thesis. - Repository TU Delft
Master of Science Thesis
Optimization strategy for conceptual
airplane design
P.T. Vasseur B.Sc.
May 9, 2014
Faculty of Aerospace Engineering
·
Delft University of Technology
Optimization strategy for conceptual
airplane design
Master of Science Thesis
For obtaining the degree of Master of Science in Aerospace
Engineering at Delft University of Technology
P.T. Vasseur B.Sc.
May 9, 2014
Faculty of Aerospace Engineering
·
Delft University of Technology
Copyright © P.T. Vasseur B.Sc.
All rights reserved.
Delft University Of Technology
Department Of
Flight Performance and Propulsion
The undersigned hereby certify that they have read and recommend to the Faculty of
Aerospace Engineering for acceptance a thesis entitled “Optimization strategy for
conceptual airplane design” by P.T. Vasseur B.Sc. in partial fulfillment of the
requirements for the degree of Master of Science.
Dated: May 9, 2014
Head of department:
prof.dr.ir. L.L.M. Veldhuis
Supervisor:
dr.ir. R. Vos
Reader:
dr.ir. H.G. Visser
Abstract
Due to the ever growing demand for more efficient aircraft novel aircraft concepts have
to be explored. By improving design tools the potential of unconventional configurations
can be further studied. This requires improvement of conceptual design tools such that
more knowledge can be gathered on alternative solutions as early in the design process
as possible. Multidisciplinary design optimization (MDO) can support this process by
providing an environment in which the various disciplines can be designed and optimized
concurrently, while a certain level of consistency is maintained.
An optimization design tool has been created to assess the potential performance gains
of novel aircraft configurations. It connects with the Initiator design tool, which is a
conceptual design framework. As such, it can also be used as a means to expose any
analysis or design issues that may exist in the Initiator.
With the optimizer tool the following four case studies were performed: a conventional
Airbus A320, a forward-swept canard aircraft, a three-surface aircraft and an oval-fuselage
aircraft. For this purpose the genetic algorithm, sequential quadratic programming algorithm and a hybrid genetic algorithm were used. From the case studies followed that
large improvements can be obtained with unconventional aircraft configurations when
compared to the initial aircraft design proposed by the Initiator design tool. Up to 20%
improvement was found with the three-surface and canard aircraft. The oval-fuselage
aircraft could be improved by a solid 10%, while the lowest improvement was attained
with the conventional A320.
Among all cases the most contributing factors were the wing longitudinal position, sweep
angle and wing aspect ratio. There is a tendency towards lower sweep angles due to the
positive effect on the weight of the wing and an underestimation of the drag rise. With
the forward-swept canard relatively high sweep angles were found, which contradicts
the findings of the aft-swept wings. Therefore, the aerodynamics routine needs further
investigation. From the highly swept, high aspect ratio wings of the forward-swept canard
aircraft followed that the weight penalty of forward swept wings is underestimated.
In three cases the fuselage fineness ratio was involved in the optimization. The results
showed that changing the fineness ratio offered some reduction in fuselage weight due to
a more favourable structural loading.
v
vi
Abstract
The sizing routine of the control surfaces is found to be inadequate, since the Initiator
derives most parameters directly from the wing and does not properly take into account
control and stability requirements. Results have shown that this mainly regards the sweep
and dihedral angle. Especially, the sweep angle is of concern, since it changes the liftcurve slope and therefore also stall characteristics. These sizing issues also affect the static
margin. It was found that class II design information was not fed back to the control
surface sizing.
Other discrepancies were found with the wing dihedral. Due to a lack of lateral stability
analysis of the Initiator the dihedral was driven by the lift-to-drag ratio rather than its
stabilizing effect. As a result a lower dihedral was observed among the cases.
From the used optimization algorithms can be concluded that the gradient algorithm was
the least effective. It had difficulties with the uncertainties in the computed results of the
Initiator. It sometimes stopped prematurely or started oscillating. This was alleviated
by increasing the step size of the algorithm, but at the expense of accuracy. The genetic
algorithm was found to be the best option since it proved to be very robust. It is far
less sensitivity to noise, because it does not use gradient information. Its computational
cost could be significantly reduced by applying parallel optimization and using a caching
mechanism. The hybrid algorithm was found to be too computational expensive. The
obtained increase in objective value did not outweigh the added cost.
Acknowledgements
This thesis marks the final step in completing the Master Program in System Engineering
and Aircraft Design at the faculty of Aerospace Engineering. I had a great time studying
at Delft University of Technology, with the company of my fellow students and friends.
The research presented in this report would not have been possible without the support
of a number of people whom I hereby would like to thank.
Fist of all, I would like to express my gratitude to my supervisor dr.ir. Roelof Vos for his
feedback, support and valuable insights during the course of my master thesis. I would
also like to thank the members of my committee, prof.dr.ir. Leo Veldhuis and Dries Visser
for their time to assess my work. I want to express my appreciation to Reno Elmendorp
for providing feedback on the optimizer tool and helping me out with integration in the
Initiator.
Last but not least, I would like to thank my parents for their continuous support and
encouragement throughout my study.
Delft, The Netherlands
May 9, 2014
P.T. Vasseur B.Sc.
vii
viii
Acknowledgements
Contents
Abstract
v
Acknowledgements
vii
List of Figures
xi
List of Tables
xiii
Nomenclature
xv
I
Thesis
1
1 Introduction
1.1 Research question and thesis goal . . . . . . . . . . . . . . . . . . . . . . .
1.2 Report outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
4
4
2 Background information
2.1 MDO in aircraft design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Optimization strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Multidisciplinary feasible (MDF) . . . . . . . . . . . . . . . . . . .
7
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8
2.2.2
Individual discipline feasible (IDF) . . . . . . . . . . . . . . . . . .
9
2.2.3
All-at-once (AAO) . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2.4
Collaborative optimization (CO) . . . . . . . . . . . . . . . . . . .
10
2.2.5 Concurrent subspace optimization (CSSO) . . . . . . . . . . . . . .
Initiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
12
3 Sensitivity analysis
3.1 Variable screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Elementary effects method . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
15
15
2.3
ix
x
Contents
4 Optimization algorithms
4.1 Problem description . . .
4.2 Gradient-based algorithms
4.3 Genetic algorithm . . . .
4.4 Hybrid algorithm . . . . .
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7 Conclusions and recommendations
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
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II
71
5 Optimizer tool description
5.1 Optimizer workflow . . . .
5.2 Optimization strategy . .
5.3 Caching results . . . . . .
5.4 Parallel optimization . . .
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6 Optimization case studies
6.1 Key performance indicators . . . . . . . . . .
6.2 Optimization procedure . . . . . . . . . . . .
6.2.1 Sensitivity analysis . . . . . . . . . . .
6.2.2 Optimization . . . . . . . . . . . . . .
6.2.3 Initiator and aircraft settings . . . . .
6.3 Case 1: Airbus A320 . . . . . . . . . . . . . .
6.4 Case 2: Canard aircraft . . . . . . . . . . . .
6.5 Case 3: Three-surface aircraft . . . . . . . . .
6.6 Case 4: Oval-fuselage aircraft . . . . . . . . .
6.7 A comparison of the obtained aircraft designs
6.8 Evaluation of the algorithms . . . . . . . . . .
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Code documentation
8 Program structure
8.1 Optimizer class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
73
75
76
9 User manual
9.1 Requirements . . . . . . . . . . .
9.2 Setting up a problem description
9.3 Operating the optimizer . . . . .
9.3.1 Module input . . . . . . .
9.3.2 Using the module handle
81
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Contents
xi
References
89
A Example three-surface aircraft report
93
A.1 General Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
94
A.3 Optimiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
A.4 Operational Performance
. . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A.5 Weight estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
A.6 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A.7 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.8 Aircraft Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xii
Contents
List of Figures
2.1
Aircraft design process [2] . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Multidisciplinary feasible strategy . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Individual discipline feasible strategy . . . . . . . . . . . . . . . . . . . . .
10
2.4
High-level activity diagram of the Initiator . . . . . . . . . . . . . . . . . .
12
3.1
Example trajectory in 3-dimensional space using a five-level grid . . . . .
16
3.2
Sampling probability using different values for p and ∆ . . . . . . . . . . .
17
4.1
The crossover, mutation and elite selection procedures of the genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.1
High-level activity diagram of the optimizer
. . . . . . . . . . . . . . . .
26
5.2
Multidisciplinary feasible implementation of the optimizer tool . . . . . .
27
5.3
Effect of caching on the genetic algorithm . . . . . . . . . . . . . . . . . .
28
5.4
Scaling the genetic algorithm using parallel computing . . . . . . . . . . .
29
6.1
An example of a payload-range diagram . . . . . . . . . . . . . . . . . . .
32
6.2
Improvement in objective value against computation time . . . . . . . . .
34
6.3
6.4
6.5
Airfoils used in the case studies . . . . . . . . . . . . . . . . . . . . . . . .
Airbus A320 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Airbus A320 sensitivity analysis results . . . . . . . . . . . . . . . . . . .
36
37
38
6.6
Airbus A320 lift-to-drag ratio and operational empty mass . . . . . . . . .
39
6.7
Airbus A320 lift-to-drag ratio for varying parameters . . . . . . . . . . . .
39
6.8
Airbus A320 change in part mass after optimization . . . . . . . . . . . .
40
6.9
Airbus A320 geometry after genetic optimization . . . . . . . . . . . . . .
43
6.10 Airbus A320 geometry after gradient-based optimization . . . . . . . . . .
43
6.11 Airbus A320 geometry after hybrid optimization . . . . . . . . . . . . . .
43
xiii
xiv
List of Figures
6.12 Canard aircraft model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.13 Canard aircraft sensitivity analysis results . . . . . . . . . . . . . . . . . .
44
45
6.14 Effect of aspect ratio and sweep angle on the canard aircraft . . . . . . . .
46
6.15 Canard aircraft change in part mass after optimization . . . . . . . . . . .
47
6.16 Canard aircraft static margin vs. sweep angle and wing position (A = 9.4) 48
6.17 Canard aircraft geometry after genetic optimization . . . . . . . . . . . .
50
6.18 Canard aircraft geometry after gradient optimization . . . . . . . . . . . .
50
6.19 Canard aircraft geometry after hybrid optimization . . . . . . . . . . . . .
50
6.20 Three-surface equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
6.21 Three-surface aircraft model . . . . . . . . . . . . . . . . . . . . . . . . . .
6.22 Three-surface aircraft sensitivity analysis results . . . . . . . . . . . . . .
51
52
6.23 Three-surface aircraft change in part mass after optimization . . . . . . .
53
6.24 Three-surface aircraft with varying aspect ratio and sweep angle (fxw = 0.5) 54
6.25 Three-surface aircraft lift-to-drag for varying parameters . . . . . . . . . .
54
6.26 Thee-surface aircraft geometry after genetic optimization . . . . . . . . .
57
6.27 Thee-surface aircraft geometry after gradient optimization . . . . . . . . .
57
6.28 Thee-surface aircraft geometry after hybrid optimization . . . . . . . . . .
57
6.29 Oval-fuselage aircraft model . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6.30 Oval-fuselage aircraft sensitivity analysis results . . . . . . . . . . . . . . .
59
6.31 Oval-fuselage aircraft change in part mass after optimization . . . . . . .
60
6.32 Oval-fuselage aircraft operational empty mass and lift-to-drag ratio . . . .
60
6.33 Oval-fuselage aircraft fuselage slenderness ratio effects (A = 9.5) . . . . .
61
6.34 Oval-fuselage aircraft geometry after genetic optimization . . . . . . . . .
63
6.35 Oval-fuselage aircraft geometry after gradient-based optimization . . . . .
63
6.36 Oval-fuselage aircraft geometry after hybrid optimization . . . . . . . . .
63
6.37 Payload-range efficiency for increasing aspect ratio using the Initiator . .
66
8.1
UML class diagram of the optimizer . . . . . . . . . . . . . . . . . . . . .
73
8.2
Flowchart of the sensitivity analysis . . . . . . . . . . . . . . . . . . . . .
76
8.3
Top-level flowchart of the optimization process . . . . . . . . . . . . . . .
77
8.4
Flowchart of the initiatorRunner method . . . . . . . . . . . . . . . . .
78
A.1 Aircraft geometry (all dimensions in meters) . . . . . . . . . . . . . . . . .
93
A.2 Objective value history . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
A.3 Aircraft geometry changes . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A.4 Loading Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A.5 Payload-Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.6 V-n diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.7 Mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.8 Loading diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
98
List of Figures
xv
A.9 CG location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.10 Drag Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.11 Aerodynamic efficiency of the aircraft . . . . . . . . . . . . . . . . . . . . 100
A.12 Fuel tank layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.13 Fuselage geometry; (blue = cargo ULDs, purple = floors) . . . . . . . . . 103
xvi
List of Figures
List of Tables
6.1
List of design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
6.2
Airbus A320 top level requirements . . . . . . . . . . . . . . . . . . . . . .
37
6.3
Airbus A320 properties using the Initiator . . . . . . . . . . . . . . . . . .
37
6.4
Optimum design vectors for the Airbus A320 . . . . . . . . . . . . . . . .
38
6.5
Airbus A320 optimization results . . . . . . . . . . . . . . . . . . . . . . .
41
6.6
Airbus A320-200 specifications [21, 22] . . . . . . . . . . . . . . . . . . . .
42
6.7
Canard aircraft top level requirements . . . . . . . . . . . . . . . . . . . .
44
6.8
Canard aircraft properties using the Initiator . . . . . . . . . . . . . . . .
45
6.9
Optimum design vectors for the canard aircraft . . . . . . . . . . . . . . .
46
6.10 Canard aircraft optimization results . . . . . . . . . . . . . . . . . . . . .
49
6.11 Three-surface aircraft top level requirements . . . . . . . . . . . . . . . . .
51
6.12 Three-surface aircraft properties using the Initiator . . . . . . . . . . . . .
52
6.13 Optimum design parameters for the three-surface aircraft . . . . . . . . .
53
6.14 Thee-surface aircraft optimization results . . . . . . . . . . . . . . . . . .
56
6.15 Oval fuselage aircraft top level requirements . . . . . . . . . . . . . . . . .
58
6.16 Oval-fuselage properties using the Initiator . . . . . . . . . . . . . . . . .
58
6.17 Optimum design parameters for the oval-fuselage aircraft . . . . . . . . .
59
6.18 Oval-fuselage aircraft optimization results . . . . . . . . . . . . . . . . . .
62
6.19 Comparison of the case studies using the VEM parameter . . . . . . . . .
64
6.20 Comparison of the case studies using the VEO parameter . . . . . . . . .
64
8.1
Public properties of the optimizer class . . . . . . . . . . . . . . . . . . . .
74
8.2
8.3
Problem structure fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Public methods of the optimizer class . . . . . . . . . . . . . . . . . . . .
74
75
8.4
Sensitivity analysis results structure . . . . . . . . . . . . . . . . . . . . .
76
8.5
Optimization results structure . . . . . . . . . . . . . . . . . . . . . . . . .
79
xvii
xviii
List of Tables
9.1
General optimizer settings . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
9.2
Elementary effects method settings . . . . . . . . . . . . . . . . . . . . . .
86
A.1 Max payload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
A.2 Optimiser results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
A.3 Performance results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Mass summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
97
A.5 Component masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
A.6 Centre-of-gravity locations . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A.7 Aerodynamic properties at cruise . . . . . . . . . . . . . . . . . . . . . . .
99
A.8 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.9 Main Wing dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.10 Horizontal Stabiliser dimensions .
A.11 Front Stabiliser dimensions . . .
A.12 Vertical Stabiliser dimensions . .
A.13 Fuselage dimensions . . . . . . .
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101
101
102
102
Nomenclature
Latin Symbols
A
Wing aspect ratio
[−]
ŝ
Normalized sensitivity value
[−]
b
Wing span
[m]
CD
Drag coefficient
[−]
CL
Lift coefficient
[−]
cr
Root chord length
[m]
cT
Specific fuel consumption
ct
Tip chord length
[m]
di
Elementary effect
[−]
df
Fuselage diameter
[m]
e
Oswald efficiency number
[−]
L/D
Lift-to-drag ratio
[−]
lf
Fuselage length
[m]
M
Mach number
[−]
p
Grid level
[−]
R
Harmonic range
r
Number of trajectories
S
Wing surface area
[m2 ]
sL
Landing distance
[m]
sT O
Take-off distance
[m]
t/c
Thickness-over-chord ratio
[−]
[g/(s · kN )]
[km]
[−]
xix
xx
Nomenclature
Vs
Stall speed
[m]
Wfb
Block fuel mass
[kg]
Wp
Payload mass
[kg]
X
Range parameter
x
Aircraft longitudinal axis
[km]
[m]
Greek Symbols
Twist angle
[◦ ]
Γ
Dihedral angle
[◦ ]
Λ
Sweep angle
[◦ ]
λ
Taper ratio
[−]
λf
Fuselage fineness ratio, lf /df
[−]
µ
Mean
[−]
µ∗
Modified mean
[−]
σ
Standard deviation
[−]
Subscripts
c
Canard
f
Fuselage
h
Horizontal tail
k
Wing kink section
r
Wing root section
t
Wing tip section
v
Vertical tail
Abbreviations
AAO
All-at-once
ALGA
Augmented Lagrangian genetic algorithm
CO
Collaborative optimization
CSSO
Concurrent subspace optimization
EE
Elementary effects method
FM
Aircraft fuel mass
GA
Genetic algorithm
Nomenclature
HT
Horizontal tail
IDF
Individual discipline feasible
KPI
Key performance indicators
MAC
Mean aerodynamic chord
MDA
Multidisciplinary analysis
MDF
Multidisciplinary feasible
MDO
Multidisciplinary design optimization
MTOM
Aircraft maximum take-off mass
MZFM
Maximum zero-fuel mass
NLP
Non-linear optimization problem
OEM
Aircraft operational empty mass
OFAT
One factor at a time
PRE
Payload-range efficiency
SAND
Simultaneous analysis and design
SM
Static margin
SQP
Sequential quadratic programming
TSA
Three-surface aircraft
UML
Unified modeling language
VEM
Value efficiency parameter w.r.t. MTOM
VEO
Value efficiency parameter w.r.t. OEM
VT
Vertical tail
XML
Extensible markup language
xxi
xxii
Nomenclature
Part I
Thesis
1
Chapter 1
Introduction
Due to the ever growing demand for more efficient aircraft novel aircraft concepts have
to be explored. By improving design tools the potential of unconventional configurations
can be further studied. The quest for more efficient aircraft requires improvement of
conceptual design tools such that more knowledge can be gathered on alternative solutions
as early in the design process as possible.
Since in aircraft design many disciplines are involved, obtaining the optimal design that
satisfies the requirements is not an easy task. At the conceptual design level multidisciplinary design optimization (MDO) can support this process by providing an efficient
methodology in which the various disciplines can be designed and optimized concurrently,
while a certain level of consistency is maintained. As such, MDO plays an important role
in the coordination and optimization of the various disciplines.
As the fidelity of the disciplines increases and the coupling between the disciplines grows,
the more difficult and costly it becomes to develop and maintain an efficient MDO framework. Progress has been made in the field of MDO by the development of more advanced
architectures, which use system decomposition, approximation models and concurrent
optimization to reduce the computational expenses, cost and required interdisciplinary
communication in large-scale systems. For this reason implementing an efficient MDO
strategy for the advanced preliminary and detailed design phases remains a complex matter.
