Echeverria, D. (David)

Encica, L.

Hemker, P.W. (Piet)

Lahaye, D.J.P. (Domenico)

Lomonova, E.A.

Vandenput, A.J.A.

English

CWI's Institutional Repository

IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 1183Manifold-Mapping Optimization Appliedto Linear Actuator DesignD. Echeverría1, D. Lahaye1, L. Encica2, E. A. Lomonova2, P. W. Hemker1, and A. J. A. Vandenput2Centre for Mathematics and Computer Science (CWI), 1098 SJ Amsterdam, The NetherlandsEindhoven University of Technology (TU/e), 5600 MB Eindhoven, The NetherlandsOptimization procedures, in practice, are based on highly accurate models that typically have an excessive computational cost. Byexploiting auxiliary models that are less accurate, but much cheaper to compute, space mapping (SM) has been reported to acceleratesuch procedures. However, the SM solution does not always coincide with the accurate model optimum. We introduce manifold mapping,an improved version of SM that finds this precise solution with the same computational efficiency. By two examples in linear actuatordesign, we show that our technique delivers a significant speedup compared to other optimization schemes.Index Terms—Actuators, optimization methods, space mapping (SM), surrogate-based optimization.I. INTRODUCTIONMANY practical design problems in electromagnetics arenowadays solved by means of highly accurate compu-tational models. Typically, finite elements (FEs) are employed.However, this approach is computationally expensive. Thespace-mapping (SM) technique [1] aims at accelerating suchprocedures using approximate models that are faster to com-pute. The application of SM to optimization in low-frequencyelectromagnetics was considered, e.g., in [2]. The originallyconceived SM technique might fail to converge to the accuratemodel optimum. We introduce manifold mapping (MM) [3],an improved SM technique that efficiently finds the desiredoptimum. Its application is illustrated with some examples.II. SM AND MMA. SM TechniqueLet us consider an optimization problem in the design spacewith specifications . The accuratebehavior of electromechanical devices is often studied usingmodels that have large computational costs, e.g., FE models.In SM terminology, these models are called fine models. Thefine model response is denoted by , where is thedesign variable. The associated optimum is referred to as .SM needs a second, possibly less accurate but computationallymuch cheaper model. This is the coarse model; in this paper, thecoarse models are assumed to be defined over the same designspace . The coarse model response and the corresponding op-timum are denoted by and , respectively.The SM function is defined by(1)SM corrects , the coarse optimum, in order to approximate. In traditional SM, this is done either by solving the nonlinearsystem of equations(2)Digital Object Identifier 10.1109/TMAG.2006.870969or, more generally, by minimizing with the surrogate(3)However, the solution of neither of these two problems doesnecessarily coincide with the fine model optimum .B. MM TechniqueAnalyzing the conditions under which the SM solution yieldsthe correct answer, the SM technique can be improved in such away that the accurate optimal design can be computed by an iter-ative process. Each step in this scheme requires one evaluationof the fine model and one solution of a cheap minimizationproblem.MM replaces the SM function by an arbitrary bijectionand introduces the manifold mapping .With this mapping, the point is mapped to andthe tangent space for at to the tangent space forat . By this construction, thenIn the examples in this work, the function is taken equalto the identity.C. Implementation of MMDuring the iterative solution process, the manifold mappingis approximated by a sequence of affine mappings .This leads to the following algorithm (with anarbitrary bijection).1) Set and compute2) Evaluate and and finish if appropriatestopping criteria are met.3) If and with and, we defineand to be the matrices with, re-spectively, and as columns. Their singular valuedecompositions are, respectively, and.0018-9464/$20.00 © 2006 IEEE1184 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006Fig. 1. Axisymmetrical geometry and design variables of (left) the automotiveand (right) the voice-coil actuator.4) The approximate affine mapping is(4)where is a regular matrix equal to the identityfor and to if. The operator denotes the