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Encyclopedia of Aerospace Engineering, Online 2010 John Wiley & Sons, Ltd.This article is 2010 US Government in the US and 2010 John Wiley & Sons, Ltd in the rest of the world.

This article was published in the Encyclopedia of Aerospace Engineering in 2010 by John Wiley & Sons, Ltd.

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 vehicles shape. Methods for aerodynamic shape

optimization have progressed through increasingly complex

aerodynamic analysis tools roughly speaking, for linear

aerodynamics (e.g., panel methods), transonic small distur-

bance equations, and full potential equations in the 1970s; for

linear aerodynamics with boundary-layer corrections in the

1980s; for Euler equations from the 1980s through the mid-

1990s; for NavierStokes equations on structured meshesin the 1990s; for Euler and NavierStokes equations on

unstructured meshes from the late 1990s onward; and for

time-dependent flows, most recently. Although the discus-

sion in this chapter is confined to external flows, most of

the discussion is equally applicable to internal flows, suchas those through aircraft engines. Readers interested in sum-

maries of the important developments (and contributors) in

this field should consult review articles such as those by

Labrujere and Slooff (1993), Newmanet al.(1999), Reuther

et al.(1999), and Mohammadi and Pironneau (2004), as well

as the hundreds of references therein.

Figure 1 illustrates the key processes and variables for

aerodynamic shape optimization. (The gradient variables are

shown in parentheses, as they are only relevant for gradient-

based optimization methods.) The Optimization process

feeds a set of design variables x (a vector of length d) to

the Shape Definition component, which provides a mathe-matical descriptions of the surface. The surface description

is used by the Mesh Generation process, which produces the

computational mesh y used by the Aerodynamic Analysis

component. The analysis output is the state variable q (and

its gradientq). The Objective and Constraint Evaluationprocess supplies the objective functionf, 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.

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2 Aerospace System Optimization

equality constraintsg andh (perhaps along with their gradi-

ents) back to the Optimization process.

Those components of the optimization process that are of

special importance to aerodynamic shape optimization are the

optimization formulation (objectives and constraints), shape

definition, mesh generation, and gradient computation (for

those methods that utilize gradient information). These top-

ics are covered in that order. The initial expository material

on the optimization formulation is given in the context of

linear aerodynamics. Thereafter, the emphasis is on the im-

portant considerations for aerodynamic shape optimization

using the Euler or NavierStokes equations, that is, what are

commonly called computational fluid dynamics (CFD) tools.

The chapter concludes with some considerations for the use

of CFD in aerodynamic shape optimization. The discussion

is restricted to use of optimization for improving an existing

design, that is, optimization that starts from an existing base-

line design and is constrained to maintain the same topology

for 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 has

the 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 for

an appropriate conceptual model for the flow (compressible

or 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 this

chapter 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,gjandhkare inequality and equality constraint

functions, respectively, andxiL 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 to

emphasize its parametric dependence upon them. Likewise,

we now write the state equation (1) as

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

0 (6)

We necessarily havehk= rk for k= 1, . . . ,dimension ofr,as each component ofr yields a constraint. Because the state

vector is defined implicitly, as in equation (6), the objective

and constraint functions in equations (2)(3) have a depen-

dence uponq andx. For aerodynamic shape optimization, the

design variables characterize the shape of the airfoil, wing or

aircraft.

To illustrate the basic principles, we consider optimization

of airfoils in the four-digit NACA series (see, e.g., Abbott and

von Doenhoff, 1959, Chapter 6). These airfoils are described

by the maximum camber (m), the distance of the location

of the maximum camber (p) from the leading edge, and the

maximum thickness (t); all of these are given as fractions of

the chord c. These parameters are illustrated in Figure 2a.

For a four-digit NACA airfoil, say, a NACA d1d2d3d4airfoil,

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 hasa maximum camberof 0.20c located

atx= 0.10c, and a maximum thickness of 0.16c, wherec isthe airfoil chord. Figure 2b is indicative of internal structures

that constrain the geometry; such important constraints are

ignored in the present illustration.

The conceptual model used for the airfoil analysis in this

example is inviscid, irrotational, incompressible, constant-density flow. The velocity potential can be used as the

(scalar) state variable. The mathematical model is given by

the Laplace equation

r= (7)

where denotes the gradient operator (with respect to x).The Laplace equation is subject to a homogeneous Neumann

boundary condition ( n = 0, wheren is unit vector nor-mal to the surface) on the airfoil surface, an homogeneous

Dirichlet condition (= 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 some

basic concepts of aerodynamic shape optimization.

