1. INTRODUCTIONDomes have been of special interest because of beinglightweightandelegantstructuresthatprovidecost-effectivesolutionstocoverthelargeareaswithoutintermediate supports. Themainaimofthispaperistoutilizevariousmeta-heuristicalgorithmsinordertoreachanoptimumdesignofdifferenttypesofsinglelayerdomes. Since in this paper the joints of the single layerdomestructuresareconsideredtoberigid,themembers are exposed to both axial forces and bendingmoments,andconsiderationofthegeometricnonlinearityintheanalysisofthesestructuresisimportant.ThusonecanrarelyfindanefficientOptimal Design of Single LayerDomes using Meta-HeuristicAlgorithms; A ComparativeStudyA. Kaveh1,*, S. Talatahari21Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Tehran, Iran2Department of Civil Engineering, University of Tabriz, Tabriz, Iran(Received February, 4, 2010 - Revised version October, 5, 2010 - Acceptation October, 8, 2010)ABSTRACT: Domes are lightweight and elegant structures that provide cost-effectivesolutionstocoverlargeareas.Themeta-heuristicalgorithmspresented in this paper carry out the optimum design of dome structures. Theserviceability and strength requirements are considered in the design problemofnetwork,SchwedlerandlamelladomesasspecifiedinLRFD-AISC. Theoptimumsolutionsofthedesignproblemareobtainedusingparticleswarmoptimizer,antcolonyoptimization,harmonysearch,BigBang-BigCrunch,heuristicparticleswarmantcolonyoptimizationandchargedsystemsearch.Thesolutionsofthesealgorithmsarepresentedtodemonstratetheeffectiveness and the strength of each algorithm. Comparison of the results ofthedomedesignsobtainedbythesealgorithmsillustratesthegoodperformanceoftheheuristicparticleswarm,antcolonyoptimizationandchargedsystemsearchmethodcomparedtootherheuristicalgorithms. Also,the Schwedler dome is found to have the most economical configuration.KeyWords: Domeoptimization;Geometricallynonlineardesign;Meta-heuristic algorithms; Single layer dome structures.International Journal of Space Structures Vol. 25 No. 4 2010 217mathematicalprogrammingapproachtosolvesuchproblems easily, while the meta-heuristic methods aresuitabletoolsforthispurpose. Theserviceabilityandstrengthrequirementsareconsideredinthedesignproblems as specified in LRFD-AISC [1]. In this studythe nonlinear response of the dome due to the effect ofaxialforcesontheflexuralstiffnessofmembersistakenintoaccount.Here,network,Schwedlerandlamella domes are studied. The cross-sectional areas ofelements, rise for the crown, and the number of ringsareconsideredasthedesignvariables[2].Thesedomesareoptimizedbyusingvariousmeta-heuristicalgorithm containing particle swarm optimizer (PSO),ant colony optimization (ACO), harmony search (HS),*Corresponding author: alikaveh@iust.ac.ir218 International Journal of Space Structures Vol. 25 No. 4 2010Optimal Design of Single Layer Domes Using Meta-Heuristic Algorithms; A Comparative StudyhybridBigBang-BigCrunchalgorithm(HBB-BC),heuristicparticleswarm-antcolonyoptimization(HPSACO) and charged system search (CSS).The remaining sections of this paper are organizedasfollows:Section2containsthestatementofdomedesignproblem.Section3reviewstheutilizedmeta-heuristic methods. The achieved results are explainedinSection4.Finally,Section5isdevotedtotheconcluding remarks.2. STATEMENT OF OPTIMUM DESIGNOF SINGLE LAYER DOMES Optimal design of dome structures consists of findingoptimalcross-sectionsforelements,optimalriseforthecrown,andtheoptimumnumberofringsunderdeterminedloadingconditions.Themathematicalform