Thesis_Report_jan_hoogervorst.

Thesis_Report_jan_hoogervorst.
Wing aerostructural optimization
using the Individual Discipline
Feasible architecture
Delft University of Technology
Jan E.K. Hoogervorst
W ING AEROSTRUCTURAL OPTIMIZATION
USING THE I NDIVIDUAL D ISCIPLINE
F EASIBLE ARCHITECTURE
by
Jan Eduard Kornelis Hoogervorst
in partial fulfillment of the requirements for the degree of
Master of Science
in Aerospace Engineering
at the Delft University of Technology.
Supervisor:
Thesis committee:
Date:
Thesis registration number:
Student number:
Cover page image:
Dr. A. Elham
Dr. Ir. L. L. M. Veldhuis
Dr. R. P. Dwight
10-02-2015
074#16#MT#FPP
1377256
Edited. Credits: Björn, https://www.flickr.com/photos/[email protected]/
An electronic version of this thesis is available at http://repository.tudelft.nl/.
S UMMARY
This thesis presents an effort to contribute to the minimization of fuel use of aircraft. The intention is to
achieve this efficiently by using the Individual Discipline Feasible architecture for solving a gradient-based,
aerostructural Multidisciplinary Design Optimization problem for a static aeroelastic wing. This wing is then
optimized for minimal fuel consumption during the cruise phase.
The work in this paper is an effort continuing in the trend of using high-fidelity analyses for optimization of
an aeroelastic wing. However this effort makes use of the Individual Discipline Feasible architecture, which
decouples the aerodynamic and structural disciplines from each other. Using this approach the consistency
of the system as a whole is maintained by the use of equality constraints on surrogate design variables. No
coupled sensitivity information is required because of this decoupled system. This makes the system not only
simpler, but also provides more freedom in software choice. Furthermore, the time to perform optimization
is reduced with respect to the traditional Multidisciplinary Feasible architecture as the work of making the
system consistent is removed from the computationally expensive individual disciplines and put it in the
hands of the cheap optimization algorithm.
The aerodynamic and the structural disciplines hence independently calculate both the intermediate states
of the system and the partial derivatives of these states with respect to the design vector. The performance
module is dependant on the aerodynamic discipline and is therefore included in it. This module calculates
the fuel weight.
SU2 is used within the aerodynamic discipline to deform the surface grid and the volume grid of the wing
and its domain, calculate the flow properties and gain sensitivities of lift and drag with respect to surface
perturbations of the wing. The software uses the 3D Free-Form Deformation parameterization technique to
deform the surface grid. The code is originally meant to only deform the airfoil shapes of a wing, nevertheless it has been modified in order to also deform the wing according to the static aeroelastic deformation.
The sensitivity analysis is performed by a continuous adjoint solver. It is shown however that the continuous adjoint method in SU2 does not capture the sensitivities of the trailing edge well due to assumptions of
smoothness. This is why extra corrective factors are added to the false sensitivities. The results of the optimization verify the working of these corrective factors, except for the corrected sensitivities with respect to
the planform design variables.
The Euler model is used for the flow analysis, due to its speed advantages. However, because viscosity is
neglected in the Euler model, the viscous drag component and its sensitivity derivatives are estimated by a
separate module.
For the structural discipline the FEMWET software is used, providing the static aeroelastic deformation and
the aeroelastic axis of the wing. FEMWET uses equivalent panel thicknesses representing the wing box to
calculate its deformation and failure modes. Its code is slightly modified for the purpose of this work, namely
the Free-Form Deformation parameterization is included for the deformation of the airfoils.
Lastly, the weight of the wing structure is estimated by summing up the weight of the equivalent panels and
empirical data.
The optimization design variables are selected to be the angle of attack, the exterior shape of the wing, being
the airfoil and planform shapes, and the thicknesses of the equivalent panels representing the internal wing
box. The problem is constraint by compression, tension, shear, buckling and fatigue failure modes. Moreover
it is constraint by a minimum aileron effectiveness and a maximum wing loading. The aerodynamic analysis
is performed under cruise conditions while the wing structure is also analyzed under the load cases of pull
op, push over, gust load and a roll manoeuvre. The reference aircraft used throughout this report is a modern
high-speed transport aircraft, the Airbus A320. The optimization algorithm chosen is The Sparse Nonlinear
Optimizer, based on the Sequential Quadratic Programming optimization algorithm.
iii
iv
0. S UMMARY
In this report the results of three aerostructural and one pure aerodynamic optimization problem are presented.
The aerodynamic optimization successfully showed the working of the aerodynamic discipline when controlled by the optimization algorithm and the ability to reduce induced and wave drag simultaneously.
The first aerostructural optimization was performed with a fixed planform. This optimization has resulted in
a 7% reduction in the aircraft fuel weight. It is shown by this optimization that both the optimization tool and
the corrective factors for the aerodynamic sensitivities work.
The second optimization included all planform design variables. It gave disappointing results. The fuel
weight was reduced with 9%, however the equality constraint were violated too severe, making the solution
invalid. It appeared that the correction factors did not work for the the sensitivities with respect to the planform design variables. To confirm this statement, the last optimization is performed with the sensitivities
with respect to the planform design variables calculated by Finite Difference.
This last optimization resulted in a success. The optimizer managed to reduce the aircraft fuel weight with
11%, with an acceptable amount of constraint violation.
P REFACE
The present master thesis was accomplished within the faculty of Aerospace Engineering at the TU-Delft. It
is an effort to contribute to the minimization of fuel use of aircraft by efficiently making use of high-fidelity
analysis in the early stages of multidisciplinary wing design.
I would like to thank my supervisor Dr. Ali Elham for the support and input throughout this project. Furthermore I would like to thank the peers of room 6.01 for their company and interesting discussions on the
topic of aircraft design. Last but not least I would like to thank my family, friends and especially my girlfriend
for their support and interest in my work.
Jan Hoogervorst
Delft, February 2016
v
C ONTENTS
Summary
iii
Preface
v
List of Figures
ix
List of Tables
xi
Nomenclature
xiii
1 Introduction and problem description
1.1 Wing aeroelasticity . . . . . . . . . . . . . .
1.2 Prior aerostructural wing optimization efforts
1.3 Research objective . . . . . . . . . . . . . .
1.4 Thesis outline . . . . . . . . . . . . . . . . .
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1
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2 IDF architecture for wing multidisciplinary design optimization
2.1 Optimization architecture. . . . . . . . . . . . . . . . . . .
2.2 Aerostructural optimization problem formulation . . . . . .
2.2.1 XDSM . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Objective function . . . . . . . . . . . . . . . . . . .
2.2.3 Design variables . . . . . . . . . . . . . . . . . . . .
2.2.4 Constraints . . . . . . . . . . . . . . . . . . . . . .
2.3 Optimization algorithm . . . . . . . . . . . . . . . . . . . .
2.3.1 Sequential Quadratic Programming . . . . . . . . . .
2.3.2 Optimality Conditions . . . . . . . . . . . . . . . . .
2.4 Decoupled sensitivity analysis . . . . . . . . . . . . . . . .
2.4.1 Objective function sensitivity . . . . . . . . . . . . .
2.4.2 Constraint sensitivity . . . . . . . . . . . . . . . . .
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3 Aerodynamic and performance analysis
3.1 Stanford University Unstructured . . . . .
3.2 Free-Form Deformation parameterization
3.2.1 Implementation . . . . . . . . . .
3.3 Surface and volume grid . . . . . . . . .
3.3.1 Grid deformation . . . . . . . . .
3.4 Computational Fluid Dynamics . . . . . .
3.4.1 Viscous drag estimation . . . . . .
3.4.2 Validation . . . . . . . . . . . . .
3.5 Sensitivity analysis . . . . . . . . . . . .
3.5.1 Adjoint method . . . . . . . . . .
3.5.2 Sensitivity projection . . . . . . .
3.5.3 Sensitivity verification . . . . . . .
3.5.4 Sensitivity error origin . . . . . . .
3.6 Fuel weight analysis. . . . . . . . . . . .
3.7 Analysis structure . . . . . . . . . . . . .
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4 Structural analysis
4.1 EMWET/FEMWET . . . . . . . . . .
4.2 Weight estimation. . . . . . . . . . .
4.3 Static aeroelastic deformation analysis
4.3.1 Gird . . . . . . . . . . . . . .
4.3.2 Aerodynamic analysis . . . . .
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viii
C ONTENTS
4.3.3 Finite Element Analysis . . . . .
4.3.4 Aerostructural coupling . . . . .
4.4 Failure modes and aileron effectiveness.
4.5 Validation and verification . . . . . . .
4.5.1 Sensitivity verification . . . . . .
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6 Results
6.1 Aerodynamic optimization . . . . . . . . . .
6.2 Fixed planform aerostructural optimization .
6.3 Complete aerostructural optimization . . . .
6.3.1 Adjoint planform sensitivities . . . . .
6.3.2 Finite Difference planform sensitivities
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5 Test case application
5.1 Load cases . . . . . . . . . . . . . .
5.2 Initial surface and volume grid . . . .
5.2.1 Geometry . . . . . . . . . . .
5.2.2 Initial surface and volume grid.
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7 Conclusions and recommendations
65
7.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Bibliography
67
Appendix
71
L IST OF F IGURES
1.1
1.2
1.3
1.4
Wingbending of a Boeing 787 Dreamliner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wingbending of a Boeing B-52 Stratofortress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spanwise lift distribution difference between a Boeing B-47 Stratojet flexible and rigid wing . . .
Chordwise pressure distribution difference between a Boeing B-52 Stratofortress felxible and
rigid wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1 XDSM of the MDF architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 XDSM of the IDF architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 XDSM of the present fuel weight optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
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11
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
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28
Example of Free-Form Deformation: From a sphere to a blended-wing-body . . . . . . . . . . . .
Example of the usage of two FFD boxes to realize a translation in the z-direction in SU2 . . . . .
Example of an airfoil shape deformation using FFD control points . . . . . . . . . . . . . . . . . .
Example of a wing section transformation using FFD control points . . . . . . . . . . . . . . . . .
Example of a wing section rotation using FFD control points . . . . . . . . . . . . . . . . . . . . .
Definition of the planform design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a wing section chord length change using FFD control points . . . . . . . . . . . . . .
Example of a volume grid deformation as a ersult of a surface deformation . . . . . . . . . . . . .
FD sensitivity analysis for determination of ∆x . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between adjoint and FD sensitivities of lift with respect to the upper side FFD design variables at mid-span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Comparison between adjoint and FD sensitivities of drag with respect to the upper side FFD
design variables at mid-span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 An example of sectional deformation without TE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.13 Aerodynamic discipline flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1 Illustration of the wing box structure, represented by 4 equivalent panels . . . . . . . . . . . . . .
4.2 Example of the grid for the VLM and FEM analyses within FEMWET . . . . . . . . . . . . . . . . .
4.3 Verification of implementation of the FFD method in FEMWET (below) using implementation
of SU2 (above) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.1 Reference aircraft Airbus A320 wing planform view in meters . . . . . . . . . . . . . . . . . . . . .
5.2 Grid domain in meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Planform view of the converged surface grid of the upper wing surface . . . . . . . . . . . . . . .
42
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43
6.1 Lift coefficient distribution comparison between initial and optimized wing of the aerodynamic
optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Upper wing pressure coefficient distribution comparison between initial (left) and optimized
(right) wing of the aerodynamic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3 Upper wing Mach number distribution comparison between initial (left) and optimized (right)
wing of the aerodynamic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.4 Spanwise airfoil shape and pressure coefficient distribution comparison between initial and
optimized wing of the aerodynamic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.5 Lift coefficient distribution comparison between initial and optimized wing with fixed-planform 49
6.6 Comparison between the wing jig shape (blue) and the shape under 2.5g pull up load at sea level
of the initial (grey) and optimized (red) wing with fixed-planform . . . . . . . . . . . . . . . . . . 50
6.7 Twist deformation distribution comparison between initial and optimized wing with fixed-planform
of the jig shape (blue) and the 1g shape (red) wing with fixed-planform . . . . . . . . . . . . . . . 50
ix
x
L IST OF F IGURES
6.8 Upper wing pressure coefficient distribution comparison between initial (left) and optimized
(right) wing with fixed-planform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Upper wing Mach number distribution comparison between initial (left) and optimized (right)
wing with fixed-planform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Spanwise airfoil shape and pressure coefficient distribution comparison between initial and
optimized wing with fixed-planform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Planform view comparison between initial and optimized wing . . . . . . . . . . . . . . . . . . .
6.12 Upper wing pressure coefficient distribution comparison between initial (left) and optimized
(right) wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.13 Upper wing Mach number distribution comparison between initial (left) and optimized (right)
wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.14 Spanwise airfoil shape and pressure coefficient distribution comparison between initial and
optimized wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.15 Planform view comparison between initial and optimized wing . . . . . . . . . . . . . . . . . . .
6.16 Lift coefficient distribution comparison between initial and optimized wing . . . . . . . . . . . .
6.17 Comparison between the wing jig shape (blue) and the shape under 2.5g pull up load of the
initial (grey) and optimized (red) wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.18 Twist deformation distribution comparison between initial and optimized wing of the jig shape
(blue) and the 1g shape (red) wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.19 Upper wing pressure coefficient distribution comparison between initial (left) and optimized
(right) wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.20 Upper wing Mach number distribution comparison between initial (left) and optimized (right)
wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.21 Spanwise airfoil shape and pressure coefficient distribution comparison between initial and
optimized wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
51
52
54
55
56
57
58
59
59
60
61
62
63
L IST OF TABLES
2.1 Design vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 (In)equality constraint overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
14
3.1 Comparison between FD and adjoint sensitivities with respect to the P , α, U ∗ and E A ∗ design
variables at mid-span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Comparison between FD and adjoint sensitivities with respect to the U ∗ design variables at
mid-span excluding trailing edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1 Comparison between FD and AD method of FEMWET sensitivity derivatives . . . . . . . . . . . .
39
5.1 Load cases for the Airbus A320 aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Reference aircraft Airbus A320 wing geometry parameters . . . . . . . . . . . . . . . . . . . . . . .
5.3 Grid maximum element sizes on surfaces for converged grid . . . . . . . . . . . . . . . . . . . . .
41
42
43
6.1
6.2
6.3
6.4
6.5
6.6
6.7
45
49
49
53
54
58
58
Top-level aerodynamic optimization results . . . . . . . . . . . . .
Top-level fixed-planform optimization results . . . . . . . . . . . .
Aerodynamic and structural fixed-planform optimization results .
Top-level complete optimization results . . . . . . . . . . . . . . .
Aerodynamic and structural complete optimization results . . . .
Top-level complete optimization results . . . . . . . . . . . . . . .
Aerodynamic and structural complete optimization results . . . .
xi
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29
N OMENCLATURE
L ATIN SYMBOLS
Variable
J
A
AIC
b
B
B
c
c
c eq
cj
CD
CD f
Cf
C f l am
C f t ur b
CL
C L cr ui se
Cp
D
~
e
EA
E Ajig
f
F
F
FFD
g
H
H
Hz
K
K f or m
L
L
L
M
M
Mf f
~
n
n
P
q
R
R
Re
Description
Cost-function
Aerodynamic residual
Aerodynamic Influence Coefficient matrix
Wing span
Bernstein polynomials
Byte
Airfoil chord length
Inequality constraint
Equality constraint
Cruise specific fuel consumption
Drag coefficient
Viscous drag coefficient
Flat-plate friction coefficient
Laminar flat-plate friction coefficient
Turbulent flat-plate friction coefficient
Lift coefficient
Cruise lift coefficient
Pressure coefficient
Drag force
Unit vector of grid edge
Vector of elastic axis positions
Vector of elastic axis positions of the jig-wing
Objective function
Failure modes
Force matrix
Vector of Free-From Deformation control point movements
Gravitational acceleration of the Earth
Height
Hessian matrix
Hertz
Stiffness matrix
Form-factor correction
Lagrangian
Lift force
Rolling moment
Mach number
Merit function
Fuel-fraction
Vector of the unit normal to a surface
Load factor
Vector of planform parameters
Dynamic pressure
Cruise Range
Residual
Reynolds number
xiii
Unit
[−]
[−]
[−]
[m]
[−]
[−]
[m]
[−]
[−]
g
[ N ·s ]
[−]
[−]
[−]
[−]
[−]
[−]
[−]
[−]
[N ]
[−]
[m]
[m]
[−]
[−]
[N ]
[m]
[m
]
s2
[m]
[−]
[−]
N
[m
]
[−]
[−]
[N ]
[N m]
[−]
[−]
[−]
[−]
[g ]
[−]
[P a]
[m]
[−]
[−]
xiv
0. N OMENCLATURE
s
S
S
Sr e f
S wet
SF
t
T
~
u
U
V
Vi n f
W
W f uel
WM T O
Wr est
Wwbox
Wwi ng
Slack variable
Structural residual
Surface
Reference wing area
Wetted wing area
Safety Factor
Airfoil thicknesses
Vector of equivalent panel thicknesses
Vector of the displacement of a grid node
Vector of aeroelastic deformation
Aerodynamic right hand side vector
Undisturbed airspeed
Weight residual
Fuel weight
Maximum take-off weight
Maximum take-off weight minus wing and fuel weight
Equivalent panel wing box structural weight
Wing structural weight
[−]
[−]
[m 2 ]
[m 2 ]
[m 2 ]
[−]
[m]
[m]
[m]
[−]
[ ms ]
[ ms ]
[−]
[kg ]
[kg ]
[kg ]
[kg ]
[kg ]
WL
Wing loading
[ m2 ]
W L i ni t
x
X
Y
Initial wing loading
Displacement
Vector of design variables
Vector of state variables
[ m2 ]
[m]
[−]
[−]
kg
kg
G REEK S YMBOLS
Variable
α
Γ
Γ
δ
²
η
θ
λ
λ
µ
Description
Angle of attack
Wing sweep angle
Vortex strength
Aileron deflection angle
Airfoil twist angle
Spanwise position
Airfoil twist due to aeroelastic deformation
Equality constraint Lagrange multiplier
Wing taper
Inequality constraint Lagrange multiplier
Unit
[d eg ]
[d eg ]
[s −1 ]
[d eg ]
[d eg ]
[−]
[d eg ]
[−]
[−]
[−]
ρ
ρ
σ
τ
φ
φ
ψ
Air density
Penalty parameter
Stress
Shear stress
Adjoint vector
Rotation around x-axis due to aeroelastic deformation
Rotation around z-axis due to aeroelastic deformation
[ m3 ]
[−]
[P a]
[P a]
[−]
[d eg ]
[d eg ]
kg
1
I NTRODUCTION AND PROBLEM
DESCRIPTION
At present, on the aviation market a need exists for lighter and more efficient aircraft than the ones dominating the airspace today. These new generation aircraft consume less fuel per passenger for a given mission,
which does not only reduce costs of an airline operating the aircraft, but also reduces air pollutants like carbon dioxide (C02), nitrogen oxides (NOx) and soot. It is shown by Ruijgrok and van Paassen [1] that CO2
increases linearly and that NOx and soot even increase exponentially with increasing fuel flow. While the
reasoning behind reduction in airline operating costs is obvious, that has not always been the case for the
reduction of air pollutants. It is only in the modern age that people are aware of the scale of the potential
damage of these pollutants. An example of this is the growing concern that NOx increases the greenhouse
effect when dumped in the upper region of the troposphere, see Ruijgrok and van Paassen [1] for more details.
One example of the demand for light and efficient aircraft is the orders for the Airbus A350 XWB. The amount
of firm orders is 777, by 41 worldwide costumers, with 15 aircraft already deliveredi . This is while Airbus
is struggling to break even on the total programme cost basis of the much larger and less efficient Airbus
A380ii . The Airbus A350 presents a 25% decrease in CO2 exhaust with respect to the previous generation of
long-range aircrafti . Apart from the new engines used for the Airbus A350, this reduction is mainly gained
by reducing the weight of the aircraft by using composite materials and by improving the aerodynamics by
designing the wing from scratch.i .
When weight of an aircraft is reduced the necessary lift, mainly generated by the wings, to keep the aircraft
level is also reduced. This also reduces the drag generated by the aircraft wings, as part of the total drag is
dependent on the lift. Reduction of drag translates directly in a reduction of thrust required, reducing fuel
use. If on top of this the wing is aerodynamically optimized the fuel use is even further reduced. Beside the
above named reduction in operating costs and air pollutants, this reduction in fuel use can result in several
advantages with respect to the performance of the aircraft, see McLean [2]:
• Increased range
• Increased payload capacity
• Decreased structural weight, which can decrease the fuel use even further
• Decreased of take-off field length
• Decreased take-off noise
The present thesis is an effort to contribute to this reduction of fuel use by optimizing both the internal wing
box structure and external wing shape of a modern high-speed transport aircraft for minimal necessary fuel
i A350 XWB by Airbus. http://www.a350xwb.com/, Last accessed: Jan 2016
ii Telegraph, "Is Airbus’s A380 a ’superjumbo’ with a future or an aerospace white elephant?" by Alan Tovey. http://www.telegraph.co.uk/,
Last accessed: Okt 2015
1
2
1. I NTRODUCTION AND PROBLEM DESCRIPTION
weight while maintaining its range specification. The novelty of this work is the use of the Individual Discipline Feasible (IDF) architecture to decouple the disciplines within the aerostructural optimization. This is
further explained in section 1.3 Research objective and chapter 2.
1.1 W ING AEROELASTICITY
When designing a wing one needs to keep in mind that a wing is flexible, meaning that the wing bends and
twists when there are forces acting on it. The manufactured, unloaded shape of the wing is called the jig
shape or 0-g shape. What happens in flight is that the wing structure is strained until an equilibrium of elastic
and aerodynamic forces. This static aeroelastic equilibrium for the cruise condition is called the 1g-shape. In
figure 1.1 and 1.2 several static aeroelastic equilibria are visible. As becomes clear from the figures, the wing
bending can be quite high in cases of high aspect ratio, like the Boeing B-52 Stratofortress, or when flexible
material like Carbon Fiber Reinforced Plastic (CFRP) is used, in the case of the Boeing 787 Dreamliner. When
discussing flexible wings two deformation types are most pronounced: the up or downwards bending of the
wing, the z-direction, and twisting of the wing around the lateral axis of the aircraft, the y-axis.
Figure 1.1: Wingbending of a Boeing 787 Dreamlineri .
Figure 1.2: Wingbending of a Boeing B-52 Stratofortress[3].
The importance of taking wing bending and twist into account becomes apparent when comparing the spanwise lift distribution and chordwise pressure distributions of both a rigid wing and a flexible wing. Due to the
twist the effective angle of attack of a wingspan section changes directly, influencing the pressure distribution
and hence the local lift. Furthermore, when the wing is aft-swept, the effective angle of attack is also changed
by pure vertical wing bending, without any twist. This is because the bending axis is at an approximately right
angle to the elastic axis of the wing, not to the direction of the incoming airflow, see Vos and Farokhi [3] for
more details.
i C. Raezer. https://www.flickr.com/photos/arlingtonpics/, Last accessed: September 2015
1.2. P RIOR AEROSTRUCTURAL WING OPTIMIZATION EFFORTS
3