For this example, the design variablesx= (m,p,t), withthese three NACA airfoil parameters treated as continuous

variables; their upper and lower bounds are taken to be

xL= (0.0, 0.1, 0.0) andxU= (0.1, 0.5, 0.2), respectively. In

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Airfoil/Wing Optimization 3

Chord c(a) (b)

Mean camberline Chord

line

p c

m c

t 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 = 1. The keyperformance outputs are the (section) lift, drag, and pitching

moment 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-based

optimizations employ Sequential Quadratic Programming,

using finite differences to compute the gradients and the

BroydenFletcherGoldfarbShanno (BFGS) method to ap-

proximate the Hessian2. The starting point for the optimiza-

tions isx0= (xL+ xU)/2.The optimal solutions to these four problems are given in

Table 1. The pitching moment constraint is active in cases (ii)

and (iv). The drag constraint is also active in case (ii). The lift

constraint is active in case (iii). Thecorresponding airfoils are

shown in Figure 3. The key performance parameters are also

included in the table. The impact of constraints is especially

dramatic in the contrast between the results for the first two

cases (maximization of lift). The conceptual model for this

simple 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 [g

i:c

mandc

d]

min cd [g

i:c

mandc

l]

min cl/ cd [gi:cmand 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 pressure

distribution 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 surface

S of interest, p is the pressure in the flow produced by an

airfoil described by the design variables x, andptarget is the

target 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 cmand 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

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

years for choosing target pressure distributions to achieve

desired performance measures. Methods for matching partic-

ular flow-field characteristics rather than directly minimizing

a global objective function such as drag are referred to as

inverse design methods; the more conventional alternative is

called adirect optimization method. One could apply formal

optimization to the inverse problem with the objective func-

tion given by equation (8). However, there is a wide variety

of inverse design methods that minimize the objective in

equation (8) much faster than direct optimization methods.

An example of one such method is provided in Section 6 (see

Labrujere and Slooff, 1993 for descriptions of many others).

In practical applications, numerous geometric constraints

are 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-

ingedge, andminimal angle of thetrailingedge.Furthermore,

aerodynamic shapes are required to operate over a range of

conditions, especially in Mach number. This is addressed

with a multi-objective optimization formulation (commonly

referred to as multipoint optimization in the aerodynamic

optimization community) (see Drela, 1998 for an extensive

discussion of constraints and objectives for airfoil optimiza-

tion).

The NACA airfoil family used for the example above was

convenient for this demonstration because the parameteriza-

tion guaranteed a simple airfoil shape. Furthermore, the de-sign variables corresponded directly with physical quantities

for which experienced aerodynamicists have considerable in-

tuition. However, the optimization problems of interest for

many decades now require consideration of a much broader

range of shapes than those covered by the NACA airfoil fam-

ilies. In the following section, we cover the fundamentals of

parameterizations that can represent a more general range of

shapes.

3 SHAPE DEFINITION

Aerodynamic shape optimization methods using nonlinear

analysis tools are faced with significant challenges in shape

parameterization, volume mesh generation, and sensitivity

analysis. Although these components are routine for manyoptimization problems in other disciplines, they are nontriv-

ial for aerodynamic shape optimization methods that yield

realistic shapes robustly and efficiently. The surface of an

aircraft, a wing, or even an airfoil is an infinite-dimensional

object, but it must be parameterized as a finite-dimensional

object. (The parameters of the shape correspond to the design

variables for aerodynamic shape optimization.) The shape

parameterization should compactly cover the design space

of interest while presenting no undue difficulties. In partic-

ular, parameterization-induced waviness and discontinuities

in (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 are

quite sensitive to such discontinuities). Sometimes, optimiz-

ing the shape of the entire surface is desired, but other times

optimization is only applied to a portion of the surface. The

alternatives summarized below apply in both cases.

The usual custom for aerodynamic shape parameterization

is 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() depend upon a single

coordinate. The expansion is given by

s(;x) =d

k=1xkbk() (9)

with the coefficients xk in the expansion serving as the de-

sign variables. The perturbation s(;x) may be applied to

whatever functions are used for the surface definition, for

example, 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 trailing

edges of airfoils are fixed in most design problems, having

basis functions that vanish at both end points is desirable.

