Multi-Fidelity Multi-Objective Efficient Global Optimization Applied to Airfoil Design Problems

21 Pages·2017·6.45 MB·English

applied
sciences
Article
Multi-Fidelity Multi-Objective Efficient Global
Optimization Applied to Airfoil Design Problems
AtthaphonAriyarit*,† ID andMasahiroKanazaki
DepartmentofAerospaceEngineering,GraduateSchoolofSystemDesign,TokyoMetropolitanUniversity,
Hino-shi,Tokyo191-0065,Japan;[email protected]
* Correspondence:[email protected];Tel.:[phone]
† Theseauthorscontributedequallytothiswork.
Received:30October2017;Accepted:15December2017;Published:18December2017
Abstract: Inthisstudy,efficientglobaloptimization(EGO)withamulti-fidelityhybridsurrogate
model for multi-objective optimization is proposed to solve multi-objective real-world design
problems. Intheproposedapproach,adesignexplorationiscarriedoutassistedbysurrogatemodels,
whichareconstructedbyaddingalocaldeviationestimatedbythekrigingmethodandaglobalmodel
approximatedbyaradialbasisfunction. Anexpectedhypervolumeimprovementisthencomputed
onthebasisofthemodeluncertaintytodetermineadditionalsamplesthatcouldimprovethemodel
accuracy. Intheinvestigation,theproposedapproachisappliedtotwo-objectiveandthree-objective
optimization test functions. Then, it is applied to aerodynamic airfoil design optimization with
two objective functions, namely minimization of aerodynamic drag and maximization of airfoil
thickness at the trailing edge. Finally, the proposed method is applied to aerodynamic airfoil
designoptimizationwiththreeobjectivefunctions,namelyminimizationofaerodynamicdragat
cruisingspeed,maximizationofairfoilthicknessatthetrialingedgeandmaximizationofliftatlow
speedassumingalandingattitude. XFOILisusedtoinvestigatethelow-fidelityaerodynamicforce,
and a Reynolds-averaged Navier–Stokes simulation is applied for high-fidelity aerodynamics in
conjunctionwithahigh-costapproach. Forcomparison,multi-objectiveoptimizationiscarriedout
usingakrigingmodelonlywithahigh-fidelitysolver(singlefidelity). Thedesignresultsindicate
thatthenon-dominatedsolutionsoftheproposedmethodachievegreaterdatadiversitythanthe
optimalsolutionsofthekrigingmethod. Moreover,theproposedmethodgivesasmallererrorthan
thekrigingmethod.
Keywords: multi-fidelityoptimization;efficientglobaloptimization;multi-objectiveoptimization;
airfoildesign
1. Introduction
Ahigh-costcomputationfunctionisrequiredtosolveaerodynamicdesignproblems. Inaddition,
real-worlddesignproblemsofteninvolveseveralobjectivefunctions[1,2]. Forexample, inaircraft
design,itisnecessaryforadesignertoaccountfornotonlytheperformanceataspecificcruisecondition,
butalsotheperformancesatalloperatingspeeds,includingthoseduringtake-offandlanding.Owing
to these problems, several researchers have explored methods to reduce the computational costs of
designoptimizationalgorithmsformulti-ormany-objectiveoptimizationproblems.
Inaerodynamicevaluation,itispossibletoselectvariousphysicalcomputationmodelstosolvea
designproblem[3]. Forexample,forcesaroundanairfoilcanbeevaluatedbyapotentialequationasa
low-levelcomputation[4]. Moreover,theNavier–Stokesequationcanbeemployedasahigh-level
computation[5]. Giventheadvantageoftheabove-mentionedapproach,amulti-fidelityapproach
combinestwo-fidelitydataforoptimizationinordertoimprovetheefficiencyoftheoptimization
Appl.Sci. 2017,7,1318;doi:10.3390/app7121318 www.mdpi.com/journal/applsci
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process.Thus,approachesthatincludeamulti-fidelitysurrogatemodelhavebeenwidelystudied[6–8]
inaerospaceengineering. Multi-fidelitymethodsforsingle-objectiveoptimizationbasedontheerror
estimationofresponsesurfaceshavebeensuccessfullyappliedtodesignlow-boomsupersonicjets[6].
