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Robust Design of Absorbers Using Genetic Algorithms and the Finite Element-Boundary Integral Method Suomin Cui, Member, IEEE, and Daniel S. Weile, Member, IEEE

Abstract—A new method for the genetic algorithm (GA) based design of broadband, high-performance electromagnetic absorbers is discussed. The method gives rise to novel absorber designs with a geometrical complexity greater than that of absorbers typically in use today. The finite element-boundary integral method is applied to efficiently analyze the scattering from complex geometries occupied by given lossy material, and genetic algorithms are adopted to optimize the geometry parameters to minimize the overall reflection coefficients. In addition, a method is proposed for accelerating the convergence of the GA. Numerical results for absorbers are presented for wide-angle incidence over a broad frequency range considering both polarizations, and demonstrate the new technique’s power and robustness. Index Terms—Absorbers, finite element-boundary integral (FE-BI) method, genetic algorithms, periodical structures, simulated annealing.

I. INTRODUCTION

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ROPERLY designed anechoic chambers are crucial to the experimental characterization of antennas and scatterers. The walls of an anechoic chamber are lined with lossy material to simulate a free space environment, and backed with a good conductor to avoid outside interference. The front surface of the absorber is often a periodic structure that reduces reflection from the air-absorber interface. The analysis and design of such periodic absorber structures has been an active research topic for several years [1]–[3]. Previous design efforts have focused on the several different approaches to reduce the reflection from the absorbers, including: 1) shaping geometries [1] and [2]; 2) choosing different materials (i.e., materials with different electrical properties) assuming a given geometry [3]; 3) using hybrid structures composed of ferrite materials and lossy dielectrics [3]; and 4) using multiple layers of lossy materials [4], [5]. These methods have been successful in designing many different types of absorbers, and are responsible for the design of most absorbers in use today. Despite the past successes of the above methods, they have some shortcomings. For instance, the first approach uses simple

Manuscript received June 1, 2002; revised February 20, 2003. This work was supported by the Office of Naval Research, Code 334, under the Seaborne Structures Materials Program Grant PE 0602234N under the guidance of Dr. Ignacio Perez as part of the Advanced Materials Intellegent Processing Center at the University of Delaware Center for Composite Materials. The authors are with the Department of Electrical and Computer Engineering, Center for Composite Materials, University of Delaware, Newark, DE 19716 USA. Digital Object Identifier 10.1109/TAP.2003.820971

geometrical surfaces to design the absorber, and thus has difficulty lowering the reflection for large angles of incidence. The second approach can lead to excellent results, but it assumes that large numbers of materials are at hand or can be fabricated easily. The use of ferrite materials, as in the third approach, is probably the best way of absorbing low frequency waves, but leaves open the question of how to design the surface of the absorber to absorb high frequency waves. The fourth approach, taken in [4] and [5], is quite general and allows for the design of an absorber using only a discrete set of materials at hand, but is limited to flat absorber shapes that are generally suboptimal. For these reasons, this work presents a new approach for creating high performance periodic absorbers backed with perfect electrical conductors (PECs) for use in wide-band frequency and wide-angle incidence applications. In particular, the approach used here assumes that a database of available materials is provided, and uses numerical optimization techniques to select materials and shape the absorber surface. The actual application of the technique requires both an electromagnetic analysis method and an optimization method. In this work, the analysis is carried out by the finite element-boundary integral (FE-BI) method, and the synthesis is accomplished by genetic algorithm (GA). The reasons for these choices are discussed below. Periodic structures have been extensively analyzed using various numerical methods such as the moment method (MM) [6], [7], the finite-difference time-domain (FDTD) method [8], and the finite element method (FEM) [9]–[13]. The MM is not efficient for problems involving material inhomogeneities. The FDTD method handles material inhomogeneities admirably but is extremely difficult to implement for the oblique incidence case although many attempts have been made [14]. The FEM can model material inhomogeneities efficiently and can simulate periodic boundaries without any difficulty, and so is the choice adopted in this work. For synthesis, this work uses GAs [15]. GAs, as one representative of a host of new global optimization techniques, have become very popular for the solution of electromagnetic design problems [4], [5], [16], [17]. In comparison with local optimization techniques (including conjugate gradient methods, quasi-Newton methods and the simplex method) GAs have the following features: robustness, capacity to search without gradient information, independence of the initial guess, and ability to operate on discrete and continuous parameters simultaneously. GAs are especially powerful for problems that have a large number of dimensions and many local optima [4],

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Fig. 1.

