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Optimal design of magnetic devices
Journal of Magnetism and Magnetic Materials 209 (2000) 42}44
Optimal design of magnetic devices N. Takahashi* Department of Electrical and Electronic Engineering, Okayama University, Okayama 700-8530, Japan
Abstract The e!ect of number of "nite elements on the obtained optimal value is examined, and the applicability of optimization method to practical magnetic devices is investigated. It is shown that the model should be subdivided into a su$cient number of "nite elements in order to get the optimal value in good accuracy. Rosenbrock's method can be applicable to the optimization of practical 3-D eddy current device. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Optimal design; Magnetic devices; Finite element method
1. Introduction Various optimization methods, such as a Rosenbrock's method (RBM) [1], an evolution strategy (ES) [2], and a simulated annealing method (SAM) [3], are recently applied to the optimal design of magnetic devices. RBM, which is a kind of direct search method, has an advantage that the number of iterations is small, but it cannot search the global minimum of objective function, in general. In the case of ES and SAM, the number of iterations is large, but the global minimum can be obtained. Therefore, RBM may be suitable to the optimal design of actual 3-D machines although only a local minimum is obtained, and ES and SAM may be suitable for searching a global minimum of 2-D nonlinear problems. In this paper, the e!ect of number of "nite elements on the obtained results is discussed [4] using the benchmark model (TEAM Workshop Problem 25) [5]. The usefulness of some typical optimization methods are examined by applying them to actual models from the viewpoint of practical application. 2. E4ect of number of 5nite elements on optimal value The result obtained by the optimal design method using FEM is not always the same for di!erent "nite * Tel.: #81-86-251-8115; fax: #81-86-251-8258. E-mail address: [email protected] (N. Takahashi)
element meshes. Then, the e!ect of number of elements on optimal value of problem 25 (die press model) is investigated using contour lines of objective function. Fig. 1 shows the die press model [5]. The objective function = is de"ned by the following equation: n =" + M(B !B )2#(B !B )2N, (1) x*1 xi0 y*1 yi0 i/1 where n ("10) is the number of the speci"ed points on the line e}f in the cavity in Fig. 1. The subscripts p and 0 mean the calculated and speci"ed values, respectively. The radius R of inner die and the long axis ¸ of ellipse 1 2 of outer die are "xed to 9.3 and 18.0 mm, respectively. The short axis ¸ of ellipse and the length ¸ of outer die 3 4 are chosen as design variables. Fig. 2 shows the contour lines of the objective function. The "gure is obtained using the ordinary "nite element method for di!erent die molds for which the dimensions ¸ and ¸ are changed at the step size of 0.1 mm in the 3 4 range of 14(¸ (15.9 mm and 16.5(¸ (18.4 mm. 3 4 The number `nea of "nite elements is changed from 1130 to 7876. The "gure denotes that there exists several local minima in this problem. Table 1 shows the results obtained using the simulated annealing method (SAM). The obtained dimensions ¸ and ¸ are shown in Fig. 2 by 3 4 the mark (m). The error e and e are de"ned by B.!9 h.!9 B !B 0 ]100 (%), (2) e "max 1 B.!9 B 0 e "maxDh !h D (deg). (3) h.!9 1 0
0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 6 4 1 - 1
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N. Takahashi / Journal of Magnetism and Magnetic Materials 209 (2000) 42}44
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Table 1 E!ect of number of elements on the obtained result
Fig. 1. Die press model.
The subscripts p and 0 denote the calculated and speci"ed values of the amplitude B of #ux density vector and angle h, respectively. In Fig. 2, the shapes of contour line of small number of elements (ne"1130) is di!erent from that of large number of elements (ne"3970, 7876). If the number of elements are increased, almost the same results can be obtained. This is because the dimensions ¸ and ¸ con3 4 verge to the values which deviate from the optimal value due to the large error of #ux density when ne is not su$ciently large. On the contrary, when ne is large, ¸ and ¸ converge to the same values because the 3 4 accurate #ux density can be obtained. Fig. 2 and Table 1 suggest that this model should be analyzed using the mesh having about 4000 elements.
