Airfoil/Wing Optimization

Thomas A. ZangSystems Analysis and Concepts Directorate, NASA Langley Research Center, Hampton, VA, USA

1 Introduction 1

2 Optimization Formulation 2

3 Shape Definition 4

4 Mesh Generation 7

5 Gradient Computation 7

6 Optimization Using CFD 8

7 Disclaimer 10

Acknowledgments 10

Notes 10

References 10

1 INTRODUCTION

This chapter focuses on techniques for improving the aerody-namic characteristics of an aerospace vehicle through refine-ment of the vehicle’s shape. Methods for aerodynamic shapeoptimization have progressed through increasingly complexaerodynamic analysis tools – roughly speaking, for linearaerodynamics (e.g., panel methods), transonic small distur-bance equations, and full potential equations in the 1970s; forlinear aerodynamics with boundary-layer corrections in the1980s; for Euler equations from the 1980s through the mid-1990s; for Navier–Stokes equations on structured meshesin the 1990s; for Euler and Navier–Stokes equations onunstructured meshes from the late 1990s onward; and for

Encyclopedia of Aerospace Engineering.Edited by Richard Blockley and Wei ShyyThis article is a US Government work and is in the public domainin the United States of America. Copyright c© 2010 John Wiley& Sons Ltd in the rest of the world.

time-dependent flows, most recently. Although the discus-sion in this chapter is confined to external flows, most ofthe discussion is equally applicable to internal flows, suchas those through aircraft engines. Readers interested in sum-maries of the important developments (and contributors) inthis field should consult review articles such as those byLabrujere and Slooff (1993), Newman et al. (1999), Reutheret al. (1999), and Mohammadi and Pironneau (2004), as wellas the hundreds of references therein.

Figure 1 illustrates the key processes and variables foraerodynamic shape optimization. (The gradient variables areshown in parentheses, as they are only relevant for gradient-based optimization methods.) The Optimization processfeeds a set of design variables x (a vector of length d) tothe Shape Definition component, which provides a mathe-matical description s of the surface. The surface descriptionis used by the Mesh Generation process, which produces thecomputational mesh y used by the Aerodynamic Analysiscomponent. The analysis output is the state variable q (andits gradient ∇q). The Objective and Constraint Evaluationprocess supplies the objective function f , the inequality and

Objectiveand constraint

evaluation

Shapedefinition

Aerodynamicanalysis

Optimizationx

y

q

f, gj, hk

(∇f, ∇gj ,∇hk)

Meshgeneration

s

(∇q)

Figure 1. Key processes and variables for aerodynamic shapeoptimization.

2 Aerospace System Optimization

equality constraints g and h (perhaps along with their gradi-ents) back to the Optimization process.

Those components of the optimization process that are ofspecial importance to aerodynamic shape optimization are theoptimization formulation (objectives and constraints), shapedefinition, mesh generation, and gradient computation (forthose methods that utilize gradient information). These top-ics are covered in that order. The initial expository materialon the optimization formulation is given in the context oflinear aerodynamics. Thereafter, the emphasis is on the im-portant considerations for aerodynamic shape optimizationusing the Euler or Navier–Stokes equations, that is, what arecommonly called computational fluid dynamics (CFD) tools.The chapter concludes with some considerations for the useof CFD in aerodynamic shape optimization. The discussionis restricted to use of optimization for improving an existingdesign, that is, optimization that starts from an existing base-line design and is constrained to maintain the same topologyfor the shape.