of the objective function can be expressed as(1)where X is the design vector containing the all designvariables;Aiistheareaoftheithsizedesigngroupwhichisselectedfrom37steelpipesectionsfromLRFD-AISC[1]asshownin Table1;ng denotesthetotal number of size groups; h is the rise of the dome;Nr represents the total number of the rings; W(X) is theweight of the structure; Ljis the length of element j; niis the total number of elements in group i; and iis thematerial mass density. The effect of the type of cross-sectionsontheobjective(weightofthestructure)isconsidered by Ai. The rise of the dome and the numberof the rings can alter the magnitude of Ljand thus canaffectthevalueoftheweightofthedome.Thepermittedminimum,maximumandtheincreasedamounts of the crown rise are taken as 1.00 m, 8.75 mand0.25m,respectively.Thetotalnumberofrings(Nr)ischosenas3,4or5.SinceselectingdifferentnumbersforNr changesthenumberofsizedesignvariables,thetotalnumberofsizevariablesissettothe maximum allowable number obtained by using Nr 5,andwhenasmallvalueforNr isselected,theunnecessarysizevariables(Ai)willbeconsideredaszero.TheLRFDspecificationandthedriftlimitationsareconsideredasconstraintsforthesestructures.Themodulusofelasticityforthesteelistakenas 205 kN/mm2. The diameter of the domes is selectedas20 m.Thesestructuresareconsideredtobesubjected to only the equipment loading of 1000 kNat their crown. The limitations imposed on the jointFindtX { , , , : } A h Nr i ngi1oo minimize W A Li i jjniing( ) X 1 1displacements are 28 mm in the all direction for thenodes. The constraints can be formulated as follows:Displacement constraint(2)Interaction formula constraints(3)(4)Shear constraint(5)inwhichiisthedisplacementofnodei;imaxisthepermitted displacement for the ith node; nn is the totalnumber of nodes; Puis the required strength; Pnis thenominal axial strength; cis the resistance factor (c 0.9fortension,c0.85forcompression);MuxandMuyaretherequiredflexuralstrengthsinthex andydirections,respectively;MnxandMnyarethenominalflexuralstrengthsinthex andy directions;andbisthe flexural resistance reduction factor (b0.90); Vuisthefactoredserviceloadshear;Vnisthenominalstrengthinshear;andvrepresentstheresistancefactor for shear (v0.90).3. REVIEW OF THE META-HEURISTICALGORITHMS3.1. Particle swarm optimizationThe particle swarm optimization involves a number ofparticles,whichareinitializedrandomlyinthespaceof the design variables. These particles fly through thesearch space and their positions are updated based onthebestpositionsofindividualparticlesandthebestposition among all particles in the search space whichcorresponds to a particle with the smallest weight [3].Alsoconsideringthepositionofaparticleselectedrandomlyfromtheswarm(passivecongregation)[4]can improve the performance of the algorithm.Theupdatemovesaparticlebyaddingachangevelocity Vjk+1to the current position Xkjas follows(6)(7)V V P Pjkjkjkjkgkjkc r c r+ + + 11 1 2 2 ( ) ( ) X X( ) + c rjkjk3 3R XX X Vjkjkjk + + +1 1V Vu v n PPMMMMuc nuxb nxuyb ny + +

_,

891 ForPPuc n 0 2 .PPMMMMuc nuxb nxuyb ny21 + +

_,