ne
¸ 3 (mm)
¸ 4 (mm)
= (]10~4)
e B.!9 (%)
e h.!9 (deg)
1130 2037 2980 3970 5934 7876
14.27 14.68 15.00 14.73 14.71 14.75
18.03 17.33 16.95 17.09 17.06 17.12
9.758 7.128 5.634 5.135 5.139 5.191
4.026 3.424 2.702 2.803 1.964 1.935
1.856 0.834 1.845 1.793 1.404 1.397
3. Optimal design of 3-D retarder using Rosenbrock:s method Fig. 3 shows a model of permanent magnet type of retarder, which is used as an auxiliary brake for heavy vehicle [6]. The inner side having pole piece, permanent magnet and yoke are stationary and outer rotor rotates. When rotor rotates in the counterclockwise direction as shown in Fig. 3, the eddy current #ows in the rotor and electromagnetic force is induced due to the #ux of permanent magnet and the eddy current in the clockwise direction. The outer rotor rotates with a constant speed at 1000 rpm. The outer rotor, pole piece and yoke are made of carbon steel (S15C) and the nonlinearity is taken into account. The permanent magnet (Sm Co ) is assumed 2 17
Fig. 2. Contour lines of objective function.
44
N. Takahashi / Journal of Magnetism and Magnetic Materials 209 (2000) 42}44 Table 2 Design variables and braking torque! Shape
Initial Optimal Fig. 3. Permanent magnet type of retarder (thickness: 16 mm).
Fig. 4. Flux distributions (1000 rpm).
Design variable ¸ 1 (mm)
¸ 2 (mm)
h 1 (deg)
h 2 (deg)
Braking torque (N m)
6.3 4.3
3.4 6.3
75 25.3
105 146.3
1.92 (1.0) 2.66 (1.39)
!Computer used: HPC160(140MFLOPS); convergence criterion: 0.01 mm, 0.013; number of iterations: 84; CPU time: 31.5 h.
10 000 rpm [7]. The obtained shape is di!erent from that in Fig. 4. The "gure denotes that it is not easy to obtain the optimal shape shown in Fig. 5 by only the parameter survey (changing the design variables) which is used in the conventional design process, because the shape is fairly di!erent from that in Fig. 4. The optimal value (although the local minimum) can be obtained within 100 iterations by using RBM. As the CPU time for one iteration is long in the case of 3-D analysis, it is necessary to adopt the fast method like RBM.
4. Conclusions The obtained results can be summarized as follows: (1) As the contour line is a!ected by the number of "nite elements, the search of optimal value may reach a di!erent value if the number of elements is not su$cient. (2) The Rosenbrock's method is applicable to 3-D optimal design from the viewpoint of practical application. Fig. 5. Flux distribution (10 000 rpm).
References to be magnetized in parallel direction and the magnetization is equal to 1 T. The conductivity of the outer rotor is equal to 7]106 S/m. The angles h and h of pole piece and the width ¸ of 1 2 1 yoke and the length ¸ of magnet are chosen as design 2 variables. The optimization is carried out using Rosenbrock's method (RBM) in order to increase the braking torque. Fig. 4 shows the #ux distributions for initial and optimal shapes. The braking torque of the optimal shape is 1.39 times larger than that of the initial shape as shown in Table 2. Fig. 5 shows the obtained optimal shape at
[1] D.M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, New York, 1972. [2] T. BaK ck, Evolutionary Algorithm in Theory and Practice, Oxford University Press, Oxford, 1996. [3] J. Simkin, C.W. Trowbridge, IEEE Trans. Magn. 28 (6) (1992) 1545. [4] N. Takahashi, M. Otoshi, Proceedings of TEAM Workshop, 7th Round, Tucson, 1998, p. 19. [5] N. Takahashi, Proceedings of the TEAM Workshop in the Sixth Round, Okayama, 1996, p. 61. [6] N. Takahashi et al., IEEE Trans. Magn. 34 (5) (1998) 2996. [7] N. Takahashi et al., Proceedings of LIDIA'98, 1998, p. 375.