The sine functions are one option for a complete set of

global basis functions. Drela (1998) recommends using them

in the form

bk() =1

ksin(k) (10)

They form an orthogonal set for [0, 1]. The sine basisfunctions are illustrated in Figure 5a. Suitable combinations

of orthogonal polynomials can also be chosen to ensure that

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0 0.2 0.4 0.6 0.8 11

0.5

0

0.5

1

x x

bk

(a)

k=1

k=2

k=3

k=4

1 0.5 0 0.5 11

0.5

0

0.5

1

bk

(b)

k=2

k=3

k=4

k=5

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

the basis functions vanish at the end points. For example, one

can use

bk() =1

2

(k + 1) (Lk1() Lk+1()) for k 1 (11)

for [1, 1], where Lk() is the Legendre polynomial ofdegreek. The set of basis functions given by equation (11) is

nearly orthogonal the inner product ofbk andbl vanishes

except forl= k 2, k , k + 2. The Legendre basis functionsare illustrated in Figure 5b. These form a complete set of

basis functions. They are advantageous for approximating

functions with a high degree of smoothness (more than, say,

fourcontinuous derivatives), because theyexhibitmuch fasterconvergencethan thesine functionsin such cases (see Canuto

et al., 2006, Chapter 2).

A noteworthy set of local basis functions specifically

chosen for airfoil parameterization are the HicksHenne

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

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

with the domain now again normalized to [0, 1];t1 con-trols the location of the peak and t2 its width. Figure 6a

illustrates some of these functions. Although these functions

lack the completeness property possessed by the sine func-

tions and the orthogonal polynomials, they have proven very

useful in the hands of skilled designers.

A more conventional choice of local basis functions are

splines, which are piecewise polynomials. Figure 6b illus-

trates the case of cubic B-splines on uniformly distributed

knots (denoted by the dots on the x-axis); the curves are la-beled by the xcoordinate of the center of the spline. Unlike

the global basis functions shown in Figure 5, the underly-

ing B-spline basis functions effect only local changes in the

shape. 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=0.3

c=0.5

c=0.7

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

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0 0.2 0.4 0.6 0.8 10.2

0.1

0

0.1

0.2

x x

z z

(a) (b)

Baseline

Camber perturbation

0 0.2 0.4 0.6 0.8 10.2

0.1

0

0.1

0.2

Baseline

Thickness 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, ifp >2, then the shape

perturbation, along with its first and second derivatives, is

continuous. 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 widely

used parameterization, particularly for complex shapes, such

as full aircraft configurations. NURBS represent functions

(in this case surfaces) as a rational function, with both the

numerator 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 of

their mathematical description and a short survey of vari-

ous uses in aerodynamic shape optimization. An important

advantage of a NURBS representation for the shape is thatthey are compatible with most computer-aided design (CAD)

systems. Figure 7 illustrates airfoil deformations based on

NURBS expansions of the airfoil camber and thickness.

Using the locations of the surface mesh points as design

variables is yet another approach. The main challenge here

is picking appropriate geometric constraints and/or filtering

procedures 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 approaches

apply, but, of course, there are now additional types of design

variables. Some representative wing parameters with a direct

aerodynamic interpretation are illustrated in Figure 8. The

symbols indicate locations where the parameters are defined.

The root chord, tip chord, semi-span, and leading edge

sweep angle are planform variables; these are usually fixedearly in the design process. The airfoil section parameters

Camber and thickness

Twist and shear

Leading edgesweep angle

Root chord

Tip chord

Semi-span

y

x

Figure 8. Typical design variables for wings.

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(camber and thickness) as well as the wing twist and shear

are often the focus of wing shape design. The twist angle at

a given airfoil section is the difference between the airfoil

section incident angle at the root and the incident angle of

that airfoil section. Similarly, the shear (dihedral) is the dif-

ference between the airfoil leading edge zcoordinate for the

root and the zcoordinate 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 solution

of the state equation. For compressible flow, the state variable

qis given by

q= (, u, E)T (13)

where is the density, uthe velocity, and the (specific) total

energyE= e + 12u u, whereedenotes the (specific) inter-

nal energy. For steady, inviscid flow described by the Euler

equations, we have

r(q(x),x) = F (14)

where the flux F= (u, uuT, uE + pu)T. The NavierStokes equations have the same form as equation (14) but

with additional terms in the flux function. Figure 9 illustrates

the 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 wingfuselage configuration. The

coordinates of these volume mesh-points are denoted by yin

Figure 1.

