The design of a shaped beam reflectarray is a challenging issues. The problem can be hardly addressed by deterministic techniques or standard optimization due to the elevated number of design variables and its non-convex nature while can be much easily solved adopting Evolutionary Optimization Algorithms. In particular, in this paper a recently introduced evolutionary approach, named Social Network Optimization (SNO) has been applied to the design of a reduced size shaped beam reflectarray: the obtained numerical results are promising and prove the effectiveness of the adopted method

Beccaria, Michele

Niccolai, Alessandro

Pirinoli, Paola

Zich, Riccardo

English

The design of a shaped beam reflectarray is a
challenging issues. The problem can be hardly addressed by
deterministic techniques or standard optimization due to the elevated number of design variables and its non-convex nature while
can be much easily solved adopting Evolutionary Optimization Algorithms. In particular, in this paper a recently introduced evolutionary approach, named Social Network Optimization (SNO) has been applied to the design of a reduced size shaped beam reflectarray: the obtained numerical results are promising and prove the effectiveness of the adopted method

Alessandro Niccolai

RIccardo Zich

Michele Beccaria

Paola Pirinoli

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALESNO Based Optimization for Shaped Beam Reflectarray Antennas / Niccolai, Alessandro; Zich, Riccardo; Beccaria,Michele; Pirinoli, Paola. - (2019). ((Intervento presentato al convegno EuCAP 2019 tenutosi a Cracovia nel 31 Marzo - 5Aprile.OriginalSNO Based Optimization for Shaped Beam Reflectarray AntennasieeePublisher:PublishedDOI:Terms of use:openAccessPublisher copyrightcopyright 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all otheruses, in any current or future media, including reprinting/republishing this material for advertising or promotionalpurposes, creating .(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2738196 since: 2019-06-28T17:40:05ZIEEE1SNO Based Optimization for Shaped BeamReflectarray AntennasAlessandro Niccolai 1, Riccardo Zich 1, Michele Beccaria2, Paola Pirinoli21Dipartimento di Energia, Politecnico di MilanoVia La Masa 34, 20156 Milano, Italy2Department of Electronics and Telecommunications Politecnico di TorinoCorso Duca degli Abruzzi 24, 10129 Torino, Italyalessandro.niccolai@polimi.it, riccardo.zich@polimi.it,michele.beccaria@polito.it, paola.pirinoli@polito.itAbstract—The design of a shaped beam reflectarray is achallenging issues. The problem can be hardly addressed bydeterministic techniques or standard optimization due to the ele-vated number of design variables and its non-convex nature whilecan be much easily solved adopting Evolutionary OptimizationAlgorithms. In particular, in this paper a recently introducedevolutionary approach, named Social Network Optimization(SNO) has been applied to the design of a reduced size shapedbeam reflectarray: the obtained numerical results are promisingand prove the effectiveness of the adopted method.Index Terms—Shaped beam, reflectarray, Social Network Op-timization.I. INTRODUCTIONEvolutionary Algorithms (EAs) are effective and flexibletechniques able to optimize multi-modal and non-convex costfunctions like those modelling in general many engineeringproblems, and more specifically antenna ones: the Genetic Al-gorithm (GA), has been widely applied to antenna optimization[1], as well as the Particle Swarm optimization (PSO) [2]- [3]-[4]. Even if both of them produced satisfactory results, theincreasing size of the problems and their complexity demandfor new algorithms with higher efficiency.A promising recently-developed algorithm is the SocialNetwork Optimization [5], [6], that mimics the behaviour afthe people when interacting right through a social network.It has been tested on several problems, from the optimizationof a linear permanent magnet generator [7] to the design ofpencil-beam Reflectrray (RA) antennas [8].The optimization of a reflectarrays represents a good bench-mark for the testing of an optimization technique. In fact,the surface of a RA is generally discretized with hundreds orthousands of re-radiating elements, each of one characterizedby one or more geometrical parameters, that are use to controland properly adjust the phase of the incident field in orderto obtain the desired radiation pattern: as a consequence, theassociate mathematical problem present a very high number ofdegrees of freedom, that could be difficult to manage even byan evolutionary approach. As a consequence, the algorithmhas to work hard to converge, in a reasonable number ofiterations an without being trapped in local minima. Moreover,slow convergence means that the cost function modelling theproblem has to be computed thousands of times, with a strongincrease of the computational cost. For all these reasons, it isevident that an efficient optimization algorithm is necessaryfor the design of a reflectarray antenna.SNO has been already applied successfully to the designof