Optimalo

AAiT,MechanicalEngineeringDepartment

CourseObjectiveThecourseintroduces:

y Understandingofprinciplesandpossibilities ofoptimizationinEngineeringandinparticularindesigny Understandhowtoformulateanoptimumdesignproblembyidentifyingcriticalelementsy knowledgeofoptimizationalgorithms,abilitytochooseproperalgorithmforgivenproblemy Practicalexperiencewithoptimizationalgorithmsy Practicalexperienceinapplicationofoptimizationtodesignproblems

CourseoutlineChapter1:IntroductiontoEngineeringOptimizationofDesigny Introduction: Historicalbackground,Definitionofterms,Basicconcepts,Classificationofoptimizationsproblems,y Applications:Designoptimization,benefitsofoptimization,automateddesignoptimization,whentouseoptimization,examples

Chapter2:OptimumDesignFormulationy Designmodels,Mathematicalmodels,Definingoptimizationproblem,Multiobjective

designproblems,applicationsofoptimizationindesign

Chapter3ClassicalOptimizationtechniquesy Singlevariableoptimizationy Multivariableoptimizationwithequalityandinequalityconstraints

Chapter4:Onedimensionalunconstrainedoptimizationtechniquesy Eliminationmethods:Exhaustivesearch,Intervalhalvingmethod,FibonacciMethod,GoldenSectionmethod.

y Interpolationmethods:quadraticinterpolation,cubicinterpolationy Directrootmethods: Newton'smethod,QuasiNewtonmethod,Secantmethod

CourseoutlineChapter5:UnconstrainedOptimizationtechniquesy Directsearchmethods: Randomsearch,GridsearchMethod,Powellmethody Indirectsearch(Descent)methods: Steepestdescent(Cauchy)method,Conjugate

gradient(FletcherReeves)method,Newtonsmethod,y UnconstrainedoptimizationusingMatlab

Chapter6:ConstrainedOptimizationtechniquesy Directsearchmethods:Randomsearch,complexsearchMethod,Quadratic

programmingy Indirectmethods:Penaltyfunctionmethod,Lagrangemultipliermethody ConstrainedoptimizationusingMatlab

Chapter7:DynamicProgrammingy Introduction,Multistagedecisionprocesses,Applicationsofdynamicprogramming.

Chapter8:GeneticAlgorithmbasedOptimizationy IntroductiontoGeneticAlgorithm,ApplicationsofGAbasedoptimizationtechniques,GAbasedOptimizationusingMatlab

ReferenceMaterials1. S.S.Rao,EngineeringOptimization,3rd edition,WileyEastern,20092. Papalambros andWilde,PrincipleofoptimalDesign,modelingand

computation,CambridgeUniversitypress,20003. Kalyanmoy Deb,EngineeringDesignforoptimization, PHI,20054. FredvanKeulen andMatthiis Langelaar,LecturenotesinEngineering

Optimization,TechnicalUniversityofDelft5. Ravindran,Ragsdell andRekalaitis,EngineeringOptimizationMethodsand

application,2nd edition,Willey,20066. Arora,IntroductiontoOptimumdesign,2nd edition,ElsevierAcademicPress,

20047. Forst andHoffmann,Optimizationtheoryandpractice,Springer,20108. Haftka andGurdal,ElementsofStructuralOptimization,3rd edition,Kluwer

academic,19919. Belegundu andChandrupatla,Optimizationconceptsandapplicationsin

Engineering,2nd edition,CambridgeUniversitypress,201110. Kalyanmoy Deb,MultiobjectiveOptimizationusingEvolutionary

Algorithms,Wiley,200211. Bendose,Sigmund,Topologyoptimizationtheoryandmethodsand

applications, Springer,2003

PrerequisitesMathematicalandComputerbackgroundneededtounderstandthecourse:y Familiaritywithlinearalgebra(vectorandmatrixoperations)andy basiccalculusisessentialandCalculusoffunctionsofsingleandmultiplevariablesmustalsobeunderstoody FamiliaritywithMatlab andEXCELisalsoessential

