Pareto Optimal Design of Absorbers Using a Parallel Elitist Nondominated Sorting Genetic Algorithm and the Finite Element–Boundary Integral Method Suomin Cui, Anuraag Mohan, and Daniel S. Weile Department of Electrical & Computer Engineering Center for Composite Materials University of Delaware, Newark, DE 19716 Email: [email protected]

Abstract Microwave absorbing structures have many applications including the lining of anechoic chambers and the reduction of electromagnetic interference. Pareto optimization is an important tool in the design of absorbers, since most absorbers must be designed keeping both performance and economy in mind. In this paper, a new elitist strategy is implemented into the nondominated sorting genetic algorithm (NSGA) to effectively and efficiently design broadband high performance electromagnetic absorbers.

The absorbers are analyzed using the finite

element boundary integral method, and the optimization is accelerated with parallel processing. Numerical tests demonstrate that the elitist NSGA proposed in this paper converges faster that the standard NSGA and other classical techniques for a wide variety of absorber design problems. Finally, this efficient elitist NSGA is applied to design complex polygonal absorbers. Numerical results not only demonstrate the robustness of the design algorithm, but also reveal some important information and advantages related to absorber designs based on Pareto optimization. Index terms: electromagnetic absorbers, Pareto optimization, genetic algorithms, periodic structures, finite element-boundary integral method.

Acknowledgements The work described in this report was performed for under a small business innovation research grant from the department of the Army through Applied EM, Inc of Hampton, VA. The authors also would like to thank Prof. John Xiao of the department of physics of the University of Delaware for measuring the permittivity of the materials used in section 4.

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1. Introduction Electromagnetic absorbers are used in applications ranging from electromagnetic interference mitigation to anechoic chamber construction. Very often, absorbers are constructed by tiling material in a periodic fashion to simplify construction. Because of their range of application, the analysis and design of periodic absorber structures have been active research topics for several decades. Most previous design efforts have focused on different approaches to reduce the reflection from the absorbers without regard to other considerations [1-5]. (An exception to this general rule has been the multilayer microwave absorptive coating [6-8].) Despite the excellent results achieved in these works, practical results are rarely achieved without considering other factors.

Depending on the application, absorber weight, thickness, and profile may be

mechanically (or even aesthetically) restricted, and fabrication costs are often an important consideration. While changing the geometrical model of the absorber may affect any or all of these considerations, profoundly different results can be obtained even while holding to a specific model. For all of these reasons, absorber designers should use multi-objective Pareto optimization algorithms to design high performance absorbers: such algorithms return a set of solutions (the Pareto optimal set or Pareto front) representing the best tradeoffs between conflicting design goals, and the best absorber for a particular application can be chosen once the best tradeoffs are found. Indeed, even given different geometric absorber models, a multiobjective optimizer run several times can illuminate the best choice. In short, even given a particular geometry for an absorber design, there exists no single best absorber; only a set of absorbers that represents optimal tradeoffs between conflicting design goals. Traditional multi-objective optimization approaches usually combine all objectives to form a scalar fitness function by using a weighted aggregation approach, the method of distance functions, or the Min-Max formulation [7, 9-11].

Any optimization algorithm (linear

programming, gradient search, standard genetic algorithms (GAs), random search, simulated annealing, etc.) can then optimize this scalar fitness function. However, these classic approaches can be very sensitive to the precise aggregation of the goals and tend to be ineffective and inefficient [7, 11]. The nondominated sorting genetic algorithm (NSGA) [7, 11] is a variant of the simple GA [12] that uses a nondominated sorting algorithm to evaluate relative Pareto optimality and a

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sharing operator to ensure a good spread of points on the front. The NSGA is an effective method for determining the Pareto front in multi-objective optimization problems too complex to be solved by traditional methods.

