INFLUENCE OF PARAMETERIZATION ANDOPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

B.G. MARINUSRoyal Military Academy/von Karman Institute for Fluid Dynamics,

Brussels/Sint-Genesius-Rhode, Belgium

Keywords: airfoil, parameterization, optimization, aerodynamics

Abstract

The potential influence of different combinationsof parameterization methods and optimizationstrategies on the attainable optimum and the opti-mization cost is evaluated by comparisons of theoptimized airfoils. The optimization strategiesconsist of a single-objective metamodel assistedGenetic Algorithm and multi-objective Differen-tial Evolution both assisted and non-assisted. B-spline and Bézier formulations are used togetherwith a direct and standard description of the air-foil shape. The multi-point optimizations are ap-plied to airfoils in 2D cascades corresponding topropeller geometries.

Nomenclature

Bi,q(.) Basis functionc Airfoil chordC(.) Curve coordinate vectorCd Airfoil drag coefficientCl Airfoil lift coefficientCp Pressure coefficientdD Elemental dragdL Elemental liftdT Elemental thrustdU Elemental torque forcef1, f2 Weighting factorJ Advance ratioM∞ Free-stream Mach numbern, N, q Variablesr RadiusRtip Propeller tip radius

u Running coordinatev∞ Free-stream velocityw. WeightX Coordinate vector

α Local angle of attackβ Local blade angleε, γ Angleηel Elemental efficiencyΣ Overall performanceτ Constraint penaltyΩ. Objective functionω Angular velocityρ Air specific mass

AbreviationsANN Artificial Neural NetworkDOE Design Of ExperimentsGA Genetic AlgorithmMODE Multi-Objective Differential EvolutionRANS Reynolds Averaged Navier-Stokes

1 Introduction

The airfoils selected to shape propeller bladesare of primary importance. The key role in trans-forming the engineering problem of optimizingan airfoil shape in a mathematical problemlies within the parameterization technique usedto represent the airfoil. Most works in airfoiloptimization, ranging from pure aerodynamicto multidisciplinary optimization for turboma-chinery applications or wings, rely on spline

1

B.G. MARINUS

parameterization [1, 2, 3, 4, 5, 6, 7, 8] or Bézierparameterization [9, 10, 11, 12, 13, 14, 15] withtypically 10 to 50 parameters. Unfortunately,few papers assess the influence of the parameter-ization technique [16, 17, 18, 19], in particularb-spline versus Bézier parameterization. More-over, the influence of the combination of aparameterization with an optimization techniqueis seldom analyzed [20, 21]. For these reasons,the present effort investigates the effect of boththe blade parameterization and optimizationtechnique on the attainable optimum and on thecomputational cost of the process. Comparisonsare made in a realistic multidisciplinary opti-mization environment for complex engineeringapplications with a moderate total number ofdesign variables -O(30∼ 40)- so that only a lim-ited amount of parameters (< 10) are dedicatedto the airfoil shape for the remaining parametersare assigned to other design variables such asradial distributions of sweep, twist, thickness andchord in the case of propellers.

2 Airfoil shape parameterizations

The b-spline and Bézier representations havebeen selected among the various formulationsavailable for the designer because they are aflexible technique, providing smooth shapeswith few design parameters. Both are basedon a set of control points that define the shapeof the curve and the convex hull of the controlpolygon, formed by joining neighboring controlpoints, contains the curve. This implies thatthe design parameters (i.e. the position of thecontrol points) control important features of thedesign problem and explains their popularity foroptimization purposes.

Bézier and b-spline representations rely on aset of n+1 control points X0, X1,...,Xn and the ex-pression of the curve C(u) is given in both casesby:

