276 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 1, JANUARY 2004

Fast Parameter Optimization of Large-ScaleElectromagnetic Objects Using DIRECT

with Kriging MetamodelingEng Swee Siah, Student Member, IEEE, Micheal Sasena, John L. Volakis, Fellow, IEEE, Panos Y. Papalambros,

and Rich W. Wiese

Abstract—With the advent of fast methods to significantly speedup numerical computation of large-scale realistic electromagnetic(EM) structures, EM design and optimization is becoming increas-ingly attractive. In recent years, genetic algorithms, neural net-work and evolutionary optimization methods have become increas-ingly popular for EM optimization. However, these methods areusually associated with a slow convergence bound and, further-more, may not yield a deterministic optimal solution. In this paper,a new hybrid method using Kriging metamodeling in conjunctionwith the divided rectangles (DIRECT) global-optimization algo-rithm is used to yield a globally optimal solution efficiently. Thelatter yields a deterministic answer with fast convergence boundsand inherits both local and global-optimization properties. Threeexamples are given to illustrate the applicability of the method, i.e.,shape optimization for a slot-array frequency-selective surface, an-tenna location optimization to minimize EM coupling from the an-tenna to RF devices in automobile structures, and multisensor op-timization to satisfy RF coupling constraints on a vehicular chassisin the presence of a wire harness. In the first example, DIRECTwith Kriging surrogate modeling was employed. In the latter twoexamples, the adaptive hybrid optimizer, superEGO, was used. Inall three examples, emphasis is placed on the speed of convergence,as well as on the flexibility of the optimization algorithms.

Index Terms—Coupling, DIRECT optimization algorithm,electromagnetic compatibility (EMC), electromagneticinterference (EMI), finite element boundary integral (FE–BI),frequency-selective surface (FSS), Kriging metamodeling, Krigingsurrogate modeling, multilevel fast multipole moment method(MLFMM), superEGO.

Manuscript received December 4, 2002; revised June 23, 2003. This work wassupported by the U.S. Air Force under the Multiuniversity Research InitiativeGrant F49620-01-1-0436 and by the General Motors Electromagnetic Compat-ibility Laboratory.

E. S. Siah is with the Radiation Laboratory, Department of ElectricalEngineering and Computer Science, The University of Michigan at Ann Arbor,Ann Arbor, MI 48109-2122 USA (e-mail: esiah@engin.umich.edu).

M. Sasena was with the Optimal Design Laboratory, Department of Mechan-ical Engineering, The University of Michigan at Ann Arbor, Ann Arbor, MI48109-2122 USA. He is now with Emmeskay Inc., Plymouth, MI 48170 USA.

J. L. Volakis is with the Radiation Laboratory, Department of ElectricalEngineering and Computer Science, The University of Michigan at AnnArbor, Ann Arbor, MI 48109-2122 USA and also with the ElectroScienceLaboratory, Ohio State University, Columbus, OH 43212 USA (e-mail:volakis@engin.umich.edu).

P. Y. Papalambros is with the Optimal Design Laboratory, Department of Me-chanical Engineering, The University of Michigan at Ann Arbor, Ann Arbor, MI48109-2122 USA (e-mail: pyp@engin.umich.edu).

R. W. Wiese is with the General Motors Corporation, Milford, MI 48340USA.

Digital Object Identifier 10.1109/TMTT.2003.820891

I. INTRODUCTION

RECENT developments on fast algorithms, such as themultilevel fast multipole moment method (MLFMM)

[15]–[17] and the hybrid finite-element boundary-integral(FE–BI) method [18], [19], have allowed for significantreduction in CPU time while retaining geometrical adapt-ability and material generality. This makes the application ofdesign optimization a realistic possibility. Previous work inRF design (antennas, RF circuits, etc.) has primarily focusedon optimizing specific problems [2] and involved the use ofevolutionary schemes, like genetic algorithms (GAs) [1]–[3],least squares optimization, and physically modeled processeslike simulated annealing (SA) [4], [5].

The GA is a relatively robust stochastic global-optimizationalgorithm modeled after the Darwinian process of natural selec-tion to produce the best-fit design. As such, it lacks efficiency inits optimization routine and requires typically hundreds or thou-sands of solver evaluations. In addition, the tuning parametersinvolved in using a GA, such as population size, crossover, mu-tation operators, and the fundamental aspects of natural selec-tion (by using random numbers to do the mutation crossovers),causes GAs to yield possibly nonconclusive solutions. This isnot necessarily bad, as it may allow the GA to find an acceptablesolution, albeit within a longer time frame. In practice, differentruns with a GA would yield different answers and a measure ofluck is involved in producing the optimal solution.

