multidisciplinary design optimization of a morphing
TRANSCRIPT
Multidisciplinary Design Optimization of a Morphing Wingtip
Concept with Multiple Morphing Stages at Cruise
by
Michael Leahy
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Mechanical and Industrial Engineering
University of Toronto
© Copyright by Michael Leahy 2013
ii
Multidisciplinary Design Optimization of a Morphing Wingtip
Concept with Multiple Morphing Stages at Cruise
Michael Leahy
Master of Applied Science
Mechanical and Industrial
University of Toronto
2013
Abstract
Morphing an aircraft wingtip can provide substantial performance improvement. Most civil
transport aircraft are optimized for range but for other flight conditions such as take-off and
climb they are used as constraints. These constraints could potentially reduce the performance of
an aircraft at cruise. By altering the shape of the wingtip, we can force the load distribution to
adapt to the required flight condition to improve performance. Using a Variable Geometry Truss
Mechanism (VGTM) concept to morph the wingtip of an aircraft with a Multidisciplinary Design
Optimization (MDO) framework, the current work will attempt to find an optimal wing and
wingtip shape to minimize fuel consumption for multiple morphing stages during cruise. This
optimization routine was conducted with a Particle Swarm Optimization (PSO) algorithm using
different fidelity tools to analyze the aerodynamic and structural disciplines.
iii
Acknowledgments
I would like to thank my supervisor Dr. Kamran Behdinan for his continuous help and guidance
throughout this work. I would also like to thank my parents and my friend Susan for their
encouragement.
I also appreciate the support from Dr. Fengfeng (Jeff) Xi from Ryerson University and the
financial support from Bombardier and the Natural Sciences and Engineering Research Council
of Canada is also greatly acknowledged.
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Table of Contents
Acknowledgments .......................................................................................................................... iii
Table of Contents ........................................................................................................................... iv
List of Tables ............................................................................................................................... viii
List of Figures ................................................................................................................................ xi
Nomenclature ................................................................................................................................ xv
List of Appendices ....................................................................................................................... xvi
Chapter 1 The Morphing Aircraft ................................................................................................... 1
1.1 Introduction ......................................................................................................................... 1
1.2 Defining a Morphing Aircraft ............................................................................................. 1
1.3 Current Morphing Aircraft Research .................................................................................. 2
1.3.1 Span Variation ........................................................................................................ 4
1.3.2 Sweep-Span ............................................................................................................. 6
1.3.3 Wing Twist .............................................................................................................. 8
1.4 Wingtip Design ................................................................................................................. 10
1.4.1 Induced Drag ......................................................................................................... 11
1.4.2 Current Morphing Wingtip Research .................................................................... 14
1.5 Thesis Motivation and Overview ...................................................................................... 15
Chapter 2 Variable Geometry Truss Mechanism .......................................................................... 17
2.1 Introduction ....................................................................................................................... 17
2.2 Kinematics ........................................................................................................................ 18
2.3 Modular Discretization ..................................................................................................... 20
2.4 VGTM Structural Layout .................................................................................................. 21
Chapter 3 Multidisciplinary Design Optimization Architectures and Problem Definition .......... 24
3.1 Introduction ....................................................................................................................... 24
v
3.1.1 Multi-Disciplinary Feasible .................................................................................. 25
3.1.2 Individual Discipline Feasible .............................................................................. 26
3.1.3 All-At-Once .......................................................................................................... 28
3.1.4 Additional MDO Architectures ............................................................................. 28
3.2 MDO Test Cases ............................................................................................................... 30
3.3 Objective Function ............................................................................................................ 32
3.4 Constraints ........................................................................................................................ 33
3.5 Design Variables ............................................................................................................... 35
Chapter 4 Particle Swarm Optimization Algorithm ...................................................................... 40
4.1 Introduction ....................................................................................................................... 40
4.2 Literature Review .............................................................................................................. 41
4.3 Social and Cognitive Parameter Variations ...................................................................... 42
4.3.1 Linear variation of acceleration constants ............................................................ 42
4.3.2 Step variation of acceleration constants ................................................................ 43
4.3.3 Sinusoidal variation of acceleration constants ...................................................... 44
4.4 Unconstrained Test Cases ................................................................................................. 45
4.5 Constrained Test Case Results .......................................................................................... 46
4.5.1 Problem Description [64] ...................................................................................... 47
4.5.2 Constrained Structural Optimization Results ........................................................ 49
Chapter 5 Discipline Computer Models ....................................................................................... 52
5.1 Introduction ....................................................................................................................... 52
5.2 Low Fidelity Models ......................................................................................................... 52
5.2.1 Aerodynamic Model ............................................................................................. 53
5.2.2 Structural Model ................................................................................................... 54
5.2.3 Aero-Structural Model .......................................................................................... 57
5.2.4 Fluid-Structure Interaction .................................................................................... 59
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5.3 High Fidelity Models ........................................................................................................ 61
5.3.1 Aerodynamic Model ............................................................................................. 61
5.3.2 Structure Model .................................................................................................... 63
5.3.3 Finite Element Mesh ............................................................................................. 65
5.3.4 VGTM Model ....................................................................................................... 66
5.3.5 Aero-Structural Analysis ...................................................................................... 67
Chapter 6 Multidisciplinary Design Optimization Results ........................................................... 69
6.1 Introduction ....................................................................................................................... 69
6.2 Low Fidelity Results ......................................................................................................... 69
6.2.1 Optimal Design for Two Morphing Stages ........................................................... 72
6.2.3 Optimal Design for Four Morphing Stages .......................................................... 76
6.2.4 Low Fidelity Summary ......................................................................................... 81
6.3 High Fidelity Results ........................................................................................................ 82
Chapter 7 Conclusion .................................................................................................................... 89
7.1 Thesis Conclusion ............................................................................................................. 89
7.2 Future Work ...................................................................................................................... 90
Bibliography ................................................................................................................................ 92
Appendix A – PSO Results ......................................................................................................... 100
A.1 Linear Test Case Results for Static and Dynamic .......................................................... 100
A.2 Step Test Case Results for Static and Dynamic .............................................................. 102
A.3 Sinusoidal Test Case Results for Static and Dynamic .................................................... 104
Appendix B – Beam Finite Element Model Validation .............................................................. 108
Appendix C – MDO Result Extras ............................................................................................. 111
C.1 Optimal Reference Wing Mass .................................................................................. 111
C.2 Two Morphing Stages .................................................................................................. 112
C.3 Four Morphing Stages ................................................................................................. 114
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C.4 High Fidelity Results .................................................................................................... 118
viii
List of Tables
Table 1.1 - Advantages and disadvantages of different morphing degrees of freedom.................. 2
Table 3.1 - Common MDO frameworks ....................................................................................... 28
Table 3.2 - Possible objective for each mission leg ...................................................................... 31
Table 3.3 - Design variable list for low fidelity models ............................................................... 35
Table 3.4 - Design variable list for high fidelity models .............................................................. 37
Table 4.1 - Unconstrained benchmark problems .......................................................................... 45
Table 4.2 - Load conditions for the 25-bar truss ........................................................................... 47
Table 4.3 – Test cases for 25-bar space truss ................................................................................ 49
Table 4.4 - Constrained test case results ....................................................................................... 50
Table 5.1 - Reference wing geometry ........................................................................................... 52
Table 5.2 - Comparing cross-sectional data from ANSYS to current FE model (X and Y Scaling
are set to 0.5) ................................................................................................................................. 56
Table 6.1 - Optimal design for wing structure of reference wing ................................................. 70
Table 6.2 - Performance results for the 2 Morphing stage case .................................................... 74
Table 6.3 - Performance results for 4 morphing stages ................................................................ 76
ix
Table 6.4 - Structural Optimization Results Comparison ............................................................. 82
Table 6.5 - First and second morphing stage performance results ............................................... 87
Table A.1 - Linear test case for unconstraint benchmark problems ........................................... 100
Table A.2 - Linear test case results for unconstraint benchmark problems ................................ 101
Table A.3 - Step test case for unconstraint benchmark problems ............................................... 102
Table A.4 - Step test case results for unconstraint benchmark problems ................................... 103
Table A.5 - Sinusoidal test case for unconstraint benchmark problems ..................................... 104
Table A.6 – Sinusoidal test case results for unconstraint benchmark problems ......................... 105
Table A.7 – Best overall social and cognitive distributions ....................................................... 107
Table B.1 - Material and geometric properties for problem 5.2 ................................................. 108
Table B.2 - Nodal displacement and nodal reaction comparison ............................................... 108
Table B.3 - Material and geometric properties for problem 5.8 ................................................. 109
Table B.4 - Nodal displacement comparisons ............................................................................ 110
Table C.1 - Optimal module design for 2 morphing stage case study ........................................ 112
Table C.2 – Optimal main wing and wingtip design for 2 morphing stage case study .............. 112
Table C.3 - Optimal module design for 4 morphing stage case study ........................................ 114
Table C.4 - Optimal main wing and wingtip design for 4 morphing stage case study ............... 115
x
Table C.5 - High fidelity first and second stage module results ................................................. 118
Table C.6 - High fidelity main wing and wingtip results ........................................................... 119
xi
List of Figures
Figure 1.1 - Airfoil section with pressure and shear forces .......................................................... 10
Figure 1.2 Escape of air around the wing tip [31] ........................................................................ 11
Figure 1.3 - Shed vortex sheet from trailing edge ........................................................................ 11
Figure 1.4 - Effect of downwash on the local flow over a local airfoil section of a finite wing [32]
....................................................................................................................................................... 12
Figure 1.5 - Different wingtip designs [31] .................................................................................. 13
Figure 1.6 - Boeing 767-400 with raked wingtip .......................................................................... 14
Figure 2.1 - VGTM like structure (left) and traditional truss structure (right) ............................. 17
Figure 2.2 - Base configuration of the modular VGTM morphing wing [45] .............................. 18
Figure 2.3 - Two coordinate systems arbitrarily orientated in 3D space ...................................... 19
Figure 2.4 - Discretized morphing wing used for the current work .............................................. 21
Figure 2.5 - Module structural layout ........................................................................................... 22
Figure 2.6 - Module degree of freedoms ...................................................................................... 23
Figure 3.1 - MDO structure for three disciplines .......................................................................... 24
Figure 3.2 - MDF architecture for aeroelastic analysis ................................................................. 25
Figure 3.3 - IDF architecture for aeroelastic analysis ................................................................... 27
Figure 3.4 - Aircraft mission showing mission legs ..................................................................... 31
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Figure 3.5 - Design space without constraint on objective (left) design space with constraint on
objective (right) ............................................................................................................................. 34
Figure 4.1 - Linear variation of particle acceleration constants .................................................... 43
Figure 4.2 - Step variation of particle acceleration constants ....................................................... 44
Figure 4.3 - Sinusoidal variation of particle acceleration constants ............................................. 45
Figure 4.4 - 25-bar space truss ...................................................................................................... 47
Figure 5.1 - Wingbox geometry confined within airfoil ............................................................... 55
Figure 5.2 - FE nodes (dots) with aerodynamic model ................................................................. 57
Figure 5.3 - Equivalent nodal loads for concentrated point loads ................................................ 58
Figure 5.4 - Deformed and undeformed wing shape .................................................................... 58
Figure 5.5 - Convergence plots for an elastic wing at constant angle of attack ........................... 60
Figure 5.6 - Convergence plots for an elastic wing while simultaneously being trimmed to
specified weight requirement ........................................................................................................ 61
Figure 5.7 - ICEM CFD unstructured surface mesh ..................................................................... 62
Figure 5.8 - Internal Structure layout ............................................................................................ 64
Figure 5.9 - Wing geometry without top skin panels .................................................................... 64
Figure 5.10 - FE mesh of wing ..................................................................................................... 66
Figure 5. 11 - FE mesh of VGTM structure .................................................................................. 67
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Figure 6.1 - Mass convergence plot for reference wing ............................................................... 70
Figure 6.2 - Stress distribution for reference wing ....................................................................... 71
Figure 6.3 - Bending stiffness distribution for reference wing ..................................................... 71
Figure 6.4 - Fuel convergence for 2 morphing stage case ............................................................ 72
Figure 6.5 - Morphed wingtip for the first stage ........................................................................... 73
Figure 6.6 - Morphed wingtip for the second stage ...................................................................... 74
Figure 6.7 – Stress constraint distribution for 2 morphing stages ................................................ 75
Figure 6.8 - Fuel convergence for 4 morphing stages .................................................................. 76
Figure 6.9 - Aerodynamic performance of the 4 morphing stages ............................................... 78
Figure 6.10 - First morphing stage wingtip shape for the 4 morphing stage case ........................ 79
Figure 6.11 - Second morphing stage wingtip shape for the 4 morphing stage case .................... 79
Figure 6.12 - Third morphing stage wingtip shape for the 4 morphing stage case ....................... 80
Figure 6.13 - Fourth morphing stage wingtip shape for the 4 morphing stage case ..................... 80
Figure 6.14 - Stress Constraint distribution for 4 morphing stage case ........................................ 81
Figure 6.15 - Von-Mises stress contours for optimal reference wing structure ............................ 83
Figure 6.16 – High fidelity first morphing stage optimal wing shape .......................................... 84
Figure 6.17 – High fidelity first morphing stage optimal wingtip shape ...................................... 84
Figure 6.18 - High fidelity second morphing stage optimal wingtip shape .................................. 85
Figure 6.19 - Pressure contours of wingtip region for first and second stage .............................. 86
Figure 6.20 - Von-Mises stress contours for first morphing stage ............................................... 87
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Figure 6.21 - Von-Mises stress contours for second morphing stage ........................................... 87
Figure B.1 - Problem 5.2 ............................................................................................................. 108
Figure B.2 - Problem 5.8 ............................................................................................................. 109
Figure C.1- Main wing and wing-module junction cross-sectional areas .................................. 111
Figure C.2 - Modules and wingtip cross-sectional area .............................................................. 111
Figure C.3 - First morphing stage for 2 morphing stage case ..................................................... 113
Figure C.4 - Second morphing stage for 2 morphing stage case ................................................ 114
Figure C.5 - First morphing stage for 4 morphing stage case ..................................................... 116
Figure C.6 – Second morphing stage for 4 morphing stage case ................................................ 117
Figure C.7 - Third morphing stage for 4 morphing stage case ................................................... 117
Figure C.8 - Fourth morphing stage for 4 morphing stage case ................................................. 118
Figure C.9 - Pressure distribution of first morphing stage wing showing shockwave at trailing
edge ............................................................................................................................................. 120
Figure C.10 - Pressure distribution of second morphing stage wing showing shockwave at
trailing edge ................................................................................................................................ 121
Figure C.11 - Comparing deformed to undeformed wing for the first morphing stage .............. 121
Figure C.12 - Comparing deformed and undeformed for second morphing stage ..................... 121
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Nomenclature
R Range ΔW Fuel Consumed
N Morphing Stage or Swarm Size σ Stress
W Weight σY Yield Stress
L Lift c1 Cognitive Acceleration Constant
D Drag c2 Social Acceleration Constant
V Free Stream Velocity A Cross-Sectional Area
CL Lift Coefficient M Mach Number or Bending
Moment
CD Drag Coefficient Ix Out-Of-Plane Second Area
Moment of Inertia
CL/CD Lift to Drag Ratio Iy In-Plane Second Area Moment
of Inertia
Ri,i-1 Rotation Matrix Γ Dihedral/Cant Angle
Λ Sweep Angle θ Twist Angle
b span t Airfoil thickness
U,M,L Upper, Middle, Lower Group
xvi
List of Appendices
Appendix A – PSO Results ………………………………………………………….100
Appendix B – Beam Finite Element Model Validation ……………………..……....108
Appendix C – MDO Result Extras ………………………………………………..…111
1
Chapter 1
The Morphing Aircraft
1.1 Introduction
Morphing vehicles are classified as structures exploiting actuation mechanisms to achieve
substantial wing area change resulting in an improved flight envelope. The concept behind
morphing vehicles stems from the desire to expand the mission capabilities of aircraft platforms.
