Investigation of Topology Optimization of Magnetic Circuit Using Density Method

YOSHIFUMI OKAMOTO and NORIO TAKAHASHIOkayama University, Japan

SUMMARY

Topology optimization using the density method,which determines the optimal topology by distributing themagnetic material in the design domain, is attractive fordesigners of magnetic devices, because an initial concep-tual design can be obtained. As the number of designvariables in the design domain is so huge, a sufficientsolution cannot be obtained by using the direct searchmethod. Then, the steepest descent method is adopted toobtain fast convergence. In this paper, some problems inapplying the density method to the practical topology opti-mization are investigated. The accuracy of calculating thesensitivity using the adjoint variable method and the effectof initial material density on the obtained results are exam-ined. The effectiveness of the proposed optimizationmethod is illustrated by applying it to the determination ofthe optimal topology of a permanent magnet which gener-ates a uniform magnetic field. © 2006 Wiley Periodicals,Inc. Electr Eng Jpn, 155(2): 53–63, 2006; Published onlinein Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/eej.20259

Key words: topology optimization; densitymethod; sensitivity analysis; steepest descent method; fi-nite element method.

1. Introduction

Size and shape optimization [1–4] of magnetic cir-cuits in electric and electronic devices has come into wide-spread use recently. The shapes of iron cores, magnets, andother internal components are recalculated so as to improvethe device characteristics under specified constraints. Whensuch calculations are performed, a device must be predes-igned to some extent prior to optimization. On the otherhand, the rough shape of the magnetic circuit can be derivedby topology optimization algorithms, such as the densitymethod [5–7], which treat the material constants of thecircuit components as parameters. When developing a

novel device, this not only offers a considerable saving ofthe designer’s labor in the prototype stage, but makespossible design of magnetic circuits based on entirely newconcepts.

In topology optimization using genetic algorithms[8], the design variables are the control points of splinefunctions representing the numbers of holes and the bound-ary shapes. In this case, however, holes must not intersectother holes or boundaries, which implies complicated con-straints in modeling. In addition, the finite element method(FEM) is combined with genetic algorithms in Refs. 9 and10. A motor’s rotor topology is determined with the differ-ent materials assigned to every cell as design parameters.This method appears versatile because multiple materialsare handled easily, and differential coefficients of the ob-jective function are not required. However, the materialsthus determined are not continuously distributed, and fur-ther research is therefore necessary for application to realmachines.

In contrast to the discrete allocation of materials todesign elements, the density method produces the roughshape of the magnetic circuit, with the material density ofthe elements treated as a continuous value. In the densitymethod, the number of design variables is the same as thenumber of elements in the design domain. Therefore, thenumber of elements in the design domain must be largeenough to obtain a specific topology. In direct search algo-rithms based on random numbers such as the evolutionarystrategy, discontinuous topologies of magnetic circuits areobtained, and hence it is difficult to determine a specificlayout [11].

In this study, we examine topology optimization bythe density method. In particular, we deal with variousdesign variables by the steepest descent method [12, 13].Provided that the objective function is selected properly,this method can find practicable minimum values, eventhough they may not be global optimal solutions. Thesteepest descent method requires the first derivatives of theobjective function with respect to the design variables.Therefore, we used the adjoint variable method [14], which

© 2006 Wiley Periodicals, Inc.

Electrical Engineering in Japan, Vol. 155, No. 2, 2006Translated from Denki Gakkai Ronbunshi, Vol. 124-D, No. 12, December 2004, pp. 1228–1236

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can calculate the sensitivity to all design variables in onedirect analysis. For the purpose of a fundamental investiga-tion of topology optimization based on the density method,we compared the accuracy of the sensitivity calculated bythe adjoint variable method and by the forward differencemethod, and considered the dependence of resulting topol-ogy on the initial settings. In addition, as an applicationexample, we used a permanent magnet model to verify theproposed method.

2. Density Method

2.1 Relations between magnetic materialdensity and permeability

The design variable in the density method is thematerial density ρ in every element of the design domain.The density ρi and the permeability µi of the i-th elementare related as follows:

Here µ0 is the permeability of a vacuum, and µr is thepermeability of the magnetic material, here assumed to be1000. The magnetic characteristic is assumed linear. Thelarger the exponent n, the more effectively grayscale valuesare eliminated. However, this does not necessarily assure abetter solution, because the design space is limited. Regard-ing this trade-off, a value of n from 2 to 4 is assumedappropriate [15]. In this study, n is set to 2, thus allowingfor grayscale and aiming at better solutions.