For aerodynamic shape optimization, changes to the sur-

face of the airfoil resulting from design variable changes

require adjustments to the volume mesh as well, and these

adjustments need to occur automatically. Nowadays, com-

putational meshes suitable for even viscous CFD analysis

can be generated automatically from the surface shape defi-

nition for airfoils and wings, but this state has proven to be

very elusive for complex aircraft configurations subjected to

NavierStokes analyses some type of user intervention is

typically needed if the mesh generation must be performed

ab initio. Thompson, Soni and Weatherill (1998) provide an

extensive description of methods for CFD mesh generation.

Moreover, as discussed in the next section, analytically based

gradients are highly desirable, and these are not generally

available 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 upon

the mesh associated with the baseline shape. Samareh (2001)

contains an overview of the various approaches to volume

mesh generation.

5 GRADIENT COMPUTATION

For gradient-based optimization methods, the derivatives of

the 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 using

finite differences has several disadvantages: (i) computation

of each gradient requires another full CFD solution since

the equations are nonlinear; (ii) extensive trial and error is

necessary to choose the appropriate step size for each de-

sign variable; and (iii) for some difficult CFD problems, the

analysis code may be simply unable to converge to the level

needed for accurate finite-difference gradients. Computation

of these derivatives through quasi-analytical means is pre-

ferred. (The adjective quasi-analytical is used because the

equations for the gradients are derived analytically but solvednumerically.) In order to distinguish the state variable and the

state equations, which are functions, from their discrete rep-

resentations, which are vectors, we use the symbols q and

r, respectively, for the latter; similarly y is the vector of the

mesh-point coordinates. For a problem withNmesh-points,

the length ofq and r is 5Nsince q has five components at

each point, and the length ofyis 3N. (Boundary conditions

may slightly alter these lengths.)

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Recall that a change in a design variable is propagated

through the shape definition and mesh generation processes

(Figure 1), with the computational mesh y depending upon

x. In particular, the gradient of, say, the objective function

with respect to the particular design variable xjis given by

df

dxj= f

xj+

5Nk=1

f

qk

qk

xj

+

3Nl=1

f

yl

yl

xj

(15)

Evaluation of this requires determination of the three explicit

partial derivatives off plus the qk/xj andyl/xjterms.

The first term on the right-hand side is usually straightfor-

ward to compute. (For the airfoil example in Section 2, all

these derivatives vanish, as the design variables do not appear

explicitly in the expressions for the lift, drag, and pitching

moment.) Likewise, an analytical expression can typically

be derived straightforwardly and then evaluated numerically

forf/qk and f/yl. Theyl/xjterms represent the influ-ence of the design variables upon the computational mesh.

The surface definition techniques illustrated for airfoils in

Figures 57, as well as some mesh perturbation techniques,

lend themselves to analytical evaluation of these terms.

Computation of theqk/xjterms is more involved. Con-

sider the state equation given by equation (6). Its solution

q, 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 linearin thedesired qk/xj term. Thesize of

the 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, the

linear systemis extremely large in three dimensions so large

that direct solution methods are impractical. An efficient it-

erative solution method is surveyed in Newmanet al.(1999)

and described in detail in Koriviet al.(1994).

For most aerodynamic shape optimization problems, the

number 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 direct

approach described above. Details may be found, for exam-

ple, in Newmanet al.(1999) and Reutheret al.(1999). Both

continuous and discrete adjoints have been used in aero-

dynamics. (The phrase continuous adjoint refers to one in

which the continuous adjoint equations are first derived from

the governing partial differential equations (PDEs) and then

discretized; a discrete adjoint is one in which the adjoint

equations are derived from the algebraic equations resulting

from the discretization of the PDEs.) See studies by Newman

et al. (1999) and Reuther et al. (1999) for extended lists of

references to the early experiences of various groups with

both 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 Giles

et al., 2003; Nielsen and Park, 2006 for detailed recommen-

dations). An approach to obtaining second-order derivatives

by combining direct and adjoint techniques is described by

Shermanet al.(1996).

Yet another alternative to computing the gradients is the

use of complex variables, which we illustrate on a simple

real-valued scalar functionf. In this method, the function is

evaluated at the complex point x + i, where i denotes theimaginary unit, for small. A Taylor expansion yields

f(x + i) = f(x) + (i)df

dx + O(2

) (17)

Thus, for small , the real part of f(x + i) is a good ap-proximation tof(x), and the imaginary part divided by is

a good approximation to the gradient df/dx. Note that this

approach does not involve the subtraction of nearly equal

quantities, as occurs for gradients computed via finite dif-

ferences. Hence, round-off errors are not a concern for the

complex variable approach. Provided that the source code is

available, implementation of this approach can require little

more than changing the type of variables from real to com-

plex. A prototypical use of this approach for aerodynamic

shape optimization is given by Nielsen and Kleb (2006).

