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Journal of Magnetism and Magnetic Materials 322 (2010) 1651–1654

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Topology optimization of the permanent magnet type MRI considering themagnetic field homogeneity

Junghoon Lee a, Jeonghoon Yoo b,�

a Graduate School of Mechanical Engineering Yonsei University, 262 Seongsanno, Seodaemun-Gu, Seoul 120-749, Republic of Koreab School of Mechanical Engineering, Yonsei University, 262 Seongsanno, Seodaemun-Gu, Seoul 120-749, Republic of Korea

a r t i c l e i n f o

Available online 7 May 2009

Keywords:

Magnetic resonance imaging

Permanent magnet

Magnetic field

Eddy current

Topology optimization

Density method

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.04.078

esponding author. Tel.: +82 2 2123 2859; fax:

ail address: yoojh@yonsei.ac.kr (J. Yoo).

a b s t r a c t

This study is to suggest a concept design of the permanent magnet (PM) type magnetic resonance

imaging (MRI) device based on the topology optimization method. The pulse currents in the gradient

coils in the MRI device will introduce the effect of eddy currents in ferromagnetic material and it may

worsen the quality of imaging. To equalize the magnetic flux in the PM type MRI device for good

imaging, the eddy current effect in the ferromagnetic material must be reduced. This study attempts to

use the topology optimization scheme for equalizing the magnetic flux in the measuring domain of the

PM type MRI device using that the magnetic flux can be calculated directly by a commercial finite

element analysis package. The density method is adopted for topology optimization and the sensitivity

of the objective function is computed according to the density change of each finite element in the

design domain. As a result, optimal shapes of the pole of the PM type MRI device can be obtained. The

commercial package, ANSYS, is used for analyzing the magnetic field problem and obtaining the

resultant magnetic flux.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The magnetic resonance imaging (MRI) device is one of theimportant diagnostic and analytical tools for biomedical studies.The MRI device provides sub-millimeter resolution of soft tissuesin the body. The majority of the MRI device use superconductivemagnet and majority of the superconductive magnets are basedon niobium–titanium (NbTi) alloys, which are very reliableand can guarantee extremely uniform fields and extreme stabilityover the working period; however, a special system to keep theconductors at approximately 4.2 K (equal to �268.8 1C) isrequired. To maintain the temperature, the magnet is enclosedand cooled by a cryogen containing liquid helium (sometimes alsonitrogen). The permanent magnet (PM) type MRI device is a viablealternative to the resistive and superconducting MRI devicebecause it has advantages of compact assembly, small leakageflux and low operating cost [1]. Therefore, PM type MRI deviceshave become more popular as a diagnostic tool in recent years.

Usually, the main magnetic field built up by the PM must behomogeneous and the gradient magnetic field built up by gradientcoils should have well gradient strength and linearity [2].Combining the main magnetic field with the well-tuned gradientcoil can provide the good image condition. However, pulse

ll rights reserved.

+82 2 362 2736.

currents in gradient coils will introduce the eddy current effectin ferromagnetic materials such as the pole and the yoke in theMRI device. The eddy current may worsen the linearity of thegradient magnetic field and have a bad influence on the imagequality. Moreover, it can cause muscle contraction and nervestimulation. Therefore, the eddy current effect must be consideredin the MRI device design.

Topology optimization method in electromagnetic fieldsoriginates from topology optimization applications in structuraldesign based on the so-called density method [3] or thehomogenization design method [4]. Topology optimization in-tends to obtain the optimal material distribution according to thedensity variation of each element composing the design domain.Dyck and Lowther’s work [5] can be regarded as a pioneering workof the topology optimization application to MRI design and it isextended using the genetic-algorithm-based topology optimiza-tion method [6].

This study focuses on the conceptual design of the PM typeMRI device based on the topology optimization scheme. The mainpurpose of the design is finding the optimal shape of the pole ofthe PM type MRI device considering the eddy current effect. It isobjected to equalize the magnetic flux in the measuring domainfor good imaging by applying the topology optimization schemeto the pole shape design. It seems to be similar to previousstudies; however, this study is focused on equalizing the magnetflux considering the eddy current effect using the topologyoptimization scheme instead of parameter or shape optimization

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375

15

18

27

Yoke

PolePM

y

J. Lee, J. Yoo / Journal of Magnetism and Magnetic Materials 322 (2010) 1651–16541652

schemes as in previous works. The magnetic field analysis isperformed using the commercial finite element method (FEM)package ANSYS ver. 10.0 and it is used for obtaining the resultantmagnetic flux as well as for analyzing the magnetic field. As aresult, the optimal design of the PM type MRI device is suggested.

x

x

122148

27Designdomain

Measuringdomain

x

2 3

5 6

y

y

1

4

7 8 9

2. Problem definition

The initial geometry of the PM type MRI device is shown inFig. 1. The pole is made of ferrous material steel (SS400) withrelative permeability value of 2000 and the conductivity value of7.51�106 S/m. The yoke is also made of the same material. Thegradient coil having 15 turns and 5 A is located on the surface ofpole piece. The magnetization of the Nd-Fe-B magnet is 1.21 T.