2.2 Relation between material density ofmagnet and the magnetization

When designing the magnet’s topology, the follow-ing relation is assumed for the density ρi and the magneti-zation Mdi of the i-th element:

Here Md is the magnetization intensity of the magnet used,and n is set to 2. Air is assumed in Eqs. (1) and (2) when ρis zero. On the other hand, when ρ is 1, the specifiedmagnetic material is allocated to such elements.

3. Topology Optimization Design Using SensitivityAnalysis

3.1 Topology optimization design

In topology optimization using the density method,the design variable is the number of elements in the designdomain. For this reason, a magnetic circuit obtained by

direct search of the magnetic parameters and material den-sities so as to maximize or minimize the objective functionwill contain discontinuities, which is impracticable. In ad-dition, since unconstrained problems are considered, thesteepest descent method is applied to the first derivative ofthe objective function. On the other hand, Newton’s methodusing the second derivative seems inappropriate due to thegreat computational complexity involved in this problem.In the steepest descent method, the density r is corrected asfollows:

Here ∆r(k) is the correction vector for the design variable atthe k-th iteration:

Here α is a correction coefficient, and W is the objectivefunction. Since the convergence value of the objectivefunction also depends on correction coefficient α, the coef-ficient is set automatically to a value that is assumed opti-mal. Subjecting the objective function W

(k+1) in the (k +1)-th iteration to Taylor expansion, and setting it to zero, weobtain

If the initial value and optimal value are extremely far fromeach other, terms of second and higher order prove effec-tive. In this case, however, the computational complexityincreases, and we therefore use a first-order approximation.Substituting Eq. (4) into Eq. (5), the correction coefficientis obtained as follows:

Therefore, Eq. (3) takes the following form:

If the correction |∆r(k)| of the density ρ is less than 1.0 ×10–2 for all elements, then a minimum value is recognized,and the calculation is terminated.

3.2 Sensitivity analysis

An additional calculation of the sensitivity vector s isnecessary in order to use Eq. (7) for steepest descent. Thematrix representation of the sensitivity vector s is

(2)

(1)

(4)

(5)

(6)

(7)

(3)

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Here N is the number of elements in the design domain, andT stands for the transpose. There are several methods ofcalculating Eq. (8), such as difference-based algorithms, theadjoint variable method, the mutual energy method of Byunand colleagues [7], and the Telegen’s theorem algorithm ofDyck and Lowther [16]. Here we consider the forwarddifference method and the adjoint variable method, whichoffer great versatility.

(a) Sensitivity calculation using forward difference

Applying the forward difference method, the differ-ential terms of s are obtained as follows:

Here a direct analysis must be performed N times. There-fore, the computing time increases rapidly with the numberof elements in the design domain in detailed layout design.

(b) Sensitivity calculation using adjoint variablemethod

In this method, the sensitivity is calculated by usingdiscrete algebraic equations of the FEM. Assume the fol-lowing FEM equation:

Here H is the overall coefficient matrix, A is an unknownvector potential, and G is a right-hand column vector.Differentiating with respect to the density r gives

Now, formulation for component ρm gives

The following total differential equation is obtained if theobjective function W is a function of the design variablerm and the vector potential A:

In addition, Eq. (12) can be transformed to

Here A~

is the potential vector A found beforehand by theFEM. Incidentally, the first term on the right-hand side in

Eq. (13) has a value if the objective function is explicit withrespect to ρm. In other words, if there exists an objectivefunction W composed of only rm, without involving A, itmay be treated here as an objective function for the mag-netic flux, and hence the first term in Eq. (13) may beignored. Another example, namely, the calculation of thesensitivity when the objective is the maximization of motoroutput and the minimization of torque ripple, is given in theAppendix. Substituting Eq. (14) into Eq. (13) and introduc-ing adjoint variable λ, we obtain

From the above equation, the adjoint variable is

The adjoint equation can ultimately be written as followsby applying transposition to both sides and considering thesymmetry of H:

Solving this equation for λ and substituting the result intoEq. (15), we can find the sensitivity dW/drm of the objectivefunction.

When finding the sensitivity, the forward differencemethod requires excessive amounts of calculation, but theadjoint variable method requires only one-time directanalysis for all design variables.

3.3 Topology optimization algorithm

The above optimization process can be presented interms of a flowchart as shown in Fig. 1.

Step 1: choice of initial topology

The initial value of material density ρ is specified,and the initial topology is determined by using Eqs. (1) and(2).

