Lift Superposition and Aerodynamic Twist
Optimization for Achieving Desired Lift Distributions
Kevin A. Lane
∗
, David D. Marshall
†
, and Rob A. McDonald
‡
California Polytechnic State University, San Luis Obispo, CA, 93407-0352
A method for achieving an arbitrary lift distribution with an arbitrary planform is pre-
sented. This is accomplished through optimizing aerodynamic twist for a given number of
either known airfoils or airfoils to be designed. The spanwise locations of these airfoils are
optimized to get as close to the desired lift distribution as possible. Airfoils are linearly
interpolated between these points. After aerodynamic twist, the planform is twisted geo-
metrically using radial basis functions to model the twist distribution. The aerodynamic
influence of each twist distribution is determined and all are superimposed to determine
the function weights of each twist function, yielding the optimal twist to match the given
lift. This method has been shown to match both an elliptical and a triangular lift distri-
bution for an arbitrary planform. This method can also be used with any fidelity model,
creating a powerful design tool.
Nomenclature
A aerodynamic influence matrix
b required aerodynamic change vector
c chord length
C
l
section lift coefficient
i twist function index
M number of spanwise wing segments
N number of twist basis functions
r radius from origin or center point in radial basis function
r
0
reference radius in radial basis function
w basis function weight
x basis function weighting vector
y non-dimensional semispan location
α angle of attack
∆ increment
δ magnitude of twist basis function
η center point in radial basis function
λ wing taper ratio
φ radial basis function or velocity potential
ω geometric twist angle
\ gradient
I. Introduction
F
or increasing the aerodynamic efficiency of wings, it is desirable to reduce their induced drag. One way
this can be done is through achieving an elliptical lift distribution. This is the theoretical minimum
∗
Graduate Student, Aerospace Engineering Department, Student Member AIAA
†
Associate Professor, Aerospace Engineering Department, Senior Member AIAA
‡
Assistant Professor, Aerospace Engineering Department, Member AIAA
1 of 9
American Institute of Aeronautics and Astronautics
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2010, Orlando, Florida
AIAA 2010-1227
Copyright © 2010 by Kevin A. Lane and David D. Marshall. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
for induced drag as shown by Prandtl’s classical lifting-line theory.
1
Phillips
2
has introduced an expression
for the optimum washout distribution for a wing of arbitrary planform. It comes from a “more practical
form of the analytical solution for the effects of geometric and aerodynamic twist” on a wing of arbitrary
planform that is based on Prandtl’s lifting-line theory. This is a good low-fidelity model that works well
for a clean wing. However, if a fuselage and/or a nacelle/pylon combination were added to the wing, the
optimal washout distribution based on lifting-line theory is no longer valid.
What is required is a more general method that does not depend on geometry, but rather simply examines
lift distributions. This can be accomplished by determining the required change in the lift distribution of
an untwisted wing in order to achieve the desired lift distribution. The wing can then be twisted according
to a set of known basis functions and each twist distribution’s influence on the untwisted lift distribution
can be obtained. Any set of basis functions used for function approximations should work to represent the
wing twist. However, radial basis functions will be used here. All the resulting ∆C
l
·c distributions can
be superimposed to attain all the necessary twist function weights, which yields a wing that achieves the
desired lift distribution. The reason superposition works is noted by examining Laplace’s equation. For
incompressible, irrotational fluid flow, Laplace’s equation reduces to
\
2
φ = 0 (1)
This is a second order linear partial differential equation. Since Laplace’s equation is linear, superposition
can be used to add together the effects of all the twist distributions on the lift distribution. Since this is
a general method that does not depend on geometry, influences from additional bodies such as a fuselage
and/or a nacelle/pylon combination can be included. Also, since this method does not specify an elliptical
lift distribution, it will work for any desired lift distribution.
II. RBF Wing Twist
Radial basis functions (RBFs) are functions whose values depend solely on the distance from the origin or
from some other point to be taken as the center. Therefore, RBFs take the form of φ(r). For representation
of wing twist, the radius of an RBF is represented as:
r = y − η (2)
where y is the semispan location and η is the point to be used
as the center of the RBF. There are many common types of RBFs
including Gaussian, multiquadric, and the polyharmonic spline. The
form used in this method to model the twist distribution of an arbi-
trary wing is the multiquadric RBF, shown below,
φ(r) = r
2
+ r
0
2
, r
0
≥ 0 (3)
where r
0
is a reference radius. Figure 1 displays several multi-
quadric RBFs for differing centers. The center of each RBF corre-
sponds to the y-location of the minimum φ point for each function.