Currently, a conceptual design framework is being developed and maintained by the Flight
Performance and Propulsion (FPP) group at Delft University of Technology. This framework uses a multidisciplinary design approach to support the conceptual design, analysis
and evaluation of conventional and novel aircraft configurations. Based on a set of top
level requirements a first aircraft design can be generated, which can serve as input for
higher fidelity analysis tools.
3
4
Introduction
1.1
Research question and thesis goal
As has been previously discussed, the use of a multidisciplinary optimization framework
in the conceptual design phase can give valuable insights in the potential of novel aircraft
configurations. Therefore, the research presented in this thesis is aimed at assessing the
potential performance gain of unconventional aircraft configurations through the use of
an optimization tool. This leads to the following research question:
What effect has the developed optimization strategy on the key performance indicators of unconventional aircraft configurations?
In order to answer the research question subquestions have to be established. They are
formulated as follows:
ˆ How can the sensitivity of the design variables be determined efficiently in order to
reduce the computational cost of the optimization?
ˆ Which optimization strategies and algorithms are most suitable for implementation
in the Initiator design tool?
ˆ What is the impact of the optimization on the design of the aircraft configurations?
In order to answer the subquestions and subsequently the main research question an
optimization tool has to be developed, which will be connected to the Initiator design
framework. Therefore the thesis goal can be formulated as follows:
The development of an optimization tool for the conceptual design of conventional
and unconventional aircraft that connects with the Initiator design tool.
1.2
Report outline
This report is dived into two parts. In the first part of the report the content with respect
to the thesis is presented. In the second part the implementation details of the optimizer
tool are described.
In Chapter 2 background information is given regarding the role of multidisciplinary
optimization in aircraft design and various optimization strategies. Also, a brief overview
of the Initiator design tool is given. The used sensitivity analysis methodology is explained
in Chapter 3. It provides a screening technique, which is used to find the most important
design variables. Chapter 4 elaborates on the implemented algorithms in the optimizer
tool. The weak and strong points of the algorithms are discussed. The optimizer tool
is described in Chapter 5. It involves the optimizer workflow, implemented optimization
strategy and parallel optimization. In Chapter 6 the results of the optimizer are evaluated
by means of four case studies. The chapter also present some key performance indicators,
1.2 Report outline
5
which are used to compare the obtained designs. The conclusions and recommendations
are given in Chapter 7. This chapter concludes the thesis part of this report.
The second part of the report starts with the program structure of the optimizer in
Chapter 8. It provides descriptions for the tool methods and properties. The user manual
is described in Chapter 9. It explains how the optimizer tool should be operated.
6
Introduction
Chapter 2
Background information
This chapter contains background information regarding the thesis, which has been collected as part of the preliminary research. In the first section the multidisciplinary design optimization process is explained with respect to aircraft design. In Section 2.2 an
overview of MDO strategies is given. In Section 2.3 a brief description of the Initiator
design tool is presented.
2.1
MDO in aircraft design
Aircraft design involves many disciplines such as aerodynamics, propulsion, structure
and cost. The disciplines often dependent on each other, which results in a complex
and iterative design process. For instance, the required strength and thus weight of
the wing depends on the aerodynamic loads and total weight of the aircraft, while the
latter depend on the weight of the wing. Therefore the coordination between the various
disciplines plays an important role. Multidisciplinary design optimization supports this
design process by providing an efficient methodology in which the various disciplines can
be designed concurrently while a certain level of consistency is maintained. Decomposition
of large coupled problems into smaller subproblems may positively benefit the design time
by reducing the computational complexity and design groups no longer have to wait for
the results of other groups [29].
The aircraft design process can be divided into three phases: conceptual design, preliminary design and detailed design. In the conceptual design the requirements are established
and an initial aircraft design is created. In the preliminary design phase the concept is
further developed. At this stage calculations are done using high-fidelity models and tests
are performed. Based on this information the design is refined. If the decision is made
to manufacture the aircraft the detailed design phase is entered. In this last phase the
fabrication details of the aircraft are determined like the placement of rivets, spars and
other structural elements.
7
8
Background information
These design processes are depicted in Figure 2.1. As can be seen the goal is to gather
as much knowledge on the design as possible in the early design stages, while keeping
a high level of freedom in the design. This can be realized by the application of MDO
techniques.
Figure 2.1: Aircraft design process [2]
2.2
Optimization strategies
The design of aircraft requires collaboration between many disciplines as described in the
previous section. Because of the coupling between the various disciplines an optimization
strategy has to be developed to ensure that all constraints are satisfied and that the
interdisciplinary coupling variables have converged.
The MDO strategies can be categorized in two groups: the monolithic and distributed
architectures [17]. The monolithic architecture uses a single optimization problem to solve
the system. In the distributed approach the optimization problem is divided into smaller
subproblems. Though an optimization problem can be solved by many optimization
strategies, a suitable choice has to be made such that the most efficient strategy is used
for the problem at hand.
2.2.1
Multidisciplinary feasible (MDF)
The multidisciplinary feasible strategy is a monolithic architecture. In this strategy the
optimizer only controls the design variables an global design constraints. At the system level a multidisciplinary analysis (MDA) is performed to solve the coupling between
the various disciplines. A representation of the strategy is shown in Figure 2.2. The
optimization problem at the system level is as follows:
min f (x)
x
subject to
(2.1)
g(x) ≤ 0
2.2 Optimization strategies
9
The main advantage of MDF is that it always results in a consistent system at every
design point. A second benefit is that complexity at the optimizer is reduced. A key
disadvantage is that at each every evaluation of a design point a full MDA has to be
performed. When using a gradient-based algorithm, the computation of the gradient
requires a complete MDA run, which can be rather expensive in large problems [20].
Another downside is the high degree of coupling between the disciplines. Disciplines are
likely to vary in computational difficulty, but they are run the same number of times.
Optimizer
f, g
x
System analyzer
Discipline 1
Discipline 2
Discipline 3
Figure 2.2: Multidisciplinary feasible strategy
2.2.2
Individual discipline feasible (IDF)
The individual discipline feasible strategy uses the optimizer to enforce compatibility
between the disciplines. Like MDF, IDF is also a monolithic architecture. The optimizer
provides a guess for the coupling variables to each discipline. Based on the guess the
disciplines are solved individually. Convergence of the system is obtained by putting an
equality constraint on the actual and guessed values of the coupling variables. A schematic
representation of the IDF strategy is given in Figure 2.3. The optimization problem can
be formulated as follows:
min
x,y,y 0
subject to
f (x, y(x, y 0 ))
g(x, y(x, y 0 )) ≤ 0
(2.2)
y0 − y = 0
In Equation 2.2 the coupling variables are denoted by y and the optimizer guess by y 0 .
The IDF strategy allows the individual disciplines to be solved in parallel, since each
discipline is supplied with a guess for the coupling variables. This can speed up the
analysis. Generally, the strategy works well for relatively small problems. When the size
of the problem grows, the number of coupling variables can become large which adversely
affects its performance.
10
Background information
Optimizer
x, y’
f, g, y
Discipline 1
Discipline 2
Discipline 3
Figure 2.3: Individual discipline feasible strategy
2.2.3
All-at-once (AAO)
The all-at-once strategy is also known by the name simultaneous analysis and design
(SAND). It belongs to the same category as MDF and IDF. The AAO approach further
decomposes the system by simultaneously solving the state equations and the optimization
problem. The state equations are formulated as equality constraints in the optimization
problem. The optimization problem for AAO can be defined as follows [17]:
min f (x, y)
x,y
subject to
(2.3)
g(x, y) ≤ 0
Ri (x0 , xi , y, ŷi ) = 0
for i = 1, .., N
In Equation 2.3 R refers to the residuals of the state equations and N denotes the number
of disciplines. A major disadvantage of the AAO strategy is that it quickly becomes
impractical, because it requires all state equations and variables to be combined in the
problem statement.
2.2.4
Collaborative optimization (CO)
Collaborative optimization is a distributed architecture. The optimization problem is
solved at the system level and at the discipline level. For each discipline a optimization
subproblem is formulated in which the discipline governs its own design variables and
local constraints. This reduces the communication requirements in the system [1]. The
role of the system-level optimizer is to minimize the design objective and the disciplinelevel optimizers are responsible for minimizing interdisciplinary inconsistency. The CO
system-level problem can be formulated as follows:
min
x0 ,x̂,ŷ
subject to
f (x0 , x̂i , .., x̂N , ŷ)
(2.4)
g(x0 , x̂i , .., x̂N , ŷ) ≤ 0
Ji∗ (x0 , x̂i , .., x̂N , ŷ) = 0
for i = 1, .., N
2.2 Optimization strategies
11
In Equation 2.4 J ∗ symbolizes the interdisciplinary inconsistency of the system. The local
design variables are denoted by x̂ and the target values for the coupling variables by ŷ.
The subproblem for each discipline is defined in Equation 2.5.
min
x0 ,xˆi ,yˆi
subject to
Ji (x0 , xi , yi (x0 , xi , ŷ))
(2.5)
c(x0 , xi , yi (x0 , xi , ŷ)) ≤ 0
The advantage of the CO approach follows from its fully separated disciplines. This
strategy is useful for problems which have a low degree of coupling, since a high number
of coupling variables leads to an increase in complexity and computational effort at the
system level.
2.2.5
Concurrent subspace optimization (CSSO)
Concurrent subspace optimization belongs to the category of distributed architectures
and decomposes the system into several independent subproblems, typically one for each
discipline. Each subproblem tries to minimize the global objective with respect to its
local design variables, while keeping the coupling variables constant.
The strategy starts with a full MDA of the system to obtain a consistent design. Using
this design point the subspace optimizations are carried out concurrently. Each subsystem
optimization results in a different design. These designs are used to generate an approximation model of the objective function, which is used by the system-level optimizer to
solve the coordination problem and obtain convergence among the disciplines. After each
iteration the approximation model is updated. The system-level problem can be defined
as follows:
min f (x, ỹ)
x,ỹ
subject to
(2.6)
g(x, ỹ) ≤ 0
In Equation 2.6 ỹ denotes the state of the coupling variables of the other subspaces. Each
subspace optimization can be formulated using Equation 2.7.
min
x,yi ,ỹ
subject to
f (x, yi , ỹj6=i )
(2.7)
g(x, yi , ỹj6=i ) ≤ 0
The main advantage of CSSO is the separation of the disciplines into subspace optimization problems, which can be evaluated in parallel. A downside is that the accuracy of the
approximation models needs to be checked and validated [7]. Also, extensive tuning may
be required in order to run CSSO efficiently, especially on large non-linear problems [17].
12
2.3
Background information
Initiator
The optimizer tool that is designed for the purpose of this thesis uses the Initiator design
tool. It is a conceptual aircraft design tool and is mainly written in Matlab . It uses
a modular structure to represent the components and disciplines of the system. The
advantage this approach is that the components and analysis routines can be easily added
or changed. The tool is driven by top level requirements, which are specified through a
configuration file.
A simplified workflow of the Initiator is shown in Figure 2.4. As can be seen in this
diagram the Initiator starts with the top level requirements. Using these requirements
the sizing modules are called, which execute the class I design methods. At this point
the initial geometry of the aircraft is generated and rough estimates for the weight and
performance are obtained.
Next, the Initiator advances to the analysis modules which calculate the properties and
characteristics of the aircraft in more detail by using class II and class II.V design methods. Amongst these modules is EMWET, which estimates the weight of the wing. The
method has been developed by Elham as part of his PhD thesis [13]. The fuselage weight
estimation module is designed by Schmidt for his Master’s thesis [25], which can handle
both conventional and novel fuselage shapes. The aerodynamics module is based on AVL,
which is a vortex-lattice method developed by Drela [11].
Read top level requirements
and settings
Run preliminary sizing
modules
Run analysis modules
Class II to II.V not converged?
Class I to II.V not converged?
Return converged aircraft
Figure 2.4: High-level activity diagram of the Initiator
2.3 Initiator
13
When all analysis modules have been run, the results of the class II.V weight estimation
are checked against the class II method. If the error is too large an iteration is performed
until the results converge. Next, the results of the class II.V methods are compared to
the class I estimates. An iteration of the design is performed when the results are too far
off. At the end a fully converged aircraft is obtained for the specified requirements.
Besides the sizing and analysis modules, there are also design and workflow modules. The
design modules involve the more detailed design of some part of the aircraft. They are
placed outside the analysis workflow. Examples are the design of the cabin, the design
of control surfaces like ailerons and elevators or the design of the landing gear. The
workflow modules are used for tools or routines that control the Initiator workflow or to
process module results. The optimizer tool will be part of this category. For an in-depth
description of the Initiator design tool the reader is referred to Elmendorp [14].
14
Background information
Chapter 3
Sensitivity analysis
Aircraft design is a multidisciplinary design process which involves many design parameters. Due to the complex nature of the analysis routines and the couplings between the
various disciplines it is often difficult to predict what impact each design variable has on
the aircraft characteristics. This is where sensitivity analysis comes into play. In Section 3.1 variable screening is explained and in Section 3.2 a description of the elementary
effects method is given.
3.1
Variable screening
Variable screening is a subcategory in the area of sensitivity analysis and is used to identify
the contribution of input variables to the outputs of a model. This way the most influential
parameters can be selected, such that optimization complexity and computational cost
can be reduced.
In screening the aim is to qualify the measure of importance of the input factors rather
than quantifying the exact sensitivity values. As such, screening is a useful addition to a
design optimization strategy.
Once the sensitivity data has been obtained, the input factors can be ranked based on their
importance. By selecting only the most important design variables, the dimensionality of
the optimization problem can be reduced leading to faster optimization.
3.2
Elementary effects method
One of the most commonly used screening approaches is the elementary effects (EE)
method. It employs the one-factor-at-a-time (OFAT) principle and provides a global
sensitivity analysis. The computational cost of this approach is relatively low compared
to other screening methods [8], which makes it a prime candidate when computationally
expensive models are involved, like in the Initiator.
15
16
Sensitivity analysis
The elementary effects method is based on the work of Morris [18]. His method provides
two sensitivity measures to determine the importance of input variables based on a series
of experiments: the mean µ, which signifies the overall importance of an input factor,
and the standard deviation σ, which indicates non-linear effects and interactions. These
sensitivity measures are obtained by conducting a series of experiments in which the
inputs are changed one at a time.
The sensitivity measures are obtained by changing the k-dimensional input vector x one
component at a time in random order. This creates a so-called trajectory through the
input space. The more trajectories are used the more reliable the sensitivity measures
become as more input space is sampled. An example of a trajectory in 3-dimensional
space is shown in Figure 3.1.
k
x(3)
x(2)
j
x(0)
x(1)
i
Figure 3.1: Example trajectory in 3-dimensional space using a five-level grid
In the example it can be seen that in each subsequent step only one input is changed.
The start of the trajectory x(0) is obtained by selecting a random point in the input
space [0, 1]k , which is discretized into a p-level grid Ω. The next point x(1) is acquired by
increasing or decreasing one component from x(0) with ∆ such that x(1) is still in Ω. This
is done until all components of x have been displaced with ∆. It follows that k+1 model
runs are required to compute a single trajectory. So, for each input xi the elementary
effect di can be defined as follows:
di (x) =
y(x1 , .., xi−1 , xi + ∆, xi+1 , .., xk ) − y(x)
∆
(3.1)
The step size ∆ must be a predefined multiple of 1/(p−1). Though different combinations
of p and ∆ can be chosen, there exists some values for p and ∆ for which the grid points
have equal probability of being sampled. This occurs when p is an even number and ∆ is
3.2 Elementary effects method
17
calculated using Equation 3.2.
∆=
p
2(p − 1)
(3.2)
The effect of choosing different combinations of p and ∆ is illustrated using the examples
in Figure 3.2. Figure 3.2a shows that for p = 4 and ∆ = 1/3 the two outer points are less
likely to be sampled. When ∆ is changed to 2/3 in Figure 3.2b, it can be seen that the
sampling probability is equal for all grid points. Using p = 5 and ∆ = 1/4 as shown in
Figure 3.2c the same problem arises as with example 3.2a. The two outer points have a
lower sampling probability. Example 3.2d uses p = 5 and ∆ = 3/4. Here the center grid
point is never sampled.
0
1/3
1
2/3
0
(a) p = 4 and ∆ = 1/3
0
1/4
2/4
3/4
1/3
1
2/3
(b) p = 4 and ∆ = 2/3
1
0
(c) p = 5 and ∆ = 1/4
1/4
2/4
3/4
1
(d) p = 5 and ∆ = 3/4
Figure 3.2: Sampling probability using different values for p and ∆
The choice of p is also related to the number of trajectories r. When a high-level grid is
used more trajectories are required to make sure that all possible levels are explored. In
this thesis a four-level grid (p = 4) is used with a ∆ of 2/3 and a total of 4 trajectories.
According to Morris [18] a sample size of at least 4 is needed to obtain a reasonably
reliable result.
With the calculated elementary effects the sensitivity measures can be determined. The
mean µi of each input parameter follows from Equation 3.3.
r
µi =
1X
di (x)
r
(3.3)
j=1
In this equation the parameter r refers to the number of trajectories. The corresponding
18
Sensitivity analysis
standard deviation σi is given by Equation 3.4.
v
u
r
u 1 X
t
σi =
(di (x) − µi )2
r−1
(3.4)
j=1
An improved version was developed by Campolongo et al.[9], who added a modified mean
µ∗ . It uses the absolute values of the elementary effects to avoid cancellation effects when
the function is non-monotonic. The formula is displayed in Equation 3.5.
r
µ∗i =
1X
|di (x)|
r
(3.5)
j=1
In order to rank the input parameters by importance the Euclidean distance with respect
to modified mean µ∗ and standard deviation σ is used. Though the value of µ∗ alone
would suffice to rank the parameters, results show that inputs with a high value for µ∗
generally have a high value for σ as well [24].
q
si = σi2 + (µ∗i )2
(3.6)
The elementary effects method is demonstrated in Chapter 6. In this chapter four case
studies are presented for which the screening method is used to reduce the number of
design variables.
Chapter 4
Optimization algorithms
In this chapter the algorithms used in the optimizer tool are presented. Multidisciplinary
optimization problems are generally very costly in terms of computation time, so it is
important to find a suitable algorithm that offers the most gain while keeping the computational effort as low as possible. Since each optimization problem has different requirements and characteristics, there is no algorithm that fits every case. Aspects like
the available resources, required accuracy, model noise and chosen optimization strategy
may affect the selection of an algorithm.
For this thesis a gradient-based algorithm, a genetic algorithm and a genetic hybrid
algorithm is used. In the first section a formulation of the general optimization problem
is given. In Section 4.2 gradient-based algorithms are discussed. This is followed by a
description of the genetic algorithm in Section 4.3. The hybrid algorithm is explained in
Section 4.4.
4.1
Problem description
The general non-linear optimization problem (NLP) can be defined as follows:
min f (x)
x
x ∈ Rn
subject to gi (x) ≤ 0, i = 1, .., j
(4.1)
hi (x) = 0, i = 1, .., k
xl ≤ x ≤ xu
In Equation 4.1 f (x) is the objective function, which is a measure for the optimality of
the design. An example of an objective function could be the payload-range efficiency or
the lift-to-drag ratio. The problem is subject to inequality constraints gi (x) and equality
19
20
Optimization algorithms
constraints hi (x). Examples of constraints could be noise restrictions, emission regulations
or coupling variables. Furthermore, the design vector x is restricted by an upper and lower
bound.
4.2
Gradient-based algorithms
Gradient-based methods rely on first and second-order derivatives of the objective function to compute the search directions. One of the primary advantages of gradient-based
algorithms is that tthey tend to convergence rather rapidly, especially near the optimum.
In general the computational cost scales linearly with the number of design variables [30].
