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Static Aeroelastic Optimization of Aircraft Wing with Multiple Surfaces
Conference Paper · June 2017
DOI: 10.2514/6.2017-4320
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Virginia Polytechnic Institute and State University
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Static Aeroelastic Shape Optimization of
Aircraft Wings with Multiple Control
Surfaces
Wei Zhao∗and Rakesh K. Kapania†
Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061
This paper presents a static aeroelastic shape optimization framework for
aircraft wings, having multiple control surfaces, for optimizing several performance measures, all using gradient based optimization. This approach
can be integrated into a multidisciplinary design, analysis and optimization
framework for aircraft wing design at the conceptual/preliminary design
stages for an efficient optimization analysis. For a linear aeroelastic system, aeroelastic performance indices such as the hinge moment and the
lift distribution can all be expressed as a linear function of the wing shape
including the flap rotations and the initial twist distribution. Similarly, the
induced drag can be represented in a quadratic form of the wing shape. By
using these approximations, one can use the gradient-based optimization
approach to determine the optimal wing shape for achieving the optimal
aeroelastic performances, such as the minimum drag during cruise, the minimum wing root bending moment for maneuver flight, and the maximum
lift coefficient during takeoff and landing subjected to various constraints
including the hinge moment, lift stall, etc. The static aeroelastic shape
optimization framework is employed for performing optimization in two
different applications, the blended-wing-body and the Goland wing, for different optimal performance indices. This is done both for a verification
study and for an efficiency comparison study. Results for all the studied
cases demonstrate the accuracy and efficiency of using the linear approximations in expressing the performance indices of interest for an efficient
optimization of aircraft wings with multiple control surfaces.
∗
Graduate Research Assistant, Kevin T. Crofton Department of Aerospace and Ocean Engineering, AIAA
Student Member
†
Norris and Wendy Mitchell Endowed Professor, Kevin T. Crofton Department of Aerospace and Ocean
Engineering, AIAA Associate Fellow
1
Nomenclature
b
= full span
c
= chord for aerodynamic strip
c̄
= reference chord
cH
= reference chord for control surface
cl
= aerodynamic strip lift coefficient
cn
= aerodynamic strip normal load coefficient, cn = cl for horizontal wing
N
= total strips number for full aerodynamic model
HM
= hinge moment
WRBM = wing root bending moment
k
= number of flaps for half model
S
= reference aerodynamic area
Si
= area for i−th aerodynamic strip
SH,i
= reference area for i−th control surface
q
= dynamic pressure
α
= angle of attack
β
= elevator rotation
γ0
= initial wingtip twist
Matrix and vector
C LM
= stability derivatives matrix of total lift and aerodynamic moment with respect
to trim variables U
CH
= stability derivatives matrix of hinge moment with respect to all trim variables U
Cl
= sensitivity matrix of local lift coefficient with respect to design variables X
CX
= sensitivity matrix of output trim variables with respect to design variables X
LM
= a vector for total lift and aerodynamic moment
S
= a diagonal matrix for aerodynamic strip areas
T
U
= a vector for design variables and output trim variables, U = X T X Tout
X
= a vector for design variables, such flap rotations and initial twist
X out
= a vector for output trim variables in trim analysis
δ
= a vector for control surface rotations
I.
Introduction
Thin, swept wings with high-aspect ratio have attracted many studies for their use in
future commercial civil transport aircraft design due to their high aerodynamic efficiency
in both drag reduction and weight savings [1]. However, the increase in the flexibility of
wings due to the large aspect ratio can lead to various aeroelastic problems [2, 3], such as
a large tip deflection, a decrease in the flutter speed and a lose of control authority, among
others. To avoid these aeroelastic problems, the Active Aeroelastic Wing (AAW) concept
[4] has been considered as a technology to integrate aerodynamics, structural flexibility and
2
active control to maximize air vehicle performance. This concept utilizes the benefit of the
aeroelastic wing flexibility to enable the use of large aspect ratio wings by changing their wing
shape for optimal performances using multiple control surfaces, such as wing root bending
moment reduction [5, 6], load alleviation [7–9], drag reduction [10, 11], minimization of elastic
deformation[12], and so on. Additionally, this concept can be used for flutter suppression
and the roll control through the use of many control surfaces to further improve the aircraft
performances [13–16].
Induced drag reduction and maneuver load alleviation are two main benefits of using the
AAW concept for aircraft wing design. Both the trailing-edge and leading-edge flaps can
tailor the lift distribution close to elliptical shape for a minimum induced drag and can move
the lift inboard to reduce the wing root bending moment. The reduction in the wing root
bending moment can be considered as a surrogate to represent the weight reduction.
Kolonay and Eastep [11] studied the induced drag minimization of the Goland wing
with twenty discrete trailing-edge flaps. These flaps were scheduled to tailor an optimal lift
distribution close to being an elliptical shape resulting in a minimum induced drag. Zink et
al. [6, 17] studied wing root bending moment minimization for active aeroelastic wings by
using gradient-based optimization. The sensitivity matrix of the wing root bending moment
with respect to flap rotations was obtained by applying a unit rotation for each flap and
obtaining the resulting bending moment acting at the wing root. This study ignores the
possible lift stall close to the wing root in the wing root bending moment minimization.
Recent study by Lin [18] summarizes several “inconveniences” in various sensitivity-based
approaches that are being used for the induced drag [11] and the wing root bending moment
minimization [6]. An analytical solution was presented for evaluating sensitivity of the wing
root bending moment and the hinge moment with respect to various flap rotations. Both
the sensitivity analysis and the trim analysis were developed in ASTROS. All of this work
used ASTRO Matrix Analysis Problem Oriented Language, which itself maybe somewhat
difficult to learn and use for many researchers and engineers.
Our own recent work [19] used an MDAO framework for designing a composite flyingwing with many control surfaces. This MDAO framework includes both shape and size design
variables for designing a light-weight aircraft for flight test to demonstrate the active flutter
suppression technology. To obtain the optimal lift distribution in optimizing the aircraft
composite laminate configurations and the internal structural layout, the wing root bending
moment minimization was considered as a surrogate for weight reduction by tailoring the
flap rotations. Linear programming was used for this optimization in the presence of the
hinge moment, stall lift, stall AoA and the maximum flap rotation as constraints. To save
fuel/power consumption, total drag minimization was also conducted using a constrained
nonlinear programming by scheduling rotations for all the flaps during cruise. The hinge
moment was found to be a critical constraint in the optimization studies for both wing root
bending moment minimization and the total drag reduction.
Stanford et al. [20] conducted a weight reduction study for subsonic transport wing
by integrating the distributed trailing-edge control effectors to aeroelastic tailoring scheme.
Results show that there is up to 46% weight reduction of using tow steered laminates and
3
distributed trailing-edge control surfaces for transport wings than metallic wing only with
various thickness distribution. Later, Stanford [16] used distributed multiple control surfaces
for aeroservoelastic wing optimization study. That study shows that there is around 31%
weight reduction for stiffened wing when optimizing the skin thickness, stiffeners dimensions
and the scheduling of control surface rotations. However, when 20% open-loop flutter or
closed-loop flutter constraint is enforced, the weight penalty to the original optimal design
without the flutter constraint depends on the control cost. The higher the control cost values,
the lesser the weight penalty.
Additionally, the multiple surfaces can be used to change the spanwise shape for the wing
along with the elastic deformed shape resulting in a maximum lift coefficient. Fujiwara and
Nguyen [21] conducted multidisciplinary optimization studies of flexible wings using variable
camber continuous trailing edge flaps (VCCTEF) during takeoff and landing. Their research
studies show that increasing the number of flaps along the span has the potential to yield
higher maximum lift coefficient. In their work, a 3D linear vortex lattice method with known
2D airfoil viscous data is used to predict the lift near stall. For a high fidelity aerodynamic
analysis in lift maximization study, Kim et al. [22] used an adjoint-based Navier-Stokes
design and optimization method to study the lift maximization for two-dimensional multielement high-lift configurations at a fixed angle of attack and a fixed drag coefficient. For an
efficient determination of maximum lift coefficient for jig-shaped wings, Phillips and Alley
[23] used lift-line theory to optimize the wing twist distribution for a maximum lift coefficient
while keeping the wingtip twist at a practical level.
For complex aircraft wings with multiple control surfaces at the conceptual/preliminary
design stages, such as the composite flying-wings aircraft, mAEWing2, studied in the previous work [19], different flap rotations leading to a different load distribution significantly
influences the optimal aircraft sizing study results. To determine the optimal flap rotations
for any given sized configuration within the optimization framework, this paper presents a
unified framework for optimizing the flap rotations directly for optimal aeroelastic performances that were approximated explicitly in terms of flap rotations. By taking advantage of
the linearity of the aeroelastic system, one can use the gradient-based optimization approach
to rapidly obtain the optimal flap rotations for the desired performances in aircraft wings
MDAO study.
The remainder of the paper is organized as follows: Sections III-VI presents the approximations for the hinge moment, the lift distribution, the wing root bending moment and the
total drag in terms of the flap rotations. Section VII validates the accuracy and examines
the efficiency of the developed unified aeroelastic optimization framework for different applications. Section VII-A verifies and studies the control power minimization and the wing
root bending moment minimization using the blended-wing-body aircraft. Section VII-B
validates the present method for the minimal induced drag optimization study and presents
results for the minimal induced and total drag. Section VII-C studies the lift maximization
for the Goland wing at a fixed angle of attack and having a fixed drag coefficient. The last
section VIII concludes the developed unified framework for static aeroelastic optimization of
aircraft wings with multiple surfaces for various aeroelastic performances and demonstrates
4
the necessity to integrate this unified framework for an efficient sizing study of wings with
multiple surfaces.
II.
Aeroelastic Equation of Motion
The static aeroelastic equation of motion is expressed as [24]:
[K − q · Qae ] · ue = q · Qax · ux + P 0
e
Qae = GT · S · A−1
aic · D · G
x
Qax = GT · S · A−1
aic · D
(1)
g
P 0 = q · GT · S · A−1
aic · w
where K is the structural stiffness matrix; vectors ue and ux are structural elastic deformation and the aerodynamic degrees of freedom (such as flap rotations, angle of attack, and
etc.), respectively; matrices G, S and Aaic are aero/structural coupling matrix, aerodynamic
panel area integration matrix and aerodynamic influence coefficient (AIC) matrix, respectively; matrices D e and D x are substantial derivative matrices which are used to convert
the aerodynamic model deformation to the downwash for computing aerodynamic force; the
force P 0 is due to the initial downwash, wg , which is caused by the camber effect, the initial
twist and the initial angle of attack. The gravity can be integrated to the force vector P 0
as a constant term; and q is the dynamic pressure.
Based on Eq. (1), it is observed that the aerodynamic forces are linear function of the
flap rotations given in ux and the initial twist given in wg . For flexible aircraft wings, the
aeroelastic effect changes the aeroelastic performances, such as the lift distribution, the hinge
moment, etc., because of the elastic structural deformation. The sensitivities of the elastic
deformation ue with respect to the flap rotations and the initial twist are expressed as:
∂ue
= q · Qax
∂ux
∂ue
[K − q · Qae ] ·
= q · GT · S · A−1
aic
∂wg
[K − q · Qae ] ·
(2a)
(2b)
Equation (2) show the elastic deformation, ue , changes linearly with the flap rotations and
the initial twist when the structural model, aerodynamic model and the dynamic pressure are
given. This linear relationship enables us to use the gradient-based optimization approach to
optimize the wing shape by changing the flap rotations and the initial twist for achieving the
desired structural/aeroelastic performances without using significant computational effort.
The two equations in Eq. (2) also show that the elastic deformation depends on the
dynamic pressure for a given aeroelastic model at different altitudes and airspeeds, which
leads the change in the aeroelastic responses. In other words, the stability derivatives change
with the dynamic pressure when the elastic effect is considered [25]. For a given aerodynamic
5
model, the stability derivatives can be determined for a rigid body. The elastic effect to the
stability derivatives can be obtained using the two equations in Eq. (2) when there is any
change in the dynamic pressure and the structural model.
III.
Hinge Moment
For a linear aeroelastic system, non-dimensional derivative for the hinge moment, the
total lift coefficient and the total aerodynamic moment coefficient with respect to all the
trimmed variables can be obtained. For a wing with k control surfaces, the hinge moment
for each of them can be expressed as:






HM1 
HM1,0 
















HM
HM



2 
2,0 



..
.
.
=
+
.
.










 HMk−1 



HMk−1,0 












HMk
HM
k,0
k×1
k×1

∂CH,1
∂CH,1
...
cH,1 SH,1
cH,1 SH,1

∂β1
∂βk

∂CH,2
∂CH,2


cH,2 SH,2
...
cH,2 SH,2

∂β1
∂βk

.

.
q
.

∂C
∂CH,k−1
H,k−1
 c
... cH,k−1 SH,k−1
 H,k−1 SH,k−1
∂β1
∂βk


∂CH,k
∂CH,k
cH,k SH,k
...
cH,k SH,k
∂β1
∂βk
∂CH,1
cH,1 SH,1
∂α
∂CH,2
cH,2 SH,2
∂α
∂CH,k−1
cH,k−1 SH,k−1
∂α
∂CH,k
cH,k SH,k
∂α

... 


... 





... 



...









β1
..
.









βk






α







 .. 

.
n×1
k×n
(3)
where cH and SH are, respectively, the reference chord and the reference area for the control
∂()
are the partial derivatives of the hinge moment coefficient with respect to all the
surface;
∂()
trim variables, such as flap rotations, β, angle of attack, α, etc. HM 0 is the hinge moment
for each flap due to the initial resultant applied load.
The above equation (3) can be expressed in a matrix form as:
HM = HM 0 + qC H U
(4)
where HM and U are vectors for hinge moment of each control surface and all trim variables,
respectively. The matrix C H is the stability derivative of hinge moment for each control
surface with respect to all trim variables. For complex structural/aerodynamic models,
this matrix C H for the flexible stability derivatives can be obtained from NASTRAN static
aeroelastic analysis SOL 144. For fixed structural parameters, aerodynamic model and
dynamic pressure, these stability derivatives can be assumed to be constant regardless of the
change in the values of trim variables.
6
Similarly, the total lift and aerodynamic moment for static longitudinal stability are[26]:
(
L
M
)
(
=
L0
M0
)
∂CL
 ∂β1
+ qS  ∂C
M
c̄
∂β1

∂CL
...
∂βk
∂CM
... c̄
∂βk
∂CL
∂α
∂CM
c̄
∂α
...
...









β1
..
.








βk






α



2×n 

 .. 

.
n×1


(5)
= L0 + qSC LM U
where L0 and M0 refer to the lift and aerodynamic moment due to the initial downwash,
such as the initial twist distribution; c̄ and S are the reference chord and the reference area,
respectively; matrix C LM includes the stability derivatives of the total lift and the total
aerodynamic moment with respect to all trim variables.
IV.
Wing Root Bending Moment
The wing root bending moment (WRBM) minimization is considered as a surrogate for
the aircraft structural weight. The reduction of WRBM can help to reduce the maximum
stress and thus achieve weight savings. The numerical expression the WRBM is given as:
Z
WRBM =
b/2
F~n (y) × ~r(y)dy
0
=q
N/2
X
Si cn,i (cos Λi yi + sin Λi zi ) = q
i=1
N/2
X
(6)
Si cn,i ŷi
i=1
where b is the span of the full aircraft wing model, and N is number of the total strips for a
full model including left and right wings. yi and zi are, respectively, the distances from the
i−th strip aerodynamic center to the wing root in spanwise and thickness directions. Si and
cn,i is the reference area and the normal load coefficient for the i−th strip, respectively. For
a horizontal wing, cl = cn . WRBM can be obtained by multiplying the strip lift, li , with the
distance between the strip aerodynamic center and the wing root, yi . For dihedral wings or
vertical aerodynamic surfaces, such as the rudder, the wing root bending moment due to the
normal load components with the moment arm measured from both spanwise and thickness
directions is included. Λ is the local dihedral angle for the aerodynamic strip. The positive
value for Λ is considered when the strip bends upward.
For a linear aeroelastic system, the approximation of WRBM can be expressed as:
WRBM(X) = WRBM(x1,0 , x2,0 , ..., xm,0 )+
∂WRBM
∂WRBM
(x1 −x1,0 )+
(x2 −x2,0 )+... (7)
∂x1
∂x2
7
Expressing Eq. (7) in matrix format:
WRBM(X) = WRBM0 +
∂WRBM
∂x1


x1 







x
∂WRBM
∂WRBM
2
...
.

∂x2
∂xm
 .. 





xm
(8)
The constant term, WRBM0 , in Eq. (8) is
WRBM0 = WRBM(x1,0 , x2,0 , ..., xm,0 ) −
k
X
∂WRBM
∂xi
i=1
xi,0
(9)
where WRBM(x1,0 , x2,0 , ..., xm,0 ) can be obtained using Eq. (6) where the spanwise load
coefficient for each strip, cn,i , is obtained with the initial design variables {x1,0 , x2,0 , ..., xm,0 }T .
∂WRBM
The derivatives of WRBM with respect to the design variables,
, in Eq. (8) can
∂xi
be further expanded to the derivative of the spanwise load distribution cn,i for each strip
with respect to the design variables, X.
N/2
X ∂cn,i
∂WRBM
=q
Si
ŷi
∂xi
∂xi
i=1
(10)
The vector for the derivative of WRBM with respect to each design variable can be
expressed as:



∂WRBM
∂cn,1











∂x1
∂x1


 ∂c



n,1


 ∂WRBM 

∂x2
= q  ∂x2
..








.