SA is also a stochastic global-optimization algorithm thatmodels the physical process of annealing, defined as a thermalprocess for obtaining low-energy states of a solid in a heat bath.In SA, the objective function is analogous to temperature. Thesolid is heated until it begins to melt (a high objective function)to a liquid. Following this, the temperature is allowed to cooland the particles in the liquid arrange themselves randomly. InSA, crystallization of the liquid occurs when the temperatureis sufficiently cool and this is referred to as the ground state ofthe solid. The ground state is analogous to the global minimumsolution and the current state of the thermodynamic system isanalogous to the current iterate. To achieve the ground state (theoptimum solution), the starting temperature must be sufficientlyhigh (large objective function) and cooling has to be sufficientlyslow. Thus, SA suffers from the same drawbacks as the GA inthat convergence is slow and the optimized solution is not al-ways repeatable. In addition, the performance of SA depends

0018-9480/04$20.00 © 2004 IEEE

SIAH et al.: FAST PARAMETER OPTIMIZATION OF LARGE-SCALE EM OBJECTS USING DIRECT WITH KRIGING METAMODELING 277

on proper initialization of program parameters used within SA.The difficulty of finding suitable parameters values is a weak-ness of the GA and SA.

In this paper, we propose two hybrid global-optimizationschemes that converge quickly and yield a deterministic opti-mized solution. The first method involves use of the DIRECTglobal optimizer in conjunction with Kriging surrogate mod-eling. In comparison, the second method (i.e., superEGO) usesthe DIRECT global optimizer to predict a candidate optimalsolution by solving an auxiliary problem based on a Krigingmetamodel. In the second method, the Kriging metamodel isadaptively improved and updated by each simulation iterationthat does not meet the termination criteria according to someinfill sampling criteria (ISC). Both of these hybrid schemesinvolves use of Kriging [8]–[12] to interpolate between datapoints and employ the divided rectangles (DIRECT) as theglobal optimizer. The DIRECT algorithm [13], [14] is a deriva-tive-free global algorithm that reaches a deterministic solutionand does not require selecting values for any parameters. Inaddition, DIRECT has the added benefit of possessing bothlocal and global-optimization properties. Hybridizing theDIRECT search algorithm with Kriging metamodel parametersproduces an efficient global optimizer that converges quickly.These attributes make the proposed hybrid optimizer idealfor optimizing complex large-scale electromagnetic (EM)structures within an acceptable time frame.

The theory and pertinent aspects of Kriging metamodelingand the DIRECT optimizer are explained in Section II. In Sec-tion III, we apply the proposed hybrid optimizer consisting ofDIRECT with Kriging surrogate modeling to optimize the sizeof a slot-array frequency-selective surface (FSS) with respectto a pre-specified reflection coefficient and bandwidth. For thisexample, Kriging surrogate modeling is applied to the problemwhereby the entire design space is split into a finely sampledmesh and the analyzer code is applied to each separate point tocreate the Kriging model. In this instance, the EM modelingtool is the hybrid FE–BI method. In Section IV, we employthe adaptive hybrid optimizer algorithm to solve an auxiliaryproblem, constructed with Kriging metamodeling in conjunc-tion with the DIRECT global optimizer (superEGO). For thisexample, the Kriging metamodel is initially created using asparse number of points. Furthermore, this metamodel is con-tinually updated using the current data point for each optimiza-tion iteration that does not satisfy the termination criteria. Also,the flexibility of the hybrid optimizer is improved by allowingDIRECT to optimize on other ISC. This allows the user tochange the emphasis the optimizer places on a local versusglobal search. As an application, this hybrid optimizer is usedto optimize the antenna position on an automobile (Section V).The antenna location on the automobile is selected to mini-mize EM coupling on the chip pins housed within a resonantcavity in the automobile. In Section VI, the same optimizationscheme is applied to determine the maximum allowed excita-tions that can be applied at ports of a harness (running over thefloor of the automobile body) given the maximum allowableinterference to an FM antenna printed on the back glass of theautomobile.

Analyzer Code such as

MLFMM or FE-BI.

Optimization Schemes such

as SQP, GAs or SuperEGO

with DIRECT on kriging

surrogate modeling.

NoYes --

Update newvariable

values

Input variablevalues

CodeConvergence?

Yes

No CheckMesh /

Geometry file

Output Optimized variable

values

Termination

tolerance met?

Terminate

–

Fig. 1. Flow diagram showing the interaction between the optimizer andanalyzer.

II. OPTIMIZATION METHODS—KRIGING AND DIRECT

Gradient-based optimization algorithms such as sequen-tial quadratic programming (SQP) and generalized reducedgradient (GRG) have fast convergence rates. However, theyrequire information on the gradients of the objective functionswith respect to all design variables at each iteration step. For alarge problem with many variables, the process of evaluatingthese gradients numerically at each iteration step is compu-tationally expensive. Furthermore, gradient-based algorithmsfind only local minima within the problem domain and the finaloptimized solution may depend on the starting point specifiedfor the search process if multiple optima exist. On the otherhand, gradient-free optimization methods rely primarily on theobjective function values and are suitable for problem domainseither with many design variables or fewer design variables,but with computationally expensive objective functions. Someof these algorithms have the desirable properties of being ableto sift through multiple local minima to achieve a more optimalsolution. However, since global algorithms sift the entire searchspace, their convergence rate tends to be rather slow, usuallyin the hundreds or thousands of solver iterations. Also, theycannot deal with a large number of design variables efficiently.

The interaction between the optimizer and EM analyzer codecan be seen in Fig. 1. Overall convergence depends on both theconvergence rates of the optimizer, as well as the analyzer. Ouraim in this paper is to use an efficient global optimizer that uti-lizes a statistical model in its search for a global minimum so-lution and is also capable of exhibiting local searching prop-erties. As is essential, a fast EM solver is employed to designlarge-scale EM structures. The statistical model is derived fromKriging interpolation metamodeling and DIRECT is used as theglobal-optimization algorithm.

