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ORIGINALARTICLEPseudo-constructaltheoryforshapeoptimizationofmechanicalstructuresJeanLucMarcelinReceived:10January2007/Accepted:1May2007/Publishedonline:25May2007#Springer-VerlagLondonLimited2007AbstractThisworkgivessomeapplicationsofapseudo-constructaltechniqueforshapeoptimizationofmechanicalstructures.Inthepseudo-constructaltheorydevelopedinthispaper,themainobjectiveofoptimizationisonlytheminimizationoftotalpotentialenergy.Theotherobjectivesusuallyusedinmechanicalstructuresoptimizationaretreatedlikelimitationsoroptimizationconstraints.Twoapplicationsarepresented；thefirstonedealswiththeoptimizationoftheshapeofadropofwaterbyusingageneticalgorithmwiththepseudo-constructaltechnique,andthesecondonedealswiththeoptimizationoftheshapeofahydraulichammersrearbearing.KeywordsShapeoptimization.Constructal.Geneticalgorithms1IntroductionThispaperintroducesapseudo-constructalapproachtoshapeoptimizationbasedontheminimizationofthetotalpotentialenergy.Wearegoingtoshowthatminimizingthetotalpotentialenergyofastructuretofindtheoptimalshapemightbeagoodideainsomecases.Thereferencetotheconstructaltheorycanbejustifiedinsomewayforthefollowingreasons.AccordingtoBejan1,shapeandstructurespringfromthestruggleforbetterperformanceinbothengineeringandnature；theobjectiveandconstraintsprincipleusedinengineeringisthesamemechanismfromwhichthegeometryinnaturalflowsystemsemerges.Bejan1startswiththedesignandoptimizationofengineeringsystemsanddiscoversadeterministicprincipleforthegenerationofgeometricforminnaturalsystems.Thisobservationisthebasisofthenewconstructaltheory.Optimaldistributionofimperfectionisdestinedtoremainimperfect.Thesystemworksbestwhenitsimperfectionsarespreadaroundsothatmoreandmoreinternalpointsarestressedasmuchasthehardestworkingparts.Seeminglyuniversalgeometricformsunitetheflowsystemsofengineeringandnature.Bejan1advancesanewtheoryinwhichheunabashedlyhintsthathislawisinthesameleagueasthesecondlawofthermodynamics,becauseasimplelawispurportedtopredictthegeometricformofanythingaliveonearth.Manyapplicationsoftheconstructaltheoryweredevelopedinfluidsmechanics,inparticularfortheoptimizationofflows210.Ontheotherhand,thereexists,toourknowledge,littleexamplesofapplicationsinsolidsorstructuresmechanics.Sowehaveatleasthalfofthereferencestopapersinfluiddynamics(mostofthesameauthor),becausetheconstructalmethodwasdevelopedfirstbythesameauthor,AdrianBejan,withonlyreferencestopapersinfluiddynamics.Theconstructaltheoryrestsontheassumptionthatallcreationsofnatureareoveralloptimalcomparedtothelawswhichcontroltheevolutionandtheadaptationofthenaturalsystems.Theconstructalprincipleconsistsofdistributingtheimperfectionsaswellaspossible,startingfromthesmallestscalestothelargest.Theconstructaltheoryworkswiththetotalmacroscopicstructurestartingfromtheassemblyofelementarystruc-tures,bycomplyingwiththenaturalrulesofoptimaldistributionoftheimperfections.Theobjectiveistheresearchoflowercost.IntJAdvManufTechnol(2008)38:16DOI10.1007/s00170-007-1080-2J.L.Marcelin(*)LaboratorieSolsSolidesStructures3S,UMRCNRSC5521,DomaineUniversitaire,BPn53,38041GrenobleCedex9,Francee-mail:Jean-Luc.Marcelinujf-grenoble.frHowever,aglobalandmacroscopicsolutionfortheoptimizationofmechanicalstructureshavingleastcostastheobjectivecanbeveryclosetotheconstructaltheory,fromwherethetermpseudo-constructalcomes.Theconstructaltheoryisapredictivetheory,withonlyonesingleprincipleofoptimizationfromwhichallrises.Thesameappliestothepseudo-constructalstepwhichisthesubjectofthisarticle.Thesingleprincipleofoptimiza-tionofthepseudo-constructaltheoryistheminimizationoftotalpotentialenergy.Moreover,inourexamplespresentedhereafter,thepseudo-constructalprinciplewillbeassociatedwithageneticalgorithm,withtheresultthatouroptimizationwillbeveryclosetothenaturallaws.Theobjectiveofthispaperisthustoshowhowthepseudo-constructalstepcanapplytothemechanicsofthestructures,andinparticulartotheshapeoptimizationofmechanicalstructures.Thebasicideaisverysimple:amechanicalstructureinabalancedstatecorrespondstoaminimaltotalpotentialenergy.Inthesameway,anoptimalmechanicalstructuremustalsocorrespondtoaminimaltotalpotentialenergy,anditisthisobjectivewhichmustintervenefirstoveralltheothers.Itisthisideawhichwillbedevelopedinthisarticle.Twoexampleswillbepresentedthereafter.Theideatominimizetotalpotentialenergyinordertooptimizeamechanicalstructureisnotbrandnew.Manypapersalreadydealwiththisproblem.Whatisnew,istomakethisapproachsystematic.Theonlyobjectiveofoptimizationbecomestheminimizationofenergy.InGosling11,asimplemethodisproposedforthedifficultcaseofform-findingofcablenetandmembranestructures.Thismethodisbaseduponbasicenergyconcepts.Atruncatedstrainexpressionisusedtodefinethetotalpotentialenergy.ThefinalenergyformisminimizedusingthePowellalgorithm.InKannoandOhsaki12,theminimumprincipleofcomplementaryenergyisestablishedforcablenetworksinvolvingonlystresscomponentsasvariablesingeometricallynonlinearelasticity.Inordertoshowthestrongdualitybetweentheminimizationproblemsoftotalpotentialenergyandcomplementaryenergy,theconvexformulationsoftheseproblemsareinvestigated,whichcanbeembeddedintoaprimal-dualpairofsecond-orderprogrammingproblems.InTaroco13,shapesensitivityanalysisofanelasticsolidinequilibriumispresented.Thedomainandboundaryintegralexpressionsofthefirstandsecond-ordershapederivativesofthetotalpotentialenergyareestablished.InWarner14,anoptimaldesignproblemissolvedforanelasticrodhangingunderitsownweight.Thedistributionofthecross-sectionalareathatminimizesthetotalpotentialenergystoredinanequilibriumstateisfound.Thecompanionproblemofthedesignthatstoresthemaximumpotentialenergyunderthesameconstraintconditionsisalsosolved.InVentura15,theproblemofboundaryconditionsenforcementinmeshlessmethodsissolved.InVentura15,themovingleast-squaresapproximationisintroducedinthetotalpotentialenergyfunctionalfortheelasticsolidproblemandanaugmentedLagrangiantermisaddedtosatisfyessentialboundaryconditions.Theprincipleofminimizationoftotalpotentialenergyisinadditionatthebaseofthegeneralfiniteelementsformulation,withanaimoffindingtheunknownoptimalnodalfactors16.2ThemethodsusedInthepseudo-constructaltheorydevelopedinthispaper,themainobjectiveofoptimizationisonlytheminimizationoftotalpotentialenergy.Theotherobjectivesusuallyusedinmechanicalstructuresoptimizationaretreatedherelikelimitationsoroptimizationconstraints.Forexample,onemayhavelimitationsontheweight,ortonotexceedthevalueofastress.Theideawhichwillbedevelopedinthispaperisthusverysimple.Amechanicalstructureisdescribedbytwotypesofparameters:variablesknownasdiscretizationvariables(forexample,degreesoffreedomindisplacementforfiniteelementsmethod),andgeometricalvariablesofdesign(forexampleparameterswhichmakeitpossibletodescribethemechanicalstructureshape).Totalpotentialenergydependsonanimplicitorexplicitwayofdetermin-ingdiscretizationanddesignvariablesatthesametime.Onethuswillcarryoutadoubleoptimizationofthemechanicalstructure,comparedtothediscretizationanddesignvariables；theobjectivebeingtominimizetotalpotentialenergyoverall.Clearly,theproblemofoptimiza-tionofamechanicalstructurewillbeaddressedbythefollowingapproach:Objective:tominimizetotalpotentialenergyVariablesofoptimization:concurrentlydeterminingdiscretizationvariables(inthecaseofatraditionaluseofthefiniteelementmethodinmechanicsofstruc-tures),anddesignvariablesdescribingtheshapeofthestructureOptimizationlimitations:WeightorvolumeDisplacementsorstrainsStressesFrequenciesTheproblemofoptimizationofamechanicalstructurewillbesolvedinthefollowingway,whilereiteratingon2IntJAdvManufTechnol(2008)38:16thesestages,ifneeded(accordingtothenatureoftheproblem):Stage1Minimizationofthetotalpotentialenergyofthemechanicalstructurecomparedtotheonlydis-cretizationvariablesofthestructure(degreesoffreedominfiniteelements).Itactshereasanoptimizationwithoutoptimizationlimitations.Theonlylimitationsatthisstageareofpurelymechanicalorigin,andrelatetotheboundaryconditionsandtotheexternaleffortsappliedtothestructure.Inthisstage1,thedesignvariablesremainfixed,andoneobtainstheimplicitorexplicitexpressionsofthedegreesoffreedomaccordingtothedesignvariables(whichcanbethevariableswhichmakeitpossibletodescribetheshape,inthecaseofashapeoptimization,forexample).Onewillseeintheexamplesofthefollowingpartthattheseexpressionscanbeexplicitorimplicitandwhichisthesuitabletreatmentfollowingthecases.Inthecaseofafiniteelementsmethodofcalculation,thisstage1isthebasisoffiniteelementscalculationtoobtainthedegreesoffreedomofthemechanicalstructure.Indeed,infiniteelements,displacementswiththenodesofthemechanicalstructuremeshareobtainedbyminimizationoftotalpotentialenergy16.Stage2Theexpressionsofthedegreesoffreedomofthemechanicalstructureaccordingtothedesignvariablesobtainedpreviouslyaretheninjectedintothetotalpotentialenergyofthemechanicalstructure(onewillseeinthesecondexampleofthefollowingparthowonetreatsthecasewherethedegreesoffreedomareimplicitfunctionsofthedesignvariables).Onethenobtainsanexpressionofthetotalpotentialenergywhichdependsonlyonthedesignvariables(inexplicitorimplicitform).Stage3Onethencarriesoutasecondandnewminimi-zationofthetotalpotentialenergyobtainedintheprecedingform,butthistimecomparedtothedesignvariableswhilerespectingthetechnolog-icallimitationsortheoptimizationconstraintsoftheproblem.Thismethodcanbeappliedwithmoreorlessfacilityaccordingtothenatureoftheproblem.Itisclear,forexample,thatifthediscretizationvariablescanbeexpressedinanexplicitwayaccordingtothedesignvariables,thesettinginofstages2to3isimmediate,andwithoutiterations.Ifthediscretizationvariablescannotbeexpressedinanexplicitwayaccordingtothedesignvariables,orifthetopologyofthestructureisnotfixed,orifthebehaviorisnotlinear,itwillbenecessarytoproceedbysuccessiveiterationsonstages1to3.Itisthecaseoftheexamplespresentedinthefollowingpart,andonewillseeonthisoccasionwhichtypeofstrategyonecanadoptfortheseiterations.Tosummarize,inthepseudo-constructalstep,themainobjectiveisonlytheminimizationoftotalpotentialenergy,theotherpossibleobjectivesaretreatedlikelimitationsoroptimizationconstraints.TheoptimizationmethodusedforourexamplesisGA(geneticalgorithm),asdescribedin17.Exampleswithsimilarinstructionalvaluecanalsobefoundinmanybooks,e.g.in18.Thisevolutionarymethodisveryconvenientforourpseudo-constructalmethod.TheauthorhasworkedextensivelyinGAsandpublishedinsomereputedjournalsonthistopic1931.AsthetopicofGAsisstillrelativelynewinthestructuralmechanicscommu-nity,weprovideheresomedetailsofexactlywhatisusedinthisGA.Amultiplepointcrossoverisusedratherthanasinglepointcrossover.Theselectionschemeusedateachgenerationisentirelystochastic.Forourexamples,thenumberofgenerationsisequaltothatusedforconver-gence.TheresultsprovidedforourexampleswereconsistentlyreproducedbyusingdifferentseedsintheGA.Ithasbeenprovedthataratherstandardgeneticalgorithmissufficientforourexamples.3ExamplesEventhoughpotentialenergymaybeagoodmeasureforsomeoptimizations,potentialenergyisnotwhatgivestheshapetoawaterdroplet,nordefinestheoptimalshapeforahammer,whichiswhypotentialenergyisnottheonlyobjective；buttheoptimizationproblemisamultiobjectiveoneandtheobjectivefunctionsforthetwoexamplesarethenclearlyformulated.3.1Example1:optimizationoftheshapeofadropofwaterThefirsttestexampleistheoptimizationoftheshapeofadropofwater(Fig.1).Thisproblemisequivalenttoanequalresistancetankcalculatedbythemembranetheory.Theobjectiveistoseeifthepseudo-constructaltheorygivesthenaturesoptimumdesign.3.1.1ThemethodsusedThegeometryofthedropofwaterisdefinedbythegeneratinglineofathinaxisymmetricshell.Thislineisdescribedbysuccessivestraightorcircularsegmentsdescribedinagivensenseanddefinedbyinputdataofmasterpointcoordinatesandradiusvalues.Theinitialdataareasetofnodalpointsconnectedbystraightsegments.EachnodalpointisidentifiedbyitstwocylindricalIntJAdvManufTechnol(2008)38:163coordinates(r,z),andarealRwhichrepresentstheradiusofthecircletangenttothetwostraightsegmentsintersect-ingatthepoint.Theothercomputercalculationsgivethecoordinatesofanyboundarypointandespeciallythetangentpointsnecessarytodefinethecirculararclengths.ThedesignofthedropofwaterisdescribedbythreearcsofcirclesasindicatedinFig.1.Analysisisperformedbythefiniteelementmethodwiththree-nodeparabolicelementsusingtheclassicalLove-Kirchoffshelltheory.Anautomaticmeshgeneratorcreatesthefiniteelementmeshofeachstraightorcircularsegmentconsideredasamacrofiniteelement.Theobjectiveistoobtainashapeforthedropofwatergivingrisetoaminimumtotalpotentialenergy(whichisthemainobjective)andanequalresistancetank(whichistheonlyconstraintorlimitationoftheproblem).Infact,forthedropofwaterproblem,thegoalisamulti-objectiveone,thetwoobjectives(f1=minimumtotalpotentialenergyandf2=equalresistance)arecombinedinamulti-objective:f=f1+f2.TheconstraintorlimitationoftheproblemistakenintoaccountbyapenalizationofthetotalpotentialenergyasindicatedinMarcelinetal.TheresultsThedesignofthedropofwaterisdescribedbythreearcsofacircle(Fig.1).Th