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Plane wave incident on a periodic lossy surface backed by metal and the propagating Floquet harmonics.

[15]–[17]. In particular, because GAs yield global or near local optima robustly, they are especially applicable to problems in which the nature of the solution space is unknown. In this paper, we propose to design high performance absorbers by increasing the complexity of the absorber interface. Because the envisioned absorbers have never been studied before, the nature of the search space (its multimodality, its sensitivity to the design parameters, etc.) is far from clear. For this reason primarily, GAs are quite well suited to the current study. Of course, GAs are not the only global search algorithm. Simulated annealing (SA) [18], [19] is an older global optimization method that has been applied in many disciplines. While in general an a priori comparison of the two methods for a given problem is impossible, GAs are often faster than SA as they use the information contained in a group of designs to perform a search, while SA uses only one design at a time. In many of the design results presented here, GA results are compared to SA results for two reasons: First, the results will show that both algorithms, when run to completion, return the same design for the problems discussed herein. Because optimality proofs for solutions to such general optimization problems are difficult if not impossible to generate, such results are the best testament to the robustness of the proposed method. Second, the comparison between GA and SA runs will show that GAs are faster than SA in finding solutions to this problem. The rest of this paper will proceed as follows: In Sections II and III, we briefly review the FE-BI method and GAs, respectively. Section IV presents the numerical results of the study. In doing so, Section IV also describes the shape description employed by the optimization and the materials used in this study. It also compares the proposed method with methods using other optimization techniques and different objective functions. The conclusions of the work are given in Section V.

II. FE-BI METHOD FOR PERIODIC STRUCTURES In this section, the FE-BI method for periodic structures is briefly reviewed. In particular, analysis variables (which will be used in the optimization) are defined, and the specific implementation used to analyze the absorbers under study is elucidated. A. Definition of Reflection Coefficient From a Periodic Structure A generic 2-D periodic structure is shown in Fig. 1. The structure is infinite and periodic with a period along the direction, finite along the direction, and uniform along the direction. The structure is illuminated by a plane wave given by (1) is the wave number in free space, and determines where is the the direction of travel of the incident wave. The field electric field for the TM case and the magnetic field for the TE case, respectively. For simplicity, we will assume that a function describes the shape of the absorber surface, that the is occupied by lossy materials whose permitregion and respectivity and permeability are is free space and characterized tively, and the region by permittivity and permeability . Floquet’s theorem states that when a plane wave is incident on a periodic structure the fields observed at two arbitrary points separated by a distance along a line parallel to the axis are . identical, except for a constant phase shift Thus, the plane wave scattering from a periodic structure can be characterized once the equivalent currents over a single unit cell are determined. In particular, the scattered

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field can be expressed as a superposition of Floquet harmonics as

(2) where is an integer. Note that the scattered field is composed of an infinite number of plane waves, each having its own ampliand direction of propagation. For any given frequency tude of the incident wave, only a finite number of these will carry power to the far field, and the rest are evanescent. The angle of departure of the th propagating order is determined (Fig. 1). by The fact that the different modes propagate at discrete, different angles implies that care must be taken in defining the reflection coefficient. Because the Floquet boundary condition forces the field magnitude to be periodic in , any energy traveling in the direction enters one side of the periodic cell, and exits the other side of the periodic cell with no change in magnitude. This happens for all periodic structures, regardless of the arrangement of materials in the unit cell. Because of this, the reflection coefficient is defined to be the ratio of all of the power reflected in the direction to the power incident in the direction, and power in the two orthogonal directions is conand to served independently. Specifically, defining be the lowest and the highest propagating orders, respectively, (for all propagating an overall power reflection coefficient modes) is defined by [7] (3) This coefficient takes into account reflection into all angles, but takes only the component of the power traveling normal to the absorber. B. Solution for the Reflection Coefficient Using the FE-BI Method The basic unit cell for the FE-BI formulation is shown in Fig. 2. The interior region associated with a single unit cell is . Boundary bounded by the flat surface is a PEC on the axis, is parallel to the axis, and and are parallel to the axis, so is a rectangular region. The field inside is governed by the Helmholtz equation

Fig. 2. Basic cell of periodic structure enclosed by the contour 0 0 +0 .