278 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 1, JANUARY 2004

A. Kriging Interpolation Metamodeling

Interpolation among the sampled data points can be accom-plished using polynomial fitting or least squares fit. However,for a relatively higher order polynomial, this method exhibits ahighly oscillatory curve-fitting function at some locations be-tween the sampled data points. This may manifest itself if thereis a large number of data points available for fitting. On theother hand, Kriging interpolation functions and neural networksexhibit much less oscillation and have been shown to providebetter fitting in multidimensional domains. Kriging is a specialform of interpolation function that employs the correlation be-tween neighboring points to determine the overall function at anarbitrary point. The concept of utilizing Kriging as interpolationfunctions originated in the 1950s, where it was first used to an-alyze mining data [10]. Consider the following decompositionfor a single dimension:

(1)

where is a random variable on the -parameter.is the interpolated point via Kriging corresponding to the truefunction with denoting the error deviation of thepredicted value from the true function . Polynomialand least squares interpolation function regards as inde-pendent. However, Kriging metamodels consider the errors inthe predicted values as dependent values and are modeled asa zero-mean Gaussian process. With this in mind, for a thdimension problem, (1) can be written as

(2)

where are the basis functions, are the correspondingcoefficients, and is the zero-mean Gaussian-distributederror function that models the deviation from . The covari-ance of the error function is, in turn, modeled as

(3)

(4)

in which is a scale factor known as the process variance thatcan be tuned to fit the given data and is the spatialcorrelation function (SCF). The vector refers to the vectorof given neighboring data points with respect to the vector ,which refers to the stationary data point in the th dimension.The value of in (4) relates to the influence of the surroundingdata points on the predicted point, with larger values indicatinga smaller degree of influence and, thus, a weaker covariancevalue. Finally, the -parameter in (4) determines the continuityof the function and the superscripts in (4) refers to one of the

dimensions in the multidimensional model. The covarianceand SCF increases in complexity with respect to the numberof design variables. The basis functions are chosen tobe an th-order polynomial function and by default are setto a linear function. Before the application of the Krigingalgorithm, the values of , , , and are determined

Kriging model

True function

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9

10

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8

8.5

9

9.5

10

7.5

8.5

9.5

Sample A Sample B

Sampled Point

Fig. 2. Comparison between the Kriging approximation and the true functionfor a given number of sample points. Sample A: Badly fitted Kriging metamodel.Sample B: Well-fitted Kriging metamodel.

from an auxiliary optimization problem where the differencebetween the function values of the predicted and the givendata points is minimized (maximum-likelihood estimation).For this, gradient-based SQP, sequential linear programming(SLP), and any other optimization algorithms can be used.The DIRECT algorithm is employed in our implementation.This process is referred to as “fitting” and is essential forconstructing the Kriging metamodel. An example comparinga Kriging approximated function with that of the true functionfor a specified number of data points for a well and badly fittedmodel is shown in Fig. 2. In the case of sample A, either abetter fit of the Kriging parameters has to be obtained or moredata samples are required. The reader interested in more detailsof Kriging is referred to the literature [8]–[10].

B. DIRECT Algorithm

The DIRECT optimization algorithm is a derivative-freeglobal algorithm that yields a deterministic and unique solution.Its attribute of possessing both local and global properties makeit ideal for fast convergence. An essential aspect of the DIRECTalgorithm is the subdivision of the entire design space intohyper-rectangles or hyper-cubes for multidimensional prob-lems. The iteration starts by choosing the center of the designspace as the starting point. Subsequently, at each iteration step,DIRECT selects and subdivides the set of hyper-cubes thatare most likely to produce the lowest objective function. Thisdecision is based upon the Lipschitzian optimization theory,specifically the manipulation of the Lipschitzian constant.Mathematically, the Lipschitzian constant satisfies therelation

domain

(5)

where and lie within the entire design space andrefers to the objective function for the optimization problem.The Lipschitzian function finds the global minimum point pro-vided the constant is specified to be greater than the largestrate of change of the objective function within the design spaceand that the objective function value is continuous. Within DI-RECT, all possible values of the Lipschitzian constant are

SIAH et al.: FAST PARAMETER OPTIMIZATION OF LARGE-SCALE EM OBJECTS USING DIRECT WITH KRIGING METAMODELING 279

1st Iteration

2nd Iteration

3rd Iteration

4th Iteration

Obje

ctive F

unction V

alu

e

Design Space

Of Variable x

Fig. 3. One-dimensional optimization using the DIRECT routine.

used with the larger values of chosen for global optimiza-tion (to find the basin of convergence of the optimum) followedby smaller values of for local optimizations within this basinof convergence. As mentioned above, DIRECT divides the do-main into multiple rectangles at each iteration. Thus, the conver-gence process is greatly sped up and the optimization algorithmachieves both local and global searching properties.

