Optimization Methods

1 1 Optimization Introduction:Inoptimizationofa design, satisfactorysolutionis ,optimizationhasbecomea partofcomputer-aideddesignactivities. Therearetwodistincttypesofoptimizational gorithmswidelyusedtoday.(a) (b) :Anaiveoptimaldesignisachievedbycomparin gafew(limiteduptotenorso) (cost,profit,etc., ) ofeachsolutionis comparedandbestsolutionis isimpossibletoapplysingleformulationproc edureforallengineeringdesignproblems,sin cetheobjectiveinadesignproblemandassocia tedtherefore,designparametersvaryproduct to ,whichthencanbesolvedusinganoptimization algorithm. Figure1 :Theformulationofanoptimizationproblembe ginswithidentifyingtheunderlyingdesignva riables,whichareprimarilyvariedduringthe optimizationprocess. A designproblemusuallyinvolvesmanydesignpa rameters, Other(notsoimportant)designparametersusu allyremainfixedorvaryin :Theconstraintsrepresentsomefunctionalre lationshipsamongthedesignvariablesandoth erdesignparameterssatisfyingcertainphysi calphenomenonandcertainresourcelimitatio ns.

2 Thenatureandnumberofconstraintstobeinclu dedin theformulationdependontheuser. ,maximumstressis a constraintofa astructurehasregularshapetheyhaveanexact mathematicalrelationofmaximumstresswithd imensions. Butincaseirregularshape, ,smallerthanorequalto,a :Thestress (x)developedanywhereina componentmustbesmallerthanorequaltotheal lowablestrength(Sallowable)ofthematerial . (x) SallowableSomeconstraintsmaybeofgreater- than/ ,thenaturalfrequency(f(x)) ofa systemtobegreaterthan2 Hzorbynotationf(x) :Thedeflection (x)ofa pointinthecomponentmustbeexactlyequalto5 (x)= isverydifficulttohandletheequalityconstr aintsinthealgorithms. Insuchcases, :Previously (x)= 5 Nowit is changedtoinequalityconstraintsasgivenbel ow: (x) 4, (x) :Thenexttaskintheformulationprocedureist ofindtheobjectivefunctionintermsofthedes ignvariablesandotherproblemparameters.

3 Thecommonengineeringobjectivesinvolvemin imizationofoverallcostofmanufacturingorm inimizationofoverallweightofa (expressedinmathematicalform),thereareso meobjectives(suchasaestheticaspectofades ign,ridecharacteristicsofacarsuspensiond esignandreliabilityofa design) , , minimizingtheoverallweightofthestructure andsimultaneouslybeconcernedinminimizing thedeflectionofa specificpointinthetruss. Intheoptimalproblemformulation,thedesign ermayliketousetheweightofthetruss(asafun ctionofthecrosssectionsofthemembers)asth eobjectivefunctionandhavea constraintwiththedeflectionoftheconcerne dpointtobelessthana Eitherit is tobemaximizedorit hastobeminimized. Usuallytheoptimizationalgorithmswerewrit tenforminimizationproblemsormaximization problems. Althoughinsomealgorithms,someminorstruct uralchangeswouldenabletoperformeithermin imization(or)maximization; minorchangeintheobjectivefunctioninstead ofa changein thealgorithmisforsolvinga minimizationproblem,it canbeeasilychangedtoamaximizationproblem bymultiplyingtheobjectivefunctionby 1 , , = 1, 2, 3.

