Application of Bio-Inspired Algorithms and Neural Networks for Optimal Design of Fractal Frequency Selective Surfaces

Technological advances in the field of microwave and communication systems and the increase of their commercial applications in recent years have resulted in more stringent requirements for innovative design of microwave passive devices, such as: antennas, filters, power splitters and couplers, frequency selective surfaces, etc. To be competitive in the commercial marketplace, microwave engineers may be using computer-aided design (CAD) tools to minimize cost and design cycle times. Modern CAD tools have become an integral part of the microwave product cycle and demand powerful optimization techniques combined with fast and accurate models so that the optimal solutions can be achieved, eventually guaranteeing first-pass design success. The target of microwave device design is to determine a set of physical parameters to satisfy certain design specifications (Mohamed, 2005).


Introduction
Technological advances in the field of microwave and communication systems and the increase of their commercial applications in recent years have resulted in more stringent requirements for innovative design of microwave passive devices, such as: antennas, filters, power splitters and couplers, frequency selective surfaces, etc.To be competitive in the commercial marketplace, microwave engineers may be using computer-aided design (CAD) tools to minimize cost and design cycle times.Modern CAD tools have become an integral part of the microwave product cycle and demand powerful optimization techniques combined with fast and accurate models so that the optimal solutions can be achieved, eventually guaranteeing first-pass design success.The target of microwave device design is to determine a set of physical parameters to satisfy certain design specifications (Mohamed, 2005).
Early methods of designing and optimizing microwave devices by hand are time and labor intensive, limit complexity, and require significant expertise and experience.Many of the important developments in microwave engineering were made possible when complex electromagnetic characteristics of microwave devices were represented in terms of circuit equivalents, lumped elements and transmission lines.Circuit simulators using empirical/analytical models are simple and efficient, reduce optimization time, but have limited accuracy or validity region.Although circuit simulator is still used today it suffers from some severe limitations (the most serious of them is that it considers only fundamental mode interactions) and requires corrections in the form of post manufacturing tuning (Fahmi, 2007).
While developments in circuit simulators were taking place, numerical electromagnetic (EM) techniques were also emerging.With the computational power provided by modern computers, the use of accurate full-wave electromagnetic models by EM simulators for design and optimization of microwave devices became possible.By using full-wave electromagnetic methods higher order modes are taken into consideration and microwave devices can be rigorously characterized in the designs so that simulation and experimental results are in close agreement.This is particularly of interest for the rapid large scale production of low-cost high performance microwave devices reducing or eliminating the need of post manufacturing tuning (Bandler et al., 1994;Fahmi, 2007).
The EM simulators can simulate microwave device structures of arbitrary geometrical shapes and ensure a satisfactory degree of accuracy up to millimeter wave frequencies (Mohamed, 2005).These simulators are based on EM field solvers whose function is to solve the EM problem of the structure under analysis, which is described by the Maxwell´s equations.Thus, the design of electromagnetic structures is usually a very challenging task due to the complexity of the models involved.In the majority of cases, there are no simple analytical formulas to describe the performance of new microwave devices.However, the use of EM field solver for device optimization is still a time consuming procedure and need heavy computations.For complex problems, resulting in very long design cycles, this computational cost may be prohibitive (Haupt & Werner, 2007).
Actually, many approaches are available to implement optimization using full-wave methods.For instance, the exploitation of commercial EM software packages inside the optimization loop of a general purpose optimization program.New techniques, such as geometry capture (Bandler et al., 1996) (suitable for automated EM design of arbitrary threedimensional structures), space mapping (Bandler et al., 1994) (alternative design schemes combining the speed of circuit simulators with the accuracy of EM solvers), adjoint network concept (Nikolova et al., 2004), global optimization techniques based on bio-inspired algorithms, knowledge based methods, and artificial neural networks (ANNs), establish a solid foundation for efficient optimization of microwave device structures (Haupt & Werner, 2007;Zhang & Gupta, 2000;Silva et al., 2010a).This chapter presents a new fast and accurate EM optimization technique combining fullwave method of moments (MoM), bio-inspired algorithms, continuous genetic algorithm (GA) and particle swarm optimization (PSO), and multilayer perceptrons (MLP) artificial neural networks.The proposed optimization technique is applied for optimal design of frequency selective surfaces with fractal patch elements.A fixed FSS screen geometry is choose a priori and then optimizing a smaller subset of FSS design variables to achieve a desired bandstop filter specification.