Therefore, the η values displayed in Fig. 1 are 0, 0.2, 0.4, 0.6, 0.8,
and 1. Also, the minimum φ value corresponds to the r
0
value of
each function. Therefore, each RBF has an r
0
value of 1 in this
example.
RBFs can be used to build up function approximations. This is how a twist distribution is modeled in
this method. Multiquadric RBFs using different centers are summed up and scaled by their corresponding
function weights to yield the twist angle at a given semispan location. This is expressed as:
N
N
ω(y) = δ · w
i
· φ( y − η
i
) (4)
i=1
where δ is the magnitude of the basis functions used prior to calculating the weights required to match
a given lift distribution.
2 of 9
Figure 1. General Multiquadric RBFs
American Institute of Aeronautics and Astronautics
III. Aerodynamic Twist Optimization
A wing can be twisted aerodynamically according to a given number of airfoils. These airfoils are placed
optimally in order to achieve as close to the desired lift distribution as possible. These airfoils are then
designed to achieve the required lift coefficient found from the desired lift distribution. Airfoils are linearly
interpolated between these designed airfoils to yield the entire lift distribution. This is done by finding the
set of spanwise locations and corresponding design lift coefficients that minimize the error between the C
l
distribution achieved by aerodynamic twist and the C
l
distribution resulting from the desired lift distribution
and planform shape. The spanwise placement of each airfoil was optimized using the MATLAB
3
constrained
optimizer fmincon. A constraint was placed on the location of airfoils such that each one must be placed
further outboard on the span than the previous airfoil. At each iteration in the optimization, when an airfoil
was placed, its lift coefficient was taken to be the same as that from the desired lift distribution. Figure 2
shows the C
l
distribution corresponding to a desired elliptical lift distribution for two different taper ratios.
They also show the C
l
distribution for the optimized airfoil placement. Figure 2(a) shows the C
l
distribution
for a wing with a taper ratio of 1 and Fig. 2(b) shows the C
l
distribution for a wing with a taper ratio of 0.3.
This was done at a Mach number of 0.8 at an altitude of 35,000 feet. With just four airfoils between the root
and the tip, linear interpolation of the design lift coefficient gets fairly close to the desired lift distribution.
(a) λ = 1 (b) λ = 0.3
Figure 2. Optimized Airfoil Placement
Another way to optimize the aerodynamic twist of a wing is with a set of given airfoils. Again, their
placement can be optimized with fmincon. However, with this method the C
l
of each airfoil is already
determined, not taken from the desired lift distribution. While this does not get as close to the desired lift
distribution as the previous method, it does save on computational time because the airfoils do not have to
be designed.
IV. Geometric Twist Optimization
IV.A. Overview
A wing can also be twisted geometrically to achieve a desired lift distribution. This is done by superimposing
all the ∆C
l
·c distributions due to each twist distribution together to calculate the required twist function
weights. First, all of the ∆C
l
·c distributions can be organized in matrix form into an aerodynamic influence
matrix. It is an M x N+1 matrix where M is the number of spanwise sections the wing is broken up into and
N is the number of twist basis functions. The extra column comes from an angle of attack function used in
the process. The aerodynamic influence matrix is expressed as:
3 of 9
American Institute of Aeronautics and Astronautics
⎤⎡
⎢
⎢
⎢
⎢
⎣
(∆C
l
· c)
α,1
(∆C
l
· c)
ω
1
,1
(∆C
l
· c)
ω
2
,1
· · · (∆C
l
· c)
ω
N
,1
(∆C
l
· c)
α,2
(∆C
l
· c)
ω
1
,2
(∆C
l
· c)
ω
2
,2
(∆C
l
· c)
ω
N
,2
⎥
⎥
⎥
⎥
⎦
(5)A =
. .
.
. . .
.
(∆C
l
· c)
α,M
(∆C
l
· c)
ω
1
,M
(∆C
l
· c)
ω
2
,M
· · · (∆C
l
· c)
ω
N
,M
. .
where each column represents a particular twist distribution’s influence on the lift distribution along the
span of the wing. In order to calculate the necessary twist to achieve the desired lift distribution, the required
∆C
l
·c distribution must be known. This is equivalent to subtracting the untwisted wing lift distribution
from the desired lift distribution. This is expressed in a required aerodynamic performance change vector
as:
⎤⎡
b =
⎢
⎢
⎢
⎢
⎣
(C
l
· c)
1,desired
− (C
l
· c)
1,base
(C
l
· c)
2,desired
− (C
l
· c)
2,base
.
.
.
(C
l
· c)
M,desired
− (C
l
· c)
M,base
⎥
⎥
⎥
⎥
⎦
(6)
The final components in this system are all the weights of the twist basis functions and angle of attack.
