Proposal for a Benchmark Model of a Laminated Japan2010

8/8/2019 Proposal for a Benchmark Model of a Laminated Japan2010

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of a reference solution is preferable. It is considered signifi-

cant to verify the computational accuracy of various ap-

proximate 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 otherhand, the hybrid finite element and boundary element (FE-

BE) method [68] 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) [1013], we

can analyze large-scale problems such as the detailed mod-

eling 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 analy-

sis 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 de-tailed 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 homogeniza-

tion 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 homog-

enization method with the reference solutions and investi-

gate the accuracy.

2. 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-

ing of Japan (IEEJ). The configuration is the same as the

benchmark model for magnetostatic field analysis men-

tioned in Ref. 14. The coil is excited by a 3000 AT DC

current. The core is constructed by laminating 200 nonori-

ented 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 mag-

netic 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 ironcore, we investigated nonlinear magnetostatic field prob-

lems to clarify the effectiveness of the various modeling

methods in this paper.

3. Calculation of Reference Solution by Hybrid

Finite ElementBoundary Element Method

3.1 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].

Table 1. BH curve of 50A1300

Fig. 1. Benchmark model of laminated iron core.

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In this paper, two regions are considered. Region m,with boundary m, consists of both a nonconducting mag-netic substance, whose permeabilitym may have nonlinearcharacteristics, and insulation gaps, whose magnetic prop-

erties are assumed to be the same as a vacuum. In this region

with no supply current, the magnetic scalar potentialcanbe defined and the FEM is applied as follows:

where N is the scalar shape function for a nodal finite

element and n is the unit normal vector on .The region0 is free space, which extends to infinity,

with a supply current. The BEM is applied to this region

and is considered as the physical quantity. The integralequation is

where is the boundary between the FEM and BEM

regions (= m), CP is the solid angle enclosed by region 0,

andJis the magnetic scalar potential produced by a supply

current [6].

The interface conditions between the free space and

the region m 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 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:

where Ynm denotes the spherical harmonics defined in Ref.

10 and (r, , ) are spherical coordinates. The correspond-ing multipole expansions are as follows:

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 [1619], a method which is not re-

quired to take into account the singularity ofYnm on the

z-axis is fairly useful (see the Appendix for more informa-

tion). 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 simul-

taneously in the FMM process. Therefore, the translations

of multipole and local expansions such as the multipole-to-

multipole (M2M), multipole-to-local (M2L), and local-to-local (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 un-

knowns [20].

3.3 Calculation of reference solution by

hybrid FE-BE method with FMM

In order to obtain a reference solution of the bench-

mark model, a large-scale nonlinear magnetostatic field

analysis is carried out by the hybrid FE-BE method com-bined 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 New-

tonRaphson (NR) method. When the change of the flux

density is less than 103

T for each element, the NR iterationis 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

(1)

(2)

(5)

(6)

(3)

(4)

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convergence criterion for the GMRES method is set to 108.

In the minor iteration of the MIP, we adopt the Bi-

CGSTAB2 method [22], and its convergence criterion is setto 103. Table 2 shows the specifications of the analysis. The

CPU time is about 15 hours, because we utilize an ex-

tremely 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 compu-

tational 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 Bz along line A (0 < x

< 50,y = 49 mm, z = 50 mm) and line B (0


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The homogenized magnetic reluctivities parallel and

perpendicular to the lamination N|| and N are obtained as

follows:

where is the space factor, ns(bs) and n0 are the magnetic

reluctivities of the steel sheet and the vacuum, and bs is themagnetic flux density in the steel sheet. These equations are

easily derived from the continuity of the tangential compo-

nent of the magnetic field and the normal component of the

flux density.

On the other hand, bs is given by

where B(BX, BY, BZ) is the homogenized magnetic flux

density, obtained by the FEM, and the Zdirection is per-

pendicular 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:

whereA is the magnetic vector potential,His the magnetic

field, J is the current density, and Ni is the vector shape

function. Because Eq. (10) is a nonlinear equation, it is

solved by using the NR method as follows:

The Jacobi matrix can be derived from Eqs. (7) to (9) and

is given as follows:

In applying this homogenization method to the FEM, we

replace the original constitutive relations with the homoge-

nized one represented by Eq. (12).

