DOC

Optimal_0

By Annette Lawrence,2014-07-24 00:52
8 views 0
Optimal_0Optima

    Optimal

1264

    ;JournalofZhejiangUniversitySCIENCEA

    ;ISSN1673565X(Print);ISSN18621775(Online)

    ;wwwzjueducn~zus;wwwspringerlinkcom

    ;E?mail:jZUS@zjueduca

    ;Liueta1./JZhejiangUnivSciA20089f9J:1264?1269

    ;Optimaldesignofpressurevesselusinganimproved

    ;geneticalgorithm

    ;Peng.feiLIU”,PingxuShu.xinHAN,Jin.yangZHENG

    ;(InstituteofChemicalMachineryandProcessEquipment,Zh~iangUniversity,Hangzhou31002LChina)

    ;(InstituteofAppliedMechanics,Zh~iangUniversity,Hangzhou31002LChina) ;(3HangzhouSpecialEquipmentInspectionInstitute,Hangzhou310003,China) ;Email:pfliul980@yahoo.com;pingxu@zju.edu.cn

    ;ReceivedMar.5,2008;revisionacceptedMay22,2008

    ;Abstract:Astheideaofsimulatedannealing(SA)isintroducedintothefitnessfunction,animprovedgeneticalgorithm(GA)is

    ;proposedtoperforlTltheoptimaldesignofapressurevesse1whichaimstoattaintheminimumweightunderburstpressurecon.

    ;straint.Theactua1burstpressureiscalculatedusingthearc.1engthandrestartanalysisinfiniteelementanalysis(FEA).Apenalty

    ;functioninthefitnessfunctionisproposedtodea1withtheconstrainedproblem.Theeffectsofthepopulationsizeandthenumber

    ;ofgenerationsintheGAontheweightandburstpressureofthevesse1areexplored.Theoptimizationresultsusingtheproposed

    ;GAarealsocomparedwiththoseusingthesimpleGAandtheconventiona1MonteCarlomethod.

    ;Keywords:Pressurevessel,Optimaldesign,Geneticalgorithm(GA),Simulatedannealing(SA),Finiteelementanalysis(FEA)

    ;doi:10.1631/jzus.A0820217Documentcode:ACLCnumber:TH12

    ;INTR0DUCT10N

    ;ItiSwellknownthatthepressurevesselhasbeen

    ;widelyusedinavarietyofareassuchaschemical

    ;engineering,medicaltreatment.aviationandastro.

    ;nauticsaswellasnuclearengineering.Currentlythe

    ;pressurevesseltendstobedevelopinginlarge-scale

    ;andhigh-parameterdirections,especiallyinchemical

    ;industry.However.thepressurevesselisgenerally

    ;subjectedtoacomplexenvironmentsuchashigh

    ;pressureandhightemperature.Thismeansnotonlya ;strongchallengeregardingtheperformanceofthe ;materialandstructure.butalsoconcerningthedesign ;Ofthepressurevesse1.Howtoachieveaperfect ;combinationofexcellentperformanceandlowcostin ;thedesignofapressurevesselundercertaindesign ;conditionsisanimportanttopic.

    ;Severalstudieshaveconcentratedontheoptimal ;designforthestructuralshapeofapressurevessel ;Correspondingauthor

    ;Project(Nos2006BAK04A0202and2006BAK02B02o8)sup

    ;portedbytheNationalKeyTechnologyR&DProgram,China ;understrengthandstiffnessconstraints.However, ;cheyareallbasedonconventionaloptimization ;methodssuchassimplemathematicalcalculations ;(Liueta1.,2001;Magnuckieta1.,2004)andthegra- ;dientsearchmethod(HyderandAsif,2008).These ;methodsmayeasilylcadtolocaloptimizationresults ;andlargelydependontheinitialvaluesoftheopti. ;mizedvariables.Besides.thesearchcapacityisalso ;limitedforabigdesignspacewithalargenumberof ;discretevariables.

    ;Currently,twointelligentoptimizationalgo. ;rithmsaresuitableforaspecificoptimizationprob. ;1emwithalarge.scalediscretedesignspace:genetic ;algorithm(GA)(Holland,1975;Schmitt,2001; ;BlachutandEschenauer.20011andsimulatedan. ;nealing(SA)(Kirkpa~icketa1.,1983;deVicenteet ;a1.,2003).Usingthesetwomethodstheoptimization ;solutionsareobtainedfromthestochasticsamplingof ;thedesignedspace.TheGAwasoriginallyemployed ;tosimulatetheevolutionofbiologyandexhibitsa ;goodcapacityforachievingglobaloptimizationre- ;sults.However.the”hil1.climbing”capacitV昂科拉hatis

    ;

    ;Lmeta1./JZhejiangUnivSciA20089l9).”1264—1269

    ;the1oca1searchcapacity,isrelativelyweak.Incon

    ;trast,theSAwasoriginallyusedtosimulatethean

    ;nealingprocessofthephysicalmultiparticlesystem

    ;instatisticalthermodynamicsandprovedtohavea ;strongabilityinthe1oca1searchofdiscretespace. ;Unfortunately,noreportabouttheapplicationofthese ;twoalgorithmsintheoptimaldesignofapressure ;vesselhasbeenfoundsofar.