The model of the PM type MRI device is axisymmetric. Therefore,a quarter model is used for the optimization process. The schematicshape and design domain is shown in Fig. 2(a) and the right cornerof the pole is defined as the design domain. The measuring domainis located at the center of the full model of the PM type MRI devicewith the area of 100�100 mm2. Since the quarter model is used, themeasuring domain is designated as the square of 50�50 mm2 areaat the bottom left corner in Fig. 2(a). The nine points of themeasuring domain are selected to measure the magnet flux density.Fig. 2(b) and (c) show the flux flow of the model of the PM type MRIdevice without and with considering eddy current, respectively. It isconfirmed that the eddy current causes the skin effect on the poleand the yoke. The optimization process is focused to obtain theoptimal shape of the design domain for equalizing the magnet field,especially to the y-directional magnetic flux.

Fig. 2. Axisymmetric model of the PM type MRI device. (a) Geometry of the

quarter model and design domain (unit: mm), (b) magnetic flux plot of the model

without considering eddy current, and (c) magnetic flux plot of the model

considering eddy current.

3. Computation of the magnetic field

The governing equation of the magnetic field considering eddycurrents is as follows:

r �1

mðr � AÞ þ s @A

@t¼ Js (1)

where Js is the source current density in gradient coils. In theprocess of electromagnetic field analysis considering eddycurrents, the time stepping method is necessary. Therefore, thefinite difference time domain (FDTD) method is a prospectivesolution method for a wide frequency range analysis while theFEM is generally used in the analysis of electromagnetic fieldproblems with a specific frequency range. In this study, since theperformance at the specific frequency of o ¼ 60 Hz is focused, themagnetic vector potential can be represented as A ¼ A0ejot. Then,the linear FEM analysis of electromagnetic field can be usedconsidering the eddy current effect.

Gradient coil

Magnet Hole Yokey

xPole

14 27

75

140

R30R136.5R165 374.8

60

Fig. 1. Schematic shape of the PM type MRI device (unit: mm).

The design domain is discretized by finite elements for theanalysis and the design variable is set as the density of eachelement in topology optimization. The magnetic property of eachelement is changed according to the change of the elementdensity. The so-called density method [3] is applied based on thereluctivity-based or the permeability-based optimization schemefor the comparison [7]. The relative magnetic permeability andreluctivity of the ferromagnetic material can be represented withthe element density as follows:

mer ðrÞ ¼ rpðmr � 1Þ þ 1; ð0prp1Þ (2)

ner ðrÞ ¼ rpðnr � 1Þ þ 1; ð0prp1Þ (3)

where the relative permeability mr is set to 2000 while nr is set to1/2000 and r and p represent the element density and thepenalization parameter, respectively. Formulations expressedabove are objected to avoid the gray scale element by using thepenalization parameter. Large permeability value means the yieldof the magnetic flux flow while the large reluctivity works as aresistance of the flow. The permeability is located at thedenominator of the material property in the governing equationshown in Eq. (1) and it may cause bad convergence during theoptimization process. Therefore, the reluctivity-based optimiza-tion can give better results than the permeability-based one [7]. Ifthe element density is 1, the element is filled with material whilethe element is just the same as air with the element density of 0.The material property (i.e., the permeability or the reluctivity,based on the method used) is changed using Eq. (2) or (3) and it isused to calculate the value of the design objective.

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V f F N

x

y

Design domain

Iteration number0 10 20 30 40

0.5

0.55

0.6

0.65

1.5

2.0

2.5

3.0

3.5

Volume fraction

Objective function0.45

J. Lee, J. Yoo / Journal of Magnetism and Magnetic Materials 322 (2010) 1651–1654 1653

4. Optimization problem and sensitivity analysis

The design objective is equalizing the y-directional magnetfield and design constraints are the upper limit of the total volumefraction in the design domain and the upper and the lower boundof the design variable, i.e., the element density.

maximize FN ¼ ðjsyjÞ0=ðjsyjÞ þX9

i¼1

jBxij

!0

=X9

i¼1

jBxij

!

subject toXNE

e¼1

reVe ¼ Vf ; 0orminprep1: (4)

where Vf is the specified volume fraction predefined in theoptimization process. sy represents standard deviation of y-directional magnetic flux density at all measuring points Bxi andrepresents x-directional magnetic flux density at the measuringpoint i in Fig. 2(a).