These scale all of the ∆C
l
·c distributions so that it reflects the final angle of attack and twist distribution,
not the basis function twist distributions. The basis function weights are expressed in a weighting vector as:
⎤⎡
x =
⎢
⎢
⎢
⎢
⎢
⎢
⎣
w
α
w
ω
1
w
ω
2
.
.
.
w
ω
N
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(7)
Knowing A and x allows the ∆C
l
·c distribution of the twisted wing to be calculated, which when added
to the untwisted wing lift distribution yields the twisted wing C
l
·c distribution. This is expressed as simply
the matrix multiplication problem:
A · x =
b (8)
However, if the final twist distribution is not yet known, it must be solved for. The values in x must be
found such that the ∆C
l
·c distributions are scaled by the appropriate amounts. Therefore, when added to the
untwisted lift distribution, the C
l
·c value at every spanwise location on the wing matches the corresponding
value from the desired lift distribution. This is accomplished by building up
b as shown in Eq. (6) and
solving for x using the pseudo inverse of A. The pseudo inverse is used because Eq. (8) is a rectangular
overdetermined system unless the number of basis functions used is one less than the number of spanwise
sections the wing is broken up into. It is one less due to the angle of attack term present in the system. This
is typically far more basis functions than is required.
Figures 3 on the next page and 4 on the following page give a more visual representation of the geometric
twist optimization. Figure 3 on the next page represents all the basis functions used in this example. The
first function represents an angle of attack, followed by a linear twist and four sinusoidal basis functions.
Even though this is currently being performed with multiquadric RBFs, sinusoidal functions are easier to
distinguish for the purposes of this visualization. Sinusoidal basis functions provide a good fit of an arbitrary
lift distribution, but oscillations occur in the resulting twist distribution. Multiquadric RBFs also provide
a good fit, but the twist distribution is much smoother than that resulting from sinusoidal basis functions.
Therefore, multiquadric RBFs are the method of choice. Figure 4 on the following page represents the ∆C
l
·c
distributions that correspond to each basis function.
4 of 9
American Institute of Aeronautics and Astronautics
(a) α (b) Linear (c) 1/4 Period (d) 1/2 Period (e) 3/4 Period (f ) 1 Period
Figure 3. Example Basis Functions
(a) α (b) Linear (c) 1/4 Period (d) 1/2 Period (e) 3/4 Period (f ) 1 Period
Figure 4. ΔC
l
·c of Example Basis Functions
IV.B. Fuselage Effects
The inclusion of a fuselage into this process requires a slight modification to the matrix math. Adding a
fuselage changes the lift distribution on a wing. Therefore, Eq. (6) must be modified in order for the required
aerodynamic performance change to account for how the fuselage affects the lift distribution of the wing.
The required lift change now equals the fuselage effect on the lift distribution subtracted from Eq. (6).
⎤⎡
b =
⎢
⎢
⎢
⎢
⎣
(C
l
· c)
1,desired
− (C
l
· c)
1,base
− (∆C
l
· c)
1,f uselage
(C
l
· c)
2,desired
− (C
l
· c)
2,base
− (∆C
l
· c)
2,f uselage
.
.
.
(C
l
· c)
M,desired
− (C
l
· c)
M,base
− (∆C
l
· c)
M,f uselage
⎥
⎥
⎥
⎥
⎦
(9)
IV.C. Advantages
What makes this method so powerful is not just that it can match any lift distribution with an arbitrary
planform, but also because of its computational efficiency. It is very inexpensive compared to a numerical
optimization process. The reason for this is that the user determines how many simulations to perform.
Also, once the aerodynamic performance change vector and the aerodynamic influence matrix are filled in, it
is simply an analytical solution for the optimal twist distribution. However, a numerical optimizer performs
a search technique that is very expensive when an aerodynamic analysis is in the objective function. The
optimizer must perform many simulations to calculate derivatives in order to determine what direction to
travel in the design space. Another benefit to this method is that the aerodynamic analysis can be kept as
a “black box.” It is a multifidelity method that can be used during all phases of the design process as the
analysis tools become more computationally intensive.
V. Solution Techniques
V.A. Athena Vortex Lattice
Athena Vortex Lattice (AVL)
4
is a vortex lattice model that uses horseshoe vortices (vortex sheet) for the
lifting surfaces and a slender-body model for fuselages and nacelles. Fuselages and nacelles are modeled with
source and doublet lines.
V.B. Panel Method Ames Research Center
Panel Method Ames Research Center (PMARC)
5
is a NASA panel code that computes the potential flow
field around complex three-dimensional bodies. It is a low order panel method, using constant strength
5 of 9
American Institute of Aeronautics and Astronautics