Another advantage is that they have a straightforward termination criterion. When the
step size has been reduced by a certain order of magnitude it can be said with certainty
that at least a local minimum has been found.
A disadvantage of gradient methods is its intolerance towards noise in the objective function. The algorithm might get stuck and stop prematurely or start to oscillate around a
certain point. Also, there is no guarantee that a global minimum is found. Furthermore,
the starting point may influence the outcome, because a different starting location might
direct the algorithm towards another basin of attraction yielding a different optimum.
For this thesis the sequential quadratic programming (SQP) method is used. It is one of
the more popular gradient methods and it is quite robust [15]. The idea behind SQP is
that an approximation is made for the Hessian using a quasi-Newton updating method.
Therefore this method can be seen as an extension to the Newton methods to the field
of constraint optimization. SQP solves the non-linear problem by creating a quadratic
programming (QP) subproblem at each iteration. The results of each QP subproblem are
used to approximate the next search step. The QP subproblem can be set up using a
Taylor expansion [6]:
1
min f (xk ) + ∇f (xk )T d + dT ∇2 L(xk , λk , µk )d
d
2
T
subject to g(xk ) + ∇g(xk ) d = 0
(4.2)
h(xk ) + ∇h(xk )T d ≤ 0
where
d = x − xk
In Equation 4.2 xk is the approximation at the current iteration and L denotes the
Lagrangian function of the problem. This function is given in Equation 4.3. Here λ and
µ are the Lagrange multipliers.
L(xk , λk , µk ) = f (x) + λT g(x) + µT h(x)
(4.3)
The optimizer tool that is designed for the purpose of this thesis uses the built-in SQP
algorithm from Matlab by means of the fmincon function. This function uses the pop-
4.3 Genetic algorithm
21
ular Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to approximate the Hessian
H = ∇2 L. This approximation is shown in Equation 4.4.
Hk+1 = Hk +
where
and
qk qkT
Hk sk sTk HkT
−
qkT sk
sTk Hk sk
sk = xk+1 − xk
(4.4)
qk = ∇L(xk+1 , λk+1 ) − ∇L(xk , λk )
When the QP subproblem is solved the new iterate xk+1 can be computed:
xk+1 = xk + ak xk
(4.5)
The step length parameter ak in Equation 4.5 follows from a line search to determine an
appropriate step size. Finally, the algorithm advances to the new iterate xk+1 and a new
QP subproblem is generated. This procedure is repeated until a termination criterion
halts the algorithm. For more information regarding the algorithm implementation the
reader is referred to Matlab manual [4].
4.3
Genetic algorithm
The genetic algorithm (GA) is an evolutionary algorithm. Instead of relying on derivative
information like gradient-based methods, it uses the principle of natural selection. The
algorithm uses a population of individual solutions. Usually, the algorithm starts by
initializing a randomly generated population. At each iteration all individuals of the
current generation are ranked according to their fitness value, which follows from the
objective function. Then the following selection rules are applied to create the next
generation:
ˆ Crossover : children are created by combining the design vectors of a pair of parents
ˆ Mutation: children are created by making random changes to a single parent
ˆ Elite: individuals with the best fitness values survive to the next generation
In Figure 4.1 these operations are visualized. Fitter solutions are more likely to be
selected to create children. As the algorithm progresses the average fitness value of the
population will increase, because only the best solutions survive to the next generation.
The algorithm terminates when the best fitness value is not increasing anymore for a
number of generations.
Compared to gradient-based methods, the genetic algorithm is more robust [28]. It can
operate in noisy environments and is able to solve non-smooth optimization problems. The
algorithm is less likely to be trapped in a local optimum, since multiple solutions are used
to explore the design space. However, there is no guarantee that the global optimum is
found. The algorithm may suffer from early convergence leading to a suboptimal solution
22
Optimization algorithms
A
B
A
D
A
B
E
B
A
B
A
B
C
D
Figure 4.1: The crossover, mutation and elite selection procedures of the genetic algorithm
[26]. Increasing the mutation rate may alleviate the problem. Another disadvantage
of the genetic algorithm is that convergence tends to be slow near the optimum [30].
Furthermore, the algorithm requires many function evaluations, because each individual
within the population has to be computed every generation. This can be overcome by
caching all individual solutions, which is explained in Section 5.3. The process can be
further speed up by computing individual solutions in parallel as is described in Section
5.4.
In the optimizer tool the built-in function ga from Matlab is employed. It uses the
augmented Lagrangian genetic algorithm (ALGA) and is based on the work of Conn et
al.[10]. The ALGA algorithm treats the bounds and linear constraints separately from
the non-linear constraints. The non-linear constraints and the fitness function f (x) are
combined into a subproblem, which is shown in Equation 4.6
Φ(x, λ, s, ρ) = f (x) −
m
X
i=1
[λi si log (si − ci (x))] −
mt
X
[λi log ceqi (x)]
i=m+1
mt
X
ρ
+
2
(4.6)
2
[λi ceqi (x) ]
i=m+1
In this equation λ are Lagrange multiplier estimates, s is a vector containing non-negative
shifts and ρ is a positive penalty parameter. Each subproblem uses fixed values for λ, s
and ρ. The Lagrangian estimates are updated when the subproblem converges to feasible
conditions. When the constraints cannot be satisfied the penalty parameter is increased.
In both cases this leads to a new subproblem formulation. This procedure is repeated
until the termination criteria are reached. Examples of termination criteria for the genetic
algorithm are: no improvement in fitness value over a specified number of generations,
exceeding the maximum number of generations or an imposed time limit.
4.4
Hybrid algorithm
A hybrid algorithm combines two or more distinct optimization routines to solve the same
optimization problem. The idea behind this approach is to use the strong points of each
4.4 Hybrid algorithm
23
method such that the combined algorithm is better than the individual algorithms.
A well-known example is the hybrid genetic algorithm in which the genetic algorithm is
paired with a more fine-grained solver, which is often gradient-based. This type is also
used for the optimizer tool. It uses the previously discussed genetic and SQP algorithm
to form a hybrid algorithm.
The primary role of the genetic algorithm in this setup is to explore the design space in
order to find the region with the most promising optimum, which is the global optimum
in the ideal case. The region discovered by the genetic algorithm can then be used by the
gradient method to hit the exact optimum in that region. The difficulty of the hybrid
algorithm lies in determining the decision criterion that dictates when the algorithm must
switch to the local solver. Studies showed that there are multiple criteria that can be used
for this purpose, but the effectiveness of these criteria depends on the type of problem
and requires tuning of the termination parameters [12].
24
Optimization algorithms
Chapter 5
Optimizer tool description
Since many disciplines are involved in aircraft design a multidisciplinary design optimization (MDO) tool is required. The interactions between the disciplines greatly increase
the complexity of the system. Since analysis routines can be computationally expensive,
finding a feasible design within a reasonable amount of time can be more valuable than
finding the optimal design from a mathematical point of view.
The optimizer tool has been written in Matlab . It is implemented as a workflow module
in the Initiator framework, which has previously been described in Section 2.3. This way
the optimizer can be operated through the established routes in the Initiator and it also
facilitates easier implementation and linking with the design tool.
In this chapter first the workflow of the optimizer is explained. This is followed by a
description of the implemented optimization strategy. In Section 5.3 the caching technique
is discussed. In the final section the application of parallel optimization is presented.
5.1
Optimizer workflow
The high-level workflow of the optimizer tool is illustrated in Figure 5.1 and can be described as follows. When the optimizer the tool starts with reading the optimization
problem and settings. The optimization problem contains information regarding the objective function, design variables, constraints and selected optimization algorithm. Next,
the optimizer uses the Initiator to compute the initial design point. The analysis results
of this baseline aircraft and its geometry data are saved for later use. Then the number
of design variables is checked. If this number exceeds the specified maximum number of
design variables, a sensitivity analysis is performed first. The sensitivity analysis routine
uses the elementary effects method described in Chapter 3. When the analysis has completed, the design variables are ranked according to their computed sensitivity value and
the most important variables are selected. By default the top 5 variables are taken. Using
the selected variables the optimization problem is set up. After the provided objective,
25
26
Optimizer tool description
constraints and algorithm options have been set, the specified algorithm is called. Depending on the options a parallel Matlab session may be started. Once the algorithm
has completed, the resulting optimum design vector is used to compute the final aircraft.
Its properties and analysis results are saved to disk, along with the optimization and
baseline aircraft data.
Load settings and problem
description
Calculate properties of
baseline aircraft
Too many design variables?
Run sensitivity analysis
Select most important
design variables
Run optimization
Calculate properties of final
aircraft
Save data to disk
Figure 5.1: High-level activity diagram of the optimizer
A more detailed description of the routines and functions can be found in Chapter 8. In
this chapter flowcharts are given of the sensitivity routine and the optimization functions.
5.2
Optimization strategy
For the optimizer tool the multidisciplinary feasible optimization strategy was chosen.
Due to the existing structure of the Initiator, which already featured a design convergence
module that performs the multidisciplinary analysis, the MDF architecture was adopted.
It is a traditional MDO strategy in which the optimizer is in charge of the design variables
and global design constraints.
Figure 5.2 shows the MDF implementation of the optimizer tool. Here it can be seen
that the optimizer passes the design vector x to the cache layer (see Section 5.3). If
there are already results for the passed design vector the cache immediately returns the
5.3 Caching results
27
corresponding objective and constraint values. Otherwise the design vector is forwarded
to the Initiator and a full MDA is done. The MDA is performed by the Initiator’s design
convergence module. This module takes care of obtaining consistency across all disciplines
and compliance with system constraints.
Optimizer tool
f, g
x
Cache
f, g
x
Design convergence
module
Module 1
Module 2
Module N
Figure 5.2: Multidisciplinary feasible implementation of the optimizer tool
As already has been discussed in Chapter 2, the major advantage of MDF is that it always
returns a consistent system for a given design vector. Another benefit is that when changes
are made in the Initiator, which require adjustments in the MDA routine, the optimizer
does not have to be modified. The loose coupling promotes maximum flexibility. When a
strategy like IDF, CO or CSSO was chosen the optimizer would have to be updated every
time someone decides to make a change to a particular module. This requires knowledge
of the whole optimization structure, which is an undesirable situation. For the same
reason approximation models or response surface techniques are not incorporated in the
implementation. Since the Initiator is subject to changes maintaining and updating such
models would be too costly. As such MDF is a robust and intuitive strategy, which is
most suitable for this environment.
5.3
Caching results
In order to speed up the optimization the effect of adding a caching mechanism at the
system level has been investigated. The caching mechanism allows to quickly retrieve
the results of design points that have already been processed. This way an expensive
recalculation is avoided.
This technique is only beneficial for algorithms that may re-examine prior results. A
perfect example is the genetic algorithm. In each generation part of the population
28
Optimizer tool description
survives to the next generation. Therefore the results of the surviving design vectors will
be needed again. The effect of caching using the genetic algorithm is shown in Figure 5.3.
Genetic algorithm performance using results caching
350
300
Function evaluations
250
200
Cache off
Cache on
150
100
50
0
0
5
10
15
20
25
Time [min]
30
35
40
45
Figure 5.3: Effect of caching on the genetic algorithm
In this figure it can be seen that with caching the algorithm is able to evaluate much
more solutions in the same period of time. The steps in the line indicate that a match
in the cache has been found. The big steps can be explained by the adjustment of the
Lagrangian estimates of the genetic algorithm. The Lagrangian multipliers are used to
solve the fitness function with respect to the nonlinear constraints and are updated after
every couple of optimization steps, as was described in Section 4.3. When this happens
previous solutions are checked against the new multiplier estimates. The results can
immediately be pulled from the cache, which saves a lot of time.
Generally, gradient methods do not benefit from caching, because along its gradient-based
search path it is unlikely that the same point will be requested again. Nevertheless, the
caching layer is always on by default, because the overhead of this mechanism is very
small compared to a full MDA computation.
5.4
Parallel optimization
Since multidisciplinary analysis can be very expensive, the use of parallel computing has
been researched. In order for parallelization to be effective, there should be multiple tasks
that can be performed simultaneously and have to be of sufficient workload. If the tasks
are too small, the communication overhead will become too large resulting in parallel
slowdown. Also, attention must be paid to race conditions, because two processes may
request the same resource at the same time.
Not all algorithms are suitable for this technique. Algorithms at which the next solution
depends on previous iterate, the solutions can only be computed one at a time. Sometimes only particular subroutines of an algorithm may be suitable for parallelization. An
5.4 Parallel optimization
29
example is a gradient-based algorithm which estimates the gradients simultaneously, but
can only evaluate one design point at a time.
The multi-solution approach of the genetic algorithm makes it an excellent candidate
for parallel optimization. Each individual solution of a particular generation can be
computed independently. A benchmark has been performed to determine how well the
genetic algorithm scales with the number of workers. The test has been performed on
a modern quadcore processor, which allows up to four parallel workers. The population
size is set to 12 to ensure that the workload is equally distributed across the available
workers in each test case. The caching layer has been disabled for benchmark to test the
raw performance. The results of this benchmark are shown in Figure 5.4. A trend line
has been fitted through each set of data points.
Genetic algorithm performance using parallelization
100
90
Function evaluations
80
70
60
1 worker
2 workers
3 workers
4 workers
50
40
30
20
10
0
0
10
20
30
Time [min]
40
50
60
Figure 5.4: Scaling the genetic algorithm using parallel computing
From the results it can be seen that the algorithm scales linearly up to three parallel
workers. By using a single worker about 0.76 function evaluations per minute are performed. By adding a second and third worker this number increases to 1.53 and 2.39
respectively. The performance gain diminishes when the system’s maximum of four workers is used. Because the system processes also require processor time, the fourth available
thread cannot be fully utilized. A maximum of 2.68 function evaluations per minute is
reached, which is 2.5 times more than with a single worker. So parallelization of the
genetic algorithm proves to be very beneficial.
30
Optimizer tool description
Chapter 6
Optimization case studies
In this chapter four aircraft configurations are evaluated to assess the effect of the optimization on their performance and characteristics. For this purpose the following aircraft
configurations are selected: a conventional Airbus A320, a canard aircraft with forward
swept wings, a three-surface aircraft and an oval-fuselage aircraft.
In Section 6.1 the key performance indicators are established. Next, the optimization
procedure is explained in Section 6.2. Using this procedure the case studies are performed.
Their results can be found in Section 6.3 through 6.6. In the final section the cases are
compared and an evaluation of the algorithms is done.
6.1
Key performance indicators
The performance of an aircraft can be expressed in many ways. The key performance
indicators (KPI) are likely to vary depending on the perspective. Aircraft manufacturers,
consumers, airline companies and legislators all have different opinions with respect to
performance and efficiency parameters. For commercial airliners fuel efficiency and operating cost are very important factors. Also, environmental issues like noise and emissions
are becoming more and more important, which put constraints on the design space.
The selection of a key performance indicator is very important as it defines which objective
has to be considered for optimization. This choice will have consequences for the entire
design of the aircraft, so a sensible parameter must be used. A suitable measure of aircraft
performance is the payload-range efficiency (PRE) [19]. It is defined as follows:
PRE =
Wp · R
Wfb
(6.1)
In Equation 6.1 Wp is the payload mass, R is the harmonic range and Wfb is the block
fuel load. The block fuel is the total fuel minus any reserve fuel. The harmonic range is
31
32
Optimization case studies
the furthest distance the aircraft can fly with maximum payload. Other ranges are the
maximum fuel range and ferry range and they are typically indicated in a payload-range
diagram. An example of such a diagram is given in Figure 6.1, which is based on the
Airbus A320 from the first test case.
MTOM
Aircraft mass [metric tons]
60
55
MZFM
50
Ferry
range
Max. fuel
range
45
Harmonic
range
40
35
OEM
0
2000
4000
6000
Range [km]
8000
10000
Figure 6.1: An example of a payload-range diagram
Another important performance indicator is the range parameter X shown in Equation
6.2. The range parameter follows from the first part of the Breguet range equation. It
denotes the aerodynamic and propulsion efficiency of an aircraft.
X=
V · L/D
cT
(6.2)
In this equation L/D is the lift-to-drag ratio and cT is the specific fuel consumption.
6.2
Optimization procedure
In this section the optimization procedure is laid out. It consists of two main steps. The
first step is running the sensitivity analysis. This is explained in Section 6.2.1 and is
conducted once for each aircraft case. Next, the actual optimization is performed. This
is done three times, once for each algorithm. It is described in Section 6.2.2. In the last
section the used Initiator and aircraft settings are given.
6.2.1
Sensitivity analysis
For each case a sensitivity analysis is performed to identify the most important design
variables. This is done using the elementary effects method described in Chapter 3.
6.2 Optimization procedure
33
According to Morris [18] a sample size of at least 4 is needed to obtain a reliable result.
Therefore the number of trajectories is set to 4, which is a fair trade-off with respect to
the computational cost. Each design variable is varied across four levels on the grid.
After the screening has been performed the most influential parameters are selected based
on their sensitivity value. Therefore a maximum of 5 design variables is established. The
sensitivity value s that is used in this chapter is the normalized version of Equation 3.6:
ŝi =
6.2.2
si
· 100
smax
(6.3)
Optimization
The top 5 design variables that follow from the sensitivity analysis are used in the optimization routine. Three runs will be done per aircraft, each time using a different
optimization algorithm. For this purpose the genetic algorithm, gradient-based SQP algorithm and the genetic–SQP hybrid algorithm are used. The results of the optimization
are evaluated afterwards.
As already explained earlier in the key performance indicators section, the payload-range
efficiency is used as the optimization objective. The following objective function has been
set up:
min f (x) = −
PRE
1000
(6.4)
As can be seen in Equation 6.4 a minus sign is added and a scaling factor is introduced
to normalize its value. Though most constraints are handled by the Initiator itself, the
nose loading constraint is not forced. To solve this problem the following two inequality
constraints are defined:
g1 (x) = −
x̂cgmin − x̂fwd
≤0
MAC
(6.5)
g2 (x) = −
x̂aft − x̂cgmax
≤0
MAC
(6.6)
Equation 6.5 denotes the maximum nose loading constraint and Equation 6.6 the minimum nose loading constraint. Note that x̂, which refers to the longitudinal position, is
not the same as the design vector x. Both constraints are expressed in terms of the MAC.
A maximum runtime of two hours is maintained for the genetic and gradient algorithm.
From running the design tool multiple times it followed that after two hours there was no
significant gain in objective value. In some cases the algorithms even stopped before the
two hour limit. In Figure 6.2 the improvement in objective value against the computation
time is shown.
34
Optimization case studies
Improvement in objective against computation time
Improvement in objective w.r.t baseline [%]
25
Gradient run 1
Gradient run 2
Genetic run 1
Genetic run 2
20
15
10
5
0
0
20
40
60
80
Time [min]
100
120
140
160
Figure 6.2: Improvement in objective value against computation time
The hybrid method is given a maximum of three hours. The run time of the first stage,
the genetic algorithm, is limited to two hours. Then the local solver is run, which is
limited to 1 hour.
For the genetic and gradient-based SQP algorithm most options were left at their default
values. The population size of the genetic algorithm was adjusted to 12 with an elite
count of 2. Though a higher population count yields more diverse individuals, it takes
considerable more time to compute a single generation and therefore less generations can
be evaluated in the same amount of time. It was found that 12 individuals is a fair
trade-off. This way it takes about 5 to 10 minutes per generation depending on the
computational difficulty of the aircraft configuration. The algorithm stops when there is
no improvement in best fitness value after 4 generations.
The minimal step length of the SQP algorithm has been set to 10−3 to overcome the
output noise of the design tool. The default objective tolerance of 10−6 is maintained.