 ∂c


∂WRBM 
n,1





∂x
∂xm
m
m×1
∂cn,N/2
∂cn,2
...
∂x1
∂x1
∂cn,N/2
∂cn,2
...
∂x2
∂x2
...
∂cn,N/2
∂cn,2
...
∂xm
∂xm


















S1 ŷ1
S2 ŷ2
..
.
SN/2 ŷN/2





= qC Tn S Ŷ




N/2×1
m× N
2
(11)
where S is a diagonal matrix for strip areas, and Ŷ is a vector for ŷi given in Eq. (6).
Substituting Eq. (11) in Eq. (8), we have
T
WRBM = WRBM0 + q Ŷ S T C n X
(12)
where X is a vector for all the wing shape design variables X = {x1 , x2 , ..., xm }T , which
includes the flap rotations and the wing initial twist. C n is Jacobian of the spanwise normal
load coefficient distribution with respect to the design variables.
Substituting Eq. (11) in Eq. (9),
8
T
WRBM0 = WRBM(x1,0 , x2,0 , ..., xm,0 ) − q Ŷ S T C n X 0
(13)
where X 0 is the initial state of all the design variables, X 0 = {x1,0 , x2,0 , ..., xm,0 }T .
The spanwise normal load distribution cn is obtained from solving linear equation for
vortex strength γ using Vortex Lattice Method (VLM)[27]:
γ
=α
(14)
V
The local normal load coefficient for the j−th aerodynamic element can be obtained
using Kutta-Joukowski theorem[27].
Aaic ·
∆Fn,j
ρV γj ∆yj
=
qSj
1/2ρV 2 ∆yj ∆xj
ρV 2 ∆yj
γj
=
2
1/2ρV ∆yj ∆xj V
1 γj
=2
∆xj V
cn,j =
(15)
where ∆xj and ∆yj are the chord and the span for each aerodynamic element or each strip,
respectively.
Similarly, we can obtain the load coefficient and its derivative with respect to the design
variables for each aerodynamic element by substituting Eq. (14) into Eq. (15):
1
) · A−1
aic · α
∆x
∂cn
1
∂α
Cn =
= 2 · diag(
) · A−1
aic ·
∂X
∆x
∂X
cn = 2 · diag(
(16a)
(16b)
where α is vector for the total downwash applied at each aerodynamic element.
The downwash vector, α, for aerodynamic elements and its sensitivity to the design
variables can be further expanded as:
α = α0 + αi + αe + δ
∂α
∂α0 ∂αe
∂β
=
+
+
∂X
∂X
∂X
∂X
(17a)
(17b)
where α0 and αi are vectors for the trimmed angle of attack and the initial wing twist,
respectively; the vector δ is flap rotations; the vector αe is the local angle of attack due to
the structural elastic deformation. Based on Eqs. (2) and (5), the local elastic twist and the
trimmed angle of attack change linearly with the flap rotations and the initial wing twist.
9
∂α
Hence, the derivative of the total downwash with respect to flap rotations,
, is constant
∂X
when structural model, aerodynamic model and dynamic pressure are given. In other words,
the downwash, α, changes linearly with the flap rotations. Therefore, the spanwise normal
load coefficient, cn , changes linearly with the flap rotations. For a horizontal wing segment,
the spanwise lift distribution, cl , is a linear function in terms of flap rotations.
Figure 1: Multiple control surfaces for a flying-wing type aircraft
To use the same vector of variables for the hinge moment, as given in Eq. (4), and the
total lift and aerodynamic moment, as given in Eq. (5), and the wing root bending moment,
given in Eq. (12), the vector U given in Eq. (4) is expressed in terms of the design variables,
X. For example, the angle of attack, α, and the body flap rotation for the inner wing,
δIW , as shown in Fig. 1, can be obtained from a static aeroelastic equation with the given
set of design variables, X. Vectors X out and X are used to denote the trim variables
obtained from the static aeroelastic equations and the designs variables, respectively. For
the flying-wing case for longitudinal stability, design variables are X = {δ1 , δ2 , δ3 , δ4 , δLE }T
T
and X out = {α, δIW }T . The vector for the total trim variables is U = X T X Tout . The
load factor and the pitch rate, are considered as known variables in trim analysis.
Based on Eq. (5),
(
) (
)
(
)
L
L0
X
=
+ qSC LM
(18)
M
M0
X out
Writing Eq. (18) in matrix form,
(
LM = LM 0 + qS [C LM,inp C LM,out ]
X
X out
)
(19)
where C LM,inp and C LM,out are the stability derivatives for the total lift and the aerodynamic moment with respect to the design variables, X, and the output trim variables, X out ,
10
respectively.
Hence, X out can be obtained:
X out =
=
1 −1
(LM − LM 0 − qSC LM,inp X)
C
qS LM,out
1 −1
C LM,out (LM − LM 0 ) − C −1
LM,out C LM,inp X
qS
(20)
Recall that the vector U is for all variables including the design variables, X and the
output trim variables, X out . Based on Eq. (20), the vector U can be expressed as:
(
) "
#
(
)
X
I m×m
0
U=
=
X+
(21)
X out
CX
X out,0
where
C X = −C −1
LM,out C LM,inp
1 −1
X out,0 =
C
(LM − LM 0 )
qS LM,out
(22a)
(22b)
Hence, the hinge moment for each control surface in Eq. (4) can be further expressed as:
#
(
)
"
0
I m×m
X + qC H
(23)
HM = HM 0 + qC H U = HM 0 + qC H
CX
X out,0
Recall that the wing root bending moment from Eq. (12) expressed as:
T
WRBM = WRBM0 + q Ŷ S T C n X
(24)
Based on Eq. (19), the vector, LM , for the total lift and the aerodynamic moment can
be expressed as
ˆ 0 + qS Ĉ LM X
LM = LM
(25)
where
(
ˆ 0 = LM 0 + qSC LM
LM
"
Ĉ LM = C LM
I m×m
CX
0
X out,0
)
(26a)
#
(26b)
Equations for the hinge moment given in Eq. (23), the wing root bending moment given
11
in Eq. (24), and total lift and aerodynamic moment given in Eq. (25) are all expressed in
terms of the design variables, X.
V.
Drag Minimization
The total drag is often divided into two categories [28], parasite drag and induced drag.
The parasite drag includes the profile drag (skin friction and pressure drag), interference
drag and the wave drag. Since the profile drag and the induced drag contributes around
80% of the total drag [29], this paper focuses on the drag reduction for the profile drag and
the induced drag.
A.
Trefftz Plane Induced Drag
By virtue of the Munk’s theorem[30], the induced drag can be obtained in the Trefftz plane
rather than in the physical plane. Considering the aerodynamics to be symmetric about the
aircraft mid-axis during cruise, the total induced drag coefficient can be obtained numerically
for the half model case as[31]:
CDi =
N/2 N/2
X
X (cn c)i (cn c)j
i=1 j=1
c̄
c̄
Ai,j si
(27)
where N is the number of strips for the complete aircraft model; s is the nondimensionalized
lifting element width (s = l/b); l is the lifting line element length; cn is the normal load
coefficient; c is the local chord for each aerodynamic strip. Ai,j is known as “influence
coefficient matrix ” depending on the geometry for the lifting-line at the Trefftz plane, whose
expression can be found in Ref. [31].
Writing the above equation in the matrix form,
CDi


γ1



γ2
=

...



γN/2
T 



 







A1,1 s1
A2,1 s2
..
.
A1,2 s1
A2,2 s2
..
.
AN/2,1 sN/2 AN/2 sN/2
...
...
..
.
A1,N/2 s1
A2,N/2 s2
..
.
...
AN/2,N/2 sN/2


 γ1

 γ
2


..

.



γN/2





(cn c)i
where Âi,j = Ai,j si and the normalized load coefficient, γi =
.
c̄

 c1




0
...
0


γ1 
cn,1 
c̄












  0 c2 ...

0
γ2
cn,2


c̄
γ=
=
= Ĉcn




... 
... 




...









cN/2  c
γN/2
n,N/2
0 0 ...
c̄
12
= γ T Âγ (28)




(29)
Based on Eq. (17), the local normal load coefficient cn for each strip can be approximated
linearly in terms of the design variables, X = {x1 , x2 , ..., xm }T :
cn = cn,0 +
m
X
∂cn
i=1
∂xi
(xi − xi,0 )
(30)
Expand the equation for the local lift coefficient for each strip and the resulting vector
can be written as:


cn,1



 c
n,2
cn =
..

.



cn,N/2





∂cn,1
∂x1
∂cn,2
∂x1




= cn,0 +







 ∂cn,N/2
∂x1
∂cn,1
∂cn,1
···
∂x2
∂xm
∂cn,2
∂cn,2
···
∂x2
∂xm
...
∂cn,N/2
∂cn,N/2
···
∂x2
∂xm



x


1



 x 

2
= cn,0 +C n X
.. 

.






xm m×1









(N/2)×m
(31)
where the expression for the constant term, cn,0 , is given as:

∂cn,1
∂cn,1
∂cn,1
···

 
∂x2
∂xm
cn,1,0 
 ∂x1



∂c
∂c
∂c



n,2
n,2
n,2
 c
 
···
n,2,0

∂x2
∂xm
−  ∂x1
cn,0 =
..


...
.