Lectureoutliney Introductiony Historicalperspectivey Whatcanbeachievedbyoptimization?y Optimizationofthedesignprocessy Basicterminology,notations,anddefinitionsy Engineeringoptimizationy Popularityandpitfallsofoptimizationy Classificationofoptimizationproblemsy Designoptimizationy Benefitsofdesignoptimizationy Automateddesignoptimizationy Examples

IntroductionOptimizationisderivedfromtheLatinwordoptimus,thebest.Thusoptimizationfocuseson

Makingthingsbetter

Generatingmoreprofit

Determiningthebest

Domorewithless

Thedeterminationofvaluesfordesignvariables whichminimize(maximize)theobjective,whilesatisfyingallconstraints

Introductiony Optimizationisdefinedasamathematicalprocessofobtainingthesetofconditionstoproducethemaximumortheminimumvalueofafunction

y Itisidealtoobtaintheperfectsolutiontoadesignsituation.

y Usuallyallofusmustalwaysworkwithintheconstraintsofthetime andfundsavailable,wecanonlyhopeforthebestsolutionpossible.

y Optimizationissimplyatechniquethataidsindecisionmaking butdoesnotreplacesoundjudgmentandtechnicalknowhow

Historicalperspectivey AncientGreekphilosophers:geometricaloptimizationproblems

y Zenodorus,200B.C.:Asphereenclosesthegreatestvolumeforagivensurfacearea

y Newton,Leibniz,Bernoulli,DelHospital (1697):Brachistochrone Problem:

Historicalperspectivey Peoplehavebeenoptimizingforever,buttherootsformoderndayoptimizationcanbetracedtotheSecondWorldWar.y AncientGreekphilosophers:geometricaloptimizationproblems

y Zenodorus,200B.C.:Asphereenclosesthegreatestvolumeforagivensurfacearea

y Newton,Leibniz,Bernoulli,DelHospital (1697):Brachistochrone Problem:y Lagrange(1750):constrainedminimizationy Cauchy(1847):steepestdescenty Dantzig (1947):Simplexmethod(LP)y Kuhn,Tucker(1951):optimalityconditionsy Karmakar (1984):interiorpointmethod(LP)y Bendsoe,Kikuchi(1988):topologyoptimization

Historicalperspectivey Oneofthefirstproblemsposedinthecalculusofvariations.y Galileoconsideredtheproblemin1638,buthisanswerwasy incorrect.y JohannBernoulliposedtheproblemin1696toagroupofy elitemathematicians:y I,JohannBernoulli...hopetogainthegratitudeofthewholescientificcommunitybyplacingbeforethefinestmathematiciansofourtimeaproblemwhichwilltesttheirmethodsandthestrengthoftheirintellect.Ifsomeonecommunicatestomethesolutionoftheproposedproblem,Ishallpubliclydeclarehimworthyofpraise.

y Newtonsolvedtheproblemtheverynextday,butproclaimedIdonotlovetobedunned[pestered]andteasedbyforeignersaboutmathematicalthings."

Whatcanbeachievedbyoptimization?

y Optimizationtechniquescanbeusedfor:y Gettingadesign/systemtoworky Reachingtheoptimalperformancey Makingadesign/systemreliableandrobust

y Alsoprovideinsightiny Designproblemy Underlyingphysicsy Modelweaknesses

Whatcanbeachievedbyoptimization?Engineeringdesignistocreateartifactstoperformdesiredfunctionsundergivenconstraintsy Commongoalsforengineeringdesigny Functionalityy Betterperformance:Moreefficientoreffectivewaystoexecutetasksy Multiplefunctions:Capabilitiestoexecutetwoormoretaskssimultaneously

y Valuey Higherperceivedvalue:Morefeatureswithlesspricey Lowertotalcost:Sameorbetterownershipandsustainabilitywithlowercost