The NSGA overcomes the deficiencies of traditional

aggregation-based multi-objective optimizations, and is capable of obtaining a uniformly dense sampling of the Pareto front, whether or not it is convex. Better still, because the NSGA (like all GAs) uses a population for searching the objective function space, , it finds the entire front in a single run of the algorithm. Moreover, many studies have shown that the NSGA outperforms other Pareto GAs and other evolutionary algorithms for multi-objective functions for a wide variety of vector objective functions [7, 11, 13]. Finally, like all GAs, the NSGA has the ability to optimize simultaneous functions of discrete and continuous parameters and possesses inherent parallelism. These attributes are extremely important in practical absorber design: the use of continuous and discrete parameters allows the design of absorbers from a database of materials, and the inherent parallelism allows for efficient optimization of computationally difficult problems. Indeed, the NSGA was recently applied to shaping a corrugated coating to absorb waves at grazing incidence [14]. The study minimized the average reflection coefficient over several frequencies and several sampling angles and was limited to the use of a single absorbing material, but it produced several practical designs for the problem at hand [5, 14]. (On the other hand, the use of average reflection coefficient as an objective function can lead to suboptimal designs, see e.g. [5].) This study uses the NSGA to design absorbers with several different models to achieve low absorption for both polarizations, for wide-angle incidence, and over a wide frequency band. The designs presented here include not only shaping of absorbers but also choosing materials for absorbers from a predefined database. Due to the complexity of absorber models optimized, the absorbers can only be analyzed with time-consuming numerical techniques. (In particular, the approach presented here is based on the finite element boundary integral (FE-BI) method [15-17], which is relatively efficient. Nonetheless, because the GA requires many thousands of objective function evaluations, all methods are “time-consuming.”) Because this bottleneck is worsened by the need to analyze the absorbers over multiple polarizations, angles of incidence, and frequencies, the NSGA is too slow for this application.

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This work presents two approaches to overcome this issue. First, we present a new elitist NSGA (ENSGA) constructed by including a heuristic elitism scheme into the standard NSGA to speed up the convergence. Note that the implementation of elitist strategies in a Pareto optimization algorithm is quite complex: After all, there is no single optimal design to carry forward to the next population in a PGA. Second, we parallelize the NSGA to calculate the objective function for each individual in NSGA. Both approaches vastly improve the efficiency of the process. The designs results for three geometrical models confirm the advantages of the design method presented in this paper. The rest of this paper will proceed as follows:

Section 2 describes the NSGA,

introduces the incorporation of elitism into the NSGA, and finally discusses the implementation of the parallelization of the ENSGA. Section 3 presents designs for planar absorbers to confirm the effectiveness and efficiency of the ENSGA for this design problem. Section 4 integrates the ENSGA with the FE-BI method to find Pareto fronts for several more complicated design problems, and discuss the some applications of Pareto optimization in absorber designs. The conclusions of the work are given in Section 5.

2. Description of the optimization algorithm A complete description of a given absorber for a given problem will be assumed to depend on n decision variables.

For the purposes of this discussion, m of these values,

t1 , t2 ,…, t m , are assumed to be real, and the remaining n − m values l1 , l2 ,… , ln − m are assumed to be integers. For instance, if a four-layer absorber is to be designed by choosing the thickness of the layers and the material composition of each layer from a database, then n = 8 ; since four of the design variables are real and continuous, m = 4. These decision variables will be aggregated into a decision vector t = [t1

tm

l1

ln − m ] that describes an entire design. This work

considers the minimization two goals: the thickness T (t ) and the reflectance R(t ) . (Specifically, R(t) is the maximum power reflection coefficient over all frequencies and angles of interest. This ensures that the objective function is a measure of the worst-case reflection [5].) The goal space is thus the two-dimensional space with T (t ) on one axis and R(t ) on the other. Pareto optimization algorithms seek all absorber designs t that have minimum reflectance R(t) for a given thickness T (t) . This set of points is known as the Pareto front. (This is only a rough

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definition of the Pareto front; if two absorbers have the same maximum reflectance and different thicknesses, only the thinner one is Pareto optimal. Moreover, the Pareto criterion can be extended to include more than two goals. Nonetheless, the current definition suffices for current purposes; see [11,13] for more details.) Now that the goal of Pareto optimization has been defined, the NSGA is described. Like all genetic algorithms, the NSGA begins with an initial population of coded solutions to a problem, and evolves this population using the genetic operators selection, crossover, and mutation. The only difference between the NSGA and a simple GA is in the implementation of the selection operator. Each individual in the population is ranked based on its relative Pareto optimality. This Pareto rank is then modified by a sharing operator that penalizes chromosomes for being similar to other chromosomes in the population. This ensures a broad representation of the front. Given a population, the Pareto ranking part of the selection operator works by first locating the population’s nondominated members; that is, those members that have the minimum reflectance for a given thickness [7,11,12]. These currently nondominated designs are assigned a rank of 1, and temporarily removed from the population. The rest of the population is then searched for nondominated members (after removal of the rank 1 members).