Because the response surfaces [9] use simple concepts to construct the function, the multi-fidelity
functionbasedontheresponsesurfaceiseasytoconstruct.However,suchmethodsareoftenassociated
withlowaccuracybecausetheresponsesurfacemodelcannotbeoptimizedinsituationsinvolving
anextremelysmallnumberofdatapoints. Aco-krigingmodel[7,8]hasbeenextensivelyappliedto
combinemulti-fidelityfunctions. Inaddition,ithasbeenemployedtosolveanoptimalsingle-objective
airfoildesignproblem[7]. Thismethodisbeneficialforpredictingacomplicatedlandscapefunction.
However,itislessoptimalinpredictingasmoothlandscapefunction.Rethoreproposedamulti-fidelity
single-objectiveoptimizationprocessforwindturbinedesign[10,11]. Thisoptimizationprocessbegins
with the location of an optimal point of a low-fidelity function by using a genetic algorithm (GA).
Then, an optimization with a high-fidelity function is performed using a gradient-based method.
Theoptimumpointofthelow-fidelityfunctionisusedasastartingpoint.Thismethodisbeneficial,asit
usesabasicoptimizationtooltosolvethemulti-fidelityoptimizationproblem.However,itcouldfailto
findanoptimumhigh-fidelityfunctioniftheerrorbetweenthelow-fidelityandhigh-fidelityfunctions
islarge.Becausethegradient-basedmethodrequiredmanyevolutionfunctionstosolvemulti-objective
optimizationproblems,themulti-fidelityoptimizationbasedonevolutionarycomputationcombined
with the gradient method would not reduce the computation time of the high-fidelity function.
Thus,multi-fidelityapproachesareexpectedtoimprovetheefficiency.
An approach to multi-fidelity/multi-objective optimization involves a model reduction
technique [12,13]. This technique has been applied to helicopter rotor blade design and airfoil
design[13,14]. Theapplicationofthistechniquereducesthedesignparameterspacefordefiningthe
possibledesignrangesbasedonalow-fidelityfunction. Ahigh-fidelityfunctionisthenusedtofind
theoptimumdesignintheprimarydefineddesignrange. Thehigh-fidelityfunctionissampledby
selectingthepreferreddesignpoints,wherethebladeshowsimprovedperformancewithrespectto
thebaselinedesign,inordertoensuregooddiversityofthesampledata,inadditiontotheinitialpoint
thatisgeneratedwiththeprimarydefineddesignrange. However,thisapproachhasthepotentialto
obtainanunexpectedoptimalsolutionoutsidetheparameterspacegiventhatthedesignrangesof
thelow-fidelityfunctionarenotalwaysappropriateforthehigh-fidelityfunction. Fusi[15]usedGA
tofindtheoptimumsolutionoflow-fidelitydataandselectedtheinterestingpointsoftheoptimum
solutionusingthelow-fidelitydatatofindtheoptimumpointofhigh-fidelitydataofahoveringrotor
airfoildesign.
Efficientglobaloptimization(EGO)[16]isawidely-usedmethodthatconsistsofkriging-model-based
explorations. EGO involves additional sampling-procedure-based expected improvements (EIs) for
improving the model accuracy. The EIs are defined for single-objective optimization. The expected
hypervolumeimprovement(EHVI)algorithm[17,18]hasbeenproposedtoimprovethenon-dominated
front defining the expectations of hypervolume improvements (HVI); it considers the model
uncertainty, as well as the EIs. However, existing studies have not investigated the applicability
ofEHVItomulti-fidelitytechniques. Inthisarticle,amulti-objectiveoptimizationprocessinvolving
multi-fidelity/multi-objectiveEGOisproposedandinvestigatedthroughthesolutionsoftestfunctions
andairfoildesignproblems. Withrespecttoairfoildesignproblems,alow-fidelityfunctionisusedto
constructaglobalmodelthatcanprovidethegloballandscapeofthefunction. Further,ahigh-fidelity
functionisusedtoevaluatelocaldeviations. Theglobalmodel,constructedbyaradialbasisfunction
(RBF)[19,20]basedonadatabase,isevaluatedusinglow-fidelitymodels.Localvariancesarepredicted
using a correlation term of the kriging method. The study involves airfoil design problems with
two and three objectives. The results of the optimization are compared with those of an ordinary
kriging-basedEHVIinvolvingasingle-fidelityapproach.