ForPPuc n< 0 2 . i ii nn max, , ..., 1 2A. Kaveh, S. Talatahariwhere is an inertia weight to control the influence ofthepreviousvelocity;r1,r2andr3arethreerandomnumbers uniformly distributed in the range of (0,1); c1and c2are two acceleration constants; c3is the passivecongregation coefficient; Pkjis the best position of thejthparticleuptoiterationk;Pkgisthebestpositionamong all particles in the swarm up to iteration k; andInternational Journal of Space Structures Vol. 25 No. 4 2010 219Rkis a particle selected randomly from the swarm. Inordertoincreasetheexplorationability,inthispaperthe velocity of the particles is defined as [5,6](8)V V P X P Xjkjkjkjkgkjkc r c r+ + + 11 1 2 2 ( ) ( )+ + c r c rjkjkjkjk3 3 4 4( ) ( ) R X Rd XTable 1. The allowable steel pipe sections taken from LRFD-AISCNominal Weight perDiameter ft Area I S J ZType in. (lb) in.2in.4in.3in.4in.31 ST 1/2 0.85 0.250 0.017 0.041 0.082 0.0592 EST 1/2 1.09 0.320 0.020 0.048 0.096 0.0723 ST 3/4 1.13 0.333 0.037 0.071 0.142 0.1004 EST 3/4 1.47 0.433 0.045 0.085 0.170 0.1255 ST 1 1.68 0.494 0.087 0.133 0.266 0.1876 EST 1 2.17 0.639 0.106 0.161 0.322 0.2337 ST 11/42.27 0.669 0.195 0.235 0.470 0.3248 ST 11/22.72 0.799 0.310 0.326 0.652 0.4489 EST 11/43.00 0.881 0.242 0.291 0.582 0.41410 EST 11/23.63 1.07 0.666 0.561 1.122 0.76111 ST 2 3.65 1.07 0.391 0.412 0.824 0.58112 EST 2 5.02 1.48 0.868 0.731 1.462 1.0213 ST 21/25.79 1.70 1.53 1.06 2.12 1.4514 ST 3 7.58 2.23 3.02 1.72 3.44 2.3315 EST 21/27.66 2.25 1.92 1.34 2.68 1.8716 DEST 2 9.03 2.66 1.31 1.10 2.2 1.6717 ST 31/29.11 2.68 4.79 2.39 4.78 3.2218 EST 3 10.25 3.02 3.89 2.23 4.46 3.0819 ST 4 10.79 3.17 7.23 3.21 6.42 4.3120 EST 31/212.50 3.68 6.28 3.14 6.28 4.3221 DEST 21/213.69 4.03 2.87 2.00 4.00 3.0422 ST 5 14.62 4.30 15.2 5.45 10.9 7.2723 EST 4 14.98 4.41 9.61 4.27 8.54 5.8524 DEST 3 18.58 5.47 5.99 3.42 6.84 5.1225 ST 6 18.97 5.58 28.1 8.50 17.0 11.226 EST 5 20.78 6.11 20.7 7.43 14.86 10.127 DEST 4 27.54 8.10 15.3 6.79 13.58 9.9728 ST 8 28.55 8.40 72.5 16.8 33.6 22.229 EST 6 28.57 8.40 40.5 12.2 24.4 16.630 DEST 5 38.59 11.3 33.6 12.1 24.2 17.531 ST 10 40.48 11.9 161 29.9 59.8 39.432 EST 8 43.39 12.8 106 24.5 49.0 33.033 ST 12 49.56 14.6 279 43.8 87.6 57.434 DEST 6 53.16 15.6 66.3 20.0 40.0 28.935 EST 10 54.74 16.1 212 39.4 78.8 52.636 EST 12 65.42 19.2 362 56.7 113.4 75.137 DEST 8 72.42 21.3 162 37.6 75.2 52.8STstandard weight, ESTextra strong, DESTdouble-extra strong.wherec4istheexplorationcoefficient;r4isauniformlydistributedrandomnumberintherangeof(0,1); and Rdkis a vector generated randomly from thesearch domain.3.2. Ant colony optimizationAntcolonyoptimizationisacooperativesearchtechniquethatmimicstheforagingbehaviourofreallifeantcolonies[7].Theantalgorithmsmimicthetechniquesemployedbyrealantstorapidlyestablishthe shortest path. Real ants use their pheromone trailsas a medium for communication of information amongthemselves.TheinitializationoftheACOalgorithmincludesplacinganumberofantsarbitrarilyonthesearch space. This method is basically a discrete one,andthusthealgorithmcanselectadiscretevaluedirectly.Thisselectionforantk isperformedaccording to a probabilistic state transition rule, wheretheprobabilityofselectingtheallowablejthdiscretevalue for variable i is as follows:(9)where ij(t) is the intensity of the pheromone; ijis thevisibilitywhichhasreverserelationwiththeselectedvalue; and and are the control parameters. Once allthe ants have constructed their solutions, the intensityofpheromonetrailsoneachedgeisupdated(globalupdating rule) as(10)Where (0< < 1)representsthepersistenceofpheromone trails, (1 ) is the evaporation rate; N isthetotalnumberofants;and+ijistheamountofpheromone increase for the elitist ant which is determinedconsidering the quality of the solution, as follows [8]:(11)WhereWbestistheweightofthebestresult.Inaddition,aftereachselection,thepheromonevaluerelatedtotheselectedvalueisupdatedbythelocalupdating rule as [9](12)Where is an adjustable parameter between 0 and1,representingthepersistenceofthetrail.DetaileddescriptionoftheACOalgorithmcanbefoundinRef. [8]. ij ijt t + ( )() 1 .ij bestt W+() 1/ ij ij ijt N t ( ) ( ) ( ) + + +1 P tttijk ij ijil ill( )[ ( )] [ ][ ( )] [ ]