Radical wing area changes enable a single aircraft to perform the roll of multiple aircraft. The
recent advances in new materials and structures are allowing engineers to design aircraft that can
meet specific environment requirements. Much of these technologies and applications allow
large changes in shape to maximize performance and efficiency of aircraft. The problems with
large shape changes for large aircraft are the actuation power requirements and structural stress
constraints. To address this problem, shape changing near the wingtip, where the loading is the
lowest is preferred. With rising fuel costs, aircraft designers are motivated to improve the fuel
efficiency of existing aircraft. By morphing the wingtip region we can redistribute the
aerodynamic loads thus improving vehicle performance. Using a Variable Geometry Truss
Mechanism (VGTM) concept the shape the wingtip, the question is what is the optimal shape
subjected to structural and performance constraints? Using a Particle Swarm Optimization (PSO)
algorithm with a Multidisciplinary Design Optimization (MDO) framework an optimal wing and
wingtip shape will be determined and compared to a conventional wing.
1.2 Defining a Morphing Aircraft
The evolution of birds has inspired humans to fly for centuries. The idea of “Morphing Aircraft"
is a direct result of this inspiration. According to [1] avian morphology permits a wide range of
wing configurations that can be used in a variety of flight conditions which are used for a
particular flight task. A bird will loiter with a high aspect ratio wing until it detects its prey, upon
detection it will retract its wings to reduce wing area and dive towards its prey. Morphing wings
have the potential to improve aircraft performance, however there is no agreed upon definition of
"Morphing Aircraft".
2
In the mid-1990s NASA initialed the "aircraft morphing" project. This project defined morphing
as "efficient multi-point adaptability". Efficient implied lighter and/or more energy efficient
conventional systems, multi-point implied the accommodation of different flight requirements,
and adaptability implied increased versatility [2]. This is a general definition that includes small
and large shape changes, and virtual shapes through micro flow control. The Defense Advanced
Research Projects Agency (DARPA) initiated the "Morphing Aircraft Structures" program in
2002. In summary the goal of the program was to develop aircraft that can change their shape
substantially, provide superior system capability, and uses designs that integrates innovative
technology [3]. This definition applies mostly to Unmanned Aerial Vehicles (UAV), since large
shape changes would be difficult for larger transport aircraft. In [4] the authors defined a
morphing wing as a bird like wing that has the ability to adapt to accommodate multiple flight
regimes or to obtain better flight performance. This definition implies seamless large shape
changes to achieve its goal. In [5], morphing aircraft were defined as a set of technologies that
increase a vehicles performance by manipulating certain characteristics to better match the
vehicle state to the environment and task at hand. This problem with this definition is it allows
existing technologies such as flaps and retractable landing gear to be considered morphing. The
NATO Research and Technology Organization, Applied Vehicle Technology Technical Team on
Morphing Vehicles developed the following definition: real-time adaptation to enable multi-
point optimized performance [2]. This definition points towards the desired capability for
morphing vehicles that is distinct from conventional vehicles.
1.3 Current Morphing Aircraft Research
There have been several approaches to morphing an aircraft wing structure but the objective of
the morphing process has been the same, which is to improve the aerodynamic performance by
morphing the wing span, sweep, and twist. Other morphing options include the dihedral/cant or
the chord length of the wing. Each of these morphing degrees of freedom has their own
particular influence on aerodynamic performance which is summarized in the table below.
Table 1.1 - Advantages and disadvantages of different morphing degrees of freedom
Morphing
Degree of
Freedom
Description Advantages Disadvantages
3
Span
Increased Range,
Endurance
Decreased fuel
consumption
Increased
bending
stress at
wing root
Increased
structural
weight
Slower roll
control
authority
Sweep
Reduced takeoff
length
Increased
maneuverability at
low speeds
Increased
weight
from
sweeping
mechanis
m
Aircraft
longitudina
l stability
will be
affected
Twist
Increased roll
control (reduces
the chances of
stall at the
wingtip)
Increase in
drag is a
possibility
4
Dihedral
Improved lateral
stability
Lower
maneuvera
bility
Chord/Airfoil
Potential increase
in lift
Potential
increase in
drag
This report is primarily concerned with the span, sweep, twist, and dihedral morphing degrees of
freedom. Chord (or airfoil morphing) is a subject of detailed research in academia but it is not the
main focus here. A literature review is presented in the proceeding sections.
1.3.1 Span Variation
Increasing the wing span will improve the aerodynamic efficiency by decreasing the induced
drag. This translates into increased range and endurance by improved fuel efficiency. However
aircraft with large wing spans have poor maneuverability (increases roll damping). This creates a
very low roll rate, which would be unattractive to high performance aircraft designers. The other
problem with an increased span is the increase in root bending moment which creates larger
stresses.
The concept of a variable span is not a new one. In 1931 Ivan Makonine developed a telescoping
span aircraft. An inner wing section was extended from the outer wing section using pneumatic
actuation to increase wing span and area. The span increased by 62% and area by 57% [5]. In [6]
they created a telescopic wing, which could increase wing span by 10 inches. They used a radio
controlled aircraft, and an increase of 19% in range was achieved compared to the baseline wing.
A change in static stability of about 5% was also determined. In [7] and [8], they developed a
morphing wing concept that used a pressurized telescopic spar to create large spanwise changes.
5
Their telescopic spar design consisted of three concentric circular tubes that can support the
aerodynamic loading. Their initial design in 2003 suffered from parasitic drag increase due to the
seams on the wing which resulted in a 25% reduction in lift to drag ratio compared to its fixed
wing counterpart. In 2007 they improved their design by adding a second telescopic spar. Their
design was able to achieve a 230% increase in aspect ratio while supporting aerodynamic loads.
The spar was tested under different loading conditions and different air pressure for the actuator.
The response of the wingtip in the longitudinal direction was measured during extension and
retraction. High pressurization was able to achieve the maximum span in the quickest time. Their
Wind tunnel test results were compared to theoretical results and a solid wing with the same
dimensions at 40%and 100% span extension. For each span length, the morphing wing
underperformed the solid wing. The authors mentioned that this was most likely caused by lower
skin stiffness and the seams on the wing. Their design was able to achieve lift to drag ratios as
high as 16. In [9] they investigated the aerodynamic and aeroelastic characteristics of a variable
span morphing wing applied to a long range cruise missile. A finite element model consisting of
four wingbox sections for the main wing and two wingbox sections for the telescoping wing was
used. Their results showed a reduction in spanwise lift and induced drag but increased wing
deflections with increasing span. The wing divergent natural frequencies were also determined
and showed that the natural frequencies would decrease with increasing span. In [10] they tested
a morphing telescopic wing in a channel studying its aerodynamic and roll effects for wing-in-
ground effects. The change in aspect ratio increased the lift to drag ratio compared to only the
ground and side wall effects. It was found that the extension of the wing tip was inefficient at
controlling the rolling moment; however the aspect ratio change was small. In [11] they used
asymmetric span extension to determine roll performance. They were able to achieve a roll
moment coefficient of 1.5. They extended their model in [12] to include the effects of morphing
into the aircraft dynamics model and it was shown that span extension was effective for roll
control. Span extension induces roll damping that is greater than aileron control. In [13] they
invested asymmetric span extension for roll control of a cruise missile. They showed that the
control of the missile can be larger compared to a conventional tail surface. They presented a
nonlinear control model to control the roll, angle of attack, and side slip. In [14] they developed a
morphing wing for a UAV with span extension. It was built and tested in a wind tunnel and their
design could increase the wing area by 100%. A silicone elastomer skin which had embedded
unidirectional carbon fibre to minimize poission's ratio effects and increase chord wise in-plane
6
stiffness. A morphing core substructure with zero poission ratio was also used. The honeycomb
substructure has low stiffness in the direction of actuation allowing for large displacements.
Wind tunnel tests at two different speeds at three different angles of attack were conducted to
determine the amount of out of plane skin deformation which was only 0.5mm.
Lockheed Martin developed a folding wing approach that incorporated flexible skins along the
fold line. Their concept has two main configurations: a loiter configuration where the wing span
is extended by unfolding the wing and a dash configuration where the wing is folded up against
the fuselage to reduce the wing span. Testing had shown that the model can with stand the
aerodynamic loads at each configuration [15] and [16].
1.3.2 Sweep-Span
For high speed aircraft, their wings are designed with a sweep angle to delay the drag rise at high
Mach numbers. However at low speeds maneuverability is poor and longer take-off distance is
required. Varying the sweep reduces the take-off and landing distance and improves low sweep
performance by decreasing the sweep angle. Changing the wing sweep also moves the centre of
gravity and aerodynamic centre, thus effecting longitudinal stability. The sweep angle can also
affect the lateral stability by creating an effective dihedral. Also, a variable sweep wing will
change its span length. The application of variable sweep wing has been in use for several
decades. The Northrop Grumman F-14 Tomcat has the ability to change its sweep angle from 20
to 68 degrees. The B-1 Lancer bomber can also change sweep for high speed cruise. Other swept
wing concepts have been designed and tested in the Soviet Union. One of the main issues for
these swept wing designs is the heavy weight penalty associated with the sweeping mechanism.
In [17] they performed an experimental analysis of a variable geometry aircraft that can change
both sweep and span. The goal of the wind tunnel test was to determine the influence of platform
variables on the models aerodynamic characteristics. As the wing sweep was increased the
aerodynamic centre shifted more towards the tail section of the model, however since the wing is
being swept back the centre of gravity is also shifting towards the tail section. The aircraft is
more stable since there is an increase in the static margin. At high lift coefficients the induced
drag is dominant over profile drag, a full span extension and full sweep showed better drag
performance at lift coefficients greater than 0.4 compared to an unmorphed baseline. However at
lower lift coefficients the baseline wing performed better. In [18] they presented a bird like
7
design with an inner and outer section that can increase its aspect ratio from 4.7 to 8.5. Using a
lightweight piezo-composite actuator their design could change camber as well. They were able
to achieve a significant increase in lift coefficient for an increasing aspect ratio and a reduction in
drag at different angles of attack.
NextGen Aeronautics developed a UAV referred to as the ‘batwing’, which can change its sweep
and span [19] and [20]. The wing can change its aspect ratio by 200%, it span by 40% and wing
area by 70%. Wing sweep and area are controlled through four bar linkages, revolute, and slider
mechanisms. Their morphing UAV successfully completed flight tests in 2006. Another larger
morphing UAV was built and tested in 2007. This design was capable of 40% change in wing
area, 177% change in aspect ratio, and 73% change in span.
In [21] they performed an optimization problem based on the static equilibrium and principle of
virtual work to determine the optimal location of an actuator in a scissor mechanism.
Optimizations were conducted for single and multiple cell mechanism. They used an Sequential
Quadratic Programming (SQP) algorithm for their optimization with the location of the actuator
as the design variable. The wing skin was modeled using linear spring elements. Optimizations
were conducted with different link thicknesses. Their results showed that a distributed actuation
system was more efficient with a flexible mechanism. This work was extended in [22]. A
nonlinear finite element model was used to determine the nodal displacements in a scissor like
mechanism in a two stage optimization routine. The first stage used a GA to determine the
existence of an actuator. The actuators were either active or in active in the cell using a discreet
variable. Constraints were imposed on the number of active actuators and the extension of the
mechanism was restricted. The second stage optimization used an SQP approach to determine
the position of the actuator within each cell. For both stages the objective was to maximize the
efficiency of the mechanism defined as the ration of output energy to input energy. The skin was
modeled by linear springs connecting opposite nodes. a high and low aerodynamic load case
were examined with skin representing isotropic and orthotropic stiffness. Their results showed
that the efficiency is greatly dependent on the skin stiffness and applied external loading. Also
the morphing efficiency under high loads were better than lower loads. In [23] they presented a
methodology that employed planar unit cells as the layout as the internal structure. The sizing of
this mechanism was based on simple beam theory to determine the bending moments assuming a
8
linearly distributed aerodynamic load. A prototype was designed and built using a cable
actuation mechanism. This mechanism can change the span and sweep angle. In [24] they
extended the NextGen concept by determining the optimal actuator orientation for rigid and
flexible structures.
1.3.3 Wing Twist
Creating a wash-in or wash-out of the wing can significantly alter the span wise loading of the
aerodynamic loads. This can improve aircraft performance and control authority. The Wright
brothers used a wing warping approach to vary the twist of their wing for roll control. Increased
aircraft speed forced designers to build stiffer wings to prevent wing divergence, aileron reversal,
and flutter. This came at with a significant weight penalty.
Taking advantage of wing flexibility to reduce structural weight was the goal of Rockwell
International Active Flexible Wing program in the 1980s. The objective was to use the control
forces to twist the wing structure to enhance roll performance. Other programs such as the Active
Aeroelastic Wing program funded by NASA and the USAF, and the Active Aeroelastic Aircraft
Structures in Europe explored other approaches to active aeroelastic concepts for roll control.
In [25] they proposed a Variable Stiffness Spar (VSS) concept to improve the aircraft roll
performance. The mechanism consists of a segmented spar having articulated joints at the
connections with wing ribs and an electric actuator capable of rotating the spar. With the spar in
the horizontal position there is very little bending stiffness compared to the vertical position.