+0 +

2) The Dirichlet condition is enforced on the surface of the PEC for TM-polarization. 3) The Neumann boundary is enforced on the surface of the PEC for TE-polarization. is not available explicitly, The boundary condition along for now it will be set as (5) where is another variable to be determined. triangular elements The interior region is divided into is broken into short segments. The and, consequently, field in each element is expanded using linear interpolation functions. By applying the standard FEM [5]–[7], the following equation can be obtained to describe the fields interior to (6) , an integral equation is derived that couples the inciOver dent field to the Helmholtz equation as

(7) where

is the 2-D periodic Green’s function

(4) denotes the total field. For TM polarization and , and for TE polarization and . The field is also subject to appropriate boundary conditions as follows. 1) The periodic boundary condition must be enforced on and for both TM and TE-polarizations, respectively.

where

(8) Discretizing (7) gives rise to another matrix equation for and (9)

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Equations (6) and (9) can be solved efficiently using an iterative technique. Once this is done, the coefficients of the Floquet expansion (2) can be obtained using the equation (10) and the overall reflection coefficient can be found from (3). The specific FE-BI technique used here employs a sparse matrix storage scheme and the transpose-free quasiminimal residual (TFQMR) method for the solution of the matrix system. The numerical code has been confirmed by other numerical and experimental results [20]. In all the cases presented in this paper, the mesh density was approximately 400 nodes as is almost standard in FEM computations, where is per the wavelength in the material studied.

Fig. 3. Shaping mechanism used in this paper.

III. GAs GAs are iterative optimization algorithms that have been widely applied in electromagnetics [16], [17]. In contrast to most optimization methods that operate directly on a single design candidate, GAs operate on a population of candidate designs encoded in binary strings of a given length called chromosomes. Each such chromosome describes a single candidate design fully. GAs act on the population of chromosomes to cause an evolution toward an optimal solution given the definition of an objective function that provides a measure of the fitness of a given chromosome. A flow chart of a standard GA optimizer is illustrated in [16], to which readers unfamiliar with GAs are referred for a more complete introduction to the subject. The GA begins by randomly creating a population of chromosomes, and evaluating their objective function. The GA works by iteratively applying three genetic operators (selection, crossover, and mutation) to the population in turn as discussed below. The selection operator generates a new population of chromosomes from the existing population based on fitness values. Different selection operators are available in the literature; this study adopts elitist binary tournament selection. In binary tournament selection, two chromosomes are chosen at random from the old population, and the better of the two is inserted into the new population. (Since GAs can be applied to maximization or minimization, the definition of “better” is problem dependent, but known in advance. For instance, in a maximization problem, the chromosome with the higher objective function value is the better chromosome.) This process is chromosomes. repeated until the new population contains If the selection process does not choose the best member of the old generation, it is inserted in a random position in the new population. (This is what makes the selection “elitist”.) The crossover operator hybridizes chromosomes in the population. This work uses a crossover variant called uniform crossover. Under uniform crossover, the chromosomes are paired, and each pair is mated with a given probability, usually between 0.6 and 0.9. A pair of chromosomes selected for crossover creates a new pair of chromosomes by exchanging the bits in each position with a 0.5 probability. Pairs not chosen

for crossover are simply left alone. This results in another chromosomes. population of After crossover, mutation is applied to prevent premature convergence. (Premature convergence is defined as convergence to a suboptimal solution before a good solution is found. If the absolute optimum solution is not required, the GA can be stopped at any time before it converges, and a suboptimal, but perhaps good enough, solution can be used. But, if the GA converges prematurely before an acceptable solution is found, all of its work is wasted.) Mutation simply negates each bit in the population with a given probability, usually between 0.001 and 0.1. After the mutation operator is applied, the chromosomes are reevaluated, and the process begins anew with selection. The algorithm is terminated when either a design goal is reached, or no progress is observed in the population for several generations. IV. NUMERICAL RESULTS The specific geometry optimized by the GA is shown in Fig. 3. It is divided into two regions: a substrate region and a shaped region. The height of substrate is denoted as . To into design the shaped region, we divide the period segments of equal length, and denote the center points of these , for . segments by is a that denotes the height of the Associated with each shaped region at that abscissa. Assuming that the structure is , so the GA operates by choosing and symmetric, for . A carbon foam material is used for all designs in this paper. The measured relative dielectric permittivity of a given material to be used for the absorber is plotted in Fig. 4. Since the material is nonmagnetic, it has a relative permeability of unity. The material is a sensible choice for absorber design as it is lightweight, mechanically strong, and easily machined into any desired shape. In this study, the goal is to construct absorbers that can reduce reflections to a minimum for all incident angles and frequencies of interest. Because we wish to construct an absorber that robustly absorbs over a range of frequencies and incidence angles, we seek to minimize the maximum reflection coefficient over all angles and frequencies of interest. The specific objective

CUI AND WEILE: DESIGN OF ABSORBERS USING GENETIC ALGORITHMS AND FE-BI METHOD

Fig. 5.