Those

nondominated members of the remaining population are assigned a rank of 2, and temporarily removed.

This process of ranking nondominated individuals and removing them from

consideration is continued until all individuals are assigned a rank. The sharing operator ensures diversity on the front [7, 11, 12]. First, a metric is defined to compute the distance between competing designs in goal space. Based on this metric, each design is assigned a penalty for each design near it in design space. (Our implementation uses a linearly decreasing function of the distance that is unity for coincident designs, and reduces to zero for designs a predefined distance away.) To compute the objective function used by the GA, each design in the first rank is given unity fitness. This fitness is then divided by the sum of the penalties computed by the sharing operator, resulting in a single objective function value for each design in the first rank. The lowest of these values is taken as the pre-sharing fitness for rank two designs, and this value is then divided by the sharing penalty to assign an objective function value to all of the rank two designs. This process is continued until all designs have

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been assigned an objective function value, and roulette wheel selection proceeds as usual. Other than these modifications to the selection operator, NSGA proceeds exactly like a simple GA [12]. 2.1 Implementation of the elitism in the NSGA: The ENSGA In single objective GAs, elitism is the operator that ensures that the best chromosome found so far exists in every GA iteration. This can be realized by simply copying the best individual to the next generation. For multiobjective optimization, however, the elitism strategy no longer remains trivial, as there is no longer a single best design to copy. Because of this, various elitist strategies have been proposed [13]. This paper will adopt a novel and simple elitist approach in which parents compete with offspring after each generation of the NSGA. Specifically, consider a population of N p parent chromosomes with known objective function vectors (that is, values of reflectivity and thickness). Applying the Pareto ranking and sharing operators to this population, and performing selection, crossover, and mutation results in a child population of N p chromosomes. Finally, this child population is combined with the original parent population, and the entire group of 2N p chromosomes is ranked. The best N p of these chromosomes (in terms of Pareto rank) are retained as the parents for the next generation. Since it preserves the best designs from a former generation, this operation is a form of elitism, and adding it to the NSGA gives rise to the ENSGA. Fig.1 shows a flowchart of the ENSGA. 2.2 Implementation of parallel architecture into ENSGA Like most GAs, the ENSGA can be easily parallelized, and that parallelization results in a near linear acceleration of the algorithm. The method for parallelizing ENSGA used in this paper is to have one processor (the master) send objective function evaluations to each of the other (slave) processors only when requested to do so. In other words, if L (slave) computers are

to be used in the optimization, the master node sends out L objective function evaluations, one to each processor, at the beginning of each generation. When each processor finishes its evaluation, it sends its objective function values back to the master node, and receives a new chromosome to evaluate. In this way, all processors are kept busy until the generation ends and a new generation starts.

Given that the objective function evaluations do not differ much in terms of

computational expense, this method usually results in acceleration by nearly a factor of L.

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3. Planar absorbers In this section, the ENSGA described in the previous sections is compared with the NSGA and two classical optimization methods for the design of flat multilayer absorbers. The multilayer absorber is a good first test case since the reflection coefficient can be efficiently calculated using an exact analytical method. Fig. 2(a) shows the geometry of the multilayer coating under investigation.