2 OPTIMIZATION FORMULATION

The state variable q is a function of the spatial coordinate,which is denoted by the vector x. (The spatial variable hasthe unusual hat because in this volume the symbol x – with-out the hat – is reserved for the vector of design variables.)The state variable is a solution of the governing equations foran appropriate conceptual model for the flow (compressibleor incompressible, viscous or inviscid, turbulent or laminar,nonlinear or linear). We write the governing equations gener-ically as

r(q(x), x) = 0 (1)

The optimization problems that are considered in thischapter have the form

minx

f (x; q(x; x), x) (2)

subject to

gj(x; q(x; x), x) ≤ 0 , j = 1, . . . , m (3)

hk(x; q(x; x), x) = 0 , k = 1, . . . , p (4)

xiL ≤ xi ≤ xiU , i = 1, . . . , n (5)

where x is the vector of design variables, f is the objec-tive function, gj and hk are inequality and equality constraintfunctions, respectively, and xiL and xiU are upper and lower

bounds on the i-th design variable. The design variables x

have been added to the argument list of the state variable toemphasize its parametric dependence upon them. Likewise,we now write the state equation (1) as

r(q(x; x), x; x) = 0 (6)

We necessarily have hk = rk for k = 1, . . . , dimension of r,as each component of r yields a constraint. Because the statevector is defined implicitly, as in equation (6), the objectiveand constraint functions in equations (2)–(3) have a depen-dence upon q and x. For aerodynamic shape optimization, thedesign variables characterize the shape of the airfoil, wing oraircraft.

To illustrate the basic principles, we consider optimizationof airfoils in the four-digit NACA series (see, e.g., Abbott andvon Doenhoff, 1959, Chapter 6). These airfoils are describedby the maximum camber (m), the distance of the locationof the maximum camber (p) from the leading edge, and themaximum thickness (t); all of these are given as fractions ofthe chord c. These parameters are illustrated in Figure 2a.For a four-digit NACA airfoil, say, a NACA d1d2d3d4 airfoil,the first digit d1 is m (in hundredths), the second digit is p

(in tenths), and the last two digits are t (in hundredths). Thus,a NACA 2416 airfoil has a maximum camber of 0.20c locatedat x = 0.10c, and a maximum thickness of 0.16c, where c isthe airfoil chord. Figure 2b is indicative of internal structuresthat constrain the geometry; such important constraints areignored in the present illustration.

The conceptual model used for the airfoil analysis in thisexample is inviscid, irrotational, incompressible, constant-density flow. The velocity potential φ can be used as the(scalar) state variable. The mathematical model is given bythe Laplace equation

r = ∇ · ∇φ (7)

where ∇ denotes the gradient operator (with respect to x).The Laplace equation is subject to a homogeneous Neumannboundary condition (∇φ · n = 0, where n is unit vector nor-mal to the surface) on the airfoil surface, an homogeneousDirichlet condition (φ = 0) at infinity, and a Kutta condition(upper and lower surface pressures match) at the trailing edgeThe computational model is based upon a panel (boundary-element) method1. This very simple model ignores many im-portant effects such as viscosity, compressibility, vorticity,and nonlinearity but provides a useful illustration of somebasic concepts of aerodynamic shape optimization.

For this example, the design variables x = (m, p, t), withthese three NACA airfoil parameters treated as continuousvariables; their upper and lower bounds are taken to bexL = (0.0, 0.1, 0.0) and xU = (0.1, 0.5, 0.2), respectively. In

Airfoil/Wing Optimization 3

Chord c(a) (b)

Mean camberline Chord

line

p c

m ct cLeading edge

Trailingedge

z

x

Fuel tank Stiffeners and ribs

Spars

Figure 2. (a) National Advisory Committee for Aeronautics (NACA) four-digit airfoil parameters; (b) representative internal structures(courtesy of J. A. Samareh).

all cases, the angle of attack is chosen to be α = 1◦. The keyperformance outputs are the (section) lift, drag, and pitchingmoment coefficients, cl, cd and cm, respectively. (The mo-ment reference center is the quarter-chord point.) Four rep-resentative cases are considered: (i) maximize lift (f = −cl)with no constraints, (ii) maximize lift with inequality con-straints on the pitching moment (g1 = −cm − 0.04) and drag(g2 = cd − 0.02), (iii) minimize drag (f = cd) with inequal-ity constraints on the pitching moment (g1 = −cm − 0.04)and lift (g2 = cm − 0.30), and (iv) minimize lift/drag (f =−cl/cd) with inequality constraints on the pitching moment(g1 = −cm − 0.04) and lift (g2 = cl − 0.30. (There are noequality constraints in these examples.) The gradient-basedoptimizations employ Sequential Quadratic Programming,using finite differences to compute the gradients and theBroyden–Fletcher–Goldfarb–Shanno (BFGS) method to ap-proximate the Hessian2. The starting point for the optimiza-tions is x0 = (xL + xU)/2.