C(u) =n

∑i=0

Bi,q(u)Xi (1)

over the interval u ∈ [0,1] with q computedfrom the required degree of the curve in order toachieve the desired continuity properties. Theydiffer by the basis functions used to compute theBi,q(u) coefficients. Bézier curves with n + 1control points are of degree q = n whereas b-spline curves require a more complex theory withmore information (i.e. the degree of the curveq must be chosen and a knot vector comprisingn+q+1 elements must be defined over the sameinterval with multiplicity of the q + 1 first andlast elements to define a clamped b-spline). Thisdisadvantage is compensated by the fact thatb-splines have the local modification propertytogether with all important properties of Béziercurves such as the convex hull property andat least C 2-continuity if the number of controlpoints n + 1 is chosen accordingly. The localmodification property implies that changinga control point does not globally change theshape of the curve and allows a more localizedshape control. This interesting property [19]is illustrated on figure 1 where curves for bothrepresentations are drawn for two sets of controlpoints differing only in the y-coordinate of onesingle point. Irrespectively of the difference inshape due to the particular formulation, the localcharacter of the modification is clearly illustratedby the absence of any impact of the change of asingle parameter on the trailing part of the curve.

In combination with the previous two formu-lations, two approaches to describe the airfoilshape were investigated . The first one has thecontrol points directly controlling the shape ofthe entire airfoil as in [22, 23] and [7]. As shownon figure 2, this direct parameterization uses 7control points for the suction side and 6 pointsfor the pressure side. This results in only 9parameters because the leading edge and trailingedge locations are fixed and C 2-continuity isensured at these locations through a carefulhandling of endpoint curvature by clamping therelative position of the first and last two controlpoints on the pressure-side with respect to thecontrol points on the suction-side as is suggestedon figure 2. This leaves only 2 parameters con-

2

INFLUENCE OF PARAMETERIZATION AND OPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

(a) Bézier parameterization.

(b) B-spline parameterization.

Fig. 1 Local modification property illustratedwith two sets of control points differing only ina single coordinate ( and −− or and −−).

trolling the ordinate difference between the thirdand fourth control points on the pressure-side.

The second approach consists of a thicknessdistribution that is set upon a camberline inthe way proposed by Abbott et al [24]. Thisstandard NACA-type parameterization is il-lustrated on figure 3. It uses 6 control pointswith 6 parameters to control the shape of thethickness distribution and 4 additional pointswith 2 parameters to control the shape of thecamberline. Although the direct parameteriza-tion has inherently more freedom to generateradical new shapes than the standard one, it isexpected to be less suitable to produce efficientdesigns precisely because of the larger searchspace in terms of shape [19]. Indeed, for acomparable search space in terms of parameters,the direct parameterization delivers more nonairfoil-like shapes even if the parameters arechosen such that a minimum thickness is guaran-teed. Moreover, the standard description, on topof its intrinsic capability of delivering airfoil-likeshapes, has demonstrated its efficiency [16, 25]thanks to a close relationship between the aero-dynamic behavior and the thickness and camberdistributions.

Fig. 2 Direct parameterization and degrees offreedom for the control points .

(a) Thickness and camberline distributions with con-trol points and their respective degrees of freedom.

(b) Airfoil shape and camberline.

Fig. 3 Standard parameterization and degrees offreedom for the control points.

3

B.G. MARINUS

3 Elemental efficiency

The airfoil obtained by one of the parameteri-zations is used to define a propeller blade (fig-ure 4(a)) with a prescribed radial distribution ofchord, thickness and twist. The chord and thick-ness are constant up to 80% radius and the bladehas no sweep. From blade element theory, it iseasily shown that the efficiency of a blade ele-ment (see figure 4(b)) that produces the elemen-tal thrust dT and the elemental torque force dU ,is given by:

ηel =tanγ

tan(γ+ ε)(2)

where

γ = arctanv∞

ωr= arctan

JRtip

πr

and

ε = arctan|dD||dL|

(3)

Equations 2 and 3 reveal the importance ofincreasing the lift-to-drag ratio |dL|/|dD| of allblade elements in order to achieve better overallefficiency. Note that as the twist distributionis prescribed, so is also the angle of attackα(r) = β(r)−γ(r) of the airfoil. The engineeringproblem of finding the optimum airfoil shapeunder transonic conditions at M∞ = 0.75 with aconstant advance ratio J = 3.3, is simplified byoptimizing one single airfoil for three radii (50%,75% and 97.5%). Each radius defines a 2D highpitch cascade at fixed angle of attack as shownon figure 4. The objective of the optimization isto reduce Cd and 1/Cl concurrently for the threecascades.