Specifically, the coating consists of four layers of different

materials. The thickness of each layer is denoted by ti and the permittivity of each layer is defined to be ε i = ε i′ − jε i′′ for i = 1, 2, 3, and 4. The material for the each layer is selected from a predefined database of eight materials compiled from reference [4]. The material constants are shown in Table I. Since the permittivity of these materials did not vary much with frequency over the design range (19-37 GHz), they have been approximated as constant. This was done for simplicity of presentation and in no way constitutes a limitation of the technique. The objectives in this problem are the minimization of the maximum reflectivity in decibels between 19 and 37 GHz for normal incidence, and the minimization of the total thickness of the absorber. Classical methods for bi-objective Pareto optimization convert the two objective functions into a single objective using a scalar parameter. The Pareto front is found by iteratively optimizing the single objective function for different values of the parameter. The optimization can be performed with any optimization technique; a simple GA is used here. Two classical methods are used: ● In the weighted Tchebycheff procedure [7] the combined objective function is given by

F = max α T , 1 − α 2 ( R − R G )   

(1)

where α ranges from 0 to 1 and R G is an ideal value for the reflection coefficient. It can be shown that this iterated minimization is mathematically equivalent to the Pareto definition. ● In the exact penalty method [18], the function to be optimized is of the form

F = R + Ξ (T − α )

2

(2)

where Ξ is a very large number. Minimization of this function forces T to equal α , so that allowing α to range over all possible thicknesses returns the front.

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In the first experiment presented here, each GA ran with a population size of 1000 for 500 generations.

Fig. 2(b) compares the results obtained by the weighted Tchebycheff

procedure, the exact penalty method, and the NSGA for the four-layer coating. We see that the results obtained by the exact penalty method and NSGA are almost identical, and that the Tchebycheff procedure returns may suboptimal points. The NSGA converges at generation 200 as shown in Fig. 2(c), requiring 200000 evaluations of the objective function to find 48 points on the front. The exact penalty method with the simple GA conducts 175000000 evaluations of function to get the almost same number of points on the front. The ENSGA can further speed up the convergence as shown in Fig. 2(c), which shows that the ENSGA converges at generation 50, and thus only 50000 evaluations of the objective function. (This experiment was conducted several times with nearly identical results.

Thus, the result presented here is

intended to be representative of a larger trend.) The reader may complain that the foregoing experiment is unfair; after all, the NSGA and ENSGA are designed to find Pareto fronts in a single run, whereas standard GAs operating on classical formulas return only one point per run regardless of the population size. Thus, another experiment was run in which the population size in the classical methods was reduced to level the playing field. Specifically, we ran the exact penalty method again, with the value of T obtained by the ENSGA. In the simple GA, we set the population size and number of

generations to 100 and 200 respectively, so that the total number of objective function evaluations is 100*200*48=960000. The results returned by this procedure are shown in Fig. 2(d). Note that the Pareto front returned by the exact penalty method is inferior to that returned by the NSGA, even though it was allowed more objective function evaluations.

This

conclusion is reached not matter how the total objective function count is assigned in the algorithm; that is, the result of returned by the exact penalty method is inferior to that returned by the NSGA if the population size is taken as 200 and the number of generations is taken as 100 as well. In short, the exact penalty method can return a result as good as that returned by the NSGA, but only at a much greater cost. The Tchebycheff procedure returns a front inferior to that returned by the exact penalty method. It finds only two points on the front in the concave region where T ∈ [0, 0.15 cm] . For the convex region of the front ( T ∈ [0.15 cm, 0.35 cm] ), the Tchebycheff procedure finds a number of points on the front, but the points are distributed around the ends of the front.

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4. Polygonal absorbers In this section, we apply the parallel ENSGA technique to find the Pareto fronts for absorbers with polygonal geometry models.

Three models have been tested for several

requirements to demonstrate the flexibility, effectiveness, and robustness of the ENSGA discussed in this paper for absorber design problems. Fig. 3 shows the geometries under consideration. The absorber is divided into two regions: a substrate region and a shaped region. All of the models studied here are periodic and possess mirror symmetry with respect to the y-axis. The reflection coefficients for these three models depend on the materials used in the both regions, and geometrical parameters ti described below. The shaped regions for the three models are: 1) Model 1 shown in Fig.3 (a) is a standard conical absorber containing a single cone (per period) of height t2 atop a substrate of thickness t1 . Thus, the total thickness of a model 1 absorber is given by T = t1 + t2 . 2) Model 2 absorbers are slightly more complex. The period D is centered on the x-axis and divided into 2N segments of equal length. The endpoints of these segments are located at the points xi = − D / 2 + D (i − 1) / 2 N . The variable t1 represents the height of the substrate. The profile of the absorber is further defined by the variables t2 through t N +1 . Specifically, the height of the cone at the point x j for j = 2,..., N + 1 is taken to be j

yj = ∑tj .