The optimal solutions to these four problems are given inTable 1. The pitching moment constraint is active in cases (ii)and (iv). The drag constraint is also active in case (ii). The liftconstraint is active in case (iii). The corresponding airfoils areshown in Figure 3. The key performance parameters are alsoincluded in the table. The impact of constraints is especiallydramatic in the contrast between the results for the first twocases (maximization of lift). The conceptual model for thissimple example has neglected several important physical ef-fects, most noticeably those of viscosity and compressibility.The resulting “optimal” designs are by no means representa-tive of airfoils that would be useful in practice.

0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2

x

z

max cl

max cl [gi: cm and cd ]

min cd [gi: cm and cl ]

min cl / cd [gi: cm and cl ]

Figure 3. Optimal shapes for the NACA four-digit airfoil example.

Another widely used objective function is the difference(in the least squares sense) between the surface pressuredistribution and a target pressure distribution:

f (x) =∫

S

|| p(x; x) − ptarget(x) ||2 (8)

where the integral is taken over the airfoil or wing surfaceS of interest, p is the pressure in the flow produced by anairfoil described by the design variables x, and ptarget is thetarget pressure. The basic concept is illustrated in Figure 4.The solid line is the pressure coefficient (Cp) for the present(baseline) airfoil. The dotted line is the desired (target) pres-sure distribution. Many rules have been built up over the

Table 1. Design variables and optimal solutions for the airfoil examples.

Case f gi m p t min f cl cd cm

1 −cl — 0.10 0.50 0.20 −1.53 1.529 0.0030 −0.3212 −cl cm and cd 0.029 0.10 0.15 −0.44 0.436 0.0020 −0.0403 cd cm and cl 0.016 0.30 0.10 0.0010 0.300 0.0019 −0.0364 −cl/cd cm and cl 0.028 0.11 0.10 −369 0.403 0.0011 −0.040

4 Aerospace System Optimization

Figure 4. Baseline and target pressure distributions (courtesy ofR.L. Campbell).

years for choosing target pressure distributions to achievedesired performance measures. Methods for matching partic-ular flow-field characteristics rather than directly minimizinga global objective function such as drag are referred to asinverse design methods; the more conventional alternative iscalled a direct optimization method. One could apply formaloptimization to the inverse problem with the objective func-tion given by equation (8). However, there is a wide varietyof inverse design methods that “minimize” the objective inequation (8) much faster than direct optimization methods.An example of one such method is provided in Section 6 (seeLabrujere and Slooff, 1993 for descriptions of many others).

In practical applications, numerous geometric constraintsare needed for obtaining an acceptable result, for example,requiring the external shape to accommodate internal struc-tures (see Figure 2), minimal radius of curvature of the lead-ing edge, and minimal angle of the trailing edge. Furthermore,aerodynamic shapes are required to operate over a range ofconditions, especially in Mach number. This is addressedwith a multi-objective optimization formulation (commonlyreferred to as multipoint optimization in the aerodynamicoptimization community) (see Drela, 1998 for an extensivediscussion of constraints and objectives for airfoil optimiza-tion).

The NACA airfoil family used for the example above wasconvenient for this demonstration because the parameteriza-tion guaranteed a simple airfoil shape. Furthermore, the de-sign variables corresponded directly with physical quantitiesfor which experienced aerodynamicists have considerable in-tuition. However, the optimization problems of interest formany decades now require consideration of a much broaderrange of shapes than those covered by the NACA airfoil fam-ilies. In the following section, we cover the fundamentals ofparameterizations that can represent a more general range ofshapes.