A frequency selective surface (FSS) is a two-dimensional array of periodic metallic elements on a dielectric layer or two-dimensional arrays of apertures within a metallic screen.This surface exhibits total reflection or transmission for patch and aperture elements, respectively.The most important parameters that will determine the overall frequency response of a FSS are: element shape, cell size, orientation, and dielectric layer properties.FSSs have been widely used as spatial filters for plane waves in a variety of applications, such as: microwave, optical, and infrared filters, bandpass radomes, microwave absorbers, polarizers, dichroic subreflectors, antenna systems, etc. (Munk, 2000).
Several authors proposed the design of FSS using fractals.In this chapter, different fractal geometries are considered, such as: Koch, Dürer's pentagon, and Sierpinski.While the use of space-filling fractal properties (e.g., Koch, Minkowski, Hilbert) reduce the overall size of the FSS elements (Oliveira et al., 2009;Campos et al., 2010), the attractive features of certain selfsimilar fractals (e.g., Sierpinski, Gosper, fractal tree, etc.) have received attention of microwave engineers to design multiband FSS.Many others self-similar geometries have been explored in the design of dual-band and dual polarized FSS (Gianvittorio et al., 2001).
The self-similarity property of these fractals enables the design of multiband fractal elements or fractal screens (Gianvittorio et al., 2003).Furthermore, as the number of fractal iterations increases, the resonant frequencies of these periodic structures decrease, allowing the construction of compact FSSs (Cruz et al., 2009).In addition, an FSS with fractal elements present resonant frequency that is almost independent of the plane-wave incidence angle.
There is no closed form solution directly from a given desired frequency response to the corresponding FSS with fractal elements.The analysis of scattering characteristics from FSS devices requires the application of rigorous full-wave techniques.Besides that, due to the computational complexity of using a full-wave simulator to evaluate the FSS scattering variables, many electromagnetic engineers still use trial-and-error process until to achieve a given design criteria.Obviously this procedure is very laborious and human dependent.On the other hand, calculating the gradient of the scattering coefficients in terms of the FSS design variables is quite difficult.Therefore, optimization techniques are required to design practical FSSs with desired filter specifications.Some authors have been employed neural networks, PSO, and GA for FSS design and optimization (Manara et al., 1999;Hussein & El-Ghazaly, 2004;Silva et al., 2010b).
The main computational drawback for EM optimization of FSSs based on bio-inspired algorithms relies on the repetitive evaluation of numerically expensive fitness functions.Due the expensive computation to calculate the scattering variables for every population member at multiple frequencies over many generations, several schemes are available to improve the GA performance for optimal design of FSSs, such as: the use of fast full-wave methods, micro-genetic algorithm, which aims to reduce the population size, and parallel GA using parallel computation.However, despite of these improvements done on the EM optimization using genetic algorithms, all the same several hours are required for expensive computational simulations of GA optimization (Haupt & Werner, 2007;Silva et al., 2010b).
The application of ANNs as approximate fitness evaluation tools for genetic algorithms, though suggest often, had seldom been put to practice.The combination of ANNs and GAs has been applied mainly for the construction of optimized neural networks through GAbased optimization techniques.Few applications of ANNs to GA processing have been reported for EM optimization of microwave devices.
The advantages of the MoM-ANN-GA/PSO optimization technique are discussed in terms of convergence and computational cost.This technique is applied for optimal design of bandstop FSS spatial filters with fractal elements considering the resonant frequency (fr) and bandwidth (BW) bandstop specifications.Some FSS prototypes with fractal elements are built and measured.The accuracy of the proposed optimization technique is verified by means of comparisons between theoretical and experimental results.

An overview of bio-inspired optimization technique
The idea of blending full-wave methods, artificial neural networks, and bio-inspired optimization algorithms for electromagnetic optimization of FSS spatial filters was first proposed in 2007 (Silva et al., 2007).This optimization technique named MoM-ANN-GA replaces the computational intensive full-wave method of moments simulations by a fast and accurate MLP neural network model of FSS spatial filter, which is used to compute the cost (or fitness) function in the genetic algorithm iterations.
The proposed bio-inspired EM optimization technique starts with the definition of a FSS screen geometry that is choose a priori.A full-wave parametric analysis is carried out for accurate EM characterization of FSS spatial filter scattering properties.From obtained EM dataset, a MLP network is trained to establish the complicated relationships between FSS design variables and frequency response.Then, in order to overcome the computational requirements associated with full-wave numerical simulations, the developed MLP model is used for fast and accurate evaluation of fitness function into bio-inspired algorithm simulations.From the optimal design of FSS parameters, FSS prototypes are fabricated and measured for verification of optimization methodology.Fig. 1 gives a "big picture" overview of proposed bio-inspired EM optimization technique.