(3)

i =1

The height of the cone at points x j for j = N + 2,..., 2 N + 1 is given by the formula y j = y2 N + 2− j .

The cone absorbing material lies in the polygonal region

between the substrate and the line segments connecting the points ( x j , y j ) for j = 2,..., 2 N . The total height of a model 2 design is thus T = yN +1 . A sample model 2 absorber is shown in Fig. 3(b) with N = 4. This model is very similar to the model used in [14].

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3) Model 3 is the same as the model 2 except that the ti ( i = 1, 2...N + 1) are defined as shown in Fig.3 (c). That is, the ti now describe the total height of the cone at the points xi instead of the increment in the height for the points x2 ,..., xN +1 . Thus, the thickness of a model 3 absorber is given by T = t1 + max(t2 , t3 ...t N +1 ) . Fig. 3(c) shows a model 3 geometry for N = 4 . The ENSGA will be used to simultaneously minimize the reflection coefficient and thickness of the absorber design. The thickness T for each model is defined above. The reflection coefficient used in this work is the maximum reflection coefficient (in dB) over all incidence angles and frequencies. The reason for choosing this criterion was explained in detail in reference [5]. The first designs presented here are created with the following parameter choices: (1) N = 4 for both models 2 and 3, (2) the operating frequency band is 19 to 37 GHz, (3) TM

polarization, (4) normal incidence, and (5) the period D for the three models is set to be 0.6 cm. To ensure that the maximum allowable T is the same for the each of the three models, the lower and upper bounds of the geometrical parameters (in cm) for the three models are given as follows: 1) Model 1: t1 ∈ [0,1] and t2 ∈ [0,3] 2) Model 2: ti ∈ [0,1] (i = 1, 2..5) 3) Model 3: t1 ∈ [0,1] and ti ∈ [0, 3] (i = 2,3..5) The length of the binary string used in the ENSGA to denote each geometrical parameter is set to 10 bits for ti varying from 0 to 1cm, and 12 bits for ti varying from 0 to 3cm. Fig. 4 shows the Pareto fronts for the three absorbers. Each point in this figure represents a design that is optimal in the combination of performance and thickness it delivers under different conditions. (Of course, the design parameters of each design can be retrieved from the ENSGA.) From this figure, we can conclude that both models 2 and 3 are better than model 1 for following reasons: 1) For the same reflection coefficient, models 2 and 3 require much smaller profiles than model 1; for instance if the reflection coefficient is required to -28dB for the frequency band under consideration, the minimum thicknesses for the model 1, 2 and 3 are 2.97 cm, 0.867 cm and 0.915 cm respectively.

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2) Model 2 is more absorptive than model 1, and model 3 is more absorptive than model 2. The lowest reflection coefficients for models 1, 2, and 3 are -27.6dB, -31.1dB, and -41.7 dB respectively. These facts are reasonable and obvious, because model 2 is special case of model 3 and model 1 is a special case of model 2. This comparison suggests that Pareto front can be a way to evaluate the performance of absorber models by comparing their Pareto fronts. Of course, model 1 absorbers are easier to fabricate than model 2 absorbers, which are easier to fabricate than model 3 absorbers. Increasing N can further improve models 2 and 3 absorbers. This improvement can manifest itself both as reduction of profile and reflection coefficient. Fig. 5(a) compares the Pareto fronts for model 2 with N = 4 and N = 10 . It is clear that for a given minimum thickness, model 2 with higher value of N can improve the absorber’s performance. Fig. 5(b) depicts the Pareto fronts for model 2 with N = 10 for the following conditions: (1) only TM polarization is considered for normal incidence; (2) both TM and TE polarizations are considered together for normal incidence; and (3) TM polarization with incident angles varied from 0 degrees (normal incidence) to 30 degrees. From Fig.5 (b), it is obvious that if more restraints are enforced, thicker absorbers are required, as could reasonably be expected. This fact strongly demonstrates the effectiveness and robustness of the ENSGA. While the stated application of Pareto genetic algorithms is the location of the best tradeoffs between conflicting design goals, they also can be used if one goal is of paramount interest. In particular, if the minimization of reflection coefficient is much more important than profile minimization the ENSGA will find a design as absorbent as a single objective GA. Moreover, it will almost certainly find the thinnest design such design as well. To emphasize this advantage, we use the standard GA to find the profile for model 2 with N = 10 that minimizes the reflection coefficients for both polarizations at normal incidence. We ran the simple GA twice, and found two optimized absorbers denoted as design 1 and design 2. For comparison, the most absorptive point on the front for both TM and TE polarizations of Fig. 5(b) is denoted as design 3. The corresponding geometries for the three designs are shown in Fig. 6(a), the variation of the reflection coefficients of these three designs as a function of frequency are shown in Fig. 6(b). The reflection coefficients, the required total thicknesses, and material makeup for the three designs are listed in Table II. Table II and Fig. 6 show that both ENSGA