 
 ∂cn,N/2 ∂cn,N/2
cn,N/2,0
∂cn,N/2
···
∂x1
∂x2
∂xm


x1,0





x2,0

  ..
.



 xm,0









where cn,i,0 is the spanwise normal load distribution at the initial state of the design variables,
X 0 = {x1,0 , x2,0 , ..., xm,0 }T .
Substituting Eq. (31) into Eq. (29), we have:
γ = Ĉ (cn,0 + C n X) = γ 0 + ĈC n X
(32)
Substituting Eq. (32) into Eq. (28),
T CDi = γ 0 + ĈC n X Â γ 0 + ĈC n X
CDi =
γ T0 Âγ 0
+X
T
T
C Tn Ĉ Âγ 0
+
γ T0 ÂĈC n X
+X
T
T
C Tn Ĉ ÂĈC n X
(33)
T
Because X T C Tn Ĉ Âγ 0 = γ T0 ÂĈC n X, Eq. (33) can be re-written as:
T
T
CDi = γ T0 Âγ 0 + γ T0 ÂĈC n + γ T0 Â ĈC n X + X T C Tn Ĉ ÂĈC n X
13
(34)
When the uniform lifting-line element length is considered for the induced drag analysis,
 = Â, Eq. (34) can be further simplified as:
T
T
CDi = γ T0 Âγ 0 + 2γ T0 ÂĈC n X + X T C Tn Ĉ ÂĈC n X
B.
(35)
Profile Drag
Profile drag includes both the pressure drag and the skin-friction drag. The former, also
named form drag, is due to the pressure distribution acting on the surface and thus depends
on the airfoil shape. The skin friction drag is wall shear force in the boundary layer due to
the viscosity of the air and the resulting friction against the airfoil surface. The turbulent
flow would result in an increased skin friction compared to the laminar flow[32]. There are
several ways to estimate the profile drag in MDAO study. A quadratic function for the profile
drag coefficient in terms of the local lift coefficient has been used a lot due to its explicit
expression [33–35].
However, a quadratic function representation is not valid for all airfoils. For natural
laminar flow airfoil, there exists a drag “bucket” in the drag polar [36]. The quadratic
function thus cannot be used to approximate the drag polar for a natural laminar flow
airfoil. Instead, the profile drag can be estimated using XFOIL for a given airfoil shape
at a given local angle of attack, by considering the viscous effects. Due to the possible
difficulties in using XFOIL for profile drag estimation within MDAO, a response surface for
the profile drag in terms of the airfoil shape through changing the flap rotation and the local
lift coefficient can be used instead of running XFOIL directly for profile drag evaluation
[19, 37, 38].
Instead of using response surface as a blackbox for drag estimation in this paper, we
employ a quadratic function for approximating the profile drag for use within MDAO. This
is acceptable when the studied airfoil is not a natural laminar flow airfoil. Also, the use of
the quadratic function allows us to develop an explicit function for expressing the profile
drag in terms of flap rotations. The profile drag for each strip can be written as a quadratic
function in terms of the local lift coefficient [28, 34],
cdp = cd0 + cd2 · c2n
(36)
The coefficient cd0 is based on the skin friction estimate:
cd0 = Fc · cf
where Fc is a profile drag form-factor, which includes the airfoil thickness effect.
4
2
t
t
+ 100
Fc = 1.0 + 2.7
c
c
(37)
(38)
The determination for the skin-friction factor, cf , depends on the laminar or turbulent
flow properties. For turbulent flow, the van Driest II method is employed to determine the
14
skin-friction factor, cf . The skin-friction factor, cf , is constant when Mach number and
the ratio of the wall temperature to adiabatic wall temperature, TW /TAW are fixed. The
program turbc, developed by Prof. William H. Mason at Virginia Tech, is used to determine
the value for cf [39]. When the airfoil thickness ratio is fixed, the coefficient cd0 remains
constant regardless of flap rotations. The details of calculating the coefficient cd0 can be
found in Ref. [28].
Another skin friction coefficient, cd2 , given in Eq. (36) was computed using the approach
by Wakayama and Kroo [40],
0.38
cd0
cd2 =
cos2 Λ
where Λ is the swept angle.
The total profile drag coefficient, CDp , can be obtained by summing the profile drag over
all strips. The profile drag coefficient for the half wing model is:
CDp
N/2
N/2
1X
1 X
qcdp,i Si =
(cd0 + cd2 c2n )i Si
=
qS i=1
S i=1
(39)
Since cd0 is independent on the flap rotations, Eq. (39) can be re-written as:
CDp
N/2
1X
cd2,i c2n,i Si
= CDp,0 +
S i=1
(40)
where
CDp,0
N/2
1X
=
cd0,i Si
S i=1
The second term in Eq. (40) can be written in matrix format:

cn,1



N/2

cn,2
1
1X
cd2,i c2n,i Si =
..
S i=1
S
.



cn,N/2
T 

cd2,1 S1
0
...
0


 
0
cd2,2 S2 ...
0




...



0
0
... cd2,N/2 SN/2


 cn,1

 c
n,2


..

.