pseudoinverse.5) Compute6) Set k = k + 1 and go to 2.Step 5) can be implemented in a simpler way since it can beshown that it is asymptotically equivalent to the following:5) ComputewhereThe matrix satisfiesfor and , and, thus, in thelimit, the desirable properties of the manifold mapping arefulfilled. Under convergence of the scheme given, the limit isthe fine model optimum .III. AUTOMOTIVE MAGNETIC ACTUATORFor this first actuator design problem, SM, MM, and someother efficient optimization techniques are compared. Automo-tive actuators typically generate high levels of force and canbe used in devices, such as electromagnetic switches, relays,valves, etc. Fig. 1 is a schematic view of the cylindrical plungerelectromagnet. It was originally introduced in [4], and it con-sists of a core and a plunger, both made of iron, and a coppercoil. The sizes have to be optimized in order toassure a maximum magnetic thrust force versus displacementFig. 2. Force responses for the fine and coarse model in two points ofX .TABLE IFORCE OPTIMIZATION OF THE AUTOMOTIVE ACTUATORIN THE ORIGINAL DESIGN SPACEin the axis. The volume of the device is kept constant. Thespecifications are a constant force response of 100 , for sixvertical plunger displacements. The design space proposedin [4] is the rectangle , where all the boundshave been specified in millimeters. More details on the problemcan be found in [4], [5].The fine model is based on a static force computationfor the six plunger displacements. This force is evaluated nu-merically by second-order Lagrangian FEs [6] with one levelof adaptive refinement. The number of degrees of freedom inevery case is around 5000–7000 yielding three digits of ac-curacy in the force. The objective function here is, i.e., a relative 2–norm of the modeldiscrepancy.The coarse model is based on a magnetic equivalent cir-cuit (MEC) [7] derived from the actuator. The MEC is nonlinear,in the sense that the – curve is considered for the plunger.The core magnetic permeability is taken as infinite and neitherfringing nor leakage effects are included. A linear interpola-tion for the force is taken between the points for maximum andminimum displacement. This coarse model is around 400 timesfaster than the fine one. In Fig. 2, the fine and coarse model re-sponses are shown for two points in the design space : the ge-ometrical center and the optimum obtained with MM. Eachoptimization involving the coarse model is solved by sequentialquadratic programming (SQP) [8].Table I shows the performance of seven minimizationmethods when applied to the mentioned fine model. The givenfigures (amount of work expressed in the number of equivalentfine model evaluations) are approximately proportional to thetotal computing time. All seven techniques yield solutions withsimilar objective function values. The first two (MMA: methodECHEVERRÍA et al.: MANIFOLD-MAPPING OPTIMIZATION APPLIED TO LINEAR ACTUATOR DESIGN 1185TABLE IIFORCE OPTIMIZATION OF THE AUTOMOTIVE ACTUATORIN THE MODIFIED DESIGN SPACEof moving asymptotes and PM: penalty method) correspondto methods analyzed in [4] and the next two (EM: ellipsoidmethod and RQP: recursive quadratic programming) to otherschemes also studied by the same author [5]. The coarse modeloptimum mm mm has a lower fine cost functionthan any of the solutions given by the methods above. SM andMM converge to after just one fine model evaluation (thenumber of coarse model evaluations is also taken into accountin the amount of work shown in Table I). SQP, applied tothe fine model and starting from the coarse model optimum,performs also very efficiently.Next, the space is slightly modified to(again, in millimeters) in order to have a design in which thecoarse optimum does not end the optimization process.1 Thus, abetter comparison can be made between the three methods thatperform similarly above. Now, the coarse optimum is the point[4.68 mm, 17.72 mm] which has an associated cost function of. The optimization results are given in Table II. MM andSQP yield both a solution with the same quality, but MM isalmost four times faster. SM improves also SQP in efficiencybut the result obtained is not the optimal design. This experimentcorroborates MM as a method as efficient as SM, but with theadditional property of convergence toward the fine optimum.IV. CYLINDRICAL VOICE COIL ACTUATORA. Actuator DescriptionLinear voice coil actuators [9] are noncommutated electro-mechanical