Electrical Engineering in Japan, Vol. 170, No. 1, 2010 Translated from Denki Gakkai Ronbunshi, Vol. 127-B, No. 8, August 2007, pp. 894–901
Proposal for a Benchmark Model of a Laminated Iron Core and a Large-Scale and Highly Accurate Magnetic Analysis 1
2
3
4
YASUHITO TAKAHASHI, SHINJI WAKAO, KOJI FUJIWARA, HIROYUKI KAIMORI, and 4 AKIHISA KAMEARI 1
Kyoto University, Japan Waseda University, Japan 3 Doshisha University, Japan 4 Science Solutions International Laboratory, Inc., Japan 2
cores by the ordinary finite element method (FEM), a fairly fine mesh division of the laminated structure is necessary. From a practical point of view, however, it is impossible to divide not only the steel but also the extremely thin insulation gaps between steel sheets microscopically into hexahedra or tetrahedra, because of the extremely large computational costs. As a simple method of modeling, the laminated core is frequently modeled as a solid one. In the case of skewed motors or motors with an overhanging magnet, the eddy current induced by the magnetic flux penetrating perpendicular to the steel sheets generates nonnegligible large eddy-current losses and the simple modeling cannot estimate the losses because of the difference difference in magnetic properties between the solid core and the laminated one. Therefore, the development of an accurate and fast method for modeling laminated iron cores is highly desirable. Recently, various approximate methods for modeling laminated iron cores, such as gap elements, double nodes, and homogenization methods, have been proposed, and research is energetically proceeding toward the practical use of these methods [1–5]. However, However, there are still ambiguities about their advantages and disadvantages, and it is necessary to compare various modeling methods by using appropriate benchmark models. With this background, in order to clarify the features of various modeling methods, we propose a benchmark model of the laminated iron core. In addition, we calculate the reference solution required in order to estimate the performance of various modeling methods. Using the reference solutions, we investigate the accuracy of the homogenization method, which is considered to be fairly effective from the viewpoint of the computational cost. In order to compare the various approximate modeling methods based on the benchmark model, the existence
SUMMARY This paper describes a benchmark model proposed for the clarification of the characteristic of various methods for modeling the laminated iron core. In order to obtain a reference solution of the benchmark model, a large-scale nonlinear magnetostatic field analysis with a mesh fine enough to represent the microscopic structure of the laminated iron core is carried out by using the hybrid finite element–boundary element (FE-BE) method combined with the fast multipole method (FMM) based on diagonal forms for translation operators. The computational costs and accuracy of two kinds of homogenization methods are discussed, comparing them with the reference solution. As a consequence, it is verified that the homogenization methods can analyze magnetic fields in laminated iron core within acceptable computational costs. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 170(1): 26–35, 2010; Published online in Wiley InterScience (www.interscience.wiley. com). DOI 10.1002/eej.20809 Key words: laminated iron core; nonlinear analysis; hybrid finite element–boundary element method; fast multipole method; homogenization method.
1. Introduction In order to reduce the losses and improve the efficiency of electric machinery, a highly accurate magnetic field analysis of the laminated iron core is desirable. The thickness of one silicon steel sheet is about 0.5 mm and the number of sheets in a laminated core is extremely large. large. In addition, steel sheets have complex and nonlinear magnetic properties. For highly accurate analysis of laminated iron
© 2009 Wiley Periodicals, Inc. 26
Table 1.
of a reference solution is preferable. It is considered significant to verify the computational accuracy of various approximate modeling methods by comparing them with a reference solution obtained without any approximation, based on clearly defined conditions of analysis. However, it is impossible to accurately analyze the magnetic field in a laminated iron core, while allowing for surrounding free space with an open boundary, by using the ordinary FEM, because the computational costs are extremely large due to detailed modeling of the laminated structure. On the other hand, the hybrid finite element and boundary element (FEBE) method [6–8] and the magnetic moment method [9] can reduce the number of elements drastically because no mesh division is required for free space. Furthermore, by introducing the fast multipole method (FMM) [10–13], we can analyze large-scale problems such as the detailed modeling of laminated structures. In this paper, we divide not only steel but also extremely thin insulation gaps into multiple layers of elements and perform a large-scale analysis of the benchmark model by the hybrid FE-BE method combined with the FMM. Because there is no precedent for extremely large-scale three-dimensional analysis with detailed modeling of laminated iron, it can be utilized as a reference solution for the validation of other approximate modeling methods. Next, we verify the effectiveness of the homogenization method. In the homogenization method, the mesh is not restricted by the laminated structures. Therefore, the homogenization method is superior to gap elements and double nodes from the standpoint of computational cost. We compare the numerical results obtained by the homogenization method with the reference solutions and investigate the accuracy.