;InthisPaDertheGAisusedtooptimizethe

    ;weightofthepressurevesse1undertheburstpressure ;constraint.TheconceptofSAisintroducedintothe ;fitnessfunctiontoimprovethesearchefciencyof

    ;theGA.Apenaltyfunctionisproposedtoensurethe ;bidirectiona1searchinbothfeasibleandinfeasible ;directionsforthediscretedesignvariables.Efiectsof

     ;variOUSparametersinthegeneticiterationoftheop

    ;timizationresultsareexplored.Also,theresultsob

    ;tainedusingtheproposedGAarecomparedwith ;thoseusingthesimpleGAandtheconventional ;MonteCarlomethod.

    ;FINITEELEMENTMODELING0FPRESSURE

    ;VESSEL

    ;Thefollowinganalysisconcentratesonavesse1 ;asshowninFig.1.inwhichaCartesianCOOrdinate ;systemisdefined.Itincludesacylinder,aspherica1 ;headandnozzlesAandB.Thefiniteelementmodel ;andmaterialpropertiesareobtainedbvLiuPf ;a1.(2008).

    ;N

    ;Fig.1Schematicrepresentationofvessel(unit:mm) ;BASICDESIGNCONDITIONS

    ;1265

    ;Thedesignofthepressurevesse1aimstoachieve ;theminimumweightundertheburstpressurecon

    ;straint.Thedesignvariablesarethethicknessesf1and ;t2ofthecylinderandthehead,respectively.The ;statusvariableistheactualburstpressureP.Herethe ;basicdesignconditionsaresummarizedas

    ;minimize(tl,f2)(tlED1,t2?D2),

    ;P(tl,t2)>nXPd,

    ;wheretheweightofvesselWandtheactualburst ;pressureParefunctionsoftlandt2.POrepresentsthe ;designpressure.D1andD2arethevariablerangesfor ;tlandf2,respectively.Theconstant/,1isthesafety ;toefficientofthevesse1.

    ;0PTIMIZATIONM0DEL

    ;Inthefollowing,animprovedGAisproposed ;anditincludessevensteps:

    ;(1)Binaryencodingofthedesignvariables.If ;thestringlengthofbitsforeachvariablefisLandthe ;numberofthedesignvariablesisNum.the1engthof ;thechromosomeforeachindividualisLxNum.Ifthe

;rangeofthevariablefiswithin[tmin,tmax],therela

    ;tionshipbetweenthedecimalvaluefandbinaryvalue ;tbiSgivenby

    ;+(fm/min)

    ;(2)Generationoftheinitia1population.The ;populationwithPopindividualsisgeneratedusinga ;completelystochasticmethod.

    ;(3)Calculationsoftheobjectivefunctionand ;burstpressure.Intermsoftheparametricmodel,the ;weightofthevesse1iscalculatedusing

    ;ANSYSAPDLandtheburstpressureofthevesselis ;calculatedbyususingtheproposedalgorithm,in ;whichthearc..1engthalgorithmandtherestartanaly.. ;sisinthefiniteelementanalysis(FEA)areemployed ;(Liueta1.,2008).A1so,otherresearchwasperformed ;tocalculatetheburstpressureofthepressurevessel, ;suchasthemethodsproposedbyBlachutandVu ;(2007).

    ;

    ;1266Liuetal/dZhejiangUnivSciA20089(~:1264-1269 ;f4)Calculationsofthefitnessvaluesforthein. ;dividuals.Ingeneral,thecalculatedburstpressure ;maybesmallerthantheburstpressureconstraintand ;thebinarycodesleadtoinfeasibleresults.Herea ;penaltyfunctionisproposedtodealwimthecon. ;strainedproblem.ThefimessfunctionFforthe ;minimumprobleminthesimpleGAiswrittenas ;f(,t2)=fitl,t2)+×max[n×P,0],

    ;F=1/f(fl,t2),

    ;whereisthepenaltyfactorandgenerallyisalarge ;positivenumber.Thus,thepenaltyfunctionforcesthe ;geneticsearchtoapproachtheoptimizationsolution ;fromboththefeasibleandinfeasibledirections.The ;largerthefitnessvalueis.themoreexcellentthein. ;dividualis.