In figure 1.3 the comparison between a rigid and a flexible Boeing B-47 Stratojet wing in spanwise lift distribution is plotted. It is visible that in this case the flexibility makes the distribution deviate further from the
optimal elliptic distribution, increasing induced drag.
Figure 1.4 gives the chordwise pressure distributions for a rigid and a flexible Boeing B-52 wing at several
spanwise positions. In these plots it is visible that the shockwave at the outboard sections of the wing is reduced in strength, reducing wave drag. It has to be noted that these are not general effects of wing flexibility,
just examples of what the effect can be. In case of the Boeing B-47, the wing was designed without taking flexibility into account. However for the Boeing B-52 the wing bending was used to reduce drag at it high-speed
cruise condition, see Vos and Farokhi [3].
Figure 1.3: Spanwise lift distribution difference between a Boeing B-47 Stratojet flexible and rigid wing[3]
Figure 1.4: Chordwise pressure distribution difference between a Boeing B-52 Stratofortress felxible and rigid wing[3] (a) Flexible wing
(b) Rigid jig-wing
In this project a flexible wing will be optimized for its cruise condition, hence for the 1-g shape of the wing.
Apart from the effect on the lift and pressure distributions, one more effect which is a consequence of aeroelasticity will be taken into account: Reduced aileron effectiveness. The effect of dynamic aeroelasticity is not
taken into account in this work due to time constraints.
1.2 P RIOR AEROSTRUCTURAL WING OPTIMIZATION EFFORTS
When the wing structure and aerodynamic shape is to be optimized for minimal fuel weight one arrives at
a the field of Multidisciplinary Design Optimization (MDO), with at least two disciplines: The aerodynamic
and the structural discipline. Within such an optimization problem the aerostructural system of various disciplines has to be consistent, meaning that the values of the data flowing to one discipline has to be in agreement with the data flowing back. Furthermore, the wing used for analysis has to be flexible and therefore the
computational grid is to be deformed automatically. These issues and more have been addressed in the past.
Below a short summary of these efforts is given.
4
1. I NTRODUCTION AND PROBLEM DESCRIPTION
The first efforts of a MDO of an aeroelastic wing are found to be in the 70s. An example of these is performed
by Hafta [4], who sequentially solved the coupled aerostructural system. This system existed of a lifting line
aerodynamic model and a simple structural Finite Element Analysis (FEA). This simple model for combined
aerodynamic and structural optimization was also used by McGeer [5]. Afterwards, Grossman et al. [6], [7]
found that when a wing is sequentially optimized the disciplines individually do not take into account the
effect of changes to the other discipline. Therefore that way the optimization will not result in the optimal
point in the design space. Wakayma and Kroo [8] showed that this optimum is strongly dependent by compressibility drag, aeroelasticity and structural design conditions.
Borland et al. [9], Chattopadhyay and Pagaldipti [10] and Baker and Giesing [11] started using high-fidelity
methods within the MDO problem, but were only able to afford a limited amount of design variables. In continuation, Manning [12] demonstrates large-scale design using industry-standard analyses and Barcelos and
Maute [13] present the effects of viscous drag inclusion in high-fidelity aerodynamic analysis.
Nowadays, the use of high-fidelity analyses within an aerostructural MDO is facilitated by the ever increasing computational power available. These high-fidelity analyses are especially valuable for wing-root and
wing-tip optimization, as these areas are ruled by interference and 3D phenomena. Furthermore, for the
analysis of unconventional design concepts high-fidelity analyses are of high value, as no empirical data exists. Brezillon et al. [14] presents methods for wing shape, planform and aero-acoustic MDO. The paper states
that grid-deformation techniques are to be used instead of regenerating the grid for every design iteration,
for simplification of the wing deformation process. Furthermore the paper confirms that the non-linear interactions between the disciplines are only captured by high-fidelity analyses.
Even though the advantages of high-fidelity aerodynamic analysis is clear, recent examples of aerostructural
optimization efforts continue using panel methods, like Liem et al. [15] and Kennedy and Martins [16], [17].
The latest examples of higher fidelity aerostructural MDO’s are given by Kenway et al. [18], [19], [20] and Liem
et al. [21].
The reason why methods based on the panel method, full-potential or quasi-3D analyses, see Mariens et al.
[22], are still used is because using higher fidelity CFD analyses within a MDO can be extremely time-consuming.
This is because in the prior aerostructural MDO efforts the aerodynamic and structural disciplines have been
coupled directly, meaning that consistency of the system has to be reached before moving to the next optimization iteration. This method for MDO is not optimal, see chapter 2 for more details.
1.3 R ESEARCH OBJECTIVE
The work in this report is an effort to make more efficient use of high-fidelity analyses for the optimization of
an aeroelastic wing, hereby reducing the computational effort. This is achieved by using the IDF MDO architecture in stead of using the prior used Multidisciplinary Feasible (MDF) architecture. The IDF architecture
decouples the aerodynamic from the structural discipline. By decoupling the disciplines the work of making
the system consistent is removed from the computationally expensive individual disciplines and put it in the
hands of the cheap optimization algorithm. This is why the architecture is able to maintain consistency of
the multidisciplinary system while searching for an optimum in the design space. Besides this advantage, the
IDF architecture implies less demands on the sensitivity analyses of the disciplines. This gives more freedom
for choosing software for the analyses. The details of this architecture are explained in chapter 2.
The complete research objective of the present thesis can now be formulated. It is presented below:
To contribute to the minimization of fuel use of aircraft, by using the Individual Discipline Feasible architecture
for solving a gradient-based aerostructural Multidisciplinary Design Optimization problem and by optimizing
a static aeroelastic wing for minimal fuel consumption during the cruise phase.
In order to achieve this objective, several sub-goals have been set:
1.4. T HESIS OUTLINE
5
• Set-up and formulate the aerostructural optimization problem using the IDF architecture.
• Gather and adapt the available software and methods so they can be used within the optimization.
• Verify the sensitivities gained from the software.
• Run the optimization for minimal fuel weight of the test case aircraft.
The last item is part of the test case application performed in this report. For this test case the reference aircraft used is a modern high-speed transport aircraft, the Airbus A320. This model is chosen because of its
wide usage by airliners and the availability of its geometrical, aerodynamic and structural data.
Next to the above objective of this report, the work forms a basis for further high-fidelity optimization of
for example a wing tip extension or a complete aircraft.
1.4 T HESIS OUTLINE
This report is structured in the following way: Chapter 2 elaborates on the IDF implementation in the optimization problem and describes the optimization formulation. Afterwards in chapter 3 and 4 the aerodynamic and structural disciplines are unfolded and explained. Chapter 5 provides the details and the set-up of
the reference aircraft model. Finally chapter 6 presents the results gained after optimization and 7 provides
the conclusions and recommendations of this master thesis project.
2
IDF ARCHITECTURE FOR WING
MULTIDISCIPLINARY DESIGN OPTIMIZATION
Optimization of a wing is truly a multidisciplinary task, as the physics of a wing of an aircraft depend on both
aerodynamic and structural parameters. To illustrate this, an example of the interconnection is as follows:
The shape of the wing defines the spanwise lift and moment distribution. The wing cannot fail under this
loading, hence the structure is a function of the wing loading. However, the wing box structure defines the
wing deformation due to the wing bending of the flexible wing. This means that the wing loading is also dependent on the wing box structure. Much like this interdependency, there exists an interdependency between
necessary fuel weight needed for an aircrafts range and the wing structural weight.
These are examples of a system with multidisciplinary interdependencies and there are several ways to assure
its consistency. The area of research of optimization of these systems is called MDO, as becomes clear from
the definition given by Alexandrov [23]: "[MDO is] an area of research concerned with developing systematic
approaches to the design of complex engineering artifacts and systems governed by interacting physical phenomena".
This chapter focuses and elaborates first of all on the IDF architecture within this MDO problem. Afterwards
the optimization problem is formulated and the objective function, the design variables and the constraints
are given. Then the optimization algorithm used in this study will be explained. Finally it is shown how the
sensitivity analysis is performed in a strictly decoupled way.
2.1 O PTIMIZATION ARCHITECTURE
The optimization architecture has a great influence in how efficient the sometimes expensive discipline computations are used. This is because it controls the interactions between disciplines. This hence has an effect
on how often these disciplines are called to perform their analyses.
The publication of Martins and Lambe [24] describe four single-level, or monolithic, architectures: The
MDF, IDF, Simultaneous Analysis and Design (SAND) and All-At-Once (AAO) architecture. Multi-level, or distributed, algorithms are not of interest for this work, as no sub-level optimizations are performed inside the
main optimization. The MDF and IDF architectures will be visualized using the eXtended Design Structure
Matrix (XDSM), which shows not only the data dependency but is also extended with the flow and numbering
of the process, see Lambe and Martins [25].
The MDF architecture works as follows: Within each optimization iteration it loops all separate disciplines
until consistency is reached for all state variables. The earlier aerostructural wing optimization effort as discussed in chapter 1 incorporate this architecture. Many of these efforts include the Newton method to update
and steer the state variables to convergence. Figure 2.1 provides the XDSM of the following general optimization problem:
min f (X , Y (X , Y ))
s.t . c 0 (X , Y (X , Y )) ≤ 0
(2.1)
c i (X 0 , X i , Yi (X 0 , X i , Y j 6=i )) ≤ 0
7
8
2. IDF ARCHITECTURE FOR WING MULTIDISCIPLINARY DESIGN OPTIMIZATION
X (0)
X opt
Y t ,(0)
0, 7→1:
Optimization
2 : X0, X1
3 : X0, X2
4 : X0, X3
6:X
1, 5→2:
MDA
2 : Y2t , Y3t
3 : Y3t
Y1opt
5 : Y1
2:
Analysis 1
3 : Y1
4 : Y1
6 : Y1
Y2opt
5 : Y2
3:
Analysis 2
4 : Y2
6 : Y2
Y3opt
5 : Y3
4:
Analysis 3
6 : Y3
6:
Functions
7 : f ,c
Figure 2.1: XDSM[25] of the MDF architecture
In the above figure and equation X represents the design vector, Y the state variables, f the objective function and c the constraints. Superscript t means that it is a copy of the variable and subscripts i and j give the
number of the disciplinary analysis in question.
Advantages of this architecture are that the optimization formulation is as compact as possible; only the
design variables, objective function and constraint values and their derivatives are needed. Furthermore, the
consistency of the system is guaranteed at each optimization iteration. This means that the optimization will
provide a valid output of the multidisciplinary system even when it is stopped halfway.
However, one disadvantage exists when using the MDF architecture. The internal Multidisciplinary Analysis
(MDA) loop needs to result in a consistent system before the next optimization iteration can be performed.
This means that the disciplines and their sensitivity calculations are called multiple times before each step in
the direction of an optimum. Especially when computationally expensive disciplines are used, the optimization itself becomes very computationally expensive as well.
The second architecture, the IDF architecture, is different due to the removal of the MDA loop within the
optimization. In stead of this loop, consistency is realized by imposing equality constraints on the interdisciplinary state variables. These state variables are constraint to surrogate variables, which are included in the
design vector. Equation 2.2 provides the general formulation for the optimization problem to be solved using
the IDF architecture.
min f (X , Y (X , Y ∗ ))
s.t . c 0 (X , Y (X , Y ∗ )) ≤ 0
(2.2)
c i (X 0 , X i , Yi (X 0 , X i , Y j∗6=i )) ≤ 0
∗
∗
c eqi = Yi − Yi (X 0 , X i , Y j 6=i ) = 0
Where the superscript ∗ flags the surrogate design variables and c eq are the equality constraints. Figure 2.2
provides the XDSM for the IDF architecture.
The drawback of this architecture is that the amount of design variables and constraints increase. If this increase is too large the efficiency of the architecture will be compromised.
2.2. A EROSTRUCTURAL OPTIMIZATION PROBLEM FORMULATION
9
X (0) , Y ∗,(0)
X opt
0, 3→1:
Optimization i
Yi opt
1 : X 0 , X i , Y j∗6=i
2 : X ,Y ∗
1:
Analysis i
2 : Yi
3 : f , c, c eq
2:
Functions
Figure 2.2: XDSM[25] of the IDF architecture
Nevertheless, the removal of the MDA loop results in two large advantages. First of all, according to Cramer
et al. [26] the IDF architecture results in overall less computational costs than the MDF architecture, when the
increase in design variables is kept low. By removing the MDA loop the responsibility of consistency of the
system is removed from the disciplines themselves. Instead, it is placed in the hands of the optimization algorithm. This means that the optimizer can steer the design variables to an optimum and to multidisciplinary
consistency simultaneously. Furthermore, it facilitates the ability to run the different decoupled analyses in
parallel.
Another large advantage is that the calculation of design sensitivities is much more simple. For the MDF
architecture the sensitivities of one discipline need to include the effect of all other disciplines having an effect on it. These sensitivities can efficiently be calculated using the coupled adjoint method, see for example
Kenway and Martins [20]. However, this method requires the calculation of the sensitivities of the residual
of one discipline with respect to the state variables of another. For the IDF architecture this information is
not needed as the disciplines are completely decoupled. All inputs of a discipline are included in the design
vector and the effect of one disciple to another is captured through the equality constraints.
The last two architecture, the SAND and AAO architectures, include the residuals of the individual disciplines
and constrain them to 0. The difference between the two is that for the SAND architecture the consistency
constraints as used for the IDF architecture are removed. The job of consistency is therefore fully carried by
the residual constraints. The advantage of both architectures is that they are computationally the fastest of
all, see Cramer et al. [26]. However, they are difficult or impossible to implement for black-box components.
This is because for these architectures analysis the values of the residuals and their sensitivities are needed.
The software used for the optimization in this report does not provide this information and are in essence
black-boxes.
From the above overview between the available architectures one can conclude that for this optimization
where external software is used only the MDF and IDF architectures are valid options. Between these two
architecture IDF has a great advantage due to its efficiency and simpler sensitivity calculations. In the section
below the two architectures will be compared using the formulation of the problem at hand.
2.2 A EROSTRUCTURAL OPTIMIZATION PROBLEM FORMULATION
Now that the MDF and IDF architecture have been explained, one can go along and formulate the aerostructural optimization problem for both architectures.
The goal of this study is to minimize the fuel weight W f uel used for the defined range of the reference aircraft
by changing the planform shape P , the airfoil shapes F F D, the wing box structure thickesses T and the angle
of attack α of a flexible wing under constraints governed by structural failure modes F , aileron effectiveness
L δ , wing loading W L and the cruise lift coefficient C L cr ui se . Equation 2.3 provides the formulation of this
10
2. IDF ARCHITECTURE FOR WING MULTIDISCIPLINARY DESIGN OPTIMIZATION
problem, in MDF form.
min W f uel (X )
X = [F F D P T α]
s.t . F i ≤ 0
1 − L δ /L δ0 ≤ 0
W L/W L i ni t − 1 ≤ 0
C L /C L cr ui se − 1 = 0
(2.3)
In order to make this formulation in IDF form, one needs to add surrogate design variables and equality constraints, as mentioned in the section above. Besides the original design variables, the aerodynamic analysis
requires the aeroelastic deformation U and the elastic axis E A variables. The fuel weight analysis and structural analyses require W f uel and the maximum take-off weight WM T O besides the original design variables.
This results in the formulation of the present aerostructural optimization problem in IDF form as given in
equation 2.4.
min W f∗uel (X )
∗
X = [F F D P T α U ∗ E A ∗ W f∗uel WM
TO]
s.t . F i ≤ 0
1 − L δ /L δ0 ≤ 0
W L/W L i ni t − 1 ≤ 0
U /U ∗ − 1 = 0
E A/E A ∗ − 1 = 0
C L /C L cr ui se − 1 = 0
W f uel /W f∗uel − 1 = 0
(2.4)
∗
WM T O /WM
TO − 1 = 0
In the section above the advantages for using IDF were given. It was stated that the disadvantage of this architecture was that its efficiency could be compromised when the amount of surrogate design variables is too
large. In this case, the efficiency will not be compromised. This is because for this optimization the amount
of additional design variables is well under the amount of actual design variables.
However, the largest reason for choosing the IDF architecture for this aerostructural optimization problem is
the simplicity of the sensitivity calculation. For high-fidelity aerodynamic analysis it saves a lot of work to use
an off-the-shelve CFD solver, instead of creating one yourself. These CFD solvers usually do not provide the
sensitivities of the aerodynamic residual with respect to the structural state variables. This is also the case for
the CFD solver used in this project, see chapter 3. This means that when IDF is used for aerostructural optimization the requirements on the software for the disciplinary analyses are much lower, giving more freedom
of choice and enhances the ability to treat the disciplines as interchangeable black-boxes.
It is for these reasons that IDF has been chosen as the architecture of this aerostructural optimization problem. Below, the frameworks provided by this architecture is filled in and presented using the XDSM diagram.
Further in this section the objective function, design variables and constraints of the current optimization
will be elaborated on.
2.2.1 XDSM
Figure 2.3 gives the XDSM of the IDF architecture for the optimization in this report. In this figure the design
variables are stated above. The decoupled disciplines are the aerodynamic and the structural discipline. Because the Fuel weight analysis is dependent on the aerodynamic analysis, it is included in the aerodynamic
discipline. The design vector includes all the inputs of the two disciplines, realizing the decoupled structure.
The aerodynamic analysis provides the C L value, as well as the C D variable which is fed to the fuel weight
analysis. The elaboration on the method of the analysis is given at the end of chapter 3.
The structural discipline calculates the U , E A, F and L δ values. See chapter 4 for further details on the method
of calculation. In that chapter the calculation of the values of WM T O and W L are also given. In the present
XDSM they are part of the Functions block.
These disciplinary outputs are constraint to their corresponding surrogate variables. However, the outputs
C L , F , L δ and W L are constraint using the original constraints of the optimization problem, as explained in
section 2.2.4 Constraints.
X opt
0, 4→1:
Optimization
{C L ,C D }opt
{W f uel }opt
1: F F D, P, α,U ∗ , E A ∗
∗
2: P,W f∗uel ,WM
TO
∗
1: F F D, P, T,W f∗uel ,WM
TO
1:
Aerodynamic
analysis
2: C D
3: C L
2: Fuel weight
analysis
3: W f uel
1: Structural
analysis
{U , E A, F, L δ }opt
∗
3: U ∗ , E A ∗ ,W f∗uel ,WM
TO
3: U , E A, F, L δ
2.2. A EROSTRUCTURAL OPTIMIZATION PROBLEM FORMULATION
X (0) , X ∗(0)
3:
Functions
4 : f , c, c eq
Figure 2.3: XDSM[25] of the present fuel weight optimization
11
12
2. IDF ARCHITECTURE FOR WING MULTIDISCIPLINARY DESIGN OPTIMIZATION
For simplicity, the sensitivities are not presented in this overview. A complete overview and explanation on
their calculation is given in section 2.4 Decoupled sensitivity analysis.
2.2.2 O BJECTIVE FUNCTION
The objective function for this optimization problem is the reduction of fuel weight needed for the range of
the reference aircraft, as stated and argued in chapter 1. See chapter 3 for the explanation of how this fuel
weight is determined. The used method dictates that the total fuel weight is dependant on the lift over drag
ratio L/D of the aircraft and its maximum take-off weight WM T O . The planform and airfoil shape design
variables have a direct influence on the first parameter, while the wing box structure design variables have a
direct influence on the latter. However, as was illustrated in the beginning of this chapter, the influences of
the design variables also cross-over through indirect dependencies.
In this report the fuel weight W f uel itself is not used as objective function. In stead, the surrogate design
variable W f∗uel is used. This is possible as these two parameters are equated through an equality constraint.
2.2.3 D ESIGN VARIABLES
The design variables of the optimization in this study can be divided into three parts. Firstly the design variables which determine the exterior shape of the wing, namely the planform and airfoil shapes. Secondly the
design variables which determine the internal structure, namely the wing box panel thicknesses. The last
design variable is the angle of attack.
The wing exterior shape is defined by 9 2D airfoils. For a wing shape optimization performed by Palacios
et al. [27] 5 sections are used. For this optimization extra sections are added, for example at the kink position.
This position is needed because the planform change is realized by translation and scaling of the same 2D
sections. More about the implementation of the wing shape design variables is given in chapter 3.
The planform is defined using the following 8 planform parameters: Root chord length c r , taper from root to
kink λ1 , taper from kink to tip λ2 , wing span from root to kink b 1 , wing span form kink to tip b 2 , leading edge
sweep angle from root to tip Γ, kink twist angle ²1 and tip twist angle ²2 . Using one constant leading edge
sweep angle is not considered an underestimation, as recent passenger aircraft also have a wing design with
a constant leading edge sweep angle. The design variables for the airfoil shapes are defined as the displacement of the Free-Form Deformation (FFD) box control points. This optimization uses 9 control points per
airfoil side. Also the concept of the FFD box and its control points is explained in chapter 3.
The internal structure design variables are defined as 40 equivalent panel thicknesses. This number comes
from the fact that the wing box is divided into 10 elements and that each of these elements has an upper,
lower, front and rear equivalent panel. This same number of sections representing the wing box is used and
validated for the same reference aircraft in the publication of Elham and van Tooren [28]. More details on the
equivalent panel thickness design variables can be found in chapter 4.
The last design variable, the angle of attack, is included to facilitate the change in lift generation necessary to
maintain level flight in cruise.
Table 2.1 provides the complete overview of the design vector.
Table 2.1: Design vector
Parameter
Design Variable
FFD
P
T
α
U∗
E A∗
W f∗uel
Free-Form Deformation control points
Planform parameters
Equivalent panel thickness
Angle of attack
Deformation (4 DOF: y, z, φ, θ)
Elastic Axis
Fuel weight
162
8
40
1
32
24
1
Maximum take-off weight
Total
1
269
∗
WM
TO
number
2.3. O PTIMIZATION ALGORITHM
13
It is visible that beside the 211 design variables 58 more surrogate variables are present at the lower side of
the table, denoted with the superscript ∗. These surrogate variables exist of the deformation in 4 Degrees Of
Freedom (DOF) of the aeroelastic wing, defined at 8 of the 9 airfoils as the root is not deformed, the elastic axis
position of the 1-g shape of the wing at the same 8 airfoils, the fuel weight and the maximum take-off weight.
The reasoning behind taking 4 DOF for the deformation of the wing in stead of 6 is because the translation in
x direction and the rotation ψ, around the z axis, are negligible in comparison to the other 4 DOF. The used
DOF are: Translation in y and z direction and rotation around the x and y axis, φ and θ, respectively.
2.2.4 C ONSTRAINTS
Two types of constraints are used in this optimization routine: equality and inequality constraints. First of
all, the equality constraints are used to make sure the surrogate design variables are equal to the outputs of
the disciplines, like explained in section 2.1 Optimization architecture. Of these equality constraints the only
exception is the constraint on the lift coefficient C L , as this constraint does not constrain surrogate design