220 International Journal of Space Structures Vol. 25 No. 4 2010Optimal Design of Single Layer Domes Using Meta-Heuristic Algorithms; A Comparative Study3.3. Harmony searchTheharmonysearchmethodisanotherheuristicoptimizationtechniquethatimitatesthemusicalperformance process which takes place when a musiciansearches for a better state of harmony. The pitch of eachmusicalinstrumentdeterminestheaestheticquality,just as the objective function value is determined by thesetofvaluesassignedtoeachdecisionvariable.Thisapproach is suggested by Geem et al. [10,11] and firstappliedtothedesignofwaterdistributionnetworks.Sincethen,thealgorithmhasattractedmanyresearchers due to its simplicity and effectiveness.TheHSalgorithmincludesanumberofoptimizationoperators,suchasamemory,calledharmonymemory(HM),andstoresthefeasiblevectors,whichareallinthefeasiblespace.Theharmony memory size (HMS) determines the numberofvectorstobestored.TheHarmonyMemoryConsideringRate(HMCR)varyingbetween0and1sets the rate of choosing a value in the new vector fromthehistoricvaluesstoredintheHM,and(1-HMCR)sets the rate of randomly choosing one value from thepossible range of values. The pitch adjusting process isperformedonlyafteravalueischosenfromtheHMusingthePitch AdjustingRate(PAR).Thevalue(1-PAR)setstherateofdoingnothing.A PARof0.1indicates that the algorithm will choose a neighbouringvalue with 10% HMCR probabilityA newharmonyvectorisgeneratedfromtheHM,based on the memory considerations, pitch adjustments,and randomization(13)WherexkiistheithvariableofthevectorX intheiteration k. Here, w.p. means with probability. If thefirstcase(selectingfromHM)ischosen,thepitchadjusting process will be performed as(14)Ifanewharmonyvectorisbetterthantheworstharmony in the HM, then the new harmony is includedin the HM and the existing worst harmony is excludedfromtheHM.Theabovementionedprocessisrepeated until a terminating criterion is satisfied. xik

Do nothing ww.p. PARSelect from the neighboringof thee best of HM w.p. (1PAR)),xik

Select from HM w w.p. HMCRSelect from the possible range ww.p.(1 HMCR) ,A. Kaveh, S. Talatahari3.4. A hybrid Big Bang-Big CrunchalgorithmThe BB-BC method consists of two phases [2,12]: a BigBang phase, and a Big Crunch phase. In the Big Bangphase,candidatesolutionsarerandomlydistributedover the search space. The Big Crunch is a convergenceoperatorthathasmanyinputsbutonlyoneoutput,which is named as the centre of mass. The centre ofmass is denoted by Xc(k)iand is calculated according to(15)where N is the population size in Big Bang phase. AftertheBigCrunchphase,thealgorithmcreatesthenewsolutionstobeusedastheBigBangofthenextiterationstep,byusingthepreviousknowledge.The hybrid BB-BC approach uses the centre of mass,thebestpositionofeachcandidate(Pkj)andthebestglobalposition(Pkg)asdefinedbythePSOalgorithmto generate a new solution as(16)Wherer1isarandomnumberfromastandardnormal distribution which changes for each candidate;1isaparameterforlimitingthesizeofthesearchspace; 2and 3are adjustable parameters controllingtheinfluenceoftheglobalbestandlocalbestonthenewpositionofthecandidates,respectively.Forsizedesignvariables,Amin1.61cm2,andAmax137.43cm2, and for h the minimum and maximum values are1.00 m and 8.75 m as