BasicTerminology,notationsanddefinitionsRn ndimensionalEuclidean(real)spacex columnvectorofvariables,apointinRn

x=[x1,x2,..,xn]T

f(x),f objectivefunctionx* localoptimizerf(x*) optimumfunctionvaluegj(x),gj jth equalityconstraintfunctiong(x) vectorofinequalityconstrainthj(x),hj jth equalityconstraintfunctionh(h(x) vectorofequalityconstraintfunctionC1 setofcontinuousdifferentiablefunctionsC2 setofcontinuousandtwicedifferentiabledifferentiable

continuousfunctions

Norm/Lengthofavectory Ifweletxandybetwondimensionalvectors,thentheirdot

productisdefinedas

y Thus,thedotproductisasumoftheproductofcorrespondingelementsofthevectorsxandy.

y Twovectorsaresaidtobeorthogonal(normal)iftheirdotproductiszero,i.e.,xandy areorthogonalifxy=0.

y Ifthevectorsarenotorthogonal,theanglebetweenthemcanbecalculatedfromthedefinitionofthedotproduct:

y where istheanglebetweenvectorsxandy,and||x||representsthelengthofthevectorx.Thisisalsocalledthenormofthevector

Norm/Lengthofavectory Thelengthofavectorxisdefinedasthesquarerootofthe

sumofsquaresofthecomponents,i.e.,

y ThedoublesumofEq.(1.11)canbewritteninthematrixformasfollows

y SinceAxrepresentsavector,thetripleproductoftheaboveEq.willbealsowrittenasadotproduct:

BasicTerminologyandnotationsDesignvariables

y Parameterswhosenumericalvaluesaretobedeterminedtoachievetheoptimumdesign.

y Theyincludesuchvaluessuchas;sizeorweight,orthenumberofteethinagear,coilsinaspring,ortubesinaheatexchanger,oretc.

y Designparametersrepresentanynumberofvariablesthemayberequiredtoquantifyorcompletelydescribeanengineeringsystem.

y Thenumberofvariablesdependsuponthetypeofdesigninvolved.Asthisnumberincreases,sodoesthecomplexityofthesolutiontothedesignproblems.

BasicTerminologyandnotationsConstraintsy Numericalvaluesofidentifiedconditionsthatmustbesatisfiedtoachieveafeasiblesolutiontoagivenproblem.y Externalconstraintsy Uncontrolledrestrictionsorspecificationsimposedonasystembyanoutsideagency.

y Ex.:Lawsandregulationssetbygovernmentalagencies,allowablematerialsforhouseconstruction

y Internalconstraintsy Restrictionsimposedbythedesignerwithakeenunderstandingofthephysicalsystem.

y Ex.:Fundamentallawsofconservationofmass,momentum,andenergy

Whatismathematical/EngineeringOptimization?Mathematicaloptimizationistheprocessof1. Theformulationand2. Thesolutionofaconstrainedoptimizationproblemofthe

generalmathematicalformMinimize f(x),x=[x1,x2,,xn]T subjecttoconstraints

gj(x) 0,j=1,2,,mhj(x)=0,j=1,2,.,r

Wheref(x),gj(x)andhj(x)arescalarfunctionsoftherealcolumnvectory Thecontinuouscomponentsofxiofx=[x1,x2,,xn]T arecalled

the(design)variablesy f(x) istheobjectivefunction,y gj(x) denotestherespectiveinequalityconstraints,andy hj(x)theequalityconstraintfunction

Whatismathematical/EngineeringOptimization?y Theoptimumvectorxthatsolvestheformerlydefinedproblemisdenotedbyx*withthecorrespondingoptimumfunctionvaluef(x*).

y Ifnoconstraintsarespecified,theproblemiscalledanunconstrainedminimizationproblem

y OthernamesofMathematicalOptimizationy Mathematicalprogrammingy Numericaloptimization