variables. To make sure enough lift is maintained the C L is constraint to the lift coefficient needed to maintain
horizontal flight in cruise C L cr ui se .
In equation 2.5 the calculation of C L cr ui se is given. This equation is extracted from the method by Torenbeek
[29] for mid-cruise weight calculation.
p
9.80665 WM T O (WM T O − W f uel )
(2.5)
C L cr ui se =
0.5ρVi2n f S r e f
Where WM T O is one of the outputs of the structural analysis and W f uel is the output of the fuel weight analysis, they are hence not the surrogate design variables. S r e f is the reference wing surface area, ρ the density of
air and Vi n f the undisturbed airspeed.
Most of the inequality constraints are present to make sure the wing box structure does not fail under static
load or fatigue. Moreover a constraint is set so that the value of L δ , being the roll moment L with respect to the
aileron deflection δ, is equal or higher than the value of the reference aircraft. Lastly a constraint is present to
limit the wing loading W L at take-off conditions, which should be equal or lower than the value of the reference aircraft. This is done to meet take-off and landing requirements and is also used in the optimization by
Elham and van Tooren [28]. More on these inequality constraints can be found in chapter 4.
A total overview of the equality and inequality constraints can be found in table 2.2. The calculation of the
failure modes F can be found in equations 4.7 to 4.9.
2.3 O PTIMIZATION ALGORITHM
The optimizer running the MDO architecture decides which next point in the design space is evaluated, taking into account the value of the objective function, the violation of the (in-)equality constraints and their
sensitivities with respect to the design vector. A great variety of optimization algorithms exist, ranging from
for example Genetic Programming or Particle Swarm Optimization which can find a global optimum to Sequential Linear Programming or Sequential Quadratic Programming (SQP), which are able to find a local
optimum, see Langelaar and van Keulen [30] for a greater overview. The aerodynamic discipline of this optimization is computationally expensive, which is why it is preferred that the amount of discipline evaluations
is as low as possible while still achieving a satisfactory optimum.
It was found by Mariens et al. [22] that for wing shape optimizations in the conceptual design phase the SQP is
superior over the global exploration algorithm Local Optima Smoothing (LOS). LOS is an example of a global
algorithm which uses random exploration of the design space to find the region of attraction. In the cited
study SQP is found to be superior because the LOS algorithm required 22 times more computational time
while the reduction in objective function was less then 0.5% higher. This proves that a local optimum can be
sufficient for a wing shape optimization as the one in the present thesis. Moreover Langelaar and van Keulen
[30] states that SQP is generally seen as the best general-purpose method for constraint optimization problems. These are the reasons for the choice of the SQP algorithm for the present optimization problem.
There are multiple implementations of the SQP algorithm available for MATLAB. The Sparse Nonlinear Optimizer (SNOPT) is well-suited for smooth, nonlinear problems with a high number of constraints and vari-
14
2. IDF ARCHITECTURE FOR WING MULTIDISCIPLINARY DESIGN OPTIMIZATION
Table 2.2: (In)equality constraint overview
Load Case
Implementation
Number
Compression upper panel
Buckling upper panel
Tension lower panel
Shear front panel
Buckling front panel
Shear rear panel
Buckling rear panel
Compression upper panel
Buckling upper panel
Tension lower panel
Shear front panel
Buckling front panel
Shear rear panel
Buckling rear panel
Tension upper panel
Compression lower panel
Buckling lower panel
Shear front panel
Buckling front panel
Shear rear panel
Buckling rear panel
Fatigue lower panel
Aileron effectiveness
Wing Loading
CL
U
EA
W f uel
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
4
5
-
c = F yi el d
c = F buckl i ng
c = F yi el d
c = F yi el d
c = F shear buckl i ng
c = F yi el d
c = F shear buckl i ng
c = F yi el d
c = F buckl i ng
c = F yi el d
c = F yi el d
c = F shear buckl i ng
c = F yi el d
c = F shear buckl i ng
c = F yi el d
c = F yi el d
c = F buckl i ng
c = F yi el d
c = F shear buckl i ng
c = F yi el d
c = F shear buckl i ng
c = F yi el dLP + 0.58
c = 1 − L δ /L δ0
c = W L/W L i ni t − 1
c eq = C L /C L cr ui se − 1
c eq = U /U ∗ − 1
c eq = E A/E A ∗ − 1
c eq = W f uel /W f∗uel − 1
64
64
64
32
32
32
32
64
64
64
32
32
32
32
64
64
64
32
32
32
32
64
1
1
1
32
24
1
WM T O
-
Type
Constraint
Inequality
Equality
Total
∗
c eq = WM T O /WM
TO − 1
1
1085
ables, see Gill et al. [31]. This is why it is chosen over other, less powerful, MATLAB based optimization functions like fmincon or SQPlab. The working of the SQP algorithm used by SNOPT is shortly explained in the
section below.
As stated, the optimization algorithm decides the direction of search and the next point in the design space. It
continues its search until the stopping criterion is reached or the optimality conditions are met. The stopping
criterion used in this project is 50 major iterations of SNOPT. The optimality conditions are also explained
separately below.
2.3.1 S EQUENTIAL QUADRATIC P ROGRAMMING
The SQP algorithm uses quadratic sub-optimization problems minimizing the quadratically approximated
objective function, subjected to the linearised original constraints. The objective function f k+1 and constraints c k+1 and c eqk+1 are approximated using Newton’s method, where k +1 stands for the updated location
in the design space. This results in a the quadratic optimization sub-problem as given in equation 2.6
½
¾
1
df T
min f +
∆x + ∆x T H ∆x
(2.6)
∆x
dx
2
s.t . c + dd xc ∆x ≥ 0
c eq +
d c eq
dx
∆x = 0
Where ∆x the search direction towards an optimum and H the approximation of the Hessian, which is gained
using the quasi-Newton approach Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. This quadratic
2.4. D ECOUPLED SENSITIVITY ANALYSIS
15
sub-problem is solved using an active-set algorithm, see Langelaar and van Keulen [30] and Gill et al. [31]
for more information on these algorithms.
After the search direction ∆x is gained, a line search is performed to find a sufficient decrease of the Merit
function M . The calculation of this function is given in equation 2.7.
M = f − µT (c − s) + λT (c eq − s) +
m
n
1X
1X
ρ ci (c i − s i )2 +
ρ c (c eq j − s j )2
2 i =1
2 j =1 eq j
(2.7)
Where µ and λ are the Lagrange multipliers, s the slack variables and ρ the penalty parameters, see Gill et al.
[31] for further details.
2.3.2 O PTIMALITY C ONDITIONS
In order to check if a candidate for an optimum is a real local optimum the Karush-Kuhn-Tucker (KKT) conditions are one of the conditions that need to be satisfied. They are given by equations 2.8 to 2.10. They are
the necessary conditions for a local optimum, the sufficient condition is given further below.
∂ f X ∂c X ∂c eq
∂L
=
+ µ
+ λ
= 0T
∂X
∂X
∂X
∂X
(2.8)
c ≤ 0, c eq = 0
(2.9)
λ 6= 0, µ ≥ 0, µc = 0
(2.10)
In these equations L is the Lagrangian as given in equation 2.11.
L(X , µ) = f (X ) + µT c(X ) + λT ceq(X )
(2.11)
In short, equation 2.8 gives the optimality condition, stating that a stationary point in the design space is
reached. This condition provides the Lagrange multipliers. The feasibility condition in equation 2.9 checks
if the point in the design space is actually feasible. The last condition is the complementarity condition in
equation 2.10 which checks if the Lagrange multipliers do not help in satisfying the optimality condition.
Lastly, if all the KKT conditions are satisfied, the sufficient conditions need to be satisfied in order to verify
that the constraint is not only a stationary point put also an optimum. This is done by checking if the objective
and the feasible domain are locally convex. The condition is given in equation 2.12.
(∂X )T
∂2 L
∂X > 0
∂X 2
(2.12)
2.4 D ECOUPLED SENSITIVITY ANALYSIS
As visible in the equation 2.6 in the prior section, the SQP algorithm needs sensitivity information of the objective function and the constraints with respect to the design vector. However, as the disciplines within the
IDF MDO architecture are fully decoupled, the sensitivities also need to be decoupled. This means sensitivities of the objective function and intermediate states of the optimization with respect to the design variables
and surrogate design variables are required. This section will expand the objective function and constraint
sensitivities with respect to the design variables until a fully decoupled system is obtained. For extra clarity,
the design vector X is given again by equation 2.13
X=
h
FFD
P
T
α
U∗
E A∗
W f∗uel
∗
WM
TO
i
(2.13)
All sensitivities of the equality, non-equality and objective function are normalized by multiplying with the
normalization vector X0. This normalization vector consist of the starting values of all (surrogate) design
variables. The value for normalization of the FFD design variables is taken to be 0.1, as stated in section 2.2.3
Design Variables. Furthermore, the addition of 1 to the FFD control point design variables has no influence
on the sensitivities, as it it only an addition.
16
2. IDF ARCHITECTURE FOR WING MULTIDISCIPLINARY DESIGN OPTIMIZATION
2.4.1 O BJECTIVE FUNCTION SENSITIVITY
The sensitivity of the objective function of the present optimization with respect to the design variables is
given by equation 2.14:
dW f∗uel
(2.14)
dX
The surrogate design variable W f∗uel is also present inside the design vector. Hence its sensitivity with respect
to X is a vector of zeros except for the sensitivity with respect to itself, which is 1.
2.4.2 C ONSTRAINT SENSITIVITY
The sensitivities of the equality constraints with respect to the design variables all have the same form. They
are given by equations 2.15 to 2.19. Of which only the first one is fully worked out.
dU ∗ 1
d ceqU ∂ceqU ∂ceqU dU ∂ceqU dU ∗ dU 1
−
U
=
+
+
=
dX
∂X
∂U d X
∂U ∗ d X
dX U∗
d X U ∗2
(2.15)
d ceq E A d E A 1
d E A∗ 1
−
EA
=
dX
d X E A∗
d X E A ∗2
(2.16)
d ceqW f uel
=
dX
d ceqWM T O
dX
dW f uel
1
dX
W f∗uel
−
dW f∗uel
1
dX
W f∗2
uel
W f uel
(2.17)
∗
=
d ceqC L
dWM T O
dWM T O
1
1
−
WM T O
∗
∗2
d X WM T O
d X WM
TO
=
dX
dC L cr ui se
1
1
dC L
−
2
d x C L cr ui se
dx
CL
CL
(2.18)
(2.19)
cr ui se
Within these expanded sensitivities there are still many unknown sensitivities. The way of calculating each
one of them is given below.
∗
∗
dW f∗uel
dW ∗
dE A
First of all, the sensitivities dU
and dMXT O are sparse matrices with a diagonal line of ones for
dX , dX , dX
the sensitivity of a surrogate design variable to itself, being 1. The sensitivity ddEXA is a summation of dU
d X and
dE Ajig
dX
, because the output of the structural discipline is the elastic axis of the jig wing and E A ∗ is defined as
the deformed elastic axis. Both sensitivities
dU
dX
and
dE Ajig
dX
dW f uel
dX
are calculated by the structural discipline using
dW
Automatic Differentiation (AD). The sensitivities
and dMXT O are expanded as shown in equations 2.20
and 2.21.
dW f uel ∂W f uel ∂W f uel dC D
=
+
(2.20)
dX
∂X
∂C D d X
dWM T O ∂WM T O ∂WM T O dWwi ng ∂WM T O dW f uel
=
+
+
dX
∂X
∂Wwi ng
dX
∂W f uel d X
Where Wwi ng is the wing structural weight. All unknown partial derivatives of the above equations and
dC L cr ui se
dx
are calculated using AD. The sensitivity
dC L cr ui se
dX
Where
∂C L cr ui se
∂W f uel
and
∂C L cr ui se
∂WM T O
=
(2.21)
dWwi ng
dX
is expanded as given in equation 2.22.
∂C L cr ui se d M T OW ∂C L cr ui se dW f uel
+
∂M T OW
dX
∂W f uel
dX
(2.22)
are given in equations 2.23 and 2.24.
∂C L cr ui se
∂W f uel
∂C L cr ui se
∂WM T O
=
=
9.80665
0.5ρVi2n f
Sr e f
9.80665
0.5ρVi2n f
Sr e f
−WM T O
p
2 WM T O (WM T O − W f uel )
(2.23)
2WM T O − W f uel
p
2 WM T O (WM T O − W f uel )
(2.24)
2.4. D ECOUPLED SENSITIVITY ANALYSIS
The last unknown sensitivities are the
Of the partial derivatives
∂C L
∂X
and
∂C D
∂X
17
dC L
dX
and
dC D
dX
, which are given by equations 2.25 and 2.26 respectively.
dC L ∂C L
∂C L d S r e f
=
+
dX
∂X
∂S r e f d X
(2.25)
∂C D d S r e f
dC D ∂C D
=
+
dX
∂X
∂S r e f d X
(2.26)
∗
the sensitivities with respect to T , W f∗uel and WM
T O are 0. The rest is
calculated using the aerodynamic solver implemented in the aerodynamic discipline.
to − SC L
re f
and − SC D
re f
, respectively, and
d Sr e f
dX
∂C L
∂S r e f
and
∂C D
∂S r e f
is equal
is calculated using AD.
Most of the sensitivities of the inequality constraints only depend on the structural discipline and are therefore calculated by the structural discipline, using AD. This is also the case for the aileron effectiveness, of
which the sensitivity is given by equation 2.27
d cLδ
dX
=−
d Lδ 1
d X L δ0
(2.27)
dL
Where d Xδ is again calculated by the structural discipline using AD.
The sensitivity of the constraint on wing loading is given by equation 2.28.
µ
¶
1
∂W L
∂W L dWM T O
1
d cW L dW L
=
=
+
dX
d X W L i ni t
∂X
∂WM T O d X
W L i ni t
of which
∂W L
∂X
is provided using AD and
∂W L
∂WM T O
is given by
1
Sr e f
.
(2.28)
3
A ERODYNAMIC AND PERFORMANCE
ANALYSIS
The present chapter will elaborate on two separate analyses: The aerodynamic and the performance analyses.
The aerodynamic analysis is responsible for calculating the drag and lift forces of the wing. The analysis
uses the same distinct steps as described by Samareh [32] for aerodynamic shape optimization (ASO):
• Geometry parameterization.
• Surface grid generation.
• Volume grid generation, regeneration, or deformation.
• Computational fluid dynamics (CFD) function and sensitivity analyses.
This chapter elaborates on all these steps and the used tool, being Stanford University Unstructured (SU2).
The performance analysis is responsible for calculation of fuel weight necessary for the range specification
of the reference aircraft. This module receives the wing drag directly from the aerodynamic calculations. The
details of the performance module calculation is shown in section 3.6 Fuel weight analysis.
Furthermore, at the end of this chapter an overview of the analysis within the complete discipline is given.
3.1 S TANFORD U NIVERSITY U NSTRUCTURED
SU2 is an open-source computational analysis and design software collection which has been released in
January 2012 by Palacios et al. [27] at Stanford University. This tool is chosen to perform the aerodynamic
analysis for the present optimization problem because of its abilities and its availability. Moreover, the TUDelft has grown to be one of the most active contributors to the code. This thesis can therefore also be seen
as a competence check for further usage of SU2.
The software collection of SU2 is able to perform all the named steps in the introduction of this chapter
and more using different modules downloadable as a complete package. The first three steps are performed
by a volume mesh deformation script, controlled by design variables deforming the surface grid. This surface
gird deformation is performed using the FFD parameterization. For the last step, the flow solver of SU2 is
able solve the governing equations of laminar Navier-Stokes, Reynolds-averaged Navier-Stokes (RANS), Euler
and/or Full Potential equations for unstructured grids. The sensitivity analysis is performed by a continuous
adjoint solver, which calculates gradients efficiently of a potentially very large number of design variables.
This collection of abilities make SU2 especially suitable for performing aerodynamic analysis for an optimization problem.
In the next sections, the modules of SU2 and the application to the present optimization problem are explained in more depth.
19
20
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
3.2 F REE -F ORM D EFORMATION PARAMETERIZATION
As stated, one of the tasks of the aerodynamic discipline is to change the wing shape and planform before the
flow analysis is performed. However, it would be very inefficient to have all x, y and z coordinates of all grid
points of the wing shape as design variables. This is why an efficient parametrization is needed. Samareh [33]
provides an overview of eight available geometry parameterization techniques, of which the Bezier curves,
the analytical shape functions and FFD stand out as compact and hence efficient approaches. The technique used by SU2 to parametrize and deform the 3D wing is the FFD parametrization. This will hence be
the approach used for both the aerodynamic as the structural discipline to deform the airfoil shape. For the
aerodynamic discipline the FFD approach will also be used for planform changes.
FFD is a parametrization technique which parametrizes deformation instead of the geometry itself. It has its
origins in computer graphics where it is used for the deformation of solid geometric models, see Sederberg
and Parry [34]. It uses a so-called FFD-box which encloses an initial geometry which is to be deformed. On
the edges of this FFD-box control points are defined, which are mapped onto the geometry through a trivariate tensor product Bernstein polynomial. These control points can be used as design variables to change the
geometry in a free-from manner. The Bernstein polynomials localize the deformation and ensure continuity.
In figure 3.1 an example is given of the freedom of the user of FFD. In this figure a sphere is morphed into a
blended wingbody.
Figure 3.1: Example of Free-Form Deformation: From a sphere to a blended-wing-body[35]
Due to this large deformation freedom with only relatively few design variables FFD is a very efficient parametrization technique which is increasingly being used in aerospace design by for example Nielsen and Anderson
[36], Anderson et al. [37], Samareh [32] and Palacios et al. [27]. For example, Samareh [38] uses the FFD not
only to change the airfoil shape in the deformation to aerodynamic shape design variables such as thickness,camber, twist, shear, and planform.
3.2.1 I MPLEMENTATION
The FFD parametrization has already been implemented in SU2 as given in equation 3.1 [27].
x(u, v, w) =
lX
−1 m−1
X n−1
X
i =0 j =0 k=0
n
F F D i , j ,k B il (u)B m
j (v)B k (w)
(3.1)
Where x is the deformation of a point on the geometry and u, v, w ∈ [0, 1] are the coordinates of that point.
F F D i , j ,k is the control point movement vector in all directions. All coordinates and movements are normalized w.r.t. the FFD box. i,j,k are the indices of the control points and l,m,n are the number of control points.
B are the Bernstein polynomials. They are calculated as given by equation 3.2.
B vn (t ) =
n!
t v (1 − t )n−v
v!(n − v)!
(3.2)
3.2. F REE -F ORM D EFORMATION PARAMETERIZATION
21
As this parametrization is 3-dimensional, any sectional deformation will have an effect, however small, on all
other sections of the wing. This is not desirable, as span-wise sections are isolated within the structural discipline, see chapter 4. This discrepancy is solved by using several FFD-boxes to cover the wing surface as the
deformation within one FFD box does not influence the geometry in another box. These boxes are placed in
between the spanwise sections. When a section is deformed by a design variable the two adjacent FFD-boxes
will realize this deformation together. An example of how sectional deformation is realized by two separate
FFD boxes is given in figure 3.2, where the mid-span section is translated in the z-direction.
Figure 3.2: Example of the usage of two FFD boxes to realize a translation in the z-direction in SU2
The algorithm as explained above is also included within the the structural discipline for deformation of the
airfoils. For this purpose the 2D version of equation 3.1 is used.
Deforming airfoils is however not the only application of the FFD parametrization. Also the aeroelastic deformation and planform of the wing is parametrized using the FFD method. In order to realize deformations
as for example sectional translation, twist or planform changes the FFD control points are moved as a group,
not individually. Using groups of FFD control points for aeroelastic or planform deformation was however
not yet implemented in SU2. This extension to the code was created by the author and can be found in the
appendix. This piece of C++ code is inserted in the file grid_movement_structure.cpp of the downloadable
source code of SU2. Links to the new design variables have been inserted throughout the SU2 source code.
In the same piece of code a part is created to calculate the sensitivity with respect to E A ∗ . This calculation
is performed in two steps. First a reverse implementation of the rotational sectional deformation using the
grouped FFD control points is performed, after which the same sectional deformation is applied again but
with a slightly changed position of E A ∗ . In section 3.5.2 Sensitivity projection it is further explained how a
sectional deformation is projected onto the design variables to gain the right sensitivity. The amount with
which E A ∗ is changed is taken to be the same as for all other sensitivity calculations within SU2: 0.001[m],
again see section 3.5.2 Sensitivity projection.
All applications of the FFD parametrization method are further explained in the sections below.
A IRFOIL SHAPE
Airfoil shape deformation is realized by 9 2D airfoil sections. The shape deformation of each seciotn is
parametrized by the movement of 18 FFD control points distributed over the upper and lower side of each
FFD-box. These control points are constraint to motion along the z-axis. Figure 3.3 gives an example of an airfoil shape deformation using the FFD control points. It can be seen that as the control points of the FFD-box
move the airfoil shape changes accordingly. The airfoil plotted is the normalized root-airfoil of the reference
wing.
It has to be stated that no investigation is performed towards the effect of adding more or reducing the
amount of 2D airfoil sections or FFD control points per section. This is because at this moment, SU2 has
a limit of 10 FFD boxes. Furthermore, the amount of FFD control points per section was taken to be similar
as in the wing shape optimization as performed by [39]. In this publication 16 FFD control points were used
per section.
The amount of FFD control points in figure 3.3 is different than the amount of control points used as design
variables, 24 in stead of 18. The two leading edge control points and the last four trailing edge control points
are not included in the design vector and will hence not move to deform the airfoil shape. The reason for
the decision to keep the corner control points at the leading edge and trailing edge constant is to keep the
design variables separated. In theory, the FFD control points could make the airfoil increase its incidence
angle in stead of deforming the airfoil. This could be realized by moving all control points in the front half of
22
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
Undeformed FFD box
Deformed FFD box
Undeformed airfoil
Deformed airfoil
0.2
z/c
0.1
0
−0.1
−0.2
0
0.2
0.4
0.6
0.8
1
x/c
Figure 3.3: Example of an airfoil shape deformation using FFD control points
the airfoil upwards and all the point at the rear half of the airfoil downwards. However, this incidence angle
is separately controlled by the planform design variables. To leave no doubt about separation of the different
types of design variables, the corner FFD control points are held constant.
On top of this, the one-to-last couple of trailing edge control points are also held constant. The reason for
this decision is given later in this chapter, in section 3.5 Sensitivity analysis.
W ING AEROELASTIC DEFORMATION
Wing aeroelastic deformation knows 4 degrees of freedom, namely translation in the y and z direction and
rotations φ and θ. These degrees of freedom are per spanwise section, together making up the total wing
aeroelastic deformation. The translation in x direction and rotation ψ are neglected, as explained in chapter
2.
Translation of a section is done by displacing all sectional control points with the same displacement. This
way the shape of the airfoil is not changed. An example of this is given in figure 3.4, where a section is translated in the z direction with 0.025.
Undeformed FFD box
Deformed FFD box
Undeformed airfoil
Deformed airfoil
0.2
z/c
0.1
0
−0.1
−0.2
0
0.2
0.4
0.6
0.8
1
x/c
Figure 3.4: Example of a wing section transformation using FFD control points
Undeformed FFD box
Deformed FFD box
Undeformed airfoil
Deformed airfoil
0.2
z/c
0.1
0
−0.1
−0.2
0
0.2
0.4
0.6
0.8
1
x/c
Figure 3.5: Example of a wing section rotation using FFD control points
3.2. F REE -F ORM D EFORMATION PARAMETERIZATION
23
The two rotational degrees of freedom need an axis around which the section is rotated. This is the elastic
axis. The deformation is done by rotating all sectional control points with the same amount around the elastic
axis. Again, the shape is not changed with this deformation. Figure 3.5 illustrates an example where the airfoil
is rotated around the y axis with −5◦ . The elastic axis is in this example equal to the quarter chord point.
P LANFORM
The planform design variables are listed in chapter 2. For extra clarity they are plotted in figure 3.6.
z
Γ
y
x
cr
²1
λ1 = c k /c r
λ2 = c t /c k
ck
²2
b1
ct
b2
Figure 3.6: Definition of the planform design variables
Where the subscripts r , k and t stand for root, kink and tip, respectively.
For parametrization these design variables are transformed in the following parameters per spanwise section:
x,y and z location of the leading edge, twist and chord length. Four of these parameters are implemented in
the same way as for wing aeroelastic deformation, see the section above. For twist the quarter chord position is taken as point around which the section is rotated. The last parameter to be implemented is the chord
length. This time, the FFD implementation is to be performed in reverse, because the geometry is known, but
the accompanying location of the FFD control points is not. In order to obtain this control point deformation
with the geometry deformation as an input equation 3.1 has to be manipulated. Writing this equation in its
2D and matrix form it can be written as shown in equation 3.3.