Their concept allows the stiffness to be controlled as a function of flight conditions. An
optimization was performed at different transonic Mach numbers to maximize the roll rate
subjected to flutter, control surface hinge moment, and maximum deflection constraints. Two
VSS concepts were tested: for the first concept a single VSS was used and for the second, a
torsion-free concept where two stiff spars positioned close together carried most of the bending
moment with two VSS (one VSS near the leading edge and the other near the trailing edge). The
torsion-free concept allows for more control over the torsion stiffness. The single VSS was able
to achieve a 6-22% increase in roll rate and the second approach was able to achieve a 29-126%
improvement. In [26] they performed wind tunnel tests on a wing with multiple control surfaces
and VSS mechanism. The VSS was located at 60% of the chord extending to 58% of the wing
span. The effects of the VSS on modal frequencies, lift curve lope, and control surface
9
effectiveness were determined. Their predicted modal frequencies were lower than the measured
frequencies suggesting that the wind tunnel model was stiffer than the FE model. The horizontal
orientation of the VSS shows a decrease in lift curve slope compared to the vertical position
however the difference between them is small at subsonic speeds compared to transonic and
supersonic. Similarly the differences in control surface derivatives for the horizontal and vertical
position were only apparent at subsonic speeds. In [27] they used a Shape Memory Alloy (SMA)
spar approach for the enhancement of aircraft roll performance has been presented to replace the
VSS concept which was mechanical actuation system with a large weight penalty. It was
observed that the baseline aircraft has about 65 % of loss in the roll performance due to wing
flexibility at Mach number 0.95. With activation of SMA spar located at the leading edge, a 59
% roll rate improvement was achieved. And 61% roll rate improvement was achieved by using a
SMA spar located at the trailing edge.
In [28] they introduced the Flexspar configuration which employs an aerodynamic shell that can
pivot around a high strength main spar. A piezoelectric actuator is mounted inside the
aerodynamic shell with one end bonded to the spar and the other to the shell. As the piezoelectric
actuator is energized there is a change in pitch angle. Wind tunnel tests showed deflections of +/-
11o with collocated pitch axis, aerodynamic centre, and centre of gravity. With the quarter chord
moved ahead of the pitch axis the deflections are magnified to +/-16o. In [29] they built a series
of piezoelectric specimens to demonstrate the utility of the new post-buckled pre-compressed
(PBP) actuator scheme. These actuator elements were designed as actuators for grid fins for a
new class of subscale UAVs (vertical take-off and landing capable). An extensive series of tests
with varying levels of axial pre-compression showed that static deflections could be controllably
magnified approximately 4.5-fold over conventional levels, with excellent agreement between
theory and experiment. In [30] they presented a new class of flight control actuators integrated
into a flexible wing, allowing it to be deformed upon actuation. By using PBP actuators their aim
was to reduce weight, complexity and power consumption with respect to conventional actuators,
while increasing control bandwidth and control authority. By applying axial compression to
piezoelectric bimorph bender actuators, significantly higher deflections can be achieved than for
conventional piezoelectric bender actuators. It was shown during flight with a 1.4m span UAV
that wing morphing could produce 38% more roll control and 3.7 times greater control
derivatives than conventional approaches. Because PBP actuators are solid state and do not
10
employ any linkages, pushrods or gears, they operate very efficiently This efficiency lead to a
99.6% decrease in power consumption, 87% reduction in flight control system-related weight, an
order of magnitude increase in control actuation bandwidth, and an order of magnitude fewer
parts than conventional electromechanical servoactuators [30].
1.4 Wingtip Design
The intended purpose of all wingtip devices is to reduce the overall drag of the aircraft. Drag is
just the flight-direction component of the total aerodynamic force. This aerodynamic force can
be separated into a component parallel to the local surface (shear force) and a component
perpendicular to the surface (pressure force).
Figure 1.1 - Airfoil section with pressure and shear forces
When these two components are integrated over the entire wing surface, the resulting forces are
referred to as the "skin-friction" drag and the pressure drag. The skin-friction drag is a result of
viscous effects in the boundary layers on the airplane’s surfaces. The pressure drag is a result of
a more complicated combination of flow mechanisms, including viscous effects, shocks, and the
global effects of lift. Induced drag is a significant part of the drag due to the global effects of lift
and it will be discussed in more detail.
11
1.4.1 Induced Drag
When a wing generates aerodynamic lift, the air on the top surface has lower pressure relative to
the bottom surface. The air will ‘escape’ from below the wing and out around the tip to the top of
the wing in a circular fashion. This generates a tip vortex shown in Figure 1.2.
Figure 1.2 - Escape of air around the wing tip [31]
In actuality the vortices that feeds into the vortex cores generally comes from the entire span of
the trailing edge, not just from the wingtips. The shed vortices from the trailing edge roll up near
the outer edges of the wing creating two distinct vortex cores. This is show in Figure 1.3.
Figure 1.3 - Shed vortex sheet from trailing edge
12
The rotary motion of the air within the shed wing vortices reduces the effective angle of attack of
the air on the wing. This is best illustrated with the 2D airfoil in Figure 1.4. For a 2D airfoil the
lift and drag forces are taken relative to the global frame of reference. The airfoil is rotated with
a geometric angle of attack (α) relative to the direction of travel of the wing velocity (V). Since
the aircraft wing is finite there is an induced angle of attack (αi) caused by the downwash effect
(w) of the shed vortices. The air is deflected by the downwash which causes the total lift vector
to tilt back. The aft component of this lift vector is the induced drag (Di). The magnitude of the
induced drag is determined by the spanwise distribution of vortices shed downstream of the wing
trailing edge, which is related in turn to the spanwise lift distribution.
Figure 1.4 - Effect of downwash on the local flow over a local airfoil section of a finite wing
[32]
For high-lift low-speed conditions such as during takeoff or for high subsonic cruise speed, the
induced drag from the vortices can account for up to half of all drag [33]. To reduce the induced
drag effect several wingtip concepts have been implemented as shown in Figure 1.5. The first
studies on wingtip devices were conducted in 1897 by English engineer Frederick W.
Lanchester. His design called for endplates to be attached to the wingtip, and this design showed
a decrease in induced drag at high lift coefficient. However during cruise the airflow would
13
separate near the tip increasing the profile drag of the wing [33]. The most common wingtip
device is the Hoerner wingtip [34] due to its relative ease of construction, however it can only
achieve modest improvements in the lift to drag ratio. For commercial transports and business
jets, the winglet is the preferred wingtip device. First developed by Richard T. Whitcomb in the
1970’s, his design was specifically intended for lifting and subsonic Mach numbers. Wind tunnel
tests at Langley showed a reduction in induced drag of almost 20 percent and an increase in wing
lift-drag ratio of roughly 9 percent. This was twice the performance enhancement given by
simple tip extensions [35]. The problem with winglets is the increase in bending moment at the
wing mount attached to the fuselage. This increase in bending moment is attributed to the
increase in wing loading and added weight at the wingtip. The wings flutter speed can also be
adversely affected. Also due to the higher aerodynamic loading at the wingtip, roll performance
can be greatly affected.
Figure 1.5 - Different wingtip designs [31]
Another wingtip device used mainly by Boeing is the raked wingtip shown in Figure 1.6. They
are described as an integrated wingtip extension, as they are horizontal additions to the existing
wing, rather than the previously described vertical solutions. The main difference between a
traditional winglet and a raked wingtip is the amount of cant. They are another wingtip
technology to increase efficiency, to improve climb performance, and to shorten takeoff distance,
all by increasing the effective aspect ratio of the wing by disrupting the detrimental wingtip
14
vortices. Such technologies and retrofits promise between 6 and 12 percent increases in range
[36]. In testing by Boeing and NASA, raked wingtips have been shown to reduce drag by as
much as 5.5%, as opposed to improvements of 3.5% to 4.5% from conventional winglets [37].
Figure 1.6 - Boeing 767-400 with raked wingtip
The benefit of induced drag reduction is an increase in range, fuel savings, increased payload
capacity, and reduced takeoff length. However such improvements only occur at the design lift
coefficient that the winglet has been optimized for. Morphing technology presents the possibility
of actively reconfiguring the outer wing system geometry for the given flight condition to
provide maximum benefit throughout the entire aircraft operational envelope.
1.4.2 Current Morphing Wingtip Research
The concept of wing with a large area change is suitable for small aircraft since the aerodynamic
loading requirements is relatively small. Business jets and commercial transport aircraft
however, require very high aerodynamic loads to achieve flight and with these high loads come
large structural stresses and aeroelastic instabilities. To achieve large shape changes for large
aircraft would require significant actuator power and will increase the structural weight.
In [38], [39], [40] and [41] they investigated the use of variable cant wingtips for morphing
aircraft control. Their concept consisted of a pair of winglets with adjustable cant angle,
independently actuated and mounted at the tips of a baseline flying wing. Computations with a
15
vortex lattice model and wind-tunnel tests demonstrate the viability of the concept, with
individual and/or dual winglet deflection producing multi-axis coupled control moments.
Comparisons between the experimental and computational results showed reasonable to good
agreement. However they concluded that a single pair of adjustable winglets cannot substitute for
all the conventional control surfaces at the same time if one wants a full control envelope.
In [42] they proposed the Morphing Winglet (MORPHLET) concept. The objective of the
MORPHLET project was to investigate the use of adaptive materials and structures technology
to dynamically tailor the external shape of the wingtip devices in order to improve multi-phase
mission performance, maneuverability and integrated economics. The twist, cant, and span of
four modules at the wing tip were adjusted to maximize the wings performance at three different
points in the aircrafts mission: start of cruise, start of final cruise and start of descent. A
significant increase in the specific air range was achieved at the cost of a heavier wing structure.
The aerodynamic analysis was performed using a low fidelity vortex lattice approach. In [43]
they performed a multidisciplinary design optimization to the problem of maximizing the
performance benefit through the retrofit of a morphing non-planar outer wing system to a
commercial narrowbody aircraft. Their work involved a two-level combined aero-structural-
control-performance optimization that requires the coupling of aerodynamic and structural
design modules. However their analysis was performed using low fidelity tools which are
sufficient for conceptual design only. They were able to achieve a substantial improvement in
specific air range of 4-6%.
1.5 Thesis Motivation and Overview
Researchers have been inspired by the morphing capabilities of insects and birds that improve
their performance at a variety of flight conditions. Using advanced adaptive structural design and
smart materials, imitating these features for aircraft performance improvement is becoming a
reality. Much of the work being done has been applied to UAVs for military applications. Areas
of particular interest for morphing technology are the wingtips of civil aircraft.
Modern commercial aircraft are optimized only for a narrow range of flight conditions. As a
consequence, off-design conditions results in poor performance. Most civil transport aircraft are
optimized for range but for other flight conditions such as take-off and climb they are used as
16
constraints. These constraints could potentially reduce the performance of an aircraft at cruise.
By morphing the wingtip to different shapes, we can alter the load distribution and adapt the
wing performance to the required flight condition.
Most of the work done with the current morphing design from [44], [45] and [46] have focused
largely on either aerodynamics or module kinematics or structural layout. The intent of the
current work is to analyze the combined aerodynamic and structural analyses in a
Multidisciplinary Optimization (MDO) routine. This MDO routine will use either low fidelity or
high fidelity computer models to determine a superior design compared to a baseline aircraft
wing. There are three main objectives:
1. Develop and test an optimization algorithm
2. Integrate aerodynamic and structural discipline codes.
3. Determine the optimal wingtip shape for different MDO architectures
This thesis is organized as follows: the second chapter will describe the VGTM concept to be
used as the morphing mechanism, the third chapter will describe the MDO architecture, the
fourth chapter will describe the PSO algorithm and it will test is performance against several
benchmark functions, the fifth chapter will describe the discipline models used for the low and
high fidelity models, the sixth chapter will present the results of the MDO analysis, and the
seventh chapter will be the conclusion.
17
Chapter 2
Variable Geometry Truss Mechanism
2.1 Introduction
The current work does not focus on the design of a morphing mechanism; however it is
important to have an understanding of how the mechanism works. The mechanism that has been
adopted for the current work has been developed at Ryerson University, [44] and [45]. Their
concept follows a Variable Geometry Truss Mechanism (VGTM), which are truss structures that
use active and passive prismatic actuators instead of rigid links to achieve kinematic motion.
This is shown in the figure below.
Figure 2.1 - VGTM like structure (left) and traditional truss structure (right)
VGTM structures are capable of large, rigid displacements and possess high stiffness to mass
ratios [45]. Unlike a traditional VGTM, any truss mechanism used for wing morphing is subject
to considerable size constraints. As wings are narrow at the wingtip, a VGTM would either have
very limited motion, or disrupt the integrity of the wing surface, both of which are undesirable
[44].
18
2.2 Kinematics
VGTMs can provide the morphing wingtip with kinematic capabilities and can also support
aerodynamic and structural loading as they are sufficiently rigid. The spar and stringer
arrangement of conventional wingtips is replaced in favor of VGTMs that kinematically connect
airfoil ribs. The truss architecture shown in Figure 2.2 below consists of seven active and passive
kinematic branches.
Figure 2.2 - Base configuration of the modular VGTM morphing wing [45]
The four active branches relate to a specific morphing degree of freedom; qΓ, qb, qΛ, and qα
correspond to the branch actuators that control module cant, span, sweep, and twist, respectively
[45]. That is, the VGTM modules are capable of individual or simultaneous cant, span, sweep,
and twist morphing. The modules also maintain the flexibility to adapt to changes in morphing
requirements through reconfiguration of the base module.
The position and orientation of the module platform relative to the global coordinate system can
be described by:
[ ] (2.1)
19
The location of the reference point of the ith
module platform, denoted by , relative to the
global coordinates system, is the summation of the ith module base and the position of the
reference point of the module platform relative to the module base . Before summation,
must be converted to the global reference using the rotation matrix [ ]. The two subscripts
represent the rotation between two coordinate systems attached to two adjacent bodies. For this
case ‘o’ represents the global coordinate system and ‘i’ is the ith module orientation. The
orientation of a reference point relative to the global orientation is the product of the rotation
matrices up to the current coordinate system ‘n’.
∏ ( )
(2.2)
For example if there were two coordinate systems (n=2), shown in Figure 2.3 below, we can
calculate the position of the module platform’s reference point using (2.1).
[ ][ ] (2.3)
Figure 2.3 - Two coordinate systems arbitrarily orientated in 3D space
The rotation matrix uses the desired orientation of the module platforms relative to the base
module as input. These orientations are the dihedral (Γ), twist (θ), and sweep (Λ):
20
( ) [
]
( ) [
]
( ) [
]
(2.4)
The product of these three matrices will give the final rotation matrix:
( ) ( ) ( ) (2.5)
2.3 Modular Discretization
The VGTM can be applied in a modular way to meet the kinematic requirements of the
morphing system. Modular discretization has been examined by [45]. Their objective was to
determine the number of morphing wing modules and the respective spacing required to emulate
a known wing shape and satisfy a corresponding flight requirement. The curvature and twist
distribution from the reference wing shape quarter chord line were extracted and used to
determine the spacing of the discretized wing modules. Using ANSYS © Fluent ™, each
modular morphing wing configuration was evaluated for its lift and drag performance. At a
constant angle of attack and incompressible flow, their results showed maximum lift to drag ratio
with seven modules. With additional modules, there was a decline in the lift to drag ratios caused
by adverse flow conditions in regions near the module tips due to the discontinuity between two
adjacent module surfaces [45]. This work was extended to find an optimal wingtip shape for
climb, cruise, and descent flight conditions by [46]. Using the number of modules and the sweep,
span, dihedral, and twist for each module, a Harmony Search algorithm was used to find an
optimal shape for each flight condition (cruise, climb, and descent). A figure of the discretized
wing is shown below used for this work. For the current work the wing will have three main
21
components: the main wing, the modules, and the wingtip. The main wing and wingtip are
inactive sections (i.e. no morphing) and modules which are active.