Design variables
The next step is to establish a set of design variables. The selected variables are gathered
in Table 6.1. Most parameters are related to the wing geometry, since it is expected that
they have the most influence on the objective value. Also, the longitudinal position of
the wing and the diameter of the fuselage are added.
The wing reference area must be kept constant when setting its geometry after a design
point has been chosen in the preliminary sizing. Therefore the root chord of the wing is
treated as a dependent variable and is calculated based on the given aspect ratio, taper
ratio and sweep angle. Changes in wing area due to twist, dihedral and thickness ratio
are assumed to be small. The wing position is selected because of its important role in
the longitudinal stability of the aircraft. It influences the size of the control surfaces and
the position of the landing gear. It is expressed in terms of the fuselage length. For the
6.2 Optimization procedure
35
same reasons the canard position is added, but it is only used for the three-surface aircraft
case.
Variable
Aspect ratio
Sweep angle1
Taper ratio
Root thickness ratio
Kink thickness ratio2
Tip thickness ratio
Dihedral angle
Root twist angle
Kink twist angle2
Tip twist angle
Wing longitudinal position
Canard longitudinal position3
Fuselage diameter
Symbol
Min.
Max.
A
Λ
λ
( ct )r
( ct )k
( ct )t
Γ
r
k
t
fxw
fxc
df
8.0
0.0◦
0.0
−25%
−25%
−25%
−5◦
−5◦
−5◦
−5◦
−0.15
0.05
−25%
15.0
35.0◦
1.0
+25%
+25%
+25%
+5◦
+5◦
+5◦
+5◦
+0.15
0.20
+25%
Table 6.1: List of design variables
1 For a forward swept wing the bounds are inversed
2
Not applicable to the canard aircraft due to its forward swept wing
3
Three-surface aircraft only
The fuselage diameter is added due to its effect on its structural weight and moment
arm with respect to the control surfaces. Based on the given diameter the design tool
calculates the required length to make sure that enough seats can be placed to carry the
required number of passengers. In case of the oval-fuselage aircraft the width is controlled
instead of the diameter.
6.2.3
Initiator and aircraft settings
The Initiator settings are mostly kept at their default values. The most important settings
are mentioned here. The allowed convergence error between the class II and II.V weight
estimation is kept at 1%. The weight error between the class I and II.V estimation is
0.5%. The minimum nose gear loading is set to 5% and the maximum nose loading is set
to 20%. The passenger mass is 80 kg and the luggage mass per passenger is 25 kg.
For all cases the Boeing 737 airfoils are used for the main wing. The airfoil shown in
Figure 6.3a is used for the root section, while the airfoil from 6.3b is placed at the kink
and tip section. The kink location is fixed at 30% of the wing semi-span.
36
Optimization case studies
(a) Boeing 737-based root
section airfoil
(b) Boeing 737-based kink
and tip section airfoil
(c) NACA0012 airfoil
Figure 6.3: Airfoils used in the case studies
The horizontal tails have an aspect ratio of 5.0 and a taper ratio of 0.35. The other
properties are automatically sized by the Initiator.
The vertical tail has an aspect ratio of 1.0 and a taper ratio of 0.35. In case of a T-tail
configuration the values are 1.6 and 0.7 respectively. For the canard an aspect ratio of 5
and taper ratio of 0.60 is used. The sweep and dihedral are derived from the main wing
as well. For all control surfaces the NACA0012 airfoil is used, which is shown in Figure
6.3c.
6.3 Case 1: Airbus A320
6.3
37
Case 1: Airbus A320
The first case that is considered is an aircraft which has similar requirements as the Airbus
A320-family. It is a conventional aircraft for short to medium range. It must be able to
carry 150 passengers over a range of 2870 km at a cruise speed of Mach 0.78. All top level
requirements are gathered in Table 6.2.
Pax.
Payload mass
MC
Altitude
Range
sTO
sL
150
20.5 tons
0.78
11.3 km
2870 km
2180 m
1440 m
Table 6.2: Airbus A320 top level requirements
The Initiator uses this information to create an aircraft that fulfils these requirements.
The resulting aircraft properties are shown in Table 6.3. A 3-dimensional model of the
aircraft is illustrated in Figure 6.4.
A
S
b
Λ
cr
λ
9.4
126
34.5
26.2
7.6
0.16
0
−
m2
m
◦
m
−
◦
Γ
fxw
( ct )r
( ct )k
( ct )t
lf
df
6.0
0.45
0.151
0.104
0.104
40.6
4.2
◦
−
−
−
−
m
m
Ah
Sh
bh
Av
Sv
bv
4.9
25
11.1
1.6
18
5.2
−
m2
m
−
m2
m
MTOM
OEM
FM
Rh
PRE
L/D
CLmax,clean
58.9
30.8
7.6
2900
7880
17.6
1.24
tons
tons
tons
km
km
−
−
Table 6.3: Airbus A320 properties using the Initiator
Figure 6.4: Airbus A320 model
First a sensitivity analysis is performed using the design variables mentioned in Table
6.1. In total 52 runs had to be performed, which took 62 minutes to complete. The
resulting mean and standard deviation of each parameter are shown in Figure 6.5a and
the sensitivity index is given in Figure 6.5b. The variables are numbered according to
their importance.
38
Optimization case studies
A320 design vector sensitivity index w.r.t PRE
Airbus A320 sensitivity analysis w.r.t PRE
Standard deviation σ (non−linearity/interaction)
2
5
4
1
3
2
Wing x−position (1)
Sweep angle (2)
t/c tip section (3)
Dihedral angle (4)
Design variables
1. Wing x−position
2. Sweep angle
3. t/c tip section
4. Dihedral angle
5. Aspect ratio
6. Taper ratio
7. t/c kink section
8. Fuselage diameter
9. t/c root section
10. Kink twist
11. Root twist
12. Tip twist
6
Aspect ratio (5)
Taper ratio (6)
t/c kink section (7)
Fuselage diameter (8)
t/c root section (9)
4
Kink twist (10)
3
1
Root twist (11)
5
Tip twist (12)
0
0
200
400
600
800
0
1000
10
*
Mean µ (overall importance)
(a) Mean vs. standard deviation
20
30
40
50
60
70
80
Normalized sensitivity value [−]
90
100
(b) Sensitivity index
Figure 6.5: Airbus A320 sensitivity analysis results
In the graphs it can be seen that two parameters stand out, which are the wing position
and the sweep angle. The other variables are grouped in the left bottom corner. The
section twist angles have the least impact on the payload-range efficiency. The top 5
design variables are selected for optimization.
The optimizer is run for the genetic, gradient-based and hybrid algorithm. The resulting
optimum design vector of each algorithm is shown in Table 6.4. A detailed overview of
the changes in geometry and performance with respect to the initial design are given in
Table 6.5.
Initiator
Genetic
Gradient
Hybrid
A
fxw
Λ
( ct )t
Γ
9.4
11.6
14.4
13.1
0.45
0.53
0.48
0.45
26.2◦
0.1◦
6.2◦
14.5◦
0.104
0.081
0.078
0.079
6.0◦
1.9◦
6.0◦
1.5◦
PRE
7880
8240
8250
8400
km
km
km
km
∆PRE
t
−
+4.6%
+4.6%
+6.6%
−
108 min
101 min
169 min
Table 6.4: Optimum design vectors for the Airbus A320
In Table 6.4 it can be seen that the hybrid algorithm obtained the best payload-range
efficiency. It improved by 6.6% with respect to the reference aircraft, but it also took the
most time. The payload-range found by the genetic and gradient algorithm is about 2%
less, but both completed within 2 hours. They all agree on a slightly thinner wing tip
section, but there are significant differences when comparing the other design parameters.
Looking at the genetically optimized A320 in Figure 6.9 the very low sweep angle of
the wing immediately stands out. At a cruise speed of Mach 0.78 one expects a higher
sweep angle to reduce the drag rise due to compressibility effects. Therefore it seems
that the drag is underestimated. In Figure 6.6a it can be seen that up to Mach 0.80 a
low sweep angle is beneficial for the lift-to-drag ratio. The optimizer takes advantage
of this by trading sweep angle for a higher aspect ratio. From Figure 6.6b follows that
at lower sweep angles the aspect ratio has less impact on the operational empty mass.
6.3 Case 1: Airbus A320
39
Though the structural weight increases at higher aspect ratios, it is compensated by a
better lift-to-drag ratio due to lower induced drag. This in turn benefits the payload-range
efficiency.
20
42
18
40
OEM [metric tons]
L/D [−]
16
14
12
10
8
38
36
34
32
30
30
6
0.5
28
15
0
0.6
10
0.7
20
14
13
12
20
0.8
0.9
30
Sweep angle [deg]
11
10
10
Aspect ratio [−]
Mach number [−]
(a) Lift-over-drag ratio for varying sweep angle and Mach number (A = 9.4)
9
0
Sweep angle [deg]
(b) Operational empty mass for varying aspect ratio and sweep angle
Figure 6.6: Airbus A320 lift-to-drag ratio and operational empty mass
The wing position of the genetic solution is a bit more aft than the other designs. The
very low sweep angle causes the wing to shift a bit aft. This effect is shown in Figure
6.7a. There is a strong correlation between the sweep angle, wing position and lift-to-drag
ratio. With increasing sweep angle the wing has to shift forward to attain a better L/D,
but nose loading constraints limit this movement.
20
19.5
19
16
L/D [−]
L/D [−]
18
14
18.5
18
17.5
12
13
0.4
10
0.45
30
0.5
20
0
0.6
12
2
11
4
6
0.55
10
17
0
10
8
10
Wing position [−]
Sweep angle [deg]
(a) Lift-over-drag ratio for varying sweep angle and wing position (A = 9.4)
9
Aspect ratio [−]
Dihedral angle [deg]
(b) Lift-over-drag ratio for varying dihedral
angle and aspect ratio
Figure 6.7: Airbus A320 lift-to-drag ratio for varying parameters
The hybrid algorithm improved on the genetic solution by a few percent. Though it started
at the optimum design vector of the genetic algorithm, it came up with a rather different
40
Optimization case studies
combination of aspect ratio, sweep angle and wing position. The resulting aircraft is shown
in Figure 6.11. The aspect ratio increased from 11.6 to 13.1 and the sweep angle increased
to 14.5◦ . Up to about 15◦ sweep the aspect ratio weight penalty remains roughly the same
when observing Figure 6.6b. In combination with the more forward wing position a better
optimum was found with these parameters.
Both the genetic and hybrid solution have a lower wing dihedral angle. A lower dihedral
angle yields a higher effective planform area, which leads to a slight increase in lift [23].
This is also in accordance with Figure 6.7b. The dihedral angle largely depends on the
trade-off between lateral stability and roll control. Especially low-wing aircraft like the
A320 require some dihedral due to the wing-fuselage interaction, which is usually in the
range of 5◦ to 7◦ [27]. The dihedral angle is also constraint by the engine ground clearance
requirement and tip clearance during take-off rotation and landing. The Initiator does not
yet cover lateral stability and therefore the dihedral is entirely driven by the lift-to-drag
ratio.
The gradient-based solution resulted in the heaviest aircraft. It is depicted in Figure 6.10.
The operational empty weight increased by almost 12%. This can be mainly attributed
to wing as can be observed in Figure 6.8. The aspect ratio of 14.4 allowed it to reach a
lift-to-drag ratio of 20.3 at the cost of a significant increase in wing weight. Therefore it
does not outperform the genetic solution with an L/D of 19.1.
Aicraft part mass comparison
Empenage
Engines
Aircraft part
Furnishing
Fuselage
Landing gear
Operational items
Systems
Wing
0
1
2
3
Initiator
4
5
6
Part mass [metric tons]
Genetic
Gradient
7
8
9
10
Hybrid
Figure 6.8: Airbus A320 change in part mass after optimization
When looking at the tail surfaces of each design it can be observed that their sweep
and dihedral angles are coupled to the main wing. The sweep angle alters the lift curve
slope, which has consequences for the stall angle of attack and maximum lift coefficient.
Therefore the shape of the tail surfaces are likely to be far from optimal with respect to
the control and stability of the aircraft. Also, the high trailing edge sweep angle of the
genetic and hybrid solution create an unfavourable condition for the placement of elevator
and rudder control surfaces.
All solutions show a rather high static margin. This means that the center of gravity of
the aircraft is relatively far ahead of the neutral point. A lower static margin allows a
6.3 Case 1: Airbus A320
41
Genetic
Gradient
Hybrid
km
km
km
km
tons
tons
tons
−
−
m/s
kg/m2
−
%MAC
%MAC
7880
7060
2900
9560
58.9
30.8
7.6
17.6
1.24
77.7
466
0.27
68
34
8240
7630
2890
6440
59.3
31.6
7.2
19.1
1.39
73.2
465
0.24
69
37
8240
8140
2890
6710
62.2
34.5
7.2
20.3
1.37
73.7
464
0.23
59
37
8400
8050
2890
7420
60.5
32.9
7.0
20.1
1.30
75.5
464
0.23
56
38
Aspect ratio
Planform area
Span
Root chord
Mean aerodynamic chord
Taper ratio
Sweep angle
Dihedral angle
Wing position fraction
−
m2
m
m
m
−
−
9.4
126
34.5
7.6
4.5
0.16
26.2
6.0
0.45
11.6
127
38.5
4.1
3.5
0.45
0.1
1.9
0.53
14.4
134
43.9
4.6
3.3
0.36
6.2
6.0
0.48
13.1
130
41.4
5.7
3.6
0.26
14.5
1.5
0.45
Fuselage
Parameter
lf
df
λf
Fuselage length
Fuselage width
Fineness ratio
m
m
−
40.6
4.2
9.6
40.6
4.2
9.6
40.6
4.2
9.6
40.6
4.2
9.6
Ah
Sh
bh
crh
Λh
Γh
Aspect ratio
Planform area
Span
Root chord
Sweep angle
Dihedral angle
−
m2
m
m
4.9
25
11.1
3.3
29.3
6.0
4.9
20
9.9
2.9
0.1
1.9
4.8
20
9.8
2.9
6.9
6.0
4.9
20
10.0
3.0
16.2
1.5
Av
Sv
bv
crv
Λv
Aspect ratio
Planform area
Span
Root chord
Sweep angle
−
m2
m
m
1.6
18
5.2
4.8
39.2
1.6
20
5.6
5.2
0.1
1.6
24
6.1
5.6
9.3
1.6
21
5.8
5.3
21.7
VT
Wing
Key performance indicators
Initiator
HT
reduction in tail size and requires a lower download from the tail. A smaller tail could
reduce the static margin. The design tool does not use the class II design information to
update the tail size, so this may be a point of improvement. Another option is to move
the wing more aft. However, the constraint on minimum nose loading limits its position.
PRE
X
Rh
Rmax.fuel
MTOM
OEM
FM
L/D
CLmax,clean
Vsclean
W/S
T /W
SM
c.g. travel
A
S
b
cr
MAC
λ
Λ
Γ
fx w
Description
Unit
Payload-range efficiency
Range parameter
Harmonic range
Max. fuel range
Maximum take-off mass
Operational empty mass
Fuel mass
Lift-to-drag ratio
Max. lift coefficient
Stall speed clean
Wing loading
Thrust loading
Static margin
Center of gravity travel
◦
◦
◦
◦
◦
Table 6.5: Airbus A320 optimization results
The results of this optimization case can also be viewed with respect to the actual Airbus
A320-200. Its specifications are given in Table 6.6. The aircraft has a harmonic range
of 2870 km and a maximum payload of 20.5 tons, which is the same as the top-level
requirements of this case. At maximum payload the fuel mass is 12.5 tons, which yields
a payload-range efficiency of approximately 4700 km. This value is significantly lower
than the Initiator reference aircraft and the three optimizations. This stems from the
42
Optimization case studies
lower estimated aircraft mass and the underestimated drag penalty due to the onset of
compressibility effects. Both affect the lift-to-drag ratio, which plays a major role in this
key performance indicator.
A
S
b
Λ
cr
λ
Γ
9.5
122
34.1
25.0
6.1
0.24
5.0
−
m2
m
◦
m
−
◦
lf
df
λf
Ah
Sh
bh
Λh
37.6
4.1
9.1
5
31
12.5
28
m
m
−
−
m2
m
Av
Sv
bv
Λv
W/S
T /W
1.8
22
6.3
35
600
0.31
−
m2
m
◦
kg/m2
−
MTOM
OEM
PM
Rh
PRE
73.5
39.7
20.5
2870
4710
tons
tons
tons
km
km
◦
Table 6.6: Airbus A320-200 specifications [21, 22]
The geometry of the A320-200 has most resemblance with the Initiator reference design.
For all obtained designs the horizontal tail planform area is considerably smaller than the
actual A320. This also indicates that the sizing routine of the tail surfaces needs further
investigation.
It can be concluded that with the conventional A320 only relatively small improvements
can be found with respect to the reference aircraft. It seems that the drag rise is underestimated, which followed from analyzing the optimizations and comparing the results with
the actual A320-200. The beneficial weight effect of lower sweep angles outweighs the
drag penalty. Also, the sizing method of the tail surfaces could use some improvement.
The sizing should be based on stability and control requirements, rather than only using
the main wing as reference. Purely looking at the objective value the hybrid algorithm
found the best aircraft. Its computational time is a bit higher, but in this case it can be
justified.
6.3 Case 1: Airbus A320
(a) Top view
43
(b) Front view
(c) Side view
Figure 6.9: Airbus A320 geometry after genetic optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.10: Airbus A320 geometry after gradient-based optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.11: Airbus A320 geometry after hybrid optimization
44
6.4
Optimization case studies
Case 2: Canard aircraft
The canard aircraft shares its top level requirements with the Airbus A320, which are
repeated in Table 6.7. It has a forward swept wing and a canard instead of a horizontal
tail. The engines are mounted to the rear of fuselage.
Pax.
Payload mass
MC
Altitude
Range
sTO
sL
150
20.5 tons
0.78
11.3 km
2870 km
2180 m
1440 m
Table 6.7: Canard aircraft top level requirements
For this aircraft the control-canard is used. The primary role of a control-canard is to
provide longitudinal control for the aircraft. The other variant is the lifting-canard, which
also carries part of the lift during normal flight. This type of canard usually has a higher
aspect ratio to reduce its lift-induced drag.
The canard generates an upward force to control the aircraft, while a horizontal tail
produces negative lift that must be compensated by the wing. This seems to make the
canard configuration the better choice due to the improved lift capability. However, the
downwash of the canard affects the airflow over the main wing which may worsen its
aerodynamic performance. Also, the canard must always stall first to ensure that the
aircraft pitches down during such event. Therefore the main wing can never reach its
maximum lift coefficient.
The forward swept wing has some advantages over an aft swept wing. It generally requires
a lower leading edge sweep angle to cope with the compressibility effects at high Mach
numbers. The aerodynamics model is not capable of fully computing these effects [11], so
this will not be reflected in the results. The downside is that the structure must be rigid
enough to withstand bending and torsion, especially at high sweep angles. This may lead
to a serious weight penalty. In addition, aeroelasticity effects can be problematic as the
tip may have flutter tendencies. The design tool does not evaluate the aeroelasticity, so
these effects are not taken into account.
Using the aforementioned requirements the canard aircraft is generated using the Initiator.
The resulting aircraft properties and performance figures are listed in Table 6.8. A 3dimensional representation of the model is shown in Figure 6.12.