Step 2, Step 6: FEM → W

The objective function for the obtained topology iscalculated by the FEM.

Step 3: sensitivity analysis by the adjoint variablemethod

The sensitivity is calculated by using the adjointvariable. First, the adjoint variable λ is found by solving Eq.

(8)

(10)

(11)

(12)

(13)

(14)

(9)

(15)

(16)

(17)

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(17), and is then substituted into Eq. (15) to calculate thesensitivity.

Step 4: calculation of step size

The correction coefficient of steepest descent methodis calculated by substituting the sensitivity found in Step 3and the objective function into Eq. (6).

Step 5: updating of design variables

The material density is updated by substituting thecorrection coefficient and the sensitivity vector into Eq. (7).

Step 7: convergence?

The optimization calculation is terminated if the cor-rection |∆r(k)| is less than 1.0 × 10–2 for all design variables.

4. Application of Topology Optimization Algorithmto Various Models

Here we optimize the topology of a C-core model asan example of the optimization of magnetic materials. Inaddition, we deal with the permanent magnet model as anexample of the optimization of magnet topology. In bothcases, we perform a two-dimensional static magnetic fieldanalysis by FEM using rectangular first-order elements.

4.1 Topology optimization of C-core model

(a) Analytic model and objective function

The basic model of the C-core is shown in Fig. 2. Themagnetic flux generated by the exciting coil (current den-

sity J0 = 2.0 × 106 A/m2) produces a pull-in force Fx on thearmature. The corresponding analytic model is shown inFig. 3. Since the basic model is symmetric about the x axis,we consider only the upper half in the analytic model. Theshaded area in Fig. 3 is considered as the design domain,and the density r in this area is treated as the designvariable. The number of elements is 2400 in the designdomain, and 13,650 in the whole model.

Consider the topology of the C-core providing themaximum force pulling in the armature. Assuming that thiscorresponds to the largest magnetic flux in the armature,consider topology that will minimize the following objec-tive function W:

Here By(e) is the y-axis component of the magnetic flux

density of armature element e, as shown in Fig. 3. In thiscase, according to Eq. (17), the right-hand side of theadjoint equation is

Fig. 1. Flowchart of topology optimization.

Fig. 2. C-core model.

Fig. 3. Design domain of C-core model.

(18)

(19)

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Here Nie is an interpolation function for rectangular first-order elements.

(b) Accuracy of sensitivity calculation

We examined the accuracy of calculation of the sen-sitivity of the C-core model by the forward difference andadjoint variable methods. In particular, we compared thesensitivity for elements 100, 780, 1642, and 2236 in Fig. 4with the initial value of ρ set to 0.2.

Table 1 presents a comparison between the forwarddifference method with a perturbation value ∆ρ of 10–3 and10–5 (FDM_1 and FDM_2), and the adjoint variablemethod (AVM). As is evident from the table, the results ofthe adjoint variable method agree well with those of theforward difference method. In addition, the disagreementbetween the two methods (Error_1, Error_2) decreases asthe size of the perturbation ∆ρ is set smaller. Since theadjoint variable method calculates the sensitivity analyti-cally by using the FEM, precision sensitivity analysis ispossible. The computing time per sensitivity calculation iscompared in Table 2. It is evident that the adjoint variablemethod offers considerable time saving.

(c) Convergence to optimal solution

Topology optimization of the C-core was carried outassuming that the initial value of the density was ρ0 = 0.2.The convergence to the optimal solution is illustrated in Fig.5.

As is evident from the diagram, the topology is modi-fied gradually from the initial state. In the initial state, thedensity correction is significant around the angular pointswhere the magnetic flux density is high. In addition, whena closed magnetic path is formed inside the design domain,most of the magnetic flux does not reach the armature, andhence such magnetic paths are opened. In order to direct asmuch flux as possible to the armature, the cross section ofthe core end facing the armature is made larger than in thebasic model in Fig. 2. Thus, a more appropriate core isdesigned. The optimal solution is reached at the 17th itera-tion and constitutes a definite topology free of grayscaleareas (elements with a density between 0 and 1).

The magnetic flux distribution for the optimal solu-tion is shown in Fig. 6. Since the core surface facing thearmature became larger, the flux arrives at the armatureefficiently, without leakage.

Fig. 4. Elements for comparing sensitivity.

Table 1. Comparison of accuracy of forward differencemethod (FDM) and adjoint variable method (AVM)

Fig. 5. Convergence process of C-core (ρ0 = 0.2).