Optimal design of magnetic devices N. Takahashi* Department of Electrical and Electronic Engineering, Okayama University, Okayama 700-8530, Japan
Abstract The e!ect of number of "nite elements on the obtained optimal value is examined, and the applicability of optimization method to practical magnetic devices is investigated. It is shown that the model should be subdivided into a su$cient number of "nite elements in order to get the optimal value in good accuracy. Rosenbrock's method can be applicable to the optimization of practical 3-D eddy current device. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Optimal design; Magnetic devices; Finite element method
1. Introduction Various optimization methods, such as a Rosenbrock's method (RBM) [1], an evolution strategy (ES) [2], and a simulated annealing method (SAM) [3], are recently applied to the optimal design of magnetic devices. RBM, which is a kind of direct search method, has an advantage that the number of iterations is small, but it cannot search the global minimum of objective function, in general. In the case of ES and SAM, the number of iterations is large, but the global minimum can be obtained. Therefore, RBM may be suitable to the optimal design of actual 3-D machines although only a local minimum is obtained, and ES and SAM may be suitable for searching a global minimum of 2-D nonlinear problems. In this paper, the e!ect of number of "nite elements on the obtained results is discussed [4] using the benchmark model (TEAM Workshop Problem 25) [5]. The usefulness of some typical optimization methods are examined by applying them to actual models from the viewpoint of practical application. 2. E4ect of number of 5nite elements on optimal value The result obtained by the optimal design method using FEM is not always the same for di!erent "nite * Tel.: #81-86-251-8115; fax: #81-86-251-8258. E-mail address: [email protected] (N. Takahashi)
element meshes. Then, the e!ect of number of elements on optimal value of problem 25 (die press model) is investigated using contour lines of objective function. Fig. 1 shows the die press model [5]. The objective function = is de"ned by the following equation: n =" + M(B !B )2#(B !B )2N, (1) x*1 xi0 y*1 yi0 i/1 where n ("10) is the number of the speci"ed points on the line e}f in the cavity in Fig. 1. The subscripts p and 0 mean the calculated and speci"ed values, respectively. The radius R of inner die and the long axis ¸ of ellipse 1 2 of outer die are "xed to 9.3 and 18.0 mm, respectively. The short axis ¸ of ellipse and the length ¸ of outer die 3 4 are chosen as design variables. Fig. 2 shows the contour lines of the objective function. The "gure is obtained using the ordinary "nite element method for di!erent die molds for which the dimensions ¸ and ¸ are changed at the step size of 0.1 mm in the 3 4 range of 14(¸ (15.9 mm and 16.5(¸ (18.4 mm. 3 4 The number `nea of "nite elements is changed from 1130 to 7876. The "gure denotes that there exists several local minima in this problem. Table 1 shows the results obtained using the simulated annealing method (SAM). The obtained dimensions ¸ and ¸ are shown in Fig. 2 by 3 4 the mark (m). The error e and e are de"ned by B.!9 h.!9 B !B 0 ]100 (%), (2) e "max 1 B.!9 B 0 e "maxDh !h D (deg). (3) h.!9 1 0
0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 6 4 1 - 1
K
K
N. Takahashi / Journal of Magnetism and Magnetic Materials 209 (2000) 42}44
43
Table 1 E!ect of number of elements on the obtained result
Fig. 1. Die press model.
The subscripts p and 0 denote the calculated and speci"ed values of the amplitude B of #ux density vector and angle h, respectively. In Fig. 2, the shapes of contour line of small number of elements (ne"1130) is di!erent from that of large number of elements (ne"3970, 7876). If the number of elements are increased, almost the same results can be obtained. This is because the dimensions ¸ and ¸ con3 4 verge to the values which deviate from the optimal value due to the large error of #ux density when ne is not su$ciently large. On the contrary, when ne is large, ¸ and ¸ converge to the same values because the 3 4 accurate #ux density can be obtained. Fig. 2 and Table 1 suggest that this model should be analyzed using the mesh having about 4000 elements.