Fig. 1. An overview of proposed bio-inspired optimization technique
This section is a brief introduction that provides an overview of the proposed optimization technique to be presented.The overview includes fundamentals of multilayer perceptrons, continuous genetic algorithm, and particle swarm optimization.

Artificial neural networks
Since the beginning of the 1990s, the artificial neural networks have been used as a flexible numerical tool, which are efficient for modeling of microwave devices.In the CAD applications related to microwave engineering, the use of ANNs as nonlinear models becomes very common.Neural network models trained by accurate EM data (obtained through measurements or by EM simulations) are used for fast and accurate design/optimization of microwave devices.In addition, the use of previously established knowledge in the microwave area (as empirical models) combined with the neural networks, results in a major reliability of the resulting hybrid model -with a major ability to learn nonlinear input-output mappings, as well as to generalize responses, when new values of the input design variables are presented.Another important advantage is the data amount reduction necessary for the neural networks training.Some hybrid modeling techniques have been proposed for the use with empirical models and neural networks, such as: Source Difference Method, PKI (Prior Knowledge Input), KBNN (Knowledge Based Neural Network), and SMANN (Space Mapping Artificial Neural Network) (Zhang & Gupta, 2000).
Versatility, efficient computation, reduced memory occupation, stability of learning algorithms, and generalization from representative data, are some characteristics that have motivated the use of neural networks in many areas of microwave engineering as models for complex ill-defined input-output mappings in new, not well-known microwave devices (Santos et al., 1997;Patnaik & Mishra, 2000;Zhang & Gupta, 2000).As mentioned previously, the electromagnetic behavior of a microwave device is extremely complex and simple empirical model cannot accurately describe its behavior under all conditions.Only with a detailed full-wave device model, more accurate results can be found.In general, the quality of simulation is decided by the accuracy of device models.On the other hand, a very detailed model would naturally slow down the program.A compromise between accuracy and speed of computation has to be struck.Using neural networks enables to overcome this problem (Silva et al., 2010a).
The multilayer perceptrons is the most used artificial neural network for neuromodeling applications.Multilayer perceptrons artificial neurons are based on the nonlinear model proposed by (McCulloch & Pitts, 1943;Rosenblatt, 1958, as cited in Haykin, 1999).In this model, neurons are signal processing units composed by a linear combiner and an activation function, that can be linear or nonlinear, as shown in Fig. 2.

Fig. 2. Nonlinear model of an artificial neuron
The input signals are defined as i x , where N i is the number of input units.The output of linear combiner corresponds to the neuron level of internal activity j net , as defined in (1).The information processed by neuron is storage in weights ji w , 1, ,N j j =  , where N j is the number of neurons in a given neural network layer; 0 1 x =± is the polarization potential (or threshold) applied to the neurons.The neuron output signal j y is the value of the activation function () Multilayer perceptrons presents a feed forward neural network (FNN) configuration with neurons set into layers.Each neuron of a layer is connected to those of the previous layer, as illustrated in Fig. 3. Signal propagation occurs from input to output layers, passing through the hidden layers of the FNN.Hidden neurons represent the input characteristics, while output neurons generate the neural network responses (Haykin, 1999).

Fig. 3. Feed forward neural network configuration with two hidden layers
The design of a MLP model consists by three main steps: i) configuration -how layers are organized and connected; ii) supervised learning -how information is stored in neural network; iii) generalization test -how neural network produces reasonable outputs for inputs not found in the training set (Haykin, 1999).In this work, we use feed forward neural networks and supervised learning to develop MLP neural network models.
In the computational simulation of supervised error-correcting learning, a training algorithm is used for the adaptation of neural network synaptic weights.The instantaneous error (n) e , as defined in (3), represents the difference between the desired response (n) d , and the neural network output () n y , at the n-th iteration, corresponding to the presentation of the n-th training example, ( ) Training examples variables are normalized to present unitary maximum absolute value.So, when using a given MLP model, prior scaling and de-scaling operations may be performed into input and output signals of MLP neural network, according to (4) and ( 5), respectively.Haykin, 1999).Among the family of back-propagation algorithms, the RPROP algorithm shows to be very efficient in solving complex modeling learning tasks.
After neural network training, we hope that MLP weights will storage the representative information contained on training dataset.The trained neural network is tested in order to verify its capability of generalizing to new values that do not belong to the training dataset.Therefore, the MLP neural network operates like a "black box" model inside a given region of interest, which was previously defined when the training dataset was generated.