F F D 0,0


 

..
−1
−1
m−1

B l −1 (u 0 )B 0m−1 (v 0 ) ... B ll−1
(u 0 )B 0m−1 (v 0 ) ... B ll−1
(u 0 )B m−1
(v 0 ) 
x 0,0
.


 0

 

..
..
..
..
..


F F D l −1,0  =  ...  (3.3)
.
.
.
.
.




..

−1
−1
m−1
x g ,g
B 0l −1 (u g )B 0m−1 (v g ) ... B ll−1
(u g )B 0m−1 (v g ) ... B ll−1
(u g )B m−1
(v g ) 


.
F F D l −1,m−1
Where the first matrix on the left hand side is the matrix made by the Bernstein polynomials, the second
matrix the movement of the FFD control points and the matrix on the right hand side of the equation the
deformation of the geometry itself. The last matrix is gained by simply stretching and thickening the airfoil
to the required chord length. The equation implies that the control point deformation is given by the inverse of the Bernstein polynomial matrix times the geometry deformation. However, the resulting system is
overdetermined. The solution is found by using the normal equations. The solution is given in equation 3.4.
F F D = (B T B )−1 B T x
(3.4)
Where B is the matrix filled with Berstein Polynomials, FFD is the vector of FFD displacements and x the
vector of geometry displacements, see equation 3.3.
The result of this inversion can be seen in figure 3.7. In the figure the chord is extended by 0.2. It is found
that the percentage increase of the chord is equal to the percentage increase of the FFD box. However, the
leading edge of the airfoil need to remain at its original position. This is why the whole FFD-box is moved
slightly to the left. This last point makes the use of the inverse FFD implementation as shown in equation 3.4
necessary.
24
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
Undeformed FFD box
Deformed FFD box
Undeformed airfoil
Deformed airfoil
0.2
z/c
0.1
0
−0.1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x/c
Figure 3.7: Example of a wing section chord length change using FFD control points
3.3 S URFACE AND VOLUME GRID
For any numerical aerodynamic analysis the grid can be seen as the most important part, as its quality directly influences the quality of the simulation. In general, grids can be divided in two types: Structured and
unstructured grids. Structured grid cells are quadrilaterals is 2D and hexahedra in 3D, while unstructured
grids typically use triangles in 2D and tetrahedral in 3D. Mixed or hybrid grid approaches are unstructured
but also consist of quadrilaterals in 2D and hexahedra and tetrahedra in 3D to properly simulate the boundary layer, see Rizzi [40]. SU2 is however only able to solve the flow for unstructured grids. Moreover, as the
Euler CFD model is used no mixed grid is needed for boundary layer simulation. The reasoning behind the
CFD model choice is given in section 3.4 Computational Fluid Dynamics.
The unstructured grid method produces a random collection of elements, filling the area. This can be performed almost completely automatically. The unstructured methods are fast and enable the solution of very
large and detailed problems in a relative short period of time, enhancing flexibility. However, the user lacks of
control when setting up the mesh. For more detail on mesh characteristics see Rizzi [40]. ANSYS ICEM is used
to create the initial unstructured grid used for this optimization, see the ANSYS ICEM CFD User Manual[41]
for details on the software.
The initial wing geometry, surface and volume grid used for the test case application will be given in chapter
5. Note that this is only the initial definition. During the course of the optimization the grid and thereby also
the geometry of the wing will be deformed. This deformation process is explained in the subsection below.
3.3.1 G RID DEFORMATION
Earlier in this chapter the implementation of the FFD parametrization to deform the geometry surface is explained. One of the large advantages of SU2 is that it also deforms the volume mesh to conform the deformed
surface. This releases the user from a lot of work as meshing can be a time-consuming job.
The responsive volume deformation within SU2 is solved iteratively. Within each step the volume grid is deformed based on the classical spring method. In this spring method a stiffness is applied to the connections
between two corners of a volume element. This effectively means that every edge of a volume element acts
~i is therefore calculated using the displacements of its neighbours
as a spring. The displacement of a corner u
u~j using equation 3.5, as published by Palacios et al. [27].
Ã
!
X
j ∈N (i )
Ki j ~
ei j ~
e iTj
~i =
u
X
j ∈N (i )
Ki j ~
ei j ~
e iTj u~j
(3.5)
Where N (i ) is the set of neighbouring corners, ~
e i j the unit vector in the direction of the edge connecting the
two corners and K i j is the stiffness matrix. The stiffness of the edges are a function of the distance to a wall.
By making the volumes which are closest to the wall the stiffest it is less likely that the module creates negative volumes. These negative volumes occur if the edges of a volume switch positions, which happens when
the deformation is large and the stiffness is too low. These large deformations occur for the deformation according to the chord length changes, se for the root chord length and taper design variables. As it could not
be prevented that the grid deformation module creates negative volumes, these design variables have been
bounded. Only lower bounds were found to be necessary. The values of these bounds can be found in chapter
3.4. C OMPUTATIONAL F LUID DYNAMICS
25
6.
Beside the variables of taper, this method is able to handle all deformations of the grid which are encountered during this optimization. An example of a surface deformation with the accompanying volume grid
deformation is illustrated in figure 3.8.
Figure 3.8: Example of a volume grid deformation as a result of a surface deformation[42]
3.4 C OMPUTATIONAL F LUID DYNAMICS
In the introduction of this thesis it is already stated that high-fidelity CFD analyses are especially valuable
for wing-tip and root optimization and for unconventional design concepts. This is the case in the present
optimization as the shape and planform of the wing tip and root are included in the design variables. Furthermore, this thesis forms a basis for other optimization effort, like for example the wing tip or unconventional
design concepts.
The RANS CFD model would capture all main physical phenomena of the geometry at hand accurately, as
presented by Lyu et al. [43]. The RANS model simplifies the Navier Stokes equations by decomposing them
into time-averaged and fluctuating quantities. RANS uses turbulence models to gain the turbulent stresses,
being the so-called Reynolds stresses. Because this way the turbulence is not modelled directly by the NavierStokes equations the grid can be coarser than for more fundamental methods like the Large Eddy Simulation
(LES) and the Direct Numerical Simulation (DNS). This makes RANS method usable for larger scale CFD applications, like aircraft, see Rizzi [40] for more details. A computationally cheaper option is the Euler model.
The Euler equations are gained by neglecting viscosity completely within the Navier-Stokes equations. This
simplifies the equations greatly, resulting in reduction of computational time but also in accuracy. Lyu et al.
[43] performed a wing shape optimization using both the Euler model and the RANS model. It presented that
using the Euler model the total drag coefficient at the optimum point was only 3.7 count higher then when
using the RANS model. While the same amount of iterations was used, the Euler model saved considerable
computational time. It used 26% of the computational time of the RANS solution. This is why the Euler model
is used for the aerodynamic and sensitivity analyses in this thesis.
Because viscosity is neglected in the Euler model, the viscous drag component and its sensitivity derivatives
are not included. This is why this component has to be calculated by a separate module. This viscous drag
estimation module is explained below.
3.4.1 V ISCOUS DRAG ESTIMATION
For this optimization the viscous drag contribution cannot be assumed to be constant because the wing surface area will change during the routine. This naturally has a large effect on the viscous drag component. The
airfoil shape also has an effect on the viscous drag, but significantly less. The viscous drag estimation used
in this report is based on the viscous drag estimation published by Raymer [44]. This estimation uses the
flat-plate turbulent skin friction corrected using form factor corrections to include the airfoil thickness effect,
see equation 3.6.
P
(K f or m s C f s S wet s )
CD f =
(3.6)
Sr e f
Where C D f is the viscous drag component and S wet the wetted surface area, approximated using equation
3.7. The subscript s indicates that the parameters are taken per spanwise wing section. K f or m is the form-
26
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
factor correction calculated using equation 3.8 and C f is the flat-plate skin friction coefficient.
S wet = S r e f (1.977 + 0.52(t /c))
µ
K f or m = 1 +
µ ¶
µ ¶4 ¶
¢
¡
0.6
t
t
+ 100
1.34M 0.18 (cosΓm )0.28
(x/c)m c
c
(3.7)
(3.8)
In the above equations t /c is the thickness-to-chord ratio, (x/c)m is the chordwise location of the airfoil maximum thickness point, M the Mach number, Γm the sweep of the maximum thickness line.
According to the method of Raymer [44], C f can be calculated as a weighted average between the laminar
flat-plate skin friction coefficient C f l am and the turbulent flat-plate skin friction coefficient C f t ur b . The approach of reversed engineering is used to find the weighting factors. The friction drag is aimed at a value
of 0.0049 for the wing of the reference aircraft, taken from Elham and van Tooren [28]. This method found
that 29% of laminar flow over the wing is to be used to gain the same friction drag as used in the reference
stated. However, in the reference a full turbulent flow over the wing is used. This means that the viscous drag
estimation is overpredicted by the present method for the given reference aircraft. However, as a form of correction for this overprediction the unrealistic portion of laminar flow of 29% maintained in use. In chapter 7
a recommendation is added for a better calculation of the boundary layer and the resulting friction drag.
The coefficients C f l am and C f t ur b are calculated using equation 3.9 and 3.10 respectively.
1.328
C f l am = p
Re
C f t ur b =
0.455
(log10
Re)2.58 (1 + 0.144M 2 )0.65
(3.9)
(3.10)
In the above equations Re is the Reynolds number.
3.4.2 VALIDATION
Validation of the SU2 CFD code is provided by Palacios et al. [27] and the viscous drag model has been corrected to provide the friction drag values as found in literature. Furthermore, a grid and domain convergence
study is performed in section 5.2 Surface and volume grid to make sure the grid and domain has no or little
influence on the flow solution. Lastly, the improvements resulting from the optimization are not compared
to other models than the initial model. Because of these reasons the CFD analysis is not validated within this
project.
Aside form this, the sensitivities provided by SU2 are verified. This, together with the explanation on how
these sensitivities are acquired, is explained in the next section.
3.5 S ENSITIVITY ANALYSIS
One of the reasons for using SU2 as a CFD solver is the availability of design sensitivities. When these sensitivities are accurate they can speed up a optimization significantly as they provide the direction towards
an optimum. In addition to this, the computational time of the whole process can be greatly reduced when
these sensitivities are calculated in an efficient way. A method for efficient calculation of the sensitivities is
the adjoint method. The section below will explain this further. Moreover, this section will elaborate on how
SU2 obtains the sensitivities with respect to the design vector using the surface sensitivities provided by the
adjoint method. Finally the obtained sensitivities will be verified using Finite Differencing (FD).
3.5.1 A DJOINT METHOD
It is known that the adjoint method is a very efficient method for calculation of sensitivities of a cost-function
with respect to a large number of design variables. This is because the calculations only have to be performed
once per cost-function, instead of once per design variable. The basics of the adjoint approach is explained
below.
The derivative to be calculated is given by equation 3.11.
dJ
dX
=
∂J
∂X
+
∂J d Y
∂Y d X
(3.11)
3.5. S ENSITIVITY ANALYSIS
27
In this equation J is a cost-function, X is the design vector and Y the vector of state-variables. Of the above
equation dd YX is difficult to calculate as its calculation would require one converged flow solution per design
variable. In order to solve this in a more efficient way another sensitivity equation is used, being the sensitivity
with regard to the residual R. The equation for this sensitivity is given in equation 3.12.
∂R ∂R d Y
dR
=
+
=0
dX
∂X ∂Y d X
(3.12)
Substituting equation 3.12 into equation 3.11 results in equation 3.13.
dJ
dX
=
∂J
∂X
−
∂J
∂Y
µ
∂R
∂Y
¶−1
∂R
∂X
(3.13)
The adjoint vector φ is given as in equation 3.14, which is called the adjoint equation.
∂J
∂R
φ=
∂Y
∂Y
(3.14)
The partial derivatives in equation 3.13 are easier to be calculated then the full derivative dd YX . Using the
adjoint method the computational effort of calculating the sensitivities is comparable to the computational
effort of one converged flow solution. The output of the adjoint method as implemented in SU2 are the sensitivity of C L and C D with respect to all grid nodes on the surface of the wing.
Adjoint methods can be divided into continuous and discrete adjoint methods. The continuous adjoint
method is implemented in SU2, whereas the discrete is not yet implemented at the date of writing. The difference between the two methods is that for the continuous method the adjoint method is applied at the level
of the governing equations and afterwards discretized. Using the discrete method the governing equations
are discretized before the adjoint equation is derived. For a derivation of both methods see Palacios et al. [27].
3.5.2 S ENSITIVITY PROJECTION
Using the adjoint method described above, SU2 is able to calculate the sensitivity of C L and C D with respect to
a perturbation of each surzface grid node of the wing surface in the direction normal to that surface. Having
each node on the surface as a design variable is however not optimal, as stated before. This is why a projection
method is implemented in SU2. This method projects the surface sensitivity onto a certain design variable,
in this case the FFD control points. The projection is performed using the approximation given in equation
3.15, as published by Economon et al. [45].
¾
½
¾
Z ½
X
∂J
∂J
∂J
~i · ∆~
∂S
n
xi
=
ds ≈
∆S i
(3.15)
∂F F D
∂S ∂F F D
S
i ∈N (S) ∂S i ∆F F D
Where J is the cost-function, in this case
n being
o C L or C D and FFD the design variable. S is the affected surface, N(S) is the amount of nodes on S,
∂f
∂S i
the surface sensitivity on node i as calculated by the adjoint
~i is the unit normal to the surface at i , ∆~
method, n
x i is the change of node i in Cartesian coordinates after a perturbation of the design variable, and ∆S i is the area surrounding node i . The geometrical values of
the above equation are gained by deforming the FFD coordinates with a small amount. During this project
0.001[m] or 0.001[d eg ] is taken as perturbation. It was found that an increase or decrease in this value did
not have an effect on the sensitivity.
Within SU2 the projection onto FFD control points individually was already implemented. However for this
project also the projection on the other (surrogate) design variables is necessary, like the aeroelastic deformation or planform changes. For this purpose the same extension to the SU2 code as described in section 3.2
Free-Form Deformation parameterization was used. This is possible because the values of ∆~
x i in equation
3.15 are in essence a grid deformation as discussed in section 3.3 Surface and volume grid.
3.5.3 S ENSITIVITY VERIFICATION
Having the correct sensitivity values is important for a smooth optimization process. If these values are not
accurate they will have an adverse effect on the optimization process, especially near a minimum, according
28
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
to Nielsen and Anderson [36]. This is why it is of high importance to verify the sensitivity derivatives gained
by SU2. This section will show this verification.
The sensitivity derivatives gained by the disciplines will be compared with the same derivatives approximated
using Finite Differencing (FD). FD is a straight-forward method for derivative comparison, but it is prone to
truncation or linearization errors. This is why first an analysis has to be performed to determine the value
with which the design variable of interest is changed, being ∆x. This analysis has been performed for two
sensitivities: The sensitivity of C L with respect to a vertical translation of the mid-span section in meters and
the sensitivity of C L with respect to the rotation θ of the mid-span section in degrees, see figure 3.9 for the
results. The value of 0.0001m is selected as the ∆x value for the FD analysis of the aerodynamic sensitivities
with respect to the translations and the value of 0.1d eg is chosen for the sensitivities with respect to rotations.
For these values the sensitivities are not affected by truncation nor linearisation errors.
·10−3
1.2
1.15
4
1.1
3.8
FD dC L /dU z sensitivity
FD dC L /dUθ sensitivity
10−6
10−5
10−4
dC L /dUθ
dC L /dU z
4.2
1.05
10−3
∆x [m] or [d eg ]
10−2
10−1
100
Figure 3.9: FD sensitivity analysis for determination of ∆x
In order to perform the sensitivity verification for all aerodynamic sensitivities with respect to 269 design variables one would need to run 270 CFD analyses. In order to save effort only a selection of the sensitivities is
compared for the present verification. The design sensitivities of C L and C D with respect to the upper airfoil
shape design variables, planform variables, angle of attack, aeroelastic deformation surrogate variables and
elastic axis position surrogate variables at the mid-span section are verified in this section.
Figures 3.10 and 3.11 and table 3.1 show the comparison between the adjoint and FD method for the sensitivity derivatives provided by SU2.
0.1
dC L /d F F D
0
−0.1
−0.2
−0.3
Adjoint sensitivity
FD sensitivity
0
0.2
0.4
0.6
0.8
1
x/c
Figure 3.10: Comparison between adjoint and FD sensitivities of lift with respect to the upper side FFD design variables at mid-span
In the figures the sensitivity is plotted against the chordwise positions of the FFD control points. The definition of the planform design variables as used in the table is given in figure 3.6. The subscripts y and z stand
for transformations in the y- and z-direction, respectively. The subscripts φ and θ stand for rotations around
the x-axis and the y-axis, respectively.
3.5. S ENSITIVITY ANALYSIS
29
dC D /d F F D
0.5
·10−2
0
−0.5
Adjoint sensitivity
FD sensitivity
−1
0
0.2
0.4
0.6
0.8
1
x/c
Figure 3.11: Comparison between adjoint and FD sensitivities of drag with respect to the upper side FFD design variables at mid-span
Table 3.1: Comparison between FD and adjoint sensitivities with respect to the P , α, U ∗ and E A ∗ design variables at mid-span
Sensitivity
Adjoint
Finite Difference
Difference ratio
dC L /d P cr
dC L /d P λ1
dC L /d P λ2
dC L /d P b1
dC L /d P b2
dC L /d P Γ
dC L /d P ²1
dC L /d P ²2
dC L /dU y∗
dC L /dU z∗
dC L /dUφ∗
dC L /dUθ∗
dC L /d E A ∗x
dC L /d E A ∗y
dC L /d E A ∗z
dC L /d α
0.204800
1.262100
0.136430
-0.663952
-0.200557
-1.267325
-7.086881
-0.895321
-0.128709
1.092200
0.015439
-0.965147
-0.014966
-0.037037
-0.003868
0.131180
0.049411
0.548132
0.073995
0.073718
0.022879
-0.091545
4.367452
1.390726
0.002426
0.004124
0.003278
1.052352
-0.000058
-0.000141
0.000131
0.138230
0.241264
0.434302
0.542368
-0.111029
-0.114077
0.072235
-0.616273
-1.553327
-0.018849
0.003776
0.212328
-1.090354
0.003876
0.003807
-0.033864
1.053743
dC D /d PC r
dC D /d P λ1
dC D /d P λ2
dC D /d P b1
dC D /d P b2
dC D /d P Γ
