It is necessary to develop a recording head having large recording field and small stray field to adjacent tracks and adjacent bits in perpendicular magnetic recording system. In this paper, in order to decrease the leakage flux in the adjacent bit, the approach of magnetic shield design of perpendicular magnetic recording head for 200 Gb/in/sup 2/ is performed by using the topology optimization technique.</p

Ohtake, Masaya

Okamoto, Yoshifumi

Takahashi, Norio

Okayama University Scientific Achievement Repository

PhysicsElectricity & Magnetism fieldsOkayama University Year 2004Magnetic shield design of perpendicularmagnetic recording head by usingtopology optimization techniqueYoshifumi Okamoto Masaya OhtakeOkayama University Okayama UniversityNorio TakahashiOkayama UniversityThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/electricity and magnetism/291788 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY 2005Magnetic Shield Design of Perpendicular MagneticRecording Head by Using TopologyOptimization TechniqueYoshifumi Okamoto, Masaya Ohtake, and Norio Takahashi, Fellow, IEEEDepartment of Electrical and Electronics Engineering, Okayama University, Okayama 700-8530, JapanIt is necessary to develop a recording head having large recording field and small stray field to adjacent tracks and adjacent bits inperpendicular magnetic recording system. In this paper, in order to decrease the leakage flux in the adjacent bit, the approach of mag-netic shield design of perpendicular magnetic recording head for 200 Gb/in2 is performed by using the topology optimization technique.Index Terms—Perpendicular magnetic recording head, sensitivity analysis, shield design, topology optimization.I. INTRODUCTIONTHE perpendicular magnetic recording system is consid-ered as an important technique for the high density mag-netic recording exceeding 100 Gb/in [1]. On the other hand,when the linear recording density (BPI) is increased, the eraseof bits due to the stray field becomes remarkable. In order to de-crease the leakage flux to adjacent bits, it is considered to set themagnetic shield along the down-track direction. The optimiza-tion technique utilizing the finite-element method (FEM) is ef-fective to get an optimal magnetic shield. Recently, the topologyoptimization technique [2] has been investigated to design themagnetic circuit of new concept.In the topology optimization by using the density method [3],[4] combined with a descent method, the relationship betweenthe material constant and the material density is continuous,then, there occurs a grayscale element. It is not clear whetherthis region can be treated as magnetic material or not [5]. There-fore, the topology optimization method using the genetic algo-rithm is presented [6]. This method is superior to the densitymethod from the viewpoint that no grayscale element appearsand the calculation of sensitivity is not required. However, acontinuous shape cannot be often obtained, because the mag-netic material is distributed discretely due to huge design vari-ables. More investigation is desired for the practical utilizationof this method.In this paper, a topology optimization algorithm (ON/OFFmethod [7]) considering the existence of material (ON/OFF) ineach element is adopted. In order to prevent the discretizedtopology, the existence of magnetic material in each elementis decided by using the sensitivity. As the fast calculation ofsensitivity is desired in order to decrease the total CPU time,the adjoint variable method [8], [9], in which only one extracalculation is required, is applied. Moreover, to decrease thenumber of Newton–Raphson iteration in the nonlinear analysis,the line search method [10] is introduced. As a result, aneffective write head and magnetic shield is designed quickly byusing the ON/OFF method.Digital Object Identifier 10.1109/TMAG.2005.845985II. TOPOLOGY OPTIMIZATION TECHNIQUEA. Sensitivity Analysis MethodThe sensitivity is accurately calculated by using the adjointvariable method [8], [9]. The equation for FEM is given as(1)where is the coefficient matrix, is the column vector ofmagnetic vector potential, and is the right-hand side vector.Taking the derivative of (1) with design variable in an element(2)(2) can be rewritten as follows:(3)where is the value obtained by solving (1). If the objectivefunction is expressed as the function of the perme-ability in design domain and the magnetic vector potential, thesensitivity is given by(4)Substituting (3) into (4), (5) can be obtained:(5)In order to avoid the calculation of the inverse of , an adjointvector is introduced [8]. The adjoint equation is