4 Optimization techniques and objectives

Evolutionary methods have excellent robustnessand a high potential to find the global opti-mum of complex problems involving a largenumber of design parameters associated with adiscontinuous and non-convex objective domain

(a) View of the three radii (50%, 75% and 97.5%)chosen for the multi-point optimization of the airfoilin the corresponding blade passage.

(b) 2D geometry of a blade passage at given radius r.

Fig. 4 Airfoil cascade geometry.

4

INFLUENCE OF PARAMETERIZATION AND OPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

[26, 20, 27]. Two techniques have been investi-gated for the present multi-point optimization: asingle-objective binary-coded Genetic Algorithm(GA) and Multi-Objective Differential Evolution(MODE).

The binary-coded GA [28, 29] closely mim-ics natural evolution by coding the parameters ofan individual (i.e. a blade design) in a binarystring or genotype. The genotype translates intoa particular shape (the phenotype), and its per-formance is evaluated. Members of a generationare obtained through reproduction of two par-ents selected by a tournament in which only thebest of any combination of two individuals is re-tained. The reproduction mechanism consists ofan exchange of parts from their genotype throughcrossover. Then a mutation operator changes arandom bit in the binary string of the offspring.Ultimately elitism is applied as the best individ-ual from the parent population is carried over ifit outperforms all offspring. The selection proce-dure relies on performance estimates cast into anaggregated objective ΩGA to be minimized. It iscomputed by the weighted sum of performanceestimates for the three radii:

ΩGA =3

∑i=1

wi( f1Cd,i + f21

Cl,i)+ τ (4)

where the weights wi are chosen such that the75% radius section has twice more impact on theoptimization result and the factors f1 and f2 en-sure that a drag reduction or a lift increase affectthe objective value in a comparable way. τ is anexternal penalty triggered only if the airfoil hasinsufficient thickness so that the feasability ofthe design is ensured. The estimates are providedby an Artificial Neural Network (ANN). Thepurpose of this model is to reduce the computa-tional cost so that the evolutionary process canbe spread over a large amount of generations.The ANN is trained using a database of RANSsimulations initially obtained for geometriesdefined by a Design Of Experiments (DOE).The purpose of the DOE is to fill the searchspace with carefully selected samples to allowextraction of maximum information about the

Fig. 5 Layout of the single objective binarycoded GA-optimizer with its metamodel.

underlying input-output relationships. The accu-racy of the surrogate model is improved duringthe evolutionary process by recursive training ofthe ANN each time the database is augmentedwith the most promising individual of every1000 generations. The process is repeated for100 iterations using each time the new ANN for1000 generations comprising each 50 individuals(see figure 5).

Differential Evolution has been developed byPrice and Storn [30] and does not require param-eters to be coded in a binary string. A new gener-ation of N individuals is formed by mutation andrecombination among the N parents. Indeed, thevector of parameters of an offspring is a recombi-nation of parameters that are randomly chosen ei-ther from one parent or from a linear combinationof parameters from three other individuals cho-sen randomly in the Pareto front. This last step isalso responsible for mutation. Multi-objective se-lection is performed in the way proposed by Ma-davan [31] through Pareto ranking of parent andchild populations together with respect to the two

5

B.G. MARINUS

objectives to be minimized:Ω1 = ∑

3i=1 wiCd,i

Ω2 = ∑3i=1 wi

1Cl,i

(5)

in which the weights wi have the same value asfor the GA. Note that with this approach, thetrouble of choosing adequately the f factors isrelieved and the external penalty on thicknessis dealt with by a separate constraint function.The population size for the next evolutionarystep is restored to its original size because onlythe N best ranked individuals are passed whilealso looking after the diversity of the populationby rejecting individuals with a too high degreeof similarity. With respect to performanceassessment, two cases have been considered:a two-level case in which the performance isassessed through an ANN-metamodel that istrained with RANS simulations every 1000 gen-erations (figure 6(a)), and a single-level case forwhich only accurate RANS simulations are used(figure 6(b)). The first approach is highly similarto the one used for the GA with the differencethat now the whole population of N individualsis passed through the iteration loop and henceanalyzed by the RANS solver, as suggested onfigure 6. The second approach consists of asingle generation loop and as no metamodel isused, it offers the advantage that it is not subjectto the inaccuracy of the performance estimates.However, less generations (100) are consideredin this case because of the elevated numericalcost associated with the RANS evaluation of thecomplete generation. Nevertheless, it providesa convenient comparison to assess the effect ofthe metamodel. The population size for bothapproaches is 20 and the individual RANSsimulations can be performed concurrently.