Figure 2.4 - Discretized morphing wing used for the current work
2.4 VGTM Structural Layout
In a conventional wing of a commercial or business aircraft, the combination of spars and skins
typically provides the necessary torsional stiffness, whereas the combination of spars and the
skin-stringers provide the necessary bending stiffness. By replacing the primary structural
elements with a series of linear actuators and passive members that provide only axial stiffness;
it will require a locking and unlocking at an appropriate sequence to achieve the dual purpose of
a structure and mechanism [47]. A simplified representation of the morphing mechanism is
shown below.
22
Figure 2.5 - Module structural layout
The passive members exist to provide structural stiffness and the active members provide the
necessary kinematic motion. Figure 2.6 below shows the mechanism shaping the platform airfoil
into different motions.
23
Figure 2.6 - Module degree of freedoms
There are three main design issues using conventional VGTMs as an internal mechanism for
morphing wings:
1. The primary structural elements are replaced by actuators which will add weight to the
structure
2. Lack of stiffness due to the geometric restrictions at the wingtip
3. Lack of fault tolerance
To address the second issue, actuators with larger diameters could provide the extra stiffness, but
this of course will add more weight. The third issue could be addressed by adding redundant
actuation but this will again increase weight and it will increase the complexity of the design.
24
Chapter 3
Multidisciplinary Design Optimization Architectures and
Problem Definition
3.1 Introduction
Multidisciplinary Design Optimization, MDO, "can be described as a methodology for design of
complex engineering systems that are governed by mutually interacting physical phenomena and
made up of distinct interacting subsystems" [48]. In other words, MDO utilizes optimization
techniques to solve problems that feature various disciplines as shown in Figure 3.1. These
discipline platforms are incorporated simultaneously by passing information between them and
an optimal solution is provided. Since the interaction between disciplines is taken into
consideration, the multidisciplinary solution is superior when compared to the individual
optimization of each discipline. The complexity of engineering problems has sparked a growing
interest in multidisciplinary optimization.
Figure 3.1 - MDO structure for three disciplines
Multidisciplinary Analysis (MDA) is a process by which equilibrium is established by
simultaneously solving state equations of disciplinary analysis. For example aerodynamics and
structural performance (stress, deflection …etc.) are both highly coupled. The deformation of the
Discipline 1
Discipline 2
Discipline 3
OPTIMIZER
25
wing structure is dependent on the aerodynamic loading which is in turn dependent on the wings
deformation. This makes the MDA an iterative process. Fixed-point iteration is the easiest to
implement but convergence is dependent on the initial guess at the beginning. Newton iteration
is another approach but it can be computationally expensive with the determination of the
sensitivity derivatives.
3.1.1 Multi-Disciplinary Feasible
Multi-Disciplinary Feasible (MDF) method is the most common way of solving an MDO
problem and it is the easiest to implement. The MDF method works to achieve multidisciplinary
feasibility at each iteration of the optimization. Multidisciplinary feasibility essentially means
that at each iteration all of the design variables satisfy each discipline and there are no conflicting
variables. The MDF method links a single optimizer with a MDA. The optimizer is responsible
for selecting both global and local design variables for the MDA. Achieving a multidisciplinary
feasible design essentially provides the coupling variables to the system. Once the feasibility is
achieved all design variables are known and the analysis calculates the values of the objective
function and the constraints for use in the optimizer. A figure of the MDF architecture is shown
below:
Figure 3.2 - MDF architecture for aeroelastic analysis
26
For the above architecture we are using an aeroelastic analysis of an aircraft wing as an example.
Some of the notations have been adopted from [49]. The purpose for MDO is to optimize an
objective function ‘f’ that is dependent on the global and local design variables (XD) which are
the input to the MDA. The objective function is also dependent on the aerodynamic analysis
A(XD) and structural analysis S(XD). The design variables are passed to the disciplines to execute
a MDA, the aerodynamic pressure forces from the wing are turned into structural loads by a
mapping process (MAS), and after the structural analysis the displacements are used to reshape
the aerodynamic grid using another mapping process (MSA). The first output of the MDA is the
convergence of the aerodynamic discipline (A) which is dependent on the design variables XD
and the coupling between the aerodynamic and structure (GAS) discipline which are also
dependent on the design variables and the structural performance. Likewise the second output is
the convergence of the structure (S) which is dependent on the structure and aerodynamic
coupling (GSA). Constraints are imposed on each discipline (CD) such as the limits on the design
variables and maximum allowable stress of the wing structure.
A primary advantage of the MDF architecture is that, at each iteration, multidisciplinary
feasibility is achieved by only treating global and local design variables as optimization
variables, and leaving the coupling variables to be solved by the MDA. Another advantage of
this method is that it is easy to implement and is suitable for small to medium sized engineering
problems. However there are several disadvantages that need to be mentioned. The main
disadvantage of the MDF method is that it is computationally expensive. This is because a MDA
must be performed for every iteration of the optimization and for large engineering problem this
can add a significant amount of computational time. Finding a feasible point for the MDA might
be difficult as well. Depending on the solution method for the MDA a feasible point might take
several iterations or it may not converge at all.
3.1.2 Individual Discipline Feasible
The Individual Discipline Feasible (IDF) method is an alternative to the possibly
computationally expensive MDF method because it avoids a complete MDA upon each iteration.
Instead of achieving multidisciplinary feasible design at every iteration of the optimization, IDF
enforces individual discipline feasibility for each discipline and iteration. A discipline is
considered to be feasible when the equations the discipline code is intended to satisfy are solved
27
[49]. For example when the CFD model converges to find the pressures given an input shape, the
aerodynamic discipline is considered to be feasible. Likewise for the FE model, when the stress
and displacements are determined the structural discipline is considered to be feasible. The IDF
method only achieves a complete multidisciplinary feasible design when the solution has
converged.
The architecture links a single optimizer and a decoupled disciplinary analysis for each
discipline. The specific analysis variables that have been promoted are those that represent
communication, or coupling, between analysis disciplines via interdisciplinary mappings. This
changes the MDF architecture by forcing the introductions of surrogate design variables that will
allow the coupling variables to be explicitly optimized by the optimizer [49]. The IDF
architecture is shown below.
Figure 3.3 - IDF architecture for aeroelastic analysis
The extra constraints must be included in order to drive the optimization to a feasible design
upon convergence. The optimizer selects the surrogate design variables (XMAS and XMSA) and the
global/local variables (XD). These are then introduced to each discipline for analysis where the
actual values of the coupling variables can be calculated. These actual coupling variables are the
additional equality constraints (GAS for aerodynamics and GSA for structure). XMAS is the
28
surrogate design variable for the aero-structural coupling and XMSA is the surrogate design
variable for the structural-aero coupling. These surrogate variables could be the coefficients of a
spline curve or a spline surface. For example the surrogate design variables for the structural
discipline could use spline coefficients to create the aerodynamic loading for the FE model.
The advantage of this method is that a complete MDA is avoided upon each iteration which will
improve computational time. However, this is not always the case since satisfying the coupling
variables equality constraint can be difficult and there is an increase in the problem dimension.
3.1.3 All-At-Once
Another interesting MDO architecture is the All-At-Once (AAO) method. It is the exact opposite
extreme of the MDF method. For the AAO, we do not seek to obtain feasibility for the analysis
problem. In other words discipline feasibility, multidiscipline feasibility, and equations within
the discipline itself are not satisfying feasibility. For example an uncoverged CFD or nonlinear
FE model are considered infeasible. This approach attempts to reduce the computational time by
avoiding feasibility when the design is far from the optimum. The problem with this approach is
that it introduces several constraints and increases the dimension of the problem. The AAO
approach will not be used in the current work.
3.1.4 Additional MDO Architectures
Table 3.1 below list several common MDO frameworks with a brief description of the technique
with some of the advantages and disadvantages associated with each approach. Each method has
been summarized from [50].
Table 3.1 - Common MDO frameworks
Framework Description Advantages Disadvantages
Multidisciplinary feasible
(MDF)
This approach couples an
optimization code to the
MDA process.
Excellent for problems
with strong coupling. State
equations are satisfied
during the optimization.
Computationally
expensive. Disciplinary
code integration can be
difficult.
Individual discipline
feasible (IDF)
Coupling variables are
treated as optimization
Only one cycle of the
disciplinary analysis code
Increase in problem
dimensionality. Coupling
29
variables and auxiliary
variables are used to
satisfy individual
disciplines
is required. variables may not satisfy
the physics of the problem.
Disciplinary analysis
optimization (DAO)
Each discipline chooses its
own system level auxiliary
variables in disciplinary
analysis. Convergence is
ensured by placing
equality constraints on the
system variables.
Ensures disciplinary
constraints are satisfied at
each iteration of the
optimization cycle.
Increased problem
dimensionality. Several
equality constraints must
be satisfied.
All-at-once (AAO) The state equations and
optimization problem are
solved simultaneously.
State equations are
considered equality
constraints on the system
level optimization.
Eliminate iterative
disciplinary analysis.
Reduced computation cost.
Difficult to satisfy state
equations’ equality
constraints. Difficult to
obtain a global optimum.
Collaborative
Optimization (CO)
The optimization problem
is solved at two levels:
system level and discipline
level. Each discipline
selects its own set of
design variables and then
each discipline-specific
optimization minimizes
the difference between
discipline-chosen design
variables and design
variables provided by the
system-level optimizer
Independent execution of
disciplines allows for
concurrent processing of
disciplinary optimization.
Subsystem optimization
could have a reduced
dimensionality. Each
discipline has its own
autonomy to select its own
optimization process.
System-level constraints
can have discontinuous
derivatives making this
difficult to solve with
gradient optimizers. Large
number of coupling
variables can lead to
excessive computational
costs and convergence
difficulties.
Concurrent subspace
optimization (CSSO)
The system-level is treated
as another discipline and
each discipline has its own
optimizer. This method
Subsystem and system-
level are solved
independently from each
other reducing
Uses approximation
models for subsystem and
system-level optimization.
Approximation models are
30
uses decomposition if the
design variables and
approximations so as to
ensure multidisciplinary
fidelity in each step of the
optimization problem.
computational time. accurate within small
regions surrounding the
point of interest.
Bilevel integrated system
synthesis (BLISS)
Formulates the
disciplinary objective
function from the given
system-level objective
function such that
minimization of the
disciplinary objective
function will result in the
reduction of the system-
level objective function
value.
System-level optimization
process guides subsystem-
level optimization to
improve overall system
design. Reduction of the
optimization problem to a
subsystem-level
optimization process
reduces problem
dimensionality
Individual disciplines do
not have autonomy to
select their own objective
function. Difficult to solve
using gradient optimizers
Disciplinary interaction
variable elimination
(DIVE)
Reduce optimization
complexity by elimination
certain variables. Each
discipline constructs a
metamodel.
Variable elimination
reduces model dimension.
Construction of
metamodels can be
computationally
expensive.
Which method to use will be highly dependent on the problem to be solved, problems with high
coupling between disciplines and separation of these disciplines proves to be difficult, the MDF
approach will be the most appropriate but more computationally expensive.
3.2 MDO Test Cases
The overall objective of the VGTM concept is to find an optimal wing shape for each mission
leg of an aircraft’s flight. A simple mission is presented in Figure 3.4. Each circle represents a
shape change in the wing. The hollow circle represents the start of a mission leg and a filled
circle represents a shape change during its respective mission leg. More specific details on each
mission leg will be given later. A morphing wing should change its shape during each mission
leg since the weight of the aircraft is decreasing due to fuel being consumed. Conventional wings
are optimized for beginning of cruise but at subsequent intervals during the mission legs the
31
wing will be suboptimal. The work done in [51] has shown that a continuous shape changing of a
morphing wing concept during each mission leg will provide the best results. Using a multi-level
optimization routine, they broke down each mission leg into different segments and optimized
the wings performance to reduce the fuel consumption. They determined that an aircraft whose
wing area is shrinking, they referred to this as “cruise-shrink”, will give optimal results.
Take-Off
Cruise Loiter
Descent
Land
Climb
Figure 3.4 - Aircraft mission showing mission legs
Table 3.2 below shows a list of possible objectives for each mission leg.
Table 3.2 - Possible objective for each mission leg
Mission Leg Possible Objectives
Take-Off Minimize take-off distance
Minimize noise
Climb Maximize rate of climb
Maximize climb angle
Cruise Maximize range
Minimize fuel consumption
Loiter Maximize endurance
Descent Maximize sink rate
Land Minimize landing distance
32
Minimize stall speed
3.3 Objective Function
Attempting to perform an optimization for all of the mission legs simultaneously would require
significant computational resources. Commercial and business jet aircraft will devote most of
their flight time to cruise. Therefore it will be selected as the dominant mission leg that should be
optimized. For commercial aircraft, the fuel totals amount 38% of operations and maintenance
costs [31, p. 570]. With rising fuel costs, aircraft designers have been motivated towards
improving aircraft fuel efficiency. Since business jets or commercial aircraft will consume most
of its fuel during cruise, minimizing the fuel consumed for this mission leg was selected to be the
objective. Maximizing range are other objectives that can be considered, however the amount of
fuel consumed will need to be known aprioi, and since aircraft are typically designed with a
range requirement with fuel efficiency in mind, minimizing fuel would be a better option.
The objective of the low and higher fidelity test cases is to minimize the fuel consumption for a
known thrust specific fuel consumption (ct) and range (R) at a given flight speed (V). The range
for an aircraft is given below [31]:
(
) (3.1)
L is the lift, D is the drag, Wi is the fuel at the end of cruise and Wi-1 is the fuel at the beginning.
Since we are interested in minimizing fuel, the equation can be rewritten as:
( ) (3.2)
From the equation above, if we maximize the velocity and lift, and minimize the specific fuel
consumption, range and drag, we will minimize fuel consumption. This equation is
contradictory: high speed means high drag, high lift means high drag, and high drag means high
thrust (higher specific fuel consumption). However, for the optimization problems below the
specific fuel consumption, Mach number, and range will be held constant at 2e-4 /s, 0.85, and
6000 km respectively. With these constants, the only variables that will influence the objective
are the lift and drag.
33
With multiple morphing transitions during cruise, there is now a multi-objective problem. The
objective will be to minimize the fuel consumed for each morphing stage at cruise.
