Reduced-order modeling of multiscreen frequency-selective ...€¦ · IEEE TRANSACTIONS ON ANTENNAS...

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 5, MAY 2001 801

Reduced-Order Modeling of MultiscreenFrequency-Selective Surfaces Using Krylov-Based

Rational InterpolationDaniel S. Weile, Member, IEEE, Eric Michielssen, Senior Member, IEEE, and Kyle Gallivan

Abstract—A method is presented for generating a broad-bandrational interpolant approximation of the reflection coefficient ofmultiple-screen frequency-selective surfaces (FSSs). The techniqueis structured around a linearization of the system provided by aspectral domain moment method-based analysis of the FSS, fol-lowed by a model-order reduction of the linearized system usingthe dual rational Arnoldi method. This process creates a rationalinterpolant of the linearized system that matches its transfer func-tion and its derivatives at several expansion points in the Laplacedomain. Numerical results indicate that a reduced-order modelwith a system matrix of dimension less than20 20 can accuratelyreproduce the broad-band behavior of multiscreen FSSs originallymodeled with several hundreds or thousands of unknowns.

Index Terms—Frequency selective surfaces, model-order reduc-tion, spectral Galerkin methods.

I. INTRODUCTION

M ULTISCREEN frequency-selective surfaces (FSSs) areused as frequency and angular filters or polarizers over a

broad band of the electromagnetic spectrum [1]–[7]. Composedof thin sheets of periodic, perfectly conducting planar patchesor apertures in a thin perfectly conducting sheet, FSSs are oftendesigned to reflect or transmit plane waves at the resonancesof the patches or apertures. This study outlines a fast methodfor approximating the plane-wave reflection coefficient of mul-tiscreen FSSs over a large band of frequencies.

The most popular method for analyzing plane-wave reflec-tion from FSSs is the method of moments (MoM) [8] or,more specifically, its spectral Galerkin variant [1], [2]. In itsmost general incarnation, this method involves expansion of thescreen currents in terms of subdomain basis functions, thoughfull-domain basis functions are available for a large variety ofcommonly used simple elements [1], [2]. Application of thistechnique results in a linear system of equations for the basis

Manuscript received March 15, 1998; revised February 5, 1999.D. S. Weile was with the Department of Electrical and Computer Engineering,

Center for Computational Electromagnetics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. He is now with the Department of Elec-trical and Computer Engineering, University of Delaware, Newark, DE 19716USA (e-mail: [email protected]).

E. Michielssen is with the Department of Electrical and Computer Engi-neering, Center for Computational Electromagnetics, University of Illinois atUrbana-Champaign, Urbana, IL 61801 USA.

K. Gallivan is with Computational Science and Engineering Program andComputer Science Department, Florida State University, Tallahassee, FL 32306USA.

Publisher Item Identifier S 0018-926X(01)03582-7.

function weighting parameters and a linear relationship be-tween these parameters and the FSS plane-wave reflection co-efficient. For most FSS scattering problems, hundreds or thou-sands of subdomain basis functions must be used to model thescreen currents accurately.

Because of the summations involved and the number of basesrequired, the application of the spectral Galerkin technique tothe analysis of FSSs can be rather tedious, especially whenbroad-band information is required. This heavy computationalload arises from a variety of sources. First, the construction ofthe MoM matrix requires the summation of slowly convergentinfinite series, which must be recomputed for each frequencyand incidence angle of interest. Second, because the MoMsystem is usually large, the matrix inversions required toreconstruct the screen currents are costly. Third, FSS reflectioncharacteristics may exhibit very narrow-band resonances thatcan only be discovered with a very dense frequency sweep,necessitating the construction and inversion of the MoM matrixat hundreds of different frequencies.

This paper introduces a rational Krylov model order re-duction (MOR) technique that accelerates the broad-bandcharacterization of FSS plane-wave reflection coefficients.Specifically, the method reduces the previously mentionedMoM system to a small state-space model with fewer thantwenty degrees of freedom. To arrive at a model of this type,the MoM matrix is first approximated by a Hermite PolynomialInterpolant (HPI) [9], [10], so that the system can be expressedin a companion form with a linear frequency dependence. Thisstep both facilitates the MOR and accelerates the constructionof the matrix. The reduced-order model is constructed tomatch both the reflection coefficient of the HPI model and itsderivatives (moments) at selected Laplace domain interpolationpoints [9]–[13]. The method presented here will be seen toremedy all three contributions to the heavy computational loaddescribed above.