4 N.(1 )Inanygivenproblem, tomakea guessabouttheoptimalsolutionandsetthemin imumandmaximumboundssothattheoptimalsolu tionlieswithinthesetwobounds()( )LUiiix xx ()Lix()Uix13 Ifanydesignvariablecorrespondingtotheopt imalsolutionisfoundtolieonorneartheminim umormaximumbound, ,theoptimizationproblemcanbemathematical lywritteninaspecialformat,knownasnonline arprogramming(NLP) :Denotingthedesignvariablesasa columnvectorx = (x1, )T-,theobjectivefunctionasa scalarquantityf(x),Jinequalityconstraint sasgj(x) 0andKequalityconstraintsashk(x)= 0,wewritetheNLPproblem:Minimizef(x)Subje ctto,gj(x) 0j = 1, 2, 3, ..J;hk(x)= 0k = 1, 2, 3, ..K;i = 1, 2, 3, ..N.()( )LUiiix xx 14 Example:1 Optimaldesignofa Theloadingisalsoshownin themembersAC= CE= l =1mOptimize,1. Topologyofthetrussstructure(theconnectiv ityoftheelementsinatruss).2. Onceoptimallayoutis known,crosssectionofeveryelementis , Usingthesymmetryofthetruss,A7= A1;A6= A2;A3= A5 Thus,therearepracticallyfourdesignvariab les(A1toA4).

5 Formulationoftheconstraints:Thetrusscarr ythegivenloadP= 2 kN, ,Syt= Syc= 500 MPaandmodulusofelasticityE = 200 GPa. Axialforcesin eachmembersofthetrussare16 MemberAB Pcsc ;MemberBC+ Pcsc ;MemberAC+ Pcot ;MemberBD P(cot + cot );Now,theaxialstresscanbecalculatedbydiv idingtheaxialloadbythecross-sectionalare aofthatmember. Thus,thefirstsetof constraintscanbewrittenas1csc,2ycPSA 2cot,2ytPSA 3csc,2ytPSA 4(cotcot ).2ycPSA + 17In most structures, deflection is a major consideration. In the above truss structure, let us assume that the maximum vertical deflection at C is max = 2 mm. By using Castigliano s theorem, we obtain the deflection constraint:Intheabovestructure,tan = = 2/3. Theothersetofconstraintsarisesfromthesta bilityconsiderationofthecompressionmembe rsAB,BD, membersABandBD:212,2 242(cotcot ).

6 + A A A +++ 18 Inthisproblem, , ,wewritetheobjectivefunctionasThenexttas kistosetlowerandupperboundsforthefourcro sssectionalareas. Wemaychoosetomakeallfourareasliebetween1 0and500mm2. ThusthevariableboundsareasInthefollowing , lA lA lA l++ +19 Subject lA lA lA l++ +4(cotcot ) 0,2ycPSA + ,0sin21 APSyc,0cot22 APSyt,0sin23 APSyt202120, sinEAPl 242(cotcot ) 0, + ,PlEA A A A +++ 21 Example:2 Optimaldesignofa A two-dimensional model of a car suspension systemThecomfortinridinga damperateachwheel(Figure4). Inordertoformulatetheoptimaldesignproble m,thefirsttaskis ,Frontcoilstiffnesskfs,Frontunsprungmass mfu,Rearcoilstiffnesskrs,Rearunsprungmas smru,Fronttyrestiffnesskft,Reardampercoe fficient rReartyrestiffnesskrt,Frontdampercoeffic ientAxle-to-axledistancel,Polarmomentofi nertiaofthecarJ,Aslongtimeis takenfortheconvergenceoftheoptimizationw ithallparametersasdesignvariables,onlyfo urimportantparameters-frontcoilstiffness kfs,rearcoilstiffnesskrs, frontdampercoefficient,andreardampercoef ficient rareconsideredasdesignvariables.

7 Otherdesignparametersarekeptconstant:ms= 1000kgl= mmfu= 70kgl1= mmru= 150kgl2= mkft= 20kg/mmJ= 550kg-m2krt= 20kg/mmf f 23 Usingtheseparameters,differentialequatio nsgoverningtheverticalmotionoftheunsprun gmassatthefrontaxle(q1), thesprungmass(q2), andtheunsprungmassattherearaxle(q4), andtheangularmotionofthesprungmass(q3) arewritten(Fig. 5) The dynamic model of the car suspension system. The above model has four degrees-of-freedom (q1to q4)24(9)(10)(11)(12)Where the forces F1 to F6are calculated as follows:(13)Theparametersd1, d2, d3, andd4aretherelativedeformationsinthefron ttyre,thefrontspring,thereartyre,andther earspringrespectively. Figure5 showsallthefourdegreesoffreedomoftheabov esystem(q1toq4). Therelativedeformationsinspringsandtyres canbewrittenasfollows:2112 2344 4563,,,, ,.ftfsfrsrrtF kdF kdFdF kdFdF kd ======25(14)Thetimevaryingfunctionsf1(t) andf2(t)areroadirregularitiesasfunctions oftime.