Figure 6.12: Canard aircraft model
6.4 Case 2: Canard aircraft
A
S
b
Λ
cr
λ
9.4
121
33.7
-26.2
5.4
0.16
0
−
m2
m
◦
m
−
Γ
fxw
( ct )r
( ct )t
lf
df
6.0
0.60
0.151
0.104
40.6
4.2
45
◦
−
−
−
m
m
Av
Sv
bv
Ac
Sc
bc
1.6
17
5.3
5.1
14
8.5
◦
−
m2
m
−
m2
m
MTOM
OEM
FM
Rh
PRE
L/D
CLmax,clean
57.1
28.5
8.0
2900
7420
15.6
1.84
tons
tons
tons
km
km
−
−
Table 6.8: Canard aircraft properties using the Initiator
Next, a sensitivity analysis is conducted to reduce the number of design variables. Since
the wing is swept forward there is no kink section. So from the design variables listed
Table 6.1 the thickness ratio and twist angle at the kink are left out. Furthermore the
sweep angle boundaries are inversed, giving it a lower bound of −35◦ and an upper bound
of 0◦ . In total 10 design variables are sampled, requiring 44 analysis runs. The sensitivity
analysis took 54 minutes to complete.
The results of the sensitivity analysis are shown in Figure 6.13. The top 5 variables are
indicated in the graph. It can be seen that the wing position, sweep angle and fuselage
diameter have the most influence on the objective when observing their µ∗ and σ values.
This is also reflected in the sensitivity index in Figure 6.13b.
Canard aircraft design vector sensitivity index w.r.t PRE
Canard aircraft sensitivity analysis w.r.t PRE
1
3
2.5
2
4
5
2
1.5
Wing x−position (1)
Sweep angle (2)
Fuselage diameter (3)
Design variables
Standard deviation σ (non−linearity/interaction)
3
3.5
1. Wing x−position
2. Sweep angle
3. Fuselage diameter
4. t/c root section
5. Aspect ratio
6. Dihedral angle
7. t/c tip section
8. Taper ratio
9. Tip twist
10. Root twist
t/c root section (4)
Aspect ratio (5)
Dihedral angle (6)
t/c tip section (7)
Taper ratio (8)
1
Tip twist (9)
0.5
Root twist (10)
0
0
500
1000
1500
2000
2500
Mean µ* (overall importance)
(a) Mean vs. standard deviation
0
10
20
30
40
50
60
70
80
Normalized sensitivity value [−]
90
100
(b) Sensitivity index
Figure 6.13: Canard aircraft sensitivity analysis results
Using all three optimization algorithms the results shown in Table 6.9 and 6.10 are obtained. The latter gives a more extensive overview the aircraft properties. It can be seen
that the payload-range efficiency has been greatly improved, especially for the genetic and
hybrid algorithm. The gradient algorithm performed the worst as it attained the lowest
payload-range efficiency using the same amount of time as the genetic algorithm.
46
Optimization case studies
A
Initiator
Genetic
Gradient
Hybrid
9.4
10.0
10.1
12.3
fxw
Λ
( ct )r
0.60
0.56
0.59
0.59
−26.2◦
0.151
0.118
0.119
0.116
−19.2◦
−31.9◦
−23.4◦
df
4.2
4.3
4.8
4.7
PRE
m
m
m
m
7420
8590
8260
8850
km
km
km
km
∆PRE
t
−
+15.8%
+11.3%
+19.3%
−
124 min
129 min
170 min
Table 6.9: Optimum design vectors for the canard aircraft
All algorithms remain close to the initial wing position of 0.60. In contrast to the aft-swept
A320 where the sweep was drastically reduced among all optimizations, the algorithms
maintained a higher sweep angle with a forward swept wing in canard configuration,
which is curious. The relation between the sweep angle, aspect ratio and operational
empty mass becomes clear when viewing Figure 6.14a. From this graph follows that a
moderate sweep angle of 20◦ to 25◦ results in the lowest aircraft weight. The wing mass
seems to be underestimated at high aspect ratios and sweep angles as their is no severe
increase in structural mass required to resists the large bending stresses of a forward swept
wing.
20
34
18
L/D [−]
OEM [metric tons]
36
32
30
15
28
16
14
40
14
13
26
0
12
5
10
11
15
20
25
10
30
35
9
12
15
30
14
20
13
12
Aspect ratio [−]
Aspect ratio [−]
11
10
10
9
0
Sweep angle [deg]
Sweep angle [deg]
(a) Operational empty mass for varying aspect ratio and sweep angle
(b) Lift-over-drag ratio for varying aspect ratio and sweep angle
Figure 6.14: Effect of aspect ratio and sweep angle on the canard aircraft
The gain in lift-to-drag ratio with respect to aspect ratio and sweep angle is depicted in
Figure 6.14b. From this graph can be observed that their relation with lift-to-drag ratio
is much stronger than witnessed in the A320 case. Also, with increasing sweep angle the
effect on the lift-to-drag ratio becomes more pronounced. Results showed that this effect
mainly stems from a decrease in drag from the vortex-lattice based AVLVLM module.
This contradicts the findings with the aft-swept wing of the A320. Therefore further
investigation is required in the aerodynamics routines.
As such, the algorithms tried to find a compromise between a higher lift-to-drag by increasing the sweep angle and aspect ratio, while keeping the weight increase to a minimum
such that the fuel consumption is kept as low as possible.
6.4 Case 2: Canard aircraft
47
Looking at Figure 6.15 there are significant differences in the aircraft part masses between
the four designs. The heaviest design follows from the hybrid solution, which has an
operational empty mass of 31.1 metric tons. The largest contributor is the main wing as
it gained over 3 tons in mass. Due to its relatively large aspect ratio a heavier structure
is required.
Aicraft part mass comparison
Empenage
Engines
Aircraft part
Furnishing
Fuselage
Landing gear
Operational items
Systems
Wing
0
1
2
3
Initiator
4
5
Part mass [metric tons]
Genetic
Gradient
6
7
8
Hybrid
Figure 6.15: Canard aircraft change in part mass after optimization
The gradient and hybrid algorithm tried to save weight on the fuselage by decreasing
its fineness ratio. This is visualized in Figure 6.18 and 6.19. The genetic algorithm
maintained roughly the same ratio (Figure 6.17). Due to the shorter moment arm a slight
increase in empenage weight is observed. The improved aerodynamic efficiency allowed
for a better thrust-to-weight ratio, thereby saving on engine weight.
The three optimization solutions all show a negative static margin. For a canard aircraft
to be longitudinally stable the static margin must be positive. In order to get a positive
static margin the wing or canard can be moved aft, or the canard size can be decreased
for example. The static margin as a function of sweep angle and wing position is given in
Figure 6.16. Here it can be seen that the margin becomes less with increasing sweep and
a more forward wing position. So for the moderate sweep angles of the optimized designs
a wing position fraction of around 0.65 is required to reach the feasible static margin
region. However, at such aft position the minimum nose loading constraint is violated.
The location of the canard is fixed, so the option that remains is reducing its size. After
investigating the sizing method of the canard it followed that it is linearly scaled with
the main wing’s planform area, MAC and position from the class I design methods. To
obtain a better static margin the information from the class II methods should be used
to adjust the canard size.
Like in the A320 case, the same tail–wing sizing relation is found. The vertical tail sweep
is heavily affected by the main wing sweep. Also, the sweep angle of the canard is sized
according to the main wing. As mentioned earlier, the control surface sizing should be
improvement in order to meet stability and control requirements instead of depending on
empirical geometric functions.
48
Optimization case studies
Static margin [%MAC]
100
50
0
−50
−100
0
0.7
10
0.65
20
0.6
30
Sweep angle [deg]
0.55
Wing position [−]
Figure 6.16: Canard aircraft static margin vs. sweep angle and wing position (A = 9.4)
From the optimization of the canard aircraft it can be concluded that a large improvement in objective value can be obtained. A slightly higher wing aspect ratio and lower
sweep angle resulted in a nearly 20% higher payload-range efficiency, at the cost of a
5% to 10% heavier aircraft. Some weight savings are achieved with the engine and fuselage. Improvements with respect to the static margin could be done by feeding class
II design information back into the canard sizing. Compared to the A320, the baseline
canard aircraft performed worse than the conventional aircraft, but after optimization
the canard configuration obtained a superior payload-range efficiency. Lastly, it must be
noted that some characteristics of the forward swept wing, as mentioned earlier in the
case description, could not been taken into account as it is not covered in the design tool.
Genetic
Gradient
Hybrid
km
km
km
km
tons
tons
tons
−
−
m/s
kg/m2
−
%MAC
%MAC
7420
6250
2900
7310
57.1
28.5
8.0
15.6
1.84
64.1
471
0.30
33
67
8590
7700
2900
6470
57.4
29.9
6.9
19.2
1.22
78.1
464
0.25
-57
53
8260
7520
2890
6670
58.7
31.0
7.2
18.8
0.77
98.4
465
0.25
-76
55
8850
8240
2890
6230
58.4
31.1
6.7
20.6
1.25
77.0
462
0.23
-47
43
Aspect ratio
Planform area
Span
Root chord
Mean aerodynamic chord
Taper ratio
Sweep angle
Dihedral angle
Wing position fraction
−
m2
m
m
m
−
−
9.4
121
33.7
5.4
3.7
0.16
-26.2
6.0
0.60
10.0
124
35.2
5.1
3.6
0.22
-19.2
6.0
0.56
10.1
126
35.7
5.4
3.7
0.13
-31.9
6.0
0.59
12.3
126
39.4
4.8
3.3
0.18
-23.4
6.0
0.59
lf
df
λf
Fuselage length
Fuselage width
Fineness ratio
m
m
−
40.6
4.2
9.6
40.2
4.3
9.5
35.9
4.8
7.5
36.3
4.7
7.7
Ac
Sc
bc
crc
Λc
Γc
Aspect ratio
Planform area
Span
Root chord
Sweep angle
Dihedral angle
−
m2
m
m
5.1
14
8.5
2.1
23.5
-3.0
5.1
18
9.7
2.4
17.3
-3.0
5.1
20
10.1
2.5
28.7
-3.0
5.1
18
9.6
2.4
21.0
-3.0
Av
Sv
bv
crv
Λv
Aspect ratio
Planform area
Span
Root chord
Sweep angle
−
m2
m
m
1.6
17
5.3
4.9
39.2
1.6
15
4.9
4.5
28.8
1.6
16
5.1
4.7
47.8
1.6
19
5.5
5.1
35.1
VT
Wing
Key performance indicators
Initiator
Fuselage
Parameter
49
Canard
6.4 Case 2: Canard aircraft
PRE
X
Rh
Rmax.fuel
MTOM
OEM
FM
L/D
CLmax,clean
Vsclean
W/S
T /W
SM
c.g. travel
A
S
b
cr
MAC
λ
Λ
Γ
fx w
Description
Unit
Payload-range efficiency
Range parameter
Harmonic range
Max. fuel range
Maximum take-off mass
Operational empty mass
Fuel mass
Lift-to-drag ratio
Max. lift coefficient
Stall speed clean
Wing loading
Thrust loading
Static margin
Center of gravity travel
◦
◦
◦
◦
◦
Table 6.10: Canard aircraft optimization results
50
Optimization case studies
(a) Top view
(b) Front view
(c) Side view
Figure 6.17: Canard aircraft geometry after genetic optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.18: Canard aircraft geometry after gradient optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.19: Canard aircraft geometry after hybrid optimization
6.5 Case 3: Three-surface aircraft
6.5
51
Case 3: Three-surface aircraft
The three-surface aircraft features three horizontal surfaces: a canard, main wing and
horizontal tail. A well-known example is the Piaggio P.180 Avanti, which achieved lower
weight and drag thanks to its three-surface configuration [3].
Traditional aircraft with only a horizontal tail rely on the tailplane to balance and control
the aircraft. The tailplane provides a negative lift to counteract the moment due to the
lift of the wing which. This in turn must be compensated by additional lift of the wing.
By adding a canard the required counteracting moment can be shared with the horizontal
tail. Because the canard provides an upward force, the wing loading becomes lower and
therefore the wing size can be reduced. A schematic overview of the equilibrium of a
three-surface aircraft is shown in Figure 6.20.
90% W
15% W
5% W
W
Figure 6.20: Three-surface equilibrium
The three-surface aircraft has similar requirements as the A320 and the canard aircraft.
It has a high-wing configuration, low canard and a T-tail, such that the surfaces are not
in each other’s wake. The requirements are shown in Table 6.11.
Pax.
Payload mass
MC
Altitude
Range
sTO
sL
150
20.5 tons
0.78
11.3 km
2870 km
2180 m
1440 m
Table 6.11: Three-surface aircraft top level requirements
From these requirements an aircraft is generated using the Initiator. The resulting design
is depicted in Figure 6.21. The aircraft properties are listed in Table 6.12.
Figure 6.21: Three-surface aircraft model
52
Optimization case studies
A
S
b
Λ
cr
λ
Γ
−
m2
m
9.4
136
35.8
26.2
7.1
0.16
0
0.0
◦
m
−
◦
◦
−
−
−
−
m
m
−
−
0.60
0.151
0.104
0.104
40.6
4.2
0.10
4.9
fxw
( ct )r
( ct )k
( ct )t
lf
df
fxc
Ac
Sc
bc
Ah
Sh
bh
Av
Sv
bv
19
9.7
4.9
30
12.1
1.0
24
4.8
m2
m
−
m2
m
−
m2
m
MTOM
OEM
FM
Rh
PRE
L/D
CLmax,clean
64.5
34.5
9.5
2900
6270
14.7
1.02
tons
tons
tons
km
km
−
−
Table 6.12: Three-surface aircraft properties using the Initiator
The sensitivity analysis was done using all 13 design variables from Table 6.1. It took
89 minutes to perform all 56 runs. The results from the sensitivity analysis are shown in
Figure 6.22a in which the top 5 variables are indicated. Using the mean and standard
deviation of the parameters the sensitivity index is composed. This index is given in
Figure 6.22b.
TSA sensitivity analysis w.r.t PRE
TSA design vector sensitivity index w.r.t PRE
1
12
10
1. Wing x−position
2. Sweep angle
3. Aspect ratio
4. Fuselage diameter
5. Dihedral angle
6. Canard x−position
7. Taper ratio
8. t/c kink section
9. t/c root section
10. t/c tip section
11. Kink twist
12. Tip twist
13. Root twist
Design variables
Standard deviation σ (non−linearity/interaction)
14
8
2
6
3
4
Sweep angle (2)
Aspect ratio (3)
Fuselage diameter (4)
Dihedral angle (5)
Canard x−position (6)
Taper ratio (7)
t/c kink section (8)
t/c root section (9)
t/c tip section (10)
5
Kink twist (11)
4
2
Wing x−position (1)
Tip twist (12)
Root twist (13)
0
0
500
1000
1500
2000
Mean µ* (overall importance)
(a) Mean vs. standard deviation
0
10
20
30
40
50
60
70
80
Normalized sensitivity value [−]
90
100
(b) Sensitivity index
Figure 6.22: Three-surface aircraft sensitivity analysis results
It can be observed that the wing position is by far the most influential parameter. The
objective is also very sensitive to the sweep angle. The top 5 is concluded by the aspect
ratio, fuselage diameter and dihedral angle. The longitudinal position of the canard did
not make it to the selection. It was a near tie with the dihedral angle. Apparently, canard
sizing benefits due to better positioning with respect to the wing and tail surfaces does
not change the payload-range efficiency very much.
The selected design variables are used for optimization of which the results are displayed
in Table 6.13. A more detailed overview of the aircraft properties is gathered in Table
6.14. For the gradient optimized three-surface aircraft also an example report is generated
by the Initiator. This report is shown in Appendix A.
6.5 Case 3: Three-surface aircraft
A
Initiator
Genetic
Gradient
Hybrid
9.4
11.4
11.7
11.5
53
fxw
Λ
0.60
0.50
0.47
0.50
26.2◦
Γ
0.3◦
11.1◦
−0.7◦
12.9◦
11.5◦
−3.6◦
−0.8◦
df
4.2
4.1
4.7
4.1
m
m
m
m
PRE
6270
7550
7570
7550
km
km
km
km
∆PRE
t
−
+20.4%
+20.7%
+20.4%
−
114 min
152 min
172 min
Table 6.13: Optimum design parameters for the three-surface aircraft
A large gain in payload-range efficiency is obtained through optimization. The best
objective value is achieved by the gradient-based algorithm, but the other two algorithms
are not far behind. The initial value of 6270 is increased by roughly 20% for all algorithms.
When the computation time is taken into account, it can be said that the genetic algorithm
performed best. Looking at the design vectors there is a trend towards a slightly higher
aspect ratio and a more forward wing position. These notable differences in geometry are
clearly visible in the top views of Figure 6.26, 6.27 and 6.28. There are also some notable
differences with respect to the fuselage diameter and sweep angle.
The weight decreased for all solutions with respect to the baseline version. Weight savings
were mainly achieved by smaller engines and lighter wing structures as can be seen in
Figure 6.23.
Aicraft part mass comparison
Empenage
Engines
Aircraft part
Furnishing
Fuselage
Landing gear
Operational items
Systems
Wing
0
1
2
3
Initiator
4
5
Part mass [metric tons]
Genetic
Gradient
6
7
8
Hybrid
Figure 6.23: Three-surface aircraft change in part mass after optimization
Most weight was saved by the gradient-based algorithm. This follows from its low fuselage
fineness ratio. The difference with respect to the baseline geometry can be clearly noticed
in Figure 6.27. A shorter fuselage has less bending stresses and therefore the structure
can be lighter. This is also reflected in the system components mass. The high-wing
configuration resulted in a high fuselage mass when compared to the low-wing aircraft
from the first two cases.
The weight of the wing is largely influenced by the sweep angle and aspect ratio. At large
sweep angles this effect becomes more pronounced. This is shown in Figure 6.24a. The
54
Optimization case studies
lift-to-drag ratio, which plays an important role in the payload-range efficiency, benefits
from a larger aspect ratio. This is depicted in Figure 6.24b. At larger sweep angles the
L/D decreases a bit, which is likely caused by an increase in lift-dependent drag. So a
trade-off arises between weight and aerodynamic efficiency.
42
19
38
18
L/D [−]
OEM [metric tons]
40
36
34
15
13
10
13
5
12
5
14
16
0
14
30
28
0
17
15
32
11
15
20
12
10
30
9
11
15
20
10
25
Aspect ratio [−]
10
25
Aspect ratio [−]
9
30
Sweep angle [deg]
Sweep angle [deg]
(a) Operational empty mass vs. aspect ratio
and sweep angle
(b) Lift-over-drag ratio vs. aspect ratio and
sweep angle
Figure 6.24: Three-surface aircraft with varying aspect ratio and sweep angle (fxw = 0.5)
Clearly, when observing these graphs an unswept wing would be the best choice. This
conclusion does not match with the sweep angle obtained from the optimizations. As can
be seen in Figure 6.25a the optimum sweep angle is also dictated by the position of the
wing. The optimum sweep angle becomes higher as the wing is located further aft.
16
17
16.5
L/D [−]
L/D [−]
15.5
16
15.5
15
15
14.5
0.45
14.5
30
0.5
20
0.55
10
0
0.6
13
14
−4
12
−2
11
0
2
Wing position [−]
Sweep angle [deg]
(a) Lift-over-drag ratio vs. sweep angle and
wing position (A = 10)
10
4
6
9
Aspect ratio [−]
Dihedral angle [deg]
(b) Lift-over-drag ratio vs. aspect ratio and
dihedral angle
Figure 6.25: Three-surface aircraft lift-to-drag for varying parameters
The dihedral angle became lower for all three solutions. As can be seen in Figure 6.25b
the L/D improves with decreasing dihedral. A higher aspect ratio slightly enhances this
6.5 Case 3: Three-surface aircraft
55
effect. A similar effect has been witnessed in the A320 case. The high-wing nature
of the three-surface aircraft makes it more laterally stable with respect to the dihedral
effect. Therefore the obtained dihedral angles are not very unrealistic. However, since the
Initiator does not compute the lateral stability yet, the lower dihedral is purely driven by
the beneficial L/D instead of taking into account the dihedral effect.