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Comparison of the two competing objective functions.

Fig. 4. Measured dielectric constant for the Touchstone carbon foam material.

function used here measures the reflection coefficients in decibels. (Given the choice of objective function and GA selection operator, however, this makes no difference. Any monotonically increasing function of the reflection coefficient would return the same results, since the GA operates merely by comparing objective function values, and does not use the quantitative difference in objective function values in any way.) In particular, the objective function used here is (11) is the maximum reflection coefficient for all inciwhere dent angles and frequencies considered. The GA seeks to minimize this function. The objective function above is used because it is a measure of the most important absorber figure of merit—the minimal performance that can be expected over the range for which it was designed. Other performance measures are possible; but these figures of merit (such as the average reflection coefficient over a set of incident angles and frequencies) are not as relevant. For instance, an absorber may have an excellent average performance, but fail to absorb anything at a given angle of interest. As an example of this phenomenon, we compare the proposed objective function to a function attempting to minimize the average reflection. In particular, the minimization of the proposed objective function (11) is compared to the minimization of (12) is the total number of frequency and angle samples of where is the reflection coefficient at the th angle/freinterest, and quency combination. (Note, that for this objective function, the decision to measure the objective function in decibels is relevant. Choosing not to measure the objective function in decibels actually worsens the effect described below.) Both objective functions are applied to the optimization of an absorber with geometry control points over the absorber half-period. The sample interval for the angles is 10 . The period is set to

1.5 cm (0.5 wavelengths), and the maximum is limited to 4.5 cm (3 wavelengths). The designs are referred to the oper. ating frequency of 10 GHz Fig. 5 shows the reflection coefficients for both TM and TE considered together by different objective functions defined by equations and . Note that the GA did its job in both cases: inWhile the geometry returned by the objective function deed has a lower average reflection, it has a much higher reflection coefficient for the TE polarization than the design produced by the function has anywhere. In short, the design optimized has a worst-case performance using the objective function that is more than 7 dB better than that of the absorber optimized with function . In addition to considering several objective functions for the optimization, the performance of the GA was also compared with that of other optimizers for the same objective function . In particular, results are presented for the simulated annealing algorithm (SA) [18], [19]. Fig. 6 shows the convergence histories for GA and SA, and the optimized geometries for only TM polarization and for both polarizations considered together, respectively. These convergence histories were averaged over ten runs of each algorithm, using different values of the control parameters (i.e., crossover rate in GA, annealing schedule in SA, etc.). (The control parameter values in this experiment were taken to be representative of typical parameter values from the literature.) The “Number” in Fig. 6 is the number of objective function evaluations for each algorithm respectively. From Fig. 6, the following becomes clear. 1) The GA returns the result with fewer function evaluations than SA: GA converges using only half the CPU time of SA. On a 1.4 GHz Pentium 4, this translates into savings of six hours. 2) Both GA and SA find almost the same minimum reflection coefficient for both problems. This implies that the results are robust. 3) If only TM polarization is considered, the different geometries obtained by GA and SA give the almost same minimum reflection coefficients as shown in Fig. 6(a). By restricting the result more, and forcing the absorber to absorb both polarizations, GA and SA design the same absorber as shown in Fig. 6(b).

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(a) Fig. 7. Optimized geometry for the inhomogeneous material and the resulting reflection coefficients. The graph of the reflection coefficients for the optimized homogeneous absorber is included for comparison.

(b) Fig. 6. Convergence of GA and SA for (a) TM polarization and (b) Both TM and TE polarization.