    ;However,twoproblemsappearforthefitness ;functionabove:First.thedifierencebetweenthein- ;dividualsintheearlygeneticiterationislargeandthe ;excellentindividualscaneasilyoccupythewhole ;population,whichleadstothephenomenonof”pre-

    ;maturity”:Second,thefitnessvaluestendtobecon.

    ;sistentinthelatergeneticiterationandtheexcellent ;individualscannotexhibittheirpredominance,which ;resultsinthestagnancyofthewholepopulation

;evolution.

    ;Inordertosettlethetwoproblemsabove.anovel ;fitnessfunctionFisproposedusingtheideaOfSAtO ;improvethesearchemciency.TheSA.whichwas ;proposedbyKirkpatricketa1.(1983),hasbeenwell ;appliedtosolvecomplexstochasticoptimization

     ;problemswithdiscretespaceanddemonstratedtO;havegoodlocalsearchcapacity,whichisalsocalled ;the”hill—climbing’’capacity.

    ;TheSAisdescribedasfollows:ifasystemisina ;configurationAattimetandanewconfigurationBof ;mesystemattimeAtisgeneratedrandomly,the

    ;configurationBisacceptedaccordingtotheaccep- ;tanceprobabilityPf1:

    ;P()expI-()()

    ;B

    ;decreasingtemperaturewhichmeansthatthema- ;terialenergywillgraduallyapproachalowenergy ;statusandfinallyconvergewiththeminimumenergy ;status.

    ;ByintroducingtheideaofSA,thefitnessfunction ;jntheGAismodifiedas

    ;J,F=exp[_ffz)/(e),

    ;=,

    ;whereTodenotestheinitialannealingtemperature ;andTistherea1timeannealingtemperatureafter ;eachgeneticiteration.kisthecoolingscheduleand ;generally1sanumberapproximatingunlty. ;f5)Selection.Theroulettewheelselection ;proposedbyHollandr1975)isthemostfamousse. ;1ectionmethodanditsbasicprincipleistodetermine ;mebirthprobabilityofeachindividualaccordingto ;theproportionofthefimessvalueofeachindividual ;inthewholepopulation.

    ;CorrespondingtoEqs.(3)and(5),theprobabili

    ;tiesofselectionarerespectivelyexpressedas ;:?Pop,//

    ;f=l,

    ;f6)Crossover.Aonepointcrossoverisadopted ;inthiswork.Thepositionofthecrossoverisran. ;domlyselectedandthecrossovermaybecarriedout ;wimacrossoverprobability.Takinga20.bitchro. ;mosomeforexample,theoperatingrepresentationof ;thecrossoverisshowninFig.2.

    ;r71Mutation.Thebitsforeachindividualare ;randomlychangedfromzerotooneandviceversa ;withamutationprobability.Theoperatingrepresen- ;tationofmutationisshowninFig.3.

    ;Parent110001001011I100011100

    ;Parent2O01111O01101011011O01

    ;JlChild1l000l00l0lll011o110o1

    ;Child100111100110l100011100

    ;whereErepresentstheenergy,KBistheBoltzmann ;constantandTistheannealingtemperature.Inthis ;casemeacceptanceprobabiliPdecreaseswimF.g.2Operatingrepresentationofcrossover

    ;

    ;Liueta1./JZhejiangUnivSciA20089(9):12641269

    ;Parent110001001l011100011100

    ;Parent2001111O0l110011011001

    ;Child10111011010111000ll100

    ;Child111000011l11OO11O11OO1

    ;Fig.3Operatingrepresentationofmutation ;Basedonthestepsabove,theoptimaldesignof ;thepressurevesselisimplementedbyassociatingthe ;softwareMATLABwithANSYS.Theflowchartis ;showninFig.4.

    ;Fig.4Flowchartoftheimprovedgeneticalgorithm ;NUMERICALEXAMPLE

    ;1267

    ;Theconstantparametersintheoptimization ;analysisofthepressurevesselarechosenasfollows: ;thethicknessesfmm)ofthecylinderandheadare ;tl?[36,44],f2?[20,28];theinnerradiusandlengthof ;thecylinderare3251TnTIandl2001TnTI,respectively: ;theinnerradiusofthehemisphericalheadis325mm; ;thesizetypesforthenozzlesAandBare46minxl1 ;mmand54rnlnxl51TnTI.respectively;thestring ;lengthofbitsforeachvariableis=l0:thedesign ;pressureisPd=l8MPa;thesafetycoefficientofthe ;pressurevesselisn=3:thepenaltyfactoris2=50;the ;probabilitiesofcrossoverandmutationare0.85and ;0.0lrespectively;theinitialannealingtemperatureis ;=

    ;l00K;thecoolingscheduleisk=0.99;the

    ;BoltzmannconstantisKB=1.