ObjectiveandConstraintfunctionsy Thevaluesofthefunctionsf(x),gj(x),hj(x)atanypointx=[x1,x2,,xn]T gj(x),mayinpractise beobtainedindifferentways

i. Fromanalyticallyknownformulae,e.g.,f(x)=x12+2x22+Sinx3

ii. Astheoutcomeofsomecomplicatedcomputationalprocesse.g.,g1(x)=a(x)amax,wherea(x)isthestress,computedbymeansofafiniteelementanalysis,atsomepointinstructure,thedesignofwhichisspecifiedbyx;or

iii. Frommeasurementtakenofaphysicalprocess,e.g.,h1(x)=T(x)To,whereT(x)isthetemperaturemeasuredatsomespecifiedpointinareactor,andxisthevectorofoperationalsettings.

PresenterPresentation NotesThe first two ways of function evaluation are by far the most common. The optimization principle that apply in these cased, where computed function values are used, may be carried over directly to also be applicable to the case where the function values are obtained through physical measurement.

ElementsofoptimizationDesignspacey ThetotalregionordomaindefinedbythedesignvariablesintheobjectivefunctionsUsuallylimitedbyconstraintsy Theuseofconstraintsisespeciallyimportantinrestrictingtheregionwhereoptimalvaluesofthedesignvariablescanbesearched.y Unboundeddesignspacey Notlimitedbyconstraintsy Noacceptablesolutions

Optimizationinthedesignprocess

Conventionaldesignprocess:

Collectdatatodescribethesystem

Estimateinitialdesign

Analyzethesystem

Checkperformancecriteria

Isdesignsatisfactory?

Changedesignbasedonexperience/heuristics/

wildguesses

Done

Optimizationbaseddesignprocess:

Collectdatatodescribethesystem

Estimateinitialdesign

Analyzethesystem

Checktheconstraints

Doesthedesignsatisfyconvergencecriteria?

Changethedesignusinganoptimizationmethod

Done

Identify:1. Designvariables2. Objectivefunction3. Constraints

PresenterPresentation NotesTaken from J.S. Arora Introduction to Optimum Design, fig. 1-2.

Optimizationinthedesignprocessy Isthereoneaircraftwhichisthefastest,mostefficient,quietest,mostinexpensive?

Youcanonlymakeonethingbestatatime.

OptimizationMethods

ComparisonofConventionalandOptimalDesigny TheCDprocessinvolvestheuse

ofinformationgatheredfromoneormoretrialdesignstogetherwiththedesignersexperienceanintuition

y Itsadvantageisthatthedesignersexperienceandintuitioncanbeusedinmakingconceptualchangesinthesystemortomakeadditionalspecificationsintheprocedure

y TheCDprocesscanleadtouneconomicaldesignsandcaninvolvealotofcalendartime.

y TheODprocessforcesthedesignertoidentifyexplicitlyasetofdesignvariables,anobjectivefunctiontobeoptimized,andtheconstraintfunctionsforthesystem.

y Thisrigorousformulationofthedesignproblemhelpsthedesignergainabetterunderstandingoftheproblem.

y Propermathematicalformulationofthedesignproblemisakeytogoodsolutions.

OptimizationpopularityIncreasinglypopular:y Increasingavailabilityofnumericalmodelingtechniques

y Increasingavailabilityofcheapcomputerpowery Increasedcompetition,globalmarketsy Betterandmorepowerfuloptimizationtechniquesy Increasinglyexpensiveproductionprocesses(trialanderrorapproachtooexpensive)

y Moreengineershavingoptimizationknowledge

Optimizationpitfalls!y Properproblemformulationcritical!y ChoosingtherightalgorithmforagivenproblemyManyalgorithmscontainlotsofcontrolparametersy Optimizationtendstoexploitweaknessesinmodelsy Optimizationcanresultinverysensitivedesignsy Someproblemsaresimplytoohard/large/expensive

PresenterPresentation NotesIt is generally accepted that the proper definition and formulation of a problem takes roughly 50 percent of the total effort needed to solve it. Therefore, it is critical to follow well defined procedures for formulating design optimization problems.