2.
ing of Japan (IEEJ). The configuration is the same as the benchmark model for magnetostatic field analysis mentioned in Ref. 14. The coil is excited by a 3000 AT DC current. The core is constructed by laminating 200 nonoriented electrical steel sheets (JIS grade: 50A1300) in the x direction. The thickness is 0.5 mm and the space factor is 96%. The magnetic properties of the steel sheet are assumed to be isotropic and nonhysteretic. Table 1 shows the magnetic flux density B (T) and the magnetic field H (A/m), which are average values of the longitudinal and lateral components, measured by a single sheet tester [15]. As the first step for highly accurate analysis of a laminated iron core, we investigated nonlinear magnetostatic field problems to clarify the effectiveness of the various modeling methods in this paper.
Benchmark Model of Laminated Iron Core
Figure 1 shows the proposed laminated iron core model. This is one of the benchmark models proposed by a research committee of the Institute of Electrical Engineer-
3.
Calculation of Reference Solution by Hybrid Finite Element–Boundary Element Method
3.1
Fig. 1.
B–H curve of 50A1300
Formulation of hybrid FE-BE method
We adopt the hybrid FE-BE formulation using the magnetic scalar potential and the current vector potential [6, 8]. This hybrid method has the advantage that the number of unknowns can be reduced by using the scalar potential as much as possible in the formulation. Therefore, it is considered very effective for large-scale analysis. A first-order hexahedral element is adopted for the FEM and a mixed linear and constant quadrilateral element for the boundary element method (BEM) [6].
Benchmark model of laminated iron core.
27
where Y m n denotes the spherical harmonics defined in Ref. 10 and (r , θ, φ) are spherical coordinates. The corresponding multipole expansions are as follows:
In this paper, two regions are considered. Region Ωm, with boundary Γ m, consists of both a nonconducting magnetic substance, whose permeability µm may have nonlinear characteristics, and insulation gaps, whose magnetic properties are assumed to be the same as a vacuum. In this region with no supply current, the magnetic scalar potential ψ can be defined and the FEM is applied as follows:
(5) (6) In applying the FMM to the double layer potential, the gradients of the spherical harmonics are required. Although various methods with both advantages and disadvantages have been proposed [16–19], a method which is not required to take into account the singularity of Y m n on the z-axis is fairly useful (see the Appendix for more information). In this paper, we adopt the method based on Eq. (A.3), which provides the best performance from the standpoint of CPU time. After obtaining the multipole expansions by Eqs. (5) and (6) and adding the coefficients, we can deal with the contribution of the single and double layer potential simultaneously in the FMM process. Therefore, the translations of multipole and local expansions such as the multipole-tomultipole (M2M), multipole-to-local (M2L), and local-tolocal (L2L) translations [10] are the same as the ordinary FMM. The method based on Eq. (A.3) can be easily applied to the BEM, considering vector quantities such as the magnetic vector potential and the magnetic field as unknowns [20].