∑
(3.3)
The multi-objective problem will follow a weighted sum approach. With multiple morphing
states at cruise, a weight of 1/N has been implemented (with N being the number of morphing
states). Including this weighting to the fuel objective will give an equal weight of 1. An uneven
weight will introduce bias to the objective with the large weighting. Also each objective will
need to be normalized with respect to a reference value to prevent bias.
3.4 Constraints
There are two intended purposes of constraints: the first is to provide limitations to the
optimization algorithm to make it more efficient and the second is to satisfy a design
requirement. For this MDO routine upper and lower limits have been specified for all design
variables. This constraint forces the PSO algorithm to search in a region where the optimal
design will most likely be found. Constraining the bending stress to be below the yield stress
with a factor of safety is a design requirement to ensure a safe wing structure. The material
properties for the wing are Aluminum 2014-T6 (Young’s Modulus is 73.1 GPa, Yield Stress ( )
414 GPa, and density 2790 kg/m3).A factor of safety (FS) of 1.5 was used.
(3.4)
A constraint has been imposed on the fuel consumption objective. This was done for two
reasons: the first is a design requirement because we want an improved performance compared to
a baseline wing and the second reason was to improve algorithm convergence. This is illustrated
in the figure below.
34
Figure 3.5 - Design space without constraint on objective (left) design space with constraint
on objective (right)
The figure shows an objective (F) with global and local minima with only one design variable
(V). The left figure is without an objective constraint. This constraint absence increases the size
of the feasible design space for the algorithm to search and it may end up getting stuck in the
local minima. The algorithm may find the global optimum eventually, but it will likely require
several more iterations. On the other hand, if we introduce a constraint on the objective we can
encourage exploration closer to the global optimum. This figure is meant for illustrative purposes
only.
(3.5)
The objective above is directly influenced by the aerodynamic discipline only. Therefore to keep
the mass of the structure within a reasonable limit a constraint will also be imposed on the
structural mass. This constraint will confine the mass of the wing to be below the mass of the
reference wing. However, to give some leeway to the optimizer the reference wing mass will be
increased by 10%.
(3.6)
35
In order to achieve discipline feasibility for aircraft performance, steady level flight conditions
must be satisfied (i.e. lift must equal weight).
|
| (3.7)
The aircraft needs to be trimmed using the angle of attack to find this equality constraint. This
constraint is necessary to find an equilibrium state to satisfy the aircraft performance discipline.
Depending on the method (IDF or MDF) used satisfying this constraint can be difficult, so a
tolerance (tol) of 0.01 was used. This means that the lift must be within 1% of the weight
estimation. For the IDF approach the angle of attack is a variable used for trimming. However
for the MDF a different approach is used. Since the lift for an aircraft at low angles of attack
(less than 8o) is relatively linear, an interpolation/extrapolation scheme can be used to find the
trim state. Using this approach, a trimmed state can be found in only three function calls of the
aerodynamic model as opposed to a fixed-point iterative scheme which could take several more
function calls. However there is no guarantee that achieving a trimmed state using this approach
is possible, so the trimmed constraint above is still important.
3.5 Design Variables
The number of design variables for the high and low fidelity models are dependent on the
number of morphing transitions (N) during cruise. If there is only one morphing state at the
beginning of cruise, there are a total of 31 variables. If the number of morphing states were to
increase to two, the number of variables becomes 46.
Table 3.3 – Design variable list for low fidelity models
Morphing Stage Wing
Component Discipline Design Variable
Upper
Limit
Lower
Limit
Initial Stage Main Wing Aerodynamic
Span (m) 6 9
Sweep (o) 40 20
Taper 0.3 0.1
36
Chord (m) 5 3
Main
Structure
Leading Edge
Offset (%chord)
0.3 0.1
Trailing Edge Offset
(%chord)
0.9 0.6
X Scale (Root and
Tip)
0.99 0.7
Y Scale (Root and
Tip)
0.99 0.7
Wingtip
Aerodynamic
Taper 1 0.3
Sweep (o) 20 -20
Span (m) 1 0.5
Twist (o) 5 -5
Dihedral (o) 20 -5
Structure
X-Scale 0.99 0.7
Y-Scale 0.99 0.7
VGTM (3 Mods)
Aerodynamic
Span (m) 0.5 0.25
Twist (o) 5 -5
Dihedral (o) 20 -5
Sweep (o) 20 -20
Taper 1 0.7
Structure
X Scale 0.99 0.7
Y Scale 0.99 0.7
37
Flight Condition Performance Angle of Attack (o) 3 -3
Additional
Stages
VGTM (3 Mods) Aerodynamic
Span (m) 0.5 0.25
Twist (o) 5 -5
Dihedral (o) 20 -5
Sweep (o) 20 -20
Flight Condition Performance Angle of Attack (o) 3 -3
Total Number 16+N*15
The variables for the higher fidelity case are slightly different from the low fidelity case and are
listed in the table below. From the previous low fidelity case all the variables were continuous.
For the high fidelity case two discrete variables were included. The number of stringers and ribs
are considered to be discrete. To implement the discrete variables in the PSO algorithm a simple
rounding of the number being passed to the discipline code takes place.
Table 3.4 - Design variable list for high fidelity models
Morphing
Stage
Wing
Component Discipline Design Variable
Upper
Limit Lower Limit
Initial Stage Main Wing
Aerodynamic
Span (m) 9 7
Sweep (o) 40 20
Twist (o) 8 2
Taper 0.3 0.1
Chord (m) 5 3
Main
Structure
Ribs 24 16
Stringers 8 3
Skin Thickness (m) 0.025 0.015
38
Rib Thickness (m) 0.01 0.001
Spar Thickness (m) 0.01 0.001
Stringer Radius (m) 0.01 0.001
Spar Location (m) 0.9 0.7
Structural Taper 0.6 0.2
Wingtip
Aerodynamic
Span (m) 1 0.25
Taper 1 0.7
Structure Spar Thickness (m) 0.008 0.001
VGTM (3 Mods)
Aerodynamic
Sweep (o) 15 -15
Span (m) 0.75 0.25
Twist (o) 5 -5
Dihedral (o) 20 0
Structure
Passive Member Radius (m) 0.01 0.001
Active Member Radius (m) 0.01 0.001
Skin Thickness (m) 0.008 0.001
Rib Thickness (m) 0.008 0.001
Flight Condition Performance Angle of Attack (o) 6 0
Additional
Stages VGTM (3 Mods) Aerodynamic
Sweep (o) 15 -15
Span (m) 0.75 0.25
Twist (o) 5 -5
Dihedral (o) 20 0
39
Flight Condition Performance Angle of Attack (o) 6 0
Total Number 23+15*N
40
Chapter 4
Particle Swarm Optimization Algorithm
4.1 Introduction
Particle Swarm Optimization (PSO) was developed by [52] as a stochastic optimization
algorithm based on social simulation models. The PSO algorithm uses a swarm of particles to
travel stochastically in the search space. The velocity vector is used to update the particles
current position through
(4.1)
The velocity update vector is given by (4.2).
(
) (
) (4.2)
The velocity of the ith
particle with the jth
dimension is updated based on the information from
previous time step t and an inertia weight constant given by w from [53] [54] [55]. The inertia
weight was introduced to help concentrate the swarm near the promising regions. Values greater
than 1 encourage the swarm to explore the design space which is desirable for the early stages of
optimization. Smaller values reduce the velocity step to exploit the best solutions. and are
uniformly distributed random numbers between 0 and 1. is the best position of the particle that
it has ever visited is held in memory. In other words it is the best position or combination of
design variables that yielded the best objective. In addition to the personal best position there is a
global best . This represents the best position out of all of the particles at the current iteration.
The best positions are communicated to the entire population with the intent to encourage the
particles to search the more promising regions in the search space. and are the cognitive and
social learning factors respectively (accelerations constants) supplied by the user. The values
associated with the two learning factors influence how much confidence the particle has in itself
(cognitive) or in the swarm (social). If is less than then the particles will be attracted to the
global best solution and this global best position may not be the objectives global minimum
solution, and a premature convergence is possible. If is greater than the swarm will be more
41
biased towards itself and convergence may take several iterations. Also the two constants control
the magnitude of the velocity vector step size. After the position has been updated using (4.1) it
is necessary to find the new personal best and global best. If the personal best is smaller than the
previous personal best then the personal best is updated to the current value. Otherwise the
personal best is not updated. Each value of the particles new personal best is compared to the
previous global best. If the new particle’s personal best is smaller than the current global a new
global is created.
4.2 Literature Review
Several enhancements have been used to improve the basic algorithm. Using a constant value of
inertia weight showed improvement to the PSO algorithm, however linearly varying the inertia
weight from 0.9 to 0.4 has shown substantial improvement in performance compared to a
constant inertia weight [54]. A dynamic approach to changing the inertia weight was used in [56]
and [57]. The advantage of a dynamic reduction approach is the fewer function evaluations
necessary for a satisfactory solution. This has been shown in [58] with a 10 and 25 bar truss
example. The Time Varying Acceleration Coefficient (TVAC) proposed from [59], for the early
stage of the optimization having a larger initial value for the cognitive over the social allows the
particles to have more confidence in its exploration. For the later stages of optimization, to
facilitate exploitation having a larger social factor would be more beneficial to obtain the global
optimum. A mutation operator was also introduced to improve swarm diversity.
A slight modification to the standard PSO algorithm has been introduced. The social and
cognitive acceleration constants will not vary with time but will vary with the particle elitism.
The intent of this approach is to try and jostle a particle free of a local optimum. A similar
approach has been used in [60], where they used the particles local and global best values to
determine their acceleration constants. c1 and c2 were assumed to be the same. They were able to
successfully reduce the average number of iterations and the average fitness for several
benchmark problems. Their results were improved further by incorporating mutation, crossover,
and root mean square variant to tune theirs results. In [61] they applied a self-adaptive inertia
weight (SAIW) and social acceleration coefficient (SAIC) PSO. The SAIC was tested against
high dimensionality Rastrigin, Sphere, and Griewank functions. The results were excellent for
42
the Griewank and Sphere test cases but were poor for the Rastrigin; the authors incorporated a
mutation operator from [62] to their algorithm to maintain diversity.
4.3 Social and Cognitive Parameter Variations
For this section, the three different social and cognitive parameter variations will be described.
The three variations will follow a linear, step, or sinusoidal trend.
4.3.1 Linear variation of acceleration constants
The cognitive and social parameters will follow a linear distribution from the best to worst
particle. Equations 4.3 and 4.4 are the cognitive and social acceleration constants respectively.
c1best and c1worst are the cognitive parameter for the best and worst particle with c2best and c2worst
are the social parameters for the best and worst particle. N is the number of particles in the
swarm.
( )( )
( ) (4.3)
( )( )
( ) (4.4)
For the above equations the ith
particle at the current iteration is given a rank given by j based on
its eliteness. The best partcle is identified with an index of 1 and a worst particle has a rank of N
which is the swarm size. A sample distribution is given in Figure 4.1 with c1best = c2worst = 1 and
c1worst = c2best = 3.
43
Figure 4.1 - Linear variation of particle acceleration constants
A large value for c1best over c2best promotes exploration and may require many function
evaluations to converge, however with greater social attraction exploitation of the best position is
preferred but may force a premature convergence. Values of c1worst which are greater than c2worst
encourage exploration of the particles such as the second half of the ranked swarm in Figure 4.1.
4.3.2 Step variation of acceleration constants
In addition to the linear variation described previously, a step variation will also be examined.
The step variation separates the swarm into three distinct groups: upper, middle, and lower
groups. Figure 4.2 below is an example of the step distribution. For this example the upper group
(U) has a larger cognitive learning capability compared to its social forcing a larger step towards
its own personal best, the middle group (M) is an average of the social and cognitive parameters
(this is just the SPSO algorithm), the lower group (L) has a higher social learning than cognitive
forcing the particles to take a larger step towards the more promising regions found in the search
space.
44
Figure 4.2 - Step variation of particle acceleration constants
4.3.3 Sinusoidal variation of acceleration constants
Another variation being considered is a sinusoidal variation of the particles acceleration
constants. Figure 4.3 below shows an example distribution determined from equations 4.5 and
4.6. ω is the frequency, is the phase, and j is the particle rank index.
(
) (4.5)
(
) (4.6)
45
Figure 4.3 - Sinusoidal variation of particle acceleration constants
For the three different social and cognitive variations described, the parameters will be
determined statically and dynamically. For the static case, the social and cognitive parameters
will be determined by the initial swarm and will remain constant throughout the optimization
process. However for the dynamic case, the social and cognitive parameters will be revaluated at
each iteration.
4.4 Unconstrained Test Cases
Four benchmark functions listed in Table 4.1 will be evaluated for the current work. The
Rastrigin, Griewank, and Rosenbrock problems are multimodal equations with several local
minima thus making them a good test for the algorithms exploration capability. The Sphere
problem is unimodal used to test the exploitation capabilities of the algorithm. Each function has
a global optimum of 0.
Table 4.1 - Unconstrained benchmark problems
Function Limits Vmax
Rastrigin ( ) ∑ ( ( ) )
10
46
Rosenbrock ( ) ∑ (
) ( ) 30
Sphere ( ) ∑
50
Griewank ( ) ∑
∏ (
√ )
600
Each function will be used to compare the standard PSO algorithm (SPSO) and the constriction
based PSO (CPSO) [63] to the proposed methods. Each benchmark problem will have 100 trial
runs, a dimension of 30, and a swarm size of 30. The criteria for comparison include: the mean
and standard deviation after 3000 iterations have been completed. All test cases will have a time
varying inertia weight (TVIW) from 0.9 to 0.2 (with the exception of the CPSO). The TVIW
uses the specified number of iteration to change the inertia weight in a linear fashion. A large
inertia weight is beneficial for exploration at the early stage of optimization but a lower value is
best to exploit the more promising regions in the design space. The initial population is randomly
generated within the specified bounds on the design variables. These limits on the design
variables confine the algorithm to a specified search space to improve algorithm efficiency. Also
each particle receives an initial randomly generated velocity confined to a maximum velocity.
The maximum velocity will limit the step size of the particle. If the velocity is too large the
algorithm will not converge. The results for each benchmark problems are presented in appendix
A. From the results it was concluded that a sinusoidal variation demonstrated the best
performance
4.5 Constrained Test Case Results
The previous section dealt with only unconstrained problems. Most engineering design
optimization problems require at least some constraints to be imposed. To handle constraints for
the PSO algorithm, an exterior quadratic penalty function approach will be used.
∑ { }
(4.7)
47
This equation states that the original objective is penalized with a penalty factor P set to 1e7
and the summation of all the violated constraints . A constraint is considered to be violated if
its value is greater than zero. The penalized objective is sent to the optimizer. In this section
we will use a straight forward structural optimization problem to analyze the modified PSO
algorithm’s performance. The problem description is presented next.
4.5.1 Problem Description [64]
The 25-bar space truss shown in the figure below is required to support two load conditions
given in Table 4.2. Constraints have been imposed on member stresses as well as Euler buckling.