1) The explicit construction of the spectral Galerkin matrixat each frequency is eschewed by the introduction of theHPI approximation and the subsequent construction ofthe reduced order model.

2) The computational cost inherent in the inversion of thelarge MoM matrices needed to characterize the reflectioncoefficient over a broad frequency band is all but avoidedas inversion of these matrices is only required at a small

0018–926X/01$10.00 © 2001 IEEE

802 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 5, MAY 2001

number of frequency points to create the reduced-ordermodel.

3) The computational cost associated with the required fre-quency sweep is made negligible due to the low cost ofconstructing and inverting small matrices.

In spite of the approximations introduced by the HPI and theMOR in the frequency response calculation, the reduced-ordersystem will be seen to reproduce faithfully the behavior of theFSS, including narrow-band resonances. While the examplesshown here are limited to subdomain models of FSSs residing onrectangular grids, the technique is fully applicable toanyMoMsystem, without regard to the bases used or the orthogonality ofthe axes of periodicity (see Section V).

It is emphasized that MOR methods based on momentmatching are not new to electromagnetics. Asymptotic wave-form expansion and Padé via Lanczos techniques have beenused quite successfully for a number of years in circuit mod-eling [14]–[16], and such methods have also been employedto construct reduced-order models of interconnect structures.This study differs from these previous works in several ways.First, unlike [15], the method described here relies on HPI’s inplace of Taylor series expansions for linearizing the frequencydependence of the MoM matrix. This not only leads to a morebroad-band linearization of the system, but also avoids thecomputation of derivatives of very high order, which can be ex-tremely expensive or even impossible. Second, unlike previousKrylov-based MOR procedures used in electromagnetics, theproposed technique results in a rational interpolant that matchesthe system transfer function and its derivatives at several pointsin the band of interest, not just at one point as do modelsresulting from a Padé approximation. Third, the right-handside of the matrix equation provided by the spectral Galerkinmethod varies with frequency. This frequency-dependent,right-hand side results from the peculiar periodic nature of theFSS, thus, it does not occur in circuit modeling. Fourth, tothe authors’ knowledge, this study is the first to apply modelreduction to the characterization of FSSs.

II. FORMULATION

This section outlines the application of a Krylov-basedMOR technique to the characterization of multiscreen FSSs.Section II-A reviews the spectral Galerkin method as appliedto the analysis of plane-wave reflection from an FSS, demon-strating the construction of the unreduced MoM system. Onlysingle-screen FSSs are considered in this section as their anal-ysis contains all elements relevant to the MOR of multiscreenFSSs. Section II-B then details a method for converting theequations derived in Section II-A to the form of a canonicallinear system to which Krylov-based MOR methods can beapplied. Finally, Section II-C discusses the actual MOR of thiscanonical system to a small state-space model.

A. Spectral Galerkin Analysis of FSSs

This section describes the use of the spectral Galerkin methodin the characterization of plane-wave reflection from single-screen FSSs. While this topic has been extensively covered in

the literature, it is presented here to highlight its salient featureswith respect to MOR because it results in the large system men-tioned in the introduction. Assuming that the screen current isexpanded with basis functions, this system takes the form

(1)

whereLaplace frequency parameter;

matrix;, -vectors related to the incident and reflected

waves’ direction and polarization;-vector of basis function weighing parame-

ters;screen reflection coefficient;

* denotes the complex conjugate transpose oper-ator.

Consider a single-screen version of the FSS structure de-picted in Fig. 1, situated in the plane and infinite andperiodic along the transverse directionsand with periodici-ties and , respectively. Let this screen be excited by anelectric field with transverse components given by

(2)

where , ,, and is the angular wavenumber of the

incident wave. In the above,and are the angles of incidenceof the wave, and the wave amplitude is given by

if TE

if TM

(3)

where TE TM indicates the polarization of the wave.This incident wave induces currents on the metallized portionsof the screens, which may be expressed as [1], [2]

(4)

where is the current on a singlecell of the FSS (called the primary cell)

(5)

and

(6)

WEILE et al.: REDUCED-ORDER MODELING USING KRYLOV-BASED RATIONAL INTERPOLATION 803

Fig. 1. Three-screen FSS with a typical mesh inset.