dC D /d P ²1
dC D /d P ²2
dC D /dU y∗
dC D /dU z∗
dC D /dUφ∗
dC D /dUθ∗
dC D /d E A ∗x
dC D /d E A ∗y
dC D /d E A ∗z
dC D /d α
0.014292
0.072277
0.006936
-0.024199
-0.005488
0.025027
-0.343528
-0.032725
-0.004491
0.039328
0.000655
-0.033723
-0.000539
-0.001334
-0.000138
0.006917
0.002727
0.022210
0.003262
0.002855
0.000216
-0.018208
0.209228
0.050448
0.000203
0.000591
0.000172
0.042245
-0.000008
-0.000020
0.000003
0.006974
0.190784
0.307283
0.470311
-0.117979
-0.039442
-0.727534
-0.609057
-1.541598
-0.045207
0.015027
0.262741
-1.252705
0.014845
0.014996
-0.021804
1.008241
It is observed that most sensitivities with respect to the FFD design variables are in good agreement. This
30
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
is except for the last two control points near the trailing edge. Also the sensitivity with respect to α are correct. However, the sensitivities with respect to all other design variables are completely off. Some of them
have the wrong sign and the difference ratios variate between 0.0034 and 96.1765, absolutely spoken. This is
an issue which has to be solved for the optimization algorithm to find a good optimum. In the section below
the origin of this problem is explained and a temporary solution is proposed.
It has to be stated that all the sensitivities calculated in this section using the adjoint method neglect the
sensitivities in proximity of the trailing edge. The reason for this is also given in the section below.
3.5.4 S ENSITIVITY ERROR ORIGIN
The difference in sensitivity derivatives when compared to FD is too severe to neglect, see the previous section. To gain an understanding of the origin of this error further investigation is needed. In this section this
investigation and its results are presented. After this a temporary solution is proposed.
The first striking observation is that the derivatives of lift and drag with respect to the individual FFD control points are most of the time in agreement with the FD results, except for the derivatives near the trailing
edge. In figures 3.10 and 3.11 it becomes clear that the nearer the trailing edge of the airfoil, the greater the
error in the derivative value. The last two data points are even of the wrong sign. This observation resulted in
the hypothesis that the error present at the sensitivity derivatives with respect to P , U ∗ and E A ∗ is caused by
a large error at the trailing edge of the surface sensitivity. Meaning that the error originates from the continuous adjoint module in SU2.
The above hypothesis can be tested by excluding the trailing edge from the comparison completely. This
effectively means that a new FFD box is formed around the wing, excluding the trailing edge. This is illustrated in figure 3.12.
Figure 3.12: An example of sectional deformation without TE
When this new FFD box is deformed, the surface sensitivities of the trailing edge are not included in the projection onto the (surrogate) design variables. Moreover, when the box is deformed for FD analysis, the trailing
edge will not deform or move. Nevertheless, a discontinuity is not present because the displaced nodes on the
surface grid remain connected to the non-displaced nodes of the trailing edge surface, keeping the surface
intact. The comparison of the sensitivity derivatives without trailing edge are given in table 3.2. The same
mid-span section is used for this comparison.
It can be observed that the sensitivities are now in much better agreement. This fact confirms the hypothesis
of that the trailing edge surface sensitivities are the origin of the error in the previous section.
This error origin was confirmed by the developers of the SU2 code. They state that the surface sensitivities
near the trailing edge are inaccurate due to the assumption within the derivation of the continuous adjoint
equations that the surface is smooth, see Palacios et al. [39] and CFD-Onlinei .
They have not quantified the error, however they do present a solution, which is to discard the trailing edge
i CFD-Online Forum. http://www.cfd-online.com/Forums/su2/157134-trailing-edge-sensitivity-anlaysis.html, Last accessed: Sept 2015
3.6. F UEL WEIGHT ANALYSIS
31
Table 3.2: Comparison between FD and adjoint sensitivities with respect to the U ∗ design variables at mid-span excluding trailing edge
Sensitivity
Adjoint
Finite Difference
Difference ratio
dC L /dU y∗
dC L /dU z∗
dC L /dUφ∗
dC L /dUθ∗
dC D /dU y∗
dC D /dU z∗
dC D /dUφ∗
dC D /dUθ∗
-0.160506
0.992069
0.009432
-0.757714
-0.006387
0.123954
0.001128
-0.019825
-0.129634
0.945899
0.007202
-1.044903
-0.015811
0.107068
0.001335
-0.063084
0.807660
0.953460
0.763582
1.379021
2.475318
0.863773
1.184011
3.182011
surface sensitivities completely. This discarding of the trailing edge sensitivities is used for the adjoint calculations given in figures 3.10 and 3.11 and table 3.1, because it resulted in a slight improvement of the values.
This solution however does not improve the sensitivity derivatives enough, as can be seen by the comparison
with the FD values. This comparison has quantified the error and has shown that the sensitivities at the trailing edge cannot simply be removed.
A real solution to the problem could in theory be to change from the continuous adjoint method to the discrete adjoint method. This way no assumption has to be made with regard to the smoothness of the surface.
However, the discrete adjoint method has not yet fully been implemented in the SU2 code. In the near future
this method will be available.
Because of the limited time frame of this project, a temporary solution is implemented. First of all, the 4
control points closest to the trailing edge, 2 at the top and 2 at the bottom, are held constant. This way the
false sensitivities with respect to the airfoils shape are removed. Second of all, the sensitivity derivatives with
respect to the planform variables, aeroelastic deformation surrogate variables and elastic axis position surrogate variables are multiplied with corrective factors. The difference ratios given in table 3.1 are used as the
corrective factors. This means that the corrective factors gained by comparing the mid-span section are used
for all span wise sections. The author is aware that this is a very crude solution, however it showed to give
satisfactory results, see chapter 6. In chapter 7 recommendations will be added to improve the sensitivities
for further optimization efforts based on this research.
3.6 F UEL WEIGHT ANALYSIS
The performance analysis exists of the fuel weight analysis. As can be seen in figure 2.3 this analysis receives
C D directly form the aerodynamic analysis and is therefore not a decoupled discipline.
The fuel weight is gained using the fuel-fraction method described by Roskam [46] and given by equations
3.16 and 3.17.
W f uel = 1.05 ∗ (1 − M f f )WM T O
(3.16)
R=
Vi n f L ¡
¢
l n M f f cr ui se
cj D
(3.17)
Where in the first equation the factor 1.05 makes sure that there is 5% of fuel in reserve and where M f f is
the overall mission fuel-fraction. The fuel-fraction values for all mission elements not being cruise are taken
as a constant and the fuel-fraction for cruise M f f cr ui se is calculated using the lower equation, the Brequet’s
range equation. In this equation R is the cruise range and c j the specific fuel consumption at cruise. The
lift L is taken to be equal to the mid-cruise weight of the aircraft, as given in chapter 2. The drag D is gained
by summing the wing drag with the rest drag of the reference aircraft, being the drag of the engines, pylons,
fuselage and tail section. This rest drag is gained through Obert [47].
3.7 A NALYSIS STRUCTURE
At this point it is clear what the aerodynamic discipline consists of and what the capabilities of SU2 include.
The missing part is the overall structure in which all these discipline fractions are placed. This structure is
given by the flowchart in figure 3.13.
32
3. A ERODYNAMIC AND PERFORMANCE ANALYSIS
X
Deform airfoil shapes
Run adjoint calculations
Deform wing planform
Project surface sens.
onto the F F D, U ∗ and
E A ∗ design variables
Deform wing according to U ∗
Redefine FFD box
Run CFD calculations
Project surface sens. onto
planform design variables
CD
CL
Fuel weight analysis
W f uel &
∂W f uel
∂X
&
∂W f uel
∂C D
∂C L ∂C D
∂X & ∂X
Figure 3.13: Aerodynamic discipline flowchart
It was already shown in figure 2.3 that when ’zoomed out’, the discipline receives the design vector and produces C L , W f uel and the derivatives of both with respect to the design vector. The present flowchart provides
the ’zoomed in’ view. Below a step-by-step explanation of the flowchart is given, staring with the aerodynamic
analysis.
First of all, the airfoil shape is deformed. It is important to state that the mesh at this point is untwisted.
This is done because the structural discipline deforms the airfoil shapes with untwisted airfoils. The same is
to be done in SU2. After this the wing planform is changed, of which again the airfoil changes are performed
first, being the change in chord length. This is followed by the rest of the planform, being the translation of
the kink and tip airfoil in x, y and z direction and twist. It is at this moment where also the original twist is
added to the wing. The last step in deformation of the wing is the aeroelastic deformation. As stated before,
the aeroelastic deformation is performed around the elastic axis of the wing.
After this deformation process the CFD flow analysis is performed, providing the values of C L and C D and
the complete pressure and velocity distributions on the wing surface. The C L is a direct output of the discipline and the C D is fed to the fuel weight analysis. The pressure and velocity distributions are fed to the
adjoint calculation module.
The adjoint calculation module uses CFD results to calculate the surface- and angle of attack-sensitivity of
the wing. This is done twice, once for the sensitivities of C L and once for C D . The surface sensitivity is then yet
to be projected to the FFD control points and other design variables to gain the requested sensitivity values.
This is done in the next step.
Within this last step the projection onto the F F D, U ∗ or E A ∗ design variables is performed. Firstly, it has
to be stated that the perturbation of the FFD control points to calculate the sensitivity has to be in the untwisted z direction with respect to the untwisted airfoil to comply with the deformation as performed in the
structural discipline. This is why the perturbation direction on this twisted and aeroelastically deformed wing
3.7. A NALYSIS STRUCTURE
33
is also rotated with the same amount as the twist and deformation.
After this the sensitivities with respect to P have to be projected. This actually requires the program to first
make a new FFD box around the deformed wing, as the sections of effect are not the airfoils anymore, but the
root, kink and tip section.
The fuel weight analysis module uses the design vector X and C D , as stated. The output is the fuel weight
W f and its partial derivatives with respect to X and C D . How these derivatives are used can be found in
section 2.4 Decoupled sensitivity analysis.
4
S TRUCTURAL ANALYSIS
In this chapter the method for calculating the outputs of the structural discipline is given. First of all, a short
background will be given on the tools used within this discipline. After that, some details of the method of
weight estimation is given. In the following section the aeroelastic deformation analysis is explained, including the Finite Element Analysis (FEA), the aerodynamic analysis and the coupling between the two. Lastly,
the constraints on the thicknesses of the wingbox equivalent panels and aileron effectiveness are provided.
4.1 EMWET/FEMWET
The aeroelastic analysis tool ’Finite element based Elham Modified Weight Estimation Technique’ (FEMWET),
by Elham and van Tooren [48], is used for calculation of the aeroelastic deformation and elastic axis position
within this discipline. It is developed in 2015 an was originally an extension to ’Elham Modified Weight Estimation Technique’ (EMWET), by Elham et al. [49]. This extension was essential for a more precise estimation
of the wing weight and a better distribution of this weight to cope with aeroelastic effects such as aileron effectiveness and, with a lower degree of precision, fatigue.
Within this combination of tools, EMWET is used as an initializer for sizing of the wing structure. FEMWET
uses the equivalent panel thicknesses to provide the aeroelastic analysis. However, at this stage the equivalent panel thicknesses are taken as design variables. This results in that FEMWET can be used independently
from EMWET after gaining the initial estimation. This is with one exception: When wing structural weight is
required, the module for weight estimation of EMWET is used. More detail on this module is given below.
The advantages of FEMWET is that it is computationally cheap and does not need detailed data about the
wing structure, which in this stage of design is not available. Furthermore it provides sensitivities of any
function of interest with respect to the design variables. The methodology of the aerostructural analysis of
this tool is also given below.
4.2 W EIGHT ESTIMATION
As explained above, the wing weight estimation as used in this study is equal to the method used within
EMWET. This is because it provides a good correlation between the wing weight estimate and the actual wing
weight. As shown by Elham et al. [49], the error remains within 3% for various aircraft from different manufacturers, size and category. It is a Quasi-analytical weight estimation method and can predict the structural
weight with the same accuracy as FEA based methods but with a much shorter computational time, as also
presented by Elham and van Tooren [50]. Apart from this, the module of EMWET is easily implementable
in combination with FEMWET as it also uses the equivalent panels to represent the wing box. Within this
method, the wing weight is divided in two parts : primary and secondary wing weight.
The primary wing weight consist of all the parts that are necessary to divert the forces acting on the wing.
These are the four equivalent panels of the wing box, the ribs and non-optimum structure weight. The equivalent panels represent each side of the wing box, where the upper and lower equivalent panels represent the
box upper and lower skin, stringers and spars caps. The vertical equivalent panels represent the spar webs
35
36
4. S TRUCTURAL ANALYSIS
of the wing box. The weight of the equivalent panels is gained using its summed volume and the density of
the used material. For a better understanding, figure 4.1 provides an illustration of the wing box structure,
represented by the 4 equivalent panels.
Figure 4.1: Illustration of the wing box structure, represented by 4 equivalent panels[50]
The non-optimum structure weight embodies the sheet taper, joints, cutouts, mountings, connections and
attachments. Together with the weight of the ribs its weight is estimated by Elham and van Tooren [28] to be
1.5 times the weight of the wing box existing of only the equivalent panels, Wwbox .
The secondary wing weight includes the leading and trailing edge structure, leading and/or trailing edge high
lift devices and control surfaces. Torenbeek [51] presents that the total secondary wing weight can be estimated by 15 · S r e f .
The above results in a weight for the complete wing structure Wwi ng given by equation 4.1.
Wwi ng = 1.5 · Wwbox + 15 · S r e f
(4.1)
With the wing weight and therefore WM T O known, the wing loading W L can be calculated. See chapter 2 for
its usage within the W L constraint. Equation 4.2 provides the calculation.
WL=
WM T O
Sr e f
(4.2)
4.3 S TATIC AEROELASTIC DEFORMATION ANALYSIS
In order to gain proper static aeroelastic deformation and design sensitivities of aeroelastic phenomena an
aeroelastic deformation analysis is to be performed. As can be read above, FEMWET is the appropriate choice
at an early design level, like in this study. FEMWET includes both a Vortex Lattice Method (VLM) as aerodynamic solver and a Finite ELement Method (FEM) solver to obtain the static aeroelastic equilibrium of the
wing. The details of the solvers are given below, followed by the coupling methodology.
4.3.1 G IRD
Firstly, the grids for the VLM and FEM within FEMWET have to be defined. As already stated in chapter 2,
9 airfoil sections are used as design variables for the present optimization problem. At these same positions
the aeroelastic deformation and aeroelastic axis positions have to be defined. To omit the need for interpolation, the nodes for the FEA are placed at the same spanwise positions as the 9 airfoil sections. However, to
increase the accuracy of the calculated deformation, an extra node is placed between all these airfoil sections.
The deformations of these extra nodes are not further used within the optimization problem. The FEA node
positions are also used to define the spanwise edges of the VLM mesh. An example of a grid used for the VLM
and FEM analyses is given in figure 4.2.
4.3.2 A ERODYNAMIC ANALYSIS
The aerodynamic analysis within FEMWET is based on the VLM as presented by Katz and Plotkin [52]. The
governing equation to be solved is given in equation 4.3.
AIC · Γ = V
(4.3)
Where AIC is the Aerodynamic Influence Coefficient matrix, Γ the vortex strength matrix and V the right
hand side. Aerodynamic forces are calculated based on Γ. The AIC and V matrix are calculated based on the
mesh used for the vortex lattice method. The Prandtl-Glauert compressibility correction is used to correct
for compressibility effects and the boundary conditions are applied on the wing camber line to include the
effects of airfoil shape. This airfoil shape is parametrized using the same FFD method as described in chapter
4.3. S TATIC AEROELASTIC DEFORMATION ANALYSIS
37
Figure 4.2: Example of the grid for the VLM and FEM analyses within FEMWET[28]
3. The initial airfoil shapes are given using Class Shape Transformation coefficients.
It has to be stated that this separate aerodynamic analysis is required and cannot be replaced with the highfidelity CFD solver used in the aerodynamic discipline of this optimization problem. This is because SU2 is
not able to provide sensitivities w.r.t. the moment and lift distribution.
4.3.3 F INITE E LEMENT A NALYSIS
The FEMWET tool uses a finite beam element analysis representing the wing. The elements of this beam are
placed between the nodes which are placed at the elastic axis of the wing. It is assumed that the elastic axis
is equal to the shear centre of the sections. These shear centres are calculated at each node with the method
presented by Megson [53].
The second output of FEMWET is the deformation vector U . The structural governing equation is used to
calculate this vector, see equation 4.4.
K ·U = F
(4.4)
Where K is the stiffness matrix and F the force matrix. These matrices are computed using the consistent
shape functions for a 3-D 2-node Timoshenko beam element as presented by Luo [54], see Elham and van
Tooren [48] for more details on the creation of these matrices. The solution of the coupled aerostructural
problem of equation 4.4 is gained by the Newton coupling method, explained in the section below.
Once the nodal displacement is known, the stresses within the equivalent panels can be calculated. This
is done using both the displacements and the geometry of the wing box equivalent panels. For this analysis
each of these panels, situated between two nodes, are divided into smaller elements, 4 elements for the upper
and lower panels and 2 for the front and rear panels. In section 4.4 Failure modes and aileron effectiveness it
is explained how the yield and buckling stresses and the failure modes of these panel elements are calculated.
4.3.4 A EROSTRUCTURAL COUPLING
The aerodynamic and structural data are coupled. The Newton method is used by FEMWET to solve the system. Solving the system is necessary to guarantee its consistency, meaning that all residuals of the governing
equations are 0. The system of governing equations is given in equation 4.5.
38
4. S TRUCTURAL ANALYSIS
 