3 SHAPE DEFINITION

Aerodynamic shape optimization methods using nonlinearanalysis tools are faced with significant challenges in shapeparameterization, volume mesh generation, and sensitivityanalysis. Although these components are routine for manyoptimization problems in other disciplines, they are nontriv-ial for aerodynamic shape optimization methods that yieldrealistic shapes robustly and efficiently. The surface of anaircraft, a wing, or even an airfoil is an infinite-dimensionalobject, but it must be parameterized as a finite-dimensionalobject. (The parameters of the shape correspond to the designvariables for aerodynamic shape optimization.) The shapeparameterization should compactly cover the design spaceof interest while presenting no undue difficulties. In partic-ular, parameterization-induced waviness and discontinuitiesin (the slope and curvature of) the surface are undesirable.These features are undesirable because they present manu-facturing and CFD analysis difficulties (since flow fields arequite sensitive to such discontinuities). Sometimes, optimiz-ing the shape of the entire surface is desired, but other timesoptimization is only applied to a portion of the surface. Thealternatives summarized below apply in both cases.

The usual custom for aerodynamic shape parameterizationis to represent the surface as a baseline surface plus a pertur-bation expressed as an expansion in terms of basis functions.Consider the case of airfoils. The surface S is one dimen-sional, and the basis functions bk(ξ) depend upon a singlecoordinate ξ. The expansion is given by

s(ξ; x) =d∑

k=1

xkbk(ξ) (9)

with the coefficients xk in the expansion serving as the de-sign variables. The perturbation s(ξ; x) may be applied towhatever functions are used for the surface definition, forexample, upper or lower airfoil surface, mean camber line,airfoil thickness. A variety of basis functions have been used,some global and some local. Since the leading and trailingedges of airfoils are fixed in most design problems, havingbasis functions that vanish at both end points is desirable.

The sine functions are one option for a complete set ofglobal basis functions. Drela (1998) recommends using themin the form

bk(ξ) = 1

ksin(πkξ) (10)

They form an orthogonal set for ξ ∈ [0, 1]. The sine basisfunctions are illustrated in Figure 5a. Suitable combinationsof orthogonal polynomials can also be chosen to ensure that

Airfoil/Wing Optimization 5

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x x

bk

(a)

k=1k=2k=3k=4

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

bk

(b)

k=2k=3k=4k=5

Figure 5. Examples of global basis functions: (a) sine; (b) Legendre.

the basis functions vanish at the end points. For example, onecan use

bk(ξ) = 1

2√

(k + 1)(Lk−1(ξ) − Lk+1(ξ)) for k ≥ 1 (11)

for ξ ∈ [−1, 1], where Lk(ξ) is the Legendre polynomial ofdegree k. The set of basis functions given by equation (11) isnearly orthogonal – the inner product of bk and bl vanishesexcept for l = k − 2, k, k + 2. The Legendre basis functionsare illustrated in Figure 5b. These form a complete set ofbasis functions. They are advantageous for approximatingfunctions with a high degree of smoothness (more than, say,four continuous derivatives), because they exhibit much fasterconvergence than the sine functions in such cases (see Canutoet al., 2006, Chapter 2).

A noteworthy set of local basis functions specificallychosen for airfoil parameterization are the Hicks–Henne

(Hicks and Henne, 1978) functions, which are defined by

b(ξ) = {sin[πξlog(1/2)/ log(t1)]}t2 (12)

with the domain now again normalized to ξ ∈ [0, 1]; t1 con-trols the location of the peak and t2 its width. Figure 6aillustrates some of these functions. Although these functionslack the completeness property possessed by the sine func-tions and the orthogonal polynomials, they have proven veryuseful in the hands of skilled designers.

A more conventional choice of local basis functions aresplines, which are piecewise polynomials. Figure 6b illus-trates the case of cubic B-splines on uniformly distributedknots (denoted by the dots on the x-axis); the curves are la-beled by the x coordinate of the center of the spline. Unlikethe global basis functions shown in Figure 5, the underly-ing B-spline basis functions effect only local changes in theshape. On the other hand, they have only a finite number

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

x x

bk

(a) (b)

t1=0.25, t2=0.5

t1=0.50, t2=1.0

t1=0.75, t2=1.5

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

bk

c=0.3c=0.5c=0.7

Figure 6. Examples of local basis functions: (a) Hicks–Henne; (b) B-spline.