described in Section 2. Thesesuccessiveexplosionandcontractionstepsare carried out repeatedly until a stopping criterion hasbeen met. A maximum number of iterations is utilizedas a stopping criterion.3.5. Heuristic particle swarm ant colonyoptimizationTheheuristicparticleswarmantcolonyoptimizationconsists of two stages [13]. The first stage (PSO stage)involvesanumberofparticles,whicharesimilartothose as PSO as described in Section 3.1. In the secondstage,ACOworksasalocalsearch,whereinthealgorithmhandlesN antsequaltothenumberofparticlesinthePSOstage,andeachantgeneratesasolution around Pkgwhich can be written asX X P Pjk c kgkjk + + + ( )12 2 3 31 1 ( )( ) ( )( )max min++r x xk1 11XXc k jjkjNj jNWW( )

1111International Journal of Space Structures Vol. 25 No. 4 2010 221(17)Where, Zkjis the solution constructed by ant j in thestagek;N(Pkg,)denotesarandomvectornormallydistributed with mean Pkgand variance , and(18)Here isthestepsizeandissetto0.01. Then,theobjectivefunctionvalueforeachant,W(Zkj)iscomputed, and the best one is selected to be used in thenextiterations.Inaddition,aharmonybasedhandlingmechanismisutilizedtoimprovetheperformanceofthe algorithm as described in [13].3.6. Charged system searchThechargedsystemsearch(CSS)basedonelectro-staticand Newtonian mechanicslawswasintroducedby the authors [1416]. The CSS contains an array ofagentscalledChargedParticles(CP).EachCP hasamagnitude and the initial position of charged particlesisselectedrandomly.TheCSShasamemorywhichstoresthebestsofarresults.EachCP movestothenewpositionusingthegoverninglawsoftheNewtonian mechanics as(19)(20)whereXk+1j,Vk+1j,XkjandVkjarethepositionandvelocityofthejthCP intheiterationk+1andk,respectively;r1andr2aretworandomnumbersuniformlydistributedintherangeof(0,1);kaistheacceleration coefficient; kvis the velocity coefficient tocontrol the influence of the previous velocity; and FjistheresultantforceaffectingthejthCP whichisdeterminedusingtheCoulombandGaussslawsandis equal to(21)whererijistheseparationdistancebetweentwocharged particles; Xiand Xjare the positions of the ithand jth CPs, respectively; a is the radius of the chargedsphere(CP).qidenotesthemagnitudeoftheithCPchargeandisdefinedconsideringthequalityofitssolution asF Xj jiijiij i i jij iqqar iqri p +

_,

3 1 2 2,(

_,

1 (( ) W Wj i>0 otherwise,qW WW Wii worstbest worst

222 International Journal of Space Structures Vol. 25 No. 4 2010Optimal Design of Single Layer Domes Using Meta-Heuristic Algorithms; A Comparative Studygeometricnonlinearityrequiresacomplexanalysis.Furthermore,overallstabilitycheckisnecessaryduring the analysis to ensure that the structure does notloseitsloadcarryingcapacityduetoinstability[17].DetailsofthenonlinearstiffnessmatrixofaspacememberaregiveninMajid[18]andEkhandeetal.[19].Therefore,tohavearealistictreatmentofthedome,thegeometricnonlinearityisalsoincludedinthis study.Here,network,Schwedlerandlamelladomesarestudied.Networkdomeshaverib,diagonalandringelementsasshowninFig.1.Thedistancesbetweentheringsonthemeridianlineofthesedomesaregenerally made to be identical. There are 12 joints onthe odd rings, and the even rings have 24 nodes. Thefirst joint on the first ring is on the radius of the domewhichcoincideswiththex axisandallofthefirstjoints of the rings are located on the intersection pointsof those rings and the x-axis. Schwedler domes beingone of the most popular type of braced domes, consistsof meridional ribs connected to a number of horizontalpolygonalrings.Inordertostiffentheresultingstructure,eachtrapeziumformedbyintersectingmeridional