4.2 Homogenization method 2 [24, 25]

In the case of an ordinary laminated iron core, n0 >>

ns and 1 are generally true. By using this assumption,Eqs. (7) and (9) can be simplified as follows:

In this case, we can calculate bs directly from the homoge-

nized magnetic flux. Furthermore, because is approxi-mated by 1, Eq. (12) is simplified 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 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 ac-

curacy. On the other hand, when is 0.1, the difference

becomes greater as the steel becomes magnetically satu-

(9)

(10)

(11)

(12)

(13)

(7)

(8)

(14)

(15)

(16)

Fig. 4. Homogenization of laminated iron core.(a) Laminated core; (b) Macroscopic model.

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rated. When the steel is extremely saturated, ns(bs) is aboutequal to n0. In this situation, the difference between the

homogenized magnetic reluctivities obtained by Eqs. (7)

and (14) is about 4.2% when is 0.96, and about 10% when is 0.1. Therefore, it is necessary to adopt homogenizationmethod 1 when the space factor is fairly small or the steel

is extremely saturated.

4.3 Numerical examples

We perform the finite element analysis of the lami-

nated iron core model shown in Fig. 1 by the homogeniza-

tion method. A first-order hexahedral element is used.

When the change of the flux density is less than 103 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

103. 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 compu-

tational 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 homogenizationmethod 1.

Figure 6 shows the magnetic flux density distribu-

tion. It agrees qualitatively with Fig. 3(b), which was ob-

tained 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. 5. Magnetic characteristics of laminated core.

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.]

Fig. 7. Comparison of computational results between

various methods for modeling the laminated core.

(a) Bz on line A; (b) Bz on line B.

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where the change of magnetic flux is sharp. Although the

homogenization method does not require detailed model-

ing, such as a mesh restricted by the laminated structure, a

fine mesh sufficient to approximate the change of the mag-

netic field accurately is essential.

Figure 7 shows a comparison ofBz 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 thespace 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 lami-nated 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 homogeni-

zation 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 homog-

enization method 1 is a maximum of 3. Therefore, the CPU

time for homogenization method 1 is very close to that forhomogenization method 2.

5. Conclusions

We have proposed a benchmark model for the devel-

opment of an approximate method for modeling laminated

iron cores. First, a large-scale and highly accurate analysis

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 mag-

netic fields in laminated iron cores within acceptable com-

putational costs. The results of this research should promote

progress in the practical design of electrical machines based

on accurate electromagnetic field computations, taking ac-

count 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:

(1) The numerical results obtained by homogeniza-

tion method 1 are in good agreement with the reference

solutions obtained by the hybrid FE-BE method. The com-

putational 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 simplifica-tion 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 micro-

scopic 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 homog-enization 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 homog-

enization 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 accu-

rate 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 analy-

sis of a laminated iron core. We will investigate modeling

methods for nonlinear magnetic field analysis of laminated

iron cores including eddy currents.

Table 3. CPU time and number of NR iterations to

convergence

Computer used: Pentium D/3.0 GHz

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Acknowledgment

The authors thank the members of the IEEJ Research

Committee on Advanced Computational Techniques for

Practical Electromagnetic Field Analysis for useful discus-

sions of the benchmark model.

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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 gradi-

ents 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 deriva-tives. 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 complex-

ity caused by singularities.

In Ref. 12, recurrence equations for the spherical

harmonics are described as follows:

where i is the imaginary unit. With these recurrence equa-

tions, partial derivatives of Ynm with respect to x can be

analytically obtained as follows [19]:

Partial derivatives ofYnm with respect to y and z can easily

be obtained in the same way. In this approach, the recur-

rence equations are simple and we can obtain the partial

derivatives ofYnm directly. Furthermore, in the case of BEM

considering the vector quantities as unknowns, where the

x-,y-, andz-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. 17so as to be consistent with the definition ofYn

m in Ref. 10,

the recurrence equations for rnYnm are obtained as follows:

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 expan-

sions are defined in a cell has also been proposed [18]. First,

by treating r as the origin of the coordinate system, thefirst-order coefficient of the multipole expansion is easily

obtained from Eq. (A.4). Next, the origin of the multipole

expansions, which is located atr, is converted to the centerof the cell by M2M translation. This method is also fairly

effective from the standpoint of computational cost:

(A.1)

(A.3)

(A.2)

(A.4)

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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 optimizationof 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 nonlinear 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 2000and 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.

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Proposal for a Benchmark Model of a Laminated Japan2010