电子器件 技术文件Optimization Minicourse 2015.pdf

该【电子器件 技术文件Optimization Minicourse 2015 】是由【fwang2】上传分享，文档一共【52】页，该文档可以免费在线阅读，需要了解更多关于【电子器件 技术文件Optimization Minicourse 2015 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..SOLMultiphysicsCOMSOLConference2015WalterFrei,PhDApplicationsEngineer?,text,.:..Agenda?Anintroductiontooptimization–Alotofconcepts,andalittlebitofmath–Whatdotheseoptionsmean??Demo?OverviewofExamples:..Aquickconceptualintroduction,andsometerminologies-Dimensions-PerformanceObjective:f(u(χ))“BlackBox”ConstrainedDesign-Failurecriteria&-MaterialPropertiesu(χ)Variables:χ-Constraints:etc…g(u(χ))-OperatingConditions-etc…K(χ)u=b(χ)Optimization:..minf(u(χ))χ??Moreformally,optimizationis…χ?χ?χLUp(χ)?0ObjectivefunctionSimpleboundsonthedesignvariablessuchthat:g(u(χ))=0Pointwiseconstraintsonthedesignvariablesh(u(χ))GeneralEqualityconstraints?0GeneralInequalityconstraints:..Thedesignvariables:χχ2DesignχPointwiseconstraintsSpace2,Up(χ)≤0χ1,Lχ1,Uχ1χ2,LUpperandLowerBounds:..Thedesignspacemustbecontinuousχ2DesignSpace1DesignSpace2χ1Breakthisupintotwoseparateoptimizationproblems:..Whynoequalityconstraintsforχ?χ2χ2,Up(χ)=0χ1,Lχ1,Uχ1χ2,LAnequalityconstraintisequivalenttoadifferentoptimizationproblemwithonelessdesignvariable:..Introduceanewdesignvariableinsteadχ2χ=f(χ)1Aχ2,UχA,Uχ=f(χ)2AχAχ1,Lχ1,Uχ1χA,Lχ2,L:..Thedesignspacemustbecontinuousintherealnumberspaceχ2χ1OptimizingoverasetofdiscretevaluesisanIntegerProgrammingproblemLiveLink?forMATLAB?&LiveLink?forExcel?rdcanbeusedtointerfaceto3partyoptimizers:..ItishelpfulifthedesignspaceisconvexUsuallymoredifficultEverypointcanseeeveryotherdesignpoint:..Nowletslookattheobjectivefunction:f(u(χ))orf(χ)χ2χ1χ2χ1:..WearealwaysstartingsomewherefWewanttoimprovethisTip:Alwaysstartχ2optimizingfromafeasibledesignInitialdesignχ1:..Letsfirstassumeasmoothanddifferentiableobjectivefunctionwithasingleminimumf1)Findthegradient?fχ2χ2)Searchalongtheline3)Findtheminimum4)Repeatχ1:..RepeatuntilconvergedStartfromapoint,findthedirectionofsteepestdescent(thegradient)andsearchinthatdirectionforaminimumRepeatOncethegradientiszero,ortheboundaryofthedesignspaceisreached,stop:..K(χ)u?b(χ)=0?(K(χ)u?b(χ))=0?χ?K(χ)?u?b(χ)u+K(χ)=putederivatives?χ?χ?χFiniteelementequations?u??b(χ)?K(χ)??=K(χ)??u????χ?χExpand?χ???f?f?uRe-arrange=?χ?u?,regardlessofhowmanydesignvariablesthereare:..Ifwehitaconstraint,followit…Startfromapoint,findthedirectionofsteepestdescent(thegradient)andsearchinthatdirectionforaminimumRepeatOncethegradientiszero,ortheboundaryofthedesignspaceisreached,stop:..Whatifwehavemultipleminima?Dependsonwhereyoustart!Youareneverguaranteedoffindingtheglobalminimum,butyoucanfindalocalminimum:..Whatiftheobjectivefunctionisnotsmoothordifferentiable?ApproximatetheshapebyevaluatingtheobjectivefunctionrepeatedlyForexample,Nelder-Meadevaluatesn+1pointswhenoptimizingndesignvariables:..Whatiftheobjectivefunctionisnotsmoothordifferentiable?ApproximatetheshapebyevaluatingtheobjectivefunctionrepeatedlyForexample,Nelder-Meadevaluatesn+1pointswhenoptimizingndesignvariables:..Whatifwewanttoincludegeneralequalityandinequalityconstraints?h(u(χ))≤0g(u(χ))=0χ1χ2χ1χ2StrongdependenceoninitialconditionsHighlyconstraineddesignspace:..Summary…?Designvariablesmustbecontinuousandreal-valued?Designspace:–Simple(Cartesian)bounds–Pointwiseinequalityconstraints–Ifyouwanttosetupanequalityconstraint,getridofonedesignvariable–Convexdesignspaceisbetter?Objectivefunction–Ifitissmoothanddifferentiable,?canusetheAdjointmethodandthegradient-basedoptimizationtechnique–Ifitisnon-smoothornon-differentiable?Usethegradient-freeapproach?GeneralEqualityandInequalityConstraints–plicatetheoptimizationproblem–Inequalityconstraintscanmakethedesignspacenon-continuous:..Demo:AbracketwithaholeFixedLoad:..First,minimizethemassbychangingtheholeradiusRTheradiusmustbegreaterthanzero,andnotsolargeastocutthebracketinhalf:..Next,addaconstraintonthemaximumstresswithinthepartσ<σmaxButthelocationofthepeakstressisnotknown,Soweuseamaximumcouplingoperator&aconstraint:..Whataboutmoving&resizingthehole?:..Let’slookattheconstraints...Howcanweexpressthesemathematically?:..AddonemoredesignvariableRA:..Let’staketheseconstraintsafewatatimeWithabitof(behindthescenes)trigonometry:B=(1-*A)/(1+sqrt()/2)RRAWhichleadstotheconstraint:B-R>0:..Whatabouttheotherlimits?A–R>:..SometimeswecanjustignoreaconstraintBaseduponthesimulationssofar,itslikelythisconstraintwillneverbeanissue:..TheavailableoptimizationsolversOptimizationModuleGradient-FreeGradient-BasedMethodsMethodsMonteLevenberg-MMASNOPT-CarloMarquardtCoordinateNelder-BOBYQACOBYLASearchMead:..Whentousegradient-freemethods??Non-differentiableobjectivefunction,and/orconstraints?Fewdesignvariables–Optimizationtimeincreasesexponentiallywithnumberofvariables–Aimforlessthan10designvariables?Wheneverre-ur–Re-meshingresultsinanon-smoothobjectivefunction:..Thegradient-freesolvers?COBYLAFaster–SimilartoBOBYQA,butusesalinearapproximation–Canconsiderconstraints?BOBYQA–Constructsaquadraticapproximanttotheobjectivefunction–Probablythefastest,butneedsa“reasonablysmooth”objectivefunctionUsuallythe?Nelder-Mead–Constructasimplex,andimprovetheworstpointMost–ProbablythebestiftheobjectivefunctionisrelativelynoisyRobust–Canconsiderconstraints?CoordinateSearch–Searchalongonedesignvariableatatime–Estimatethegradientsalongthatline,moveontonextvariable,repeat?Monte-CarloSlower–Randomchoicesofdesignvariablesareevaluated–Onlyaverydensestatisticalsamplingcanfindtheglobaloptimum:..Whentousegradient-basedmethods??Differentiableobjectivefunction,and/orconstraints?Manydesignvariables–Optimizationspeeddoesnotdependstronglyonnumberofvariables–100,000+designvariablesarenotunreasonable?TopologyOptimization:..Thegradient-basedsolvers?SNOPT–SequentialQuadraticProgrammingalgorithm?MMA–Linearconvergencerateneartheoptimum–munity?Levenberg-Marquardt–Onlyforunconstrainedleastsquaresminimizationproblems–Veryfast:..ScalingandTolerances?Specifyscalesforallcontrolvariables–lobalparameters–InOptimizationinterfacefeaturesforfields?Allsolversworkwithrescaledvariables–Solvertolerancesarerelativetothese?Keepobjectivesandconstraintscloseto1–Solversmayusescaledgradientfortermination:..ComparisonofAlgorithmsGradient-FreeGradient-BasedObjectiveAnyscalaroutputMustbebothsmoothanddifferentiableFunctionDesignVariablesAnything,includinggeometricAnything,butcannotresultinremeshingofdimensionsthegeometryAllowsYesNoRemeshingConstraintsCanonlyconstrainscalaroutputsConstraintsmustbesmoothanddifferentiable,butcanbeateachpointinspaceRelativeIncreasesexponentiallywiththePerformanceisnotverysensitivetothePerformancenumberofdesignvariablesnumberofdesignvariables:..Sowhatelsecanyoudo?ParameterEstimation&CurveFittingShape&DimensionTopology:..StructuralSizing?Optimizationofjointpositionsinatruck-mountedcrane.?Reducesforceonboomliftcylinderforarangeofoperationconditions?UsestheMultibodyDynamicsModuleel/optimization-of-a-crane-link-mechanism-18113:..Multi-studyStructuralSizing?Weightminimizationofamountingbracket.?Multi-studyconstraints–Maximumstressunderstaticload–Lowesteigenfrequencyel/multistudy-optimization-of-a-bracket-19761:..Estimatingthematerialpropertiesbaseduponexperimentaldatael/transient-optimization-fitting-material-properties-of-a-wall-10905:..Example:OptimizingaFlywheel,withConstraints?Makestressdistributionasuniformalongtheradiusaspossible?Constrainthemassnottochange?Constrainthemomentofinertianottochange?Gradient-basedapproachel/optimizing-a-flywheel-profile-4356:..Example:BanddispersioninamicrochannelMinimizethedifferenceistransittimebetweeninsideandoutsideGradient-Freeoptimizationel/solute-band-dispersion-in-curved-microchannel-16157:..Example:OptimizingaHorn?Maximizethesoundintensityalongtheaxisofthehorn?Theshapeofthehornisdescribedbyasumofsinewaves–TruncatedFourierseries?TheDeformedMeshfunctionalityisusedtoavoidremeshingthedomain?Gradient-Basedapproachel/optimizing-the-shape-of-a-horn-4353:..TopologyOptimization:Buythisbook:..Example:OptimizingaBeamplianceAddconstraintontotalmaterialSIMPmethodel/topology-optimization-7428:..Example:TeslaMicrovalveFlowtotheright:MinimizeΔpFlowtotheleft:MaximizeΔpLauritsH?jgaardOlesen,FridolinOkkelsandHenrikBruus,“Ahigh-levelprogramming-languageimplementationoftopologyoptimizationappliedtosteady-stateNavier–Stokesflow,”;65:975–-optimization-of-a-tesla-microvalve-14513:..COMSOLConferencePapersonTopologyOptimization?TopologyOptimizationinMultiplePhysicsProblems,,DTUMechanicalEngineering,ers/1790/?MultiphysicsTopologyOptimizationofHeatTransferandFluidFlowSystems,,ToyotaResearchInstituteofNorthAmerica,ers/6282/?SimulationofTopologyOptimizedElectrothermalMicrogrippers,,,,&,DTUMechanicalEngineering,ers/5346/?SOL,,,&,DepartmentofMicrosystemsEngineering,UniversityofFreiburg,ers/1543/?SOLMultiphysics,&,KyotoUniversity,ers/12519/?rystalWaveguideTermination,,ers/3165/:..Whentousewhichone??Earlyinthedesignprocess,whenyouhavealotoftimetorunanalyses,anddon’thaveaveryrigidideaaboutyourfinaldesign??Alreadyhavethebasictopologyfixed,butcanchangetheshape??It’sabouttogotoproduction,andwerealizeditdoesn’twork!!!:..Whentousewhichone??Earlyinthedesignprocess,whenyouhavealotoftimetorunanalyses,anddon’thaveaveryrigidideaaboutyourfinaldesign?TopologyOptimization?Alreadyhavethebasictopologyfixed,butcanchangetheshape?ShapeOptimization?It’sabouttogotoproduction,andwerealizeditdoesn’twork!!!DimensionalOptimization:..Somegeneraloptimizationtips…?Scaleyourobjectivefunctiontobeclosetounity?Donotgetfrustrated,thesoftwaredoesn’tknowthatyouhaven’taskedtherightquestion?Startassimpleaspossible,especiallywithconstraints:..Concludingremarks…minf(u(χ))χ??χ?χ?χLUp(χ)?0suchthat:g(u(χ))=0h(u(χ))?0