ribs with horizontal rings is subdivided intotwotrianglesbyintroducingadiagonalmember. ThezY XG 1G 1G 8G 3G 3G 2G 2G 5G 5G 9G 9G 4G 4G 10G 10G 7G 7G 6G 6Z(a) 3D view(c) Elevation(b) PlanX YFigure 1. A network dome with four ringsA. Kaveh, S. Talataharinumber of nodes in each ring for the Schwedler domesisconsideredasconstantanditisequalto12inthispaper. The distances between the rings in the dome onthe meridian line are generally of equal length. Fig. 2shows a typical Schwedler dome. Lamella dome havediagonals extending from the crown down towards theequatorofthedome,inbothclockwiseandanti-clockwisedirections,andhavehorizontalrings,buthavenomeridianribs.SimilartotheSchwedlerdomes, the number of nodes in each ring is equal to 12,while contrary to the two previous types, only the firstjointsoftheoddringsarelocatedontheintersectionpoints of that ring and the x-axis, and the first nodes oftheevenlynumberedringsareobtainedbyananti-clockwiserotationofthenodesalongz-axisby36.Fig. 3 shows a typical lamella dome.Thegroupingofmembersisperformedinawaythat the rib members between each consecutive pair ofringsbelongtothesamegroup,diagonalmembersbelongtoonegroupandthemembersoneachringformanothergroup.Therefore,thetotalnumberofgroups for the network and Schwedler domes is equalto3Nr 2.Forthelamelladomes,thisnumberis 2Nr 1, since no meridian ribs are present.International Journal of Space Structures Vol. 25 No. 4 2010 223A populationof20individualsisusedasthenumberofagentsforallalgorithms;ForPSOandHPSACO, the value of constants c1and c2are set to0.8,thepassivecongregationcoefficientc3andtheexplorationcoefficientc4aregiven0.6and0.1,respectively[5].Themagnitudeof issetto0.01for HPSACO [13]. The value for is set to 0.9 in thefirst iteration and it is decreased linearly to 0.4 in thelastiteration.ForACOalgorithm,thevalueofconstants and are set to 1.0 and 0.4, respectively[8].Thelocalupdatingcoefficient,,istakenas0.25 and the global updating coefficient, , is set to0.2, as in [8]. The value of HMCR is set to 0.95 andthatofPARistakenas0.10fortheHSalgorithm.Previous investigations show that 11, 20.40and30.80aresuitablevaluesforHBB-BCalgorithm [2]. For the CSS algorithm, a is set to one[14]andkvandkaaresetto0.5.Duetocontinuousnatureofthealgorithms,aroundingfunctionisutilized to create the discrete or integer values. Thisfunctionroundsacontinuousvaluetothenearestdiscrete one. Thefirstexampleisanetworkdome.Thedesignhistory of the best run by each technique is shown in(c) ElevationXZY(a) 3D view(b) Plan and the related group numberYZXYXG 1G 1G 8G 8G 3G 2G 2G 3G 9G 9G 5G 5G 4G 4G 10G 10G 6G 6G 7G 7Figure 2. A Schwedler dome with four rings224 International Journal of Space Structures Vol. 25 No. 4 2010Optimal Design of Single Layer Domes Using Meta-Heuristic Algorithms; A Comparative StudyFig.4,andtheminimumresultsobtainedbythealgorithmsaregiveninTable2.Thelightestweightdesignisattainedbytheheuristicparticleswarmantcolony optimization, and the optimum result obtainedby the charged system search algorithm is the secondbest among all the other meta-heuristics. However, theminimumweightfoundbytheCSSinthisstudyisveryclosetotheoneobtainedbytheHPSACOalgorithm.TheHBB-BC,ACO,HS,andPSOalgorithms are the next, respectively. YZXYXG 1G 1G 5G 5G 2G 2G 2 G 6G 6G 3G 3G 3 G 4G 4G 4G 7G 7YXZ(c) Elevation(a) 3D view(b) Plan and the related group numberFigure 3. A lamella dome with four rings18,00016,00014,00012,000Weight (kg)10,0008,0006,0000 