6 Aerospace System Optimization

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

x x

z z

(a) (b)

BaselineCamber perturbation

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

BaselineThickness perturbation

Figure 7. NURBS-based perturbations for (a) airfoil camber; (b) thickness.

of continuous derivatives, namely, p − 1 continuous deriva-tives for splines of order p. Hence, if p > 2, then the shapeperturbation, along with its first and second derivatives, iscontinuous. This includes the cubic B-splines (p = 3) shownin Figure 6b. A generalization of B-splines, called nonuni-form rational B-splines (NURBS), has become a fairly widelyused parameterization, particularly for complex shapes, suchas full aircraft configurations. NURBS represent functions(in this case surfaces) as a rational function, with both thenumerator and denominator consisting of B-spline expan-sions. Piegl and Tiller (1996) provide a comprehensive de-scription of NURBS. See Samareh (2001) for a summary oftheir mathematical description and a short survey of vari-ous uses in aerodynamic shape optimization. An importantadvantage of a NURBS representation for the shape is thatthey are compatible with most computer-aided design (CAD)

systems. Figure 7 illustrates airfoil deformations based onNURBS expansions of the airfoil camber and thickness.

Using the locations of the surface mesh points as designvariables is yet another approach. The main challenge hereis picking appropriate geometric constraints and/or filteringprocedures to ensure a smooth surface. See Li and Krist(2005) for one among many approaches to surface smooth-ing.

For the parameterization of wings, the same approachesapply, but, of course, there are now additional types of designvariables. Some representative wing parameters with a directaerodynamic interpretation are illustrated in Figure 8. Thesymbols indicate locations where the parameters are defined.The root chord, tip chord, semi-span, and leading edgesweep angle are planform variables; these are usually fixedearly in the design process. The airfoil section parameters

Camber and thickness

Twist and shear

Leading edge sweep angle

Root chord

Tip chord

Semi-span

y

x

Figure 8. Typical design variables for wings.

Airfoil/Wing Optimization 7

(camber and thickness) as well as the wing twist and shearare often the focus of wing shape design. The twist angle ata given airfoil section is the difference between the airfoilsection incident angle at the root and the incident angle ofthat airfoil section. Similarly, the shear (dihedral) is the dif-ference between the airfoil leading edge z coordinate for theroot and the z coordinate for the particular airfoil section.

4 MESH GENERATION

Unlike panel methods, for which a surface mesh is sufficient,CFD methods require a volume mesh for numerical solutionof the state equation. For compressible flow, the state variableq is given by

q = (ρ, ρu, ρE)T (13)

where ρ is the density, u the velocity, and the (specific) totalenergy E = e + 1

2u · u , where e denotes the (specific) inter-nal energy. For steady, inviscid flow described by the Eulerequations, we have

r(q(x), x) = ∇ · F (14)

where the flux F = (ρu, ρuuT, ρuE + pu)T. The Navier–Stokes equations have the same form as equation (14) butwith additional terms in the flux function. Figure 9 illustratesthe surface and symmetry plane portions of an unstructured

Figure 9. Surface mesh for wing optimization in the presence ofthe fuselage. Reproduced with permission from Nielsen and Park(2006) c© AIAA.

CFD volume mesh for a wing–fuselage configuration. Thecoordinates of these volume mesh-points are denoted by y inFigure 1.

For aerodynamic shape optimization, changes to the sur-face of the airfoil resulting from design variable changesrequire adjustments to the volume mesh as well, and theseadjustments need to occur automatically. Nowadays, com-putational meshes suitable for even viscous CFD analysiscan be generated automatically from the surface shape defi-nition for airfoils and wings, but this state has proven to bevery elusive for complex aircraft configurations subjected toNavier–Stokes analyses – some type of user intervention istypically needed if the mesh generation must be performedab initio. Thompson, Soni and Weatherill (1998) provide anextensive description of methods for CFD mesh generation.Moreover, as discussed in the next section, analytically basedgradients are highly desirable, and these are not generallyavailable from ab initio mesh generation packages. Hence,many aerodynamic shape optimization processes use specialprocesses that lend themselves to analytically based gradi-ents for obtaining the volume mesh as a perturbation uponthe mesh associated with the baseline shape. Samareh (2001)contains an overview of the various approaches to volumemesh generation.