An illustration of a one-dimensional optimization by DI-RECT is shown in Fig. 3. At the first iteration, DIRECTsamples the center of the design space, subdivides the domaininto two, and samples at the centers of the sub-domains duringthe next iteration. The domain with the lower sampled objectivefunction value is further subdivided and the center points withinthe new sub-domains are further sampled. This is repeated untilthe termination criterion (usually the maximum number ofiterations) has been met. Such a global process of subdividingthe domains and sampling at their centers is mathematicallyguaranteed to obtain the optimum solution in the limit providedthe Lipschitzian constant is chosen to be greater than the largestgradient of the objective function. In choosing from all possiblevalues for this constant, DIRECT has sufficient resolution tocapture the largest change of the objective function gradient toobtain the most optimal point. The multidimensional optimiza-tion process of the DIRECT algorithm can be easily extendedfrom this one-dimensional example. Fig. 4 shows DIRECToptimization in two dimensions. This is summarized by thefollowing steps.

Step 1) Begin at center of the user-supplied bounds of designspace.

Step 2) Divide the design space (into three rectangles inFig. 4).

Step 3) Evaluate the centers of new rectangles.Step 4) Use the Lipschitz constant to select which boxes will

be further divided.Step 5) Go back to Step 2 until the maximum number of

function evaluations is reached or the terminationcriterion has been met.

Fig. 4. Multidimensional optimization using the DIRECT optimizationroutine [13], [14].

t

y

x

FSS element

unity

unitx

y

x

y

x

metal

Fig. 5. Geometry of the slot-array FSS for optimization example 1.

For further information on DIRECT, the reader is referred to theliterature [13], [14].

III. APPLICATION 1—FSS OPTIMIZATION

As a first application, the hybrid DIRECT optimizer (withKriging surrogate modeling) is applied to optimize the size ofthe slot-array FSS to achieve a pre-specified reflection coeffi-cient passband. For this example, a design of experiments iscarried out over the entire design space (evaluated with smallperturbations in the variables) to create the Kriging surrogatemodel. This Kriging model approximation then replaces the an-alyzer code to obtain a good design efficiently. The Kriging sur-rogate model is, in turn, applied to both the DIRECT global op-timizer and to the gradient-based SQP. The analysis code, forthis example, is a well-validated hybrid FE–BI solver algorithmand the overall objective function is defined as

(6)

This objective function is a sum of the 10-dB reflection co-efficients of the FSS both within and outsidethe passband region, where and refer to the weights forthe th in-band and th out-of-band frequency components, re-spectively.

The geometry for the slot-array FSS is given in Fig. 5. Forthis optimization problem, there are four design variables, twoof which pertain to the size of the unit cell for the FSS and

280 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 1, JANUARY 2004

Fig. 6. Residual surface mapping of the Kriging metamodel with twovariables.

two other variables relating to the physical size of the slot. Inaddition, the optimization process must satisfy four inequalityconstraints, which relate to the physical constraints of the slot-array FSS. These are

unit (7)

unit (8)

unit mesh size (9)

unit mesh size (10)

The variables unit and unit relate to the size of the unitcell of the FSS. The thickness of the FSS substrate is fixed at1.0 cm and the size of the unit cell for the FSS is limited to within1.2 cm. The designed 10-dB reflection-coefficient bandwidth ofthe FSS is from 10.7 to 11.3 GHz centered at 11.0 GHz.

Surface mapping of the objective function via Kriging withrespect to the two variables pertaining to the size of the slot(the other variables held constant) is shown in Fig. 6. This isa highly wrinkled surface with the presence of multiple localminima. As can be expected, the presence of the valley of localminima causes gradient-based algorithms to perform poorly. In-deed, when SQP was applied to this Kriging surrogate model, alarge number of local optima were obtained for different startingpoints. This is illustrated in Fig. 7. This figure shows a histogramwhereby SQP is applied to various starting points within the de-sign domain and a history of the occurrences of the same “opti-mized” design point is collected. From this figure, we note thatthe best obtained solution occurs only when the search is startednear the optimum solution, but most of the obtained answers aresuboptimal.

In contrast, when DIRECT is applied to the Kriging surro-gate model, after 112 iterations, the following optimized vari-ables were obtained: cm, cm, unit

cm, and unit cm. To verify the acceptability ofthis design, a final simulation using these optimized parameterswith the hybrid FE–BI algorithm yields the reflection coefficientplot shown in Fig. 8. This indicates a 10 dB or less return lossfrom 10.65 to 11.33 GHz centered at 11 GHz. Clearly, this per-formance is close to the predefined return-loss bandwidth andcenter frequency stated earlier.

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

2

4

6

8

10

12

14

16

18

20Weight Factor = 1.0

Value of the Objective Function

Num

ber

of O

ccura

nces

Fig. 7. Histogram showing the variety of different optimized points withdifferent starting points characterized by gradient-based SQP algorithms.

10 10.2 10.4 10.6 10.8 11 11.2 12-22

-20

-18

-16

-14

-12

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-8

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-2

Frequency in GHz

Re

fle

ctio

n c

oe

ffic

ien

t in

dB

11.4 11.6 11.8

Fig. 8. Plot of the reflection coefficient of the final optimized FSS.