In (4)–(6), quantities without tildes are the spectral domaincounterparts of same symbol with a tilde, connected by theFourier relations

(7)

and the symbols and represent and , subject to the caveatthat . The rest of the work also adheres to these conven-tions. Imposing the condition that the total tangential electricfield vanishes on the conducting portions of the screen yields

(8)

for all position vectors on the conductors.The MoM is used to solve (8). The current is approximated

as a summation over a finite number of basis functions as

(9)

Inserting this approximation into (9) and taking the innerproduct of the result with each basis function resultsin the set of equations

(10)

for , and where the asterisk denotes the com-plex conjugate transpose. This set of equations may be solvedfor the unknown expansion coefficients. Once obtained, thesecoefficients may be used to solve for the fundamental mode FSSreflection coefficient as

(11)

Equations (10) and (11) together form the system (1) where, TE TM indicates the polarization of the desired

reflection coefficient.While not a necessity of the MOR method itself, all of the

results presented in Section III are presented in terms of a dis-cretization of the FSS into- and -directed rooftop basis func-tions lying on a uniform grid (Fig. 1). Such a dis-


cretization allows the matrices and vectors in (1) to be writtenas [1], [2]

(12)

In (12), F is the two-dimensional (2-D) fast Fourier transform(FFT) operator,u is a vector containing all ones, and

. The matricesP, , and are diag-onal with elements

(13)

where ,, and . Expressions (12) are

written to include the contribution from all possible bases in thegrid; if certain bases are not present, the rows and columns cor-responding to these bases are simply deleted. In this form, theMoM equations can be solved for the current using an iterativemethod and the FFT in matrix multiplications, which saves com-puter time and memory. The rest of this study is based on thisregular discretization, then, for two reasons.

• Sections II-B and II-C demonstrates that the transforma-tions of the system required by the MOR method do notalter the Toeplitz-block-Toeplitz-like nature of the matrixthe makes the FFT matrix multiplication possible.

• Section III shows that even with the acceleration of thereflection coefficient characterization provided by a for-tunate discretization of the cell, the MOR method remainsfaster than a straightforward analysis.

Thus, the fact that this study is based on (13) should not bemisconstrued to imply that the method is based on them. Onthe contrary, the regular discretization of the FSS presents someunique challenges for the MOR of FSS systems, which can allbe overcome with the methods presented in the next section.

B. Conversion of FSS Equations to State-Space Form

Krylov-based MOR methods are based on matching thederivatives of the transfer functions of linear, time-invariantsystems, which are expressed in a Laplace domain canonicalform as

(14)

The system described in (1) is not in this form for two rea-sons. First, the vectors and are not frequency in-dependent, and second, the matrix is not a linear func-tion of frequency. A method for rectifying this situation andconverting (1) to (14) is presented in this subsection. Becausethis transformation involves approximating matrix with anHPI of order and writing the approximation in companion

form, it will be seen that matrices and are of dimension, and the vectors , , and have length

.First, the input and output vectors of (1) must be made fre-

quency-independent. This can be accomplished by multiplyingit on the left by

(15)

and redefining the state variables with the transformation

(16)

The resulting system can be expressed as

(17)

with

(18)

where and.