A(Γ,U , α)
AIC · Γ − V
R =  S(Γ,U , α)  =  K U − F  = 0
W (Γ,U , α)
L − n · WM T O

(4.5)
Where A, S, and W stand for the residuals of the aerodynamic, structural and weight governing equations,
respectively. The first two have already been introduced in the sections above, while the last one is introduced
to maintain level flight by equating lift to the weight of the aircraft. Γ, U and α are the state variables of the
same governing equations. Within this coupled system is the AIC matrix dependent on U , V dependent on
U and α and both F and L are dependent on Γ.
The Newton method finds the update of the state variables using equation 4.6. This updating of the state
variables is repeated until the norm of the update vector is less then 10−9 .





∂A
∂A 
∂A
∆Γ
A(Γ,U , α)
∂Γ
∂U
∂α

 ∂S
∂S
∂S
∆U  = −  S(Γ,U , α) 
(4.6)
 ∂Γ
∂u
∂α 
∂W
∂W
∂W
∆α
W
(Γ,U
,
α)
∂Γ
∂U
∂α
As can be seen, for the Newton method coupled derivative information is required. These derivatives are
gained using Automatic Differentiation and analytical techniques.
4.4 FAILURE MODES AND AILERON EFFECTIVENESS
The yield stresses are dependent on the used material, while the buckling stresses are computed by the FEM in
the following way: The upper and lower panel buckling stress is calculated using the stiffened panel efficiency
method, as presented by Niu [55]. The vertical panels are subjected to shear buckling, of which the stress
is calculated using the same stiffened panel efficiency method. From these stresses the failure modes in
equations 4.7 to 4.9 are defined.
µ
¶
τ · SF 2
σ · SF
+
−1
(4.7)
F yi el d =
σ yi el d
τ yi el d
F buckl i ng =
F shear
buckl i ng
σ · SF
−1
σbuckl i ng
=
τ · SF
τbuckl i ng
−1
(4.8)
(4.9)
This results in 10 constraints per beam element. Compression, tension and buckling for the upper and lower
equivalent panel and shear load and shear buckling for each of the vertical panels. However, the amount of
used constraints are less as some failure modes are not relevant, for example the tension failure mode at a
panel under compression.
Fatigue is represented by a constraint on the stress on the lower equivalent panel: under a gust load of 1.3g
the stress must be lower than 42% of the maximum allowable stress of the material. This estimation of fatigue
is suggested by Hurlimann [56] ans is used by Kenway and Martins [20] and Elham and van Tooren [48].
A total overview of the equality and inequality constraints can be found in table 2.2.
Aside from the structural and fatigue constraints governed by the failure modes, a constraint is added on
the aileron effectiveness. It is presented by Elham and van Tooren [48] that aileron effectiveness is an active
constraint, showing its importance for an aeroelastic aerostructural optimization. The constraint makes sure
that the rolling moment due to a deflection of the aileron L δ is equal or higher than the value of the reference
aircraft. FEMWET provides this value and its sensitivity with respect to the design variables.
4.5 VALIDATION AND VERIFICATION
The validation of the output of the weight estimation module and FEMWET is published by Elham et al. [49]
and Elham and van Tooren [28]. As the methodologies of these programs are not changed for this project, no
additional validation needs to be performed.
However, one change to FEMWET is performed: The FFD parameterization is included for 2D airfoil shape
4.5. VALIDATION AND VERIFICATION
39
deformation. As SU2 already implemented this parameterization approach, the deformations can be compared. This can serve as verification of the FFD parameterization implementation in FEMWET.
Figure 4.3 shows the deformation as performed by SU2 in comparison with the deformation using the FFD
approach in FEMWET. For both deformations 24 control points were used. Two control points were moved
on the upper side and one on the lower. As can be seen, the shape deformations are a match.
Figure 4.3: Verification of implementation of the FFD method in FEMWET (below) using implementation of SU2 (above)
4.5.1 S ENSITIVITY VERIFICATION
For the same reasons named above, not all sensitivities need to be verified. This does not count for the sensitivities with respect to the F F D design variables, as they are newly implemented. The method of verification
is equal as the one in chapter 3, section 3.5 Sensitivity analysis.
In table 4.1 the sensitivity derivatives calculated by FEMWET using AD are compared with the same sensitivities calculated using FD. The ∆x is taken to be 1e − 6, the same value as used for verifying the sensitivities
of the original FEMWET publication. The section used for verification is again at mid span, with the leading
edge upper FFD control point as the design variable.
Table 4.1: Comparison between FD and AD method of FEMWET sensitivity derivatives
Sensitivity
dU z /d F F D LE
d E A z /d F F D LE
d F buckl i ng upper /d F F D LE
d F shear f r ont /d F F D LE
d η/d F F D LE
Adjoint
Finite Difference
0.00364
0.00780
0.00673
-0.01602
0.00347
0.00364
0.00780
0.00673
-0.01602
0.00347
The verification of the structural analysis is considered a success as the sensitivities are a match.
5
T EST CASE APPLICATION
By now, the optimization problem, the disciplines and their content are explained. This chapter will apply
this optimization tool to a test case. As stated in chapter 1, the reference aircraft used for this test case is a
modern high-speed transport aircraft, the Airbus A320. The geometrical, aerodynamic and structural data is
gained through the book of Obert [47].
5.1 L OAD CASES
The load cases of the reference aircraft used in this study can be found in table 5.1 and are estimated using
the load diagram of an aircraft of the same class, presented by Dillinger et al. [57].
Table 5.1: Load cases for the Airbus A320 aircraft
Load case
type
1
2
3
4
5
6
pull up
pull up
push over
gust
roll
cruise
M
q[Pa]
n[g]
H[m]
0.89
0.58
0.89
0.89
0.83
0.82
21200.0
23500.0
21200.0
21200.0
29700.0
10650.0
2.5
2.5
-1.0
1.3
1.0
1.0
7500
0
7500
7500
4000
11000
The first three of these load cases are used for evaluation of the constraints on structural failure. The fourth is
used for a constraint on fatigue of the wing structure. For the first four load cases the weight of the aircraft is
taken to be the maximum take-off weight. The fifth load case in table 5.1 is used for the aileron effectiveness
constraint. The last load case is used for the calculation of U and E A and for the aerodynamic discipline.
For the last two load cases the aircraft mid-cruise weight is taken, as given in equation 5.1 as presented by
Torenbeek [29].
q
Wcr ui se = WM T O (WM T O − W f uel )
(5.1)
5.2 I NITIAL SURFACE AND VOLUME GRID
In the present section the initial wing geometry, surface and volume grid is presented. This will only be the
initial grid as throughout the optimization the grid is deformed as explained in section 3.3 Surface and volume
grid.
5.2.1 G EOMETRY
The planform data of the reference aircraft is presented in table 5.2, which results in a planform-view of the
wing as given in figure 5.1.
The original airfoils of this wing are defined at four locations: 0, 0.33, 0.66 and 1 span length. This data is
interpolated to find the initial airfoil shapes at the airfoil sections used in this optimization.
41
42
5. T EST CASE APPLICATION
Table 5.2: Reference aircraft Airbus A320 wing geometry parameters[47]
Position
Root
Kink
Tip
x
y
z
Chord [m]
Twist at 1/4 chord [deg]
0
3.3006
8.8306
0
6.3403
16.9635
0
0.5547
1.4841
7.0518
3.7584
1.4958
0
-2.50
-2.50
z
y
x
7.05
3.76
6.34
1.50
16.96
Figure 5.1: Reference aircraft Airbus A320 wing planform view in meters
5.2.2 I NITIAL SURFACE AND VOLUME GRID
Before the grid can be created the domain size of the grid has to be determined. It is important that the
edges of the domain do not interfere with the flow solution on the surface of the wing. To confirm this independence several domain sizes were investigated and tested. For this investigation the outlet position was
handled separately, due to the presence of the wake of the wing. By visual inspection of the converged flow
solution the approximate width and height of the domain and length in front of the wing was already found.
This was done by making sure that the pressure disturbance due to the wing was relatively small in size at the
inlet, upper, lower and side plane of the domain. This estimate was however just a starting point. Afterwards
the domain was decreased in size with steps of approximately 10%. This was continued until the change in
lift and drag coefficient became higher than 0.1%. After this, the outlet position was independently decreased
until the same criterion was met. The result of this investigation is a domain of 250m wide, 650m deep and
600m high, see figure 5.2.
250
outlet
inlet
600
150
500
symmetry plane
z
y
x
Figure 5.2: Grid domain in meters
Creating a grid is always a compromise between accuracy and computational speed, as in CFD the discretized
5.2. I NITIAL SURFACE AND VOLUME GRID
43
governing flow equations are solved for each of the cells in the volume grid. This is why a grid convergence
study is performed. During this study a similar method is used as for determining the domain size as explained above. This time the maximum size of the elements on each surface is increased separately until the
change in lift and drag coefficient become higher than 0.2%. The step-size of increase varied per surface, for
the maximum element size the increase was 100%, for the upper and lower wing surfaces 0.05 and for the
wing tip and trailing edge 0.01. This resulted in the maximum grid size values as given by table 5.3 and the
surface grid as shown in figure 5.3. The total amount of elements in the converged grid is 1,134,344. This grid
is used as the initial point of the optimization.
Table 5.3: Grid maximum element sizes on surfaces for converged grid
Surface
Far-field
Wing upper surface
Wing lower surface
Wing tip surface
Wing trailing edge line
Maximum element size
16 [m]
0.3 [m]
0.3 [m]
0.08 [m]
0.03 [m]
Figure 5.3: Planform view of the converged surface grid of the upper wing surface
It has to be noted that the created surface grid representing the wing is not twisted at all. The original twist
given by the geometry of the reference aircraft is added at a later stage, see section 3.7 Analysis structure. This
is done because in the structural discipline the shape of the airfoils is deformed using untwisted airfoils. To
maintain coherence, the aerodynamic discipline also deforms the shape of the untwisted airfoils, after which
the twist is added later using grid deformation.
6
R ESULTS
After having described the decoupled disciplines and the optimization problem in which they operate, the
results of this optimization can be shown and discussed. It is chosen two perform four optimizations: One
pure aerodynamic optimization, one aerostructural optimization with a fixed planform, being the original
planform of the reference aircraft, and two complete aerostructural optimizations including the planform
design variables as described in chapter 2. Of the last three optimizations, it is expected that the third and
fourth optimization problem will show a greater reduction in fuel weight, because of the increased degrees of
freedom in the design space.
In the optimization code, the objective function and design variables of all these optimizations are normalized.
All results are gained using 8 3.50 GHz processors and 63 GB of RAM (Random Access Memory). Each single
function evaluation has a computational cost of around 1 hour, for all optimization problems.
6.1 A ERODYNAMIC OPTIMIZATION
The first optimization which is performed is a pure aerodynamic optimization. This is done to prove the
functioning of the aerodynamic discipline when it is steered by the optimization algorithm. The U value of the
initial wing is held constant and applied throughout the optimization. The objective of this optimization is to
reduce C D by using the F F D and α design variables only. C D f is held constant throughout the optimization.
The problem is constraint by two constraints, being that the C L value has to remain equal to the initial value
and that the maximum thickness of each wing section has to be equal or greater than the initial value. The
problem is formulated as given in equation 6.1.
min C D (X )
X = [F F D α]
s.t . C L /C L i ni t − 1 = 0
t maxi ni t − t max ≤ 0
(6.1)
The optimization results are shown in table 6.1
Table 6.1: Top-level aerodynamic optimization results
Parameter
Value
Objective function value
Iterations
Function evaluations
Maximum constraint violation
0.8382
15
131
1.5797e-4
The C D value reduced with 16%, the constraints on t max have not been violated and the constraint on C L is
violated with 1.5797e − 4. However, this constraint violation translates to a violation of C L of 0.02%, which is
an acceptable amount. The total computational time of this optimization is 104 hours. Further details of the
45
46
6. R ESULTS
results of this optimization are given below.
The airfoil shape changes not only reduced drag but also managed to reduce the α from 3.1584d eg to 3.0022d eg .
The lift distribution comparison between the initial wing and the optimized wing is given in figure 6.1.
4
CC l [m]
3
2
Baseline
Optimized
1
0
0
2
4
6
8
10
12
14
16
y[m]
Figure 6.1: Lift coefficient distribution comparison between initial and optimized wing of the aerodynamic optimization
As can be seen, the optimization managed to make the lift distribution much more elliptical. The result of
this is an induced drag reduction. The amount of this induced drag reduction can not be given, because SU2
is not able to divide the computed inviscid drag into induced and wave drag components.
The total drag reduction originated from a combined reduction of wave drag and induced drag. The reduction
of wave drag can be investigated using the plots of the pressure coefficient and Mach number distributions,
which are given in figures 6.2 and 6.3 respectively, and the airfoil shape changes and sectional pressure distributions, which are plotted in figure 6.4. From these plots it is found that especially at the inboard half of the
wing the shock wave is removed, or at least reduced in strength. Around the mid-span section a shock wave
remains near the trailing edge.
The change in spanwise load distribution as given above is also visible in these plots. Front loading is increased over the whole span, while the rear loading is reduced for the inboard sections of the wing.
The increased front loading together with the decreased rear loading of the inboard sections also result in that
the pressure coefficient isobars run more parallel to the leading edge. This way more of the potential advantage of wing sweep is realized, as the root and tip effects are minimized, see Obert [47] for more explanation
on straight isobars.
This described airfoil loading is a direct result of the airfoil shape changes. It is visible that for the inboard
part of the wing the leading edge thickness increased and that the overall camber decreased. For the outboard
airfoil sections the overall thickness and camber is increased.
With these results it is shown that the aerodynamic discipline reacts appropriate when it is controlled by
the optimization algorithm and that the aerodynamic analysis is not only able to reduce induced drag and
wave drag at the same time, but also to find an optimal balance between them.
6.2 F IXED PLANFORM AEROSTRUCTURAL OPTIMIZATION
For this optimization problem the planform is kept fixed throughout the optimization. This is done to show
the potential and proof of functioning of the decoupled aerostructural system.
In order to realize keeping a fixed planform, three adaptations are made to the initial problem description.
First of all, the P vector in the design variables is removed. Secondly, the wing loading constraint is removed
as the wing surface will not change. It is expected that the wing loading will only decrease, as the WM T O is
expected to decrease. The third adaptation is that even though the airfoil shape changes, the friction drag
is assumed to be constant. The last assumption is made because the wing surface has the greatest effect on
the friction drag estimation and this parameter will not change. Equation 6.2 shows the formulation of this
adapted optimization problem.
6.2. F IXED PLANFORM AEROSTRUCTURAL OPTIMIZATION
47
Figure 6.2: Upper wing pressure coefficient distribution comparison between initial (left) and optimized (right) wing of the aerodynamic
optimization
Figure 6.3: Upper wing Mach number distribution comparison between initial (left) and optimized (right) wing of the aerodynamic
optimization
Cp
48
6. R ESULTS
−1
−1
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
Cp
0.6
0.8
1
1
(b) η = 0.25
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
0.6
0.8
1
0.8
1
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
x/c
(d) η = 0.5
0.4
(c) η = 0.375
−1
0
0.2
x/c
−1
1
0
x/c
(a) η = 0.125
0.5
B asel i ne
Opt i mum
0.5
0.2
0.4
0.6
x/c
(e) η = 0.625
(f) η = 0.75
−1
Cp
−0.5
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
0.6
0.8
1
x/c
(g) η = 0.875
Figure 6.4: Spanwise airfoil shape and pressure coefficient distribution comparison between initial and optimized wing of the aerodynamic optimization
6.2. F IXED PLANFORM AEROSTRUCTURAL OPTIMIZATION
49
min W f∗uel (X )
∗
X = [F F D T α U ∗ E A ∗ W f∗uel WM
TO]
s.t . F i ≤ 0
1 − L δ /L δ0 ≤ 0
U /U ∗ − 1 = 0
E A/E A ∗ − 1 = 0
C L /C L cr ui se − 1 = 0
W f uel /W f∗uel − 1 = 0
(6.2)
∗
WM T O /WM
TO − 1 = 0
Table 6.2 provides the basic results of the fixed planform optimization.
Table 6.2: Top-level fixed-planform optimization results
Value
Parameter
Objective function value
Iterations
Function evaluations
Maximum constraint violation
0.9333
18
90
1.3952e-5
On the used hardware the computational time for this optimization is around 90 hours. This is very little
taken into account that the optimization including high-fidelity analysis has been performed on only 8 processors, while in earlier studies the amount of used processors is significantly higher and the total time to
reach the optimum was similar, see for example Brezillon et al. [14].
From the result presented above it can be seen that 7% of fuel weight is saved with a negligible maximum
constraint violation.
Table 6.3 gives the optimization results for the aerodynamic and structural properties of the aircraft.
Table 6.3: Aerodynamic and structural fixed-planform optimization results
Initial
Optimized
α[d eg ]
CL
CD
CD f
WM T O [kg ]
W f uel [kg ]
Wwi ng [kg ]
3.1584
2.5924
0.5228
0.5209
0.0195
0.0169
0.0049
0.0049
73,500
72,571
17,940
16,743
8,801
9,068
It is observed that C D is reduced with 13% while C L is only slightly reduced. The reduction of C L is a direct
result of the reduction in WM T O and W f uel . The reduction of C D is due to the improved shape of the airfoils
which not only reduced wave drag, but also the induced drag through the improved the spanwise lift distribution. This lift distribution of the initial and optimized wing is given in figure 6.5. The airfoil shape changes