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and the standard GA find the almost same lowest reflection coefficients for this absorber model, but design 3 found by the ENSGA has the smallest thickness. This fact clearly shows the advantage of the ENSGA over the simple GA for absorber design. Moreover, the fact that both methods return the same optimal reflection coefficient lends credence to both methods’ robustness and the quality of the results returned. The next design problem involves the design of absorbers constructed from a database of eight carbon foam materials for the frequency band from 3GHz to 18GHz. The materials are nonmagnetic, the measured real and imaginary parts of the relative dielectric permittivity ( ε ′ and ε ′′ ) of the eight materials are plotted in Fig. 7(a) and Fig. 7(b), respectively. The goal is to design absorbers using models 2 and 3 with N = 10 for over the entire band 3GHz to 18GHz, considering (1) both polarizations and (2) incidence angles ranging from 0 to 30 degrees. Fig. 8 shows the Pareto fronts found by the ENSGA for both models. Specially, for model 2, each of the ti was allowed to vary between 0 and 1 cm. For model 3, the substrate is allowed to be between 0 and 1 cm thick, and each of the cones had a height that ranged from 0 to 10 cm. Thus, the maximum allowed height for each case is the same. Fig.8 confirms that model 3 results in better absorbers than model 2: the lowest reflection coefficient for model 3 is −29.49dB while it is −23.69dB for model 2. Moreover, for the same required low reflection

coefficient, absorbers defined by model 3 are thinner than those from model 2 are. Two designs, which are indicated in Fig. 8, are chosen to illustrate the designed absorbers. Design 1 and design 2 are from the front for model 2 and the model 3 front respectively. Table III lists the maximum reflection coefficient, total thickness, and constituent materials of the two selected designs. Fig. 9 shows corresponding geometries of the two designs. Although 50 slave processors with 1.4 GHz CPU were used to optimize this problem, it still took 2 hours to run one generation with a population size of 1000 for wide-incidence angles, wide-frequency band and both polarizations considered together. These numbers translate into computation time of 25 days for running 300 generations. To accelerate the process, a scheme first presented in [5] was used. First, we optimized the geometries for operation in the lower part of the frequency (3~10 GHz) band for 250 generations. Because of the smaller frequency band, only 15 minutes per generation were needed to do this. Next, the designs obtained by this process were used as the starting population for an ENSGA optimizing the reflection coefficient of the absorbers over the whole band. Using this technique, the entire process only took 7 days.

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5. Conclusions This paper has introduced a new algorithm, the ENSGA, created by incorporating a novel type of elitism into the NSGA. Numerical results for the design planar absorbers have confirmed the improvement and the utility of the approach relative to the standard NSGA. This ENSGA was also combined with the FE-BI to design polygonal absorbers of differing models and performance requirements. Various numerical results have demonstrated the versatility and advantages of the approach discussed in this paper.

In comparison to conventional GA

approaches, Pareto GAs have the advantage of being able to consider more than one goal at a time; a virtual necessity for real design problems. Indeed, because Pareto GAs can incorporate any number and type of performance metrics, they may be used to evaluate the applicability of different models for different problems. Future work involves the extension of the present technique to three dimensions. Because of the greatly increased computational load, further improvements in efficiency will be necessary. Nonetheless, the elitism introduced here should be invaluable in beginning to analyze this problem.