The system described in (17) now has frequency-indepen-dent input and output vectors. To obtain a system that linearlydepends on frequency, an HPI approximation to is con-structed using a generalization of Newton’s divided differenceformula [9], [10]. Actually, this method subsumes the lineariza-tion method of Elfadelet al. [15] as a special case. Elfadeletal. used a Taylor series followed by a companion form reor-ganization of the matrix to linearize the frequency dependenceof MoM systems in interconnect modeling. The current method


reduces to that of [15] if all the interpolation points are collo-cated. Our motivations for using an interpolant function ratherthan a Taylor series are twofold. First, an HPI is not as localizedan approximant to a function as is its Taylor series, and second,very high-order derivatives of the functions appearing as entries

in are expensive to compute. Furthermore, because bothHPI and rational interpolant MOR schemes use multiple inter-polation points, an HPI is more consistent with rational interpo-lation than a Taylor series. If the interpolation points are locatedat where the are not neces-sarily distinct, the HPI generated by Newton’s method takes theform [9], [10]

(19)

Notice that the interpolation of as a function of frequencyonly involves the interpolation of diagonal matrices and

, and that each matrix can be multiplied with a vectorusing the FFT.

Of course, the use of the FFT and the need to only interpolatediagonal matrices depends on the regular, orthogonal discretiza-tion of the cell. Nonetheless, the interpolation can be carried outfor an arbitrary MoM matrix. In the general case, the process ofinterpolation is more expensive then that shown here becausethe frequency dependence may be different for each matrix ele-ment. However, it must be remembered that the cost of the con-struction of the matrix at each frequency in the general case isincreased by the same factor as is the interpolation process (seeSection V). Moreover, the model reduction algorithm describedin the next section is not affected by the structure of the MoMmatrix.

Once constructed, the matrix polynomial (19) can be con-verted into a polynomial expanded about a single pointvia inverse synthetic division [9], resulting in the system

(20)

where is chosen somewhere in the middle ofthe band of interest to enhance the accuracy of the coefficients

of the resulting polynomial, is the HPI approximation to ,

and the matrices are the coefficients of the HPI expandedabout . Defining

(21)

the system of (31) may be cast into form shown in (25) wherethe identity matrix is written as

...

... (22)

and

...

...

...(23)

The problem is now formulated in the form of (14) where. This frequency shift is ignored in the next section for

simplicity.

C. Krylov-Based Model-Order Reduction

Now that the system has been put in the form of (14), themethod for generating a rational interpolant for the system isdescribed. Specifically, rectangular matricesand of dimen-sion , are sought such that a reduced-order modelof (25) can be found in the form

(24)

where

(25)

at selected interpolation points forderivatives of order . The above expres-sions indicate that the reduced-order model hasdegrees offreedom. The examples of the next section will demonstrate thatan accurate model can typically be produced for .

The and needed to generate the above described inter-polant can be characterized with the following theorem [13]:

Let be the Krylovsubspace of dimensiongenerated by the square matrixandthe vector , and let . Furthermore, denote the


Fig. 2. Dual rational Arnoldi algorithm.

space spanned by the columns of an arbitrary rectangular matrixas colsp( ). Then, if

(26)

and

(27)

then

(28)

for and , where ,

, , and .The quantities on the left-hand side of (28) are the system

moments at interpolation points , and those on theright-hand side of (28) are the corresponding quantities forthe reduced-order system. Thus, the theorem states that areduced-order model with and as defined in (26) and (27)will match moments of the original system at the .

An algorithm is given in Fig. 2 for generating suchandfor the special case where for all so that (25)holds. This method, known as dual rational Arnoldi (DRA) [13],

Fig. 3. Meshed Jerusalem cross FSS.

also orthogonalizes and as they are generated to ensure thestability of the reduced-order model.

A few comments are in order about the application of theDRA algorithm to the FSS problem. The companion matricesformed in Section II-B can be quite large, but they are struc-tured (whether or not the FFT matrix multiplication methodis used). Thus, these matrices should not directly be used for

computing the quantities of the form and

required by the DRA algorithm for arbitrary

vectors and . Instead, the structure of the matrixfor a given can be exploited by partitioning, , and


Fig. 4. Comparison of MoM reflection coefficient to that produced by (a) polynomial interpolant, (b) second-order model, (c) fourth-order model, and (d) sixth-order model.

as , ,, and and then

noting that and may be found from

(29)

(30)

(31)

(32)

and

(33)

Note that, since the matrices in (29) and (31) are conjugate trans-poses of one another, and should be found concur-rently using an algorithm that can solve for both simultaneously.A BiCG-like algorithm is used to accomplish this. Moreover, ifthe FSS is discretized in a regular fashion (as described in Sec-

tion II-A), the matrices need not be stored in their entirety.