The third observation again implies that the results are robust, and that the solution provided by both GA and SA to the problem involving both polarizations is probably the true optimum. It also implies that the problem of designing an absorber for only one polarization has many equivalent global optima. To demonstrate the ability of the approach discussed in this paper to deal with arbitrary inhomogeneous materials, we use a fictitious material whose permittivity profile is linear. When (on the PEC) the permittivity is exactly same as that of the for 10 GHz), and material used in this paper ( cm is unity to provide an optimal match to the value at free space. (Other tapers were also attempted, including a linear cm to a permittaper from a permittivity of unity at at , and tapering the real tivity of and imaginary parts separately. The results presented for the given taper were the best results achieved among all of these attempts.) Fig. 7 shows the optimized geometry and the reflection coefficients. The reflection coefficients from the optimized geometry shown in Fig. 6(b) are included in Fig. 7 for comparison purposes. It is interesting to see that the optimized geometry with the fictitious linear profile material has larger reflection coefficients than that with the homogeneous profile. This phenomenon can also be seen in [9], Figs. 19–20. The fact that both this work and [9] find that the inhomogeneities degrade absorber performance also lends credence to the results presented here. Nonetheless, the method presented here can be extended

to allow for the inclusion of different materials or taperings in the optimization, but extensive discussion is beyond the scope of this work. The above discussion confirms that the GA optimizing the is a reasonable approach to the given objective function design of absorbers. The following subsections present some design results to show that high-performance absorbers can be obtained by using a GA to design intricate absorber air interfaces. In all of the following examples, the GA was run for 500 generations. SA results were also obtained for the final example, but at a much higher cost to return the same design. SA results are, therefore, excluded from the following discussion as redundant. A. Results for Wide-Angle Incidence and Given Frequency For the first example, we design a geometry from which the at reflection coefficient is minimized for 10 GHz. The period is set to 1.5 cm, the maximum is limited to 4.5 cm, and sample interval is 5 . Fig. 8(a) shows the optimized geometries for TM or TE inci. The reflection coefficients versus dence separately for incident angles for both optimized geometries are depicted in Fig. 8(b). For each polarization, the reflection coefficients from dB. If both TM and TE incidences 45 to 90 are lower than are considered together, the performance suffers: an optimized dB. geometry only reduces the reflection coefficient to is increased, much lower reflection On the other hand, if coefficients within the observation angles of interest can be ob, the coefficients for TM or tained. For example, when dB. For TE polarizations (alone) can be made as low as both TM and TE incidences together, reflection coefficients as dB are obtained. Fig. 9 shows the optimized geomlow as , when both polarizations are etry and coefficients for considered together. From Figs. 9(b) and 10(b), it is found that the variation of the reflection coefficient versus incident angle is smooth. This implies that larger intervals of angle (such as 10 degrees) can be considered without much loss in accuracy. This fact can reduce the times for computing objective functions in GA resulting in faster optimization.

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(a)

(b)

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Fig. 8. (a) Optimized geometries for TM and TE incidences ( (b) the reflection coefficients for the geometries in shown in (a).

= 10) and

B. Results for Wide-Band Frequency and One Given Incident Angle For the second example, the incidence angle is held fixed and the frequency varied. In particular, the frequency band of interest is 2 to 18 GHz. Unfortunately, computing the absorber reflection over a broad band of frequencies is expensive. A few observations may be used to accelerate the process: First, for a given period, the computation of reflection coefficients is much more expensive for higher frequencies. This is a direct result of the dense grid needed to represent the high frequency waves. Second, suppressing the reflection from the absorber is much more difficult at low frequencies than at high frequencies. This results from the fact that the absorber is electrically small at low frequencies. Therefore, we propose the following scheme to optimize absorbers over large bands of frequency. First, the geometry is optimized for operation in the lower part of the frequency band. In this case, the lower part of the band used for the initial optimization is 2 to 10 GHz. As the GA progresses, increasing numbers of high frequencies can be added to the function definition using the following heuristic: if the objective function does not change after several generations, one can assume that the GA has converged and add another frequency into the optimization. In this way, time is not wasted analyzing geometries with poor low frequency performance.

(b)

N = 20) and

Fig. 9. (a) Optimized geometry for both TM and TE incidence ( (b) its reflection coefficient.