5 Results

The RANS simulations are performed by meansof the TRAF 2D solver for transonic flow [32]. Ituses a multi-stage Runge-Kutta time integrationin conjunction with acceleration techniques

(a) Two level approach.

(b) Single level approach.

Fig. 6 Layout of the single and two-level MODE-optimizers.

6

INFLUENCE OF PARAMETERIZATION AND OPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

on a H-type mesh with 100 grid-points onthe suction and pressure-side and 96 points inthe wall-normal direction. The equations arediscretized using a cell-centered finite volumescheme with artificial dissipation of the Jamesontype. Acceleration techniques comprise localtime stepping, variable coefficient implicitresidual smoothing and a full multigrid method.The effect of turbulence is accounted for by theeddy-viscosity hypothesis expressed throughthe two-layer mixing length model of Baldwinand Lomax. A NACA 16-009 airfoil is usedas benchmark because it is commonly used forpropellers.

Table 1 gives an overview of the differentcases together with the number of iterations nec-essary to reach the optimum. Despite convergingearly toward their respective optimum, all opti-mizations were ran for 100 iterations (or gener-ations in the MODE single level case) in orderto ensure that absolute convergence is reached.For cases F , G and J, no solution better than thebenchmark was found after 100 iterations in spiteof modifications to the evolutionary and meta-model settings. For these cases, the optimiza-tion is considered as not having succeeded. Thesenon-optimum airfoils are shown as well for com-pleteness. In each case where the metamodel isused, the database has 64 samples thereby requir-ing 192 calls to the RANS solver. Disregardingthe parameterization type, all optimizations havean identical search domain as each parameter isallowed to fluctuate over the same interval in allcases. The overall performance of an airfoil ismeasured by its Σ-value (given in table 1) whichis the mass-flow averaged area under the Cl/Cdcurve normalized by the corresponding area forthe benchmark:

Σ =

∫ Rtip

Rroot

2πρ∞v(r)Cl(r)Cd(r)

∣∣∣∣airfoil

rdr∫ Rtip

Rroot

2πρ∞v(r)Cl(r)Cd(r)

∣∣∣∣NACA 16-009

rdr(6)

The purpose of mass-flow averaging in theΣ-value is to yield a fair overall performance thattakes the radial distribution of the mass-flow,

hence the radial distribution of the load, intoaccount. Nevertheless, the Σ-values clearly showthat significant overall gains can be achieved bybetter performance at lower radius. All success-ful cases have a better overall performance withgains ranging from 25% to 40%. Case H hasan excellent overall performance mainly due tothe high lift despite a comparatively high drag,as will be shown later. Case A, B and C havea gain close to 27% in spite of a locally worseperformance around 70% radius.

From table 1, the local modification propertyof the b-spline formulation appears to offersome advantage with a gain of over 15% withrespect to the number of calls to the RANSsolver before reaching the optimum. This isespecially true when a surrogate model is usedand a higher number of generations are assessed.These results also underline the effectiveness ofa GA combined with a surrogate model as in thebest case, only 86 calls to the RANS solver areneeded. Considering MODE, the results suggestalso the effectiveness of using a metamodelas a gain of 23% is obtained globally in thatconfiguration.

The off-design performance of the optimumairfoil is assessed by looking at the airfoillift coefficient Cl and drag coefficient Cd insuccessive 2D cascades along the entire bladespan. Each radius corresponds to a given angleof attack and Mach number that are differentfrom the optimization ones. This way, transonicto sonic Mach numbers and various angles ofattack are ascertained. First, lift-to-drag ratiofor all radii are shown in figures 7(a) and 7(b)for the b-spline and Bézier parameterizationsrespectively. These figures reveal that mostof the improvement is achieved at radii below60% and above 80%. Around 75% radius,there is either a small improvement (cases Eand I), no significant improvement (cases A, C,D and H) or even a local deterioration of theperformance (case B). From these graphs, theunsatisfactory performance of cases F , G and Jis also enlighted.