(1) where N is the scalar shape function for a nodal finite element and n is the unit normal vector on Γ . The region Ω0 is free space, which extends to infinity, with a supply current. The BEM is applied to this region and ψ is considered as the physical quantity. The integral equation is
(2) where Γ is the boundary between the FEM and BEM regions (= Γ m), C P is the solid angle enclosed by region Ω0, and ψ J is the magnetic scalar potential produced by a supply current [6]. The interface conditions between the free space and the region Ωm are based on the continuity of the potential and of the normal component of the magnetic flux density. The BEM and FEM can be combined directly without change of variables by using mixed linear and constant BE discretization, in which a linear element is used for the potential and a constant element for the normal derivative of the potential. 3.2
3.3 Calculation of reference solution by hybrid FE-BE method with FMM In order to obtain a reference solution of the benchmark model, a large-scale nonlinear magnetostatic field analysis is carried out by the hybrid FE-BE method combined with the FMM. We apply Eq. (1) to the laminated iron core including insulation gap and Eq. (2) to free space. The interface between the FEM and the BEM region is set in free space. Figure 2 shows the mesh of the laminated iron core model. An eighth part of the whole model is analyzed because of the symmetry. One sheet is divided into three layers of hexahedral elements in the laminated direction and the insulation gap between steel sheets is divided into two layers as shown in Fig. 2. As a nonlinear iteration method, we utilize the Newton–Raphson (NR) method. When the change of the flux density is less than 10 –3 T for each element, the NR iteration is terminated. We utilized the GMRES method [21] as an iterative solver in the NR method, and minor iterative preconditioning (MIP) with incomplete LU factorization [8, 23] as a preconditioning method. In the hybrid FE-BE method, the convergence characteristic of the NR iteration deteriorates when the convergence criterion of the iterative solver in the NR method is relaxed [19]. Therefore, the
Application of FMM to BEM
In order to reduce the large computational costs, the FMM based on diagonal forms for translation operators [10, 12] is introduced into the first and second terms of the right-hand side of Eq. (2), which correspond to the single and double layer potentials, respectively [13]. By using the spherical harmonic addition theorem, the first and second terms of the right-hand side of Eq. (2) can be written as follows: (3)
(4)
28
Fig. 2. Mesh of laminated core model. (a) Overall view; (b) Enlarged view.
convergence criterion for the GMRES method is set to 10 –8. In the minor iteration of the MIP, we adopt the BiCGSTAB2 method [22], and its convergence criterion is set to 10–3. Table 2 shows the specifications of the analysis. The CPU time is about 15 hours, because we utilize an extremely fine mesh to guarantee computational accuracy. Figure 3(a) shows the flux density distribution. The magnetic flux is deflected to the end of the core in the direction of the lamination. Because the homogenization method is based on the assumption of periodicity of the microscopic structure, there is a possibility that the computational accuracy may deteriorate at the end of the core, where periodicity does not apply. For this reason, the flux density is evaluated at the end of the core. Figure 3(b) shows the z-component of the flux density B z along line A (0 < x < 50, y = 49 mm, z = 50 mm) and line B (0 < x < 50, y = 49 mm, z = 99 mm) shown in Fig. 1. All the evaluation points are inside the steel sheets. On line A, B z is constant in one steel sheet and varies discontinuously between the steel sheets. On the other hand, the change of B z near the surface
Fig. 3. Reference solution obtained by the hybrid FE-BE method. (a) Distribution of magnetic flux density; (b) Z -component of magnetic flux density in laminated core.
of the core is fairly large because line B is close to the edge of the core. 4.
Homogenization Method
4.1 Table 2.
Finite Element Analysis with
Specifications of analysis
Homogenization method 1 [2]
By using the homogenization method, the laminated structure shown in Fig. 4(a) is replaced by a homogeneous magnetic substance with the equivalent magnetic anisotropy as shown in Fig. 4(b). Therefore, the homogenization method enables us to regard the laminated iron core as a solid one. In this homogenization method, the magnetic properties of the steel sheet are assumed to be isotropic and nonhysteretic. In addition, the magnetic field and the magnetic flux are assumed to be locally uniform in the steel and the insulation gap.
29
(12) Fig. 4. Homogenization of laminated iron core. (a) Laminated core; (b) Macroscopic model.
(13)
The homogenized magnetic reluctivities parallel and perpendicular to the lamination N || and N ⊥ are obtained as follows:
In applying this homogenization method to the FEM, we replace the original constitutive relations with the homogenized one represented by Eq. (12).