Bio-inspired optimization algorithms
Bio-inspired algorithms, which are stochastic population-based global search methods inspired by nature, such as simulated annealing (SA), genetic algorithm and particle swarm optimization are effective for optimization problems with a large number of design variables and inexpensive fitness function evaluation (Haupt, 1995;Haupt & Werner, 2007;Kennedy & Eberhart, 1995).However, the main computational drawback for optimization of microwave devices relies on the repetitive evaluation of numerically expensive fitness functions.Finding a way to shorten the optimization cycle is highly desirable (Silva et al., 2010b).For instance, several GA schemes are available in order to improve its performance, such as: the use of fast full-wave methods, micro-genetic algorithm, which aims to reduce the population size, and parallel GA using parallel computation (R. L. Haupt & Sue, 2004).
Bio-inspired algorithms start with an initial population of candidate individuals for the optimal solution.Assuming an optimization problem with N var input variables and N pop individuals, the population at the i-th iteration is represented as a matrix P(i) Npop×Nvar of floating-point elements, denoted by , i mn p , with each row corresponding to an individual.
Under GA and PSO jargons, the individuals are named chromosomes and particles (or agents), respectively.

Continuous genetic algorithm
Continuous genetic algorithm is very similar to the binary-GA but works with floating-point variables.Continuous-GA chromosomes are defined in (7) as a vector with N var floatingpoint optimization variables.Each chromosome is evaluated by means of its associated cost, which is computed through the cost function E given in (8).
, ( ) Based on the cost associated to each chromosome, the population evolves through generations with the application of genetic operators, such as: selection, crossover and mutation.Flow chart shown in Fig. 4(a) gives an overview of continuous-GA.
Mating step includes roulette wheel selection presented in (Haupt & Werner, 2007; R. L. Haupt & Sue, 2004).Population selection is performed after the N pop chromosomes are ranked from lowest to highest costs.Then, the N keep most-fit chromosomes are selected to form the mating pool and the rest are discarded to make room for the new offspring.
Mothers and fathers pair in a random fashion through the blending crossover method (R. L. Haupt & Sue, 2004).Each pair produces two offspring that contain traits from each parent.
In addition, the parents survive to be part of the next generation.After mating, a fraction of chromosomes in the population will suffer mutation.Then, the chromosome variable selected for real-value mutation is added to a normally distributed random number.
Most users of continuous-GA add a normally distributed random number to the variable selected for mutation with a constant standard deviation (R. L. Haupt & Sue, 2004).In particular, we propose a new real-value mutation operator for continuous-GA as given in (9), where p max and p min are constant values defined according to the limits of the region of interest composed by input parameters.Function randn() returns a normal distribution with mean equal to zero and standard deviation equal to one.
This mutation operator was inspired by simulating annealing cooling schedules (R. L. Haupt & Sue, 2004).It is used to improve continuous-GA convergence at the neighbourhood of global minimum.The quotient function Q given in ( 10) is crescent when the number of iterations increases and the global cost decreases.Thus, similar to the decrease of temperature in a simulating annealing algorithm, the standard deviation is decreased when the number of continuous-GA iterations is increased.The parameter A is a constant value and B is a value of cost function neighbour to the global minimum.The continuous-GA using the real-value mutation definition given in ( 9) and ( 10) is denominated improved genetic algorithm.

Particle swarm optimization
Particle swarm optimization was first formulated in 1995 (Kennedy & Eberhart, 1995).The thought process behind the algorithm was inspired by social behavior of animals, such as bird flocking or fish schooling.PSO is similar to continuous-GA since it begins with a random initial population.Unlike GA, PSO has no evolution operators such as crossover and mutation.Each particle moves around the cost surface with an individual velocity.The implemented PSO algorithm updates the velocities and positions of the particles based on the local and global best solutions, according to ( 11) and ( 12), respectively.