ne
¸ 3 (mm)
¸ 4 (mm)
= (]10~4)
e B.!9 (%)
e h.!9 (deg)
1130 2037 2980 3970 5934 7876
14.27 14.68 15.00 14.73 14.71 14.75
18.03 17.33 16.95 17.09 17.06 17.12
9.758 7.128 5.634 5.135 5.139 5.191
4.026 3.424 2.702 2.803 1.964 1.935
1.856 0.834 1.845 1.793 1.404 1.397
3. Optimal design of 3-D retarder using Rosenbrock:s method Fig. 3 shows a model of permanent magnet type of retarder, which is used as an auxiliary brake for heavy vehicle [6]. The inner side having pole piece, permanent magnet and yoke are stationary and outer rotor rotates. When rotor rotates in the counterclockwise direction as shown in Fig. 3, the eddy current #ows in the rotor and electromagnetic force is induced due to the #ux of permanent magnet and the eddy current in the clockwise direction. The outer rotor rotates with a constant speed at 1000 rpm. The outer rotor, pole piece and yoke are made of carbon steel (S15C) and the nonlinearity is taken into account. The permanent magnet (Sm Co ) is assumed 2 17
Fig. 2. Contour lines of objective function.
44
N. Takahashi / Journal of Magnetism and Magnetic Materials 209 (2000) 42}44 Table 2 Design variables and braking torque! Shape
Initial Optimal Fig. 3. Permanent magnet type of retarder (thickness: 16 mm).
Fig. 4. Flux distributions (1000 rpm).
Design variable ¸ 1 (mm)
¸ 2 (mm)
h 1 (deg)
h 2 (deg)
Braking torque (N m)
6.3 4.3
3.4 6.3
75 25.3
105 146.3
1.92 (1.0) 2.66 (1.39)
!Computer used: HPC160(140MFLOPS); convergence criterion: 0.01 mm, 0.013; number of iterations: 84; CPU time: 31.5 h.
10 000 rpm [7]. The obtained shape is di!erent from that in Fig. 4. The "gure denotes that it is not easy to obtain the optimal shape shown in Fig. 5 by only the parameter survey (changing the design variables) which is used in the conventional design process, because the shape is fairly di!erent from that in Fig. 4. The optimal value (although the local minimum) can be obtained within 100 iterations by using RBM. As the CPU time for one iteration is long in the case of 3-D analysis, it is necessary to adopt the fast method like RBM.
4. Conclusions The obtained results can be summarized as follows: (1) As the contour line is a!ected by the number of "nite elements, the search of optimal value may reach a di!erent value if the number of elements is not su$cient. (2) The Rosenbrock's method is applicable to 3-D optimal design from the viewpoint of practical application. Fig. 5. Flux distribution (10 000 rpm).
References to be magnetized in parallel direction and the magnetization is equal to 1 T. The conductivity of the outer rotor is equal to 7]106 S/m. The angles h and h of pole piece and the width ¸ of 1 2 1 yoke and the length ¸ of magnet are chosen as design 2 variables. The optimization is carried out using Rosenbrock's method (RBM) in order to increase the braking torque. Fig. 4 shows the #ux distributions for initial and optimal shapes. The braking torque of the optimal shape is 1.39 times larger than that of the initial shape as shown in Table 2. Fig. 5 shows the obtained optimal shape at
[1] D.M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, New York, 1972. [2] T. BaK ck, Evolutionary Algorithm in Theory and Practice, Oxford University Press, Oxford, 1996. [3] J. Simkin, C.W. Trowbridge, IEEE Trans. Magn. 28 (6) (1992) 1545. [4] N. Takahashi, M. Otoshi, Proceedings of TEAM Workshop, 7th Round, Tucson, 1998, p. 19. [5] N. Takahashi, Proceedings of the TEAM Workshop in the Sixth Round, Okayama, 1996, p. 61. [6] N. Takahashi et al., IEEE Trans. Magn. 34 (5) (1998) 2996. [7] N. Takahashi et al., Proceedings of LIDIA'98, 1998, p. 375.