Fig. 10 shows the optimized geometry for an incident angle in the 2 to 18 GHz frequency band. The period is of set to 2 cm, and the maximum is limited to 15 cm, and the . From Fig. 10(b), number of design points was set to the maximum reflection coefficient for the optimized geometry dB for all frequencies. This maximum changes strongly is dB for and with incident angle; for example, it is dB for , respectively. C. Results for Both Wide-Band Frequency and Wide-Angle Incidence As a final design, an absorber is optimized to work under the following conditions: 1) The incident angle varies from 60 to 90 , and angle step size is 10 ; 2) The frequency band varies from 2 to 18 GHz; 3) The period is set to 2 cm, and the maximum is limited to 15 cm; . 4) The initial population for the GA runs was taken as the final population from the run of Section 4.1. The design produced has a maximum reflection coefficient of dB, and its geometry is shown in Fig. 11. The reflection coefficients versus the incident angle for the selected frequency (2,

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(b) Fig. 10. (a) Optimized geometry for both TM and TE incidence at wide-band frequency ( = 20) and (b) the reflection coefficients produced by the geometry.

N

(a)

(b) Fig. 12. Reflection coefficients versus the incident angles for the selected frequencies (a) TM and (b) TE.

reflection coefficients for the optimized geometries are close as well. In other words, the geometry for wide-angle incidence and wide-band frequency depends most strongly on the minimum incidence angle and frequency. This observation can be used to get good results in much reduced time. V. CONCLUSION

Fig. 11. Optimized geometry for both TM and TE incidence at wide-band frequency and wide-angle incidence ( = 20).

N

10, 18 GHz) for TM and TE incidence are plotted as in Fig. 12 and the reflection coefficients versus the frequencies for the selected incident angles for different polarizations are shown in Fig. 13. By comparing Fig. 11 with Fig. 10(a), it is found that the two optimized geometries are very similar, and the maximum

This paper has demonstrated the use of the FE-BI method and GA to design the geometrical shape of high performance absorbers for both TM and TE polarizations for different requirements: wide-angle incidence, or wide-band frequency, or both wide-angle incidence and wide-band frequency. Procedures for speeding up the optimization were also proposed, and can be profoundly important in designing the geometries to minimize the reflection coefficients for wide-band frequency and wideangle incidence. It is found that the maximum reflection coefficient strongly depends on the shapes allowed by the optimization procedure. For the symmetrical polygonal shapes used here, the following factors are important: 1) The discretization parameter : Geometries with larger have the ability to get lower maximum reflection coefficients than those with smaller . On the other hand,

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REFERENCES

(a)

(b)

Fig. 13. Reflection coefficients versus the frequencies for the selected incident angles (a) TM and (b) TE.

a large model burdens the optimization and increases the difficulty for fabrication. 2) The limitation of the maximum thickness : It is possible to get low maximum reflection coefficient by increasing the maximum allowable absorber thickness. On the other hand, increasing the depth of the absorber can increase the optimization time, the amount of material needed, and the cost of fabricating the absorber. Future work will address 1) the minimization of construction costs; 2) the inclusion of several materials; and 3) the extension of the method to 3-D problems.

ACKNOWLEDGMENT The work described in this communication was performed for the Structures and Composites Department of the Survivability, Structures, and Materials Directorate, at the Naval