1000 2000 3000 4000 5000Number of analyses6000 7000 8000 9000 10000HPSACOCSSHBB-BCACOHSPSOFigure 4. The convergence history for the network dome using six meta-heuristic algorithmsA. Kaveh, S. TalatahariInternational Journal of Space Structures Vol. 25 No. 4 2010 225Table 2. Optimum designs of the network domeOptimum sectionsGroup number PSO HS ACO HBB-BC CSS HPSACO1 PIPST (10) PIPST (8) PIPST (10) PIPST (8) PIPST (8) PIPST (8)2 PIPEST (3) PIPST (4) PIPST (3) PIPST (31/2) PIPST (3) PIPST (3)3 PIPST (31/2) PIPST (31/2) PIPST (31/2) PIPST (31/2) PIPST (31/2) PIPST (31/2)4 PIPST (21/2) PIPST (21/2) PIPST (21/2) PIPST (21/2) PIPST (21/2) PIPST (21/2)5 PIPST (5) PIPST (3) PIPST (3) PIPST (3) PIPST (3) PIPST (3)6 PIPEST (5) PIPST (8) PIPST (5) PIPDEST (4) PIPST (6) PIPEST (5)7 PIPST (3) PIPST (3) PIPST (31/2) PIPEST (3) PIPST (31/2) PIPST (31/2)Height (m) 4.75 5.50 5.00 5.25 6.50 6.25max(i) (mm) 27.65 27.76 28.00 28.00 27.96 27.95Max. Strength 0.93 0.95 0.95 0.98 0.99 0.96Ratioweight (kg) 7,509 6,843 6,792 6,754 6,544 6,540Table 3. Optimum designs of the Schwedler domeOptimum sectionsGroup number PSO HS ACO HBB-BC CSS HPSACO1 PIPDEST (4) PIPST (8) PIPST (8) PIPST (8) PIPST (8) PIPST (8)2 PIPST (1/2) PIPST (1) PIPEST (1/2) PIPEST (1/2) PIPST (1/2) PIPST (3/4)3 PIPST (31/2) PIPEST (31/2) PIPEST (31/2) PIPST (4) PIPST (4) PIPST (4)4 PIPST (1) PIPST (3/4) PIPST (1/2) PIPST (1/2) PIPST (3/4) PIPST (1/2)5 PIPST (4) PIPST (4) PIPST (4) PIPST (4) PIPST (31/2) PIPST (4)6 PIPST (8) PIPEST (5) PIPST (6) PIPEST (5) PIPDEST (4) PIPST (6)7 PIPST (4) PIPEST (3) PIPST (4) PIPST (4) PIPEST (3) PIPEST (31/2)Height (m) 7.00 6.75 6.75 6.75 6.00 6.50max(i) (mm) 27.95 27.57 28.00 27.82 28.00 28.00Max. Strength0.99 0.96 0.96 0.89 0.97 0.96Ratioweight (kg) 5,210 5,188 5,081 5,020 4,950 5,037TheSchwedlerdomeisthesecondexample.Theoptimum design for this dome is obtained using CSS.The best weight of the CSS design is equal to 4,950 kgwhichis1.41%,1.75%,2.64%,4.80%and5.25%lighterthanthedesignsobtainedbyHBB-BC,HPSACO,ACO,HSandPSO,respectively.Table3summarizestheobtainedoptimumresults.Similartothepreviousexample,bothHPSACOandCSShavethefastestconvergencerateandthePSOhastheslowest one as shown in Fig. 5.Thelaststructureinvestigatedinthisstudyisalamella dome. The optimum design is obtained by theHPSACOas5777kg.TheCSScanfindthesecondweightwhichisonly0.8%heavierthantheresultoftheHPSACO.TheHBB-BC,HSandACOfindthenext best results, respectively. Similar to the perviousexamples, PSO is the worst result. Table 4 and Fig. 6identify the best results with the corresponding weightand the convergence history of these algorithms for thelamelladomes.Whenthenumberofringsincreases,theweightofalltypesofthedomesincreasesconsiderably,andtohaveanoptimumdomeweight,the number of the rings should be selected as small aspossible.Intheabovementionedcasestudies,theoptimumnumberofringsobtainedbythealgorithmsis equal to three. 226 International Journal of Space Structures Vol. 25 No. 4 2010Optimal Design of Single Layer Domes Using Meta-Heuristic Algorithms; A Comparative StudyTable 4. Optimum designs of the lamella domeOptimum sectionsGroup number PSO HS ACO HBB-BC CSS HPSACO1 PIPDEST (5) PIPST (10) PIPST (8) PIPST (10) PIPST (8) PIPST (8)2 PIPST (3) PIPST (3) PIPEST (3) PIPST (3) PIPST (3) PIPST (3)3 PIPST (3) PIPST (3) PIPST (3) PIPST (3) PIPST (3) PIPST (3)4 PIPDEST (4) PIPDEST (3) PIPST (8) PIPDEST (3) PIPST (8) PIPDEST (4)5 PIPST (31/2) PIPST (4) PIPST (4) PIPST (31/2) PIPST (4) PIPST (31/2)Height (m) 