FSS design considerations
Frequency selective surfaces (FSSs) are used in many commercial and military applications.Usually, conducting patches and isotropic dielectric layers are used to build these FSS structures.FSS frequency response is entirely determined by the geometry of the structure in one period called a unit cell.In this section is presented some considerations about the design of FSS with fractal elements for operation at the X-band (8-12 GHz) and Ku-band (12-18 GHz).FSS fabrication and measurement procedures are summarized.

Design of FSS using fractal geometries
FSS with fractal elements has attracted the attention of microwave engineering researchers because of its particular/special features.The design of a FSS with pre-fractal elements is a very competitive solution that enables the fabrication of compact spatial filters, with better performances when compared to conventional structures (Oliveira, et al., 2009;Campos et al., 2010).Several fractal iterations can be used to design a FSS with multiband frequency response associated to the self-similarity contained in the structure.Various self-similar fractals elements (e.g., Koch, Sierpinski, Minkowski, Dürer's pentagon) were previously used to design multiband FSSs (Gianvittorio et al., 2003;Cruz et al., 2009;Trindade et al., 2011).
Fig. 5 illustrates the considered periodic array in this chapter.The periodicity of the elements is given by t x =W c , in the x axis, and t y =L c , in the y axis, where W c is the width and L c is the length of the unit cell element; in addition, W is the width and L is the length of the patch.The design of fractal patch elements depend of desired FSS filter specifications, such as: bandstop attenuation, resonant frequency, quality factor, fabrication restrictions, etc.

Koch island fractal
The Koch island fractal patch elements were obtained assuming a rectangular construction, fractal iteration-number (or level), k=0,1,2, and a variable fractal iteration-factor 1/ ra = , where a belongs to interval 3.05 10.0 a ≤≤ .The geometry of the Koch island fractal patch elements is shown in Fig. 6, considering for k=0,1,2, and a=4.The rectangular patch element (fractal initiator) dimensions are (mm): W=4.93, L= 8.22, t x =8.22, and t y =12.32.The Koch curve begins as a straight line corresponding to each side of the conventional rectangle.Next, the Koch element in the first fractal iteration is obtained by removing four scaled rectangles (with the width and length of the initiator rectangle scaled by the fractal iteration-factor) that lies at the center of each side of the initiator rectangle.The same construction is applied for others Koch iterations (see Fig. 6).After the k-th fractal iteration, the dimensions of scaled rectangles are given in ( 13) through the substitution of the dummy variable k  by the width W k or length L k of the k-th scaled rectangle.

Dürer's pentagon fractal
The Dürer's pentagon fractal geometry was generated with the application of iterated function system (Trindade et al., 2011).From a regular pentagon patch element (L=10 mm and t x =t y =16.5 mm), that corresponds to the fractal initiator element, we use a fractal iteration-factor 0.382 r = for the generation of Dürer's pentagon elements at levels k=1,2, and 3, where

Fabrication and measurement of frequency selective surfaces
The frequency selective surfaces using fractal geometries were built as periodic arrays of patch fractal elements.FSS is mounted on a dielectric isotropic layer.FSS spatial filter prototypes were fabricated using conventional planar circuit technology, with low-cost fiberglass (FR-4) substrate with 1.5 mm of height and a relative permittivity of 4.4.
The setup to measure the FSS transmission coefficients included: two horn antennas, two waveguides (cut-off frequency, 6.8 GHz), a network analyzer (model N5230A, Agilent Technologies), which operates from 300 KHz up to 13.5 GHz, beyond coaxial/waveguide transitions, handles and connectors.A fixed distance was adopted between the horn antennas in order to guarantee the operation in the far field region.The FSS filter prototypes were placed between the horn antennas for the measurement procedure (see Fig. 9).