pencil-beam RAs [8]. Here, it is used for the optimizationof a shaped beam reflectarray: the results on a reduced sizeconfiguration prove the effectiveness of the proposed method.The paper is structured as it follows: the descriptions of thealgorithm is reported in Sect. II; in Sect. III the problem tooptimize, the adopted procedure, base on the use of the SNO,and some numerical results are shown. Finally, in Sect. IVsome conclusions are drawn.II. SOCIAL NETWORK OPTIMIZATIONSocial Network Optimization is a population-based algo-rithm that takes its inspiration from the process of informationsharing of social networks [9].The algorithm population represents the members of a socialnetwork: they are characterized by the situation that they face,posted on the social as status, and by their character. Theirinteraction with other individuals is driven by their personalinterests and by their reputation [7]. The status of an individualrepresents a candidate solution of the optimization problem,while his reputation is the cost or fitness value associated to thecandidate solution. All the other variables and mechanisms areinternal to the algorithm and force the population to explore,at each iteration, the problem domain of definition, in orderto converge to the final optimal solution.The Social Network is the virtual space in which theindividuals interact and exchange information. This modifiesthe character of the people that will then face differentsituations [6]. The social network has its storage capabilityand it collects and keeps track of the best solution posted.The individuals belong to two different kind groups: groupsof friends, collecting people with similar status, and groupsof peers, set up by persons with similar character. In eachgroup, every individual selects an influencer in the light ofthe ranking of the reputations. Having selected the influencers,each individual choses its desired option (desideratum) mixing13th European Conference on Antennas and Propagation (EuCAP 2019)21: First population creation2: while not termination criterion do3: Creation of the global ranking4: Update of stored solutions5: for all the population do6: Creation of friend and peer groups7: Selection of the two influencers8: Generation of the desired option9: Calculation of the new character10: Calculation of the new status11: Cost evaluation12: end for13: end whileFig. 1. Pseudo-code of Social Network Optimizationthe statuses of the influencers accordingly to his personalinterest.The evolution of the population is described by the twofollowing equations:~ci(t+ 1) = α · ~ci(t) + β · (~di − ~si(t)) (1)~si(t+ 1) = ~si(t) + ~ci(t+ 1) (2)where ~ci(t) and ~ci(t+ 1) are the actual and the future charac-ters of the i-th individual, ~si(t) and ~di are his actual status anddesideratum; α and β are two user-defined parameters. It canbe noticed that at each iteration the status of an individualdepends on the status at the previous iteration and by hiscurrent character; at its turn, this last is expressed in termsof the previous status and character and of the desideratum.Figure 1 reports the pseudo-code of SNO.III. SHAPED BEAM REFLECTARRAY OPTIMIZATIONThe SNO is here applied to the design of a reflectarraywith a cosine squared radiation pattern in the E-plane. In orderto test the applicability of the SNO to this kind of problem,a reduced size configuration is considered. The reflectarrayconsists in 24 × 24 re-radiating square patch elements printedon a layer of Diclad 527, with r = 2.57, losses tan δ =0.0022 and thickness of 0.8 mm. To avoid grating lobes, theunit-cell size has been chosen equal to λ0/2 and thereforethe total size of the RA surface is equal to 12λ0, where λ isthe wavelength at the design frequency f0. The phase of thereflected field is controlled varying the side d of the squarepatches: in Fig. 2, the variation of the amplitude (top) andphase (bottom) of the reflection coefficient with d is plotted.The amplitude is almost everywhere equal to 0 dB, it justdecreases a little in correspondence of the patch resonant size,while the phase varies smoothly and cover a range of almost300◦. These curves have been obtained considering the unit-cell embedded in a infinitely periodic lattice and performingthe analysis of the resulting structure with CST-MicrowaveStudio.Fig. 2. Variation of the reflection coefficient with the side d of the RA unit-cell square patch. Top: amplitude. Bottom: phase.The adopted feed is a Potter horn, center-fed, with a focaldistance f/D = 1 that allows to obtain a tapering of -10 dBat the edges of the reflectarray.The design variables of this optimization problem are obvi-ously its degrees of freedom, i.e. the size d of the RA patches.They are 576 overall, but taking advantage of the symmetrywith respect to a vertical plane the final adjustment of the RApatches have to show, the effective number of optimizationvariables reduces to 288.The objective of the optimization process, i.e. to obtaina cosine squared shaped beam in the E-plane, has beenimplemented by means of two masks. The upper mask servesto constrain