devices that provide cogging-free force outputs di-rectly proportional to the applied current. An axisymmetricalvariant consisting of a permanent magnet, a coil, and a ferro-magnetic core is presented in Fig. 1. As in the previous section,the design variables correspond to sizes in the actuator. Thedistance measures the position of the coil. For this actuator,we consider two design problems. Full details on both problemscan be found in [10].B. Force Response OptimizationIn the first problem, we allow the coil to move over a 4-mmstroke, i.e., mm. Values for have to be found suchthat the force response is as flat and as close to as pos-sible. For this force level, the nonlinear effect of the ferromag-netic core has to be taken into account in an accurate fine modeldescription. The permanent magnet demagnetization curve isassumed to be linear with . The coil movement is repre-sented at nine equidistant points in the interval mm.The cost function is . The1This modification of the design space is introduced for testing purposes. Asobserved in [5], big values for x could lead to devices with significant massand a poor dynamic regime performance.Fig. 3. Cost functions associated with the coarse (simplified FE), fine, SM,and MM surrogate models for the 1-D problem.design space is an hypercube in . Additional linear con-straints define feasible coil positions.In the fine model, the force is computed by Lorentz’s formulaat each of the nine coil positions. Second-order Lagrangian tri-angular FEs are used for that purpose. The number of degreesof freedom is between 8000 and 11 000, yielding three digits ofaccuracy in the force.The first of the two coarse models considered is an FE modelin which the nonlinear material characteristic of the core islinearized. Depending on the number of Newton iterationsrequired, this model is a factor between 30 and 50 cheaper thanthe fine one. The second coarse model is an MEC. This modelhas negligible computational cost compared to the fine model.We initially consider the one-dimensional (1-D) designproblem, considering changes in the first design variableonly. The purpose is to illustrate how the cost function as-sociated with MM closely approximates the fine model costfunction in a region close to . In Fig. 3, the followingfour cost-related functions in are plotted: the coarse model, the fine model , the SM, and the MMcost functions. This figure shows that the coarse and fine modeloptimum are clearly different, and how MM improves the costfunction associated with SM in a region close to .In this 1-D problem, both SM and MM, using either the linearFE or the MEC as coarse model, converge in four fine modelevaluations. Compared with SQP or the Nelder–Mead Simplex(NMS) method [11], that are similar in performance, SM andMM deliver a speedup of a factor between four and five.If we consider the two-dimensional (2-D) design problem,with changes in and only, we can clearly illustrate that theproblem is ill conditioned. In Fig. 4, the cost function is plotted.In this plot, the dark region shows a long and steep valley in thedesign space with approximately the same cost function value.This means that there is no unique solution. The value foundby optimization depends on factors such as starting guess andalgorithm used. Uniqueness can be restored by regularization,for example, by imposing a minimal mass constraint as will bedone in the following subsection.1186 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006Fig. 4. Logarithm of the fine model cost function for the 2-D problem. Darkshading indicates low values for the cost function.TABLE IIITWO-DIMENSIONAL OPTIMIZATION OF THE VOICE-COIL ACTUATORTABLE IVSEVEN-DIMENSIONAL OPTIMIZATION OF THE VOICE-COIL ACTUATORNumerical results comparing MM with NMS, SQP, and SMfor the 2-D problem are given in Table III. As a starting guessfor the optimization procedures, we used the values obtainedby optimizing the MEC model. To stabilize the convergence ofMM, the Levenberg–Marquardt method [8] is used. The bestresults in terms of computational efficiency (speedup by a factorof six) are obtained using MM with the MEC as coarse model.The NMS and SQP algorithms are again similar in performance.C. More Complex Optimization ProblemThe second design problem, introduced in [12], is seven-dimensional with nonlinear equality and inequality constraints.The total mass of the actuator has to be minimized, whilethe