The initial value of ∂H (B⁽⁰⁾ )/∂B is determined.

H_FP⁽⁰⁾ is set to zero.

B^{(k )} is obtained from A^{(k )} which is calculated by equation (19).

H_FP^{(k )} is obtained by equation (18).

The right hand side of equation (19) is updated and the process from (3) to (5) is repeated.

It is judged to be converged if the change of B^{(k )} is less than the specified small value.

Figure 2 [Figure omitted. See Article Image.] shows the concept of the nonlinear magnetic field analysis using the MFPM. In this method, the reluctivity ν_FP shown in Figure 2(a) [Figure omitted. See Article Image.] is given as an initial value, and the derivative ∂H /∂B is updated at each iteration. At the initial iteration, the linear magnetic field analysis is carried out using the given ∂H /∂B , and the flux density B⁽¹⁾ is obtained. Next, H (B⁽¹⁾ ) corresponding to the flux density B⁽¹⁾ on the B -H curve and ∂H (B⁽¹⁾ )/∂B · B⁽¹⁾ =V_FPB⁽¹⁾ on the line ∂H (B⁽¹⁾ )/∂B shown in Figure 2(a) [Figure omitted. See Article Image.] is obtained. At the first step, δH_FP^{(k )} =H_FP⁽¹⁾ following the definition of H_FP⁽¹⁾ in equation (17). The iteration is carried out until δH_FP becomes near to zero. δH (which corresponds to δH_FP in equation (17)) can be directly obtained by using the NRM as shown in equation (10). The MFPM needs two steps (equations (18) and (19)), but the concept is the same as that of the NRM.

III. Analyzed model

The MFPM is applied to the analysis of the magnetic field in the billet heater model ([9] Takahashi et al. , 2012) shown in Figure 3 [Figure omitted. See Article Image.]. 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 [Figure omitted. See Article Image.] 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 [Figure omitted. See Article Image.]. The comparison of the CPU time and the number of iterations are shown in Table I [Figure omitted. See Article Image.]. The convergence property is shown in Figure 5 [Figure omitted. See Article Image.]. 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:

- The formulation of the MFPM using the derivative of reluctivity is almost the same as that of the NRM.

- 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.

References

1. Hantila, F.I., Preda, G. and Vasiliu, M. (2000), "Polarization method for static fields", IEEE Trans. Magn., Vol. 36 No. 4, pp. 672-675.

2. Chiampi, M., Chiarabaglio, D. and Repetto, M. (1995), "A Jiles-Atherton and fixed-point combined technique for time periodic magnetic field problems with hysteresis", IEEE Trans. Magn., Vol. 31 No. 6, pp. 4306-4311.

3. Miyagi, D., Shimomura, K., Takahashi, N. and Kaimori, H. (2012), "Usefulness of fixed point method in electromagnetic field analysis in consideration of nonlinear magnetic anisotropy", Digest of the 15th Biennial IEEE Conference on Electromagnetic Field Computation (CEFC2012), p. 41.

4. Urata, S., Enokizono, M., Todaka, T. and Shimoji, H. (2006), "Magnetic characteristic analysis of the motor considering 2-D vector magnetic property", IEEE Trans. Magn., Vol. 42 No. 4, pp. 615-618.

5. Fujiwara, K., Adachi, T. and Takahashi, N. (2002), "A proposal of finite-element analysis considering two-dimension magnetic properties", IEEE Trans. Magn., Vol. 38 No. 2, pp. 889-892.

6. Nakata, T., Takahashi, N., Fujiwara, K. and Okamoto, N. (1992), "Improvements of convergence characteristics of Newton-Raphson method for nonlinear magnetic field analysis", IEEE Trans. Magn., Vol. 28 No. 2, pp. 1048-1051.

7. Dlala, E., Belahcen, A. and Arkkio, A. (2007), "Locally convergent fixed-point method for solving time-stepping nonlinear field problems", IEEE Trans. Magn., Vol. 43 No. 11, pp. 3969-3975.

8. Dlala, E., Belahcen, A. and Arkkio, A. (2008), "A fast fixed-point method for solving magnetic field problems in media of hysteresis", IEEE Trans. Magn., Vol. 44 No. 6, pp. 1214-1217.

9. Takahashi, N., Nakazaki, S., Miyagi, D., Uchida, N., Kawanaka, K. and Namba, H. (2012), "3-D optimal design of laminated yoke of billet heater for rolling wire rod using ON/OFF method", Archives of Electrical Engineering, Vol. 61 No. 1, pp. 115-123.

Appendix

Corresponding author

Daisuke Miyagi can be contacted at: [email protected]

  

Illustration

Equation 1

  

Equation 2

  

Equation 3

  

Equation 4

  

Equation 5

  

Equation 6

  

Equation 7

  

Equation 8

  

Equation 9

  

Equation 10

  

Equation 11

  

Equation 12

  

Equation 13

  

Equation 14

  

Equation 15

  

Equation 16

  

Equation 17

  

Equation 18

  

Equation 19

  

Equation 20

  

Figure 1: Conceptual diagram of FPM

  

Figure 2: Conceptual diagram of MFPM

  

Figure 3: Analyzed model of billet heater

  

Figure 4: Comparison of numerical results of flux distribution using NRM(B ) and MFPM

  

Figure 5: Convergence properties

  

Table I: Comparison of CPU time and iterations