the SLL, and the width of the main beam. Inthe angular region corresponding to the shaped beam, a lowermask is added, with the same behaviour of the upper one andwith the aim to limit the acceptable ripple, here equal to 2.5dB.The cost function consists in the integral of the squarederror of the radiation patter when exceeding one of the twomasks:c =∑θ∑φ(∆2up(θ, φ) + ∆2dw(θ, φ)) (3)where ∆up and ∆dw are the two error functions, defined as:∆up(θ, φ) ={E(θ, φ)−Mup(θ, φ) E(θ, φ) > Mup(θ, φ)0 E(θ, φ) ≤Mup(θ, φ)(4)∆dw(θ, φ) ={Mdw(θ, φ)− E(θ, φ) E(θ, φ) < Mdw(θ, φ)0 E(θ, φ) ≥Mdw(θ, φ)(5)being E(θ, φ) the radiated field, and Mup Mdw the upper andthe lower masks, respectively.Due to the large number of design variables, the opti-mization process requires 50,000 cost function evaluations.16 independent trials have been done and the obtained con-vergence curves are shown in Figure 3, where each colouredline represents the evolution of one trial and the black thicker13th European Conference on Antennas and Propagation (EuCAP 2019)3Fig. 3. Convergence curves of the optimization process. The black bolt lineis the average convergence. The cost is represented in a logarithmic scale.Fig. 4. Optimal RA geometry found by SNO.line is the average convergence. It is worth to notice that theoptimization process shows a very high initial speed and thenit slows down. Moreover, the curves are quite close each otherand this prove the good reliability of the algorithm.The optimal geometry found by SNO is shown in Figure 4.A 3D view of the optimized antenna radiation pattern isshown in Figs 5, while in 6 the cut in the E-plane is plotted,together with the upper and the lower masks; moreover,the radiation pattern computed with a full-wave analysis ofthe designed configuration, carried on with CST-MicrowaveStudio, has been also added. During the optimization processthe radiating features of the RA are computed with a reducedcomputational cost approach (i.e. the Aperture field Method,[10]) and therefore the obtained results have to be confirmedthrough a more accurate analysis. The comparison between theradiation patterns computed with the two methods are in goodagreement: the curve obtained with the full-wave approachsatisfies the mask almost everywhere, and this confirms thegoodness of the optimization procedure, taking also intoaccount that it has not taken into account aspects as the feedblockage, not negligible in a configuration like the consideredone.Fig. 5. 3D radiation pattern of the optimal solution.Fig. 6. Radiation pattern in the E-plane.IV. CONCLUSIONSIn this paper, the feasibility of the SNO for a shapedbeam Reflectarray Antenna optimization is proved, through thenumerical analysis of a reduced size configuration radiating acosine squared pattern in E-plane. In view of the obtained goodresults, the method will be further validated by its applicationto larger structures with off-set feed, and by the experimentalcharacterization of a prototype.REFERENCES[1] Rahmat-Samii and E. Michielssen, ”Electromagnetic optimization bygenetic algorithms,” Microwave Journal, vol. 42, no. 11, pp. 232-232,1999.[2] J. Robinson and Y. Rahmat-Samii, ”Particle Swarm Optimization inElectromagnetics,” IEEE Trans. on Antennas and Propag., vol. 2, pp.297-407, 2004.[3] D. Boeringer and D. Werner, ”Particle Swarm Optimization versus Ge-netic Algorithms for Phased Array Synthesis,” IEEE Trans. on Antennasand Propag., vol. 3, pp. 771-779, 2004.[4] D. Olcan, R. Golubovic and B. Kolundzija, ”On the Efficiency of ParticleSwarm Optimizer when Applied to Antenna Optimization”, IEEE Int.Antennas and Propag. Symposium, pp. 3297-3300, 2006[5] M. Ruello, A. Niccolai, F. Grimaccia, M. Mussetta and R. E. Zich,”Black-hole PSO and SNO for electromagnetic optimization”, Evolution-ary Computation (CEC), 2014 IEEE Congress on, pp. 1912-1916, 2014[6] A. Niccolai, F. Grimaccia, M. Mussetta, P. Pirinoli, V. H. Bui and R.E. Zich, ”Social network optimization for microwave circuits design,”Progress In Electromagnetics Research C, vol. 58, pp. 51-60, EMWPublishing, 2015.[7] F. Grimaccia, G. Gruosso, M. Mussetta, P. Pirinoli, V. H. Bui and R.E. Zich, ”Design of Tubular Permanent Magnet Generators for VehicleEnergy Harvesting by means of Social Network Optimization,” IEEETransactions on Industrial Electronics, IEEE, 2017.13th European Conference on Antennas and Propagation (EuCAP 2019)4[8] A. Niccolai, F. Grimaccia, M. Mussetta and R. E. Zich, ”Reflectarrayoptimization by means of SNO and PSO”, IEEE Int. Antennas andPropag. Symposium, IEEE, pp. 781-782, 2016[9] R. Albi, F. Grimaccia, M. Mussetta, A. Niccolai, and R. E. Zich, ”Anew algorithm for antenna optimization in aerospace applications: Thesounding rocket test case”, Electromagnetics in Advanced Applications(ICEAA), 2013 International Conference on, IEEE, pp. 1537-1540, 2013[10] J. Huang, J. A. Encinar, Reflectarray Antennas, John Wiley & Sons,2007.13th European Conference on Antennas and Propagation (EuCAP 2019)

SNO Based Optimization for Shaped Beam Reflectarray Antennas

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