5 GRADIENT COMPUTATION

For gradient-based optimization methods, the derivatives ofthe objective and constraints with respect to the design vari-ables (terms in parentheses in Figure 1) are needed. For non-linear CFD methods, computation of the gradients3 usingfinite differences has several disadvantages: (i) computationof each gradient requires another full CFD solution sincethe equations are nonlinear; (ii) extensive trial and error isnecessary to choose the appropriate step size for each de-sign variable; and (iii) for some difficult CFD problems, theanalysis code may be simply unable to converge to the levelneeded for accurate finite-difference gradients. Computationof these derivatives through quasi-analytical means is pre-ferred. (The adjective quasi-analytical is used because theequations for the gradients are derived analytically but solvednumerically.) In order to distinguish the state variable and thestate equations, which are functions, from their discrete rep-resentations, which are vectors, we use the symbols q andr, respectively, for the latter; similarly y is the vector of themesh-point coordinates. For a problem with N mesh-points,the length of q and r is 5N since q has five components ateach point, and the length of y is 3N. (Boundary conditionsmay slightly alter these lengths.)

8 Aerospace System Optimization

Recall that a change in a design variable is propagatedthrough the shape definition and mesh generation processes(Figure 1), with the computational mesh y depending uponx. In particular, the gradient of, say, the objective functionwith respect to the particular design variable xj is given by

df

dxj

= ∂f

∂xj

+5N∑k=1

(∂f

∂qk

∂qk

∂xj

)+

3N∑l=1

(∂f

∂yl

∂yl

∂xj

)(15)

Evaluation of this requires determination of the three explicitpartial derivatives of f plus the ∂qk/∂xj and ∂yl/∂xj terms.The first term on the right-hand side is usually straightfor-ward to compute. (For the airfoil example in Section 2, allthese derivatives vanish, as the design variables do not appearexplicitly in the expressions for the lift, drag, and pitchingmoment.) Likewise, an analytical expression can typicallybe derived straightforwardly and then evaluated numericallyfor ∂f/∂qk and ∂f/∂yl. The ∂yl/∂xj terms represent the influ-ence of the design variables upon the computational mesh.The surface definition techniques illustrated for airfoils inFigures 5–7, as well as some mesh perturbation techniques,lend themselves to analytical evaluation of these terms.

Computation of the ∂qk/∂xj terms is more involved. Con-sider the state equation given by equation (6). Its solutionq, considered as a function of the design variables x, satis-fies dr/dx = 0. Hence, implicit differentiation of the stateequation yields the following expression:

5N∑k=1

(∂ri

∂qk

∂qk

∂xj

)= −

(∂ri

∂xj

)−

3N∑l=1

(∂ri

∂yl

∂yl

∂xj

)(16)

Equation (16) is linear in the desired ∂qk/∂xj term. The size ofthe linear system is 5N × 5N. CFD computations for wingstypically use O(106) mesh-points, and complex configura-tions can easily take upwards of 108 mesh-points. Hence, thelinear system is extremely large in three dimensions – so largethat direct solution methods are impractical. An efficient it-erative solution method is surveyed in Newman et al. (1999)and described in detail in Korivi et al. (1994).