7

B.G. MARINUS

Calls toCase Optimization method Levels Iterations RANS solver Σ-value

Benchmark 1A GA - with ANN - standard b-spline 2 22 258 1.250B GA - with ANN - standard Bézier 2 60 372 1.246C GA - with ANN - direct b-spline 2 20 252 1.329D GA - with ANN - direct Bézier 2 34 294 1.303E MODE - with RANS - standard b-spline 1 49 3132 1.302F MODE - with RANS - standard Bézier 1 − 6192 0.950G MODE - with RANS - direct b-spline 1 − 6192 0.790H MODE - with RANS - direct Bézier 1 57 3612 1.408I MODE - with ANN - standard b-spline 2 37 2412 1.308J MODE - with ANN - standard Bézier 2 − 6192 0.801

Table 1 Cases and number of iterations to optimum solution. (Cases marked with ’−’ did not convergeto an optimum better than the benchmark after 100 iterations.)

Analysis of figures 8(a) and 8(b) revealsthat globally all airfoils obtained with b-splinerepresentation perform better than the bench-mark. Only case E in terms of Cl and cases Aand C in terms of Cd present a shortcoming at apoint where they have been optimized (namely75% radius). But this shortcoming is largelycompensated within the trade-off with the othertwo design points. Additionally, shortcomingsat off-design points, as between 58% and 72%radius for Cl or between 68% and 79% for Cd ,are overwhelmingly compensated by the globalperformance out of those intervals. Case Cdenotes himself by the very poor performancein terms of Cd at low radius combined with aterrific value at high radius, hence high Machnumber. Together with the exceptional Cl over allradii, this explains why this distorted geometry(see figure 11(b)) made its way through theoptimization process. Case G is a good exam-ple of a failed tentative of local optimizationwith the drag coefficient having characteristicminima corresponding to the optimizationpoints while the lift coefficient requirement ismet at all design points. Concerning Bézierparameterization, figure 9(a) indicates that allairfoils have a favorable behavior in terms ofCl , with the design objective met by all but caseB. Here again, the overall Cl performance of

(a) B-spline parameterization.

(b) Bézier parameterization.

Fig. 7 Off-design performance in terms of Cl/Cd .

8

INFLUENCE OF PARAMETERIZATION AND OPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

this airfoil largely offsets this minor flaw. Asclearly visible from figure 9(b), cases F and Jdid not lead to an optimum solution. Despitetheir overall performance, not a single airfoil isable to achieve lower drag at 75% radius. This iscoherent with the observations about the shockstrength (see figure 12) that will be given in thecoming sections. In any way, the lift-to-dragratios moderate the local shortcomings discussedhere and underline the possibility for additionalgains in propeller performance despite possiblylower local performance.

When Cl and Cd distributions are comparedin terms of airfoil parameterization, the strikingpatterns are the better ability of the standard de-scription in producing smooth lift distributions,and the higher potential of b-spline formulationfor a drag reduction over the entire range,including off-design points. The first suggeststhat decomposing the airfoil into a thicknessand a camberline distribution introduces ageometrical stiffness in the problem formulationthat is beneficial to the lift performance. Thiscomplies with the observations made in [16, 24]and [25]. The second underlines the influenceof the local modification property. The moredirect relationship between the parameters andthe shape entitles the optimization process tobetter identify those design variables that havean influence on lift and drag. The results arebetter airfoils with a smooth off-design dragdistribution provided the intrinsic freedom ofthe representation is counter-balanced by, forexample, the geometrical stiffness of the descrip-tion. Finally, it is noted that good improvementscan be yielded with just a few design variablesdisregarding the optimization method.

Figures 10(a), 10(b), 11(a) and 11(b) revealanother aspect of the local control property asthe shapes obtained with b-spline representationshow more local features such as a narrow frontsection or a bump as on figure 11(a). The firstfeature is also reported in the work of Li et al [3]although in cases A, E and I, it does not lead tothe shock location being moved toward the aft

(a) Lift coefficient Cl .

