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ORIGINALARTICLEPseudoconstructaltheoryforshapeoptimizationofmechanicalstructuresJeanLucMarcelinReceived10January2007/Accepted1May2007/Publishedonline25May2007SpringerVerlagLondonLimited2007AbstractThisworkgivessomeapplicationsofapseudoconstructaltechniqueforshapeoptimizationofmechanicalstructures.Inthepseudoconstructaltheorydevelopedinthispaper,themainobjectiveofoptimizationisonlytheminimizationoftotalpotentialenergy.Theotherobjectivesusuallyusedinmechanicalstructuresoptimizationaretreatedlikelimitationsoroptimizationconstraints.Twoapplicationsarepresentedthefirstonedealswiththeoptimizationoftheshapeofadropofwaterbyusingageneticalgorithmwiththepseudoconstructaltechnique,andthesecondonedealswiththeoptimizationoftheshapeofahydraulichammersrearbearing.KeywordsShapeoptimization.Constructal.Geneticalgorithms1IntroductionThispaperintroducesapseudoconstructalapproachtoshapeoptimizationbasedontheminimizationofthetotalpotentialenergy.Wearegoingtoshowthatminimizingthetotalpotentialenergyofastructuretofindtheoptimalshapemightbeagoodideainsomecases.Thereferencetotheconstructaltheorycanbejustifiedinsomewayforthefollowingreasons.AccordingtoBejan1,shapeandstructurespringfromthestruggleforbetterperformanceinbothengineeringandnaturetheobjectiveandconstraintsprincipleusedinengineeringisthesamemechanismfromwhichthegeometryinnaturalflowsystemsemerges.Bejan1startswiththedesignandoptimizationofengineeringsystemsanddiscoversadeterministicprincipleforthegenerationofgeometricforminnaturalsystems.Thisobservationisthebasisofthenewconstructaltheory.Optimaldistributionofimperfectionisdestinedtoremainimperfect.Thesystemworksbestwhenitsimperfectionsarespreadaroundsothatmoreandmoreinternalpointsarestressedasmuchasthehardestworkingparts.Seeminglyuniversalgeometricformsunitetheflowsystemsofengineeringandnature.Bejan1advancesanewtheoryinwhichheunabashedlyhintsthathislawisinthesameleagueasthesecondlawofthermodynamics,becauseasimplelawispurportedtopredictthegeometricformofanythingaliveonearth.Manyapplicationsoftheconstructaltheoryweredevelopedinfluidsmechanics,inparticularfortheoptimizationofflows2–10.Ontheotherhand,thereexists,toourknowledge,littleexamplesofapplicationsinsolidsorstructuresmechanics.Sowehaveatleasthalfofthereferencestopapersinfluiddynamicsmostofthesameauthor,becausetheconstructalmethodwasdevelopedfirstbythesameauthor,AdrianBejan,withonlyreferencestopapersinfluiddynamics.Theconstructaltheoryrestsontheassumptionthatallcreationsofnatureareoveralloptimalcomparedtothelawswhichcontroltheevolutionandtheadaptationofthenaturalsystems.Theconstructalprincipleconsistsofdistributingtheimperfectionsaswellaspossible,startingfromthesmallestscalestothelargest.Theconstructaltheoryworkswiththetotalmacroscopicstructurestartingfromtheassemblyofelementarystructures,bycomplyingwiththenaturalrulesofoptimaldistributionoftheimperfections.Theobjectiveistheresearchoflowercost.IntJAdvManufTechnol2008381–6DOI10.1007/s0017000710802J.L.MarcelinLaboratorieSolsSolidesStructures3S,UMRCNRSC5521,DomaineUniversitaire,BPn°53,38041GrenobleCedex9,FranceemailJeanLuc.Marcelinujfgrenoble.frHowever,aglobalandmacroscopicsolutionfortheoptimizationofmechanicalstructureshavingleastcostastheobjectivecanbeveryclosetotheconstructaltheory,fromwherethetermpseudoconstructalcomes.Theconstructaltheoryisapredictivetheory,withonlyonesingleprincipleofoptimizationfromwhichallrises.Thesameappliestothepseudoconstructalstepwhichisthesubjectofthisarticle.Thesingleprincipleofoptimizationofthepseudoconstructaltheoryistheminimizationoftotalpotentialenergy.Moreover,inourexamplespresentedhereafter,thepseudoconstructalprinciplewillbeassociatedwithageneticalgorithm,withtheresultthatouroptimizationwillbeveryclosetothenaturallaws.Theobjectiveofthispaperisthustoshowhowthepseudoconstructalstepcanapplytothemechanicsofthestructures,andinparticulartotheshapeoptimizationofmechanicalstructures.Thebasicideaisverysimpleamechanicalstructureinabalancedstatecorrespondstoaminimaltotalpotentialenergy.Inthesameway,anoptimalmechanicalstructuremustalsocorrespondtoaminimaltotalpotentialenergy,anditisthisobjectivewhichmustintervenefirstoveralltheothers.Itisthisideawhichwillbedevelopedinthisarticle.Twoexampleswillbepresentedthereafter.Theideatominimizetotalpotentialenergyinordertooptimizeamechanicalstructureisnotbrandnew.Manypapersalreadydealwiththisproblem.Whatisnew,istomakethisapproachsystematic.Theonlyobjectiveofoptimizationbecomestheminimizationofenergy.InGosling11,asimplemethodisproposedforthedifficultcaseofformfindingofcablenetandmembranestructures.Thismethodisbaseduponbasicenergyconcepts.Atruncatedstrainexpressionisusedtodefinethetotalpotentialenergy.ThefinalenergyformisminimizedusingthePowellalgorithm.InKannoandOhsaki12,theminimumprincipleofcomplementaryenergyisestablishedforcablenetworksinvolvingonlystresscomponentsasvariablesingeometricallynonlinearelasticity.Inordertoshowthestrongdualitybetweentheminimizationproblemsoftotalpotentialenergyandcomplementaryenergy,theconvexformulationsoftheseproblemsareinvestigated,whichcanbeembeddedintoaprimaldualpairofsecondorderprogrammingproblems.InTaroco13,shapesensitivityanalysisofanelasticsolidinequilibriumispresented.Thedomainandboundaryintegralexpressionsofthefirstandsecondordershapederivativesofthetotalpotentialenergyareestablished.InWarner14,anoptimaldesignproblemissolvedforanelasticrodhangingunderitsownweight.Thedistributionofthecrosssectionalareathatminimizesthetotalpotentialenergystoredinanequilibriumstateisfound.Thecompanionproblemofthedesignthatstoresthemaximumpotentialenergyunderthesameconstraintconditionsisalsosolved.InVentura15,theproblemofboundaryconditionsenforcementinmeshlessmethodsissolved.InVentura15,themovingleastsquaresapproximationisintroducedinthetotalpotentialenergyfunctionalfortheelasticsolidproblemandanaugmentedLagrangiantermisaddedtosatisfyessentialboundaryconditions.Theprincipleofminimizationoftotalpotentialenergyisinadditionatthebaseofthegeneralfiniteelementsformulation,withanaimoffindingtheunknownoptimalnodalfactors16.2ThemethodsusedInthepseudoconstructaltheorydevelopedinthispaper,themainobjectiveofoptimizationisonlytheminimizationoftotalpotentialenergy.Theotherobjectivesusuallyusedinmechanicalstructuresoptimizationaretreatedherelikelimitationsoroptimizationconstraints.Forexample,onemayhavelimitationsontheweight,ortonotexceedthevalueofastress.Theideawhichwillbedevelopedinthispaperisthusverysimple.Amechanicalstructureisdescribedbytwotypesofparametersvariablesknownasdiscretizationvariablesforexample,degreesoffreedomindisplacementforfiniteelementsmethod,andgeometricalvariablesofdesignforexampleparameterswhichmakeitpossibletodescribethemechanicalstructureshape.Totalpotentialenergydependsonanimplicitorexplicitwayofdeterminingdiscretizationanddesignvariablesatthesametime.Onethuswillcarryoutadoubleoptimizationofthemechanicalstructure,comparedtothediscretizationanddesignvariablestheobjectivebeingtominimizetotalpotentialenergyoverall.Clearly,theproblemofoptimizationofamechanicalstructurewillbeaddressedbythefollowingapproach–Objectivetominimizetotalpotentialenergy–Variablesofoptimizationconcurrentlydeterminingdiscretizationvariablesinthecaseofatraditionaluseofthefiniteelementmethodinmechanicsofstructures,anddesignvariablesdescribingtheshapeofthestructure–Optimizationlimitations–Weightorvolume–Displacementsorstrains–Stresses–FrequenciesTheproblemofoptimizationofamechanicalstructurewillbesolvedinthefollowingway,whilereiteratingon2IntJAdvManufTechnol2008381–6thesestages,ifneededaccordingtothenatureoftheproblemStage1Minimizationofthetotalpotentialenergyofthemechanicalstructurecomparedtotheonlydiscretizationvariablesofthestructuredegreesoffreedominfiniteelements.Itactshereasanoptimizationwithoutoptimizationlimitations.Theonlylimitationsatthisstageareofpurelymechanicalorigin,andrelatetotheboundaryconditionsandtotheexternaleffortsappliedtothestructure.Inthisstage1,thedesignvariablesremainfixed,andoneobtainstheimplicitorexplicitexpressionsofthedegreesoffreedomaccordingtothedesignvariableswhichcanbethevariableswhichmakeitpossibletodescribetheshape,inthecaseofashapeoptimization,forexample.Onewillseeintheexamplesofthefollowingpartthattheseexpressionscanbeexplicitorimplicitandwhichisthesuitabletreatmentfollowingthecases.Inthecaseofafiniteelementsmethodofcalculation,thisstage1isthebasisoffiniteelementscalculationtoobtainthedegreesoffreedomofthemechanicalstructure.Indeed,infiniteelements,displacementswiththenodesofthemechanicalstructuremeshareobtainedbyminimizationoftotalpotentialenergy16.Stage2Theexpressionsofthedegreesoffreedomofthemechanicalstructureaccordingtothedesignvariablesobtainedpreviouslyaretheninjectedintothetotalpotentialenergyofthemechanicalstructureonewillseeinthesecondexampleofthefollowingparthowonetreatsthecasewherethedegreesoffreedomareimplicitfunctionsofthedesignvariables.Onethenobtainsanexpressionofthetotalpotentialenergywhichdependsonlyonthedesignvariablesinexplicitorimplicitform.Stage3Onethencarriesoutasecondandnewminimizationofthetotalpotentialenergyobtainedintheprecedingform,butthistimecomparedtothedesignvariableswhilerespectingthetechnologicallimitationsortheoptimizationconstraintsoftheproblem.Thismethodcanbeappliedwithmoreorlessfacilityaccordingtothenatureoftheproblem.Itisclear,forexample,thatifthediscretizationvariablescanbeexpressedinanexplicitwayaccordingtothedesignvariables,thesettinginofstages2to3isimmediate,andwithoutiterations.Ifthediscretizationvariablescannotbeexpressedinanexplicitwayaccordingtothedesignvariables,orifthetopologyofthestructureisnotfixed,orifthebehaviorisnotlinear,itwillbenecessarytoproceedbysuccessiveiterationsonstages1to3.Itisthecaseoftheexamplespresentedinthefollowingpart,andonewillseeonthisoccasionwhichtypeofstrategyonecanadoptfortheseiterations.Tosummarize,inthepseudoconstructalstep,themainobjectiveisonlytheminimizationoftotalpotentialenergy,theotherpossibleobjectivesaretreatedlikelimitationsoroptimizationconstraints.TheoptimizationmethodusedforourexamplesisGAgeneticalgorithm,asdescribedin17.Exampleswithsimilarinstructionalvaluecanalsobefoundinmanybooks,e.g.in18.Thisevolutionarymethodisveryconvenientforourpseudoconstructalmethod.TheauthorhasworkedextensivelyinGAsandpublishedinsomereputedjournalsonthistopic19–31.AsthetopicofGAsisstillrelativelynewinthestructuralmechanicscommunity,weprovideheresomedetailsofexactlywhatisusedinthisGA.Amultiplepointcrossoverisusedratherthanasinglepointcrossover.Theselectionschemeusedateachgenerationisentirelystochastic.Forourexamples,thenumberofgenerationsisequaltothatusedforconvergence.TheresultsprovidedforourexampleswereconsistentlyreproducedbyusingdifferentseedsintheGA.Ithasbeenprovedthataratherstandardgeneticalgorithmissufficientforourexamples.3ExamplesEventhoughpotentialenergymaybeagoodmeasureforsomeoptimizations,potentialenergyisnotwhatgivestheshapetoawaterdroplet,nordefinestheoptimalshapeforahammer,whichiswhypotentialenergyisnottheonlyobjectivebuttheoptimizationproblemisamultiobjectiveoneandtheobjectivefunctionsforthetwoexamplesarethenclearlyformulated.3.1Example1optimizationoftheshapeofadropofwaterThefirsttestexampleistheoptimizationoftheshapeofadropofwaterFig.1.Thisproblemisequivalenttoanequalresistancetankcalculatedbythemembranetheory.Theobjectiveistoseeifthepseudoconstructaltheorygivesthenaturesoptimumdesign.3.1.1ThemethodsusedThegeometryofthedropofwaterisdefinedbythegeneratinglineofathinaxisymmetricshell.Thislineisdescribedbysuccessivestraightorcircularsegmentsdescribedinagivensenseanddefinedbyinputdataofmasterpointcoordinatesandradiusvalues.Theinitialdataareasetofnodalpointsconnectedbystraightsegments.EachnodalpointisidentifiedbyitstwocylindricalIntJAdvManufTechnol2008381–63coordinatesr,z,andarealRwhichrepresentstheradiusofthecircletangenttothetwostraightsegmentsintersectingatthepoint.Theothercomputercalculationsgivethecoordinatesofanyboundarypointandespeciallythetangentpointsnecessarytodefinethecirculararclengths.ThedesignofthedropofwaterisdescribedbythreearcsofcirclesasindicatedinFig.1.AnalysisisperformedbythefiniteelementmethodwiththreenodeparabolicelementsusingtheclassicalLoveKirchoffshelltheory.Anautomaticmeshgeneratorcreatesthefiniteelementmeshofeachstraightorcircularsegmentconsideredasamacrofiniteelement.Theobjectiveistoobtainashapeforthedropofwatergivingrisetoaminimumtotalpotentialenergywhichisthemainobjectiveandanequalresistancetankwhichistheonlyconstraintorlimitationoftheproblem.Infact,forthedropofwaterproblem,thegoalisamultiobjectiveone,thetwoobjectivesf1minimumtotalpotentialenergyandf2equalresistancearecombinedinamultiobjectiveff1f2.TheconstraintorlimitationoftheproblemistakenintoaccountbyapenalizationofthetotalpotentialenergyasindicatedinMarcelinetal.19.3.1.2TheresultsThedesignofthedropofwaterisdescribedbythreearcsofacircleFig.1.Theircentersandradiusarethedesignvariables.So,thereareninedesignvariablesr1,z1,R1forcircle1r2,z2,R2forcircle2andr3,z3,R3forcircle3.Inthegeneticalgorithm,eachofthesedesignvariablesiscodedbythreebinarydigits.ThetablesofcodingdecodingwillbethefollowingForr1Forz1ForR1Forr2Forz2ForR2Forr3Forz3ForR3Allthesebinarydigitsareputendtoendtoformachromosomelengthof27binarydigits.GAisrunforapopulationof30individuals,anumberofgenerationsof50,aprobabilityofcrossingof0.8,andaprobabilityofmutationof0.1.Theoptimalsolutioncorrespondstothechromosome100100011011010011100011101whichgivesthesolutionofFig.1,forwhich–r118,z117,andR1−0.065–r213.75,z212.2andR2−7.7–r34.1,z321.4andR3−21Itisveryclosetothenaturesoptimalsolutionfortheshapeofadropofwater.Themodelofthewaterdropmodelledbythreearcsofacircleisimperfect.However,theconstructaltheoryoptimizestheimperfections,and1052025r51020z3211515Fig.1Optimizationoftheshapeofadropofwater0000010100111001011101111616.51717.51818.51919.50000010100111001011101111515.51616.51717.51818.5000001010011100101110111−0.050−0.055−0.060−0.065−0.070−0.075−0.080−0.0850000010100111001011101111313.2513.513.751414.2514.514.750000010100111001011101111212.112.212.312.412.512.612.7000001010011100101110111−7.4−7.5−7.6−7.7−7.8−7.9−8−8.10000010100111001011101113.73.83.944.14.24.34.400000101001110010111011121.121.221.321.421.521.621.721.8000001010011100101110111−18.5−19−19.5−20−20.5−21−21.5−224IntJAdvManufTechnol2008381–6
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