(7)
4.2 Homogenization method 2 [24, 25]
(8) In the case of an ordinary laminated iron core, n0 >> ns and α ≈ 1 are generally true. By using this assumption, Eqs. (7) and (9) can be simplified as follows:
where α is the space factor, ns(bs) and n0 are the magnetic reluctivities of the steel sheet and the vacuum, and bs is the magnetic flux density in the steel sheet. These equations are easily derived from the continuity of the tangential component of the magnetic field and the normal component of the flux density. On the other hand, bs is given by
(14) (15)
In this case, we can calculate bs directly from the homogenized magnetic flux. Furthermore, because β is approximated by 1, Eq. (12) is simplified as follows:
(9)
where B( B X , BY , B Z ) is the homogenized magnetic flux density, obtained by the FEM, and the Z direction is perpendicular to the lamination. bs and ns are obtained from Eqs. (7) and (9) by the NR method. The homogenized field is solved by the usual FEM with the following equation: (16) (10) where A is the magnetic vector potential, H is the magnetic field, J is the current density, and N i is the vector shape function. Because Eq. (10) is a nonlinear equation, it is solved by using the NR method as follows:
Figure 5 shows the original magnetic properties of 50A1300 and the homogenized ones obtained by Eqs. (7) and (14). The value in parentheses is the space factor. In order to clarify the difference between homogenization methods 1 and 2 from the standpoint of versatility, we investigate two values of the space factor: 0.96, which is the original space factor, and 0.1, the extreme case. When α is 0.96, there is little difference between homogenization methods 1 and 2, which means that Eq. (14) approximates the homogenized magnetic properties with satisfactory accuracy. On the other hand, when α is 0.1, the difference becomes greater as the steel becomes magnetically satu-
(11) The Jacobi matrix can be derived from Eqs. (7) to (9) and is given as follows:
30
Fig. 5. Magnetic characteristics of laminated core.
rated. When the steel is extremely saturated, ns(bs) is about equal to n0. In this situation, the difference between the homogenized magnetic reluctivities obtained by Eqs. (7) and (14) is about 4.2% when α is 0.96, and about 10% when α is 0.1. Therefore, it is necessary to adopt homogenization method 1 when the space factor is fairly small or the steel is extremely saturated.
Fig. 6. Distribution of magnetic flux density (homogenization method). (a) Nonuniform mesh; (b) Uniform mesh. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]
4.3 Numerical examples We perform the finite element analysis of the laminated iron core model shown in Fig. 1 by the homogenization method. A first-order hexahedral element is used. When the change of the flux density is less than 10 –3 T for each element, the NR iteration is terminated. The NR iteration for bs in the process of homogenization method 1 is terminated when the relative residual norm is less than 10–3. Dirichlet boundary conditions are imposed on the surfaces at x = 1 m, y = 1 m, and z = 1 m. In order to investigate the influence of mesh division, a nonuniform mesh, in which the core is divided nonuniformly in the direction of lamination and the z direction, taking computational accuracy into consideration, and a uniform mesh, in which the core is divided uniformly, are compared. In the nonuniform mesh, the size of the elements in the direction of the lamination is 2.5 mm inside the core and becomes gradually smaller near the end of the core. The smallest size is 0.44 mm, which is thinner than the thickness of a steel sheet. In the uniform mesh, the size of all elements in the core is 2 mm. Table 2 shows the specifications of the analysis for the nonuniform mesh in homogenization method 1. Figure 6 shows the magnetic flux density distribution. It agrees qualitatively with Fig. 3(b), which was obtained by the hybrid FE-BE method as a reference solution. However, the magnetic flux density shown in Fig. 6(b) is slightly smaller at the end of the core than in Figs. 3 and 6(a), because the mesh division is too coarse in the part
Fig. 7. Comparison of computational results between various methods for modeling the laminated core. (a) B z on line A; (b) B z on line B.