given by(6)is obtained by solving (6), and is calculated by sub-stituting into (7)(7)0018-9464/$20.00 © 2005 IEEEOKAMOTO et al.: MAGNETIC SHIELD DESIGN OF PERPENDICULAR MAGNETIC RECORDING HEAD 1789Fig. 1. Method for determining step size by using linear approximation.Equation (7) suggests that only one extra solution for the adjointvector is sufficient to determine the sensitivity to all parameters,rather than obtaining each value per parameter, providing a com-putationally fast method for deriving the gradients.B. Newton–Raphson Method Using Line SearchIn order to take account of the nonlinearity of magnetizationproperty of magnetic material, the Newton–Raphson method isapplied. In the topology optimization, the convergence charac-teristic is worse because the various topology is generated. Inorder to improve the convergence characteristic and to decreasethe number of iterations, the line search [10] is applied to theNewton–Raphson method.A search direction of the magnetic vector potential ateach nonlinear iteration is calculated by using the ordinaryNewton–Raphson method. Then, a step size in the iterativeprocess is determined by thefollowing equation:(8)where is the energy functional, is the residual vector, andthe superscript means the transposition of a column vector.By assuming that can be approximated by a straightline, which satisfies (8) is calculated as shown in Fig. 1.By calculating at two points, for exampleand , the cross point of the line is obtained.When the change of the flux density in each element is less than10 T, the nonlinear iteration is terminated.C. ON/OFF MethodThe ON/OFF method [7] for avoiding the grayscale elementsis effective for designing the practical magnetic circuit from theviewpoint of material arrangement in each element (magneticmaterial or air). The flowchart of the ON/OFF method is shownin Fig. 2.Fig. 2. Flowchart of ON/OFF method.Step 1: decision of initial topologyThe initial topology in the design domain is decided by the materialarrangement in each element.Step 2 and Step 7: FEMThe objective function W of the design domain is obtained fromthe calculated value using FEM.Step 3: adjoint variable methodSolving the adjoint equation (6), the sensitivity is calculated byusing (7)Step 4: sorting of elements in order of sensitivityEach element is ranked in order of the absolute value of sensitivity.Step 5: setup of changeable elementsIf the sensitivity dW=drk with respect to permeability k inelement k is negative, the permeability in the element k shouldbe increased. Then, the magnetic material is located in the elementk. On the other hand, if the sensitivity dW 0=drk is positive, thepermeability in the element k should be decreased. Then, the air isallocated in the element k. In this step, the element, of which thestate is changeable, is selected.Step 6: change of topologyThe topology is modified following the information obtained inStep 5.Step 8: ( +1) ( )If the objective function W (k+1) is smaller than W (k), return toStep 3, otherwise go to step 9.Step 9: annealingIf the objective function is not improved at all, the changeableelements Nm is decreased using the following equation:Nm =  Nm (9)where  is the annealing factor, which is chosen as 0.9. RepeatSteps 6–9 until some improvement of the objective function isdetected.Step 10: = 0If the number of changeable elements Nm becomes zero, thecomputation is terminated. Otherwise, return to Step 6.1790 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY 2005Fig. 3. Model of perpendicular head.Fig. 4. Magnetization property of FeCoAlO.Using the algorithm mentioned above, a fast convergencecharacteristic and a good topology can be obtained.III. ANALYZED MODEL AND OBJECTIVE FUNCTIONThe analyzed model of the perpendicular magnetic recordingsystem is shown in Fig. 3. FeCoAlO (saturated magnetization:T) is adopted as the magnetic material of returnyoke, soft underlayer, and design domain (modified return yokefor shielding and main pole). The magnetization property ofFeCoAlO is shown in Fig. 4. The measured data is interpolatedby a cubic function using the Akima’s method. The quadraticapproximation is applied until the saturation magnetization, andafter saturation magnetic flux density, is treated as a straightline directly proportional to magnetic field . The current den-sity in the coil is chosen as 0.1 A. The triangular elements of firstorder are adopted as element shape. The number of elements indesign domain is set as 4002.The design goal of this optimization problem is to maximizethe flux density (recording flux) in the region , and to mini-mize the flux density (leakage flux) in the region (adjacentbit) in the media. The function and in the regionsand to be minimized are given as(10)Fig. 5. Convergence process of optimal topology. (a) Initial state (W =2:2310 ;W = 0:22). (b) First iteration (W = 2:8610 ;W = 0:049).