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References 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, August 1997. 3. C. L. Holloway, R. R. Delyser, 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. Electromag. Compat., vol. 39, pp. 33-47, February 1997. 4. S. Chakravarty, R. Mittra and N. B. Williams, “ Application of a microgenetic algorithms (MGA) to the design of broad-band microwave absorbers using multiple frequency selective surface screen buried in dielectrics,” IEEE Trans. on Microwave Theory and Techniques, vol. 50, no. 3, pp. 284-296, March 2002. 5. S. Cui and D.S. Weile, “Robust Design of Absorbers Using Genetic Algorithms and the Finite element–Boundary Integral Method,” IEEE Trans. Antennas Propagat., vol. AP-51, pp. 3249-3258, December 2003. 6. E. Michielssen, J. M. Sajer, S. Ranjithan, and R. Mittra, “Design of lightweight, broadband microwave absorbers using genetic algorithms,” IEEE Trans. Microwave Theory and

Techniques, vol. 41, pp. 1024-1031, June-July 1993. 7. D. S. Weile, E. Michielssen, and D. E. Goldberg, “Genetic algorithm design of Pareto optimal broadband microwave absorbers,” IEEE Trans. Electromag. Compat., vol. 38, pp. 518-525, August 1996. 8. D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans. on Antennas and Propagation, vol. 45, no. 3, pp. 343–353, March 1997. 9. J. L. Cohon, Multiobjective programming and planning. New York: Academic press, 1978. 10. E. R. Steuer, Multiple criteria optimization: theory, Computation, Application. Malabar, FL: Krieger, 1978. 11. N. Srinivas and K. Deb, “ Multiobjective optimization use nondominated sorting in genetic algorithms”, Evolutionary Computat., vol.2, no.3, pp.221-248, 1995.

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12. D.E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison Wesley, 1989. 13. E. Zitzler, K. Deb and L. Thiele “ Comparison of multiobjective evolutionary algorithms: empirical results,” Evolutionary Computat., vol.8, no.2, pp.173-195, 2000. 14. H. Choo, H. Ling, C. S. Liang, “Shape optimization of corrugated coatings under grazing incidence using a genetic algorithm,” IEEE Trans. on Antennas and Propagation, vol. 51, no.12, pp. 3080-3087, Nov 2003. 15. S. D. Gedney, J. Jin, and R. Mittra, “Combined FEM/MoM approach to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. on Microwave Theory and Techniques, vol. 40, no. 2, pp. 364-370, February 1992. 16. J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, 1993. 17. J. L. Volakis, A. Chatterjee, and L.C. Kempel, Finite Element Method for Electromagnetics, IEEE Press: New York, 1998. 18. J. Nocedal and S.J. Wright, Numerical Optimization, Springer, 1999.

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Table captions

Table I. Database of Materials Table II. Comparison of the performance of the three designed absorbers Table III. Performance, minimum thicknesses, and material choice of the two designed absorbers

Material ε' ε ''

1 4.48 1.87

2 5.84 1.66

Design 1 Design 2 Design 3

Design 1 Design 2

3 5.21 1.18

Table I 4 7.08 2.32

5 9.84 4.95

6 11.87 9.72

7 12.73 8.13

Table II R (dB) T (cm) -33.6 3.92 -33.4 3.10 -33.7 1.94

Material 1 2 2 8

Material 2 1 1 1

Table III R (dB) T (cm) -23.68 5.89 -29.49 9.81

Material 1 6 7

Material 2 5 5

16

8 17.97 14.57

Figure captions

Fig. 1 A flowchart of the ENSGA. Fig. 2 Results for a four layer absorber: (a) The geometry of a 4-layer coating under investigation, (b) The Pareto fronts for the model shown in Fig.4 (a), (c) the convergence history of NSGA and ENSGA (generation numbers are indicated by a pound symbol), and (d) NSGA vs. the exact penalty method. Fig. 3 Polygonal geometrical models of absorbers. Fig. 4 Comparison of the Pareto fronts for different absorber models Fig. 5 Model 2 Pareto fronts: (a) Effect of the value of N on the absorber, and (b) comparison of the fronts for different objective functions. Fig. 6 Comparison of absorbers designed by GA and ENSGA: (a) profiles, and (b) reflection coefficients. Fig. 7 The measured dielectric constants of the eight carbon foam materials Fig. 8 Pareto fronts for model 2 and model 3 (N=10) Fig. 9 Geometries of two selected designs from Fig.8

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Generate initial population Evaluate objective functions Rank individuals Set nominal fitness & evaluate shared fitness Perform selection, crossover & mutation Evaluate objective functions Rank individuals END Apply No elitism operator

Maximum number of iteration is reached ?