mass of the coil is constrained at 10 g. The force at coilposition mm (see Fig. 1) should be kept at 5and the magnetic flux density in three specified regions of thecore should not exceed 1 T. The mass computation carries nosignificant computational cost but the constraints involve timeexpensive calculations. The two-model approach is applied ina constrained sense [13]. The performance of SM and MM iscompared with that of SQP. In the fine model, the constraintsare evaluated by the same FE model employed in the previoussubsection. In the coarse model, the constraints are based on aMEC model. Each coarse model related optimization is doneusing SQP.Numerical results for this problem are given in Table IV. Theoptimization process was stopped if the relative decrease ofboth the cost function and 2–norm of the design parameters wassmaller than , provided that the constraints were met withthree digits of accuracy. SM and MM show a similar behavior:convergence is reached in seven and six fine constraints eval-uations, respectively. SQP converged within 56 fine constraintevaluations. Because, in this problem, the number of constraintsis smaller than the number of design parameters, we expect theSM solution to coincide with . We recognize this to be thecase as SQP, SM, and MM yield the same optimum, but even inthis situation, MM offers an additional advantage over SM. Thecomputation of the SM function is a very delicate issue[1]. MM replaces it by the identity, and in a realistic problemlike this one, the efficiency results are comparable.V. CONCLUSIONWe introduced the manifold-mapping technique, an optimiza-tion method that exploits approximate model information to ac-celerate time-consuming design procedures. The application ofthe method on linear actuators has given evidence of the com-putational speedup that this method can deliver.ACKNOWLEDGMENTThis work was supported by the Dutch Ministry of EconomicAffairs under Project IOP-EMVT 02201.REFERENCES[1] J. W. Bandler, Q. S. Cheng, S. A. Dakroury, A. S. Mohamed, M. H.Bakr, K. Madsen, and J. Søndergaard, “Space mapping: The state of theart,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 337–361, Jan.2004.[2] D. Echeverría, D. Lahaye, L. Encica, and P. W. Hemker, “Optimizationin electromagnetics with the space mapping technique,” COMPEL, vol.24, no. 3, pp. 952–966, 2005.[3] D. Echeverría and P. W. Hemker, “Space mapping and defect correction,”CMAM, vol. 5, no. 2, pp. 107–136, 2005.[4] R. R. Saldanha, S. Pelissier, K. Kadded, Y. P. Yonnet, and J. L. Coulomb,“Nonlinear optimization applied to magnetic actuators design,” IEEETrans. Magn., vol. 28, no. 2, pp. 1581–1584, Mar. 1992.[5] R. R. Saldanha, “Optimization en Electromagnetisme par ApplicationConjointe des Methodes de Programmation Non Lineaire et de laMethode des Elements Finis,”, These de doctorat de l’Institut NationalPolytechnique de Grenoble, Grenoble, France, 1992.[6] P. P. Sylvester and R. L. Ferrari, Finite Element for Electrical Engineers,3rd ed. New York: Cambridge Univ. Press, 1996.[7] D. K. Cheng, Field and Wave Electromagnetics. Reading, MA: Ad-dison-Wesley, 1989.[8] J. Nocedal and S. J. Wright, “Numerical Optimization,” in Springer Se-ries in Operations Research. New York: Springer, 1999.[9] I. Boldea, “Linear electromagnetic actuators and their control: A re-view,” EPE J., vol. 14, no. 1, pp. 43–50, 2004.[10] D. Echeverría, D. Lahaye, L. Encica, E. Lomonova, P. W. Hemker, andA. J. A. Vandenput, “Manifold-mapping optimization applied to linearactuator design,”, Amsterdam, The Netherlands, 2006. CWI Tech. Rep.MAS-E0606.[11] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Conver-gence properties of the Nelder-Mead simplex method in low dimen-sions,” SIAM J. Opt., vol. 9, no. 1, pp. 112–147, 2003.[12] L. Encica, D. Echeverría, E. Lomonova, A. J. A. Vandenput, P. W.Hemker, and D. Lahaye, “Efficient optimal design of electromagneticactuators using space-mapping,” presented at the 6th World Congr.Structural and Multidisciplinary Optimization, 2005.[13] S. J. Leary, A. Bhaskar, and A. J. Keane, “A constrained mapping ap-proach to the structural optimization of expensive model using surro-gates,” Optim. Eng., vol. 2, pp. 385–398, 2001.Manuscript received June 21, 2005; revised October 28, 2005 (e-mail:d.echeverria@cwi.nl).

Manifold-Mapping Optimization Applied to Linear Actuator Design

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