cn,N/2









=
1 T
c C d2 Scn
S n
(41)
where C d2 is a diagonal matrix for the second skin friction coefficient, cd2 , for each strip.
Substituting Eq. (31) and Eq. (41) into Eq. (40), we have:
CDp = CDp,0 +
1 T
1
cn C d2 Scn = CDp,0 + [cn,0 + C n X]T C d2 S [cn,0 + C n X]
S
S
Equation (42) can be further simplified as:
15
(42)
1 T
cn,0 C d2 Scn,0 + 2cTn,0 C d2 SC n X + X T C Tn C d2 SC n X
(43)
S
Therefore, the total drag coefficient for the full model (left and right wings), CD , can be
obtained from Eqs. (35) and (41), which is expressed as:
CDp = CDp,0 +
CD = CDi + 2CDp = CD,0 + 2 · C 1d · X + X T · C 2d · X
(44)
where,
T
1
CD,0 = 2CDp,0 + cTn,0 Ĉ ÂĈcn,0 + 2 cTn,0 C d2 Scn,0
S
T
1 T
1
T
C d = cn,0 Ĉ ÂĈC n + 2 cn,0 C d2 SC n
S
T
1 T
2
T
C d = C n Ĉ ÂĈC n + 2 C n C d2 SC n
S
VI.
Lift Maximization
During takeoff and landing, all flaps are rotated to generate the maximum lift coefficient
and hence to generate the maximum lift at any speed. The change in the airspeed leading
the change in the dynamic pressure causes the stability derivative not constant in the takeoff
and landing flight condition. This section studies the case of the lift maximization for a
given dynamic pressure. The elastic effect to the stability derivatives due to the change in
the dynamic pressure can be obtained using Eq. (2).
For a static maneuver flight condition, the pitch rate and the pitch acceleration are zero.
z̈
For a elastic restrained model, the influence of the vertical accelerator, , has a very small
g
influence on the total lift and aerodynamic moment coefficients. For completeness, the static
longitudinal stability equation of motion can be expressed as [26]:
z̈
CL = CL,0 + CL,α α + CL,β β + CL,z + CL,δ δ
g
z̈
CM = CM,0 + CM,α α + CM,β β + CM,z + CM,δ δ
g
(45a)
(45b)
where CL,0 and CM,0 are the lift and aerodynamic moment coefficient due to the initial wing
twist distribution, αi .
For a statically trimmed aircraft, the total aerodynamic moment can be obtained from
the moment equilibrium equations about the center of gravity:
qScCM = −qSCL (xcg − xae ) − Me
(46)
where Me is the external applied moment about the CG. The positive value of Me means it
rotates the wing nose upward while a negative value implies downward. It should note that
16
the stability derivatives for all moments are calculated about the aerodynamic center when
using Eq. (46) to determine the total aerodynamic moment, CM . When there is only gravity
load considered, Me = 0.
CM = −CL
The vertical acceleration ratio
(xcg − xae )
c
z̈
in Eq. (45) can be further expressed as:
g
z̈
L
q · S · CL
=
=
g
W
W
(47)
(48)
where L is the total lift, and W is the total weight.
Substituting Eqs. (47) and (48) into Eq. (45), the total lift coefficient can be expressed
in terms of flap rotations, δ, angle of attack α, and the initial twist distribution, αi .
(CM,β CL,0 − CL,β CM,0 ) + (CM,β CL,α − Cl,β CM,α )α + (CM,β C L,δ − Cl,β C M,δ )δ
(xc CL,β + CM,β ) + k(CL,β CM,z − CL,z CM,β )
CL = C L0 αi + CL1 α + C L2 δ
xcg − xae
xc =
c̄
qS
k=
W
CL =
(49a)
(49b)
(49c)
(49d)
where C L0 is the derivative of the total lift coefficient with respect to the initial twist
distribution.
For a fixed angle of attack, all flaps when rotated downward will lead to the maximum
lift coefficient (See Eq. (49b)). However, this is not true when elastic effects are considered. Figure 2 shows that all flaps rotate downward to their maximum value leading to the
maximum lift coefficient when using rigid stability derivatives. When the elastic effect is
considered, a few flaps near the wing root rotate downward to their maximum value and the
others rotate upward to the maximum value for the maximum lift coefficient. All flaps with
positive flap angle means rotating downward while negative implies upward. Table 1 shows
the maximum lift coefficient for the two cases. The value for the maximum lift coefficient
for the case including elastic effect is larger than that without including elastic effect, which
demonstrates the benefit of using elastic wing for improving aerodynamic efficiency.
17
Flap rotations, degrees
40
30
20
10
0
-10
-20
Rigid Stability Derivatives
Elastic Stability Derivatives
-30
-40
0
2
4
0
2
4
6
8
10
12
14
16
18
20
6
8
10
12
14
16
18
20
Hinge moment, lb-ft
4000
2000
0
-2000
-4000
Trailing-edge flap from wing root to tip
Figure 2: Optimal flap rotations for maximum lift coefficient, angle of attack α = 1o and δmax =
βmax = 30o ; Goland wing with 20 flaps studied here [11]
Table 1: Maximum lift coefficients using rigid and elastic stability derivatives for Goland wing with
20 flaps
Case
Rigid stability derivative
Elastic stability derivative
CL,max
0.826
0.904
However, the obtained maximum lift at these flap rotations for a given dynamic pressure
could violate some constraints, such as the hinge moment, lift stall, structural failure, large
tip deflection, and so on. The approximation for the constraints of the hinge moment is
given in Eq. (23) and the lift distribution is given in Eq. (31). The constraints on the
structural responses, the tip deflection, dtip , and the element stresses, σ, can be included in
the optimization using the direct method [41] to calculate their sensitivities with respect to
the flap rotations. The sensitivity matrices of the elastic deformation with respect to the
flap rotation and the initial twist are given in Eq. (2).
Till now, we have approximated those aeroelastic response quantities of interest using a
linear relation in terms of the flap rotations and initial twist such as the hinge moment given
in Eq. (23), the wing root bending moment given in Eq. (24), the total lift and aerodynamic
moment given in Eq. (25), the local lift coefficient given in Eq. (31), the induced drag
coefficient given in Eq. (34), the profile drag coefficient given in Eq. (43), the total drag
coefficient given in Eq. (44), and the total lift coefficient given in Eq. (49). We can perform
optimization for the desired performances using the gradient-based optimization approach.
18
VII.
Verifications and Applications
This section is to verify the linear approximations of the hinge moment and the lift
distribution in terms of the flap rotations and then to apply these linear approximations for
different optimization studies. Two models, blended-wing-body and the Goland wing, are
used for studying different optimal aeroelastic performances.
A.
1.
Blended Wing Body
Blend Wing Body Model
One application of the unified framework is for control power minimization of the blendedwing-body (BWB) aircraft studied previously [42, 43]. The BWB aircraft has many control
surfaces as shown in Fig. 3. The structural and aerodynamic model for the BWB aircraft is
shown in Fig. 4. The control power for all surfaces is represented by the sum of the absolute
value for the hinge moment of each surface. Previous studies [42, 43] used an evolutionary
optimization scheme to optimize the flap rotation distribution that results in the minimum
control power. In both studies, instead of running NASTRAN for the cost function evaluation, a response surface of hinge moment in terms of flap rotations was constructed. The first
work [42] developed a neural network of the hinge moment in terms of flap rotations by using NASTRAN SOL 144 results. The second work [43] developed the response surface using
the stability derivative approach and it considered the total lift and aerodynamic moment
equilibria while rotating the flaps to minimize the control power for all surfaces. The second
work is much more efficient in constructing a response surface than that using results from
a large number of NASTRAN analyses for training an artificial neural network. However,
both the works are very computationally expensive for structural optimization because a
genetic algorithm is used as the optimizer for determining the optimal flap rotations.
The model studied here flies in a 2.5G maneuver and it is subjected to several external
applied loads. Considering the aerodynamic loads to be symmetric while trimming the
aircraft for longitudinal stability, all flaps are rotated symmetrically to minimize the control
power.
19
Figure 3: Many control surfaces for X-48B blended-wing body [42, 44]
Figure 4: Structural and aerodynamic model for BWB; Blue: structural mesh; Red: Aerodynamic
mesh
2.
Control Power Minimization
The cost function used here is to minimize the sum of the absolute values of hinge moments
for all control surfaces. So, the expression for the cost function is
fˆ(X) =
k
X
|HMi (X)|
i=1
20
i = 1, 2, ..., 8
(50)
The initial hinge moment, HM 0 , and the non-dimensional stability derivatives of the
hinge moment, C H , for each control surface, with respect to flap rotations can be obtained
from NASTRAN SOL 144. One only needs to run NASTRAN once to get these values. The
expression for each surface hinge moment is given in Eq. (23).
The optimization problem becomes:
P
minimize fˆ(X) = 8i=1 |HMi (X)|
w.r.t
X
s.t
|HMi | ≤ HMmax
i = 1, 2, ..., 8
−3 ≤ α ≤ 10 degrees
|X i | ≤ 37.42 degrees
i = 1, 2, ..., 7
|δ| ≤ 37.42 degrees
|cl,j | ≤ cl,max
j = 1, 2, ..., N
We selected the bound constraints for the flap rotation to be 37.42 degrees. This is
consistent with the value of the optimal configuration obtained in other publication [43].
There are no maximum hinge moment and stall lift constraints for this optimization problem because this study is conducted to validate the accuracy and efficiency of the present
program. As a verification case, the constrained nonlinear programming solver fmincon in
MATLAB is selected as the optimizer for structural optimization although the absolute cost
function leads to a possible local minima. Table 2 shows the comparison among the results
obtained using different approaches. The four flaps for the outboard wing are active and
rotate to the largest possible value allowed, which are very close to the results obtained in
Ref. [43]. It is clearly seen that the present method is much more efficient than the other
two methods in minimizing the control power. Also, the present results improve the control
power efficiency by around 3% as compared to that obtained by Adegbindin et al. [43]. This
efficient optimization framework enables to integrate structural design variables for a further
reduction in the control power consumption for all control surfaces.