IV. SUPEREGO HYBRID OPTIMIZER

The concept of creating a design of experiments [20] on theentire design space of the optimization problem, as done withthe previous example, may be inefficient since it maps both po-tentially good, as well as bad domains within the design spaceexhaustively. For the implementation of the hybrid superEGOoptimizer, an initial sparse sample is used to map the designspace for creating and fitting the Kriging metamodel. To ensurethat only the more promising design domains are searched, oneapproach is to use the information of the current iterate to updatethe Kriging metamodel. In this manner, it endows the Krigingmetamodel to have an adaptively improving characteristic and,thus, reduce the number of iterations required before conver-gence can be found. The program flow of this improved hybridoptimization algorithm is shown in Fig. 9 and is referred to asthe superEGO hybrid optimizer.

The flexibility of the hybrid optimizer is further improved bydefining the ISC. The ISC determines which location in the de-sign space to investigate at each iteration. In the previous ver-sion, the hybrid optimizer typically optimizes for the minimum

SIAH et al.: FAST PARAMETER OPTIMIZATION OF LARGE-SCALE EM OBJECTS USING DIRECT WITH KRIGING METAMODELING 281

Take an initial

Sample

Choose appropriate

kriging coefficients

Create Infill

Sampling Criteria

Optimize criteria

using DIRECT

Termination

Criteria Met?Stop

No

Up date

Kriging

Models Simulate results

at new point

Fit Kriging

Models

Yes

Fig. 9. Flowchart of the improved hybrid optimizer: superEGO.

objective function. Here, we will define two different samplingcriteria: the regional extreme sampling criteria and the minimumobjective function criteria. The regional extreme criterion [21]is mathematically defined as

(11)

In (11), is defined as the cumulative distribution function andrefers to the probability distribution function of the Kriging

model shown in (3) and (4). Also, is the current minimumobjective function value, refers to the predicted value of theobjective function, and refers to the variance in the Krigingmodel. For this infill sampling criterion, the hybrid superEGOoptimizer minimizes both objective function values, as well asthe uncertainty in the Kriging model, giving a user-defined em-phasis on the local searching properties in addition to the localproperties of the DIRECT algorithm. The minimum objectivefunction sampling criterion simply allows DIRECT to searchfor the minimum of the Kriging model approximation withoututilizing the statistical property of the Kriging metamodel.

In essence, the superEGO hybrid optimizer starts off with avery sparse sampling of the design domain and fits this modelto derive the Kriging metamodel using the DIRECT global op-timizer. It then proceeds to solve an auxiliary problem basedon the optimization of the chosen ISC. The relationship be-tween the Kriging metamodel and ISC parameters is displayedin Fig. 10. In this instance, DIRECT is used (within this aux-iliary optimization) to predict the next iterate. This is, in turn,used by the fast analyzer code to carry out an expensive com-putational evaluation of the objective function. At the end ofeach optimization iteration, the predicted point is used to updatethe Kriging metamodel. This continuous update of the Krigingmetamodels at every iteration adaptively improves the Krigingmetamodel for fast convergence. The process is summarized asfollows (also refer to Fig. 10).

Step 1) Fit Kriging model to the given data sample.Step 2) Locate optimum of ISC.Step 3) Add point from Step 2 to data sample.Step 4) Go back to Step 1 until convergence is achieved.

V. APPLICATION 2—ANTENNA POSITION OPTIMIZATION

As an example of the hybrid superEGO optimizer, we opti-mize the antenna location on the rear of an automobile subject

true function

kriging modelsample point

ISC

Fig. 10. Diagram showing the relation between the predicted Kriging modeland the ISC used in the hybrid superEGO optimizer.

VLSI Chip with 40 pins

The Location of the crossed

magnetic dipoles is optimized

within a volume of points

located at the rear of the car.

Fig. 11. Geometry of the antenna location optimization problem for minimalEM coupling from the source antenna to 40 pins on a very large scale integration(VLSI) chip (antenna location search region is the encircled volume in thisfigure).

to minimal EM coupling at the 40 pins around the peripheralchip circumference located within a resonant enclosed cavityat 700 MHz, as shown in Fig. 11. The antenna field consistsof a pair of crossed magnetic slots with orthogonal phase ex-citation to generate a circularly polarized field at a frequencycorresponding to the cavity resonance. Consequently, a signifi-cant amplification of the incident antenna field within the cavitymay occur. The overall objective function is defined as the ratioof the equally weighted sum of the total fields at the 40 pins tothe incident field at the same 40 locations. Specifically,

(12)

where refers to the total field measured in the presenceof the automobile and cavity at the th pin location, whilerefers to the incident field in the absence of these structures atthe th pin location.

For this optimization problem, there are three variables andsix inequality constraints pertaining to planes confining the spa-tial volume at the rear of the automobile (which defines the de-sign domain for the problem). The analyzer code used for thisproblem employs the MLFMM with curvilinear basis functions

282 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 1, JANUARY 2004

Fig. 12. Diagram showing the mesh of the automobile and the initial Kriging plots with variation in two with the third variable held constant in the middle of thesearch domain.

-50

0

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jective

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nctio

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alu

e

Value along x axis Value along x axisValue along y axis

Objective Function Values Covariance of the SCF

Fig. 13. Final Kriging obje ctive function and covariance after optimization (starting with the surfaces in Fig. 12).