Appl.Sci. 2017,7,1318 3of21
2. SurrogateModelsforDesignOptimization
2.1. KrigingMethod
Anordinarykrigingmethod[21]predictstheunknownfunctionyˆ(x)as:
yˆ(x) = µ+ε(x), (1)
where µ is global model and ε(x) is a local deviation. The sample points x are interpolated with
Gaussianrandomfunction. ThecorrelationbetweenZ(x )andZ(x )isrelatedtothedistancebetween
i j
thetwocorrespondingpointx andx . Alocaldeviationε(x)isexpressedas:
i j
n
d(x,x ) = ∑ θk|xk−xk|2, (2)
i j i j
k=1
whereθk(0≤ θk ≤ ∞)isthek-thelementofthecorrelationvectorparameterandnisthenumberof
thesamplepoints. Thecorrelationbetweenthepointsx andx isdefinedas:
i j
(cid:2) (cid:3) (cid:2) (cid:3)
Corr Z(x ),Z(x ) =exp −d(x,x ) . (3)
i j i j
Thekrigingpredictioncanbeexpressedas:
yˆ(x) = µ+rTR−1(F−µˆ), (4)
where Z(x ) represents a local deviation from the global model [22], F =
i
(cid:2)f(x ), f(x ), f(x ),..., f(x )(cid:3)T is the value of the evaluated function at X = {x ,x ,x ,...,x },
1 2 3 n 1 2 3 n
(cid:2) (cid:3)
Rdenotesthen×nmatrixwhose(i,j)entryisCorr Z(x ),Z(x ) andristhevectori-thelement:
i j
r (x) =Corr(cid:2)Z(x),Z(xi)(cid:3). (5)
i
µisassumedtobeconstantintheoriginalkrigingmodel,andµˆ isgivenby:
µˆ = [µ,µ,µ,...,µ]T, (6)
whereµisdefinedas:
1TR−1F
µ = . (7)
1TR−11
The unknown parameter, θ, for the kriging model can be estimated via maximum likelihood
estimation(MLE):
n 1
Ln(µ,σ2,θ) = − ln(σ2)− ln(|R|). (8)
2 2
MLEisanm-dimensionalunconstrainednonlinearoptimizationproblem.Inthisarticle,aGA[23]
isusedtosolvethisproblem. Foragivenθ,σ2canbedefinedas:
(F−µˆ)TR−1(F−µˆ)
σ2 = . (9)
n
The mean square error s2(x) at a point x of this function can be calculated using the
followingequation:
(cid:34) (cid:35)
1−1TR−1F
s2(x) = σ 1−rTR−1r+ , (10)
1TR−11
where1denotesann-dimensionalunitvector.
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2.2. HybridSurrogateModelforMulti-FidelityApproach
Inthissection,thehybridsurrogatemodel(Figure1)isproposedforthemulti-fidelityapproach.
Theproposedapproachconstructedthelocaldeviationestimatedbythekrigingmethodandtheglobal
modelapproximatedbytheRBF.ItemploysanRBFtorepresenttheglobalmodel,µ+ fr(x),basedon
adatasetobtainedfromlow-fidelityevaluation. Theproposedapproachcombinesthekrigingmethod
andtheRBFbythefollowingequation:
yˆ(x) = [µ+ fr(x)]+rTR−1(F −µˆ−FR). (11)
h
ThelocaldeviationsrTR−1(F −µˆ−FR)areevaluatedonthebasisofahigh-fidelitydatasetusing
h
thekrigingmethod, where F = (cid:2)f (x ), f (x ), f (x ),..., f (x )(cid:3)T isthevalueofthehigh-fidelity
h h 1 h 2 h 3 h n
function at x = {x ,x ,x ,...,x }; FR = (cid:2)fr(x ), fr(x ), fr(x ),..., fr(x )(cid:3)T, fr(x) the function
1 2 3 n 1 2 3 n
predictedfromthelow-fidelitydatathatpredictedbyRBF,canbeexpressedby(12):
fr(x) = a +a f (x), (12)
0 1 l
where f (x) is a function predicted by an RBF [19,20] using low-fidelity data and a and a are
l 0 1
correlationtermsbetweenthelow-fidelitydataandthehigh-fidelitydata.Further,σ2canbedefinedas:
(F−µˆ−FR)TR−1(F−µˆ−FR)
σ2 = . (13)
n
The unknown parameters (θ,a and a ) for the hybrid surrogate multi-fidelity model can be
0 1
estimatedbyMLE,asexpressedby(8).