(c) Second iteration (W = 2:33  10 ;W = 0:020). (d) 40th iteration(optimal, W = 1:77 10 ;W = 0:0084).(11)where is the -component of flux density in an elementis its area. and are the areas of regions and. and are the starting number and the final number ofthe element in the region and are the starting numberand final number of the element in the region . Finally, inorder to convert a multiobjective function into a single-objectivefunction, the objective function is defined as(12)where and are the coefficients to adjust the order ofand , respectively. In this problem, is chosen as 1.0 andis chosen as .IV. OPTIMIZATION AND DISCUSSIONThe initial material of design domain is assumed as air. Fig. 5shows the convergence process of optimal topology in designdomain. An approximate shield topology is generated at the firstiteration in order to prevent the leakage flux in the region . Atthe second iteration, the main flux in the region is decreasedso that the additional pole is generated in order to increase theflux in the region . Finally, the optimal shape of shield andpole is generated at the 40th iteration. In this optimization, thenumber of iterations by using the line search Newton–RaphsonOKAMOTO et al.: MAGNETIC SHIELD DESIGN OF PERPENDICULAR MAGNETIC RECORDING HEAD 1791Fig. 6. Flux distributions. (a) Initial topology (B = 1:65 T, B =0:61 T). (b) Optimal topology (B = 1:90 T, B = 0:34 T).Fig. 7. Changes of magnetic flux density B on media on line a–b.method is about 9, while the number of iterations is about 13 byusing the ordinary Newton–Raphson method.Fig. 6 shows the flux distributions of the initial and optimaltopology. In the initial topology, the maximum flux densityin the region is 0.61 T, because the leakage fluxis generated from the main pole. The maximum flux densityin the region is 1.65 T. By applying the topologyoptimization, is decreased, and is increased.Fig. 7 shows the changes of on the centerline of medium– . The design goal in the region and region is per-formed. The maximum value of is 1.89 T at region ,and the minimum value of is 0.24 T at the region in theoptimized value.Fig. 8 shows the convergence process of the objective func-tion. and are oscillating by the mutual effect aroundinitial steps, but is decreased steadily. The CPU time of 40iterations is 1.6 h by using a PC (CPU: 2.8 GHz, RAM: 2.0 GB).V. CONCLUSIONThe paper presents an approach to design the perpendicularmagnetic recording head for 200 Gb/in by using the ON/OFFFig. 8. Convergence process of objective function (CPU time: 1.6 h, CPU:2.8 GHz, RAM: 1.5 GB).topology optimization technique. In order to prevent the leakageflux density and to increase the main flux, the new topologyof the magnetic shield and main pole is fully designed in twodimensions. As a result, the leakage flux is decreased, and themain flux is increased.In order to produce an actual recording head having the op-timal topology obtained by the ON/OFF method, the topologycontour should be more feasible. The development of an ef-fective topology smoother, and the 3-D topology optimizationof the perpendicular magnetic recording head considering theread/write process are the future work.REFERENCES[1] K. Ise, K. Yamakawa, N. Horita, K. Ouchi, H. Muraoka, and Y. Naka-mura, “High-field gradient cusp field single-pole writing head with frontreturn yoke,” IEEE Trans. Magn., vol. 39, no. 5, pp. 2374–2376, Sep.2003.[2] Y. Okamoto and N. Takahashi, “A novel topology optimization usingdensity method combination with modified evolution strategy,” in Proc.COMPUMAG Conf., Saratoga, CA, 2003.[3] J. K. Byun, J. H. 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Sabonnadiere, “Performancederivative calculations and optimization process,” IEEE Trans. Magn.,vol. 25, no. 4, pp. 2834–2839, Jul. 1989.[9] I. H. Park, B. T. Lee, and S. Y. Hahn, “Design sensitivity analysis fornonlinear magnetostatic problems using finite element method,” IEEETrans. Magn., vol. 28, no. 2, pp. 1553–1536, Mar. 1992.[10] K. Fujiwara, Y. Okamoto, A. Kameari, and A. Ahagon, “The Newton-Raphson method accelerated by using a line search—Comparison be-tween energy functional and residual minimization,” in Proc. CEFCConf., Seoul, Korea, 2004.Manuscript received June 8, 2004.

Magnetic shield design of perpendicular magnetic recording head by using topology optimization technique

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