The importance of properly formulating a design optimization problem must be stressedbecause the optimum solution will only be as good as the formulation.

For example, if we forget to include a critical constraint in the formulation, the optimum solution will most likelyviolate it because optimization methods tend to exploit deficiencies in design models. Also,if we have too many constraints or if they are inconsistent, there may not be a solution to thedesign problem.

Structuraloptimizationy Structuraloptimization=optimizationtechniquesappliedtostructuresy Differentcategories:y Sizingoptimizationy Materialoptimizationy Shapeoptimizationy Topologyoptimization

t

E, R

r

L

h

StructuraloptimizationInegrated optimaldesignofavehicleroadarm.y a)InitialFiniteElement

Model,y b)topologyoptimizedroadarm,y c)reconstructedsolidmodel,y d)FiniteElementmeshforshapedesigny e)VonMises stressoftheshapeoptimizeddesignandy f)comparisonofthe3DRoadarm beforeandaftershapedesign

Sizingoptimizationy Inatypicalsizingproblemthegoalmaybetofindtheoptimalthicknessdistributionofalinearlyelasticplateortheoptimalmemberareasinatrussstructure.

y Theoptimalthicknessdistributionminimizes(ormaximizes)aphysicalquantitysuchasthemeancompliance(externalwork),peakstress,deflection,etc.whileequilibriumandotherconstraintsonthestateanddesignvariablesaresatisfied.

y Thedesignvariableisthethicknessoftheplateandthestatevariablemaybeitsdeflection.

Shapeoptimizationy Shapeoptimization ispartofthefieldofoptimalcontroltheory.

y Thetypicalproblemistofindtheshapewhichisoptimalinthatitminimizesacertaincostfunctionalwhilesatisfyinggivenconstraints.

y Inmanycases,thefunctionalbeingsolveddependsonthesolutionofagivenpartialdifferentialequationdefinedonthevariabledomain.

Shapeoptimization

YamahaR1

Topologyoptimizationy Topologyoptimizationis,inaddition,concernedwiththenumberofconnectedcomponents/boundariesbelongingtothedomain.Suchusdeterminationoffeaturessuchasthenumberand locationandshapeofholesand theconnectivityofthedomain.

y Suchmethodsareneededsincetypicallyshapeoptimizationmethodsworkinasubsetofallowableshapeswhichhavefixedtopologicalproperties,suchashavingafixednumberofholesinthem.

y Topologicaloptimizationtechniquescanthenhelpworkaroundthelimitationsofpureshapeoptimization.

TopologyoptimizationTopologyoptimizationisamathematicalapproachthatoptimizesmateriallayoutwithinagivendesignspace,foragivensetofloadsandboundaryconditionssuchthattheresultinglayoutmeetsaprescribedsetofperformancetargets.

y Usingtopologyoptimization,engineerscanfindthebestconceptdesignthatmeetsthedesignrequirements

Topologyoptimizationexamples

WhyDesignOptimization?

DesignComplexity

Classificationsy Problems:y Constrainedvs.unconstrainedy Singlelevelvs.multilevely Singleobjectivevs.multiobjectivey Deterministicvs.stochastic

y Responses:y Linearvs.nonlineary Convexvs.nonconvexy Smoothvs.nonsmooth

y Variables:y Continuousvs.discrete(integer,ordered,nonordered)

TypicalDesignProcess

InitialDesignConcept

SpecificDesignCandidate

BuildAnalysisModel(s)

ExecutetheAnalyses

DesignRequirementsMet?