also increased the total lift at constant α. This last result is visible in the reduction of α, while the C L remained
almost constant. The total L/D increased with 15%, from 26.81 to 30.82.
4
CC l [m]
3
2
Baseline
Optimized
1
0
0
2
4
6
8
10
12
14
16
y[m]
Figure 6.5: Lift coefficient distribution comparison between initial and optimized wing with fixed-planform
WM T O is reduced with 1% and Wwi ng is increased with 3%. This is where the price is paid for the more elliptic
50
6. R ESULTS
lift distribution. However, it seems worth it to pay that price, as the reduction of fuel use is stronger than
the increase in structural wing weight. The fact that the outer wing sections produce more lift increases the
wing bending. This is compensated by the heavier structure, but not completely. The vertical wing bending
of the tip in the cruise condition increased with 15%, from 0.57m to 0.66m. This increase in wing flexibility
is also visible in figure 6.6, which shows the jig shape and 2.5g-shape at sea level of the optimized wing and
the initial wing. In figure 6.7 the twist distributions of the initial and optimized wing are plotted for their jig
shape and the 1g-shape. Also here the increased wing flexibility becomes visible. The tip twist deformation
at cruise condition increased with 37%, from −1.4◦ to −2.0◦ .
Figure 6.6: Comparison between the wing jig shape (blue) and the shape under 2.5g pull up load at sea level of the initial (grey) and
optimized (red) wing with fixed-planform
Twist[d eg ]
−4
−2
Baseline
Optimized
0
0
2
4
6
8
10
12
14
16
y[m]
Figure 6.7: Twist deformation distribution comparison between initial and optimized wing with fixed-planform of the jig shape (blue)
and the 1g shape (red) wing with fixed-planform
Furthermore, the comparison between the initial and optimal pressure coefficient and Mach number distribution on the upper surface of the wing are plotted in figures 6.8 and 6.9 respectively. From these figures a
couple of things can be observed. Firstly, the shock wave is reduced in strength. For the reference wing a
strong shock wave was present over almost the complete wing, whereas for the optimized wing this strong
shock wave only appears at the outboard half of the wing. A logic explanation of this outboard shock wave
is that the optimizer has to counter-act the increased negative twist. This way the outboard loading remains
high and the elliptical list distribution is maintained. The rest of the wing experiences weaker pressure jumps.
The airfoil pressure distributions given in figure 6.10 show this in more detail.
Secondly, the Mach number and pressure coefficient plots also show that the optimization resulted in isobars
which are more parallel to the leading edge. Just like as explained in the section above, this also results in
reduced wave drag. One has to be careful for shock-induced boundary layer separation. Due to the larger
aft loading at the tip-sections of the optimized wing the local Mach number has increased. Obert [47] states
that shock-induced boundary separation occurs for local Mach numbers of 1.35 to 1.45. Figure 6.9 shows that
these Mach numbers are however not reached at the outboard half of the wing. Also at the leading edge of the
inboard part of the wing the local Mach number does not reach 1.35, the maximum Mach number is however
close: 1.34. It is therefore recommended that more research is performed to make sure the boundary layer
does not separate in this area.
6.2. F IXED PLANFORM AEROSTRUCTURAL OPTIMIZATION
51
Figure 6.8: Upper wing pressure coefficient distribution comparison between initial (left) and optimized (right) wing with fixed-planform
Figure 6.9: Upper wing Mach number distribution comparison between initial (left) and optimized (right) wing with fixed-planform
Cp
52
6. R ESULTS
−1
−1
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
Cp
0.6
0.8
1
1
(b) η = 0.25
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
0.6
0.8
1
0.8
1
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
x/c
(d) η = 0.5
0.4
(c) η = 0.375
−1
0
0.2
x/c
−1
1
0
x/c
(a) η = 0.125
0.5
B asel i ne
Opt i mum
0.5
0.2
0.4
0.6
x/c
(e) η = 0.625
(f) η = 0.75
−1
Cp
−0.5
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
0.6
0.8
1
x/c
(g) η = 0.875
Figure 6.10: Spanwise airfoil shape and pressure coefficient distribution comparison between initial and optimized wing with fixedplanform
6.3. C OMPLETE AEROSTRUCTURAL OPTIMIZATION
53
Figure 6.10 provides the airfoil shapes accompanied by their pressure distributions. It is found that at the root
of the wing the camber is decreased and that the leading edge of the airfoils are thickened, which is similar to
the pure aerodynamic optimization. This is a desired result form a design point of view because the isobars
are straightened and the section pitching moment coefficient is lowered.
In the middle sections the camber is increased, resulting in the more elliptical lift distribution.
Finally, at the tip sections aft loading is generated by thickening the trailing edge of the airfoils. This generates
more lift at the outboard part of the wing.
From the presented results it can be deduced that the induced drag is reduced due to the improved lift distribution and that the wave drag is reduced due to the reduction of shock wave strength, especially at the
inboard half of the wing, and isobars which are more parallel to the wing leading edge. These changes resulted in the inviscid drag coefficient to drop 18%, from a value of 0.0146 to 0.0120.
6.3 C OMPLETE AEROSTRUCTURAL OPTIMIZATION
In the previous section it was shown that the fixed planform optimization resulted in a feasible wing design with a fuel reduction of 7%. Furthermore, the effects of design variables on the state variables and the
objective function were confirmed. In this section the results of the full optimization, including the planform design variables, are presented. However, this section is divided in two parts: Firstly, an optimization
is performed using the sensitivities of the aerodynamic discipline gained using the adjoint formulation as
described in chapter 3. However, as described in the same chapter, these sensitivities with respect to the
planform design variables had to be corrected using a simple correction factor.
The second part of this section shows the results of the same optimization problem but where the sensitivities with respect to the planform design variables are calculated using FD. Because this way of calculating the
sensitivities is computationally much more expensive, the amount of planform design variables is reduced
from 8 to 6.
6.3.1 A DJOINT PLANFORM SENSITIVITIES
This subsection provides the results of the optimization problem as described throughout this report. The
normalized lower bounds the taper and root chord design variables are 0.99 on λ1 , 0.85 on λ2 and 0.9 on c r .
These bounds have been stretched until negative volumes appeared because of the deformation module in
SU2.
Table 6.4 provides the basic results of this optimization.
Table 6.4: Top-level complete optimization results
Parameter
Objective function value
Iterations
Function evaluations
Maximum constraint violation
Value
0.8846
1
50
6.6888e-2
It can immediately be seen that the results are disappointing. The optimizer could not find a feasible optimum, which is visible through the severe constraint violation. The large amount of function evaluations
without proceeding to the second iteration shows that the optimization routine tried to find the right search
direction for an feasible better point in the design space, but failed to do so due to false sensitivities. It stopped
prematurely while giving the message that it could not find a better point in the design space.
It seems a reduction of almost 12% in fuel weight is gained, however due to the violation in equality constraints, this fuel weight saving is actually 9%. On top of this there exist more unacceptable equality constraint
violations. The largest equality constraint violation occurs at the C L constraint: a difference of 6% exists between C L cr ui se and the surrogate design variable C L . This means that the C L used for the aerodynamic analysis
is actually too low, which worsens the results even further. The second largest equality constraint violation
is at the W f uel constraint, as already named. This violation is 3.0852e − 2, meaning an absolute difference of
489kg .
Table 6.5 gives the optimization results for the aerodynamic and structural properties of the aircraft.
The optimization stopped prematurely, however the departure from the staring point is a good one. Just like
54
6. R ESULTS
Table 6.5: Aerodynamic and structural complete optimization results
Initial
Optimized
α[d eg ]
CL
CD
CD f
WM T O [kg ]
W f uel [kg ]
Wwi ng [kg ]
3.1584
1.8620
0.5228
0.3972
0.0195
0.0128
0.0049
0.0048
73,500
73,732
17,940
16,355
8,801
10,612
the fixed planform optimization in the prior section the alpha, C L , C D and W f uel decreased, Wwi ng increased
and WM T O remained close to its original value. Even though the C D reduced because of an exaggerated decrease in C L , the L/D is increased with 16%, from 26.81 to 31.03. This means that there has been an extra
reduction in drag. Because this solution is infeasible only part of the details of this optimization are given,
being the aerodynamic results.
The inviscid C D reduction is again a combination of wave and induced drag reduction. Starting with the
induced drag reduction: The initial and the optimized planform of the wing can be found in figure 6.11.
x[m]
0
5
10
0
5
10
15
20
25
y[m]
Figure 6.11: Planform view comparison between initial and optimized wing
It can be seen that the sweep angle is almost kept constant while the root chord and span length increased.
This makes the whole wing surface increase with 25%, from 62.18m 2 to 77.45m 2 . The root chord increase is
not a preferred outcome, as it arrests the aspect ratio increase. The aspect ratio only increased from 9.25 to
9.54. This minor increase will create only little induced drag reduction.
Coming to the wave drag reduction, the figures 6.12 and 6.13 plot the pressure coefficient and the Mach
number distribution for both the initial and the optimized wing. Figure 6.14 provides the airfoil shapes and
accompanying pressure distributions. It can be observed that the distributions themselves did not change
much, only that the peaks in pressure and Mach number are reduced in strength. This is a direct result of the
reduced angle of attack. The reason behind this barely changed distribution is that the FFD control points
did not move much. This can on it turn be explained with the fact that the normalized design sensitivities
with respect to the planform are significantly higher than the ones with respect to the FFD design variables.
The FFD design variables were not able to move yet as the optimizer stopped prematurely. The wave drag will
hence be reduced, but not by a satisfactory amount.
The fault of the disappointing results lies in the crude correction on the inaccurate sensitivities given by
SU2. The correction of the sensitivities with respect to U ∗ and E A ∗ proved to work as shown in the previous section, however the corrections are not valid any longer for the sensitivities with respect to P for a
changing planform. This has been confirmed by performing the same optimization, but with the sensitivities
with respect to the planform design variables calculated using FD. The results of that optimization run will
be explained in the section below.
6.3. C OMPLETE AEROSTRUCTURAL OPTIMIZATION
Figure 6.12: Upper wing pressure coefficient distribution comparison between initial (left) and optimized (right) wing
55
56
6. R ESULTS
Figure 6.13: Upper wing Mach number distribution comparison between initial (left) and optimized (right) wing
CP
6.3. C OMPLETE AEROSTRUCTURAL OPTIMIZATION
57
−1
−1
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
0.8
1
1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
0.6
0.8
1
0.8
1
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
x/c
0.2
0.4
0.6
x/c
(e) η = 0.625
(d) η = 0.5
0.4
(c) η = 0.375
−1
0
0.2
x/c
−1
1
0
(b) η = 0.25
−1
(f) η = 0.75
−1
−0.5
CP
CP
0.6
x/c
(a) η = 0.125
0.5
B asel i ne
Opt i mum
0.5
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
0.6
0.8
1
x/c
(g) η = 0.875
Figure 6.14: Spanwise airfoil shape and pressure coefficient distribution comparison between initial and optimized wing
58
6. R ESULTS
6.3.2 F INITE D IFFERENCE PLANFORM SENSITIVITIES
For the generation of these results the incorrect sensitives with respect to the planform design variables as
calculated using the adjoint formulation within SU2 are replaced with sensitivities calculated using FD. This
is done in order to gain satisfactory results for the complete optimization problem and to prove that the fault
lies in the false sensitivities.
This optimization run will cost significantly more time as the flow has to be solved for the calculation of each
sensitivity with respect to P . This is why the original 8 planform design variables are reduced to 6, being the
following: root chord length, taper ratio of the complete wing, wing span, leading edge sweep angle, kink
twist angle and tip twist angle. The normalized lower bounds on taper is taken to be 0.85 and on the root
chord length 0.9.
The top-level results given by table 6.6.
Table 6.6: Top-level complete optimization results
Value
Parameter
Objective function value
Iterations
Function evaluations
Maximum constraint violation
0.8873
7
51
4.6379e-5
Already looking at these results it can be seen that this optimization did produce satisfactory results. The
fuel weight is reduced with more than 11% with a maximum equality constraint violation of 4.6379e-5. The
maximum inequality constraint violation is found to be even less, 1.5855e-6. These values are low enough for
the design to be called feasible.
For this optimization each function evaluation has the cost of 3 hours, meaning a total of 153 hours or 6.4
days.
Table 6.7: Aerodynamic and structural complete optimization results
Initial
Optimized
α[d eg ]
CL
CD
CD f
WM T O [kg ]
W f uel [kg ]
Wwi ng [kg ]
3.1584
1.4712
0.5228
0.4793
0.0195
0.0142
0.0049
0.0050
73,500
70,948
17,940
15,914
8,801
8,270
Apart from the fuel reduction the optimization routine also managed to reduce the wing weight with 6%, see
table 6.7. Both reductions together reduced the WM T O with more than 3%.
x[m]
0
5
10
0
5
10
15
20
25
y[m]
Figure 6.15: Planform view comparison between initial and optimized wing
The wing weight reduction is a result of the decreased sweep angle, form 27.5◦ to 20.2◦ . Although this is
counter-acted by the increase in wing span, from 16.96m to 19.57m being an increase of 15%, a total wing
weight reduction is observed. The optimized wing sweep angle and span can be observed from figure 6.15,
which gives the planform view of the optimized wing in comparison with the initial wing. The aspect ratio
6.3. C OMPLETE AEROSTRUCTURAL OPTIMIZATION
59
increased from 9.25 to 11.55 and the taper ratio of the wing reduced from 0.21 to 0.18. The wing surface is increased from 62.18m 2 to 66.32m 2 , being an increase of 7%. What is not visible from the above planform view
is the kink and the tip jig-wing twist. The negative kink twist reduced from −2.5◦ to −1.8◦ and the negative
tip twist reduced from −2.5◦ to −0.3◦ . Compared to the first coefficient, the change in the latter coefficient
is large. This is one way how the optimizer realized extra lift on the outboard sections of the wing. The twist
distribution after deformation and the change in lift distribution are further discussed below.
Because of the reduced wing and fuel weight the cruise C L and α are also reduced. This reduces drag, however
the L/D is increased with 26%, from 26.81 to 33.75. Hence the total drag reduction does not only come from
the reduction in lift, but also from a combined reduction in induced and wave drag. The total drag coefficient
is reduced from 0.0195 to 0.0142, while the friction drag is only increased marginally. The small increase in
the friction drag component is due to the increase in wing wetted area. Helping with the induced drag reduction is the fact that the spanwise lift distribution has a more elliptical shape, just like the results shown in the
sections above. See figure 6.16 for the comparison of lift distributions of this optimization run.
4
CC l [m]
3
2
Baseline
Optimized
1
0
0
5
10
15
20
25
y[m]
Figure 6.16: Lift coefficient distribution comparison between initial and optimized wing
It is remarkable that the wing weight is reduced, even though the loading near the wing tip is higher and the
wing itself is longer. The effect of the reduction in sweep is hence observed to be stronger than the effect of
the other two combined. Through the reduced wing weight the wing bending of the wing tip is larger: from
0.57m to 0.93m. In figure 6.17 the comparison between the jig shape and the 2.5g-shape at sea level between
the initial and optimized wing is given.
Figure 6.17: Comparison between the wing jig shape (blue) and the shape under 2.5g pull up load of the initial (grey) and optimized (red)
wing
Figure 6.18 shows the twist distributions of the initial and optimized wing for their jig shape and the 1g-shape.
Even though the negative twist deformation at the tip increased from −1, 4◦ to −1, 6◦ , the total negative twist
of the complete wing is reduced. As found in the figure and stated above, this is because the negative twist of
the jig shape wing has been reduced, especially at the outboard half of the wing. An explanation of this that it
is done to help the outboard sections creating lift.
60
6. R ESULTS
Twist[d eg ]
−4
−3
−2
Baseline
Optimized
−1
0
0
5
10
15
20
25
y[m]
Figure 6.18: Twist deformation distribution comparison between initial and optimized wing of the jig shape (blue) and the 1g shape (red)
wing
The reduction in wave drag is realized by improving the airfoil shapes so that the shock wave is reduced
or even removed. This is counteracted by the reduction in sweep angle. However the trade-off has been
made in favour of the reduction in wing weight. The resulting airfoil shapes and pressure distributions are
further discussed below. In the figures 6.19 and 6.20 the pressure coefficient and Mach number distributions
of the initial and optimized wing are given. It is found that the isobars are straightened, making the wing
sweep more effective. This helps with the reduction in wave drag. From these images the earlier discussed
increased loading on the outboard wing sections also becomes visible. The leading edge of these sections
carry this increased loading.
Figure 6.21 presents the comparison of airfoil shapes at several spanwise positions. Furthermore, it gives
the comparison of the accompanying pressure distributions. At the inboard sections the caber has bee decreased slightly. At the outboard sections the the airfoils are mostly made thinner. The lower surfaces are