TheRBFisusedtoapproximatetheglobalmodelofthehybridsurrogate-model. AnRBFisused
topredictthelow-fidelityfunction(f (x))by:
l
n
f (x) = ∑ w Φ(x−x ) (14)
l m m
m=1
where Φ(x) is an RBF, x is a sample point and w (m = 1,2,3,...,n) is a weighting function. A
m m
multi-quadraticfunctionisappliedasanRBFinthisstudy.Theweightingfunctionw=[w ,w ,w ,...,w ]T
1 2 3 n
isdeterminedfromtheinterpolationconditions:
Aw = F (15)
 
a a ··· a
1,1 1,2 1,j
a a ··· a 
where A =  2...,1 2...,2 ... 2...,j,ai,j = Φ(xi−xj),i =1,2,3,...,n,j =1,2,3,...,n
 
a a ··· a
i,1 i,2 i,j
Here, F = (cid:2)f(x ), f(x ), f(x ),..., f(x )(cid:3)T is the value of the low-fidelity function at
1 2 3 n
X = {x ,x ,x ,...,x }.
1 2 3 n
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Figure1.Schematicillustrationofsingle-fidelityandmulti-fidelitysurrogatemodels.
3. EfficientGlobalOptimizationfortheMulti-ObjectiveProblem
TheprocedureofEGOwiththeordinarykrigingmodelisillustratedinFigure2a.TheEGOprocess
startswiththegeneratedinitialsamples.Inthisstudy,theLatinhypercubesampling(LHS)isemployed.
Sample data are evaluated, and the model is predicted using the kriging method. An arbitrary
optimizationmethodcanbeusedtofindanadditionalpointbymaximizinganEHVI[17,18].TheEHVI
isthefunctionofthehypervolumeimprovement(HVI)combinedwiththeuncertaintyoftheadditional
point. HVI is calculated from the hypervolume improvement of the additional sampling and the
non-dominatedsolutionshowninFigure3a.
TheEHVIatapointcanbeexpressedas:
EHVI[f (x), f (x),..., f (x)] =
1 2 M
(cid:90) fref1(cid:90) fref2 (cid:90) frefM (16)
... HVI[f (x), f (x),..., f (x)]×φ (F )φ (F )...φ (F )dF dF ...dF ,
1 2 M 1 1 2 2 M M 1 2 M
−∞ −∞ −∞
where F denotes the Gaussian random variable N[fˆ(x),sˆ2(x)]. φ(F) is the probability density
i i i i i
functionand f isthereferencevalueusedforcalculatingthehypervolume. Themaximizationof
refi
EHVIisconsideredastheupdatingcriteriontodeterminethelocationofanadditionalsamplepoint.
Inthisstudy,thehypervolumeiscalculatedbasedontheHypEalgorithm[24],whichisanalgorithm
thatusestheMonteCarlosimulation[25]toapproximatethehypervolumeformulti/many-objective
optimization problems. The Monte Carlo simulation is one of the simplest ways to calculate the
hypervolume(HV)formanydimensions,whichisoftendifficulttocalculate.Thus,theHypEalgorithm
usesthebenefitoftheMonteCarlosimulationtocalculatethehypervolume. Thehypervolumeisthe
volumeofthenon-dominatedsolutionsmeasuredfromthereferencepoint. Theschematicillustration
ofthehypervolumeisshowninFigure3b.
Appl.Sci. 2017,7,1318 6of21
(a) (b)
Figure2. Flowchartofefficientglobaloptimization: (a)single-fidelityefficientglobaloptimization
(EGO);(b)multi-fidelityEGO.