Again, the same control surface sizing discrepancies are witnessed. The sweep of the
horizontal tail, vertical tail and canard are based on the main wing. The same can be
said of the dihedral angle.
From this case it can be concluded that a large improvement in payload-range efficiency
can be attained with the three-surface aircraft. The initial wing position of 0.60 is too aft.
A better initial guess would be a value 0.50. The optimization resulted in two distinct
fuselage designs, but with similar payload-range efficiency. The gradient algorithm found
the highest optimum, but the genetic algorithm resulted in the most gain in the shortest
amount of time. Comparing to the Airbus A320 and canard aircraft, which have the
same top level requirements, the three-surface configuration has the worst payload-range
efficiency. Even after the optimization the aircraft is no match for the conventional A320.
Parameter
Genetic
Gradient
Hybrid
km
km
km
km
tons
tons
tons
−
−
m/s
kg/m2
−
%MAC
%MAC
6270
5890
2900
6170
64.5
34.5
9.5
14.7
1.02
86.1
473
0.32
119
69
7550
7100
2900
5450
61.1
32.7
7.9
17.7
1.41
72.8
466
0.25
34
28
7570
6920
2890
5050
59.7
31.3
7.9
17.3
1.44
72.2
468
0.25
15
23
7550
7120
2900
5470
61.3
32.8
7.9
17.8
1.40
73.1
466
0.25
38
30
Aspect ratio
Planform area
Span
Root chord
Mean aerodynamic chord
Taper ratio
Sweep angle
Dihedral angle
Wing position fraction
−
m2
m
m
m
−
−
9.4
136
35.8
7.1
4.1
0.16
26.2
0.3
0.60
11.4
131
38.8
5.0
3.3
0.29
11.1
-0.7
0.50
11.7
128
38.6
5.1
3.2
0.27
12.9
-3.6
0.47
11.5
132
38.9
5.0
3.3
0.29
11.5
-0.8
0.50
lf
df
λf
Fuselage length
Fuselage width
Fineness ratio
m
m
−
40.6
4.2
9.6
42.0
4.1
10.3
36.4
4.7
7.7
42.1
4.1
10.4
Ac
Sc
bc
crc
Λc
Γc
Aspect ratio
Planform area
Span
Root chord
Sweep angle
Dihedral angle
−
m2
m
m
4.9
19
9.7
2.4
23.5
-0.2
5.0
18
9.6
2.4
10.0
0.4
5.0
21
10.3
2.6
11.6
1.8
5.0
18
9.6
2.4
10.3
0.4
Ah
Sh
bh
crh
Λh
Γh
Aspect ratio
Planform area
Span
Root chord
Sweep angle
Dihedral angle
−
m2
m
m
4.9
30
12.1
3.6
29.3
0.3
5.0
17
9.2
2.7
12.5
-0.7
5.1
18
9.5
2.8
14.4
-3.6
5.1
17
9.3
2.8
12.8
-0.8
Av
Sv
bv
crv
Λv
Aspect ratio
Planform area
Span
Root chord
Sweep angle
−
m2
m
m
1.0
24
4.8
5.7
39.2
1.0
18
4.2
5.0
16.7
1.0
19
4.4
5.2
19.4
1.0
18
4.3
5.0
17.2
VT
HT
Wing
Key performance indicators
Initiator
Fuselage
Optimization case studies
Canard
56
PRE
X
Rh
Rmax.fuel
MTOM
OEM
FM
L/D
CLmax,clean
Vsclean
W/S
T /W
SM
c.g. travel
A
S
b
cr
MAC
λ
Λ
Γ
fx w
Description
Unit
Payload-range efficiency
Range parameter
Harmonic range
Max. fuel range
Maximum take-off mass
Operational empty mass
Fuel mass
Lift-to-drag ratio
Max. lift coefficient
Stall speed clean
Wing loading
Thrust loading
Static margin
Center of gravity travel
◦
◦
◦
◦
◦
◦
◦
Table 6.14: Thee-surface aircraft optimization results
6.5 Case 3: Three-surface aircraft
(a) Top view
57
(b) Front view
(c) Side view
Figure 6.26: Thee-surface aircraft geometry after genetic optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.27: Thee-surface aircraft geometry after gradient optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.28: Thee-surface aircraft geometry after hybrid optimization
58
6.6
Optimization case studies
Case 4: Oval-fuselage aircraft
The oval-fuselage aircraft has an ellipsoidal shaped fuselage cross-section. The wider
fuselage allows more passengers seats abreast. So for the same fuselage length more
passengers can be carried. Conventional aircraft have circular shaped fuselages, which
are structurally more efficient with respect to pressurization loads. For more information
regarding oval fuselages in conventional and novel aircraft configurations the reader is
referred to Schmidt [25].
The design requirements are somewhat different from the previous cases. The first three
cases concerned short-range aircraft. In this fourth case an aircraft for medium to long
range is considered. It must have a harmonic range of 5900 km at a cruise speed of Mach
0.78. The maximum payload is established at 42 metric tons and it must be able to carry
400 passengers. All top level requirements are gathered in Table 6.15.
Pax.
Payload mass
MC
Altitude
Range
sTO
sL
400
42 tons
0.78
11.3 km
5900 km
1960 m
1490 m
Table 6.15: Oval fuselage aircraft top level requirements
The Initiator uses this information to create an aircraft that fulfils these requirements. A
model of the oval-fuselage aircraft is displayed in Figure 6.29. The corresponding aircraft
properties are shown in Table 6.16. From these properties it can be observed that the
fuselage width is 8 meters, which is about 23% larger than its height due to its oval shape.
Figure 6.29: Oval-fuselage aircraft model
A
S
b
Λ
cr
λ
9.5
291
52.5
26.2
11.5
0.16
0
−
m2
m
◦
m
−
◦
Γ
fxw
( ct )r
( ct )k
( ct )t
lf
df
6.0
0.45
0.151
0.104
0.104
60.1
8.0
◦
−
−
−
−
m
m
hf
Ah
Sh
bh
Av
Sv
bv
6.8
5.1
57.0
17.0
1.6
40.1
8.0
m
−
m2
m
−
m2
m
MTOM
OEM
FM
Rh
PRE
L/D
CLmax,clean
Table 6.16: Oval-fuselage properties using the Initiator
159
78.2
38.9
5960
6430
16.5
1.25
tons
tons
tons
km
km
−
−
6.6 Case 4: Oval-fuselage aircraft
59
The design variables from Table 6.1 are used to perform the sensitivity analysis with.
The required 52 runs were completed in 96 minutes. The results are shown in Figure
6.30. The variables are numbered according to their sensitivity value. It can be seen that
two parameters stand out, which are the wing position and the sweep angle. The other
variables have much less impact. They are grouped on the left side of the plot due to
their low overall importance.
Oval fuselage aircraft design vector sensitivity index w.r.t PRE
1. Wing x−position
2. Sweep angle
3. Aspect ratio
4. Fuselage diameter
5. t/c kink section
6. Dihedral angle
7. t/c tip section
8. Taper ratio
9. t/c root section
10. Kink twist
11. Tip twist
12. Root twist
2
12
10
1
8
3
4
6
5
Wing x−position (1)
Sweep angle (2)
Aspect ratio (3)
Fuselage diameter (4)
Design variables
Standard deviation σ (non−linearity/interaction)
Oval−fuselage aircraft sensitivity analysis w.r.t PRE
t/c kink section (5)
Dihedral angle (6)
t/c tip section (7)
Taper ratio (8)
t/c root section (9)
4
Kink twist (10)
Tip twist (11)
2
Root twist (12)
0
0
200
400
600
800
0
1000
10
Mean µ* (overall importance)
(a) Mean vs. standard deviation
20
30
40
50
60
70
80
Normalized sensitivity value [−]
90
100
(b) Sensitivity index
Figure 6.30: Oval-fuselage aircraft sensitivity analysis results
The 5 most sensitive parameters from Figure 6.30b were selected for optimization. The
optimization results for the genetic, gradient-based and hybrid algorithm are given in
Table 6.17. A more extensive overview of the changes in geometry and performance with
respect to the initial design are given in Table 6.18.
A
Initiator
Genetic
Gradient
Hybrid
9.5
12.3
9.4
12.6
fxw
Λ
( ct )k
0.45
0.48
0.43
0.47
26.2◦
0.104
0.106
0.108
0.108
5.2◦
24.1◦
5.1◦
df
8.0
7.4
8.4
7.5
m
m
m
m
PRE
6430
7040
6720
7070
km
km
km
km
∆PRE
t
−
+9.5%
+4.5%
+10.0%
−
121 min
129 min
183 min
Table 6.17: Optimum design parameters for the oval-fuselage aircraft
The hybrid algorithm achieved the best payload-range efficiency, but it remains very close
to the genetic solution. An improvement of about 10% is obtained. The gradient-based
solution did not go beyond a meager 4.5% increase in objective value. Looking at the
computation time the genetic algorithm clearly wins. The hybrid algorithm managed to
find a sightly more optimal solution, but at higher computational expense.
The kink thickness ratio increased slightly in all optimizations. A thicker section increases
the stiffness of the structure, but negatively affects the lift-to-drag ratio. The differences
over the baseline are rather small. Another trend that follows from the optimized design
vectors is the low sweep angle. This was also seen at the Airbus A320 and three-surface
aircraft cases.
60
Optimization case studies
Significant differences in weight can be found when comparing the designs. An overview of
the weight components is given in Figure 6.31. The largest differences can be found in the
wing. The gradient-based solution obtained the lightest wing structure. The genetic and
hybrid optimized designs have the heaviest wing, but saved on engine weight by having
better thrust-to-weight ratios due to an increase in L/D.
Aicraft part mass comparison
Empenage
Engines
Aircraft part
Furnishing
Fuselage
Landing gear
Operational items
Systems
Wing
0
5
10
15
Part mass [metric tons]
Genetic
Gradient
Initiator
20
25
Hybrid
Figure 6.31: Oval-fuselage aircraft change in part mass after optimization
The weight of the wing and thus the aircraft increases with aspect ratio. The weight
penalty becomes even higher at sweep angles beyond 20◦ as can be seen in Figure 6.32a.
The choice for a low sweep angle of the genetic and hybrid algorithm allowed for a larger
aspect ratio to further improve the lift-to-drag ratio at a reduced weight penalty.
17
110
L/D [−]
OEM [metric tons]
16
100
90
15
14
13
80
30
70
15
14
20
13
12
Aspect ratio [−]
11
9
0
0.4
20
0.45
0.5
10
10
10
12
30
Sweep angle [deg]
(a) Operational empty mass vs. aspect ratio
and sweep angle (fxw = 0.45)
0.55
0
Sweep angle [deg]
0.6
0.65
Wing position [−]
(b) Lift-over-drag ratio vs. wing position
and sweep angle (A = 9.5)
Figure 6.32: Oval-fuselage aircraft operational empty mass and lift-to-drag ratio
The low sweep angle is also driven by the wing position as is shown in Figure 6.32b.
6.6 Case 4: Oval-fuselage aircraft
61
This relation has already been explained in the A320 case. The L/D deteriorates with
higher sweep angle at more aft wing positions due to an increasing tail size. This effect
is stronger for the oval-fuselage aircraft than for the A320.
The gradient-based solution saved on fuselage and systems weight thanks to its shorter
fuselage. Its lower fuselage slenderness ratio is clearly distinguishable in the top view of
Figure 6.35. The effect of the fuselage slenderness on the operational empty weight is
depicted in Figure 6.33a. Here it can be observed that the weight decreases with lower
slenderness ratios. This follows from the lower forces in the fuselage structure, leading to
a less heavy design. The benefit decreases a bit as the sweep angle becomes larger.
17
80
16.8
16.6
L/D [−]
OEM [metric tons]
78
76
74
16.4
16.2
72
16
70
15.8
9
68
9
30
20
8.5
8
7.5
7
Fuselage slenderness ratio [−]
6
5.5
0
20
7
10
6.5
30
8
10
6
Sweep angle [deg]
(a) Operational empty mass vs. slenderness
ratio and sweep angle
Fuselage slenderness ratio [−]
0
Sweep angle [deg]
(b) Lift-over-drag ratio vs. slenderness ratio
and sweep angle
Figure 6.33: Oval-fuselage aircraft fuselage slenderness ratio effects (A = 9.5)
The slenderness ratio has some effect on the lift-to-drag ratio. Less slender fuselages result
in lower L/D values, which is illustrated in Figure 6.33b. Due to the lower moment arm of
the horizontal tail it must increase in size to provide a sufficient counterbalancing moment
leading to an increase in drag and weight. When comparing the gradient-based design to
the genetic and hybrid designs, it has a 30% to 35% higher horizontal tail planform area
due to its shorter fuselage. Also, the drag of the fuselage changes with slenderness ratio.
A lower ratio resulted in a higher drag coefficient.
It can be concluded that quite some improvement in payload-range efficiency can be
achieved over the baseline design. Most benefit can be gained by lowering the sweep
angle and increasing the aspect ratio. The parameters are closely related to the wing
position. By adjusting the width of the fuselage the weight and lift-to-drag ratio can be
further tuned. Improvements in L/D also yield in better thrust-to-weight ratios leading
to smaller and lighter engines.
When comparing the algorithms, the genetic showed most gain in the shortest amount
of time. The additional gradient-based step of the hybrid algorithm only improvement
marginally on the genetic solution. The gradient algorithm got stuck on a local optimum as its objective is only increased by 4.5%, which is only half as much as the other
algorithms.
62
Optimization case studies
PRE
X
Rh
Rmax.fuel
MTOM
OEM
FM
L/D
CLmax,clean
Vsclean
W/S
T /W
SM
c.g. travel
Unit
Payload-range efficiency
Range parameter
Harmonic range
Max. fuel range
Maximum take-off mass
Operational empty mass
Fuel mass
Lift-to-drag ratio
Max. lift coefficient
Stall speed clean
Wing loading
Thrust loading
Static margin
Center of gravity travel
km
km
km
km
tons
tons
tons
−
−
m/s
kg/m2
−
%MAC
%MAC
Aspect ratio
Planform area
Span
Root chord
Mean aerodynamic chord
Taper ratio
Sweep angle
Dihedral angle
Wing position fraction
−
m2
m
m
m
−
Fuselage
Fuselage
Fuselage
Fineness
Genetic
Gradient
Hybrid
6430
6620
5960
12080
159
78.2
38.9
16.5
1.25
83.7
547
0.30
67
22
7040
7390
5950
8720
155
77.7
35.5
18.5
1.51
75.7
539
0.26
50
26
6720
6610
5960
11620
151
72.2
37.3
16.5
1.32
81.5
547
0.30
50
21
7070
7420
5940
8750
155
77.6
35.3
18.5
1.53
75.0
538
0.26
50
26
−
9.5
291
52.5
11.5
6.7
0.16
26.2
6.0
0.45
12.3
288
59.6
6.9
5.2
0.37
5.2
6.0
0.48
9.4
277
50.9
10.8
6.5
0.18
24.1
6.0
0.43
12.6
288
60.2
6.8
5.1
0.37
5.1
6.0
0.47
m
m
m
−
60.1
8.0
6.8
7.5
59.7
7.4
6.3
8.0
52.9
8.4
7.1
6.3
59.3
7.5
6.4
7.9
5.1
57
17.0
5.0
29.3
6.0
5.1
43
14.7
4.4
5.8
6.0
5.1
56
16.9
5.0
26.9
6.0
5.1
42
14.5
4.3
5.7
6.0
1.6
40
8.0
7.5
39.2
1.6
44
8.5
7.8
7.8
1.6
40
8.0
7.4
36.1
1.6
44
8.4
7.8
7.7
◦
◦
lf
df
hf
λf
Ah
Sh
bh
crh
Λh
Γh
Aspect ratio
Planform area
Span
Root chord
Sweep angle
Dihedral angle
−
m2
m
m
Av
Sv
bv
crv
Λv
Aspect ratio
Planform area
Span
Root chord
Sweep angle
−
m2
m
m
VT
length
width
height
ratio
Initiator
Fus.
A
S
b
cr
MAC
λ
Λ
Γ
fxwing
Description
HT
Wing
Key performance indicators
Parameter
◦
◦
◦
Table 6.18: Oval-fuselage aircraft optimization results
6.6 Case 4: Oval-fuselage aircraft
(a) Top view
63
(b) Front view
(c) Side view
Figure 6.34: Oval-fuselage aircraft geometry after genetic optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.35: Oval-fuselage aircraft geometry after gradient-based optimization
(a) Top view
(b) Front view
(c) Side view
Figure 6.36: Oval-fuselage aircraft geometry after hybrid optimization
64
6.7
Optimization case studies
A comparison of the obtained aircraft designs
In order to be able to compare the designs that have been obtained from the four case
studies, first some parameters have to be defined on which the comparison can be based.
For this purpose the value efficiency parameters defined by Nangia [19] are used. These
parameters are based on the payload-range efficiency, but are normalized with respect
to the weight of the aircraft. The first parameter is the value efficiency parameter with
respect to the maximum take-off mass, which is abbreviated to VEM. It is defined as
follows:
PRE
WTO
VEM =
(6.7)
The second parameter is the value efficiency with respect to the operational empty mass
and is denoted as VEO. It is shown in Equation 6.8.
VEO =
PRE
WOEM
(6.8)
The above efficiency parameters values are determined for each design and these results
are shown in Table 6.19 for the VEM parameter and in Table 6.20 for the VEO parameter.
1.
2.
3.
4.
Airbus A320
Canard aircraft
Three-surface aircraft
Oval-fuselage aircraft
Initiator
[km/kN]
Genetic
[km/kN]
Gradient
[km/kN]
Hybrid
[km/kN]
13.6
13.2
9.9
4.1
14.2
15.2
12.6
4.6
13.5
14.3
12.9
4.5
14.2
15.4
12.5
4.6
Table 6.19: Comparison of the case studies using the VEM parameter
1.
2.
3.
4.
Airbus A320
Canard aircraft
Three-surface aircraft
Oval-fuselage aircraft
Initiator
[km/kN]
Genetic
[km/kN]
Gradient
[km/kN]
Hybrid
[km/kN]
26.1
26.5
18.5
8.4
26.6
29.2
23.5
9.2
24.4
27.2
24.6
9.5
26.0
29.0
23.4
9.3
Table 6.20: Comparison of the case studies using the VEO parameter
When comparing the efficiency values of the first three cases, which all have the same
top level requirements, if follows that the canard aircraft is the most efficient design.
The canard aircraft designs obtained from the genetic and hybrid algorithm show the
best results. The three-surface aircraft designs perform significantly less compared to the
6.8 Evaluation of the algorithms
65
A320 and canard aircraft. Although the baseline TSA can be improved a lot through
optimization, it is no match against the other configurations. The conventional Airbus
A320 showed the least improvement after optimization. When looking at the VEM and
VEO values of the gradient optimized A320, it has an even worse efficiency value compared
to the initial design although it has a better payload-range efficiency. This is caused by
the relatively large increase in weight. The OEM and MTOM increased by 5.6% and
11.9% respectively, while its payload-range efficiency only increased by 4.6%.
All oval-fuselage aircraft solutions have much lower efficiency values with respect to the
first three cases. This follows from its higher range and payload requirements, which
causes the weight of the aircraft to increase more rapidly than the payload-range efficiency.
This trend is in accordance with the results obtained by Nangia [19].
