TheOptimizationModule?Copyright2014COMSOL.Anyoftheimages,text,andequationsheremaybecopiedandmodifiedforyourowninternaluse.Alltrademarksarethepropertyoftheirrespectiveowners.See/trademarks.OverviewTheCOMSOL?4.4ProductSuiteTheOptimizationModuleAdd-ontoCOMSOLMultiphysicsGeneral-purposepackage,notapplication-specificOptimizationforElectrical,Mechanical,Fluid,andChemicaldevicesandprocessesCombineswithanyCOMSOLMultiphysicsModuleTheOptimizationModuleContains:UserinterfacesforsettingupoptimizationtasksOptimizationsolversExamplemodelsfromdifferentfieldsWhatisOptimization?TheimprovementofanobjectivefunctionbychangingcontrolvariableswhilemaintainingasetofconstraintsObjectivefunction(costfunction)Anyreal-valuedscalaroutputfromasolvedCOMSOLmodelMass,displacement,pressuredrop,resistance,power,etc...Controlvariables(optimizationvariables,designvariables,...)Anycontinuoussetofreal-valuedinputstothemodelDimensions,materialdistribution,materialproperties,etc...ConstraintsAnyequalityorinequalityconditionthatcanbeexpressedintermsofthesolvedmodel,orthecontrolvariablesPeakstress,minimumsize,maximumtemperature,etc...QuickIntroductiontoOptimizationintheCOMSOLDesktopHowdoweoptimize?StartwithasolvedCOMSOLmodelDefineyourobjectfunction:Whatdoyouwanttomakebetter?Choosethedesignvariables:Whatdoyouwanttochange?Specifyyourconstraints:Whatlimitscannotbeexceeded?Optimize!Tuningfork,Desiredresonanceat440HzStartwithanexistingmodelForclarity,theCOMSOLDesktopishereshownasitappearsonalow-resolutionscreen.AddtheOptimizationStudyfeatureSelectfromasuiteofoptimizationalgorithmsSettoleranceandthemaximumnumberofmodelevaluationsIfsolvingonacluster,candistributesolutionsandruninparallelSpecifyanobjectivefunctionfreq:theresonancecomputedbytheeigenfrequencysolverTheobjectiveistomaketheresonantfrequency440HzMinimize,maximum,maximizetheminimum,minimizethemaximum,andhandlesumsofobjectivefunctionsPickcontrolvariablesandconstraintsGlobalParametersetsthetuningforklength,specifyinitialvalueChooseupperandlowerbounds(constraints)onthecontrolvariablesCanaddotherconstraints,ifdesiredSolve,andthenextractsolutionOptimizationTutorialsandExamplesGettingStartedExample:TuningForkAdjustthelengthofthearmsofatuningforksuchthatthefirstresonanceisat440HzApproximategradient,sincethemodelisremeshedduringtheoptimizationBOBYQASolverCanalsobedoneviatheLiveLink?productsandcontrolledfromvariousCADsoftware,MATLAB?,andExcel?./model/tuning-fork-computing-the-eigenfrequency-and-eigenmode-8499SizeOptimization:

BanddispersioninamicrochannelMinimizethedifferenceistransittimebetweeninsideandoutsideSizeOptimization:3DFlywheelFindholeradiiwhichminimizeflywheelmassMaximumvonMisesstressmustnotexceedyieldlimit/blogs/using-gradient-free-optimization/SizeOptimizationofaFlywheel,

withAdvancedConstraintsMakestressdistributionasuniformalongtheradiusaspossibleConstrainthemassnottochangeConstrainthemomentofinertianottochangeGradient-basedapproach/model/optimizing-a-flywheel-profile-4356SizeOptimization:

MinimizingS11ofanAntenna/model/optimizing-a-bowtie-antenna-14435ChangeFlareAngleandArmHeightGradient-FreeApproach,becauseremeshingisrequiredShapeOptimization:

OptimizinganAcousticHornMaximizethesoundintensityalongtheaxisofthehornTheshapeofthehornisdescribedbyasumofsinewavesTruncatedFourierseriesTheMovingMeshfunctionalityisusedtoavoidremeshingthedomainGradient-Basedapproach/model/optimizing-the-shape-of-a-horn-4353ParameterEstimation:Findingthematerialpropertiesbaseduponexperimentaldatahttp:///model/transient-optimization-fitting-material-properties-of-a-wall-10905http:///model/curve-fitting-material-model-data-to-experimental-data-5886http:///model/determining-arrhenius-parameters-using-parameter-estimation-10305/model/degradation-of-dna-in-plasma-1391TransientanalysisMinimizeleastsquaresdifferencewithexperimentalresultsLevenberg-MarquardtInverseProblem:ImagingImagingofsoilpropertiesbasedonpumpingexperimentsFindthepermeabilitypatternmostconsistentwithexperimentaldataReconstructinteriordatabaseduponobservationsfromtheexterior/model/aquifer-characterization-through-inverse-modeling-from-pump-tests-4410TopologyOptimization:PorousCatalystObjectiveistomaximizethereactionratewhileminimizingtheamountofcatalystwithinthereactorInitialcatalystdistributionishomogeneousTopologyoptimizationadjustsamountofcatalystwithineachmeshelement/model/optimization-of-a-catalytic-microreactor-4401TopologyOptimization:TelsaMicrovalveMaximizeratioofflowlefttorightcomparedtoflowrighttoleftforsamepressuredrop./model/topological-optimization-of-a-tesla-microvalve-14513TopologyOptimization:

MinimizeBeamCompliance/model/topology-optimization-7428MinimizethecomplianceAddconstraintontotalmaterialSIMPmethodMathematicalOptimization:MinimizeaFunctionχ1χ2

NotableCOMSOLConferencePapersonTopologyOptimizationTopologyOptimizationinMultiplePhysicsProblems,O.Sigmund,DTUMechanicalEngineering,/papers/1790/MultiphysicsTopologyOptimizationofHeatTransferandFluidFlowSystems,E.Dede,ToyotaResearchInstituteofNorthAmerica,/papers/6282/SimulationofTopologyOptimizedElectrothermalMicrogrippers,O.Sardan,D.Petersen,O.Sigmund,&P.Boggild,DTUMechanicalEngineering,/papers/5346/ImplementationofStructuralTopologyOptimizationinCOMSOL,B.Lemke,Z.Liu,&J.G.Korvink,DepartmentofMicrosystemsEngineering,UniversityofFreiburg,/papers/1543/TopologyOptimizationofDielectricMetamaterialsBasedontheLevelSetMethodUsingCOMSOLMultiphysics,M.Otomori&S.Nishiwaki,KyotoUniversity,/papers/12519/IndustrySuccessStory:TopologyOptimizationLeadstoBetterCoolingExtractfromCOMSOLNews2012.?2012COMSOL.Allrightsreserved.SolidWorksisaregisteredtrademarkofDassaultSystèmesSolidWorksCorporationoritsparent,affiliates,orsubsidiaries.AluminumcoldplatewithouthierarchicalmicrochanneltopologyAluminumcoldplatewithhierarchicalmicrochanneltopologyModeledaluminumcoldplatewithhierarchicalmicrochanneltopologyAluminumcoldplatesaremountedincarstocombatheatproblems,requiringoptimalcoolingchanneltopologyforminimizedplatesizePerformedCFDandHeatTransferanalysesinconjunctionwithLiveLinkTM

forSolidWorks?tocreateaprototypeusingoptimizedtopologyCOMSOLNews2012:EricDede,ToyotaResearchInstitute,AnnArbor,MIOptimizationModuleTheory,IntroductionTheOptimizationinterfacecandefinecontrolvariablefields,integralobjectives,andlocalconstraintsChangethespatialdistributionofmaterial,subjecttolocalbounds,tominimizemasswhileconstrainingsystemcomplianceThefamilyofoptimizationsolversOptimizationModuleSearchwithoutfindingGradientsAnalyticGradientMethodsMonte-CarloCoordinateSearchMMALevenberg-MarquardtNelder-MeadBOBYQARandomSearchSearchoneaxisatatime1storderapproximategradient2ndorderapproximategradientSNOPTLinearconvergenceQuadraticconvergenceLeast-Squaresproblemsonly(veryfast)ApproximateGradientMethodsWhatisthegradient?ThegradientisthederivativeoftheobjectivefunctionwithrespecttothecontrolvariablesGradientsoftheobjectivecanbecomputedeither:ApproximatelyCoordinatesearch:Finite-differenceinonecontrolvariableatatimeNelder-Mead:Evaluates(N+1)pointsofanN-dimensionaldesignspaceandconstructsasimplex,improvesworstpointBOBYQA:MakesprogressivelocalquadraticapproximationsAnalyticallyMMAandSNOPTusetheAdjointmethodtocomputeexactgradientCancomputegradientswithrespecttoallcontrolvariablesatonceRequiressmoothanddifferentiableobjectiveandconstraintfunctionsNoremeshingGradientbasedmethodsstartatapointwithinthedesignspaceandimprovethedesignDesignSpaceχ1χ2ObjectiveFunctionf(χ)(1)Startatanexistingdesign&computegradientdirection(2)Searchalonggradient&findminimumalongline(3)Repeatuntilnomoreimprovementispossible(1)(2)(3)ComparisonofAlgorithmsApproximateGradientAnalyticGradientObjectiveFunctionAnyscalaroutputMustbebothsmoothanddifferentiableDesignVariablesAnything,includinggeometricdimensionsAnythingthatdoesnotresultinremeshingofthegeometryRemeshingYesNoConstraintsCanonlyconstrainscalaroutputsConstraintsmustbedifferentiableandsmooth,butcanbeateachpointinspacePossibleanalysesAnycombinationofalldifferentanalysistypesAnystudywithonlyoneof:

Stationary,Transient,orFrequency-DomainRelative

PerformanceIncreasesexponentially

withthenumberofdesignvariablesPerformanceisnotverysensitivetothenumberofdesignvariablesTheMonte-Carlomethodsharesalloftheseproperties,butwillhavetheslowestconvergenceWhichoptimizationsolvertouse?Docontrolvariableschangethemesh?Constraintsonthesolution?Arethereanyconstraints?TopologyOptimization?SNOPTMMALevenberg-MarquardtBOBYQASmoothlyvaryingobjective?Nelder-Mead(orCoordinateSearch)MonteCarloYesYesYesYesYesNoNoNoNoVeryrandomNoisyOptimizationUserInterfacesTheOptimizationstudystepCentralcontrolpanelforalloptimizationChooseandtunesolversSpecifyglobalobjectivefunctions,controlparametersandconstraintsEnable/disablecontributionsfrominterfacesTheOptimizationinterfaceSetupgeneralobjectivecontributions,

includingleast-squaresDefinecontrolvariablefieldsSpecifygeneralconstraintsOptimizationModuleTheory,AdvancedDerivative-freesolversDirectsearchNelder-MeadCoordinatesearchTrustRegionBOBYQADerivative-FreeOptimizationSolversRequireonlyobjectivefunctionvalues,

noderivativesControlanything,includingtheCADgeometryRobustbutexpensiveParallelonclustersGradient-basedGeneral-purposeSNOPTMMALeast-squaresLevenberg-MarquardtGradient-BasedOptimizationSolversUsegradientinformationtocontrolsearchdirectionStationaryandtransientsolverscomputegradientsefficientlyManycontrolvariables(fields)CommonOptimizationTasksOptimizationTasksPureOptimizationOptimalDesignParameterSelectionGeometricOptimizationSizingShapeOptimizationTopologyOptimizationTargetMatchingInverseProblemsParameterEstimationImagingObjectiveFunctionsinCOMSOLGlobalObjectivesAcceptsanyglobalexpression–ofteninvolvingcouplingoperatorsExample:classicaloptimizationIntegralObjectivesIntegratesanexpressionoveradomain,boundary,edgeorpointExample:minimizingtheweightofastructureProbeObjectivesEvaluatesanexpressionatgivencoordinatesLeast-SquaresObjectivesComparesanexpressiontomeasuredvaluesinafileExample:fittingreactionconstantstomatchmeasuredconcentrationsControlVariablesinCOMSOLGlobalControlVariablesChooseanyexistingmodelparametersintheOptimizationstudystepWorkswithallsolversUsedwithderivative-freesolverstocontrolCADgeometryparametersControlVariableFieldsControlvariablesareassociatedwithpositionsinthegeometryFiniteelementinterpolationgivesmanydegreesoffreedomRequireusingtheOptimizationInterfaceOnlyworkswithgradient-basedsolversConstraintsinCOMSOLBoundsSetlimitsdirectlyonthecontrolvariables,oftenrequiredbysolversDesignconstraintsSetrelationsbetweenthecontrolvariablesDonotrequireevaluationofanyPDEsolutionPerformanceconstraintsSetconditionsonthePDEsolutionvariablesSameformatasanobjectivefunctionandasexpensivetoevaluateSensitivityEvaluationinCOMSOLSymbolicmathmachineryallowsefficientgradientevaluationAslongasallexpressionsaredifferentiableAdjointsensitivityFastevaluationofthegradientofanyobjectivefunctionwithrespecttoacontrolvariablefieldForwardsensitivityFastevaluationofthederivativeofaPDEsolutionfieldwithrespecttoindividualcontrolvariablesGradientofobjectivefunctionsiscomputedusingthechainruleNumericalgradientFall-backwhenexpressionsarenotsymbolicallydifferentiableTheSolversoftheOptimizationModuleAbouttheSNOPTSolverSNOPT=SparseNonlinearOPTimizerDevelopedbyP.E.Gill,W.MurrayandM.A.SaundersatStanfordUniversitySequentialQuadraticProgramming(SQP)methodSolvesasequenceofapproximatingquadraticprogrammingproblemswithlinearizedconstraintsOuterloopusesaquasi-NewtonstrategywhereanapproximateHessianisupdatedusinggradientsevaluatedatconsecutivestepsAbouttheMMASolverMMA=MethodofMovingAssymptotesDevelopedbyKristerSvanbergatKTH(RoyalInstituteofTechnology,Sweden)COMSOLversionismorespecificallyGCMMA=GloballyConvergentMMASolvesasequenceofconvexapproximationsAllowsgeneralnonlinearconstraintsEachapproximationisgeneratedfromlineardataatcurrentpointAllintermediatepointsarefeasible,unlessthefeasiblesetisemptyAbouttheLevenberg-MarquardtSolverSolverforleast-squaresproblemsRequiresanobjectivefunctiononleast-squaresformDoesnotallowanyconstraintsTrust-regionGauss-NewtonmethodComputesthegradientofeachterminthesumofsquaresseparatelyApproximatestheHessianfromfirst-orderderivativesonlyOften