(a) (b)
Figure3. Schematicillustrationofhypervolumeimprovementandhypervolume: (a)hypervolume
improvement;(b)hypervolume.
ThebasicideaoftheoriginalEGO,namelyEHVI-basedexplanation,canalsobeappliedtothe
hybridsurrogatemodelexpressedin(10),becauselocaldeviationsareestimatedusingthekriging
method. TheprocedureofEGOwiththehybridsurrogatemodelproposedinthisstudyisshownin
Figure2b. Theproposedhybridsurrogate-model-basedEGOstartsbyacquiringinitialsamplesfor
alow-fidelity/low-costfunction. Thelow-fidelitysampledataareusedtopredicttheglobalmodel;
then,asetofsamplesforahigh-fidelityfunctionisobtained. Thisresultisusedtoestimatethelocal
deviationsusingthekrigingmethod. Then,themulti-fidelitysurrogatemodel,whichcanpredictthe
unknownpointvalue(anapproximationofthehigh-fidelityfunction),isgenerated. AGA[23]isused
tofindthemaximumEHVIpoint,x. Inthisstudy,theroulettewheelmethodwasusedintheselection
process. Blendcrossover(BLX)-0.5[26]wasusedinthecrossoverprocess,anduniformmutation[27]
withamutationrateof0.1wasusedinthemutationprocess.
Appl.Sci. 2017,7,1318 7of21
4. InvestigationofProposedMethodbySolvingTestFunctions
4.1. Formulation
Theproposedmulti-objectivemulti-fidelityEGOisinvestigatedbysolvingtwotestfunctions;
onehastwoobjectivefunctions,andtheotherhasthreeobjectivefunctions. Theresultsarecompared
withthoseofakriging-basedmulti-objectiveEGO.Thehigh-fidelityfunctionisdenotedby f,andthe
low-fidelityfunctionisdenotedby f .
l
Thedefinitionoftwo-objectiveoptimizationproblemfromVanValedhuizen’stestsuite[28]is:
(cid:16) 5 √ (cid:17)
Minimize: f (x ,x ,...,x ) =1−exp −∑(x −1/ 5)2
1 1 2 5 i
i=1
(cid:16) 5 √ (cid:17)
Minimize: f (x ,x ,...,x ) =1−exp −∑(x +1/ 5)2
2 1 2 5 i
i=1
(17)
(cid:16) 5 √ (cid:17)
f (x ,x ,...,x ) =1−exp −∑(0.5x −0.05−1/ 5)2
l1 1 2 5 i
i=1
(cid:16) 5 √ (cid:17)
f (x ,x ,...,x ) =1−exp −∑(0.75x +0.2+1/ 5)2 .
l2 1 2 5 i
i=1
Thedesignspaceofthisproblemisx ,x ,...,x ∈ [−2.5,2.5].
1 2 5
ThedefinitionoftheDTLZ2three-objectiveoptimizationproblem[28]is:
Minimize: f (x ,x ,...,x ) = (1+g)cos(0.5x π)cos(0.5x π)
1 1 2 5 i 2
Minimize: f (x ,x ,...,x ) = (1+g)cos(0.5x π)sin(0.5x π)
2 1 2 5 1 2
Minimize: f (x ,x ,...,x ) = (1+g)sin(0.5x π)
3 1 2 5 1
5
g = ∑(x −0.5)2
i
i=3
(18)
f (x ,x ,...,x ) = (0.5+1.5g )cos(0.6x π)cos(0.4x π)
l1 1 2 5 c i 2
f (x ,x ,...,x ) = (0.3+1.2g )cos(0.4x π)sin(0.6x π)
l2 1 2 5 c 1 2
f (x ,x ,...,x ) = (0.4+1.3g )sin(0.5x π)
l3 1 2 5 c 1
5
g = ∑(0.8x −0.3)2.
c i
i=3
Thedesignspaceofthisproblemisx ,x ,...,x ∈ [0,1].
1 2 5
Ineachinvestigation,10initialhigh-fidelitypoints, f,and150low-fidelitypoints, f ,wereacquired
l
byLHS.Thenumberofiterationswassetto30foreachtestfunction.