[15]–[17] over the method of moments. The automobile is mod-eled with curvilinear biquadratic elements to reduce geometryerror and problem size. An initial mesh of the automobile withthe former elements is shown in Fig. 12. The automobile modelhas approximately 36 000 unknowns solved in approximately2 h on a Silicon Graphic Inc. (SGI) computer platform. Due tohigh computational expense, it was necessary to use functionevaluations very judiciously. Hence, superEGO was much bettersuited to this problem than SQP, a GA, or SA. SuperEGO isstarted with a very sparse initial sample of 18 points located ran-domly within the design domain. The objective function valueand the SCF covariance of the Kriging metamodel for the initial18 sampled points over the – -plane with at the middle ofthe design volume is shown in Fig. 12. Again, surface mappingshows multiple local minima making gradient-based algorithms

unsuitable. The final plots of the Kriging metamodel after con-vergence are given in Fig. 13.

The convergence history for the hybrid optimizer is shownin Fig. 14 and it can be seen that this optimizer achieved con-vergence within 30 iterations. During the first segment of theoptimization history, the regional extreme sampling criterionis used (within the auxiliary optimization problem). As canbe seen from Fig. 14, this additional local search property cancause the optimizer to be trapped within a local minimum point.Changing the sampling criterion to the minimum objectivefunction (so that the hybrid optimizer has a reduced emphasison local searching) resulted in the hybrid optimizer searchingthrough other local minimum points. Thus, convergence isachieved within tens of iterations. This is in contrast to the GAand SA, which may take hundreds of iterations to converge.

SIAH et al.: FAST PARAMETER OPTIMIZATION OF LARGE-SCALE EM OBJECTS USING DIRECT WITH KRIGING METAMODELING 283

Fig. 14. Convergence curve of the hybrid optimization algorithm.

Fig. 15. Geometry of the harness-antenna coupling reduction problem.

The final objective function value iscorresponding to the antenna position mm,

mm, and mm. We remark that thevalue of the objective function for the antenna located at thecenter of the design space is 13.3025. Thus, the hybrid optimizerhas yielded a satisfactory solution that reduced coupling by asmuch as 20.37 dB, as compared to the antenna located at thecenter of the vehicle. Moreover, the optimal solution for thisoptimizer is a deterministic answer. Further, we remark that theoptimal locations correspond to an objective function behavior(see Fig. 13) whose derivatives are rather small. Consequently,small changes to the antenna locations would results in littlechange in the value of the objective function and the solutionis robust.

VI. APPLICATION 3—OPTIMIZING

ANTENNA-HARNESS COUPLING

In the third example, we optimize the coupling from a wireharness located within an automobile onto a printed antenna atthe rear of an automobile to within a certain range of values.Constraints are imposed on four sources (sensors) located at theend of the wire harness. Fig. 15 shows the computer-aided de-sign (CAD) model of the automobile in the presence of the wireharness. For our analysis, a pseudoharness was placed just abovethe floor of the car as shown. Each port on the harness is in-dependently driven by a sensor. Here, the goal is to optimizethe complex amplitude of each sensor output voltage so that theresulting average field magnitude along the length of the FMantenna (printed on the back glass) receives a maximum fieldintensity on the order of 9.7–9.8 V/m. The problem has four

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of Iterations

Obj

ectiv

e F

unct

ion

Fig. 16. Convergence data for the harness-antenna coupling reductionoptimization problem.

variables (the sensor output voltages) and four inequality con-straints, i.e., V V and V V.The objective function for this problem is given by

(13)

where are the complex electric field amplitudes at the thsample point on the antenna location and .

The hybrid superEGO optimizer in the previous examplewas used here as well. The corresponding convergence historyis shown in Fig. 16. The optimization converged within 20 iter-ations yielding a good solution. The resulting optimal drivingvoltages are V V, V V,V V, and V V. Forthis example, we did not consider harness relocation that couldcause additional parameters within the optimization loop. Asexpected, the sensors at locations 1 and 2 must keep theiroutput voltages to low values since they are exposed towardthe antenna (even though they are physically further). Sensors3 and 4 are allowed to have greater output voltage values sincethey have a lesser influence on the field values at the printedantenna location. Part of this optimization exercise is to give

284 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 1, JANUARY 2004

some guidance on the various maximum sensor outputs and,thus, avoid excessive interference by these sensors on theprinted antenna.

VII. CONCLUSIONS

We have demonstrated that the recently proposed hybrid op-timizer, with DIRECT global optimizers in conjunction withKriging metamodeling (superEGO) and surrogate modeling, iscapable of performing rapidly converging global optimization.This alleviates the slow convergence experienced with otherglobal optimizers like GAs and SA. The new superEGO hy-brid optimizer has the additional flexibility of allowing the userto change the emphasis of local searching upon the backdropof global searching within the design space. This hybrid opti-mizer was applied in conjunction with general purpose tools for:1) shape optimization of a slot-array FSS subject to a predefinedreflection coefficient bandwidth; 2) antenna location optimiza-tion to minimize EM coupling to a device located within an au-tomobile; and 3) optimization of multisensor voltages subjectto a specified field coupling criteria between the wire harnessand antenna on an automobile. Of particular importance wasthe speed of the hybrid optimizer. The demonstration with threecomplex EM examples showed that convergence occurs withintens of iterations and yields a deterministic solution. These char-acteristics make the new hybrid optimizer ideal for large-scalecomplex EM problems.