(b) Drag coefficient Cd .

Fig. 8 Off-design performance for b-spline pa-rameterization.

9

B.G. MARINUS

(a) Lift coefficient Cl .

(b) Drag coefficient Cd .

Fig. 9 Off-design performance for Bézier param-eterization.

region of the airfoil, as can be seen from the rightside of figure 12. The second feature is knownto reduce locally the strength of the shock andmight lead to localized optimization as discussedby Drela in [33]. Fortunately, the low numberof design parameters in the present cases doesnot allow for bumps to be generated at locationscorresponding to each working point. Instead,the geometries are a trade-off and offer shockstrength reduction to some extent while retain-ing the necessary smoothness over the entirechord. In contrast, a Bézier parameterization(see figures 10(a) and 10(b)) does not alwayslead to a thinner front part (except for caseD) and is intrinsically impaired of producinglocal bumps. Accordingly, the shock strength isseldom reduced as is illustrated on the right sideof figure 12(a). Except for cases C and G, allpressure distributions at 75% radius are smoothand present the same general behavior. At 97.5%radius, the pressure distributions reveal that mostof the improvement comes from changes on thepressure-side and that schock strength on thesuction-side is never reduced in a significantway. A closer study of figure 11(b) reveals thepeculiarity of geometries C and G. Those shapesare the expression of the local control propertyat its climax because they are not constrainedby the intrinsic properties of the formulationnor by the stiffness introduced with the standarddescription. Case C is the best specimen fromits parameterization, it successfully meets thedesign objectives but comes with a high drag atlow radius. Case G interestingly succeeded inmatching some design objectives but, as will beshown later, has poor off-design performance.This underlines the need for some sort of con-straints on the regularity of the shape be it by thechoice of the formulation or by the choice of thedescription or ultimately by strong restrictionson the parameters range.

6 Conclusions

The influence of the parameterization and theoptimization method on the optimum airfoil is

10

INFLUENCE OF PARAMETERIZATION AND OPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

(a) Standard.

(b) Direct.

Fig. 10 Airfoil shapes obtained with the Bézierparameterization.

(a) Standard.

(b) Direct.

Fig. 11 Airfoil shapes obtained with the b-splineparameterization.

11

B.G. MARINUS

(a) 75% radius.

(b) 97.5% radius.

Fig. 12 Cp distributions of the optimum airfoil(’−−’ optimum airfoil and ’····’ benchmark NACA16-009 airfoil). Non-optimum solutions F , G andJ are also shown.

addressed for 10 combinations in the frameworkof a realistic engineering problem where a sparseamount of variables can be effectively dedicatedto the airfoil itself. Standard parameterizationusing a camberline and thickness distributionoutperforms direct parameterization where theairfoil suction- and pressure-sides are directlydefined. The latter is true both in terms of aero-dynamic performance at design and off-designconditions, and in terms of convergence speed.The results suggest also that b-spline representa-tion globally delivers better aerodynamic resultsthan Bézier formulations. Additionally, b-splineformulations are more efficient in terms of con-vergence because of the more direct relationshipunderlying input (the parameters) and output (theshape). The combination of b-spline formulationwith the standard thickness-camberline descrip-tion makes the parameter set more likely tohave less interactions and allows it to representthose important features of the design problemthat are relevant to lift and drag. This way, thedesign space is searched more efficiently at someexpense of the freedom to generate radically newshapes. Finally, acceleration techniques basedon surrogate models offer interesting advantagesin terms of computational cost in a populationbased approach, without significantly affectingthe attainable optimum. A requisite for this is tosafeguard the accuracy of the metamodel throughtraining in new regions of the search domainwith the best individual(s) of a certain numberof generations. These issues underline theimportance of the choice of the parameterizationmethod on the optimization outcome. Choicesregarding parameterization should not be madelight-handedly but on the contrary, should bethe result of a detailed comparative study that isaimed at the problem to be handled.