8 Anyfunctioncanbeusedforf1(t).Forexample, a bumpcanbemodeledasf1(t)= A sin, whereAistheamplitudeofthebumpandTis caris movingforward,thefrontwheelexperiencesth ebumpfirst,whiletherearwheelexperiencest hesamebumpalittlelater,dependinguponthes peedofthecar. Thus,thefunctionf2(t)canbewrittenasf2(t) = f1( t l/v), wherelis theaxle-to-axledistanceand is thespeedofthecar. Fortheabovebump,f2(t)=A sin((t l/v)/T).1 11221313 4242234( ),,( ),.d q ftdqlqqd q ftdq lq q= =+ = = /tT 26 Thecoupleddifferentialequationsspecified in equations(9 ) to(12)canbesolvedusinga numericalintegrationtechnique(forexample ,afourth-orderRunge-Kuttamethodcanbeused )toobtainthepitchingandbouncingdynamicso fthesprungmassms. Equationscanbeintegratedfora ,thenexttaskistoformulatetheconstraintsa ssociatedwiththeabovecarsuspensionproble m. Inordertosimplifytheproblem, (therateofchangeoftheverticalacceleratio nofthesprungmass)is a majorfactorconcerningthecomfortoftheridi ngpassengers.

9 Theguidelineusedincarindustriessuggestst hatthemaximumjerkexperiencedbythepasseng ersshouldnotbemorethanabout18m/s3. Mathematically,'''2max q (t) 18 27 Whenthefourcoupleddifferentialequations( 9 ) to(12)aresolved,theaboveconstraintcanbec omputedbynumericallydifferentiatingtheve rticalmovementofthesprungmass(q2) ,theprimaryobjectiveis tominimizethetransmissibilityfactorwhich iscalculatedastheratioofthebouncingampli tudeq2(t)ofthesprungmasstotheroadexcitat ionamplitudeA. Thus,wewritetheobjectivefunctionasTheabo veobjectivefunctioncanbecalculatedfromth esolutionofthefourdifferentialequationsm entionedearlier. A minimumvalueofthetransmissibilityfactors uggeststheminimumtransmissionofroadvibra tiontothepassengers. Thisfactoris abs q (t)MinimizeA28 Thus,theoptimizeddesignoftheabovecarsusp ensionsystemwouldprovidetheminimumtransm issibilityoftheroadvibrationtothepasseng erswitha ,a carsuspensiondesign,butthefollowinglimit sfortheabovecarmayincludetheoptimalsolut ion:Thus,theaboveoptimalcarsuspensiondes ignproblemcanbewrittenin NLPformasfollows:Subjectto0 , 2/ ,0,300/( / ).

10 Fsrsfrkkkg mmkg m s 2max abs q (t)MinimizeA0,0,2,0,300.'''2fsrsfr18 - max q (t)kk 29 Example:3 Optimaldesignofa transitscheduleFig. 2 Figure2 showsa typicaltransitsystemnetwork. Thesolidlinesrepresentdifferentroutes,th epointsonthelinesrepresentthestopsandthe circledintersectionsoftheroutesrepresent thetransferstations. Theproblemistodetermineschedulesforthero utessuchthatthetransitsystemprovidestheb estLevelofService(LOS)toitspassengers, thelesserthewaitingtime,thebetteristheLO S. Onanytransitnetwork, (IWT),TransitTime(TT)Schedulethevehicles suchthat(IWT+ TT)is :Arrivaltime:aikk: vehiclesDeparturetime:diki: :(dik aik) sminforalliandk(1 )Maximumstoppingtime:(dik aik) smaxforalliandk(2 )Maximumallowabletransfertime:Nopassenge ronthetransitnetworkshouldhavetowaitmore thana valueofzeromeansthatthetransferofpasseng ersbetweenthosetwovehiclesis valueofonemeansotherwise.