III. N UMERICAL RESULTS

The algorithm described above was applied to several dif-ferent test cases to demonstrate its speed and accuracy. In allof the following examples, a square periodic cell ( )is assumed. Furthermore, all results are computed in terms of a


TABLE ITIMING AND ERRORCONVERGENCE OFRATIONAL INTERPOLANT APPROXIMATIONS TO AJERUSALEM CROSSFSS

Fig. 5. Six different meshed FSS screens to be used in dual-screen FSSs.

normalized frequency related to the Laplace domain fre-quency parameter by

(34)

and all other FSS dimensions are given as normalized lengthsrelated to the true lengthas

(35)

Because all results presented here are normalized, the subscript“norm” is dropped in subsequent expressions. Finally, reflectioncoefficients are calculated assuming an incident TM (to) planewave.

The algorithm was first applied to the computation of thereflection coefficient of the single-screen Jerusalem cross FSSwith unit cell shown in Fig. 3, between normalized frequencies

and . The incident plane wave traveled

with and . The HPI was constructed using theChebyshev interpolation points

(36)

for , and . In this and all sub-sequent examples, frequency equals , as this seems toimprove the result. Two interpolation points ( ),

and were used in the MOR. Fig. 4 shows theHPI and three reduced-order models constructed withand , respectively, to show the convergence of the algorithm.Table I relates timing data for the direct MoM method, the HPImethod without MOR, and each reduced-order model. The errorlisted in Table I is the mean absolute difference between eitherthe complex reflection coefficient produced by the HPI or theMOR and that produced by the MoM over the band, and the in-terpolation time is the time needed to calculate the polynomial


Fig. 6. Reflection coefficients generated by MoM, polynomial interpolant, and reduced-order model for the six screens of Fig. 5.

interpolant coefficient matrices. The setup time per order is theamount of time needed, given the polynomial interpolant coef-ficients, to set up the reduced order model divided by the modelorder. The solve time per frequency is the amount of time needed

to set up and solve the linear system for each frequency in eachmethod, given the polynomial coefficient matrices for the HPI,or the reduced-order model for the MOR. The break-even pointis the number of frequencies for which the reflection coefficient


TABLE IITIMING AND ERRORRESULTS FORRATIONAL INTERPOLANT APPROXIMATIONS TO THEFSS SCREENS OFFIG. 5

must be computed to make the MOR faster than a straightfor-ward MoM solution at each frequency.

The results shown in Fig. 4 demonstrate that the proposedmethod converges quickly, giving very reasonable results forjust a fourth-order model. Table I shows that the method is eco-nomical, as the process breaks even after on the order of 20 fre-quencies, many fewer than are typically required for an accurateassessment of the behavior of the FSS. In addition, it shouldbe mentioned that before settling on a implementation,many different implementations were attempted to noavail. This establishes a need for a multipoint rational Krylovmethod such as described here.

The next example applies the MOR algorithm to dual screenFSSs composed of identical screens. If, as in Fig. 1, the normal-ized distance of each screenfrom the first screen is denoted by

, then for all dual screen FSSs in this example. Thescreens analyzed are shown in Fig. 5. For this example, the inci-dent wave propagates along the direction given by and

. The interpolation points were placed using (36)with and , , and .The reduction was accomplished with and andwith . Re-sults in the format described above are shown in Fig. 6 and listedin Table II.

Finally, the algorithm was applied to FSSs composed ofthree identical screens catalogued in Fig. 7 withand . The screen was illuminated with a wavepropagating normal to the screen, . Again,the interpolation points were placed using (36) with

, , , and . Thereduced-order model was produced with and andwith .Results are shown in Fig. 8 and Table III.

