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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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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.
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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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4 Aerospace System Optimization
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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Airfoil/Wing Optimization 5
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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6 Aerospace System Optimization
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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Airfoil/Wing Optimization 7
(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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8 Aerospace System Optimization
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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Airfoil/Wing Optimization 9
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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10 Aerospace System Optimization
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
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 11
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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