Table 2. Comparison of CPU time of forwarddifference method and adjoint variable method

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The convergence of the objective function is illus-trated in Fig. 7. As indicated by the diagram, the objectivefunction converges fast and steadily.

(d) Influence of initial setting on optimal solution

The initial density value ρ0 was varied in order toexamine how it affects the optimal shape, as shown in Fig.8. The value of the optimization function for the optimalshape is also given in the diagram.

There is a small effect of the initial value on theoptimal solution, which can be explained by the initialsetting dependence of the steepest descent method. How-ever, the cross section of the core facing the armatureincreases in all cases, so that a similar core shape is ob-tained. In addition, the objective function varies onlyslightly with the initial setting, and efficient minimumsearch proves possible at any initial setting.

4.2 Topology optimization of permanentmagnet model

(a) Analytic model and objective function

An analytic model of the permanent magnet is shownin Fig. 9. Since the shape is symmetric about the y-axis,only the right half is considered as the area for analysis.

The magnetization of the magnet is assumed to be1.38 T, and the number of elements is 3600 in the designdomain and 7154 in the whole analysis area. The density ρis set initially as ρ0 = 0.5. Aiming at a uniform magneticflux density in the x-direction of the magnetic field evalu-ation region (target region), and the smallest density in they-direction, the objective function is set in the followingway:

Fig. 6. Flux distribution of C-core (ρ0 = 0.2).

Fig. 7. Convergence of objective function of C-core (ρ0 = 0.2).

Fig. 8. Effect of initial value ρ0 on optimal result.

Fig. 9. Permanent magnet model.

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Here le is the last element number in the target region andne is the end point. The number of elements in the targetregion is set to 48. The target flux density in the x-directionBx0 is set to –0.22 T. In addition, the adjoint equation forthis objective function is

(b) Convergence to optimal solution

Topology optimization was applied to the permanentmagnet model shown in Fig. 9. The convergence to theoptimal solution and the variation of the flux density vectordistribution in the target region are illustrated in Fig. 10.Denoting by Bxmax and Bxmin, respectively, the largest andsmallest magnetic flux densities in the x-direction of thetarget region, the uniformity factor u of flux density withinthe target region is defined as follows:

As is evident from Fig. 10, the uniformity factor u is closeto 1 at the initial stage, while Bx deviates greatly from thetarget value of 0.22 T, being about 0.09 T in the targetregion. At the 10th iteration, the magnet topology becomesclear, and the magnetic flux density in the target regiongrows. After that, a gap develops in the magnet, so that theflux density decreases in the y-direction. As a result, the fluxdensity in the y-direction approaches zero, while the uni-formity factor of Bx improves to 1.03. In addition, theoptimal solution includes grayscale areas.

The magnetic flux diagram of the optimal solution ispresented in Fig. 11. In the diagram, elements with ρ of 0.7or more are extracted in the design domain to draw a virtualoutline of the magnet. Since the elements have differentmaterial densities ρ, the magnetization in the magnet too isnot fixed, and the potential line spacing is not uniform.

The convergence of the objective function is shownin Fig. 12. As indicated by the diagram, the objectivefunction converges while varying in spikelike fashion. Thismay be explained by the strong nonlinearity of the objectivefunction. That is, as the flux density in the x-directionapproaches its target value, y-direction flux density is mini-mized. The magnetic circuit is treated as a closed path,which is a relatively difficult problem.

(c) Influence of initial setting on optimal solution

The initial density value ρ0 was varied to examinehow it affected the optimal shape, as shown in Fig. 13.

As is evident from the diagram, the optimal solutionvaries depending on the initial setting. This may be becausethe objective function is multipeaked in the permanentmagnet model. Thus, when steepest descent is applied, thesearch converges to the minimum closest to the initial value.Therefore, when applying the proposed method to a realdevice, one should prepare multiple initial settings and

(20)

(21)

(22)

Fig. 10. Convergence process of permanent magnetmodel.

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select the minimum that is associated with the lowest valueof the objective function.

(d) Influence of exponent n on grayscale I

As is evident from Fig. 13, the optimal solutionincludes a grayscale area at any initial setting. Thus, weexamined how the exponent n in Eq. (2) affects thegrayscale area and the optimal solution.