    ;IntermsofthesimpleGA.lblelshowsthe

    ;optimizationresultsoftheweight,burstpressureand ;fitnessvaluesofeachindividualforl0populations

null

;2

    ;

    ;

    ;?

    ;NumberofgenerationsEN

    ;fb)

    ;Fig?6Effectsofthenumberofgenerationsonthe(a)weightand(b)burstpressurefor100pop

    ulations

    ;usingthesimpleGAandthoseusingtheproposedGA ;Similarly,theproposedGAdistinctlyimprovesthe ;convergenceefficiencyandavoidsthe”prematurity”.

    ;Therefore.thecapacityof”hill—climbing’’ofthe

    ;simpleGAiswellenhanced.

    ;InordertoconfirmtheefficiencyoftheGA,the ;optimizationresultsusingtheGAarealsocompared ;withthoseusingtheconventionalMonteCarlo ;method.aslistedinTlable2.

    ;assumesthatthevariables

    ;TheMonteCarlomethod

    ;tlandt2areuniformly

    ;

    ;LIueta1./dZhejiangUnivSciA200899):12641269

    ;Table2ComparisonsoftheoptimizationresultsusingthesimpleGA,theimprovedGAandt

    heMonteCarlo

    ;method

    ;Pop:populationsize;EN:numberofiterations;tl:thicknessofcylinder;t2:thicknessofhea

    d;1000iterations;10000iterations

    ;disibutedandthecompletelyrandomvaluesare ;chosenwithintheirvariableranges.TheMonteCarlo ;methodisalsocarriedoutusingMATLABandthe ;ectoftheiterationtimesontheoptimizationresults ;iscalculated.Onlywhentheiterationtimebecomes ;verylargecanthecalculationsreachrelativelyopti. ;malresultsusingtheMonteCarlomethod.This ;practicallyneglectstheectOftheevolutioninthe ;GAandbecomestime.consumingandinemcient. ;Basedonthesimple6theproposed6further ;takesintoaccountthedirentcharacteristiesofthe

    ;earlyandlateriterationsbyadjustingtheselecting ;probabilityoftheindividuals,andimprovesthe ;searchemciencyandconvergencevelocity.Therefore. ;theproposedGAexhibitsmoreadvantagesthanthe ;simpleGAandtheMonteCarlomethod.From1’able

    ;2theoptimizationresultsusingtheproposedGAare

    ;381.684kgweightand54MPaburstpressureandthe ;correspondingthicknessconfigurationsaretl=39.76 ;mmandt2=20inn’1.

    ;C0NCLUS10N

    ;Inthisanalysis.animprovedGAwhichisasso. ;ciatedwiththeconceptOfSAisproposedtooptimize ;theweightofthepressurevesselundertheburst ;pressureconstraint.Theactualburstpressureofthe ;vesseliscalculatedusingthearclengthandrestart

    ;analysisintheFEA.Apenaltyfunctionisproposedto ;dealwiththeconstrainedproblem.Efiectsofthe ;populationsizeandthenumberofgenerationsonthe ;optimizationresultsareexplored.Bycomparison,the ;followingconclusionsareobtained:

    ;(1)Intermsoftheconvergencevelocityand ;precision,theproposedGAexhibitsmoreadvantages ;thanthesimpleGAandtheconventionalMonteCarlo ;optimizationmethod.

    ;f2)TheconventionalMonteCarlomethodis

    ;moretime.consumingandinefficientthantheGA ;becausethecompletely

    ;strongcapacitytoreach

    ;tions.

    ;randomsearchlacksthe

    ;therealoptimizationsolu

    ;(3)Thebidirectionalsearchfromthefeasibleand ;infeasiblesolutionsaremorerationalthanthecom. ;pleteexclusionoftheinfeasiblesolutionsfordealing ;withtheconstraintconditionsbyusingtheGA. ;References

    ;Blachut,J.,Eschenauer,H.A.,2001.EmergingMethodsfor ;MultidisciplinaryOptimization.Springer,Wien.New ;Y_0rk.

    ;Blachut…Jvu,V,2007.Burstpressuresfortorispheresand ;shallowsphericalcaDs.Strain,43(1):26.36.doi:l0.1111,

    ;i.1475.1305.2007.00304.x1

    ;deVicente…JLanchares,J.,HermidaR.,2003.Placementby

    ;thermodynamicsimulatedannealing.PhysicsLettersA, ;317(5.61:415.423.

    ;Holland.J.H.,1975.AdaptationinNaturalandArtificialSys. ;tems.Unive~ityofMichiganPress.AnnArbor,P.1.44. ;Hyder,J.M..As?

Report this document

For any questions or suggestions please email
cust-service@docsford.com