A minimum and maximum allowable area is specified for each member. The allowable stresses
for all members are specified as σmax (40e3 psi) in both tension and compression. The Young’s
Modulus is 107 psi and the density (ρ) is 0.1 Ib/in
3.
Figure 4.4 - 25-bar space truss
Table 4.2 - Load conditions for the 25-bar truss
Joint
48
Load 1 2 3 6
Fx (1) 0 0 0 0
Fy (1) 20e3 -20e3 0 0
Fz (1) -5e3 -5e3 0 0
Fx (2) 1e3 0 500 500
Fy (2) 10e3 10e3 0 0
Fz (2) -5e3 -5e3 0 0
The members are assumed to be tubular with a nominal diameter/thickness ratio of 100, so that
the buckling stress in member i becomes
(4.8)
Where Ai and li are the cross-sectional and length, respectively, of member i. The cross-sectional
areas linked as:
A1, A2=A3=A4=A5, A6=A7=A8=A9,
A10=A11, A12=A13, A14=A15=A16=A17,
A18=A19=A20=A21, A22=A23=A24=A25
(4.9)
The objective of the problem is to minimize the weight of the 25-bar space truss
∑
(4.10)
With the following constraints:
49
| |
(4.11)
Where σi,j is the stress induced in member i under load condition j. σi,j must also be less than the
critical buckling stress pi.
4.5.2 Constrained Structural Optimization Results
The optimal mass is 233.07 Ib with the design vector X=[0.1, 0.8023, 0.748, 0.1, 0.124, 0.571,
0.978, 0.802]. The test cases that are being compared are in Table 4.3.
Table 4.3 – Test cases for 25-bar space truss
Test Case
1 Standard C1=2, C2=2 8 Sinusoidal ω=2π and φ=π/2
2 Linear
C1best=C2worst=3
C1worst=C2best=1
9 Sinusoidal ω=4π and φ=π/2
3 Linear
C1best=C2worst=1
C1worst=C2best=3
10 Sinusoidal ω=6π and φ=π/2
4 Step
0≤U≤20 C1=3 C2=1
11 Sinusoidal ω=8π and φ=π/2 20≤M≤60 C1=2 C2=2
60≤L≤100 C1=1 C2=3
5 Step
0≤U≤20 C1=1 C2=3
12 Sinusoidal ω=2π and φ=0 20≤M≤60 C1=2 C2=2
60≤L≤100 C1=3 C2=1
6 Step 0≤U≤40 C1=3 C2=1 13 Sinusoidal ω=4π and φ=0
50
40≤M≤80 C1=2 C2=2
80≤L≤100 C1=1 C2=3
7 Step
0≤U≤40 C1=1 C2=3
14 Sinusoidal ω=6π and φ=0 40≤M≤80 C1=2 C2=2
80≤L≤100 C1=3 C2=1
15 Sinusoidal ω=8π and φ=0
Each test case will be run 20 times to find the mean and standard deviation, have a swarm size of
20, a maximum iteration count of 400, and will use the TVIW. The performance metric used to
compare the test cases are the mean number of iterations/fitness required to achieve the target
weight of 233.07 Ib and the iteration/fitness standard deviation. The results are presented in
Table 4.4.
Table 4.4 - Constrained test case results
Test
Case
Mean
Fitness
STD
Fitness
Mean
Iteration
STD
Iteration
Test
Case
Mean
Fitness
STD
Fitness
Mean
Iteration
STD
Iteration
1 239.4 13.12 323.8 81.14 8 233 0.07275 278.3 26.22
2 238.4 12.8 310.4 70.38 9 233 0.05197 272.5 11.65
3 249.3 23.9 358.5 67.21 10 251.4 30.5 362.9 60.43
4 233.7 2.033 284.1 48.38 11 240.9 14.97 313.2 62.1
5 258.7 46.01 317.3 65.66 12 243.3 13.92 366.5 60.94
6 253.3 55.21 336.1 68.68 13 247 27.37 316.8 72.48
7 238.2 11.22 344.5 62.45 14 246.1 23.91 323.1 82.06
15 246.8 33.34 348.2 67.14
51
From the results above, test cases 8 and 9 achieved the best performance. The results were
consistent, since the average fitness was exactly the target weight of 233.07 Ib, with a very low
standard deviation. Test case 9 had the lowest number of iterations and the lowest standard
deviation. Therefor the 9th
test case will be used for the MDO routine.
52
Chapter 5
Aerodynamic and Structure Computer Models
5.1 Introduction
Selecting an appropriate discipline tool for the MDO process cannot be taken lightly. The
discipline tool needs to be accurate enough to model the physics of the problem. Complicated
models such as a non-linear finite element analysis can be more accurate for certain problems
compared to a linear model but the computational requirements will be a lot higher. A balance
between computational efficiency and model accuracy is required. This chapter will provide a
description of the low and high fidelity models being used for the current work.
5.2 Low Fidelity Models
Low fidelity models are typically used in the very early conceptual design phase. They are
simple models based on approximations and other simplifying assumptions. These models are
selected for their computational efficiency over their accuracy. This becomes very important for
optimization problems with a large number of function calls. This section will describe two low
fidelity approaches under consideration. A mid-sized business jet wing has been selected for
testing. Its geometric description has been listed below.
Table 5.1 - Reference wing geometry
Bombardier Challenger 300
Maximum Take-Off
Gross Weight (MTGW)
18000 kg Wing Reference Area (S) 40 m2
Cruise Mach Number 0.85 Wing Root Chord (Cr) 4 m
Cruise Altitude 10 km Taper Ratio (λ) 0.2
Wing Span (b) 18 m
53
5.2.1 Aerodynamic Model
The aerodynamic tool used for the current work is the well-known Tornado MATLAB code
which has been developed by Dr. Tomas Melin [65]. The Tornado code is a vortex lattice
method (VLM), programmed to be used in conceptual aircraft design. The aircraft geometry in
Tornado is fully three dimensional with a flexible, free-stream following wake. The Tornado
solver solves for forces and moments, from which the aerodynamic coefficients are computed.
Although Tornado is considered a low fidelity tool it has been selected for the current work for
its computational efficiency. The VLM is fast, which is a requirement of design studies that
entail a large number of function evaluations, such as the one here. Tornado is used for the
calculation of relevant aerodynamic coefficients and lift distribution over the wing. This lift
distribution is necessary for the structural model.
Tornado is a low fidelity model because it is based on potential flow theory which requires
several simplifying assumptions: incompressible, inviscid, irrotational, and airfoil thickness is
neglected. For a real aircraft at transonic flight these simplifying assumptions are completely
incorrect. At transonic speeds the air is compressible meaning its density will change with
pressure significantly. This rise in density will increase drag and lift. Failing to consider
compressibility effects for structural design may result in structural failure. Also aircraft
performance can be severely over estimated. An inviscid flow model ignores the drag effects
associated with the boundary layer on the wing surface. Ignoring the viscous effects will give an
optimistic drag prediction. An irrotational model assumes the flow remains attached to the wing
at all times. At high angles of attack the air will separate from the surface creating an increase in
drag. The airfoil thickness is strongly influenced by the Mach number. A thick airfoil will create
very large velocity gradients making such a design suitable for low speed flight. However at
transonic speeds, to avoid or reduce the presence of shockwaves, a thinner airfoil is preferred.
Tornado has the option to use correction factors and approximate models to address some of the
simplifying assumptions. The Prandlt-Glauret compressibility correction factor can be applied to
the aerodynamic coefficients to estimate the effects of transonic flight. This compressibility
correction is performed by adjusting the flows incompressible coefficients using:
54
√ (5.1)
The compressibility correction adjusts the coefficient (C) using the Mach number to obtain the
new compressibility coefficient (Cc). Using correction factors Tornado can estimate the zero-lift
drag of the wing. This drag is often referred to as the profile drag, which is determined when the
lift of the aircraft is zero. Using the coefficient of drag correction factor and the wing wetted area
the zero-lift drag coefficient for a wing section can be calculated.
In [66] Tornado was verified against wind tunnel data from an elliptical wing. At Mach numbers
of 0.3, 0.4, and 0.5, the lift slopes from the experimental and Tornado models were compared
and the errors were shown to be 0.9%, 0.2%, and -2% respectively. Slightly larger errors were
determined when comparing drag polars due to the absence of viscous drag effects. For the three
different Mach numbers there errors were -4%, -5%, and -4% respectively. Tornado has also
been used by the following papers [43], [46], [51], and [67]for their work.
5.2.2 Structural Model
The structural model of the wing is a 3D cantilever beam using linear finite elements. This 3D
beam model has 6 degrees of freedom at each node. The stiffness matrix can be found in several
textbooks on finite elements, but this was taken from [68]. The stiffness matrix in its current
form is fine for a beam that is straight, however the wing has a sweep angle and the wingtip
modules will have three different orientations.
The cross-section of the wing is shown in Figure 5.1 below. It is a simple beam that has a
polynomial shape confined inside an airfoil. This cross-section is just an approximation of a real
wingbox. A real wingbox can have several design variables and require more sophisticated
computer models, however it was selected for the following reasons; it requires only two design
variables, and its bending and torsion stiffness are dependent on the size of the wing. To create
the thickness of the wingbox a simple 2D scaling approach is used. The interior shape has its
coordinates defined by scaling the x and y coordinates of the exterior shape.
55
Figure 5.1 - Wingbox geometry confined within airfoil
The area moment of inertia and polar moment of inertia are defined respectively as:
∑(
)
( )
∑(
)
( )
∑
(5.2)
The above equations are for a solid polygon shape only; however the cross-section of the wing
being considered is hollow. There are two large polynomials that can be used to define the cross-
section; an exterior and an interior. To find the overall area moment of inertia a simple
subtraction of the two polygons is possible:
(5.3)
56
Finding the polar moment of inertia is a bit more involved. We need to find the radial location ‘r’
of the centroid of each polygon relative to the origin of the cross-section and ‘A’ is the area. In
this case the polygons refer to the shapes defining the thickness of the cross-section. To find the
centroid and area of the polygons we use:
∑( )
∑( )
( )
∑( )
( )
(5.4)
To validate the above model, a sample section was created in ANSYS© to determine the
necessary cross-sectional properties. The results are compared in table Table 5.2.
Table 5.2 - Comparing cross-sectional data from ANSYS to current FE model (X and Y
Scaling are set to 0.5)
Structural Property ANSYS Current model
2.9714e-05 3.0576e-005
0.00171 0.0018
0.0405 0.0406
1.0904e-04 1.4093e-004
-0.043704 -0.0375
-0.0045851 -0.0039
57
To validate the FE solver example problems from [68] were tested. Two examples were chosen:
the first example was used to validate the frame setup with a distributed load in 2D and the
second example was chosen to assess its ability to model in 3D. The example problems are in
appendix B. The solver was able to replicate the results from [68].
5.2.3 Aero-Structural Model
To calculate the structural displacements and bending stress, the aerodynamic loads from
Tornado needs to be mapped to the finite element model. Instead of using equations 2.1 and 2.2
from chapter 2, the structural nodes will be determined from the Tornado model. This will ensure
model consistency between the two disciplines. The location of the beam centroid will be aligned
with the axis, which is the midpoint between the leading and trailing edge offset.
Figure 5.2 - FE nodes (dots) with aerodynamic model
To map the aerodynamic forces to the finite element model, the technique of equivalent nodal
forces have been used [68]. A beam can support distributed or concentrated nodal loading. In
order to account for these loading conditions we need to find an equivalent nodal load. From
Tornado the aerodynamic forces are concentrated nodal loads located between the FE nodes.
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Using the equivalent nodal loads for a beam shown in Figure 5.3, it is possible to determine the
structural displacement and stresses.
Figure 5.3 - Equivalent nodal loads for concentrated point loads
To find the stress in the beam, we need to know the reaction forces and moments. Once the
reactions are known using the following equation it is possible to determine the bending stress:
(5.5)
The stress is dependent on the reaction moment M, the second area moment of inertia Ix and the
height of the wing section t. A wing deflection compared to its initial state is shown in the figure
below.
Figure 5.4 - Deformed and undeformed wing shape
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5.2.4 Fluid-Structure Interaction
The previous section was describing a simple aero-structure problem. It assumes that the wing
deformation has no effect on the aerodynamic loading, which is an incorrect assumption. The
aerodynamic loading of the wing is dependent in the shape of the wing which is dependent on
the aerodynamic loading. The aerodynamic performance is now highly coupled with the
structural performance. Older aircraft were designed assuming a rigid wing for aerodynamic load
prediction. A rigid aircraft wing will produce more lift compared to a flexible wing; however this
higher lift would force designers to build a stiffer wing structure, which would increase the
weight of the aircraft. Figure 5.5 and Figure 5.6 will demonstrate this. For the initial state of the
aircraft, the wing was trimmed to the MTGW assuming a rigid wing. This will give a decent
starting angle of attack for convergence compared to a random starting angle of attack.
How the coupled Tornado-FE model works is very straight forward. Tornado requires four inputs
to generate a wing section: span, sweep, dihedral, and twist. The solution to the FE model
outputs the nodal displacements and rotations. The rotations of the nodes are used to transform
the Tornado model’s degree of freedoms (sweep, dihedral, and twist) into the new deformed
shape to determine the aerodynamic loading. It is assumed however, that due to the low
aerodynamic forces acting along the span, the length of the beam will remain constant.
Through several test cases, it was determined that the convergence can be unstable and it could
take several iterations. To improve convergence, numerical damping was applied to the beam
rotations before being used by Tornado. In the equations below, D is the damping set to 0.7 and
k is the iteration count.
( )
( )
( )
(5.6)
The next two figures show the lift to drag ratio and wingtip deflection convergence. It is clearly
shown that the static deflection for the rigid wing is far greater compared to the elastic wing due
to its higher loading.
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Figure 5.5 - Convergence plots for an elastic wing at constant angle of attack
The convergence is slightly unstable at the beginning but it stabilizes quickly. There is a slight
decline in the lift to drag ratio from the initial rigid wing. For this example the rigid wing lift to
drag ratio was 25.2 and after convergence of the elastic wing the lift to drag ratio dropped to
22.8. There is a 9.5% drop in performance. The tip deflection dropped from 0.53 m to 0.29 m.
The above example was for a wing at constant angle of attack. Since the aircraft lift has been
altered due to its elastic deformation, the aircraft is no longer trimmed. Figure 5.6 below is an
example of the same wing being trimmed to the required weight.
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Figure 5.6 - Convergence plots for an elastic wing while simultaneously being trimmed to
specified weight requirement
For this case, convergence required twice as many iterations. However the trimmed lift to drag
ratio is 23.8 which is only a 5.5% drop in performance. A more flexible wing will require more
iterations to find a static deflection.
5.3 High Fidelity Models
High fidelity models are typically used during the more detailed design phase. These models are
based on fewer assumptions and can model more complicated systems. However the
computational requirements can be extensive, such as CPU time and memory. This section will
describe two high fidelity approaches under consideration.