[1] W. H. Emerson, “Electromagnetic wave absorbers and anechoic chambers through the years,” IEEE Trans. Antennas Propagat., vol. AP-21, pp. 484–490, July 1973. [2] J.-R. Gau, J. W. D. Burnside, and M. Gilreath, “Chebyshev multilevel absorber design concept,” IEEE Trans. Antennas Propagat., vol. AP-45, pp. 1286–1293, Aug. 1997. [3] C. L. Holloway, R. R. Delyzer, R. F. German, P. McKenna, and M. Kanda, “Comparison of electromagnetic absorber used in anechoic and semi-anechoic chambers for emissions and immunity test of digital devices,” IEEE Trans. Electromagn. Compat., vol. 39, pp. 33–47, Feb. 1997. [4] E. Michielssen, J. M. Sajer, S. Ranjithan, and R. Mittra, “Design of lightweight, broadband microwave absorbers using genetic algorithms,” IEEE Trans. Microwave Theory Tech., vol. 41, pp. 1024–1031, June/July 1993. [5] D. S. Weile, E. Michielssen, and D. E. Goldberg, “Genetic algorithm design of pareto optimal broadband microwave absorbers,” IEEE Trans. Electromagn. Compat., vol. 38, pp. 518–525, Aug. 1996. [6] C. F. Yang, J. W. D. Burnside, and R. C. Rudduck, “A periodic moment method solution for TM scattering from lossy dielectric bodies with application to wedge absorber,” IEEE Trans. Antennas Propagat., vol. 40, pp. 652–660, June 1992. [7] R. Janaswamy, “Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers,” IEEE Trans. Antennas Propagat., vol. 40, pp. 162–169, Feb. 1992. [8] C. L. Holloway, P. M. McKenna, P. A. Dalke, R. A. Perala, and C. L. Devor, “Time-domain modeling, characterization, and measurement of anechoic and semi-anechoic electromagnetic test chambers,” IEEE Trans. Electromag. Compat., vol. 44, pp. 102–118, Feb. 2002. [9] H. Rogier, B. Baekelant, F. Olyslager, and D. DeZutter, “The FE-BIE technique applied to some 2D problems relevant to electromagnetic compatibility: Optimal choice of mechanisms to take into account periodicity,” IEEE Trans. Electromagn. Compat., vol. 42, pp. 246–256, Aug. 2000. [10] S. D. Gedney, J. Jin, and R. Mittra, “Combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 364–370, Feb. 1992. [11] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993. [12] J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics. New York: IEEE Press, 1998. [13] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New York: IEEE Press, 1998. [14] J. A. Roden, S. D. Gedney, M. P. Kesler, J. G. Maloney, and P. H. Harms, “Time-domain analysis of periodic structures at oblique incidence: Orthogonal and nonorthogonal FDTD implementations,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 420–427, Apr. 1998. [15] D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Reading, MA: Addison Wesley, 1989. [16] D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans. Antennas Propagat., vol. 45, pp. 343–353, Mar. 1997. [17] Y. Rahmat-Samii and E. Michielssen, Eds., Electromagnetic Optimization by Genetic Algorithms. New York: Wiley, 1999. [18] S. Kirkpatrick, J. C. D. Gelatt, and M. P. Vechi, “Optimization by simulated annealing,” Sci., vol. 220, pp. 671–680, 1983. [19] W. L. Goffe, G. D. Ferrier, and J. Rogers, “Global optimization of statistical functions with simulated annealing,” J. Econometrics, vol. 60, no. 1/2, pp. 65–100, Jan./Feb. 1994. [20] S. Cui and D. S. Weile, “Analysis of electromagnetic scattering from periodic structures by FEM truncated by anisotropic PML boundary condition,” Microw. Opt. Tech. Lett., vol. 35, no. 2, pp. 106–110, 2002.

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Suomin Cui (M’01) was born on November 11, 1967 in Shaanxi, China. He received the B.S. degree in physics from Shaanxi Normal University, in 1989, and the M.S. and Ph.D. degrees in electrical engineering from Xidian University, Xidian, China, in 1992 and 1995, respectively. In September 1995, he joined the Department of Electrical Engineering, Nanjing University of Science and Technology, Nanjing, China, as a Postdoctoral Fellow where he was promoted to an Associate Professor in August 1997. From September 1997 to August 2000, he was a Research Fellow at the Tokyo Institute of Technology, Tokyo, Japan. Since May 2001, he has been a Postdoctoral Fellow at the University of Delaware, Newark. His primary fields of interest include high frequency diffraction analysis, computational electromagnetics, and optimization methods. Dr. Cui was awarded postdoctoral fellowships from the China Postdoctoral Council in 1995, the Japan Society for the Promotion of Science (JSPS) in 1997, and a research fellowship from the International Communication Foundation (ICF) for Foreign Research in Japan in 2000. He received the second and first class scientific and technical achievement awards from the Ministry of Electronic Industry of China in 1996 and 1997, and the third class national scientific and technical achievement award in 1998.

Daniel S. Weile (SM’92–M’00) received the B.S.E.E and B.S. (in mathematics) degrees from the University of Maryland at College Park, in 1992 and 1994, respectively, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign, in 1995 and 1999, respectively. Currently, he is an Assistant Professor of electrical engineering at the University of Delaware, Newark. In 1994, he worked at the Institute for Plasma Research developing interactive software for the design of depressed collectors for gyrotron beams. As a Research Assistant and Visiting Assistant Professor at the University of Illinois, he worked on the efficient design of electromagnetic devices using stochastic optimization techniques, and fast time-domain integral equation methods for the solution of scattering problems. His current research interests include computational electromagnetics (especially time-domain integral equations), periodic structures, and the use of evolutionary optimization in electromagnetic design. Dr. Weile is a Member of Eta Kappa Nu, Tau Beta Pi, Phi Beta Kappa, and the International Scientific Radio Union (URSI) Commission B.