Figure 14 presents the results obtained with the initialdensity value ρ set to 0.2, and with the exponent n givenvalues of 1 and 3. For n = 1, there are many grayscaleelements; on the other hand, for n = 3, grayscale elementsdisappear. In the former case, the number of iterations was1295; in the latter case, convergence was not achieved andthe calculation was terminated at the 10,000th iteration.Thus, with n set small, convergence is achieved after a smallnumber of iterations, but the grayscale area grows. Incontrast, with n set large, the grayscale area contracts butthe convergence deteriorates. Therefore, n should be se-lected with consideration of this trade-off.

Fig. 11. Flux distribution of permanent magnet model.

Fig. 12. Convergence process of objective function ofpermanent magnet model.

Fig. 13. Effect of initial value ρ0 on optimal result inpermanent magnet model.

Fig. 14. Effect of exponent n on grayscale element (ρ0

= 0.2).

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(e) Comparison between grayscale model andcontour model

In the grayscale elements, the material allocation isnot clear, and hence the actual design is difficult. Therefore,we created a magnet model assuming the presence of mag-netic material when ρ was greater than 0.8, and air when ρwas less than 0.8, and compared it to the topology includinggrayscale areas. The results of a comparison using Fig. 13are shown in Fig. 15.

When the grayscale areas are removed, the specificgeometry of the magnetic circuit can be obtained, but theuniformity factor u of Bx deteriorates. Therefore, when thecontour model is applied, one should prepare solutions formultiple patterns using the density method, and select thecontour model offering the best results.

5. Conclusions

This paper has presented a fundamental study of thetopology optimization of magnetic circuits by the density

method. The results of the study may be summarized asfollows.

(1) The forward difference method and the adjointvariable method were compared in the calculation of thesensitivity of the objective function. It was shown that theadjoint variable method offers more accurate calculation ofthe sensitivity, while requiring only a one-time direct analy-sis.

(2) The dependence of the proposed method on initialsetting was examined using a C-core model. The obtainedoptimal solution proved to depend on the initial setting, butobjective function was nearly unaffected, and good mini-mum search could be performed with a small number ofiterations.

(3) The proposed topology optimization was used indesign to achieve uniform flux density of the permanentmagnet inside the target region. The solutions obtainedincluded many grayscale elements but were usable as de-sign guidelines for magnet layout.

In the future, we plan to explore effective methods ofeliminating grayscale, and to apply the proposed method tothe actual design of linear elevators and other devices. Inaddition, practical design methods for handling grayscaleelements should be studied.

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Fig. 15. Comparison of grayscale model and contourmodel.

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APPENDIX

Calculation of Sensitivity When the ObjectivesAre Maximization of Motor Output andMinimization of Torque Ripple

(a) Maximization of motor output

When the motor output Wout is to be maximized, theobjective function may be designed as follows:

The right-hand side of the adjoint equation for this objectivefunction is

Using Eq. (A.3), Eq. (A.2) can be written as Eq. (A.4):

Here T is the torque applied to an object, and n is the motorspeed. In a two-dimensional field, the torque T acts only inthe z direction, and this z-component is

Here xi, yi are the coordinates of node i, and fxi, fyi are theelectromagnetic force components applied to node i. Inaddition, tor is the torque calculation area. Here the nodalforce method [17] is employed to find the torque. Thus,∂T/∂A on the right-hand side of Eq. (A.4) is

Now the sensitivity of the objective function can be calcu-lated by solving adjoint equation (17) and substitutingadjoint variable λ into Eq. (15):

(b) Minimization of torque ripple

When torque ripple is to be minimized, the objectivefunction can be designed in the following way:

Here nposi is the number of rotor positions, and T0 is thetarget torque. The right-hand side of the adjoint equationfor this objective function is

The sensitivity can be calculated by finding ∂Ti/∂A with Eq.(A.6), then finding the adjoint variable λ and substituting itinto Eq. (15).

(A.1)

(A.3)

(A.4)

(A.2)

(A.5)

(A.6)

(A.7)

(A.8)

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AUTHORS (from left to right)

Yoshifumi Okamoto (student member) graduated from Kansai University (electrical engineering) in 2000, completed thefirst stage of his doctoral program at Kansai University in 2002, and is now in the second stage at Okayama University. Hisresearch interests are FEM-based optimal design of magnetic circuits, and fast algorithms for electromagnetic field analysis.

Norio Takahashi (member) graduated from Okayama University (electrical engineering) in 1974, completed the M.E.program in 1976, and joined the faculty as a research associate. He has been a professor there since 1993. His research interestsare FEM-based magnetic field analysis of electrical and electronic devices, and optimal design. He is an IEEE Fellow.

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