Acknowledgments

The author wishes to acknowledge the BelgianDefence Scientific and Technologic Research ini-tiative for its funding of the present study, aswell as the von Karman Institute, in particular

12

INFLUENCE OF PARAMETERIZATION AND OPTIMIZATION METHOD ON THE OPTIMUM AIRFOIL

R. Van den Braembussche and T. Verstraete, forthe resources and expertise made available for thepresent work.

References

[1] M. Drela and M.B. Giles. Viscous-inviscid anal-ysis of transonic and low Reynolds number air-foils. AIAA Journal, 25(10):1347–1355, Octo-ber 1987.

[2] L. Huyse and R.M. Lewis. Aerodynamic shapeoptimization of two-dimensional airfoils underuncertain conditions. Contractor Report CR2001-210648, NASA - ICASE, NASA LangleyResearch Center (USA), January 2001.

[3] W. Li, L. Huyse, and S. Padula. Robust airfoiloptimization to achieve consistent drag reduc-tion over a mach range. Contractor Report CR2001-211042, NASA - ICASE, NASA LangleyResearch Center (USA), August 2001.

[4] T. Ray and H.M. Tsai. Numerical Grid Genera-tion in Computational Field Simulations, chap-ter Some issues in NURBS representation ofairfoil shapes for optimization, pages 815–825.The International Society of Grid Generation,Mississippi (USA), June 2002.

[5] W. Li and S. Padula. Performance trades studyfor robust airfoil shape optimization. In 21stAIAA Applied Aerodynamics Conference, num-ber AIAA 2003-3790, Orlando (USA), June 23-26 2003. AIAA.

[6] M. Nemec and D.W. Zingg. Multipoint andmulti-objective aerodynamic shape optimiza-tion. AIAA Journal, 42(6):1057–1065, June2004.

[7] M.K. Karakasis, K.C. Giannakoglou, and D.G.Koubogiannis. Aerodynamic design of com-pressor airfoils using hierarchical, distributed,metamodel-assisted evolutionary algorithms. In7th European Conference on Turbomachinery,Fluid Dynamics and Thermodynamics, Athens(Greece), March 5-8 2007.

[8] A.M. Morris, C.B. Allen, and T.C.S. Ren-dall. CFD-based optimization of aerofoils us-ing radial basis functions for domain elementparametrization and mesh deformation. Interna-tional Journal for Numerical Methods in Fluids,58:827–860, March 2008.

[9] R.A.E. Mäkinen, J. Periaux, and J. Toivanen.Multidisciplinary shape optimization in aerody-namics and electromagnetics using genetic al-gorithms. International Journal for NumericalMethods in Fluids, 30:149–159, 1999.

[10] C. Poloni. Notes on numerical fluid mechan-ics, volume 68, chapter Multi-objective aerody-namic optimisation by means of robust and ef-ficient genetic algorithm, pages 1–24. Vieweg,Wiesbaden (Germany), 1999.

[11] Z.L. Tang and J-A. Désidéri. Towards self-adaptive parametrization of Bézier curves forairfoil aerodynamic design. Research ReportINRIA 4572, Institut National de Recherche enInformatique et en Automatique, Sophia An-tipolis, September 2002.

[12] S. Peigin and B. Epstein. Robust handling ofnon-linear constraints for GA optimization ofaerodynamic shapes. International Journal forNumerical Methods in Fluids, 45:1339–1362,2004.

[13] S. Peigin and B. Epstein. Robust optimiza-tion of 2D airfoils driven by full Navier-Stokescomputations. Journal of Computer and Fluids,33:1175–1200, February 2004.

[14] B. Epstein, S. Peigin, and S. Tsach. A new ef-ficient technology of aerodynamic design basedon CFD driven optimization. Aerospace Scienceand Technology, 10:100–110, 2006.

[15] D. Lopez, C. Angulo, and L. Macareno. Animproved meshing method for shape optimiza-tion of aerodynamic profiles using genetic al-gorithms. International Journal for NumericalMethods in Fluids, 56:1383–1389, 2008.

[16] S. Obayashi. Airfoil shape parameterization forevolutionary computation. In Lecture Series onGenetic Algorithms for Optimisation in Aero-nautics and Turbomachinery, number LS 2000-07, Sint-Genesius-Rode (Belgium), 2000. vonKarman Institute for Fluid Dynamics.