6 OPTIMIZATION USING CFD

Over the past several decades, a broad assortment of direct

optimization and inverse design methods has been applied to

aerodynamic shape optimization using CFD. The principal

distinctions between the approaches are (i) use of global per-

formance measures such as lift, drag, and pitching moment in

the objectives and constraints versus inverse objectives such

as 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 models

constructed off-line.

Most of the review articles cited in Section 1 concentrate

on direct optimization of global performance measures using

gradient-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

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objective functions and constraints that are contaminated by

numerical noise. Such noise gives the appearance of many

local extrema and can render gradient-based methods un-

workable. Another potential difficulty is that there may be

portions of the desired design space for which the mesh gen-

eration or the analysis process simply fails. At an even more

fundamental level, the problem formulation (objective and

constraints) itself can omit important considerations with the

result that the optimization produces a design that is unreal-

istic from the perspectives of other disciplines. See the cited

review articles for extensive discussion of these issues.

Nevertheless, many groups have used gradient-based

optimization of global performance measures without re-

course to surrogate models to produce impressive CFD-based

optimization 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 of

the lift-to-drag ratio of the vehicle illustrated in Figure 9 attransonic conditions using a NavierStokes CFD code. The

optimum 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 (see

Figure 8) perturbations from the baseline shape. (The fuse-

lage was taken as fixed.) Gradients were computed with a

quasi-analytical adjoint method.

One particular approach to inverse design that has

seen considerable use in industry applications is based on

Function evaluation

L/Dratio

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 desired

change in surface pressure. The surface geometry is modified

at the same time that the flow solver is converging; hence, this

method costs little more than a single CFD analysis. Details

can be found in the study by Campbell (1992). Figure 11,

taken from joint Lockheed-Air Force-NASA work, illustrates

the various regions of the aircraft to which this inverse de-

sign method was applied. (PAI refers to propulsion-airframe

integration.) Collectively, these improvements in the local

aircraft shape enabled the design to meet its performance

goals.

The most common non-gradient-based optimization

methods 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 of

greater likelihood of finding the global optimum, as well asthe accompanying disadvantage of taking significantly more

computational time for convergence than gradient-based

methods.

In aerodynamics, as in other disciplines, surrogate models

are extensively used in optimization. In some cases, these are

used 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 response

surfaces smooth the noise, but some other surrogate models

do not.) In other cases, surrogates are used to deal with por-

tions of thedesign space that cause difficulties for theanalysis

code, since the surrogate model enables the optimization toproceed even in the face of occasional analysis failures. Yet

another use is to permit efficient searches for global optima

using, say, pattern-search or genetic algorithm techniques.

For the relatively expensive Euler and NavierStokes mod-

els, use of surrogates entails the usual compromise between

accuracy (relative to the CFD model) and speed. All types of

surrogates design of experiments, kriging, neural networks,

radial basis functions have been fruitfully employed on this

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application. These types of surrogates all suffer from the curse

of dimensionality, limiting the number of design variables in

practice toO(10).

Of course, one would like optimization using a surrogate

to converge to the optimum corresponding to that for the

underlying analysis code. Several optimization methods that

judiciously mix calls to the surrogate with calls to the actual

code have been proven mathematically to converge to the

true optimum. See Bookeret al.(1999) for discussion of one

method in a general setting, and see Alexandrovet al.(2001)

for another method with an aerodynamic shape optimization

application; in practice, the latter seems to reduce the overall

cost by a factor of 35 compared with always invoking the

CFD code. When the surrogate is just a lower fidelity model,

for example, using an Euler code instead of a NavierStokes

code or a coarse-grid solution in place of a fine-grid solution,

then the number of design variables is not limited as it is for

generic 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 are

so fast on desktop computers that CPU time is rarely an

issue, 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 formulation

takesO(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 of

large regions of the design space. Gradient-based methodsfacilitate refinement of a local optimum, and inverse design

methods help fine-tune performance in local regions of the

surface.

7 DISCLAIMER

This chapter is declared a work of the U.S. Government and

is not subject to copyright protection in the United States.