Optimal design of fractal frequency selective surfaces
In this section are presented some applications of proposed optimization technique for optimal design of bandstop FSS spatial filters.Three optimization examples are described considering the use of FSS fractal patch elements: Koch island, Dürer's pentagon, and Sierpinski island.The EM characterization of these FSSs was accomplished by means of a full-wave parametric analysis through the use Ansoft Designer TM commercial software.
MLP neural network models for these FSSs were developed using the conventional EM-ANN neuromodeling technique (Zhang & Gupta, 2000).The supervised training of MLP weights was done through the well-established resilient back-propagation (RPROP) algorithm with standard training parameters (Ridmiller & Braun, 1993).Considering for design input variables (k, a, ε r a MLP model was trained to approach the resonant frequency fr(k, a, ε r ) and bandwidth BW(k, a, ε r ) of the FSS spatial filters.The minimal MLP configuration able to solve the FSS modeling problem was defined with four input units, five hidden units, and two output units.The MLP configuration is illustrated in Fig. 10.The minimum number of five hidden neurons was found by means of a trial and error procedure and training restarts (Cruz et al., 2009).
Using sigmoid activation function, the outputs of MLP model are computed by ( 14).
( ) 0.0572 0.1656 2.1679 0.1640 0.0936 0.2227 0.1059 0.2534 0.7188 0.2979 0.4961 0.2441 The MLP model for FSS filter design is CPU inexpensive, easy to implement and accurate.In addition, it requires small size EM dataset to learning the model of input/output mapping.The MLP model is used for evaluation of cost function in the bio-inspired optimization algorithm for FSSs with Koch island fractal patch elements.
The FSS set of design variables [k, f, BW] composed the input data for bio-Inspired optimization algorithms and are chosen by the user, within the region of interest of MLP model: (k=1,2), (2,7<fr<19) GHz, and (1.5<BW<4.5)GHz.Given a FSS filter desired specification (f, BW), the goal of bio-inspired optimization is to find an optimal solution (ε r , a) for a given fractal level k that minimizes the quadratic cost function as defined in (17).
( ) ( ) The individuals [ , ] ii i T mr m m a ε = p evolve from the initial population given according to (18).
The center of initial population 00 (,) rm m a ε in the searching space was chosen equal to (3.0, 8.5).When the population evolves, each individual is constrained to the region of interest using ( 19).The dummy variable ξ can be replaced by variables r ε or a .
In this first FSS optimization example, we intended to verify the execution of continuous-GA and improved-GA algorithms.The algorithms start with the same initial population with 25 individuals distributed according to (18) and subject to restriction given in ( 19).We use the following FSS design specification: k=2, f=10.0GHz, and BW=2.10 GHz.
The MLP model outputs are shown in Fig. 13  Fig.15(a) shows the evolution of the average global cost for continuous-GA and PSO populations.We observe that the PSO algorithm is limited in precision to the round off error of the computer (≈10 -31 ).In Fig. 15(b) is presented a comparison between the simulated and measured results for the transmission coefficient (dB) of the optimized FSS with Koch fractal patch elements.The simulated and measured results are in excellent agreement with desired FSS design specifications.

FSS with Sierpinski iIsland fractal patch elements
The Sierpinski island fractal geometry was used to design a FSS consisting of a periodic array of fractal patch elements.The input design variables (k, t x , ε r ) are limited to design region of interest defined by the following discrete values selected for parametric analysis: [15,16,17,18,19,20,21,22,23,24,25]  Considering for design input variables (k, ε r, t x ), a MLP model was trained to approach the resonant frequency and bandwidth of the FSS filters with Sierpinski island fractal patch elements.Fig. 16 shows the variation of sum squared error given in ( 6) as a function of the number of MLP hidden units.The minimal MLP configuration able to solve the FSS modeling problem was defined with four input units, fifteen hidden units, and two output units.

Conclusion
This chapter described a new EM optimization technique blending full-wave method of moments, MLP artificial neural networks and bio-inspired algorithms for optimal design of frequency selective surfaces with fractal patch elements.Three fractal geometries were considered: Koch island, Dürer's pentagon and Sierpinski island.The use of fractal geometries to design FSS on isotropic dielectric substrates becomes possible the control of its frequency responses without increase its overall size.The MLP models were trained with accurate EM data provided by FSS simulations based on the method of moments.The computation of cost function of bio-inspired algorithms was done throught the use of developed MLP models.This procedure results in fast and accurate bio-inspired optimization algorithms.According to the obtained results, genetic algorithms present zigzag paths at flat regions of the cost surface that contribute to slow down the algorithm convergence.The introdution of a new mutation operation avoid the oscilation of continuous-GA final population around the global minimum of cost function.The improved-GA final population closely to the global minimum improves the GA convergence.PSO algorithm showed to be faster and easier to implement.This makes the optimization through PSO a powerful tool in synthesizing FSS structures.The idea of blending bio-inspired algorithms and artificial neural networks to optimize frequency selective surfaces shows to be very interesting, due to its great flexibility and easy application to structures that do not have a direct fitness function.The MoM-ANN-GA/PSO proposed technique is accurate and CPU inexpensive, which are most desired characteristics in the development of computer-aided design tools for EM optimization.

p
is the best global solution; C is the constriction parameter(Kennedy & Eberhart, 1995).If the best local solution has a cost less than the cost of the current global solution, then the best local solution replaces the best global solution.PSO is a very simple bio-inspired algorithm, easy to implement and with few parameters to adjust.Flow chart shown in Fig.4(b) gives an overview of PSO algorithm.