For most aerodynamic shape optimization problems, thenumber of design variables far exceeds the number of ob-jectives and constraints. Hence, the adjoint approach for ob-taining the gradients is much more efficient than the directapproach described above. Details may be found, for exam-ple, in Newman et al. (1999) and Reuther et al. (1999). Bothcontinuous and discrete adjoints have been used in aero-dynamics. (The phrase continuous adjoint refers to one inwhich the continuous adjoint equations are first derived fromthe governing partial differential equations (PDEs) and thendiscretized; a discrete adjoint is one in which the adjoint

equations are derived from the algebraic equations resultingfrom the discretization of the PDEs.) See studies by Newmanet al. (1999) and Reuther et al. (1999) for extended lists ofreferences to the early experiences of various groups withboth approaches. Some of the subtle issues that must be ad-dressed for consistent aerodynamic gradients are the treat-ment of boundary conditions and mesh gradients (see Gileset al., 2003; Nielsen and Park, 2006 for detailed recommen-dations). An approach to obtaining second-order derivativesby combining direct and adjoint techniques is described bySherman et al. (1996).

Yet another alternative to computing the gradients is theuse of complex variables, which we illustrate on a simplereal-valued scalar function f . In this method, the function isevaluated at the complex point x + iε, where i denotes theimaginary unit, for small ε. A Taylor expansion yields

f (x + iε) = f (x) + (iε)df

dx+ O(ε2) (17)

Thus, for small ε, the real part of f (x + iε) is a good ap-proximation to f (x), and the imaginary part divided by ε isa good approximation to the gradient df/dx. Note that thisapproach does not involve the subtraction of nearly equalquantities, as occurs for gradients computed via finite dif-ferences. Hence, round-off errors are not a concern for thecomplex variable approach. Provided that the source code isavailable, implementation of this approach can require littlemore than changing the type of variables from real to com-plex. A prototypical use of this approach for aerodynamicshape optimization is given by Nielsen and Kleb (2006).

6 OPTIMIZATION USING CFD

Over the past several decades, a broad assortment of directoptimization and inverse design methods has been applied toaerodynamic shape optimization using CFD. The principaldistinctions between the approaches are (i) use of global per-formance measures such as lift, drag, and pitching moment inthe objectives and constraints versus inverse objectives suchas matching a prescribed pressure distribution, (ii) gradient-based methods versus methods using only function values,and (iii) having the optimizer invoke the aerodynamic anal-ysis tool itself versus having it rely upon surrogate modelsconstructed off-line.

Most of the review articles cited in Section 1 concentrateon direct optimization of global performance measures usinggradient-based methods that directly call CFD analysis tools.Several difficulties with this approach can arise. For exam-ple, the analysis code may produce solutions that result in

Airfoil/Wing Optimization 9

objective functions and constraints that are contaminated bynumerical noise. Such noise gives the appearance of manylocal extrema and can render gradient-based methods un-workable. Another potential difficulty is that there may beportions of the desired design space for which the mesh gen-eration or the analysis process simply fails. At an even morefundamental level, the problem formulation (objective andconstraints) itself can omit important considerations with theresult that the optimization produces a design that is unreal-istic from the perspectives of other disciplines. See the citedreview articles for extensive discussion of these issues.

Nevertheless, many groups have used gradient-basedoptimization of global performance measures without re-course to surrogate models to produce impressive CFD-basedoptimization results for multi-element airfoils, wings, wing–fuselage configurations, and even more complex shapes.Figure 10, from Nielsen and Park (2006), shows the objec-tive function convergence for unconstrained maximization ofthe lift-to-drag ratio of the vehicle illustrated in Figure 9 attransonic conditions using a Navier–Stokes CFD code. Theoptimum was achieved in a few dozen function evaluations.The surface definition for the wing utilized NURBS rep-resentations of the camber, thickness, twist, and shear (seeFigure 8) perturbations from the baseline shape. (The fuse-lage was taken as fixed.) Gradients were computed with aquasi-analytical adjoint method.

One particular approach to inverse design that hasseen considerable use in industry applications is based on

Function evaluation

L/D

rat

io

0 5 10 15 20 25 30 35

∆L / D = +32%

Figure 10. Lift/drag convergence from a wing optimization.Reproduced with permission from Nielsen and Park (2006)c© AIAA.