ACKNOWLEDGMENT

The authors would like to acknowledge Dr. T. Ozdemir forhis contribution in optimization application 3.

REFERENCES

[1] Z. Li, Y. E. Erdemli, J. L. Volakis, and P. Y. Papalambros, “Design op-timization of conformal antennas by integrating stochastic algorithmswith the hybrid finite element method,” IEEE Trans. Antennas Prop-agat., vol. 50, pp. 676–684, May 2002.

[2] S. Chakravarty, R. Mittra, and N. R. Williams, “On the application ofthe microgenetic algorithms to the design of broad-band microwave ab-sorbers comprising frequency selective surfaces embedded in multilayerdielectric media,” IEEE Trans. Microwave Theory Tech., vol. 49, pp.1050–1059, June 2001.

[3] J. M. Johnson and Y. Rahmat-Samii, “Genetic algorithms in engineeringelectromagnetics,” IEEE Antenna Propagat. Mag., vol. 39, pp. 7–25,Aug. 1997.

[4] M. Smith, Neural Nets for Statistical Modeling. New York: Van Nos-trand, 1993.

[5] P. Y. Papalambros and D. J. Wilde, Principles of Optimal Design: Mod-eling and Computation, 2nd ed. New York: Cambridge Univ. Press,2000.

[6] E. Aarts and J. Korst, Simulated Annealing and Boltzman Machines: AStochastic Approach to Combinational Optimization and Neural Com-puting. New York: Wiley, 1989.

[7] T. J. Ross, Fuzzy Logic With Engineering Applications. New York:McGraw-Hill, 1995.

[8] N. Cressie, “Spatial prediction and ordinary Kriging,” Math. Geol., vol.40, no. 4, pp. 405–421, 1997.

[9] P. Goovaerts, Geostatistics for Natural Resources Evaluation. NewYork: Oxford Univ. Press, 1997.

[10] D. G. Krige, “A statistical approach to some mine evaluations and alliedproblems at the Witwatersrand,” M.S. thesis, Univ. Witwatersrand, Wit-watersrand, Germany, 1951.

[11] M. J. Sasena, P. Y. Papalambros, and P. Goovaerts, “Metamodeling sam-pling criteria in a global optimization framework,” presented at the 8thAIAA/NASA/USAF/ISSMO Multidisciplinary Analysis and Optimiza-tion Symp., 2000.

[12] B. Betro, “Bayesian methods in global optimizations,” J. Global Opti-mization, vol. 1, no. 1, pp. 1–14, 1991.

[13] D. R. Jones, C. D. Perttunen, and B. E. Stuckman, “Lipschitzian opti-mization without the Lipschitz constant,” J. Optim. Theory Appl., vol.79, no. 1, pp. 157–181, Oct. 1993.

[14] D. R. Jones, “The DIRECT global optimization algorithm,” in Encyclo-pedia of Optimization. Norwell, MA: Kluwer, 2001, pp. 431–440.

[15] E. S. Siah, K. Sertel, J. L. Volakis, V. V. Liepa, and R. W. Wiese, “Cou-pling studies and shielding techniques for electromagnetic penetrationthrough apertures on complex cavities and vehicular platforms,” IEEETrans. Electromagn. Compat., vol. 45, pp. 245–257, May 2003.

[16] K. Sertel and J. L. Volakis, “Multilevel fast multipole method implemen-tation using parametric surface modeling,” in IEEE AP-S Conf. Dig., vol.4, CITY, UT, 2000, pp. 1852–1855.

[17] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole methodfor the wave equation: a pedestrian prescription,” IEEE Antennas Prop-agat. Mag., vol. 35, pp. 7–12, June 1993.

[18] T. F. Eibert, K. Sertel, and J. L. Volakis, “Hybrid finite element mod-eling of conformal antenna and array structures utilizing fast integralmethods,” Int. J. Numer. Modeling, vol. 13, pp. 2–3, Mar.–June 2000.

[19] M. A. Morgan, C. H. Chen, S. C. Hill, and P. W. Barber, “Finite el-ement-boundary integral formulation for electromagnetic scattering,”Wave Motion, vol. 6, pp. 91–103, 1984.

[20] J. Sacks, W. J. Welch, W. J. Mitchell, and H. P. Wynn, “Design andanalysis of computer experiments,” Stat. Sci., vol. 4, no. 4, pp. 409–435,1989.

[21] A. G. Watson and R. J. Barnes, “Infill sampling criteria to locate ex-tremes,” Math. Geol., vol. 27, no. 5, pp. 589–608, 1995.

Eng Swee Siah (S’99) was born on August 12, 1974, in Singapore. He receivedthe B.Eng. degree (with first-class honors) from the National University of Sin-gapore, Singapore, in 1999, the M.Sc. degree in electrical engineering from TheUniversity of Michigan at Ann Arbor, in 2002, and is currently working towardthe Ph.D. degree in electrical engineering and computer science at The Univer-sity of Michigan at Ann Arbor.