Fig. 6 .
Fig. 6.Koch Island fractal patch elements (a=4) . Therefore, six small-scale copies of the initiator element are generated in a given fractal iteration, N = 6, resulting in a fractal dimension log() / l o g(1 / ) DN r =, where D=1.8619.The geometry of the Dürer's pentagon fractal is shown in Fig.7.

Fig. 9 .
Fig. 9. Photograph of the Sierpinski FSS prototype and the measurement setup

Fig. 10 .
Fig. 10.Multilayer perceptron configuration Fig. 11(b)  shows the same results for the improved-GA.The observed zigzag paths at flat regions of the cost surface contribute to slow down the convergence of genetic algorithms.The final population of the continuous-GA algorithm oscillates around the global minimum of cost function (see Fig.11(a)), while the improved-GA final population is closely to the global minimum (seeFig.11(b)).

Fig. 12
Fig. 12(a) shows the global cost evolution for the best individual of continuous-GA and improved-GA populations.It is observed that improved-GA converges closely to global minimum.The final global cost values for improved-GA and continuous-GA were Fig. 11.Initial, intermediate, and final populations and GA paths plotted over the cost surface contours: (a) continuous-GA and (b) improved-GA.
Fig. 13.(a) FSS resonant frequency as a function of periodicity and relative permittivity.(b) FSS bandwidth as a function of periodicity and fractal iteration-numberAssuming the genetic algorithm parameters: crossover probability=0.6, mutation rate=0.2,A=30, B=10 -9 , Nvar=2 and Npop=25, we simulated up to 250 GA-iterations.Fig.14(a)shows the initial, intermediate and final populations, and the continuous-GA path plotted over the cost surface contours.The optimal solution was: ε r =4.4 and t x =t y =16.5 mm.

Fig. 16 .
Fig. 16.Variation of SSE as a function of MLP hidden units The MLP model outputs are shown in Fig. 17(a) and 17(b).Considering the limits of the desired region of interest of the input design variables, the MLP model is able to interpolate the training dataset.
(a) and 17(b).Considering the limits of the desired region of interest of the input design variables, the MLP model is able to interpolate the training dataset.In this third FSS optimization example, we intended to verify the execution of continuous-GA, improved-GA, and PSO.The algorithms start with the same initial population with Npop=100 individuals

Fig. 17
Fig. 17.(a) FSS resonant frequency as a function of periodicity and relative permittivity.(b) FSS bandwidth as a function of periodicity and fractal levelsThe FSS design variables [k, f, BW] composed the input data for bio-Inspired optimization algorithms and are chosen by the user within the region of interest of MLP model: (k=1,2,3), (3.1<fr<10.5)GHz, and (0.15<BW<1.7)GHz.We used the following input data for simulation of bio-inspired optimization: k=1, f=8.37 GHz and BW=1.44 GHz.

Fig. 18 .
Fig. 18.Bio-inspired algorithm paths plotted over the cost surface contours, initial, intermediate, and final populationsIn Fig.18is presented the initial, intermediate, and final populations, as well as, the bioinspired algorithm paths plotted over the cost surface contours.In this case, the optimal solution founded was: ε r =4.4 and t x =t y =16.0 mm.
This work was supported by CNPq, under contracts 479253/2009-9 and 307256/2009-0, by Federal Institute of Education, Science and Technology of Paraíba (IFPB) and by Federal University of Rio Grande do Norte (UFRN). , Where W and V are the MLP weight matrix, x=[-1, k-1, a, ε r ] T and y=[fr, BW] T are the MLP input and output vectors, respectively.The resultant MLP trained weight values are given by (15) (16).
12(b)is presented the simulated and measured FSS transmission.The obtained numeric results are presented in Table1, and are in excellent agreement with desired FSS design specifications.