Table 3 shows the comparisons of the hinge moments for all flaps obtained from NASTRAN and the present approximation for three different optimal flap rotations results given
in Table 2. For the three cases shown here, results for the hinge moment using the linear
approximation are identical to those obtained from NASTRAN. For the results of hinge moments for elevator and inboard 1 flap obtained from the present approximation, it is believed
that the differences between the two sets of results are due to the numerical errors because
the magnitudes of hinge moments for these two control surfaces are significantly less than
the value of hinge moment for all the other flaps.
21
Table 2: Comparisons of results using different approaches for BWB control power optimization
Variable
ANN+GA
(Chhabra et al. [42])
Stability derivatives+GA
(Adegbindin et al.[43])
Stability derivatives+GBO
Present results
AoA (degrees)
Elevator (degrees)
Inboard 1 (degrees)
Inboard 2 (degrees)
Outboard 1 (degrees)
Outboard 2 (degrees)
Outboard 3(degrees)
Outboard 4 (degrees)
Rudder (degrees)
8.44
8.34
-11.99
-13.83
14.92
37.16
36.18
32.65
0.19
7.95
2.32
-0.54
-20.76
36.33
36.55
37.42
36.79
9.41
8.37
2.23
-9.12
-19.81
37.42
37.42
37.42
37.42
17.82
Sum of abs. HM (lb-in)
2.69E+6
2.20E+6
2.13E+6
CPU Time
∼5 days
∼1 hour
∼6s
Table 3: Comparison of hinge moments between linear approximations and NASTRAN results
(unit: lb-in)
Control surface
Elevator
Inboard 1
Inboard 2
Outboard 1
Outboard 2
Outboard 3
Outboard 4
Rudder
3.
ANN+GA
(Chhabra et al. [42])
Stability derivatives+GA
(Adegbindin et al.[43])
Stability derivatives+GBO
Present results
NASTRAN
Linear approx.
NASTRAN
Linear approx.
NASTRAN
Linear approx.
-6.23E+5
1.05E+4
-3.80E+4
-1.64E+5
-2.02E+5
-1.74E+5
-1.09E+5
-2.50E+4
-6.23E+5
1.05E+4
-3.80E+4
-1.64E+5
-2.02E+5
-1.74E+5
-1.09E+5
-2.50E+4
-2.36e+4
-1.31E+5
-1.03E+4
-2.79E+5
-2.05E+5
-1.78E+5
-1.20E+5
-1.52E+5
-2.36E+4
-1.31E+5
-1.03E+4
-2.79E+5
-2.05E+5
-1.78E+5
-1.20E+5
-1.52E+5
0.03
-0.19
-2.01E+3
-2.86E+5
-2.09E+5
-1.79E+5
-1.22E+5
-2.69E+5
-0.09
-0.21
-2.01E+3
-2.86E+5
-2.09E+5
-1.79E+5
-1.22E+5
-2.69E+5
Wing Root Bending Moment Minimization
A case of minimizing the wing root bending moment, for the BWB aircraft, is next studied. Wing root bending moment minimization is of interest in MDAO study in aircraft
design because the wing root bending moment can be considered as a surrogate for weight
minimization. Based on Eq. (24), the optimization problem for this case is:
22
minimize WRBM(X)
w.r.t
X
s.t
|HMi | ≤ HMmax
−3 ≤ α ≤ 10 degrees
|δi | ≤ 30 degrees
|cl,j | ≤ cl,max
i = 1, 2, ..., 8
i = 1, 2, ..., 8
j = 1, 2, ..., N
Normal load coefficient
Normal load coefficient
Figure 5 shows a comparison of the lift distribution obtained from the linear approximation given in Eq. (31) and the NASTRAN results. These two lift distributions are identical
to each other. This demonstrates the accuracy of using a linear approximation for the lift
distribution with flap rotations. Since the wing root bending moment varies linearly as
a function of flap rotations as seen in Eq. (24), the linear programming solver linprog in
MATLAB is used for this optimization study.
0.5
0
Linear Approximation
NASRTAN results
-0.5
-1500
-1000
-500
0
500
1000
1500
Spanwise distance for main wing, inch
0
-0.05
-0.1
-0.15
-500
-400
-300
-200
-100
0
100
200
300
400
500
Spanwise distance for rudder, inch
Figure 5: Verification of linear distribution from linear approximation and NASTRAN results
cn × c
, for the minimal WRBM are shown in Fig. 6.
The normalized load distribution,
c̄
For comparison, there are two other lift distributions determined using the control surface
rotations from the control power optimization given in Table 2. The lower and upper bounds
for flap rotations for these two cases (minimal HM), Case I and Case II, are 37.42 degrees
and 30 degrees, respectively. The stall lift coefficient is assumed to be 1.5 for the control
power minimization cases. During the WRBM minimization, the maximum rotation for all
flaps is 30 degrees. To demonstrate the influence of the lift stall on the minimum WRBM,
the stall lift coefficient is assumed 1.5 and 0.5 for Case I and Case II, respectively, for the
WRBM minimization. Among these four optimization cases, it is clearly observed that the
lift was moved inboard to reduce the wing root bending moment.
23
Normalized load coefficient
0.75
0.6
0.45
0.3
0.15
0
-0.15
-1500
-1000
-500
0
500
1000
1500
Normalized load coefficient
Spanwise distance along the main wing, inch
0.2
0.1
0
minimal HM, Case I
minimal HM, Case II
minimal WRBM, Case I
minimal WRBM, Case II
-0.1
-0.2
-500
-250
0
250
500
Spanwise distance along the rudder, inch
Figure 6: Normalized load distribution for blended-wing-body along the main wing and rudder for
different optimization objectives
Table 4 shows the optimal flap rotations for different optimization objectives and their
corresponding values for the sum of the absolute hinge moments and the wing root bending
moment. For the two cases studied in the minimal WRBM, the value for the minimum
WRBM increases as the decrease in the stall lift coefficient from cl,max = 1.5 for Case I to
cl,max = 0.5 for Case II.
24
Table 4: Comparisons of flap rotations and aeroelastic performances for different objectives
Variable
Minimal HM
Minimal WRBM
Case I
Case II
Case I
Case II
AoA (degrees)
Elevator (degrees)
Inboard 1 (degrees)
Inboard 2 (degrees)
Outboard 1 (degrees)
Outboard 2 (degrees)
Outboard 3(degrees)
Outboard 4 (degrees)
Rudder (degrees)
8.37
2.23
-9.12
-19.68
37.42
37.42
37.42
37.42
17.82
8.19
2.21
-11.20
-13.17
30.00
30.00
30.00
30.00
30.00
6.11
30.00
30.00
-20.22
-30.00
-30.00
-30.00
-30.00
-30.00
7.30
30.00
23.50
-30.00
-30.00
-30.00
-30.00
-27.44
-7.79
Sum of abs. HM (lb-in)
WRBM (lb-in)
2.134E+6
1.747E+8
2.407E+6
1.683E+8
8.650E+6
1.931E+7
8.020E+6
2.233E+7
Note: positive value means flap rotates downward; negative, upward
The minimal values for the control power and WRBM for the four optimization cases are
depicted in Figure 7, which shows that the minimal control power and the minimal WRBM
cannot be achieved simultaneously. When one tailors the flap rotations for the minimal
WRBM, it needs more power consumption to deflect all flaps to the desired rotations. Note
that the control surface rotations obtained for the minimum control power leads to a much
larger wing root bending moment, which could lead to an increase in the weight to prevent
failure. Hence, there is a trade-off between the control power consumption and weight
savings. For a wing with multiple surfaces in the trailing edge or/and leading edge, one
needs to use optimization to tailor the flap rotations for the desired performance.
25
Figure 7: Pareto frontier for minimum control power and minimum WRBM
B.
Drag Minimization for Goland Wing
1.
Verification of induced drag minimization
A simple case studied by IDRAG is employed here for program validation. IDRAG [45] is
a program developed by Prof. William H. Mason at Virginia Tech for induced drag minimization by optimizing the lift distribution. The program has the capabilities of calculating
the induced drag for a given lift distribution and that of optimizing the spanwise lift distribution for the minimum induced drag. For the optimization problem in IDRAG, the total
lift coefficient and the total moment coefficient are integrated with the cost function using
the Lagrange multipliers. A test model used in IDRAG including the main wing, winglet
and a tail is employed to verify the current program in computing the induced drag and the
optimal lift distribution resulting in the minimal induced drag. The geometric dimensions
for this test model are available in Ref. [45] or the available files online a . The total lift
coefficient for this configuration is assumed to be CL = 1. There is no moment constraint in
Case I and the moment constraint about the CG is 0 in Case II. Figure 8 shows the optimal
lift distribution calculated from both IDRAG and the present method, respectively, using
the method of Lagrange multipliers and quadratic programming[41]. The two optimal lift
distributions obtained from the present method match very well with those calculated from
IDRAG. Also, the minimal induced drag coefficients calculated from these two methods are
very identical to each other, as shown in Table 5. This case demonstrates the accuracy of
using the present method in both induced drag calculation and its minimization by tailoring
the lift distribution.
a
Files are available in https://github.com/zhaowei0566/TrimOpt/tree/master/IDRAG_Test
26
Local lift coefficient distribution
1
0.8
0.6
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0.6
c n c/cbar
0.6
c n c/cbar
1
0.5
0
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.4
0.2
0
0.1
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.02
0.05
c n c/cbar
0.1
c n c/cbar
Local lift coefficient distribution
1.5
c n c/cbar
c n c/cbar
1.2
Present results
IDRAG
0
Present results
IDRAG
-0.04
-0.06
-0.08
0
0.2
0.4
0.6
0.8
1
0
Normalized locations of vortices for main wing, winglet and tail
0.2
0.4
0.6
0.8
1
Normalized locations of vortices for main wing, winglet and tail
(a) Case I: CL = 1
(b) Case II: CL = 1 and CM = 0
Figure 8: Optimal lift distribution comparison for the main wing, winglet and the tail with lift and
pitch moment constraints between IDRAG results and present results; the aerodynamic center is
aft CG location for this model
Table 5: The minimal induced drag coefficient CDi calculated from the present method and IDRAG
2.
Case
IDRAG
Present method
Diff. (%)
(a) Case I
5.144E-2
5.144E-2
0.000
(b) Case II
5.212E-2
5.207E-2
-0.096
Optimal flap rotations
The optimal lift distribution resulting in the minimum induced drag can be obtained by
scheduling the control surface rotations. The Goland wingb , studied earlier by Kolonay and
Eastep[11, 46], is used to determine the flap rotations that will result in the lowest induced
drag. The properties for the Goland wing are tabulated in Table 6. Twenty discrete uniform
straight flaps located at the trailing-edge for the half wing model are considered to tailor the
lift distribution during cruise for obtaining the lowest induced drag. An additional horizonal
elevator is added aft the main wing for the longitudinal stability. The horizontal elevator has
no structural model associated with it. It is only modeled to determine aerodynamic loads