ACKNOWLEDGMENTS

The author gratefully acknowledges numerous discussions

with hiscolleagues at theNASALangley Research Center for

alltheir contributions to hisunderstanding of this subject. The

numerical examples in Section 3 utilized publicly available

codes of L.N. Sankar. J.A. Samareh supplied the code used

for Figure 7.

NOTES

1. The underlying computations are performed with a

MatlabTM

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

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

2. Performed with the MatlabTM routinefmincon.

3. These gradients are often calledsensitivity derivatives.

REFERENCES

Abbott, I.A. andvon DoenhoffA.E. (1959) Theory of Wing Sections,Dover Publications, New York.

Alexandrov, N.M., Lewis, R.M., Gumbert, C.R., Green, L.L. and

Newman, P.A. (2001) Approximation and model management inaerodynamic optimization with variable-fidelity models. J. Air-

craft,38(6), 10931101.Booker,A.J., Dennism, J.E.,Jr.,Frank, P.D.,Serafini, D.B.,Torczon,

V. and Trosset, M.W. (1999) A rigorous framework for optimiza-tion of expensive functions by surrogates. Struct. Optim.,17(1),113.

Campbell, R.L. (1992) An approach to constrained aerodynamicdesign with application to airfoils. NASA-TP-3260.

Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. (2006)

Spectral Methods, Fundamentals in Single Domains, Springer,

New York.

Drela, M. (1998) Pros and cons of airfoil optimization, in Frontiersof Computational Fluid Dynamics (eds D.A. Caughey, and M.M.Hafez), World Scientific, Hackensack, pp. 363381.

Giles,M.B., Duta, M.C.,Muller, J.-D.and Pierce, N.A.(2003) Algo-rithm developments for discrete adjoint methods.AIAA J.,41(2),198205.

Hicks, R.M. and Henne, P.A. (1978) Wing design by numericaloptimization.J. Aircraft,15(7), 407412.

Holst, T. (2005) Genetic algorithms applied to multi-objective

aerospace shape optimization. J. Aero. Comput. Info. Comm.,2,217235.

Korivi, V.M., Taylor, A.C. Ill, Newman, P.A., Hou, G.J.-W. andJones, H.E. (1994) An approximate factored incremental strategyfor calculating consistent discrete CFD sensitivity derivatives.J.Comput. Phys.,113(2), 336346.

Labrujere, Th.E. and Slooff, J.W. (1993) Computational methodsfor the aerodynamic design of aircraft components. Ann. Rev.Fluid Mech.,25, 183214.

Li, W. and Krist,S. (2005) Spline-based airfoil curvature smoothingand its applications.J. Aircraft,42(4), 10651074.

Mohammadi, B. and Pironneau, O. (2004) Shape optimization influid mechanics.Ann. Rev. Fluid Mech.,36, 255279.

Newman, J.C. III, Taylor, A.C. III, Barnwell, R.W., Newman,P.A. and Hou, G.J.-W. (1999) Overview of sensitivity analysisand shape optimization for complex aerodynamic configurations.AIAA J.,36(1), 8796

DOI: 10.1002/9780470686652.eae500

8/12/2019 eae500

11/11


This article was published in the Encyclopedia of Aerospace Engineering in 2010 by John Wiley & Sons Ltd


Nielsen, E.J. and Kleb, W.L. (2006) Efficient construction ofdiscrete adjoint operators on unstructured grids using complexvariables.AIAA J.,44(4), 827836.

Nielsen, E.J. and Park, M.A. (2006) Using an adjoint approach toeliminate mesh sensitivities in computational design. AIAA J.,44(5), 948953.

Piegl, L. and Tiller, W. (1996) The NURBS Book, Springer,New York.

Reuther, J.J., Jameson, A., Alonso, J.J., Rimlinger, M.J. andSaunders, D. (1999) Constrained multipoint aerodynamic shapeoptimizationusing an adjoint formulationand parallel computers.Parts 1 and 2.AIAA J.,36(1), 5174.

Samareh, J.A. (2001) Survey of shape parameterization techniquesfor high-fidelity multidisciplinary shape optimization. AIAA J.,39(5), 877884.

Sherman, L.L., Taylor, A.C. III, Green, L.L., Newman, P.A., Hou,G.J.-W. and Korivi, V.M. (1996) First- and second-order aerody-namic sensitivity derivatives via automatic differentiation with

incremental iterative methods. J. Comput. Phys., 129(2), 307331.

Thompson, J.R., Soni, B.K. and Weatherill, N.P. (eds) (1998)

Handbook of Grid Generation, CRC Press, Boca Raton.

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