31
Table 3. CPU time and number of NR iterations to convergence
of the proposed benchmark model is carried out by using the hybrid FE-BE method with the FMM, and the reference solutions are calculated. Next, the computational costs and accuracies of two homogenization methods are discussed by comparing them with the reference solutions, and it is verified that the homogenization methods can analyze magnetic fields in laminated iron cores within acceptable computational costs. The results of this research should promote progress in the practical design of electrical machines based on accurate electromagnetic field computations, taking account of the laminated structure in detail, which will lead to the development of high-performance electric machines with high reliability. A summary of the conclusions is as follows:
Computer used: Pentium D/3.0 GHz
where the change of magnetic flux is sharp. Although the homogenization method does not require detailed modeling, such as a mesh restricted by the laminated structure, a fine mesh sufficient to approximate the change of the magnetic field accurately is essential. Figure 7 shows a comparison of B z along lines A and B between the reference solutions and the numerical results obtained with the nonuniform mesh. The numerical results obtained by homogenization methods 1 and 2 are both in good agreement with the reference solutions. Because the space factor is very close to 1 in this benchmark model, homogenization method 2 can achieve the same degree of computational accuracy as homogenization method 1. The homogenized magnetic flux obtained by finite element analysis is spatially smooth and the magnetic flux in the steel is calculated by Eq. (9). On line B, the numerical results obtained by the two homogenization methods are in good agreement, but they differ slightly from the reference solutions near the edge of the core. Periodicity of the microscopic structure does not persist near the edge of the core, and the magnetic field varies drastically in the surface steel sheet. Therefore, the difference indicates the precision limit of the homogenization method for modeling the laminated iron core. Table 3 shows the CPU time and the number of nonlinear iterations of the two homogenization methods when the number of DC ampere-turns is varied from 1000 to 50,000 AT. Within the limits of the investigation reported in this paper, the convergence characteristics of homogenization methods 1 and 2 are almost the same, which means that the homogenized reluctivities parallel to the lamination are accurately approximated by Eq. (14). In these analyses, the number of NR iterations with respect to bs in homogenization method 1 is a maximum of 3. Therefore, the CPU time for homogenization method 1 is very close to that for homogenization method 2.
(1) The numerical results obtained by homogenization method 1 are in good agreement with the reference solutions obtained by the hybrid FE-BE method. The computational costs for the homogenization method are almost the same as those for the analysis of a solid core, because the mesh is not restricted by the laminated structure. (2) Homogenization method 2, which is a simplification of homogenization method 1 using an approximation of the magnetic reluctivities parallel to the lamination, has the same level of accuracy as homogenization method 1. There is little difference between homogenization methods 1 and 2 from the viewpoint of CPU time and number of nonlinear iterations. (3) At the edge of the core, the numerical results obtained by the homogenization methods differ slightly from the reference solutions. The periodicity of the microscopic structure does not hold true near the edge of the core, and the magnetic field varies drastically in the surface steel sheet. Therefore, the computational accuracy of the homogenization method can deteriorate near the edge of the core. As a method of improving accuracy, we may consider dividing the surface steel sheets directly into multiple layers of elements. Furthermore, from the viewpoint of programming, it is easy to modify homogenization method 1 into homogenization method 2. Considering the case in which the space factor is fairly small or the steel is extremely saturated, homogenization method 1 enables us to obtain more accurate numerical results. In the future, we will perform the measurements on this benchmark model and compare the numerical results obtained by various modeling methods with experimental data. In this paper we dealt with nonlinear magnetostatic field problems, as a first step toward highly accurate analysis of a laminated iron core. We will investigate modeling methods for nonlinear magnetic field analysis of laminated iron cores including eddy currents.
5. Conclusions We have proposed a benchmark model for the development of an approximate method for modeling laminated iron cores. First, a large-scale and highly accurate analysis
32
Acknowledgment 12.
The authors thank the members of the IEEJ Research Committee on Advanced Computational Techniques for Practical Electromagnetic Field Analysis for useful discussions of the benchmark model.