b
NASTRAN files are available in https://github.com/zhaowei0566/TrimOpt/tree/master/Goland_
Wing
27
and the resultant loads are transferred to the main wing through a rigid transformation using
the resulting forces and moments. The structural and aerodynamic models for the Goland
wing are shown in Fig. 9.
Table 6: Properties for the Goland Wing; The values labeled with star * are obtained from Ref.[46]
Parameter
Value
Semi-span L (ft)
20.0
Semi-chord b (ft)
6.0
Mass of the wing per unit length m (lb/ft)
360.03
Aerodynamic center
25% chord
Spanwise elastic axis
33.3% chord
Center of gravity
43% chord
2
Wing bending stiffness EIxx (lb·ft )
2.365 × 107
Wing torsional stiffness GJ (lb·ft2 )
2.39 × 106
Additional mass moment of inertia distribution about CG Ic (lb·ft2 /ft)
808.11
Total weight (lb)
first mode frequency (rad/s)
second mode frequency (rad/s)
7200
12.7 (11.9*)
24.2 (23.7*)
20 ft
𝑉∞
6 ft
10 ft
6 ft
(b) 20 discrete flaps at the trailing edge
3 ft
(a) Goland wing with a horizontal elevator
Figure 9: Aerodynamics and structural model for the Goland wing with 20 discrete trailing-edge
flaps; dynamic pressure, q=322 psf and Mach number, M =0.7
The lift distribution obtained from the linear approximation in terms of flap rotations
28
is verified with those obtained using NASTRAN. The 20 flap rotations listed in Table 7
obtained from Ref. [11] are used to compute the lift distribution for verification. Figure 10
shows the spanwise lift coefficient distribution, cl , for both the main wing and the elevator
as obtained from the present approximation as well as NASTRAN. Both results are identical
to each other.
Table 7: Flap rotations (degrees) for lift distribution verification
δ1
δ2
δ3
δ4
δ5
δ6
δ7
δ8
δ9
δ10
3.06
2.84
2.61
2.36
2.08
1.79
1.51
1.21
0.917
0.627
δ11
δ12
δ13
δ14
δ15
δ16
δ17
δ18
δ19
δ20
0.346
0.0752
-0.190
-0.437
-0.670
-0.870
-1.022
-1.153
-1.377
-1.888
Note: positive value means flap rotates downward; negative upward
0.25
Lift coefficient
0.2
Present - main wing
Present - elevator
NASTRAN - main wing
NASTRAN - elevator
0.15
0.1
0.05
0
0
5
10
Spanwise distance,ft
15
20
Figure 10: Spanwise lift distribution obtained from the present approximation
Based on the results presented in preceding section explaining various verification studies
for the induced drag minimization with respect to the load distribution and the accurate
spanwise lift distribution approximation in terms of the flap rotations, we are confident
that the present framework accurately performs induced drag minimization by optimizing
the flap rotations. Instead of enforcing the lift distribution to be elliptical as in Ref. [11],
we directly optimize the flap rotations for the optimal lift distribution that results in the
minimal induced drag for the Goland wing. The design variables are X = {δ1 , δ2 , ..., δ20 }T .
The optimization problem is summarized as:
29
minimize CDi
w.r.t
X
s.t
|X| ≤ 10 degrees
|β| ≤ 10 degrees
where β is the elevator rotation angle.
There are no equality constraints for the total lift and the total aerodynamic moment
because for a trimmed aircraft the change in the control surface rotations leads to a change
in the angle of attack and elevator while keeping the total lift and aerodynamic moment
unchanged. The lift line element length, si , is same for both the main wing and the tail,
hence  = A. Quadratic nonlinear programming solver quadprog in MATLAB is used for
this optimization.
Figure 11 shows the optimal lift distribution and the corresponding flap rotations for the
wing with a horizontal tail resulting in the minimal induced drag. The positive values for
the flap rotations means that they are rotating downward while negative implies upward.
For comparisons, IDRAG is also used to obtain the optimal lift distribution for the minimal induced drag. Figure 11 shows the lift distribution, in the presence of the optimal flap
rotations, is close to that obtained using IDRAG. It was also found that the flap rotation distribution for the trailing-edge control surfaces corresponding to the optimal lift distribution
is close to being elliptical. The first 6 flap rotations are used to balance the pitch moment
from the horizontal tail. The minimal induced drag coefficient obtained from the present
method is very close to that obtained from IDRAG as shown in Table 8, which demonstrates
that the optimal lift distribution for the minimal induced drag can be obtained by scheduling
the trailing-edge flap rotations.
The obtained optimal flap rotations in Fig. 11 are not the same as those given in Ref. [11]
(Table 7). This is because different approaches are being used for induced drag minimization
in Ref. [11] and the present work. Instead of enforcing the lift distribution to be elliptical as
was done in Ref. [11], the present approach directly optimizes the flap rotations for obtaining
the minimum induced drag. Also, the cost function used in Ref. [11] is to minimize the sum
of the square difference between the spanwise lift distribution created using flaps and the
elliptical lift distribution. The use of the mean square error might be a misleading indication
of average error [47]. This is the reason why the minimal induced drag coefficient obtained
in Ref. [11] is larger than both the present result and that obtained from IDRAG as shown in
Table 8. The original value for the minimal induced drag coefficient is 4.3E-4 in Ref. [11] for
a full model, the authors in Ref. [11] used the lift for a half model to calculate the induced
drag for a full model. For comparison, we scale the lift distribution without any influence
on the optimal results in Ref. [11] and the induced drag coefficient is scaled by a factor of 4
accordingly.
30
Source
Present result
IDRAG
Ref. [11]
Value
1.64E-3
1.64E-3
1.72E-3
0.25
0.2
0.15
0.1
0.05
Present result − main wing
Present result − tail
IDRAG − main wing
IDRAG − tail
0
0
flap rotation, degrees
Normalized lift coefficient
Table 8: Induced drag coefficients obtained from the present method and IDRAG
5
10
Spanwise distance, ft
15
20
3
2
1
0
0
5
10
15
Trailing−edge flap from wing root to tip
20
Figure 11: Optimal normalized lift distribution and the corresponding flap rotations resulting in
the minimum induced drag
3.
Total Drag Minimization
To calculate the friction force due to the turbulent flow for the profile drag, the airfoil used
in previous study [19] as shown in Fig. 12 is employed for the Goland wing for the total drag
minimization study. The thickness ratio for this airfoil is t/c = 0.105. The flight altitude is
20,000 ft in determining the skin friction coefficient given in Eq. (37).
31
Figure 12: The airfoil used for the Goland wing
The optimization problem for the total drag minimization is summarized as:
minimize
w.r.t
s.t
CD
X
|X| ≤ 10 degrees
|HMi | ≤ HMmax
|β| ≤ 10 degrees
i = 1, 2, ..., 21
Figure 13 shows the optimal lift distribution for different drag minimization cases. For
comparison, the optimization study for the minimal induced drag is conducted. The first two
optimization cases are the induced drag and the total drag minimization with only the bound
constraints on the flap rotations. The bound constraints for all flaps are not active in the
first two optimization cases. All control surfaces rotate downward and their corresponding
hinge moment values are less than 300 lb-ft. As expected, when there is no active constraint,
the optimal lift distribution for the minimal induced drag is close to be elliptical.
When there is one more constraint in the hinge moment for each control surface (HMmax =
200 lb-ft) enforced for the total drag minimization, the flaps near the wing tip rotate upward
and the hinge moment for the 7-th flap becomes active as shown in Fig. 13. Table 9 shows
both the induced drag coefficient and the profile drag coefficient for the three optimization
cases. The active hinge moment constraint leads the increase in the value for the minimal
total drag. As expected, the non-elliptical lift distribution increases the induced drag coefficient. However, since the profile drag for each aerodynamic strip is independent on the lift
distribution in other strips, the non-elliptical lift distribution leading to the decrease in the
lift coefficient near the wing root for the main wing could help to reduce the profile drag.
32
0.25
Minimal induced drag, w/o HM constraints
Minimal total drag, w/o HM constraints
Minimal total drag, w/ HM constraints
4
3
0.2
2
1
0
Lift coefficient
Flap rotations, degrees
Hinge moment, lb-ft
5
-1
0
5
10
15
20
100
0.15
Minimal induced drag, w/o HM constraints
Minimal total drag, w/o HM constraints
Minimal total drag, w/ HM constraints
0.1
0
-100
0.05
-200
0
-300
0
5
10
15
0
20
Trailing-edge flap from wing root to tip
5
10
15
20
Spanwise distance, ft
(a) Optimal flap rotations
(b) Lift distribution for minimal drag coefficient
Figure 13: Optimal normalized lift distribution and the corresponding flap rotations for the minimum drag
Table 9: Minimal drag coefficient for different optimization cases
Case
Drag Coeff.
I: Minimal induced drag
II: Minimal total drag
(a) w/o HM constraints
(b) w/ HM constraints
Induced drag coefficient, CDi
Profile drag coefficient, CDp
1.6371e-3
3.2138e-3
1.6382E-3
3.2114E-3
1.6407E-3
3.2102E-3
Total drag coefficient, CD
4.8509e-3
4.8496E-3
4.8509E-3
4.
Optimal Twist Distribution for Minimal Drag
In addition to using flap rotations to tailor the spanwise lift distribution, the initial wing
twist distribution can be also used to optimize the lift distribution resulting in the minimal
total drag. A spanwise linear distribution for any washout distribution and any specified
lift coefficient resulting in the minimal induced drag is considered [48]. Both the minimal
induced drag and the minimal total drag are studied.
!
p
1 − (2y/b)2
γ(y) = γ0 1 −
(51)
c(y)/croot
where b is the span for the full wing model, γ0 is the maximum twist at the wingtip. The
negative value of γ0 means the wingtip nose twists downward while positive implies upward.
croot is the chord for the strip at the wing root.
The design variables include both flap rotations and wingtip twist, X = {δ1 , δ2 , ..., δ20 , γ0 }T .
The optimization problem for the case is summarized as:
33
minimize
w.r.t
s.t
CD
{X}
|δi | ≤ 10 degrees
|γ0 | ≤ 3 degrees
|β| ≤ 10 degrees
|HMi | ≤ HMmax
i = 1, 2, ..., 20
i = 1, 2, ..., 21