A few comments are in order about the results in general.First, due to the use of polynomial interpolation in the

linearization of the system, the frequency band to which thismethod is applied cannot include either zero frequency orthe blazing frequency where either poles or branch pointsingularities exist in the Green’s function. Specifically, thelocus of the onset of blazing is shown in Fig. 9. For a givenangle of incidence, a band of frequencies can be representedon this graph by a vertical line. If the point on this linerepresenting the highest frequency in the band is close to thecurve representing the blazing locus, a very high-order HPI isrequired to accurately approximate (1). Because the blazinglocus is concave up, this problem is most severe for incidence

Fig. 7. Six different meshed FSS screens for use in three-screen FSSs.

angles near normal. Thus, for a given FSS, if all the HPI andMOR parameters are fixed, and reduced-order models areconstructed for several incidence angles, the models for thoseincidence angles closest to normal will be accurate for a smallerpercentage of the usable band between zero frequency and theonset of blazing. These comments do not imply that the methodis useless for frequencies and angles of incidence where blazingoccurs, but only that new reduced-order models need to beconstructed for each frequency regime between singularities.

Second, as the response of the screens becomes more com-plicated, the number of frequency points for which the reflec-tion coefficient must be calculated for the MOR to break evenwith the MoM rises. This is to be expected, and does not pose aproblem because the accurate modeling of those types of screensby definition requires more frequency points.

Finally, note that in some instances, the reflection coefficientat the location of thin resonances given by the reduced-ordermodel does not exactly match the MoM result. At these points,however, the MoM is not very accurate as changing its param-eters may also alter the resonance size. The important behaviormimicked by the reduced-order model is not the magnitude ofthe reflection coefficients at small resonances, but their locationin the frequency band.


Fig. 8. Reflection coefficients generated by MoM, polynomial interpolant, and reduced-order model for the six screens of Fig. 7.

IV. EXTENSIONS TO THEMODEL-ORDER REDUCTION

TECHNIQUE

While all of the examples of the preceding Section apply toFSSs discretized on a grid amenable to the use of FFT matrix

multiplication methods, this fortunate discretization is not pre-requisite for the application of the algorithm. The MOR tech-nique is applicable to MoM systems created using any choiceof basis functions, though the resulting acceleration providedmay differ from the examples presented here.


TABLE IIITIMING AND ERRORRESULTS FORGRIDDED SQUARE LOOPFSSOF FIG. 7

Fig. 9. Locus of the onset of blazing for square-cell screens situated in free space. For simplicity, the azimuthal angle of incidence is assumed to be 0.

For instance, the technique can be applied to FSSs dis-cretized with full-domain bases. However, because the matricesproduced by such a discretization are typically small, anynoticeable acceleration provided by the MOR technique wouldno doubt spring from the interpolation step which obviates theneed for the construction of the MoM system at each frequency.This may be helpful if the infinite summations required arevery tedious, but is unlikely to produce the kind of speedupsshown in the tables.

A more important application of the MOR technique wouldbe to FSSs discretized with subdomain bases that do not con-form to any grid imposed by the periodic cell. Such problemsarise whenever such a grid is too restrictive for the fine-tuningof a design, or when the axes of periodicity of the FSS are notperpendicular. In these cases, the MOR method will show evenmore promise than in the examples presented here.

The method can also be extended to frequencies where theFSS blazes. As presented, the method is already applicable tosuch problems, as long as a new model is constructed after theonset of propagation of each new higher order Floquet mode.

Because of the HPI approximation to the MoM matrix, how-ever, the method discussed here cannot be used for models thatspan a frequency realm where a mode changes from evanescentto propagating. Additionally, the introduction of new modes re-quires the model to produce extra information, resulting in amultiple-output system. Extensions of the current technique thatsolve all of these problems are possible, and will be reported bythe authors elsewhere.

Perhaps most interestingly, the MOR method can be extendedto simplify the dependence of the problem on incidence angle.The method is based on the concept of generalized Krylov sub-spaces, introduced by the authors in [17], and eliminates theneed for the creation of a new model for each angle of incidence.

In addition to these rather sweeping changes to the work, manyminor extensions to it are underway, including the following:

• linearizing the MoM system with an orthogonal polyno-mial expansion instead of an HPI to reduce the size ofsystem (14);

• using Lanczos procedures in place of the DRA to accel-erate the MOR [13], [18];


• matching different numbers of moments at each interpola-tion point in the MOR to allocate computational resourcesmore wisely. This could also lead to an adaptive algorithmwhere interpolation points in the MOR are chosen in bandregions where the frequency response has not converged.