FinalDesign

Yes

No

ModifyDesign

(Intuition)

Time

Money

IntellectualCapital

HEEDS

$

HEEDS(HierarchicalEvolutionaryEngineeringDesignSystem)

AGeneralOptimizationSolution

Automotive CivilInfrastructure

BiomedicalAerospace

AutomatedDesignOptimization

CreateParameterizedBaselineModel

CreateHEEDSDesignModel

ExecuteHEEDSOptimization

PlanDesignStudy

BasicProcedure:


Identify: Objective(s)ConstraintsDesign VariablesAnalysis Methods

Note: These definitions affect subsequent steps




PlanDesignStudy


InputFile(s)

ExecuteSolver(s)

OutputFile(s)

ValidateModel

CreateCAD/CAEModelsforaRepresentative Design




PlanDesignStudy


DefineInputFilesandOutputFiles

DefineDesignVariablesandResponses

DefineObjectives,Constraints,andSearch

Method

TagVariablesinInputFilesand

ResponsesinOutputFiles

DefineBatchExecutionCommandsforSolvers




PlanDesignStudy





PlanDesignStudy ModifyVariablesinInputFile

ExecuteSolverinBatchMode

ExtractResultsfromOutputFile

Optimized Design(s)

Yes

NewDesign(HEEDS)

NoConverged?

CAEPortals

When

What

Where

TangibleBenefits*Crashrails: 100%increaseinenergyabsorbed

20%reductioninmass

Compositewing: 80%increaseinbucklingload15%increaseinstiffness

Bumper: 20%reductioninmasswithequivalentperformance

Coronarystent: 50%reductioninstrain

*Percentagesrelativetobestdesignsfoundbyexperiencedengineers

ReturnonInvestment

ReducedDesignCosts Time,labor,prototypes,tooling Reinvestsavingsinfutureinnovationprojects

ReducedWarrantyCosts Higherqualitydesigns Greatercustomersatisfaction

IncreasedCompetitiveAdvantage Innovativedesigns Fastertomarket Savingsonmaterial,manufacturing,mass,etc.

Suggestsmaterialplacementorlayoutbasedonloadpathefficiency

Maximizesstiffness Conceptualdesigntool UsesAbaqus StandardFEAsolver

Topology Optimization

WhentoUseTopologyOptimization

y Early in the design cycle to find shape conceptsy To suggest regions for mass reduction Topology

optimization

DesignofExperiments

Determinehowvariablesaffecttheresponseofaparticulardesign

Designsensitivities Buildmodelsrelatingtheresponsetothevariables

Surrogatemodels,responsesurfacemodels

B

A

WhentoUseDesignofExperiments

Following optimization

Toidentifyparametersthatcausegreatestvariation inyourdesign

ParameterOptimizationMinimize(ormaximize): F(x1,x2,,xn)

suchthat: Gi(x1,x2,,xn)

ParameterOptimizationObjective:Searchtheperformancedesignlandscapetofindthehighestpeakorlowestvalleywithinthefeasiblerange

Typicallydontknowthenatureofsurfacebeforesearchbegins

Searchalgorithmchoicedependsontypeofdesignlandscape

Localsearchesmayyieldonlyincrementalimprovement

Numberofparametersmaybelarge

SelectinganOptimizationMethod

DesignSpacedependson:

Number,typeandrangeofvariablesandresponses

Objectivesandconstraints

GradientBased Simplex Simulated

Annealing

ResponseSurface GeneticAlgorithm Evolutionary

Strategy

Etc.

DesignOptimizationProcedureUsingANSYSy Theoptimizationmodule(OPT)isanintegralpartoftheANSYS

programthatcanbeemployedtodeterminetheoptimumdesign.

y Whileworkingtowardsanoptimumdesign,theANSYSoptimizationroutinesemploythreetypesofvariablesthatcharacterizethedesignprocess:

y designvariables,y statevariables,andy theobjectivefunction.

y ThesevariablesarerepresentedbyscalarparametersinANSYSParametricDesignLanguage(APDL). TheuseofAPDLisanessentialstepintheoptimizationprocess.

y Theindependentvariablesinanoptimizationanalysisarethedesignvariables.