5.3.1 Aerodynamic Model
The aerodynamic model will use ANSYS CFX, which is Computational Fluid Dynamics (CFD)
software. CFX requires three steps: the first step is the Preprocessing, the second step is the
Solver, and the third step is the Postprocessor. The first step is where the problem boundary
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conditions and the solver options are set for the solver. Since we are interested in transonic flight
conditions, air compressibility is an important factor in determining the aerodynamic loading.
The compressibility effect option was enabled under the solver options tab. Also temperature
damping was enabled since experience has shown that convergence of the energy equation can
be unstable especially if there are shockwaves present. Viscous effects are another important
factor to be considered for aerodynamic performance. However this creates two problems:
execution time and mesh quality. For an MDO solution, execution time of your discipline codes
is very important; to accurately predict the boundary layer of the wing, mesh resolution near the
surface needs to be very fine which would increase the number of nodes and in the boundary
layer there are large velocity gradients which could make a solution difficult to convergence. It
was decided to use an inviscid solution with free-slip boundary conditions on all surfaces for the
current work. The purpose of a wingtip device is to alter the pressure distribution of the main
wing. An inviscid model will be able to determine the pressure distribution and flow separation.
The meshing program used was ICEM CFD. The meshing approach for the current work uses an
unstructured mesh. This mesh type was selected because it is very easy to generate compared to
a structured mesh. A more complicated geometry (such as a randomly generated one from the
PSO algorithm) would make it difficult to create a structured mesh of good quality.
Figure 5.7 - ICEM CFD unstructured surface mesh
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5.3.2 Structure Model
The finite element model that was used to calculate the structural response of the wing was
created in ANSYS. ANSYS has the capability for automatic model generation which is
necessary for the MDO process. This section details how the automatic generation of the model
is done and what types of elements are used to solve the problem.
The structural finite element model was created to contain the most basic elements of the
airframe structure. The main structural elements that will be used as variables are: skin thickness,
rib thickness, stringer areas, and spar thickness. In addition to these variables: the chordwise
location of the leading and trailing edge spars, the number of spars, the number of stringers, and
a structural taper are also considered to be variables.
The model was created by a series of keypoints, lines, and areas. The first step was to create the
ribs. Using equations 2.1 and 2.2 from chapter 2 and the airfoil coordinates it was possible to
generate the ribs. Using the spacing of the ribs given by and their orientations (Γ, θ, and Λ), it
is possible to create the airfoil keypoints in 3D space. The spacing of the ribs is dependent on the
span of the main wing and the number of ribs. Before we can mesh the wing we must first create
the areas from the keypoints. To do this we need to create the lines that will represent the edges
of the areas. The number of areas used for the rib will depend on the number of stringers.
The ribs have been orientated perpendicular to the leading edge spar. This will help improve
torsional stiffness compared to ribs that are parallel to the airflow. Also this perpendicular
orientation will make the manufacturing process easier. The same procedure for generating the
ribs was used to create the wing spars.
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Figure 5.8 - Internal Structure layout
The stringers are relatively straight forward to create since they can be modeled only by lines.
The wing skin follows again the same procedure that was done to generate the rib areas. The
completed geometry is given below.
Figure 5.9 - Wing geometry without top skin panels
Rib
Stringer
Trailing Edge Spar Leading Edge Spar
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5.3.3 Finite Element Mesh
The structures that make up the shape of the wing are curved areas and straight lines. In order to
accurately model the 3D wing properties appropriate 3D finite elements need to be chosen. Shell
elements were chosen for the curved areas because they can take into account the warping of the
element. They can also take into effect bending, shear, and torsional effects of the wing as well
as in-plane or normal loading. These shells were used to model the ribs, skin, and spars. Two
elements from ANSYS were considered: shell 281 and shell 181. Shell elements decouple the
deformation on the surface and the deformation in the normal direction, allowing for a simple
and efficient simulation of a thin structure. Both elements have 6 degrees of freedom for each
node, they can model thin to moderately thick structures, and they are suitable for linear and non-
linear applications. Shell 281 has 8 nodes which would allow for better modeling of curved
surfaces; however it would increase the total degree of freedom of the system which would
increase computational costs. For this reason shell 181 was selected. The stress for this element
can be calculated at the mid-plane, top surface, or bottom surface. All stresses to be exported are
from the mid-plane. The stringers are straight lines which can be modeled by beam elements.
Since shell 181 does not have any mid-side nodes, a beam element with only two nodes is
required, therefore beam 188 was chosen. The element is based on Timoshenko beam theory
which includes shear-deformation effects. Stringers assist with supporting the bending loads and
they provide stiffness to the skin panels. The stringers are modeled as a solid circular shaft. A
figure of the meshed model is shown below.
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Figure 5.10 - FE mesh of wing
Each of the structural elements in the model has been structurally tapered to help decrease the
overall wing mass and to help give an even stress distribution from the wing root to wingtip. The
structural taper was implemented in a piece-wise linear fashion with each jump in structural
thickness occurring at each rib.
5.3.4 VGTM Model
To create the VGTM model, the same procedure for the main wing was implemented. Four
additional ribs were created. Three were for the morphing platform for each module and the last
rib was for the final wingtip. For the morphing modules to achieve kinematic motion the spars
and stringers were replaced with 4 passive and 4 active links. This was shown in Figure 2.5 in
chapter 2. To allow for kinematic motion a series of revolute and spherical joints will need to be
modeled, however this would require a non-linear solution which would substantially increase
computational time. Such modeling can be quite complicated and is beyond the scope of the
current work. Instead only static conditions will be considered. Working under the assumption
that some sort of a flexible or sliding skin panels will be used to allow the modules to change
shape; for the second stage the skin thickness will be reduced by a certain percentage. This
percentage will be dependent on the span of each module. For example, if the first module were
to extend by 10%, the thickness of the skin panels for this module for this stage will be reduced
by 10%. This change in thickness will lower the modules stiffness and make it difficult for the
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skin to maintain a suitable aerodynamic shape. The FE model for the VGTM structure without
the skin is shown below.
Figure 5.11 - FE mesh of VGTM structure
5.3.5 Aero-Structural Analysis
The aero-structural analysis is required to determine the structural stresses from the CFD
pressure forces. For this analysis the CFD and FE model are solved independently and are
integrated in MATLAB. After the CFX solver converges the forces associated with each node
are exported during the post-processing stage. These pressure forces will be mapped to the FE
model.
In order to map the CFD mesh to the FE mesh, the wing was split into its upper and lower
sections. This makes the process less prone to errors since it guarantees that the upper or lower
surface CFD pressure forces are mapped to the FE nodes on the upper or lower surface. The
mapping process used here is simple algorithm that picks a CFD grid point and tries to find the
nearest FE node:
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‖
‖ (5.7)
Where is the location of the CFD grid point and
is the location of the FE nodes. is
the distance between the CFD grid point and the FE node. The algorithm finds the nearest
structural node to apply the necessary force. Since the CFD grid is more refined compared to the
FE mesh, some structural nodes will receive forces from multiple CFD grids points. A figure
showing the mapped pressure forces to the FE model is shown below.
Figure 5.12 - Mapped CFD pressure forces to FE mesh
For the current work only a rigid aeroelastic analysis was conducted. It is possible to execute a
fluid-structure interaction analysis using a similar approach as described earlier however the
computational expense would be prohibitive.
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Chapter 6
Low and High Fidelity Multidisciplinary Design
Optimization Results
6.1 Introduction
This chapter will present the results for the high and low fidelity test cases and these results will
be compared to a reference wing. The wing’s geometry has been listed in Table 5.1. For the low
fidelity model optimization routine two test cases have been executed; the cruise stage has been
segmented into 2 morphing stages and 4 morphing stages.
6.2 Low Fidelity Results
Before we can execute the MDO routine for the low fidelity case we need to find an optimal
structural mass of the reference wing. This step is necessary since this mass will be used as a
constraint in the MDO analysis. This constraint was described in chapter 3 section 2.2. The
aircraft loads were determined at an altitude of 10km at Mach 0.85. The wing was trimmed to a
MTGW of 18000kg. A load factor of 6 was applied to the pressure forces on the wing and a
factor of safety of 1.5 was applied to the structural stress constraint. The convergence of the
optimization routine is in Figure 6.1, with a final optimal mass of 385.7 kg.
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Figure 6.1 - Mass convergence plot for reference wing
The convergence of the mass objective is really quick up to 100 iterations. After this point the
change in the objective is very small since the optimization algorithm is refining the design. The
optimal structural design for the reference wing is listed in Table 6.1. The leading and trailing
edge offset are active with the lower bound and the scaling in the X direction (for all the
variables) is active with the upper bound.
Table 6.1 - Optimal design for wing structure of reference wing
Leading Edge Offset 0.1 Module Scale X 0.99
Trailing Edge Offset 0.6 Module Scale Y 0.979
Wing Root Scale X 0.99 Wingtip Scale X 0.99
Wing Root Scale Y 0.962 Wingtip Scale Y 0.973
Wing Tip Scale X 0.99
Wing Tip Scale Y 0.951
The stress constriant distribution is shown below. If the constraint is close to zero, this indicates
that the stress is active with the yeild limit of the material. This active constraint occurs at the
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wing root (high reaction load) and the wing-module junction (low bending stiffness). The
bending stiffness distribution is shown in Figure 6.3.
Figure 6.2 - Stress distribution for reference wing
Figure 6.3 - Bending stiffness distribution for reference wing
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Because of the wing taper and scaling of the wingbox, the bending stiffness decreases in a non-
linear fashion. This gives a design with decent stiffness at the wing root and a very low stiffness
for the modules and wingtip. Since we are minimizing the mass, it is not surprising that the
highest stress concentration occurs at the wing root and at the main wing tip. The figures for the
cross-sectional areas of the wing root, modules, and wingtip are in appendix C.
6.2.1 Optimal Design for Two Morphing Stages
The MDO routine has been split into two optimizations. The first 400 iterations will use an IDF
approach (the wing will not be trimmed at each iteration) and for an additional 400 iterations an
MDF approach (the wing will be trimmed) will be used. For the IDF approach the structure and
aerodynamic models will only need to be called twice (for each morphing stage) for each particle
(16000 function calls for each discipline). For the MDF approach however, to trim the aircraft,
three calls to the aerodynamic model is required (48000 aerodynamic calls and 16000 structure
calls). The reason for this combined IDF-MDF approach is very simple: better efficiency. The
IDF approach is faster since it does not waste time trimming a wing design that is far from the
optimum. The MDF is more appropriate for the refinement stage during the optimization. The
convergence plot is shown below.
Figure 6.4 - Fuel convergence for 2 morphing stage case
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During the IDF run, it required 73 iterations to find a feasible solution. Up to this point the
algorithm needed to find a design that would satisfy the trim constraint. There is a slight increase
in the objective at the start of the MDF routine. For the IDF the trim constraint was a loose
constraint and for the MDF the constraint was a lot tighter. This would change the load
distribution and affect the wing’s performance. The final wingtip geometry for the first and
second stage is in Figure 6.5 and Figure 6.6 respectively.
Figure 6.5 - Morphed wingtip for the first stage
For the first stage the wingtip follows a traditional winglet design. The modules are swept back
slightly with a large dihedral.
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Figure 6.6 - Morphed wingtip for the second stage
The second stage design is similar to the first, with the only significant difference being the
amount of dihedral. Figure C.3 and Figure C.4 show the first and second morphing stage wing
shapes in 3 different orientations. The performance for the optimal and reference wing designs is
in Table 6.2.
Table 6.2 - Performance results for the 2 Morphing stage case
Stage 1 Stage 2
Reference Optimal Reference Optimal
ΔWf 1.58e4 1.31e4 1.45e4 1.19e4
Mass 387.5 422.7 - -
CL 0.34 0.482 0.31 0.444
CD 0.0135 0.0158 0.0124 0.0143
CL/CD 25.2 30.6 25 31.1
αtrim 0.0023 0.4 -0.2516 0.102
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The total amount of fuel consumed for the optimal design is 2.502e4 N. This accounts for 14.2%
of the MTGW. The total amount of fuel consumed for the reference wing is 3.02e4 N which is
17.1% of the MTGW. The ratio of the optimal fuel consumed to the reference wing is 0.83,
which means a 17% fuel savings. The lift to drag ratio shows us that for the reference wing, the
ratio is declining slightly however for the optimal design the ratio is increasing. The reference
wing did not have a wingtip device to improve efficiency. If the reference wing had a wingtip
device and if we assume the lift to drag ratio of the optimal wing’s first stage is constant for the
6000 km range, the fuel consumption is 2.54e4 N. Comparing this to the optimal design for the
two morphing transitions, there is only a 1.2% reduction in fuel. The stress distribution is plotted
next in Figure 6.7.
Figure 6.7 – Stress constraint distribution for 2 morphing stages
The stress along the wing follows the same general trend as the reference wing, with the stress
peaking at the wing root and wingtip. Since the mass of the wing is not being minimized, the
constraints are not active at the root and the tip. The final mass was almost active with the
reference mass constraint. The 10% allowance on the mass constraint was more than sufficient to
confine the wing mass. A tighter constraint could have been implemented.
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6.2.3 Optimal Design for Four Morphing Stages
To improve performance of the aircraft, additional morphing stages are required. The number of
morphing stages has increased to 4, which increases the problem dimension to 76. Just like
before the MDO routine has been split into two, with an IDF approach to start and an MDF to
finish. Each approach was executed for 400 iterations. Since the number of morphing stages is 4,
this doubles the number function calls compared to the previous test case (32000 for IDF, 96000
aerodynamic and 32000 structure for MDF). The convergence plot is presented in Figure 6.8.
Figure 6.8 - Fuel convergence for 4 morphing stages
For the IDF portion of the optimization, convergence is very gradual, however when the MDF
approach is used, the convergence is much more rapid up to 550 iterations. After this point the
convergence slows and only minor improvements in the objective are obtained.
Table 6.3 - Performance results for 4 morphing stages
Stage 1 Stage 2 Stage 3 Stage 4
Reference Optimal Reference Optimal Reference Optimal Reference Optimal
ΔWf 8071 6746 7725 6464 7410 6128 7120 5837
Mass 385.7 424.26 - - - - - -
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CL 0.339 0.541 0.324 0.512 0.309 0.501 0.295 0.483
CD 0.0135 0.0179 0.0129 0.0169 0.0124 0.0162 0.0119 0.0155
CL/CD 25.2 30.3 25.1 30.4 25 30.8 24.8 31.2
αtrim 0.0023 0.76 -0.1276 0.653 -0.252 0.325 -0.371 0.266
The total fuel consumed for the entire cruise stage is 2.52e4 N, which amounts to approximately
14% of the MTGW. The total fuel consumed for the reference wing is 3.03e4 N, which amounts
to 17% of the MTGW. The ratio of the optimal to the reference fuel consumed is 0.832, which is
a fuel savings of 16.8%. This is slightly less than the fuel savings for the 2 morphing stage case.