13. REFERENCES 1. Yamazaki K, Watanabe Y, Mogi H, Mishima Y, Kaido C, Kanao S, Takahashi K, Ide K, Hattori K, Nakahara A, Watanabe T. Basic study of methods of eddy current analysis for stator core ends of turbine generators. Papers of the Technical Meeting on Rotating Machinery, IEE Japan, RM-05-154, 2005. (in Japanese) 2. Kaimori H, Kameari A, Fujiwara K. FEM computation of magnetic field and iron loss in laminated iron core using homogenization method. IEEE Trans Magn 2007;43:1405–1408. 3. Kameari A, Fujiwara K. Formulation for nonlinear transient magnetic field analysis with eddy currents in laminated iron core using homogenization method. Papers of the Joint Technical Meeting on Static Apparatus and Rotating Machinery, IEE Japan, SA-0624, RM-06-24, 2006. (in Japanese) 4. Muramatsu K, Okitsu T, Fujitsu H, Shimanoe F. Method of nonlinear magnetic field analysis taking into account eddy current in laminated core. IEEE Trans Magn 2004;40:896–899. 5. Gyselinck J, Sabariego RV, Dular P. A nonlinear time-domain homogenization technique for laminated iron cores in three-dimensional finite-element models. IEEE Trans Magn 2006;42:763–766. 6. Wakao S, Onuki T. Novel boundary element formulation in hybrid FE-BE method for electromagnetic field computation. IEEE Trans Magn 1992;28:1162– 1165. 7. Wakao S, Onuki T. Electromagnetic field computations by the hybrid FE-BE method using edge elements. IEEE Trans Magn 1993;29:1487–1490. 8. Takahashi Y, Wakao S. Large-scale analysis of eddy current problems by the hybrid finite element-boundary element method combined with the fast multipole method. IEEE Trans Magn 2006;42:671–674. 9. Takahashi Y, Wakao S, Kameari A. Large-scale and highly accurate magnetic field analysis of magnetic shield. J Appl Phys 99, 08H904, 2006. 10. Cheng H, Greengard L, Rokhlin V. A fast adaptive multipole algorithm in three dimensions. J Comput Phys 1999;155:468–498. 11. Hamada S. Basic formulas of fast multipole method for the Laplace equation in three dimensions. Papers of Joint Technical Meeting on Static Apparatus and
14. 15.
16.
17.
18.
19.
20. 21.
22.
23.
24.
25.
33
Rotating Machinery, IEE Japan, SA-07-54, RM-0770, 2007. (in Japanese) Hamada S, Kobayashi T. Analysis of electric field induced by ELF magnetic field utilizing fast-multipole surface-charge-simulation method for voxel data. Trans IEE Japan 2006;126-A:355–362. (in Japanese) Nishida T, Hayami K. The economic solution of 3D BEM using the fast multipole method. Trans JSCES 1996;1:315–318. (in Japanese) Technical Report 286, IEE Japan, 2000. (in Japanese) Fujiwara K, Nakano M, Ishihara Y. Standard test methods for measurement of magnetic properties of power magnetic materials. Trans Magns 2004;28:661–669. (in Japanese) Buchau A, Huber CJ, Rieger W, Rucker WM. Fast BEM computation with the adaptive multilevel fast multipole method. IEEE Trans Magn 2000;36:680–684. Washizu M, Jones TB. Dielectrophoretic interaction of two spherical particles calculated by equivalent multipole-moment method. IEEE Trans Magn 1996;32:233–242. Hamada S, Kobayashi T. Analysis of electric field induced by ELF magnetic field utilizing generalized equivalent multipole-moment method. Trans IEE Japan 2006;125-A:533–543. (in Japanese) Takahashi Y, Matsumoto M, Wakao S, Fujino S. Large-scale magnetic field analysis of laminated core by using the hybrid finite element-boundary element method combined with the fast multipole method. Papers of the Joint Technical Meeting on Static Apparatus and Rotating Machinery, IEE Japan, SA-0678, RM-06-80, 2006. (in Japanese) Stratton JD. Electromagnetic theory. McGraw–Hill; 1941. p 250–254. Saad Y, Schultz MH. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Comput 1986;7:856–869. Gutknecht MH. Variants of Bi-CGSTAB for matrices with complex spectrum. SIAM J Sci Comput 1993;14:1020–1033. Hamada S, Takuma T. Effective precondition technique to solve a full linear system for the fast multipole method. IEEE Trans Magn 2003;39: 1666–1669. Tokumasu T, Fujita M. Problems remained in 2 dimensional electromagnetic analyses (2). Papers of the Joint Technical Meeting on Static Apparatus and Rotating Machinery, IEE Japan, SA-06-02, RM-0602, 2005. (in Japanese) Muto H, Takahashi Y, Wakao S, Fujiwara K, Kameari A. Magnetic field analysis of laminated core by using homogenization method. J Appl Phys 99, 08H907, 2006.