DesignOptimizationProcedureUsingANSYSOrganizeANSYSprocedureintotwofiles:y Optimizationfiledescribesoptimizationvariables,andtriggertheoptimizationruns.y Analysisfileconstructs,analyses,andpostprocessesthemodel.y TypicalCommandsinanOptimizationFile

01020304050607080910111213

/CLEAR ! Clear model database... ! Initialize design variables/INPUT, ... ! Execute analysis file once

/OPT ! Enter optimization phaseOPCLEAR ! Clear optimization databaseOPVAR, ... ! Declare design variablesOPVAR, ... ! Declare state variablesOPVAR, ... ! Declare objective functionOPTYPE, ... ! Select optimization methodOPANL, ... ! Specify analysis file nameOPEXE ! Execute optimization runOPLIST, ... ! Summarize the results... ! Further examining results

DesignOptimizationProcedureUsingANSYS

010203040506070809101112

/PREP7... ! Build the model using the

! parameterized design variablesFINISH

/SOLUTION... ! Apply loads and solveFINISH

/POST1 ! or /POST26*GET, ... ! Retrieve values for state variables*GET, ... ! Retrieve value for objective

function... FINISH

TypicalCommandsinanAnalysisFile

DesignOptimizationProcedureUsingANSYSANSYSOptimizationAlgorithmsTwobuiltinalgorithmsinANSYS:y Firstordermethody Subproblemapproximationmethod(Zeroordermethod)

OtherOptimizationToolsProvidedbyANSYSy SingleIterationDesignTooly RandomDesignTooly GradientTooly SweepTooly FactorialTool

Summary

y Designvariables:variableswithwhichthedesignproblemisparameterized:y Objective:quantitythatistobeminimized(maximized)Usuallydenotedby:(costfunction)y Constraint:conditionthathastobesatisfiedy Inequalityconstraint:y Equalityconstraint:

( ) 0g x( ) 0h =x

( )f x

( )1 2, , , nx x x=x K

Summaryy Generalformofoptimizationproblem:

( )xxxxxhxg

xx

=

nX

f

0)(0)(

)(

:to subject

min

Summaryy Optimizationproblemsaretypicallysolvedusinganiterativealgorithm:

Model

Optimizer

Designvariables

Constants Responses

Derivativesofresponses(designsensitivities)

hgf ,,

iii xh

xg

xf

,,

x

Engineering Optimization Course Objective Course outlineCourse outlineReference MaterialsPrerequisites Lecture outline IntroductionIntroductionHistorical perspectiveHistorical perspectiveHistorical perspectiveWhat can be achieved by optimization ?What can be achieved by optimization ?Basic Terminology, notations and definitionsNorm/Length of a vectorNorm/Length of a vectorBasic Terminology and notations Basic Terminology and notations What is mathematical/Engineering Optimization ? What is mathematical/Engineering Optimization ? Objective and Constraint functions Elements of optimizationOptimization in the design processOptimization in the design processOptimization MethodsComparison of Conventional and Optimal DesignOptimization popularityOptimization pitfalls!Structural optimizationStructural optimizationSizing optimizationShape optimization Shape optimization Topology optimizationTopology optimizationTopology optimization examplesWhy Design Optimization ?Classifications Typical Design ProcessA General Optimization SolutionAutomated Design OptimizationAutomated Design OptimizationAutomated Design OptimizationAutomated Design OptimizationAutomated Design OptimizationCAE PortalsTangible Benefits*Return on InvestmentSlide Number 50When to Use Topology OptimizationDesign of ExperimentsWhen to Use Design of ExperimentsParameter OptimizationParameter OptimizationSelecting an Optimization MethodDesign Optimization Procedure Using ANSYSDesign Optimization Procedure Using ANSYSDesign Optimization Procedure Using ANSYSDesign Optimization Procedure Using ANSYS Summary Summary Summary

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Optimalo