Optimization results for the minimal induced drag and the minimal total drag without
hinge moment constraints show the lift distribution given in Fig. 14 and values for the drag
coefficients given in Table 10 are almost similar as those shown in Fig. 13 and Table 9. For
the case of the total drag minimization including the hinge moment constraint (HMmax =
200 lb-ft), the optimal twisted wing helps to further reduce the value of the minimal total
drag as compared to the one given in Table 9. The optimal lift distributions for the total
drag minimization with and without hinge moment constraints are almost identical.
Additionally, the optimal twist distribution reduces the value for the maximum hinge
moment in the total drag minimization. For the wing without initial twist in the total drag
minimization study, the hinge moment is an active constraint shown in Fig. 13a. When the
initial wing twist distribution is considered, there are no active constraints in the case of the
minimal total drag. This demonstrates that it is possible to use the optimal twisted wing
to relieve the torque requirement for the actuators in the drag minimization during cruise.
The optimal twisted wings for the three drag minimization cases are shown in Fig. 16.
Table 10: Minimal drag coefficient for different optimization cases
Case
Drag Coeff.
I: Minimal induced drag
II: Minimal total drag
(a) w/o HM constraints
(b) w/ HM constraints
Induced drag coefficient, CDi
Profile drag coefficient, CDp
1.6371e-3
3.2137e-3
1.6381E-3
3.2115E-3
1.6383E-3
3.2113E-3
Total drag coefficient, CD
4.8508e-3
4.8496E-3
4.8496E-3
34
0.25
Minimal induced drag, w/o HM constraints
Minimal total drag, w/o HM constraints
Minimal total drag, w/ HM constraints
4
3
0.2
2
1
Lift coefficient
Flap rotations, degrees
Hinge moment, lb-ft
5
0
0
5
10
15
20
0
0.15
Minimal induced drag, w/o HM constraints
Minimal total drag, w/o HM constraints
Minimal total drag, w/ HM constraints
0.1
-100
0.05
-200
0
-300
0
5
10
15
0
20
5
10
15
20
Spanwise distance, ft
Trailing-edge flap from wing root to tip
(a) Optimal flap rotations
(b) Lift distribution for minimal drag coefficient
Figure 14: Optimal normalized lift distribution and the corresponding flap rotations for the minimum drag
Figure 15: Optimal wing shape for Goland wing in drag minimization; Scale factor: 10
Figure 16 shows the variation of the minimal drag with flight altitude at Mach=0.7. The
dynamic pressure decreases with the altitude when Mach = 0.7 is considered for all altitudes.
The spanwise lift distribution increases with altitude while keeping the aircraft in 1G cruise
flight, and hence to cause the induced drag increases with altitude. However, the skin friction
factor decreases with the flight altitude which leads the value for the minimal profile drag to
decrease. The change of the minimal values for the profile drag and the induced drag leads
it possible to obtain the minimal value for the total drag at a certain flight altitude during
cruise.
35
0.07
Minimal total drag
Minimal profile drag
Minimal induced drag
0.06
D/L
0.05
0.04
0.03
0.02
0.01
1.5
2
2.5
3
3.5
Figure 16: Influence of altitude on ratio of
C.
4
4.5
# 104
Altitude, ft
D
during cruise at Mach=0.7
L
Lift Maximization
The Goland wing studied in previous section is employed here to conduct the lift maximization at a given angle of attack during takeoff and landing flight conditions. The dynamic
pressure is calculated using the density at sea level and Mach = 0.22. The flap rotations and
the initial twist distribution are both considered as design variables in the lift maximization
study. Since the total lift coefficient is a linear function of flap rotations as given in Eq. (49),
linear programming can be used with the objective to minimize (-CL (X)). The angle of
attack is fixed. The vector for the design variables, X, is the 20 trailing-edge flap rotations
and the wingtip twist, X = {δ1 , δ2 , ..., δ20 , γ0 }T . The optimization problem is
minimize
w.r.t
s.t
−CL
{X}
|δi | ≤ 30 degrees
|γ0 | ≤ 3 degrees
|β| ≤ 30 degrees
|HM i | ≤ HMmax
|cl,j | ≤ cl,max
i=1,2,...,20
i=1,2,...,21
j=1,2,..., N
During takeoff and landing, all flaps are required to rotate to generate a larger lift coefficient. The maximum values for the hinge moment is assumed to be HMmax =200 lb-ft for a
demonstration study, and the lift stall for each strip is cl,max =1.5 for the studied airfoil from
previous study [19]. The maximum rotation angles for both the trailing-edge flaps and the
elevator are 30 degrees.
Table 11 shows the maximum lift coefficient for the Goland wing for different optimization cases. When there is no hinge moment constraint, lift stall becomes an active constraint
36
in lift maximization for different angles of attack. The increase in the angle of attack increases the spanwise lift distribution and hence increases the maximum lift coefficient as
shown in Fig. 17. It is found that the maximum hinge moment influences the maximum lift
coefficient significantly. The maximum hinge moment becomes an active constraint in the
lift maximization study when the hinge moment constraint is enforced. It is the active hinge
moment constraint that reduces the value of the spanwise lift distribution and hence leads
the decrease in the maximum lift coefficient.
Optimization results show that the optimal twist distribution helps increase the maximum
lift coefficient but with very small increase. For all cases studied here, the wingtip twist
rotation reaches to its maximum value, γ0 = 3o . It means that the twisted wing with the
wingtip nose rotated upward increases the maximum lift coefficient for takeoff and landing
flight conditions.
Table 11: Maximum lift coefficient, CL,max , for different optimization cases; sea level, M=0.22 and
q = 69.2 psf
I: α = 2o
Case
Constraints
(a) Flaps
(b) Flaps+Twist
(a) Flaps
(b) Flaps+Twist
0.661
0.348
0.678
0.370
0.693
0.429
0.704
0.450
30
1.6
1.5
20
1.2
10
Lift coefficient
Flap rotations, degrees
w/o HM Constraints
w/ HM Constraints
0
0
2
4
6
8
10
12
14
16
18
20
0
Hinge moment, lb-ft
II: α = 4o
0.9
, = 2 o , (a) Flaps,
w/o HM constraints
, = 2 o , (b) Flaps + Twist, w/o HM constraints
0.6
, = 4 o , (a) Flaps + Twist, w/o HM constraints
, = 4 o , (b) Flaps + Twist, w/ HM constraints
0.3
-200
-400
0
, = 2o , (a) Flaps,
w/o HM constraints
, = 2o , (b) Flaps + Twist, w/o HM constraints
-600
, = 4o , (a) Flaps + Twist, w/o HM constraints
, = 4o , (b) Flaps + Twist, w/ HM constraints
-0.3
-800
0
2
4
6
8
10
12
14
16
18
0
20
5
10
15
20
Spanwise distance, ft
Trailing-edge flap from wing root to tip
(b) Lift distribution for maximal lift
(a) Optimal flap rotations
Figure 17: Optimal flap rotation and lift distribution for maximum lift at two different AoAs
VIII.
Conclusions
This paper presents a static aeroelastic shape optimization framework for aircraft wings
with multiple control surfaces, for different optimal performance measures, using gradient37
based optimization. For a linear aeroelastic system used in the conceptual/preliminary design
stages, the aeroelastic performance indices, such as the hinge moment and the spanwise
lift distribution can be approximated linearly in terms of the wing shape including the
flap rotations and the initial twist distribution. The induced drag and the total drag can
be expressed in a quadratic form of the wing shape. Gradient-based optimization is used
to optimize the wing shape for obtaining the optimal performances, such as the minimal
total drag during cruise, the minimal wing root bending moment during maneuver and
the maximum lift during takeoff and landing. The accuracy of the linear approximations
and the optimized results have been verified against results obtained from NASTRAN and
those available in literature. The linear approximation allows the use of the gradient-based
optimization leading to a very efficient structural optimization in sizing aircraft wing with
multiple control surfaces.
The optimization studies for the control power minimization and the wing root bending
moment minimization for a blend-wing-body aircraft reveal that the minimum control power
and the minimum wing root bending moment cannot be achieved simultaneously. One needs
to perform a multiobjective optimization to determine the optimal set of flap rotations or to
minimize the wing root bending moment subjected to hinge moment constraints.
The induced drag minimization study results show that the flap rotation distribution
leads to a lift distribution being close to elliptical results in a minimum induced drag. The
induced drag increases as the lift distribution becomes non-elliptical. Because the profile
drag for one aerodynamic strip is independent on the lift in other strips, the non-elliptical
lift distribution could help to decrease the profile drag. The optimal twisted wing can help
to further reduce the value for the minimal total drag.
The lift maximization with respect to the flap rotations study results show that the elastic
effect influences the optimal flap rotations for the maximum lift coefficient. During takeoff
and landing, the increase in the angle of attack help to increase the maximum lift coefficient.
The optimal twisted wing with the wingtip rotated upward increases the maximum lift
coefficient.
The proposed developments can be used for performing multidisciplinary design, analysis
and optimization of future aircraft with flexible wings and multiple control surfaces. For the
aircraft wings at the conceptual/preliminary design stages including both size and shape
design variables, this unified static aeroelastic shape optimization framework can rapidly
determine the optimal wing shape leading to an optimal lift distribution for various optimal
aeroelastic performances for any configuration within the MDAO processing.
Acknowledgements
The authors would like to acknowledge NASA NRA, “Lightweight Adaptive Aeroelastic
Wing for Enhanced Performance Across the Flight Envelope,” NRA NNX14AL36A, Mr.
John Bosworth Technical Monitor, for funding this research. Also, the authors acknowledge
the finite element model of BWB obtained from NASA, and Mr. Moustaine Adegbindin
and Mr. Nathan Love at Virginia Tech for their help with BWB model. The authors also
38
thank Dr. Raymond M. Kolonay of the AFRL for providing the ASTROS input files for the
Goland wing.
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