Partial derivatives of Y m n with respect to y and z can easily be obtained in the same way. In this approach, the recurrence equations are simple and we can obtain the partial derivatives of Y m n directly. Furthermore, in the case of BEM considering the vector quantities as unknowns, where the x-, y-, and z-components of multipole and local expansions are necessary, this method can derive the three components simultaneously. In another method which has the same advantages, by modifying the recurrence equations presented in Ref. 17 so as to be consistent with the definition of Y m n in Ref. 10, the recurrence equations for r nY m are obtained as follows: n
APPENDIX Multipole Expansion of Double Layer Potentials In calculating the multipole expansion of the double layer potential by Eq. (6), the gradients of the spherical harmonics are required. Although formulas for the gradients of spherical harmonics in Cartesian coordinates were published in Ref. 16, they have a rather complicated form due to the associated Legendre functions and their derivatives. In addition, we must take into consideration the singularity of the associated Legendre functions on the z-axis. Here, we outline the three methods for calculating the gradients of the spherical harmonics without complexity caused by singularities. In Ref. 12, recurrence equations for the spherical harmonics are described as follows:
(A.3) (A.1)
where i is the imaginary unit. With these recurrence equations, partial derivatives of Y m n with respect to x can be analytically obtained as follows [19]:
Because this method is straightforward and the multipole expansion can be calculated directly from Eq. (6), it is fairly effective from the viewpoint of computational cost. A method based on conversion of the origin of the coordinate system in which the multipole and local expansions are defined in a cell has also been proposed [18]. First, by treating r′ as the origin of the coordinate system, the first-order coefficient of the multipole expansion is easily obtained from Eq. (A.4). Next, the origin of the multipole expansions, which is located at r′, is converted to the center of the cell by M2M translation. This method is also fairly effective from the standpoint of computational cost:
(A.2)
(A.4)
34
AUTHORS (from left to right)
Yasuhito Takahashi (member) received his B.E., M.E., and Ph.D. degrees from Waseda University in 2003, 2005, and 2008. From 2006 to 2008, he was a research associate on the Faculty of Science and Engineering, Waseda University. Since 2008, he has been a GCOE assistant professor in the Department of System Science, Graduate School of Informatics, Kyoto University. His research interests are large-scale electromagnetic field computation and its applications to electric machines. Shinji Wakao (member) received his B.E., M.E., and Ph.D. degrees from Waseda University in 1989, 1991, and 1993. In 1996, he joined the Department of Electrical, Electronics and Computer Engineering, Waseda University, and became an associate professor in 1998. Since 2006, he has been a professor in the Department of Electrical Engineering and Bioscience. His research interests are electromagnetic field computation, photovoltaic power generation system, and design optimization of electric machines. Koji Fujiwara (member) received his B.S. and M.S. degrees in electrical engineering from Okayama University in 1982 and 1984 and D.Eng. degree from Waseda University in 1993. From 1985 to 1986, he was affiliated with Mitsui Engineering and Shipbuilding Co., Ltd. From 1994 to 2006, he was an associate professor in the Department of Electrical and Electronic Engineering, Okayama University. Since 2006, he has been a professor in the Department of Electrical Engineering, Doshisha University. His major fields of interest are the development of the 3D finite element method for non linear magnetic field analysis including eddy currents, and its application to electrical machines, and the development of standard methods of measurement of the magnetic properties of magnetic materials. Hiroyuki Kaimori (member) received his B.S. and M.S. degrees in mechanical engineering from Toyo University in 2000 and 2002 and joined Science Solutions International Laboratory, Inc. His major fields of interest are numerical methods for electromagnetic analysis and their applications to electrical machines. Akihisa Kameari (member) received his B.S. degree in physics from Kyoto University in 1973. From 1973 to 1996, he was affiliated with Mitsubishi Atomic Power Industries, Inc. and Mitsubishi Heavy Industries, Ltd. Since 1996, he has been affiliated with Science Solutions International Laboratory, Inc. His major field of interest is the development of numerical methods for electromagnetic analysis. He is a member of IEEE.
35
