多相材料结构拓扑优化

275Vol127No1520069ACTAAERONAUTICAETASTRONAUTICASINICASept.2006:2005205208;:2005208210:(90405016)(10372083)973(2006CB601205)(04B53080):100026893(2006)0520963206,(,710072)InvestigationofPerimeterControlMethodsforStructuralTopologyOptimizationwithMultiphaseMaterialsSUNShi2ping,ZHANGWei2hong(TheKeyLaboratoryofContemporaryDesignandIntegratedManufacturingTechnology,NorthwesternPolytechnicalUniversity,Xian710072,China):,,4,4:,:;;;;:TU318;V214119:AAbstract:Thedevelopmentofperimetercontrolofthecheckerboardinvolvedinstructuretopologyoptimizationofmultiphasematerialsispresented.Basedonthecharacteristicsofdensitydesignvariablesassociatedwithmultiphasematerials,thegeneralconceptofperimetercontrolisextendedandfourkindsofperimetercon2straintsareproposedinthispaper.Acomparativestudyoftheseperimeterconstraintsiscarriedoutbasedonseveral2Dtopologyoptimizationexamplesthatareextensivelyusedintheliteratureforthecomplianceminimi2zation.Designoptimizationproblemsaresolvedbyconvexlinearprogramming.Numericalexperiencesshowthattheperimetercontrolusingeachkindofdesignvariablesseparatelyiseffectiveandcanbedirectlyextendedtoproblemsofmultiphasematerials.Checkerboardsandintermediatevaluesofdensityvariablesareabletobeeliminatedefficiently.Keywords:multiphasematerials;topologyoptimization;perimetercontrol;checkerboard;dualalgorithm,Bends«eKikuchi[1],(),[2,3][4][5][6],,Thomsen[7],Sigmund[8210]MEMS;Yin[11],,Wang[12],,[8],[8],27,,[2,3],,,4,41112,(),,,1Fig11Atwo2dimensionalstructurewithmultiphasematerials111,SIMP[13][1][11][12]SIMP,0,10,101SIMP,y[0,1],E(y)=E1+yp(E2-E1)(1):E1,E212();p24,p=4,(x,y);x,y[0,1],xy[14]E(x,y)=xp[E1+yp(E2-E1)](2),(x,y)=(0,0)(1,0)1(1,1)2(3)22Fig12Thedistributionofmultiphasematerialsandinterfaces112,,()(),223,()(TotalVariation),P(X),XP(),,,Y[3]P(Y)=rk=1lk|yi-yj|(4)(4)4695:,[15]P(Y)=rk=1lk(yi-yj)2(5):lkij;yi,yjij,,(2,3),(1),,,x,y,,::x(22,3),(21)x,y,(5)1Px(X)=rk=1lk(xi-xj)2Px(6)223,(6)(y),21,,,2Py(Y)=rk=1lk(yi-yj)2Py(7),21,(23),3Pxy(X,Y)=rk=1lk[(xi-xj)2+(yi-yj)2]Pxy(8),,,,x,y,,,,,4x,yP(X)=rk=1lk(xi-xj)2PxP(Y)=rk=1lk(yi-yj)2Py(9):Px,Py12,3,x,y,,,,34124,113,,minC(X,Y)=FTU(X,Y)Vx(X)=ni=1vixiVxVy(Y)=ni=1viyiVyPP0xi1,0yi1(i=1,,n)(10):C;F;U;vii();Vx,Vy2;X,Y;,10-5,4569279C9xi=FT9U9xi=FTK-19F9xi-9K9xiU=-UT9K9xiU=-pxi(FTiUi)=-pxiCi(11)9C9yi=-UT9K9yiU=-pyi1-xpiE1E(xi,yi)(FTiUi)=-pyi1-xpiE1E(xi,yi)Ci(12),,[15],,23412E1=100,E2=200;1=2=013:1V1=012;2V2=Vy=0.3;V3=015;Vx=V1+V2=0.5211MBBMBBL,H613,F=10(),MBB,602043MBBFig13TheMBBstructurewithboundarycondition4,2,1,C;Px,Py,Pxy;Px,Py,Pxy534,M(a)1:C=1131499,Px=211,Px=240(b)2:C=1421437,Py=16312,Py=180(c)3:C=1191823,Pxy=34813,Pxy=350(d)4:C=1151953,Px=207,Px=240,Py=159,Py=2004MBBFig14Optimizedtopologyunderdifferentperimetercontrols5Fig15Theiterativehistoryofthestructuralcompliance212L,H21L,H,6,4F=10()603046Fig16Abridge2likestructurewithmultipleloads6695:7,834(a)1:C=911904,Px=320,Px=350(b)2:C=971667,Py=310,Py=320(c)3:C=911494,Pxy=54918,Pxy=550(d)4:C=911323,Px=330,Px=340,Py=250,Py=2607Fig17Optimizedtopologyunderdifferentperimetercontrols8Fig18Theiterativehistoryofthestructuralcompliance2139,,F=10()L=30(),H=15()6030410,11349Fig19Thecantileverbeam(a)1:C=413414,Px=154,Px=160(b)2:C=470519,Py=12715,Py=130(c)3:C=439317,Pxy=24519,Pxy=250(d)4:C=412712,Px=15212,Px=160,Py=10115,Py=12010Fig110Optimizedtopologyunderdifferentperimetercon2trols3:2,,4(b)10(b)769271,4(a)7(a)10(a),,34,,34;5811433,,,3:,,,411Fig111Theiterativehistoryofthestructuralcompliance3,4,4,,[1]Bends«eMP,KikuchiN.Generatingoptimaltopologiesinoptimaldesignusingahomogenizationmethod[J].ComputerMethodsinAppliedMechanicsandEngineer2ing,1988,71:1972224.[2]AmbrosioL,ButtazzoG.Anoptimaldesignproblemwithperimeterpenalization[J].CalculusofVariationsandPar2tialDifferentialEquations,1993(1):55269.[3]HaberRB,JogCS,Bends«eMP.Anewapproachtovariable2topologyshapedesignusingaconstraintonper2imeter[J].StructuralOptimization,1996,11:1212.[4]SigmundO.Designofmaterialstructuresusingtopologyoptimization[D].Denmark:TechnicalUniversityofDen2mark,1994.[5]PeterssonJ,SigmundO.Slopeconstrainedtopologyopti2mization[J].InternationalJournalforNumericalMethodsinEngineering,1998,41:141721434.[6]ZhouM,ShyyYK,ThomasHL.Checkerboardandminimummembersizecontrolintopologyoptimization[J].StructuralandMultidisciplinaryOptimization,2001,21:1522158.[7]ThomsenJ.Topologyoptimizationofstructurescomposedofoneortwomaterials[J].JournalofStructuralOptimi2zation,1992,5:1082115.[8]SigmundO.Designofmultiphysicsactutorsusingtopolo2gyoptimization2partII:twomaterialstructures[J].ComputerMethodsinAppliedMechanicsandEngineer2ing,2001,190:660526627.[9]GibianskyLV,SigmundO.Multiphaseelasticcompositeswithextremalbulkmodulus[J].JournaloftheMechanicsandPhysicsofSolids,2000,48(3):4612498.[10]SigmundO,TorquatoS.Designofmaterialswithextremethermalexpansionusingathree2phasetopologyoptimiza2tionmethod[J].JournaloftheMechanicsandPhysicsofSolids,1997,45(6):103721067.[11]YinL,AnanthasureshGK.Topologyoptimizationofcompliantmechanismswithmultiplematerialsusingapeakfunctionmaterialinterpolationscheme[J].Structuraland