The magnetic flux density is calculated using the commercialFEM package. Table 1 shows the analysis result of the initialquarter model of the PM type MRI device. Since the orders of thestandard deviation of y-directional magnetic flux density and thatof the summation of the absolute x-directional magnetic fluxdensity value are different, the normalized objective function isorganized as in Eq. (5).

The sensitivity, which is the ratio of the objective functionvariation according to the design variable variation, can becomputed directly using the second-order central-differenceapproximation [8]:

dFN

dre

¼FNðre þDrÞ � FNðre �DrÞ

2Dr(5)

where the element density re is design variable and Dr representsthe step size of the design variable. The optimality criteria method(OCM) is adopted to update design variables using sensitivityanalysis results.

Fig. 3. Optimization results of the PM type MRI device. (a) Permeability-based

optimization (p ¼ 3) and (b) reluctivity-based optimization (p ¼ 1).

5. Numerical results

For the finite element modeling, the commercial packageHyperMesh is used and QUAD4 elements are used for themodeling. The optimization process including the electromagneticfield analysis by ANSYS and the sensitivity analysis is controlledand performed using the in-house code programmed by C-language. Two different optimizations, one is based on themagnetic permeability and the other is based on the magneticreluctivity, have been performed for the design objective in Eq. (4)and results are compared.

5.1. Permeability-based optimization

First optimization has been performed based on the magneticpermeability. Fig. 3(a) shows the optimal shape of the designdomain with Vf ¼ 0.6 and the penalization parameter of 3 and itsconvergence history. The black and the white parts represent thesolid and the void, respectively. The optimal shape is obtained atthe 40th iteration and the objective function value is 2.5146, 25.7%

Table 1The analysis result of the initial model.

|sy|P

|Bxi|

Initial model 3.755e�05 1.393e�04

increase compared with the initial model which has the objectivefunction value of 2.

5.2. Reluctivity-based optimization

The second optimization has been performed based on themagnetic reluctivity without any change of the design domain, theobjective function and the volume fraction. Fig. 3(b) shows theoptimal shape of the design domain, with the penalizationparameter 1 and its convergence history. The optimal shape isalso obtained at the 40th iteration and the objective function is2.9315, 46.6% increase compared with the original model. Theobjective function value is increased more than the permeability-based case; however, many gray elements, which have theintermediate material property, are shown in the optimal shape.

5.3. On–off filtering

Since the obtained optimal shapes have some gray elements,the on–off filtering method defined as following is applied toobtain the clear boundary:

deX0:6 de ¼ 1

deo0:6 de ¼ 0

((6)

ARTICLE IN PRESS

Design domain

Design domain

Fig. 4. Optimal design with the on–off filtering method and magnetic flux line

plot. (a) Permeability-based optimization and (b) reluctivity-based optimization.

Table 2Comparison of optimization results with the on–off filtering.

Permeability-based result Reluctivity-based result

|sy| 3.021e�05 2.024e�05P|Bxi| 1.095e�04 0.895e�04

Objective function (FN) 2.5148 3.4109

Volume fraction (Vf) 0.5926 0.6296

J. Lee, J. Yoo / Journal of Magnetism and Magnetic Materials 322 (2010) 1651–16541654

where de is the element density. The on–off criterion is defined as0.6, because the volume fraction is set to 0.6.

Fig. 4(a) shows the result of the permeability-based design andits flux density line plot of the PM type MRI device. Fig. 4(b) shows

the final result of the reluctivity-based design and its contour ofthe flux density. Table 2 compares the standard deviation of they-directional magnetic flux density, the summation of the absolutex-directional magnetic flux density, the objective function valueand the final volume fraction. After on–off filtering, the result of thereluctivity-based optimization has the objective function value of3.4109, 70.5% increase compared with the initial model, and is alsosuperior to the permeability-based result.

6. Conclusions

The purpose of this study is to find the optimal design of thePM type MRI device. The topology optimization scheme toequalize the magnet flux considering the eddy current effect inthe PM type MRI device is suggested. The optimal designsuggested by the topology optimization method shows consider-able improvement in the magnetic field homogeneity comparedwith the initial design.

Acknowledgment

This work was supported by the Korea Research FoundationGrant funded by the Korean Government (MOEHRD, BasicResearch Promotion Fund) (KRF-2007-313-D00027).

References

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