6.8
Evaluation of the algorithms
For the case studies three optimization algorithms have been used: the genetic algorithm, the SQP gradient algorithm and the hybrid genetic–SQP algorithm. Based on
the obtained results and experience with the optimizer tool it was found that the genetic
algorithm worked best. The main arguments for choosing the genetic algorithm over
the other methods are its robustness and tolerance towards noise in the model outputs.
The computation time can be reduced significantly by using parallel optimization, which
eliminates one of the weak points of the algorithm.
The output noise of the Initiator proved to be troublesome for the gradient algorithm.
It may cause the gradient algorithm to start oscillating around a certain point in the
design space due improper gradient information. It often required tuning of the algorithm
settings like the minimum step size to overcome the noise.
In Figure 6.37 the presence of noise is demonstrated by plotting the aspect ratio against
the obtained payload-range efficiency. The aspect ratio was increased from 9 to 14 with
increments of 0.1 for the Airbus A320 aircraft. In this figure it can be seen that a small
increase in aspect ratio may result in lower payload-range efficiency, while a higher value
is expected or vice versa. In other words, the change in results due to a small change in
the design may contradict the global trend, which causes the gradient algorithm to take
a search step in the wrong direction.
The hybrid algorithm produced slightly better results in some cases, but at higher computational cost. The added benefit of the local gradient-based solver does not seem to
outweigh the required computation time. Also, the noise adversely affects the capability of gradient-based algorithm to find the exact optimum in the region provided by the
genetic algorithm.
66
Optimization case studies
Expected vs. obtained payload−range efficiency
7950
Payload−range efficiency [km]
7900
7850
7800
7750
7700
7650
7600
9
9.5
10
10.5
11
11.5
12
Aspect ratio [−]
12.5
13
13.5
14
Figure 6.37: Payload-range efficiency for increasing aspect ratio using the Initiator
Chapter 7
Conclusions and recommendations
7.1
Conclusions
The goal of the thesis was to develop an optimization tool for the conceptual design
of conventional and unconventional aircraft. This optimization tool is used to answer
the research question: What effect has the developed optimization strategy on the key
performance indicators of unconventional aircraft configurations?.
Through the years lots of data has been gathered on conventional aircraft and therefore
design rules and estimates for such aircraft became fairly accurate. However, this does not
apply to unconventional and novel configurations, for which far less data is available and
design approaches are sometimes rather crude. If certain edge cases are not covered well,
the optimizer might exploit this loophole in an attempt to find even better solutions. This
may result in strange or unrealistic designs. Therefore, the outcome of the optimization
strongly depends on the behaviour and flexibility of the analysis routines. Similarly,
limitations of analysis modules put constraints on the design space.
From the case studies it followed that large improvements can be obtained with unconventional aircraft configurations with respect to the reference aircraft proposed by the
Initiator design tool. The highest payload-range efficiency was obtained with the hybrid
optimized canard aircraft. Most improvement was found with the three-surface aircraft.
All three optimizations showed an increase of over 20% compared to the initial design.
The oval-fuselage aircraft could be improved by a solid 10%, while the lowest improvement
was obtained with the conventional A320.
When comparing the results of the first three cases, which share the same top level
requirements, it is clear that the canard aircraft is the best concept with respect to the
objective. It obtained the highest payload-range efficiency. It yielded a 5% higher payloadrange efficiency compared to the best solution from the A320 case. The three-surface
aircraft showed the least promising results. However, these statements are only valid with
respect to the output provided by the Initiator. Due to several discrepancies in the sizing
and analysis routines the actual performance of the considered aircraft configurations
67
68
Conclusions and recommendations
might be very different. Therefore the results should be interpreted with caution and
should be mainly used as an indication of the maturity and validity of the Initiator design
tool.
Among all cases the most contributing factors were the wing longitudinal position, sweep
angle and wing aspect ratio. There is a tendency towards lower sweep angles due to the
positive effect on the weight of the wing. The drag rise penalty due to the lower sweep
seems to be underestimated, which is exploited by the optimizer by trading sweep for
a higher aspect ratio to minimize the weight penalty of the latter. In the canard case
relatively high sweep angles were found. From this result followed that the weight penalty
of forward swept wings due to sweep is underestimated. It also contradicts the findings of
the aft-swept wing cases in which a lower sweep was actually more beneficial. This can be
traced back to an error in the drag estimation, especially with respect to compressibility
effects.
In three cases the fuselage fineness ratio was involved in the optimization. The results
showed that changing the ratio offered some reduction in fuselage weight due to a more
favourable structural loading at the expensive of more drag.
The uncertainties in the computed results of the Initiator were not handled well by the
gradient-based algorithm. The gradient algorithm either stopped prematurely or started
oscillating around a certain design point when too much noise was present. This was
alleviated by increasing the step size of the algorithm, but at the expense of accuracy.
Also, determining the starting point of the gradient algorithm remains difficult. Not
every starting point yields a feasible design and a change in start location might lead to
a different basin of attraction.
The genetic algorithm was found to be very robust. It is far less sensitivity to noise,
because it does not use gradient information. Its multi-solution approach allows the algorithm to explore multiple sites at the same time, which allows it to continue searching
in other sites when an infeasible region is encountered. Its computational cost was significantly reduced by applying parallel optimization and using a caching mechanism. The
hybrid algorithm was found to be too computational expensive. The obtained increase in
objective value did not outweigh the added cost.
7.2
Recommendations
The following recommendations and considerations can be made for further improvement
on the developed optimizer tool.
In order to further investigate unconventional aircraft configurations improvements in
the analysis tools are required. For instance, the current aerodynamics implementation
underestimates the compressibility effects which has consequences for the drag estimates
and therefore the overall aircraft design. This heavily affects the sweep angle. Through
the use of a better aerodynamic solver the potential of novel aircraft configurations can
be studied more accurately and different optimization solutions might be obtained.
The sizing routine of the control surfaces is found to be inadequate, since the Initiator
derives most parameters directly from the wing and does not properly take into account
7.2 Recommendations
69
control and stability requirements. Results have shown that this mainly regards the sweep
and dihedral angle. Especially, the sweep angle is of concern, since it changes the liftcurve slope and therefore also stall characteristics. These sizing issues also affect the static
margin. It was found that class II design information was not fed back to the control
surface sizing.
A related concern is the static margin. Mainly due to changes in sweep angle, the optimizer
moved the wing to make sure that the nose loading constraints were satisfied. This
effect outweighs the weight savings due to smaller control surfaces that would have been
obtained with a lower static margin. This could be solved by imposing a constraint on
the allowed static margin. In order to do this reliably, the Initiator’s control surface sizing
routine should be improved first.
The dihedral angle is driven by the lift-to-drag ratio, while it should also take into account
lateral effects. Currently, the Initiator does not compute the lateral stability yet, which
affected the outcome of the dihedral angles of the optimizations.
Another issue that currently affects the design space is the EMWET weight estimation
module. Wings with a high aspect ratio or unconventional shape are found to be problematic. It also seems to underestimate the weight penalty of forward swept wings. For
a better evaluation of the aircraft designs this module should be improved.
The design variables could be expanded by including, for instance, the engine location.
In this thesis their positions were fixed with respect to the wing span or fuselage. The
current set of design variables exposed some large discrepancies, which should be solved
first.
The Initiator has rather limited support for the blending-wing body concept and the
Prandtlplane. When the analysis with respect to these concepts have matured, optimizations of these concepts could be performed with the developed tool to discover any further
issues. At the time of this thesis the maturity level was found to be too inadequate and
therefore they were not included in the case study.
70
Conclusions and recommendations
Part II
Code documentation
71
Chapter 8
Program structure
In this chapter the program structure of the optimizer is described. The optimizer tool is
written in Matlab . It is part of the workflow modules. In the first section the optimizer
class, its properties and its methods are explained. In Section 8.2 the sensitivity routine
is described. Section 8.3 elaborates the optimization routine.
8.1
Optimizer class
The optimization routines and properties are housed in a single module class. As has been
explained in Section 2.3, this class inherits from the WorkflowModule class. Workflow
modules are placed outside the analysis chain and are used to control the workflow of the
Initiator. The relationship is shown in the UML diagram of figure 8.1.
Optimiser
Problem
Options
...
run()
optimise()
sensitivity()
...
InitiatorController
...
WorkerObjWrapper
Value
...
WorkflowModule
run()
...
Figure 8.1: UML class diagram of the optimizer
The optimizer class depends on the InitiatorController and the WorkerObjWrapper
classes. The Initiator controller is the main class of the Initiator. The optimizer uses
this controller to control the workflow and to communicate with the modules. The
73
74
Program structure
WorkerObjWrapper class has been developed by MathWorks [5] and is used during parallel
optimization. Normally, data is destroyed and recreated when a parallel worker advances
to the next iteration. This class allows to retain the data of the parallel worker, such that
expensive recreation of the Initiator instance is not required.
The optimizer class has several public properties that can be accessed. These properties
are listed in Table 8.1. The Debug property triggers debug mode when set to true. In this
mode the optimizer will output debug information to the command window. The Problem
property holds the problem structure in which the optimization problem is described. The
Options property contains the sensitivity analysis and optimization options.
Property
Description
Debug
Problem
Options
Results
ResultsDirectory
Debug mode
Problem description
Structure containing all options
Structure with sensitivity and optimization results
Directory in which the results are saved
Table 8.1: Public properties of the optimizer class
The Problem property that holds the optimization problem, which is required to perform
a sensitivity analysis or an optimization. The available fields are shown in Table 8.2.
Field
Description
ObjFcn
ObjScaling
AssignFcn
DesignVarScaling
ConFcn
ModuleList
Algorithm
LowerBound
UpperBound
Start
Selected
Labels
Cell array with one or more objective functions
Cell array containing objective scaling parameters
Cell array with an assign function per parameter
Cell array with scaling parameters for design variables
Cell array with constrain functions
List of modules to run
Optimization algorithm
Lower bound of the design variables
Upper bound of the design variables
Starting point
Selected design variables
Contains labels used for plotting
Table 8.2: Problem structure fields
The module list contains the modules that are executed during the sensitivity analysis
and optimization. By default this is the DesignConvergence module, but any module
can be used. The Algorithm field currently accepts the following three algorithms: gradient, genetic and hybrid. Through the Selected field the design variables can be easily
activated or deactivated. This is especially useful when there are many design variables.
The Label field can be used to provide names for the objective, assign and constraint
functions. This is for plotting purposes only.
8.2 Sensitivity Analysis
75
The exposed optimizer methods are listed in Table 8.3. Normally, the optimizer is run
through the Initiator controller, but by obtaining its module handle these methods can
be called. This may offer some more fine-grained control over the optimizer. The usage
of these methods is explained in Chapter 9.
Method
Description
addConstraint
addDesignVar
addObjective
elemEffects
listFiles
loadData
optimise
resetOptions
resetProblem
resetResults
resume
run
saveData
showOptimPlots
showProblem
showSensPlot
Adds a constraint to the optimization problem
Adds a design variable to the optimization problem
Adds an objective to the optimization problem
Elementary effects routine
Lists all available results files
Loads the problem, option and result data from disk
Starts the optimization
Resets all options to default
Resets the problem description
Clears the results
Resumes optimization from a previous run
Performs sensitivity analysis and optimization
Saves the problem, option en result data to disk
Shows the optimization plots
Shows the problem description in command window
Shows the sensitivity analysis plots
Table 8.3: Public methods of the optimizer class
8.2
Sensitivity Analysis
The optimizer module contains a sensitivity analysis routine to screen the design variables.
It can be called by using the method sensitivity. The screening procedure is able to
identify the design variables which have the most impact on the objective function. This
way the most influential parameters can be selected for the optimization phase, which
reduces its complexity and decreases computation time.
The screening is performed by using the elementary effects method. This method consists
of individually randomised one-at-a-time experiments. Each time a factor is changed its
impact is measured.
A flowchart of the sensitivity analysis process is shown in Figure 8.2. After initialization a copy of the Initiator controller is made. This is done such that the state of the
current session is not altered. Next, the routine calculates the elementary effect of each
variable. This is repeated for the specified number of trajectories. During this process
an estimate for the remaining time is given based on the average computation time of
previous iterations. Then, the sensitivity values are calculated. Based on these values the
optimizer automatically selects the most important design variables. The other variables
are disabled. The results are stored in the property Results.Sensitivity. A description
of the sensitivity results structure can be found in Table 8.4. The available sensitivity
analysis options are listed in Chapter 9.
76
Program structure
Initialize optimization problem
and sensitivity options
Create copy of Initiator
instance
Calculate elementary effect
Next input change?
no
yes
yes
Trajectory < max?
no
Compute mean, standard
deviation, sensitivity index
Select top X design variables
Figure 8.2: Flowchart of the sensitivity analysis
Field
Description
ObjValues
DesignVars
Method
Mu
Mu s
Sigma
Sigma n
Euclidean
Ranks
Labels
Objective values of each iteration
Design vectors of each iteration
Sensitivity method name
Mean µ
Improved mean µ∗
Corrected standard deviation σ
Uncorrected standard deviation σn
Euclidean distance for ranking the variables
Design vector ranks based on the Euclidean
Labels for plotting
Table 8.4: Sensitivity analysis results structure
8.3
Optimization
The actual optimization is governed by the optimise method. It currently supports
the genetic algorithm, gradient algorithms and a genetic–gradient hybrid algorithm. All
algorithms rely on Matlab implementations. The genetic algorithm is based on ga,
the gradient on fmincon and the hybrid on both ga and fmincon. Because the Matlab
optimization functions have different input and output formats, each algorithm is wrapped
inside a separate class method. This allows a uniform approach.
The optimizer offers the possibility to perform parallel optimization. This only applies to
the genetic and hybrid algorithm, since the gradient method cannot compute the objec-
8.3 Optimization
77
tive function in parallel. Parallel mode is turned on by setting Options.<Algorithm>.
UseParallel to always, or off by setting it to never. All other available optimization
options can be found in Section 9.3.1 of Chapter 9.
The top-level flowchart of the optimization process is shown in Figure 8.3. It starts with
running the Initiator with default values. After the initial design has been computed,
its results are saved. Then, a parallel Matlab session may be opened depending on
the aforementioned setting. Next, multiple copies of the current Initiator instance are
created. This prevents polluting the state of the current Initiator instance and avoids
race conditions during parallel optimization. In single-threaded mode a single copy will
be created. For parallel optimization this depends on the configured number of parallel
workers.
Initialize optimization problem
and options
Run Initiator for initial design
point and save results
Parallel optimization?
yes
Open parallel Matlab session
no
Create multiple Initiator
instance copies
Run optimization algorithm
Parallel optimization?
yes
Close parallel Matlab session
no
Run Initiator for final design
point and save results
Figure 8.3: Top-level flowchart of the optimization process
At this point the optimization algorithm is started. For the objective function and nonlinear constraint function internal class methods are assigned. These methods use the
initiatorRunner routine to obtain the objective and constraint values. The flowchart
of this routine is shown in Figure 8.4.
The initiatorRunner function starts with obtaining a copy of the Initiator instance
using the task number assigned by Matlab . Next, a cache lookup is performed for the
requested design vector. When it is a cache hit, the results are gathered and returned
immediately. A cache miss leads to a full multidisciplinary analysis. It begins with
resetting all modules and rescaling the design vector to the actual values. These design
78
Program structure
values are assigned to the modules and then the design convergence module is run.
Get Initiator instance copy
yes
Cache hit?
Retrieve cached data
no
Reset all Initiator modules
Rescale design variables
Apply design variables
Run Initiator modules
Mark design point as infeasible
Infeasible
design point?
yes
no
Construct empty results set
Retrieve objective, constraints,
module results
Save results to cache
Return results
Figure 8.4: Flowchart of the initiatorRunner method
When the design point is feasible, the results are collected. In case of an infeasible
design the results an empty results set is created. After storing the data corresponding
to the requested design vector, the results are returned to the calling function. Once the
optimization algorithm has finished, the parallel session is closed and the Initiator is run
for the final design point.
The results of the optimization are stored in the Results.Optimisation property. The
data is also automatically stored to a mat-file. Besides the results it also includes the
optimization problem and options structs, which allows the user to restore the current
optimizer state at a later point in time. A description of the results structure can be
found in Table 8.5.
8.3 Optimization
79
Field
Description
Algorithm
ConEvals
Final
Labels
ObjEvals
Original
StateData
Algorithm specific output data
Array containing constraint function evaluations
Final aircraft data
Labels used for plotting
Array containing objective function evaluations
Baseline aircraft data
Array containing algorithm state data
Table 8.5: Optimization results structure
80
Program structure
Chapter 9
User manual
This chapter serves as the user manual of the optimizer. This guide assumes that the
reader is already familiar with the Initiator. For more information on using the Initiator
the reader is referred to Elmendorp [14].
9.1
Requirements
The requirements for the optimizer are as follows:
ˆ The Initiator design tool
ˆ Matlab 2012a; version 2013b or higher is recommended
ˆ Windows 7 or higher, Mac OS X 10.71 or higher, Linux1
ˆ At least 8 GB memory is recommended for parallel optimization
In addition to these requirements Subversion may be useful to retrieve the latest version
from the repository.
9.2
Setting up a problem description
In order to use the optimizer module a problem description must be set up first. This
description contains the information that is required to perform a sensitivity analysis
and an optimization. A default problem statement is loaded automatically when no user
input is specified. The standard objective is the payload-range efficiency and the default
algorithm is the genetic algorithm. The default design variables are as follows:
1
Compatibility with these operating systems depends on the installed libraries [14].
81
82
User manual
1. Aspect ratio
5. Taper ratio
2. Wing x-position
6. Dihedral angle
3. Fuselage diameter
4. Sweep angle
7. Twist angle per wing section
8. Thickness-over-chord ratio per wing
section
The problem description can be changed by means of module inputs. These module inputs
can be specified in the aircraft input file. A description of the available inputs is given in
Section 9.3.1
9.3
Operating the optimizer
Once the problem description has been set up the optimizer can be run. When the
Initiator calls its run method, first the number of design variables will be checked. If this
number exceeds the configured maximum, a sensitivity analysis will be done first.
Depending on the number of design variables the sensitivity analysis may take some time.
Using the default design variables the analysis takes one to two hours. The optimizer
will show a remaining time estimate. When the sensitivity values have been obtained the
most important design variables will be selected to perform the optimization with.
The optimization starts with storing the state of the initial aircraft. Then the selected
algorithm is called. Depending on the settings and chosen algorithm a parallel Matlab
session may be opened. When the algorithm has found an optimum, the corresponding
design vector is used to compute the final aircraft. Finally, the optimization data is saved
to a mat-file and the results are shown.
9.3.1
Module input
The default problem setup can be changed by providing module inputs in the aircraft
configuration file. The user is not required to specify all elements in order for the optimizer
to work. The supplied input will simply overwrite the default values. When for instance
design variables are supplied, they will only replace the standard design variable list. An
example is given in Listing 9.1.
9.3 Operating the optimizer
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
83
<moduleInputs>
<input module="Optimiser">
<problem>
<objectives>
..
</objectives>
<designVars>
..
</designVars>
<constraints>
..
</constraints>
<algorithm>genetic</algorithm>
<moduleList>GeometryEstimation,DesignConvergence</moduleList>
</problem>
</input>
</moduleInputs>
Listing 9.1: Module inputs example for optimizer module
Inside the input section of the optimizer module there must be a main element called
problem. This element holds the entire problem description.
The objectives, design variables and constraints can be provided with the objectives,
designVars and constraints elements respectively. They are explained in the following
subsections.
The algorithm field can be used to provide the algorithm. Currently, there are three
algorithms available: genetic, gradient and hybrid. By default the genetic algorithm is
loaded.