Yes

Output Pareto front

Fig. 1

0

Material 4 ε 4

t3

Material 3

ε3

t2

Material 2

ε2

t1

Material 1

ε1

Perfect Electric Conductor

-2 Reflection coefficient (dB)

t4

NSGA Penalty function GA Tchebycheff GA

-4 -6 -8 -10 -12 0.0

0.1

0.2

0.3

Thickness (cm)

Fig. 2(a)

Fig. 2(b)

18

0.4

0.5

0

0 NSGA #001 NSGA #100 NSGA #200 NSGA #500 ENSGA #50

-4

-2 Reflection coefficient (dB)

Reflection coefficient (dB)

-2

No discrepancy

-6 -8 -10 -12 0.0

0.1

0.2

0.3 0.4 0.5 Thickness (cm)

0.6

NSGA Exact penalty method

-4 -6 -8 -10 -12 0.0

0.7

0.1

Fig. 2(c)

Fig. 2(d) y

t2 Material 2

−

t1

D 2

Material 1

D 2

x

PEC

D

Fig. 3(a) y 5

t5 4 3

6

t4

7

t3 8

2 1 −

0.2 Thickness (cm)

D 2

t2

Material 2

t1

Material 1 PEC D

Fig. 3(b )

19

9 D 2

x

0.3

0.4

y

t5 t2 −

D 2

t3 t4

t1

Material 2

D 2

Material 1

D

PEC

Fig. 3(c)

5 0

Model 1 Model 2 Model 3

Reflection coefficient (dB)

-5 -10 -15 -20 -25 -30 -35 -40 -45 0.0

0.5

1.0 1.5 2.0 Thickness (cm)

Fig. 4

20

2.5

3.0

x

0

Reflection coefficient (dB)

-5

Model 2 N=4 Model 2 N=10

-10 -15 -20 -25 -30 -35 -40 0.0

0.5 1.0 Thickness (cm)

1.5

Fig. 5(a) 0

Reflection coefficient (dB)

-5 -10

TM, normal incidence TM &TE, normal incidence TM ,multiple incidence

-15 -20 -25 -30 -35 0.0

0.5

1.0 1.5 2.0 Thickness (cm)

Fig. 5(b)

21

2.5

3.0

5.5 Geometry 1 (GA) Geometry 2 (GA) Geometry 3 (ENSGA)

5.0 4.5 4.0

y (cm)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

x (cm)

Fig. 6(a)

Reflection coefficient (dB)

-30

-35

-40

-45

Design 1 TM Design 1 TE Design 2 TM Design 2 TE Design 3 TM Design 3 TE

-50

-55 18

20

22

24

26 28 30 32 Frequency (GHz)

Fig. 6(b)

22

34

36

38

8 7 1

10 Relative permittivity

6

5 4 3 2

1 0

10

2

4

6

8

10

12

14

16

18

20

Frequency (GHz)

Fig. 7(a)

1

Relative permittivity

10

8 7 6 5

0

10

4 3 -1

10

2 1

-2

10

2

4

6

8

10 12 14 Frequency (GHz)

Fig. 7(b)

23

16

18

20

0

Reflection coefficient (dB)

-5

Model 2 (N=10) Model 3 (N=10)

-10 Design 1

-15 -20

Design 2

-25 -30 0

1

2

3

4 5 6 7 Thickness (cm)

8

9

10

Fig. 8 10 9

Geometry 1 Geometry 2

8 7

y (cm)

6 5 4 3 2 1 0 -0.8

-0.6

-0.4

-0.2

0.0 0.2 x (cm)

Fig. 9

24

0.4

0.6

0.8