V. CONCLUSION

An MOR technique has been outlined for reducing systemsgenerated by the application of the MoM to FSS scattering prob-lems. While not a strict requirement on the algorithm, the use ofa regular discretization saves enormous amounts of memory byexploiting the Toeplitz-block-Toeplitz-like nature of the MoMmatrix. Given the MoM system, the process consists of two mainsteps: system linearization and MOR using the DRA algorithm.To elucidate the process, the algorithm was applied to a host ofFSS scattering problems that demonstrated its accuracy and ef-ficiency. Many extensions to the work were proposed, and willbe discussed in greater detail elsewhere.

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Daniel S. Weile(S’92–M’99) received the B.S.E.E.and B.S. degrees in mathematics (summa cum laude)from the University of Maryland at College Park in1994 and the M.S. and Ph.D. degrees in electrical en-gineering from the University of Illinois at Urbana-Champaign in 1995 and 1999, respectively.

Currently, he is an Assistant Professor of ElectricalEngineering at the University of Delaware, Newark.In 1994, he worked at the Institute for Plasma Re-search developing interactive software for the designof depressed collectors for gyrotron beams. As a Re-

search Assistant and Visiting Assistant Professor at the University of Illinois, heworked on the efficient design of electromagnetic devices using stochastic op-timization techniques, and fast time-domain integral equation methods for thesolution of scattering problems. His current research interests include compu-tational electromagnetics (especially time-domain integral equations), periodicstructures, and the use of evolutionary optimization in electromagnetic design.

Dr. Weile was a recipient of a National Science Foundation Fellowship, andthe ACES “Best Student Paper” Award in 1997 for his work in model-orderreduction. He is a member of Eta Kappa Nu, Tau Beta Pi, and Phi Beta Kappa.

Eric Michielssen (M’95–SM’99) received the M.S.degree in electrical engineering (summa cum laude)from the Katholieke Universiteit Leuven (KUL), Bel-gium, in 1987 and the Ph.D. degree in electrical en-gineering from the University of Illinois at Urbana-Champaign (UIUC) in 1992.

He served as a Research and Teaching Assistant inthe Microwaves and Lasers Laboratory at KUL andthe Electromagnetic Communication Laboratory atUIUC from 1987 to 1988 and from 1988 to 1992, re-spectively. He joined the Faculty of the Department

of Electrical and Computer Engineering at the University of Illinois as a VisitingAssistant Professor in 1992, was appointed Assistant Professor of Electrical andComputer Engineering in 1993, and promoted to Associate Professor in 1998.Since 1995, he has served as Associate Director of the Center for ComputationalElectromagnetics at UIUC. He authored or co-authored over seventy journal pa-pers and book chapters and over one hundred and twenty papers in conferenceproceedings. His research interests include all aspects of theoretical and appliedcomputational electromagnetics. His principal research focus has been on thedevelopment of fast frequency and time domain integral-equation-based tech-niques for analyzing electromagnetic phenomena, and the development of ro-bust, genetic algorithm driven optimizers for the synthesis of electromagneticdevices.

Dr. Michielssen received a Belgian American Educational Foundation Fel-lowship in 1988 and a Schlumberger Fellowship in 1990. Furthermore, he wasthe recipient of a 1994 International Union of Radio Scientists (URSI) YoungScientist Fellowship, a 1995 National Science Foundation CAREER Award,and the 1998 Applied Computational Electromagnetics Society (ACES) ValuedService Award. Recently, he was named 1999 URSI—United States NationalCommittee Henry G. Booker Fellow and selected as the recipient of the 1999URSI Koga Gold Medal. He served as the Technical Chairman of the 1997 Ap-plied Computational Electromagnetics Society (ACES) Symposium (Reviewof Progress in Applied Computational Electromagnetics, March 1997, Mon-terey, CA), and currently serves on the ACES Board of Directors and as ACESVice-President. From 1997 to 1999, he was an Associate Editor for Radio Sci-ence, and he currently is an Associate Editor for the IEEE TRANSACTIONS ON

ANTENNAS AND PROPAGATION. He is a member of URSI Commission B.

Kyle Gallivan , photograph and biography not available at the time of publica-tion.

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