Figure 11. Design improvement regimes for an aircraft (courtesyof J. Hooker and A. Agelastos (Lockheed-Martin) and W. Milholenand R. Campbell (NASA)).

modifying the surface curvature in proportion to the desiredchange in surface pressure. The surface geometry is modifiedat the same time that the flow solver is converging; hence, thismethod costs little more than a single CFD analysis. Detailscan be found in the study by Campbell (1992). Figure 11,taken from joint Lockheed-Air Force-NASA work, illustratesthe various regions of the aircraft to which this inverse de-sign method was applied. (PAI refers to propulsion-airframeintegration.) Collectively, these improvements in the localaircraft shape enabled the design to meet its performancegoals.

The most common non-gradient-based optimizationmethods that have been applied to aerodynamic shape op-timization are genetic algorithms (see Holst, 2005 for an ex-ample). These have exhibited the customary advantage ofgreater likelihood of finding the global optimum, as well asthe accompanying disadvantage of taking significantly morecomputational time for convergence than gradient-basedmethods.

In aerodynamics, as in other disciplines, surrogate modelsare extensively used in optimization. In some cases, these areused to deal with noisy analysis results. Linear aerodynam-ics methods are particularly prone to producing noisy results.(Some surrogate models such as quadratic or cubic responsesurfaces smooth the noise, but some other surrogate modelsdo not.) In other cases, surrogates are used to deal with por-tions of the design space that cause difficulties for the analysiscode, since the surrogate model enables the optimization toproceed even in the face of occasional analysis failures. Yetanother use is to permit efficient searches for global optimausing, say, pattern-search or genetic algorithm techniques.For the relatively expensive Euler and Navier–Stokes mod-els, use of surrogates entails the usual compromise betweenaccuracy (relative to the CFD model) and speed. All types ofsurrogates – design of experiments, kriging, neural networks,radial basis functions – have been fruitfully employed on this

10 Aerospace System Optimization

application. These types of surrogates all suffer from the curseof dimensionality, limiting the number of design variables inpractice to O(10).

Of course, one would like optimization using a surrogateto converge to the optimum corresponding to that for theunderlying analysis code. Several optimization methods thatjudiciously mix calls to the surrogate with calls to the actualcode have been proven mathematically to converge to thetrue optimum. See Booker et al. (1999) for discussion of onemethod in a general setting, and see Alexandrov et al. (2001)for another method with an aerodynamic shape optimizationapplication; in practice, the latter seems to reduce the overallcost by a factor of 3–5 compared with always invoking theCFD code. When the surrogate is just a lower fidelity model,for example, using an Euler code instead of a Navier–Stokescode or a coarse-grid solution in place of a fine-grid solution,then the number of design variables is not limited as it is forgeneric surrogates.

In addition to the considerations mentioned above, oneneeds to weigh the computational costs of the various meth-ods, at least for CFD. (Linear aerodynamics methods areso fast on desktop computers that CPU time is rarely anissue, regardless of the choice of optimization method.)Roughly speaking, in terms of the cost of a single CFD anal-ysis, an inverse design method takes O(1) analysis time,gradient-based optimization using the adjoint formulationtakes O(10) analysis time, and non-gradient-based methods(including surrogate ones) take O(100 − 10 000) analysistime. All these methods have their niches. Non-gradient-based methods and surrogate models enable exploration oflarge regions of the design space. Gradient-based methodsfacilitate refinement of a local optimum, and inverse designmethods help fine-tune performance in local regions of thesurface.

7 DISCLAIMER

This chapter is declared a work of the U.S. Government andis not subject to copyright protection in the United States.

ACKNOWLEDGMENTS

The author gratefully acknowledges numerous discussionswith his colleagues at the NASA Langley Research Center forall their contributions to his understanding of this subject. Thenumerical examples in Section 3 utilized publicly availablecodes of L.N. Sankar. J.A. Samareh supplied the code usedfor Figure 7.

NOTES

1. The underlying computations are performed with aMatlab

TMcode of L.N. Sankar, found on-line at

http://www.ae.gatech.edu/people/lsankar/AE3903/.

2. Performed with the MatlabTM routine fmincon.

3. These gradients are often called sensitivity derivatives.

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