The modules that need to be run in the sensitivity analysis and optimization can be
changed with the moduleList field. The module names must be separated by a comma.
By default the geometry estimation and design convergence modules are called.
Objective functions
The objective can be specified with an objectives element. It requires a label, module
and value element. The label is used for plotting and can be any string. An example is
given in Listing 9.2.
1
2
3
4
5
6
7
8
<problem>
<objectives>
<label>PRE</label>
<module>PerformanceEstimation</module>
<value> − KPI.PRE</value>
<scaling>0.001</scaling>
</objectives>
</problem>
Listing 9.2: Module input example for objective functions
84
User manual
The module field refers to a module from the Initiator, which is the performance estimation module in this case. The value field specifies which result value must be used from
the module. In the example the payload-range efficiency result from the key performance
indicators is used. A minus sign can be added in front of the value field if necessary. A
scale factor can be added to change the order of magnitude of the objective value.
There can be multiple objectives elements. Note that only the first objective is considered during optimization. Multiple entries may be useful when a sensitivity analysis
must be performed for several objectives.
Design variables
The design variables can be specified with the designVars element. There are two variants, which are given in Listing 9.3.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
<problem>
<designVars>
<label>Wing x−position</label>
<module>GeometryEstimation</module>
<value>MainWingXPosition</value>
<lowerBound>0.30</lowerBound>
<upperBound>0.60</upperBound>
<start>0.45</start>
</designVars>
<designVars>
<label>Aspect ratio</label>
<type>ConfigurationParameter</type>
<value>WingAspectRatio</value>
<lowerBound>8</lowerBound>
<upperBound>15</upperBound>
<start>10</start>
<scaling>0.1</scaling>
</designVars>
</problem>
Listing 9.3: Module input example for design variables
The first design variable in the code example sets the longitudinal position of the wing.
Here the geometry estimation module is used to set the value of MainWingXPosition.
The lower bound and upper bound are set to 0.30 and 0.60 respectively. Optionally a
starting position can be provided. By default the mean between the bounds is used.
The second design variable controls an aircraft configuration parameter by means of the
type field. In the example the wing aspect ratio is controlled through the WingAspectRatio
value. Again, a lower bound and upper bound must be provided. A scale factor can be
added to change the order of the design variable. It also accepts the value auto, which
transforms the parameter such that it has a range of [−1, 1] [16]. The governing equation
is as follows:
x̄ =
2x
xu + xl
−
xu − xl xu − xl
(9.1)
9.3 Operating the optimizer
85
In Equation 9.1 x̄ is the scaled variable, xl represents the lower bound and xu denotes
the upper bound.
Constraint functions
The constrain functions can be specified with the constraints element. It requires a
label and function. An example is given in Listing 9.4.
1
2
3
4
5
6
<problem>
<constraints>
<label>Nose loading</label>
<function>myNoseLoadingConstraint</function>
</constraints>
</problem>
Listing 9.4: Module input example for constraints
As can be seen in the code example a Matlab function name must be supplied to the
function field. Since constraints can involve extensive code, it has been chosen to keep
the actual constraint logic in Matlab . When a custom function is built, one must
sure that it takes the main Initiator controller, worker controller and design vector as
input arguments. The output must be two vectors containing the inequality and equality
constraints.
Optimizer settings
The settings of the optimizer can be changed through the settings file of the Initiator.
Settings that are provided by the user will overwrite the default values of the optimizer.
An example is given in Listing 9.5.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
<settings>
<setting>
<name>Optimiser−General−MaxTime</name>
<value>8000</value>
</setting>
<setting>
<name>Optimiser−ElemEffects−Trajectories</name>
<value>4</value>
</setting>
<setting>
<name>Optimiser−Genetic−PopulationSize</name>
<value>10</value>
</setting>
</settings>
Listing 9.5: Optimizer settings example
As can be seen in this example each setting has a name and value element. The name of
each setting consists of three parts, which are separated by hyphens. First the name of
86
User manual
the optimizer module is provided, followed by the category and the name of the setting.
Currently there are five categories: General, Genetic, Gradient, Hybrid and ElemEffects.
The settings available in the general category are listed in Table 9.1 and control the global
parameters of the optimizer.
Setting
Description
Default
Debug
MaxDesignVars1
MaxTime2
PoolSize1
ResultsDir
SensScaleFactor1
ShowPlots
TolCache2
UseCache2
Enable or disable debug mode
Maximum number of design variables
Optimization time limit in seconds
Number of parallel workers
Directory to write the results data to
Factor reducing the range of design vectors
Enable or disable plots
Cache tolerance for matching design vectors
Enable or disable results cache
true
5
7200
-3
/Data/Optimiser
0.5
true
eps()
true
Table 9.1: General optimizer settings
1 Sensitivity analysis only 2 Optimization only 3 System dependent
The options of the elementary effects method are given in Table 9.2. The genetic, gradient
and hybrid algorithms use the options specified in the Matlab manual [4].
Setting
Description
Default
Grid
Retries
Trajectories
Grid sizing parameter
Number of trajectory retries after an error
Number of trajectories
4
5
4
Table 9.2: Elementary effects method settings
9.3.2
Using the module handle
To get more fine-grained control one can obtain the optimizer module handle from the
Initiator. This way the sensitivity analysis and optimization can be run individually and
additional functions are available. The sensitivity analysis can be started separately by
calling the sensitivity method. To only start the optimization the optimise method
can be used.
There is also the possibility to resume a previous optimization run. This is dony by running the resume method, which expects a results file name as parameter. The optimizer
will continue at the previously found optimum.
At any point in time the state of the optimizer can be saved to disk. This can be done
by calling the saveData method. It will automatically generate a file name if none is
provided. The save data includes the problem statement, options and results. By default
the data is stored in /Data/Optimiser.
9.3 Operating the optimizer
87
The data can be loaded through the loadData method. It restores the problem statement,
options and results to its previous state. A list of available files can be retrieved by calling
listFiles.
An overview of the problem statement can be printed in the command window by using
the showProblem method. This may give some extra insight in the problem setup. Plots
can be shown using the showSensPlots and showOptimPlots methods for the sensitivity
and optimization results respectively.
88
User manual
References
[1] Allison, J. Complex system optimization: A review of analytical target cascading,
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Michigan, 2004.
[2] Anon. White paper on industrial experience with mdo, 1999. AIAA Technical
Committee on Multidisiplinary Design Optimization.
[3] Anon. Piaggio P180 Avanti II Specification and Description. Piaggio Aero Industries
SpA, Januari 2005.
[4] Anon. Matlab r2013b (8.2.0.701). The MathWorks Inc., Natick, Massachusetts,
August 2013.
[5] Anon. Worker Object Wrapper. The MathWorks Inc., Natick, Massachusetts,
November 2013. http://www.mathworks.com/matlabcentral/fileexchange/31972.
[6] Boggs, P., and Tolle, J. Sequential quadratic programming. Acta Numerica
(1996).
[7] Breitkopf, P., and Coelho, R., Eds. Design Optimization in Computational
Mechanics. Wiley, 2010.
[8] Campolongo, F., and Cariboni, J. Sensitivity analysis: how to detect important
factors in large models. Tech. Rep. JRC37120, European Comission, DirectorateGeneral Join Research Centre, 2007.
[9] Campolongo, F., Cariboni, J., Saltelli, A., and Schoutens, W. Enhancing
the morris method. In Sensitivity Analysis of Model Output (2005).
[10] Conn, A., Gould, N., and Toint, P. A globally convergent augmented lagrangian
algorithm for optimization with general constraints and simple boundsle. SIAM
Journal on Numerical Analysis 28 (1991), 545–572.
[11] Drela, M. Athena Vortex Lattice (AVL). Massachusetts Institute of Technology,
http://web.mit.edu/drela/Public/web/avl/avl doc.txt. Last accessed: 16/4/2014.
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[12] El-Mihoub, T., Hopgood, A., Nolle, L., and Battersby, A. Hybrid genetic
algorithms: A review. Engineering Letters, 2006.
[13] Elham, A. Weight Indexing for Multidisciplinary Design Optimization of Liing
Surfaces. PhD thesis, Delft University of Technology, 2013.
[14] Elmendorp, R. Synthesis of novel aircraft concepts for future air travel. Master’s
thesis, Delft University of Technology, January 2014.
[15] Gill, P., Murray, W., and Saunders, M. Snopt: An sqp algorithm for largescale constrained optimization. SIAM Journal on Optimization 12 (2002), 979–1006.
[16] Gill, P., Murray, W., and Wright, M. Practical Optimization. Emerald Group,
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[17] Martins, J., and Lambe, A. Multidisciplinary design optimization: A survey of
architectures. AIAA Journal 51 (2013), 2049–2075.
[18] Morris, M. Factorial sampling plans for preliminary computational experiments.
In Technometrics (May 1991), vol. 33 No. 2, pp. 161–174.
[19] Nangia, R. Efficiency parameters for modern commercial aircraft. The Aeronautical
Journal (August 2006).
[20] Parte, Y., Auroux, D., Clément, J., Masmoudi, M., and Hermetz, J. Collaborative optimization. Multidisciplinary design optimization in computational mechanics. Wiley-ISTE, April 2010, pp. 321–368.
[21] Rhodes, D., Jenkinson, L., and Simpkin, P. Civil Jet Aircraft Design. Elsevier,
1999.
[22] Roux, É. Avions civils à réaction : plan 3 vues et données caractéristiques. É.
Roux, 2007.
[23] Sadraey, M. Aircraft Design: A Systems Engineering Approach. Wiley, 2012.
[24] Saltelli, A., Tarantola, S., Campolongo, F., and Ratto, M. Sensitivity
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[25] Schmidt, R. A semi-analytical weight estimation method for oval fuselages in novel
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[26] Taherdangkoo, M., Paziresh, M., Yazdi, M., and Bagheri, M. An efficient
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[27] Torenbeek, E. Synthesis of Subsonic Airplane Design. Springer, 1982.
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[29] Zhang, K. Concurrent subspace optimization for aircraft system design, Aeronautics
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of Computational Mechanics (2008), 103–126.
92
References
Appendix A
Example three-surface aircraft report
15
10
Butt line, y [m]
5
0
−5
Water line, z [m]
−10
−15
0
5
10
15
20
25
Fuselage station, x [m]
30
35
10
8
6
4
2
40
15
10
5
Water line, z [m]
(a) Top view
−5
−10
−15
(b) Front view
10
10
8
8
6
6
4
30
2
4
15
2
0
Butt line, y [m]
0
5
10
15
20
25
Fuselage station, x [m]
30
35
40
(c) Side view
10
20
5
0
−5
−10
10
−15
(d) 3D view
Figure A.1: Aircraft geometry (all dimensions in meters)
A.1
General Characteristics
Aircraft “TSA” generated by the Initiator version . The aircraft is a three-surface aircraft
with a high wing and an aspect ratio of 11.7. The aircraft is designed to transport 150
passengers with a total payload mass of 20536kg over 2870km.
93
94
A.2
Example three-surface aircraft report
Specification
Table A.1: Max payload
Pax
Payload Mass
Cruise Mach
Altitude
Range
Take Off Distance
Landing Distance
kg
m
km
m
m
Optimiser
Table A.2: Optimiser results
Algorithm
Objective value
Design variable 1
Design variable 2
Design variable 3
Design variable 4
Design variable 5
Gradient
PRE
Aspect ratio
Wing x-position
Sweep angle
Dihedral angle
Fuselage diameter
7570
11.7
0.47
12.9
-3.6
4.7
km
◦
◦
m
Objective value history
−6200
−6400
−6600
PRE [km]
A.3
150
20536
0.78
11278
2870
2180
1440
−6800
−7000
−7200
−7400
−7600
0
20
40
60
Function evaluation
80
Figure A.2: Objective value history
100
A.4 Operational Performance
(a) Top view
95
(b) Front view
(c) Side view
Figure A.3: Aircraft geometry changes
Operational Performance
Take−off Thrust−to−Weight ratio (T/W) [−]
A.4
Design Space
bmax = 80 m
sL = 1440 m
0.5
n = 1.3 during cruise, no buffet
(c/V)FAR 25.111c = 1.2 %
0.4
(c/V)FAR 25.119 = 3.2 %
(c/V)FAR 25.121a = 0 %
(c/V)FAR 25.121b = 2.4 %
0.3
(c/V)FAR 25.121c = 1.2 %
(c/V)FAR 25.121d = 2.1 %
0.2
sTO = 2180 m
Mcr = 0.78
0.1
0
tclimb = 10 min to h = 4000 m
Design Point
Reference Aircraft
0
2000
4000
6000
8000
10000
Take−off Wing Loading (W/S) [N/m2]
Figure A.4: Loading Diagram
Result: Wing loading at MTOM: 4597 N/m2
Thrust-to-weight ratio: 0.251 Table A.3: Performance results
L/Dcruise
Cruise altitude
Maximum take-off mass
Operational empty mass
Payload mass
Fuel mass
Harmonic range
Ferry range
Maximum fuel range
17.3
11278
59640
31280
20540
7830
2890
5730
5050
m
kg
kg
kg
kg
km
km
km
96
Example three-surface aircraft report
Maximum passengers range
Mission requirements
65
MTOM
60
PLM
50
45
40
35
30
0
1000
2000
3000
Range, R [km]
4000
5000
Figure A.5: Payload-Range
Vc
2.5
VD
2
1.5
Load factor n [−]
Mass Mpl [metric tons]
55
1
0.5
0
−0.5
−1
−1.5
0
50
100
150
V [m/s]
200
Figure A.6: V-n diagram
250
300
A.5 Weight estimation
A.5
97
Weight estimation
Table A.4: Mass summary
Pax
Cargo
DLM
Diversion FM
End Cruise Mass
Extension FM
FM
Initial Cruise Mass
Loiter FM
MLM
MRM
MTOM
Max FM
Mission FM
OEM
PLM
Reserve FM
ZFM
MainWing (11%)
12000
8540
53040
0
53570
0
7830
58160
0
54880
60860
59640
11280
6610
31280
20540
0
51810
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
Mission fuel (11%)
HorizontalStabiliser (1%)
FrontStabiliser (1%)
VerticalStabiliser (1%)
Extra fuel (2%)
Fuselage (12%)
Cargo (14%)
Engine1 (4%)
Engine2 (4%)
MainGear1 (2%)
MainGear2 (2%)
NoseGear (1%)
Pax (20%)
Systems (11%)
Furnishing
(1%)
OperationalItems
(3%)
Figure A.7: Mass distribution
98
Example three-surface aircraft report
Table A.5: Component masses
Engine1
Engine2
Front Stabiliser
Furnishing
Fuselage
Horizontal Stabiliser
Main Gear1
Main Gear2
Main Wing
Nose Gear
Vertical Stabiliser
APU
Air Conditioning
Anti Ice
Avionics
Electrical
Flight Controls
Fuel System
Handling Gear
Hydraulics
Instruments
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
kg
Loading according to mission requirements
Loading maximum fuel mass
4
6
1974
1974
457
830
6914
403
990
990
6523
317
375
1837
1166
119
766
395
242
79
18
1804
111
x 10
Fuel
5.5
5
Mass [kg]
Cargo
4.5
4
Pax
3.5
3
−0.2
−0.15
−0.1
−0.05
0
Xpos w.r.t. MAC/MAC [−]
0.05
Figure A.8: Loading diagram
0.1
0.15
A.6 Aerodynamics
99
10
CG at MTOM
CG at OEM
CG at ZFM
Xnp
8
6
Butt line, y [m]
4
2
0
−2
−4
−6
−8
−10
10
15
20
Fuselage station, x [m]
25
30
Figure A.9: CG location
Table A.6: Centre-of-gravity locations
Xcg (MTOM)
Xcg (OEM)
Xcg (ZFM)
Xnp
SM
A.6
16.4
16.2
16.3
16.9
15
m
m
m
m
%
Aerodynamics
Table A.7: Aerodynamic properties at cruise
CL,cruise
CD,cruise
L/Dcruise
CD0 (Clean)
CD0 (Take-Off)
CD0 (Landing)
Oswald factor (e) (Clean)
Oswald factor (e) (Take-Off)
Oswald factor (e) (Landing)
CLα
Cmα
CLmax,clean
CLmax,take-off
CLmax,landing
0.5
288
17.3
204
549
1049
0.799
0.849
0.899
5.42
-0.828
1.44
2.2
2.8
cts
cts
cts
cts
rad−1
rad−1
-
100
Example three-surface aircraft report
3
Clean cruise
Take−off, flaps & gear
Landing, flaps & gear
2.5
CL [−]
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
C [−]
D
Figure A.10: Drag Polars
20
Clean cruise
Take−off, flaps & gear
Landing, flaps & gear
18
16
14
L/D [−]
12
10
8
6
4
2
0
0
0.5
1
1.5
CL [−]
2
2.5
Figure A.11: Aerodynamic efficiency of the aircraft
A.7
Propulsion
Table A.8: Propulsion
Number of engines
SFCcruise
Bypass Ratio
Diameter
Length
2
0.575
6
1.6
3.13
h−1
m
m
3
A.8 Aircraft Geometry
A.8
101
Aircraft Geometry
Table A.9: Main Wing dimensions
Span
Planform area
MAC
Root Chord
Root t/c
Tip Chord
Tip t/c
Sections (root to tip)
Sweep 0.25c
Taper ratio
Twist
Dihedral
38.6
111
3.24
4.37
0.151
1.39
0.103
boeing-a, boeing-b, boeing-c
12.9
0.318
3.9e-15
-3.6
Table A.10: Horizontal Stabiliser dimensions
Span
Planform area
MAC
Root Chord
Root t/c
Tip Chord
Tip t/c
Sections (root to tip)
Sweep 0.25c
Taper ratio
Twist
Dihedral
9.44
17.65
2.03
2.76
0.118
0.979
0.118
N0012, N0012
14.4
0.355
0
-3.6
m
m2
m
m
m
◦
◦
◦
Table A.11: Front Stabiliser dimensions
Span
Planform area
MAC
Root Chord
Root t/c
Tip Chord
Tip t/c
Sections (root to tip)
Sweep 0.25c
Taper ratio
Twist
Dihedral
10.3
20.99
2.1
2.54
0.118
1.54
0.118
N0012, N0012
11.6
0.606
6.1e-15
1.8
m
m2
m
m
m
◦
◦
◦
m
m2
m
m
m
◦
◦
◦
102
Example three-surface aircraft report
Table A.12: Vertical Stabiliser dimensions
Span
Planform area
MAC
Root Chord
Root t/c
Tip Chord
Tip t/c
Sections (root to tip)
Sweep 0.25c
Taper ratio
Twist
Dihedral
4.37
18.93
4.42
5.06
0.118
3.6
0.118
N0012, N0012
19.4
0.713
0
0
m
m2
m
m
m
◦
◦
◦
Table A.13: Fuselage dimensions
WL, z [m]
Length
Floor Position
Diameter
Nose Fineness Ratio
Aft Fineness Ratio
Cabin Height
Nose Length
Aft Cutoff
Aft Ratio
36.4
-57
4.7
0.18
0.55
1.54
4.56
0.8
0.05
m
% of fuselage height
m
m
m
-
8
7
15
10
5
0
−5
−10
BL, y [m]
−15
22
20
18
16
FS, x [m]
Figure A.12: Fuel tank layout
A.8 Aircraft Geometry
103
WL, z [m]
7
6
5
4
2
0
−2
5
10
15
20
25
30
35
FS, x [m]
BL, y [m]
Figure A.13: Fuselage geometry; (blue = cargo ULDs, purple = floors)
104
Example three-surface aircraft report
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