4.2. Two-ObjectiveTestFunctionResults
Thesolutionspaceacquiredbytheproposedmulti-fidelity/multi-objectiveEGOiscompared
with that acquired by the single-fidelity/multi-objective EGO as shown in Figure 4. Because the
single-fidelity/multi-objectiveEGOfindslocaloptimumpointsatthebeginningoftheoptimization
process, the solution for additional samples stalls earlier. On the other hand, the developed
multi-fidelity/multi-objectiveEGOcanfindasolutionclosetotheglobaloptimumsolutionofthis
multi-objective optimization problem. The histories of the hypervolumes of the two methods are
comparedinFigure5. Theseresultsalsoindicatethattheproposedmulti-fidelity/multi-objective
EGO provides better solutions, which shows higher diversity because its hypervolume is larger
than that of the single-fidelity/multi-objective EGO. These results also suggest that the proposed
multi-fidelity/multi-objectiveEGOobtainsbettersolutionsthanthesingle-fidelity/multi-objective
Appl.Sci. 2017,7,1318 8of21
EGO,becausethenon-dominatedsolutionsoftheformercandominateallthenon-dominatedsolutions
ofthelatter.Inaddition,comparisonsofthehypervolumesshowthatthesolutionofadditionalsamples
bythesingle-fidelity/multi-objectiveEGOcontinuestostallearlier.
(a) (b)
Figure 4. Initial sampling data and additional sampling data of two-objective test problem:
(a)multi-fidelityapproach;(b)single-fidelityapproach.
Figure 5. Hypervolume comparison of multi-fidelity approach and single-fidelity approach of
two-objectivetestproblem.
To investigate the reason for the superiority of the proposed multi-fidelity/multi-objective
EGO,thecross-validation[29,30]of f and f wascompared, asshowninFigure6. Itcanbeseen
1 2
that the linear regression line nearly coincides with the predicted line in the case of the proposed
multi-fidelity/multi-objectiveEGO.Thus,themulti-fidelitysurrogatemodelachieveshigheraccuracy
thanthesingle-fidelitysurrogatemodel. Morespecifically,theproposedmulti-fidelity/multi-objective
EGOachieveshigheraccuracybecausethedatasetobtainedbylow-fidelityevaluationenablesthe
surrogatemodeltopredictthesolutionintheuncertaintyregion.
Appl.Sci. 2017,7,1318 9of21
Thedevelopedapproach Thesingle-fidelity
(a)
Thedevelopedapproach Thesingle-fidelity
(b)
Figure6.Cross-validationresultsofthetwo-objectivetestproblem:(a)resultof f ;(b)resultof f .
1 2
4.3. Three-ObjectiveTestFunctionResults
Further,allthesamplesacquiredbythetwomethods(theproposedmulti-fidelity/multi-objective
EGOandthesingle-fidelity/multi-objectiveEGO)arecomparedasshowninFigure7. Accordingto
theseresults,theproposedmulti-fidelity/multi-objectiveEGOandthesingle-fidelity/multi-objective
EGOhavesimilarnon-dominatedsolutions. However,theresultsofthemulti-fidelity/multi-objective
EGO are better than those of the single-fidelity/multi-objective EGO because some of the results
of additional sampling by the single-fidelity/multi-objective EGO stall at local optimum points.
The hypervolumes of the two methods are compared in Figure 8. The hypervolume comparison
of the proposed multi-fidelity/multi-objective EGO and single-fidelity/multi-objective EGO
is shown in Figure 9. It suggests that the proposed multi-fidelity/multi-objective EGO gives
better solutions than that of the single-fidelity/multi-objective EGO. These results indicate
that the proposed multi-fidelity/multi-objective EGO has a higher convergence rate than the
single-fidelity/multi-objective EGO. The solution of the proposed multi-fidelity/multi-objective
EGO converges after 18 iterations with a larger hypervolume, whereas the solution of the
single-fidelity/multi-objectiveEGOconvergesafter23iterations.
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f vs. f f vs. f
1 2 1 3
f vs. f
2 3
(a)
f vs. f f vs. f
1 2 1 3
f vs. f
2 3
(b)
Figure 7. Initial sampling data and additional sampling data of three-objective test problem:
(a)multi-fidelityapproach;(b)single-fidelityapproach.