He is currently a Research Assistant with the Electrical Engineering and Com-puter Science Department, The University of Michigan at Ann Arbor. FromOctober 1999 to July 2000, he participated in the graduate program in com-munications engineering and signal processing at the Technical University ofMunich, Munich, Germany. His research interests include EM theory, compu-tational electromagnetics, fast and hybrid EM methods, EM compatibility andinterference, antenna design and analysis, and EM optimization.

Mr. Siah was the recipient of the 2003 Best Symposium Paper Award pre-sented at the IEEE International Symposium on Electromagnetic Compatibility,Istanbul, Turkey.

Micheal Sasena received the Undergraduate degreein mechanical engineering from the Universityof Notre Dame, Notre Dame, IN, in 1996, andthe M.S. and Ph.D. degrees from The Universityof Michigan at Ann Arbor, in 1998 and 2002,respectively, where he studied design optimizationin the Mechanical Engineering Department and theCivil and Environmental Engineering Department.

He is currently with Emmeskay Inc. (a consultingcompany), Plymouth, MI.

SIAH et al.: FAST PARAMETER OPTIMIZATION OF LARGE-SCALE EM OBJECTS USING DIRECT WITH KRIGING METAMODELING 285

John L. Volakis (S’77–A’79–M’82–SM’88–F’96)was born on May 13, 1956, in Chios, Greece. Hereceived the B.E. degree (summa cum laude) fromYoungstown State University, Youngstown, OH, in1978, and the M.Sc. and Ph.D. degrees from theOhio State University, Columbus, in 1979 and 1982,respectively.

From 1982 to 1984, he was with the Aircraft Divi-sion, Rockwell International, Lakewood, CA. From1978 to 1982, he was a Graduate Research Associatewith the ElectroScience Laboratory, Ohio State Uni-

versity. Since 1984, he has been a Professor with the Electrical Engineering andComputer Science Department, The University of Michigan at Ann Arbor, MI.From 1998 to 2000, he also served as the Director of the Radiation Laboratory.Since January 2003, he has been the Roy and Lois Chope Chair Professor ofEngineering at the Ohio State University, and also serves as the Director of theElectroScience Laboratory. He has authored or coauthored over 200 papers inmajor refereed journals (nine of these have appeared in reprint volumes), over240 conference papers, and nine book chapters. In addition, he coauthored Ap-proximate Boundary Conditions in Electromagnetics (London, U.K.: IEE, 1995)and Finite Element Method for Electromagnetics(Piscataway, NJ: IEEE Press,1998). His primary research deals with computational methods, EM compati-bility and interference, design of new RF materials, multiphysics engineering,and bioelectromagnetics. From 1994 to 1997, he was an Associate Editor ofRadio Science. He currently serves as Associate Editor for the J. Electromag-netic Waves and Applications and the URSI Bulletin. He is listed in severalWho’s Who directories, including Who’s Who in America.

Dr. Volakis is a member of Sigma Xi, Tau Beta Pi, Phi Kappa Phi, and Com-mission B of the International Scientific Radio Union (URSI). He served as anAssociate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION

from 1988 to 1992. He chaired the 1993 IEEE Antennas and Propagation So-ciety (IEEE AP-S) Symposium and Radio Science Meeting and was a memberof the IEEE AP-S Administrative Committee (AdCom) from 1995 to 1998. Hecurrently serves as the President-Elect of the IEEE AP-S. He serves as associateeditor for the IEEE Antennas and Propagation Society Magazine. He was therecipient of the 1998 University of Michigan College of Engineering ResearchExcellence Award and the 2001 Department of Electrical Engineering and Com-puter Science Service Excellence Award.

Panos Y. Papalambros received the Diploma degreefrom the National Technical University of Athens,Athens, Greece, in 1974, and the M.S. and Ph.D.degrees from Stanford University, Stanford, CA, in1976 and 1979, respectively.

He is currently the Donald C. Graham Professorof Engineering and a Professor of mechanicalengineering at The University of Michigan at AnnArbor. He coauthored the textbook Principles ofOptimal Design: Modeling and Computation (NewYork: Cambridge Univ. Press, 1988, 2000). His

research interests include design methods and systems optimization withapplications to product design and automotive systems. He serves on theEditorial Boards of the Journal of Artificial Intelligence in Engineering Designand Manufacturing, Journal of Engineering Design, Journal of EngineeringOptimization, Journal of Computer-Integrated Engineering, Journal ofStructural and Multidisciplinary Optimization, and the International Journalof Engineering Simulation.

Dr. Papalambros is a Fellow of the American Society of Mechanical Engi-neers (ASME). He was the recipient of the 1998 ASME Design AutomationAward and the 1999 ASME Machine Design Award.

Rich W. Wiese was born on February 11, 1957. Hereceived the B.S.E.E. degree from The University ofMichigan at Ann Arbor, in 1979, and the M.S.E.S. de-gree from the Rensselaer Polytechnic Institute, Troy,NY, in 2001.

Since 1980, he has been with the General MotorsCorporation, Milford, MI, where he is currently theSenior Development Engineer of the Electromag-netic Compatibility, RF Systems Integration, and theTelematics Division. His interests include EMI/EMCand automotive RF systems integration.