6.25 5.50 5.75 5.75 6.00 6.50max(i) (mm) 27.79 27.39 27.48 27.27 27.48 27.78Max. Strength 0.98 0.96 0.94 1.00 0.96 1.00Ratioweight (kg) 6,327 6,149 6,152 6,088 5,823 5,77712,00011,00010,0009,000Weight (kg)8,0007,0005,0006,0000 1000 2000 3000 4000 5000Number of analyses6000 7000 8000 9000 10000HPSACOCSSHBB-BCACOHSPSOFigure 5. The convergence history for the Schwedler dome using six meta-heuristic algorithms13,00012,00011,00010,000Weight (kg)9,0008,0006,0007,0000 1000 2000 3000 4000 5000Number of analyses6000 7000 8000 9000 10000HPSACOCSSHBB-BCACOHSPSOFigure 6. The convergence history for the lamella dome using six meta-heuristic algorithmsA. Kaveh, S. Talatahari5. CONCLUDING REMARKSSome well-known meta-heuristic algorithms consistingofparticleswarmoptimizer,antcolonyoptimization,harmonysearch,BigBang-BigCrunch,heuristicparticleswarmantcolonyoptimizationandchargedsystemsearchareutilizedtofindtheoptimumnonlineardesignofthesinglelayerdomestructures.Thesealgorithmsareinspiredfromnature.Thesepopulationbasedalgorithmsdonotneedthegradientinformation,andoftentheinitialseedshavesmalleffect on the final results. Therefore, these algorithmsare classified as the global search methods. Comparing the results of the dome designs obtainedbythesealgorithmsillustratesthegoodperformanceoftheHPSACOandtheCSScomparedtoothermethods.HPSACOutilizestwoadditionalauxiliarytoolstofindtheoptimumsolutionandtheCSSalgorithm has been defined in a way that can control agoodbalancingbetweentheexplorationandexploitation.Anotherimportantpointthatshouldbeconsidered is the effect of the values of the parametersontheperformanceofthealgorithms.Inthispaper,sincethesuitablevaluesfortheseparametersareselectedbasedonthepreviousstudiesonstructuraloptimizationproblems,itisexpectedthatthealgorithmshavealmosttheirbestperformance.However, these parameters affect the efficiency of thealgorithm,andthisshouldbeconsideredwhenusingmostofthemeta-heuristicalgorithms.Sincethemathematical operations related to structural analysesmakethemainpartofthecomputationalcosts,therefore if an algorithm itself has a low computationalcost,thiswillnothaveasoundeffectonthetotaloptimizationcosts.Asaresult,usingthenumberofanalysesasthecomputationalcostmeasurementisagoodmeansforcomparisonofthealgorithmsforstructuralproblems,andforadeterminednumberofanalyses,thealgorithmwhichcanfindabetterresultcan be considered as the best.Betweenthreedifferentkindsofdomesstudiedinthis paper, the Schwedler dome is the most economicalone. The number of elements for Schwedler domes issmall compared to the other types and this may be thecauseofbeingthelightestamongotherdomes.Thenetwork dome due to having many elements comparedtolamellaandSchwedlerdomehastheheaviestweight.ACKNOWLEDGEMENTThefirstauthorisgratefultoIranNationalScienceFoundation for the support.International Journal of Space Structures Vol. 25 No. 4 2010 227REFERENCES[1] American Institute of Steel Construction (AISC) ManualofSteelConstruction-LoadResistanceFactorDesign.3rd ed., AISC, Chicago, 1991.[2] Kaveh,A.andTalatahari,S.,OptimaldesignofSchwedlerandribbeddomesviahybridBigBang-BigCrunchalgorithm.JournalofConstructionalSteelResearch, 66(3), 2010, 412419.[3] Eberhart,R.C.andKennedy,J., A newoptimizerusingparticleswarmtheory.ProceedingsoftheSixthInternational Symposium on Micro Machine and HumanScience, Nagoya, Japan, 1995.[4] He,S.,Wu,Q.H.,Wen,J.Y.,Saunders,J.R.andPaton,R.C.,A particleswarmoptimizerwithpassivecongregation. 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