[17] J.A. Samareh. Survey of parameterization tech-niques for high-fidelity multidisciplinary shapeoptimization. AIAA Journal, 39(5):877–884,May 2001.

[18] H-Y. Wu, S. Yang, F. Liu, and H-M. Tsai.Comparison of three geometric representationsof airfoils for aerodynamic optimization. In16th AIAA Computational Fluid Dynamics Con-

13

B.G. MARINUS

ference, number AIAA 2003-4095, Orlando(USA), June 23-26 2003. AIAA.

[19] W. Song and A.J. Keane. A study of shapeparameterisation methods for airfoils optimisa-tion. In 10th AIAA/ISSMO MultidisciplinaryAnalysis and Optimization Conference, numberAIAA 2004-4482, Albany (USA), August 30 -September 1 2004. AIAA.

[20] S. Obayashi and T. Tsukahara. Comparison ofoptimization algorithms for aerodynamic shapedesign. AIAA Journal, 35(8):1413–1415, Au-gust 1997.

[21] J. Hájek. Parameterization of airfoils and itsapplication in aerodynamic optimization. InProceedings of the 16th Annual Conference ofDoctoral Students - WDS 2007, pages 233–240,Prague, June 5-8 2007.

[22] L. Huyse, S.L. Padula, R.M. Lewis, and W. Li.Probabilistic approach to free-form airfoil shapeoptimization under uncertainty. AIAA Journal,40(9):1764–1772, September 2002.

[23] S. Painchaud-Ouellet, C. Tribes, J.Y. Trépanier,and D. Pelletier. Airfoil shape optimization us-ing a nonuniform rational b-splines parameter-ization under thickness constraint. AIAA Jour-nal, 44(10):2170–2178, October 2006.

[24] I.H. Abbott, A.E. von Doenhoff, and L.S.Stivers Jr. Summary of airfoil data. Report R-824, NACA, NACA-Langley aeronautical labo-ratory (USA), 1945.

[25] A.J. Keane and P.B. Nair. Computational ap-proaches for aerospace design. John Wiley &Sons, Chichester (United Kingdom), 2005.

[26] D. Dasgupta and Z. Michalewicz. Evolution-ary Algorithms - An overview, chapter Introduc-tion, pages 3–28. Springer-Verlag, Berlin (Ger-many), 1997.

[27] A. Shahrokhi and A. Jahangirian. Airfoil shapeparameterization for optimum navier-stokes de-sign with genetic algorithm. Aerospace Scienceand Technology, 11:443–450, 2007.

[28] D. Goldberg. Genetic algorithms in search,optimization and machine learning. Addison-Wesley, Stuttgart (Germany), 1989.

[29] T. Verstraete. Multidisciplinary turbomachin-ery component optimization considering perfor-mance, stress and internal heat transfer. Phdthesis, von Karman Institute for Fluid Dynamics

- Universiteit Gent, Rhode-Saint-Genèse - Gent(Belgium), June 2008.

[30] K. Price and R. Storn. Differential evolution,April 1 1997.

[31] N.K. Madavan. Multiobjective optimization us-ing a Pareto Differential Evolution Approach.In Proceedings of the Congress on Evolution-ary Computation, volume 2, pages 1145–1150,Honolulu (USA), 2002.

[32] A. Arnone and R.C. Swanson. A Navier-Stokessolver for turbomachinery applications. ASMEJournal of Turbomachinery, 115(2):305–313,April 1993.

[33] M. Drela. Frontiers of Computational Fluid Dy-namics, chapter Pros and cons of airfoil opti-mization, pages 363–381. World Scientific, Sin-gapore, 1998.

Copyright Statement

The authors confirm that they, and/or their company ororganization, hold copyright on all of the original ma-terial included in this paper. The authors also confirmthat they have obtained permission, from the copy-right holder of any third party material included in thispaper, to publish it as part of their paper. The authorsconfirm that they give permission, or have obtainedpermission from the copyright holder of this paper, forthe publication and distribution of this paper as part ofthe ICAS2010 proceedings or as individual off-printsfrom the proceedings.

14