made less convex.
Even though the wing sweep angle increased it can be seen that at all positions the shock wave is almost
completely removed, except for at the quarter span position. At this section the shock wave is weakened and
moved forward. Moreover, what is striking from these results is the fact that the optimizer did not create rear
loading at the outboard sections, like it did in the fixed planform optimization case. In stead, the increase
in lift generation is realized by the less convex lower surfaces and a more convex upper surface leading edge.
This is now possible because of the twist-tailoring of the jig wing.
Lastly, the aileron effectiveness is found to be an active constraint. This is expected from literature, see Elham
and van Tooren [48]. However, the wing loading constraint is not active. It appears that the increase in span
and thereby an increase in wing area are the preferred direction, in stead of a decrease in surface area. This
could be a result of the fact that a local optimum is found. The global optimum could in theory be a wing
with a smaller surface area. For the reasons stated in chapter 2 a global optimization is outside the scope of
this research.
6.3. C OMPLETE AEROSTRUCTURAL OPTIMIZATION
Figure 6.19: Upper wing pressure coefficient distribution comparison between initial (left) and optimized (right) wing
61
62
6. R ESULTS
Figure 6.20: Upper wing Mach number distribution comparison between initial (left) and optimized (right) wing
Cp
6.3. C OMPLETE AEROSTRUCTURAL OPTIMIZATION
63
−1
−1
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
0.8
1
1
−1
−0.5
−0.5
−0.5
0
0
0
B asel i ne
Opt i mum
0.2
0.4
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
0.2
0.4
x/c
0.6
0.8
1
0.8
1
B asel i ne
Opt i mum
0.5
0.6
0.8
1
1
0
x/c
(d) η = 0.5
0.4
(c) η = 0.375
−1
0
0.2
x/c
−1
1
0
(b) η = 0.25
0.2
0.4
0.6
x/c
(e) η = 0.625
(f) η = 0.75
−1
−0.5
Cp
Cp
0.6
x/c
(a) η = 0.125
0.5
B asel i ne
Opt i mum
0.5
0
B asel i ne
Opt i mum
0.5
1
0
0.2
0.4
0.6
0.8
1
x/c
(g) η = 0.875
Figure 6.21: Spanwise airfoil shape and pressure coefficient distribution comparison between initial and optimized wing
7
C ONCLUSIONS AND RECOMMENDATIONS
The final conclusions and recommendations for further research based on this research will be given in this
last chapter of the present thesis.
7.1 C ONCLUSIONS
In this study the IDF MDO architecture has been applied to solve a wing aerostructural optimization problem. The main advantage of IDF has been found to be that it does not require the coupled sensitivity analysis
for gradient-based optimization. This provides the user with more freedom of software choice for the disciplinary analyses. On top of this, it reduces the overall computational costs with respect to the widely used
MDF architecture. This increase in optimization efficiency comes from the fact that the IDF architecture is
able to advance towards an optimum in the design space while realizing consistency of the multidisciplinary
system simultaneously. This last argument has however not been proven in the present work.
The drawback of the IDF architecture is that if the increase in design variables due to this architecture is too
large, its efficiency is compromised. It was however found that is not the case in this application.
Throughout the development of the tool its was found that the sensitivities with respect to the displacement
of the trailing edge provided by SU2 were inaccurate. This was confirmed by its developers. This inaccuracy
resulted in the need to introduce corrective factors to the sensitivities. These corrective factors proved to
work for the aerodynamic sensitivities with respect to the aeroelastic deformation and elastic axis position,
but not for the sensitivities with respect to the planform design variables.
The functioning of the tool has been tested on a test case: The wing of the Airbus A320. Before the aerostructural optimizations an aerodynamic optimization run is performed, to make sure all modules of SU2 works
correctly when it is controlled by the optimization algorithm. This optimization is found to be a success as it
resulted in the reduction of the total drag of 16%, through the reduction of both induced and wave drag.
Afterwards, three aerostructural wing optimizations are performed. All of these optimizations optimize the
wing external shape and internal wing box structure for minimal fuel weight necessary for the range characteristics of the Airbus A320 reference aircraft. The first optimization kept the planform shape of the wing
constant while for the last two optimizations this was also included in the design vector. The difference between the last two optimizations is the way of calculating the aerodynamic sensitivities with respect to the
planform design variables. The second optimization used the continuous adjoint formulation for its calculation while the third used FD.
The fixed planform optimization has resulted in a 7% reduction in the aircraft fuel weight. The drag reduction
was achieved by reducing the wave drag and induced drag of the wing. The wave drag was reduced in two
ways. Firstly by making more use of the wing sweep by straightening the iso-bars on the wing and secondly
by reducing the strength of the shock wave at the inboard half of the wing. The induced drag was reduced
by making the spanwise lift distribution more elliptic. This last effect had the penalty of making the wing
structure heavier. The optimization results show the optimum for this trade-off.
It is shown that the optimization approach performs well. With the relatively small computational power of
65
66
7. C ONCLUSIONS AND RECOMMENDATIONS
8 processors an optimum is found within 4 days. Moreover, it is proven that the corrective factors for the
aerodynamic sensitivities with respect to the aeroelastic deformation and elastic axis position work. This is
however not the desirable method of sensitivity calculation and a better optimum should be reached if these
sensitivities were calculated in the correct way.
The complete optimization including the planform design variables provided disappointing results. Although
the fuel weight was reduced with 9%, the equality constraints were violated beyond the acceptable. The
optimization routine failed to simultaneously reduce the objective function and constraint violations and
stopped prematurely. This is due to the inaccurate sensitivities given by SU2, even after correction. The reason why this optimization failed to converge while the fixed-planform optimization did converge is because
the correction factors were not valid anymore for the sensitivities with respect to the planform shape. This
was tested by performing the same optimization but with the sensitivities with respect to the planform design variables calculated with FD in stead of the adjoint formulation. For this last optimization the amount
of planform design variables was reduced form 8 to 6 to reduce the computational cost.
The last optimization resulted in a success. The optimizer managed to reduce the aircraft fuel weight with
11%. The constraint violations remained within the acceptable. At the found optimum, the constraint on
aileron effectiveness is active, but the constraint on wing loading is not.
The wing friction drag is increased slightly, while the induced and wave drag are reduced. The induced drag
reduction is a result of the increase in aspect ratio and improved spanwise lift distribution. The wave drag is
reduced by removing the shock wave at almost all spanwise sections and by straightening the isobars over the
wing. Both are managed through airfoil shape changes.
Even while the increased wingspan and the outbound wing loading have a negative effect on the aircraft wing
loading, it is reduced. This is explained by the larger effect of the sweep angle decrease on the wing weight.
The wave drag increase accompanied by this sweep angle decrease is more than compensated by the airfoil
shape changes.
7.2 R ECOMMENDATIONS
The present section presents recommendations for further research based on this thesis.
• It its highly recommended to improve sensitivity accuracy by using the discrete adjoint method, which
will be implemented in SU2 in the near future. These new sensitivities should result in a successful
verification and a converging optimization when planform design variables are included in the design
vector.
• This research only estimated the viscous drag and neglected other boundary layer effects. It is recommended to include the boundary layer analysis in the optimization using RANS for a better estimation
of the the friction drag and shock-induced boundary layer separation.
• The airfoil shape design variables used are evenly spread over the supper and lower surface of the FFDboxes. It is recommended to construct areas with a higher density of FFD control points, like for example the leading edge. This will improve the design freedom in areas with high design sensitivities.
• Throughout this study only static aeroelastic behaviour is taken into account. When dynamic aeroelasticity is included, a more realistic picture of the boundaries of the design space is created.
• It is furthermore recommended that a multi-point optimization for the complete mission of the aircraft
is performed, as the wing shape of the optimized wing is only optimized for the cruise condition.
• The optimum found in this study is found using a gradient based, local optimization algorithm. It
is recommended to perform a global optimization in order to find the optimal wing in the complete
design space.
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A PPENDIX
C ODE FOR REALIZATION OF U ∗ AND P DEFORMATION AND E A ∗ SENSITIVITY
CALCULATION
void CSurfaceMovement::SetFFD_U_DEF(CGeometry *geometry, CConfig *config,
CFreeFormDefBox *FFDBox, unsigned short iDV, bool ResetDef) {
unsigned short iOrder, jOrder, kOrder;
double x, y, z, movement[3];
unsigned short index[3];
string design_FFDBox;
/*--- Set control points to its original value (even if the
design variable is not in this box) ---*/
if (ResetDef == true) FFDBox->SetOriginalControlPoints();
design_FFDBox = config->GetFFDTag(iDV);
if (design_FFDBox.compare(FFDBox->GetTag()) == 0) {
/*--- xyz-coordinates of a point on the line of rotation. ---*/
double
double
double
double
double
double
double
double
double
double
double
double
u_x=config->GetParamDV(iDV, 2)*config->GetDV_Value(iDV);
u_y=config->GetParamDV(iDV, 3)*config->GetDV_Value(iDV);
u_z=config->GetParamDV(iDV, 4)*config->GetDV_Value(iDV);
cosPH=cos(-1*config->GetParamDV(iDV, 5)
*config->GetDV_Value(iDV)*(PI_NUMBER/180.0));
sinPH=sin(-1*config->GetParamDV(iDV, 5)
*config->GetDV_Value(iDV)*(PI_NUMBER/180.0));
cosT=cos(-1*config->GetParamDV(iDV, 6)
*config->GetDV_Value(iDV)*(PI_NUMBER/180.0));
sinT=sin(-1*config->GetParamDV(iDV, 6)
*config->GetDV_Value(iDV)*(PI_NUMBER/180.0));
cosPS=cos(-1*config->GetParamDV(iDV, 7)
*config->GetDV_Value(iDV)*(PI_NUMBER/180.0));
sinPS=sin(-1*config->GetParamDV(iDV, 7)
*config->GetDV_Value(iDV)*(PI_NUMBER/180.0));
elas_axis_x = config->GetParamDV(iDV, 8);
elas_axis_y = config->GetParamDV(iDV, 9);
elas_axis_z = config->GetParamDV(iDV, 10);
/*--- The angle of rotation. ---*/
/*double psi = config->GetDV_Value(iDV)*PI_NUMBER/180.0; */
/*--- An intermediate value used in computations. ---*/
72
. A PPENDIX
/*--- Change the value of the control point if move is true ---*/
for (iOrder = 0; iOrder < FFDBox->GetlOrder(); iOrder++)
for (kOrder = 0; kOrder < FFDBox->GetnOrder(); kOrder++) {
index[0] = iOrder; index[1] = int(config->GetParamDV(iDV, 1)); index[2] = kOrder;
double *coord = FFDBox->GetCoordControlPoints(iOrder,
int(config->GetParamDV(iDV, 1)), kOrder);
x = coord[0]; y = coord[1]; z = coord[2];
double elas_rel_x=(x+u_x)-elas_axis_x;
double elas_rel_y=(y+u_y)-elas_axis_y;
double elas_rel_z=(z+u_z)-elas_axis_z;
double elas_rel_x_def_x=elas_rel_x;
double elas_rel_y_def_x=cosPH*elas_rel_y + sinPH*elas_rel_z;
double elas_rel_z_def_x=-sinPH*elas_rel_y + cosPH*elas_rel_z;
double elas_rel_x_def_xy=cosT*elas_rel_x_def_x - sinT*elas_rel_z_def_x;
double elas_rel_y_def_xy=elas_rel_y_def_x;
double elas_rel_z_def_xy=sinT*elas_rel_x_def_x + cosT*elas_rel_z_def_x;
double elas_rel_x_def=cosPS*elas_rel_x_def_xy + sinPS*elas_rel_y_def_xy;
double elas_rel_y_def=-sinPS*elas_rel_x_def_xy + cosPS*elas_rel_y_def_xy;
double elas_rel_z_def=elas_rel_z_def_xy;
double x_def=elas_rel_x_def+elas_axis_x;
double y_def=elas_rel_y_def+elas_axis_y;
double z_def=elas_rel_z_def+elas_axis_z;
movement[0] = x_def-x;
movement[1] = y_def-y;
movement[2] = z_def-z;
FFDBox->SetControlPoints(index, movement);
}
}
}
void CSurfaceMovement::SetFFD_EA_GRAD(CGeometry *geometry, CConfig *config,
CFreeFormDefBox *FFDBox, unsigned short iDV, bool ResetDef) {
unsigned short iOrder, jOrder, kOrder;
double x, y, z, movement[3];
unsigned short index[3];
string design_FFDBox;
/*--- Set control points to its original value (even if the
design variable is not in this box) ---*/
73
if (ResetDef == true) FFDBox->SetOriginalControlPoints();
design_FFDBox = config->GetFFDTag(iDV);
if (design_FFDBox.compare(FFDBox->GetTag()) == 0) {
/*--- xyz-coordinates of a point on the line of rotation. ---*/
double
double
double
double
double
double
double
double
double
double
double
double
double
double
double
double
double
double
cosPH=cos(-1*config->GetParamDV(iDV, 2)*(PI_NUMBER/180.0));
sinPH=sin(-1*config->GetParamDV(iDV, 2)*(PI_NUMBER/180.0));
cosT=cos(-1*config->GetParamDV(iDV, 3)*(PI_NUMBER/180.0));
sinT=sin(-1*config->GetParamDV(iDV, 3)*(PI_NUMBER/180.0));
cosPS=cos(-1*config->GetParamDV(iDV, 4)*(PI_NUMBER/180.0));
sinPS=sin(-1*config->GetParamDV(iDV, 4)*(PI_NUMBER/180.0));
cos_PH=cos(config->GetParamDV(iDV, 2)*(PI_NUMBER/180.0));
sin_PH=sin(config->GetParamDV(iDV, 2)*(PI_NUMBER/180.0));
cos_T=cos(config->GetParamDV(iDV, 3)*(PI_NUMBER/180.0));
sin_T=sin(config->GetParamDV(iDV, 3)*(PI_NUMBER/180.0));
cos_PS=cos(config->GetParamDV(iDV, 4)*(PI_NUMBER/180.0));
sin_PS=sin(config->GetParamDV(iDV, 4)*(PI_NUMBER/180.0));
elas_axis_x = config->GetParamDV(iDV, 5);
elas_axis_y = config->GetParamDV(iDV, 6);
elas_axis_z = config->GetParamDV(iDV, 7);
elas_axis_x_grad = config->GetParamDV(iDV, 8);
elas_axis_y_grad = config->GetParamDV(iDV, 9);
elas_axis_z_grad = config->GetParamDV(iDV, 10);
/*--- The angle of rotation. ---*/
/*double psi = config->GetDV_Value(iDV)*PI_NUMBER/180.0; */
/*--- An intermediate value used in computations. ---*/
/*--- Change the value of the control point if move is true ---*/
for (iOrder = 0; iOrder < FFDBox->GetlOrder(); iOrder++)
for (kOrder = 0; kOrder < FFDBox->GetnOrder(); kOrder++) {
index[0] = iOrder; index[1] = int(config->GetParamDV(iDV, 1)); index[2] = kOrder;
double *coord = FFDBox->GetCoordControlPoints(iOrder,
int(config->GetParamDV(iDV, 1)), kOrder);
x = coord[0]; y = coord[1]; z = coord[2];
/* Reverse rotation to get original coordinates*/
double elas_rel_x_orig=x-elas_axis_x;
double elas_rel_y_orig=y-elas_axis_y;
double elas_rel_z_orig=z-elas_axis_z;
double elas_rel_x_def_z_orig=cos_PS*elas_rel_x_orig + sin_PS*elas_rel_y_orig;
double elas_rel_y_def_z_orig=-sin_PS*elas_rel_x_orig + cos_PS*elas_rel_y_orig;
double elas_rel_z_def_z_orig=elas_rel_z_orig;
double elas_rel_x_def_yz_orig=cos_T*elas_rel_x_def_z_orig
- sin_T*elas_rel_z_def_z_orig;
74
. A PPENDIX
double elas_rel_y_def_yz_orig=elas_rel_y_def_z_orig;
double elas_rel_z_def_yz_orig=sin_T*elas_rel_x_def_z_orig
+ cos_T*elas_rel_z_def_z_orig;
double elas_rel_x_def_orig=elas_rel_x_def_yz_orig;
double elas_rel_y_def_orig=cos_PH*elas_rel_y_def_yz_orig
+ sin_PH*elas_rel_z_def_yz_orig;
double elas_rel_z_def_orig=-sin_PH*elas_rel_y_def_yz_orig
+ cos_PH*elas_rel_z_def_yz_orig;
double x_orig=elas_rel_x_def_orig+elas_axis_x;
double y_orig=elas_rel_y_def_orig+elas_axis_y;
double z_orig=elas_rel_z_def_orig+elas_axis_z;
/* Do rotation to get new coordinates*/
double elas_rel_x=x_orig-(elas_axis_x+(elas_axis_x_grad*config->GetDV_Value(iDV)));
double elas_rel_y=y_orig-(elas_axis_y+(elas_axis_y_grad*config->GetDV_Value(iDV)));
double elas_rel_z=z_orig-(elas_axis_z+(elas_axis_z_grad*config->GetDV_Value(iDV)));
double elas_rel_x_def_x=elas_rel_x;
double elas_rel_y_def_x=cosPH*elas_rel_y + sinPH*elas_rel_z;
double elas_rel_z_def_x=-sinPH*elas_rel_y + cosPH*elas_rel_z;
double elas_rel_x_def_xy=cosT*elas_rel_x_def_x - sinT*elas_rel_z_def_x;
double elas_rel_y_def_xy=elas_rel_y_def_x;
double elas_rel_z_def_xy=sinT*elas_rel_x_def_x + cosT*elas_rel_z_def_x;
double elas_rel_x_def=cosPS*elas_rel_x_def_xy + sinPS*elas_rel_y_def_xy;
double elas_rel_y_def=-sinPS*elas_rel_x_def_xy + cosPS*elas_rel_y_def_xy;
double elas_rel_z_def=elas_rel_z_def_xy;
double x_def=elas_rel_x_def+(elas_axis_x+(elas_axis_x_grad*config->GetDV_Value(iDV)));
double y_def=elas_rel_y_def+(elas_axis_y+(elas_axis_y_grad*config->GetDV_Value(iDV)));
double z_def=elas_rel_z_def+(elas_axis_z+(elas_axis_z_grad*config->GetDV_Value(iDV)));
/* Movement is difference between old rotation and new*/
movement[0] = x_def-x;
movement[1] = y_def-y;
movement[2] = z_def-z;
FFDBox->SetControlPoints(index, movement);
}
}
}
void CSurfaceMovement::SetFFD_PLANFORM(CGeometry *geometry, CConfig *config,
CFreeFormDefBox *FFDBox, unsigned short iDV, bool ResetDef) {
unsigned short iOrder, jOrder, kOrder;
double x, y, z, movement[3];
unsigned short index[3];
75
string design_FFDBox;
/*--- Set control points to its original value (even if the
design variable is not in this box) ---*/
if (ResetDef == true) FFDBox->SetOriginalControlPoints();
design_FFDBox = config->GetFFDTag(iDV);
if (design_FFDBox.compare(FFDBox->GetTag()) == 0) {
/*--- xyz-coordinates of a point on the line of rotation. ---*/
double
double
double
double
double
D_chord=config->GetParamDV(iDV, 2)*config->GetDV_Value(iDV);
l_chord=config->GetParamDV(iDV, 3);
D_FFD_x_0=config->GetParamDV(iDV, 4);
l_FFD_x=config->GetParamDV(iDV, 5);
l_FFD_z=config->GetParamDV(iDV, 6);
double
double
double
double
double
chord_ratio
l_FFD_x_new
l_FFD_z_new
FFD_x_mov =
FFD_z_mov =
= (D_chord+l_chord)/l_chord;
= l_FFD_x*chord_ratio;
= l_FFD_z*chord_ratio;
(l_FFD_x_new-l_FFD_x);
(l_FFD_z_new-l_FFD_z)/2;
/*--- The angle of rotation. ---*/
/*double psi = config->GetDV_Value(iDV)*PI_NUMBER/180.0; */
/*--- An intermediate value used in computations. ---*/
/*--- Change the value of the control point if move is true ---*/
for (iOrder = 0; iOrder < FFDBox->GetlOrder(); iOrder++)
for (kOrder = 0; kOrder < FFDBox->GetnOrder(); kOrder++) {
index[0] = iOrder; index[1] = int(config->GetParamDV(iDV, 1)); index[2] = kOrder;
double *coord = FFDBox->GetCoordControlPoints(iOrder,
int(config->GetParamDV(iDV, 1)), kOrder);
x = coord[0]; y = coord[1]; z = coord[2];
double movement_portion = (double) index[0]/(FFDBox->GetlOrder()-1);
movement[0] = (movement_portion*FFD_x_mov)+D_FFD_x_0;
movement[1] = 0;
double k_Order = (double) kOrder;
movement[2] = (FFD_z_mov*(k_Order-1))+(FFD_z_mov*k_Order);
FFDBox->SetControlPoints(index, movement);
}
}
}
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