Due to the increased number of design variables and the four objectives, it is possible that the
allowed number of iterations was insufficient for this case. More iteration would have provided a
better result. Again for this case, the design was compared to reference wing without a winglet.
If we assume that the lift to drag ratio for the first optimal stage is constant for the 6000 km
range, the fuel savings for a 4 morphing stage design is only 0.8%.
Comparing the figures in Figure 6.9, shows why there is a substantial improvement in fuel
consumption of the optimal wing compared to the reference wing.
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Figure 6.9 - Aerodynamic performance of the 4 morphing stages
There is a substantial increase in the lift to drag ratios at the start of cruise for the optimal wing.
The optimal wing has a lift to drag ratio that increases during cruise, but the reference wing’s lift
to drag ratio declines slightly. Another interesting observation is that both the lift and drag
coefficients are declining during cruise in an almost linear trend. This decline in the coefficients
is due to the lower lift required to maintain a trimmed state. The slope of the lift coefficient is
approximately the same; however for the drag coefficient the rate of change in drag is slightly
higher for the optimal design. The shapes of the optimal wingtips are presented in Figure 6.10 to
Figure 6.13.
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Figure 6.10 - First morphing stage wingtip shape for the 4 morphing stage case
For the first morphing stage above the wingtip, the modules have a greater sweep than the main
wing (similar to a raked wingtip) and have an increased dihedral (similar to a typical winglet).
Figure 6.11 - Second morphing stage wingtip shape for the 4 morphing stage case
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The second morphing stage is very similar to the first stage but with a much larger dihedral.
Figure 6.12 - Third morphing stage wingtip shape for the 4 morphing stage case
The third stage seems to have taken to a very different design possibility. The modules have been
swept forward with a much smaller dihedral compared to the second stage.
Figure 6.13 - Fourth morphing stage wingtip shape for the 4 morphing stage case
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The fourth stage has a similar design to the third, however the first two modules have been swept
back slightly but the third module has been swept forward. Also the span extension of the
morphing modules is much larger than that of the third stage. The stress constraint distribution
shows an active constraint at the main wing root and the junction point between the modules and
main wing. Also the mass constraint is active with the reference wing mass constraint.
Figure 6.14 - Stress Constraint distribution for 4 morphing stage case
6.2.4 Low Fidelity Summary
The previous results for the low fidelity cases do suffer from the aerodynamic and structural
dynamic model inaccuracies. For example for the two cases the sweep angle for the main wing
was 20o. This result should not be surprising since, Tornado cannot account for the wave drag
which occurs at transonic speeds. A lower swept wing would produce more lift at a lower angle
of attack. In hindsight it probably would have been better to keep the wing sweep constant. This
could also explain the swept forward shape for the last two stages for the 4 morphing stage case.
For both cases the root chord length is almost 3 m with a taper ratio of 0.3. The small chord
length greatly reduces the wing area so a much larger angle of attack is to trim the aircraft is
necessary. This will increase the lift induced drag of the aircraft but since the lift and drag do not
scale proportionately to each other this increase in drag is not that influential on the overall
objective. The mass of the wing is sensitive to only two constraints: stress and reference wing
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mass. Because of the stochastic nature of the PSO algorithm, any design variable combination
that does not violate these constraints is considered to be acceptable. This is observed with the
stress constraint on the modules and wingtip for both cases. These stresses are very far from the
yield limit. Comparing this to the optimal reference wing stress, the stress of the modules and
wingtip are much higher. This implies that the mass of the optimal design of the 2 and 4
morphing stage case could be reduced even further.
6.3 High Fidelity Results
Just as before with the low fidelity model, an optimal structural of a conventional wing needs to
be determined. The aircraft loads were determined at an altitude of 10km at Mach 0.85. The
wing was trimmed to a MTGW of 18000kg. A load factor of 6 was applied to the pressure forces
on the wing and a factor of safety of 1.5 was applied to the structural stress constraint. Two
structural optimization test cases were executed to determine the effect of spar location. The
design variables were: skin, rib, and spar thickness; structural taper; the number of ribs and
stringers; and the radius of the stringer cross-section. The final results are presented in the table
below:
Table 6.4 - Structural Optimization Results Comparison
Design Variable Spar Location as a Variable Spar Location as a Constant
Mass (Element Volume) 0.57 0.67
Skin Thickness (m) 0.0161 0.019
Rib Thickness (m) 0.005 0.005
Spar Thickness (m) 0.005 0.005
Stringer Radius (m) 0.0001 0.005
Structural Taper 0.4 0.4
Number of Ribs 20 20
Number of Stringers 4 5
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Spar Location %chord 0.8368, 0.25 0.8, 0.15
Comparing the results above it is obvious that having the spar location as a variable can
significantly reduce the mass of the structure. The skin thickness is higher for the fixed spar
location test because the skin will need to contribute more to resist the bending stress. The spar
located at the 0.25c location has a greater height due to the greater thickness of the spar which
increases its bending stiffness. The rib and spar thickness were the same for both cases, and the
stringer radius should have been the same but the lower limit on the variable was different. The
number of stringers for the second case was higher to address the bending stiffness. The number
of ribs was the same for both cases and inspection of the stress contours for the two cases
showed that the maximum stress occurs at the third rib near the trailing edge. It is important to
note that the spar, rib, stringer, and structural taper were active with the lower limit set on the
variables.
Figure 6.15 - Von-Mises stress contours for optimal reference wing structure
For the high fidelity case presented here the MDO routine was terminated early due to the
excessive computational time. After approximately three months of execution only 100 iterations
were completed. A combined IDF and MDF was used just as before; 50 iterations for the IDF
and 50 for the MDF. Even with this shortened optimization we can still learn from the results.
Figure 6.16 below shows the optimal wing shape and Figure 6.17 shows the optimal wingtip
shape for the first morphing stage. The shape of the wingtip has two obvious similarities: there is
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an increase in the wingtip cant (just like a typical winglet design) and an increased wing sweep
angle (just like a raked wingtip).
Figure 6.16 – High fidelity first morphing stage optimal wing shape
Figure 6.17 – High fidelity first morphing stage optimal wingtip shape
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Figure 6.18 shows the optimal wingtip shape for the second morphing stage. The design is very
similar to the first stage. The only notable main difference is the smaller cant angles.
Figure 6.18 - High fidelity second morphing stage optimal wingtip shape
Inspection of both morphing stage designs’ pressure contours near the wingtip has shown low
pressure on the lower surface of the wingtip shown in Figure 6.19. This implies that the wingtip
region has a negative angle of attack. This would reduce the aerodynamic loading at the wingtip,
which would decrease the stress concentration at the wing-module junction.
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Figure 6.19 - Pressure contours of wingtip region for first (left) and second stage (right)
The Von-Mises stress contours are shown Figure 6.20 and Figure 6.21 for the first and second
stage respectively. The maximum stress occurs at the second rib, for the first stage, and there are
some stress concentrations at the junction between the main wing and the first module. The
second stage maximum stress occurs at the junction between the first and second modules. This
is due to the lower cant angle at the wingtip. The forces on the wing are normal to the wing’s
surface. A larger cant angle will give much larger forces in the x and z directions compared to a
lower cant angle where the x and z forces will be small. This smaller cant will produce higher
forces in the y direction which will result in higher stresses for the modules.
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Figure 6.20 - Von-Mises stress contours for first morphing stage
Figure 6.21 - Von-Mises stress contours for second morphing stage
The results for the first and second stage results are in Table 6.5. The total fuel consumed for the
optimal design is 6.5e4 N compared to 6.7e4 N for the reference wing. This is only a 3%
reduction in fuel.
Table 6.5 - First and second morphing stage performance results
Stage 1 Stage 2
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Reference Optimal Reference Optimal
ΔWf 3.6e4 3.5e4 3.1e4 2.98e4
Mass 0.57 0.52 - -
CL/CD 10.86 12.3 10.08 10.1
The final design for the high fidelity model can be found in appendix C. Comparing the results to
the low fidelity two morphing stage case: the sweep angle for the high fidelity is much larger
(35o vs. 20
o) due to the CFD’s sensitivity to transonic conditions, the root chord is larger (3.9 m
vs. 3 m) but with a smaller taper ratio (0.19 vs. 0.3), and the span is much larger as well (7.85 m
vs. 6.85 m). From the low fidelity case it was demonstrated that the lift to drag ratio should
increase for during cruise for better performance, however this is clearly not the case for the high
fidelity. There is definitely room for improvement with more iteration. The pressure contours are
shown in Figure C.9 and Figure C.10 in the appendix for both the first and second stage. There is
a discontinuity in the contour near the trailing edge suggesting that there is a shockwave present.
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Chapter 7
Conclusion
Morphing an aircraft wingtip can provide a substantial performance improvement. Most civil
transport aircraft and business jets are optimized for one specific mission leg but for other flight
conditions they are suboptimal. By altering the shape of the wingtip, we can force the load
distribution to adapt to the required flight condition to improve performance. The concept
investigated in this thesis relies on the use of variable geometry truss mechanisms (VGTMs) to
permit wingtip morphing. The wing has been divided into two inactive components (the main
wing and wingtip) and three morphing modules, composed of VGTMs, which permit individual
or simultaneous cant, span, sweep, and twist morphing modes. The scope of the thesis work was
to prove the concepts viability to reduce fuel consumption with multiple morphing stages at
cruise. This was carried out through establishing and demonstrating an MDO design framework.
7.1 Thesis Conclusion
An aircraft with a morphing wingtip will never maximize its performance if the morphing
wingtip is not utilized to its full capability. Morphing the wingtip to a predefined shape only at
the start of a mission leg (cruise, loiter, climb, etc…), will not fully utilize the capability of a
morphing wingtip. Instead we might as well use a conventional winglet design because by using
a predefined shape we are now using a design that might be optimal for the beginning of the
mission leg but will become eventually suboptimal near the end. An aircraft will only see the
benefit of a morphing wingtip if the wingtip can change shape at different intervals during the
mission leg. For the current work the fuel consumption for cruise was minimized with two and
four morphing stages for a low fidelity model, and a two stage high fidelity model. A combined
effort using an IDF to start the optimization and an MDF to finish the optimization was
implemented. The purpose of this combined approach was to make the optimization process as
efficient as possible. An IDF approach is beneficial for starting the optimization since it requires
fewer discipline function calls and an MDF is more appropriate for the refinement stage of the
optimization routine. The fuel savings for the two and four morphing stage low fidelity cases
were 17% and 16.8% respectively. The four stage case should have produced better results since
there were more optimization points for cruise, but the defined number of iterations was
90
insufficient. The lower fidelity case was unable to predict the more complicated flow
phenomenon (wave drag, flow separation, viscous, etc…) and designed a wing with a small area
with a very low sweep angle. The higher fidelity model, using CFD, is a much better model for
the aerodynamic discipline but requires a lot more time to execute. The higher fidelity
optimization routine was terminated after 100 iterations and the design was able to achieve a 3%
reduction in fuel. This could definitely be improved if more iteration had been allowed. Stress
concentrations occurred near the wing root and wing-module junction. The MDO framework
was able to determine a design that could out perform a conventional wing; however there are a
few areas in which to improve which will be discussed next.
7.2 Future Work
The morphing concept and design framework established within this thesis has only considered a
few aspects of morphing wingtip design. Consequently, there is a lot of room to improve and
expand on the current work. Some of the possible ways in which this can be done are discussed
below.
The framework itself can be improved upon in several areas. The shortcomings of the linear
aerodynamic model used in the framework have been discussed. Consequently, one way to
expand on the current work would be to couple a more sophisticated aerodynamic tool to the
optimization routine. This was done using a CFD model however in an effort to reduce
computational time viscous effects were ignored and the convergence tolerance was not very
tight. Using a more sophisticated CFD model; viscous effects, shockwave-boundary layer
interaction, and the drag which accumulates due to the interacting junction flows between
adjacent modules can be determined. The low fidelity tool is, however, justified at the moment
due to its quick calculation of aerodynamic coefficients and wing loading. This is a requirement
of design studies that entail a large number of function evaluations such as the one here. A
combination of a low fidelity model and a high fidelity model could also be used. For example a
low fidelity model would be most beneficial during the early optimization stage and a more
advanced model for the refinement stage.
There are a few additional considerations that should be included into the current optimization
framework. For example, the number of morphing stages and the length of each could also be
considered as variables, the dynamic effects of the wingtip changing shape during cruise could
91
be used as a constraint, maintaining a stable aircraft is very important but it may require
additional variables for the tail section, and finally activation energy as an objective or constraint
for the VGTM could also be included.
In developing the discipline codes little regard was given toward skin. The skin material must
permit morphing, while maintaining some level of structural integrity and aerodynamic
consistency. For the low fidelity case the structural model completely ignored the skin, however
for the higher fidelity model, a simple attempt to include a morphing skin was employed where
the skin’s thickness would shrink with a module’s span extension. This would give the modules a
lower stiffness when extended. However this approach was only for a rigid aeroelastic analysis
of the wing. A more complete fluid-structure interaction would be a more appropriate. Since the
skin’s thickness is reduced, the ability for the skin panel to maintain aerodynamic consistently
would be a challenge. Also dynamic effect such as panel buffeting would also need to be
considered. Of course trying to meet these requirements in an optimization routine would take a
lot of time to execute.
As mentioned previously, an aircraft will not see the complete benefit of a morphing wingtip if it
does not change its shape during each mission leg. Multiple morphing stages would be required
to achieve its full potential. However there is no guarantee that the flight conditions will be the
same for an aircraft each time it is in operation. Presetting a morphing wingtip shape from wind
tunnel or CFD results will not necessarily provide optimum performance. It may be possible to
design an optimization algorithm to be included with the flight control system of an aircraft to
actively reconfigure the wing at different stages in flight. This of course will create some
difficulties for the optimizer since measurement data from the aircraft instruments could be
noisy.
These are just some of the areas in which to improve the current work. Morphing aircraft
research is a vast subject of interest in academia and in the industry. There are a lot of different
research areas to improve aircraft performance.
92
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Appendix A – PSO Results
A.1 Linear Test Case Results for Static and Dynamic
This section will present the results for the linear social and cognitive parameter distribution.
Table A.1 below lists the parameters that have been used. The best solutions for both the static
and dynamic approach are in bold.
Table A.1 - Linear test case for unconstraint benchmark problems
Test Case Parameters
SPSO TVIW, c1=c2=2
CPSO w=0.72984, c1=c2=1.49618
PSO-L1 TVIW, c1best = c2worst = 1 and c1worst = c2best = 3
PSO-L2 TVIW, c1best = c2worst = 3 and c1worst = c2best = 1
PSO-L3 TVIW, c1best = c2worst = 0 and c1worst = c2best = 2
PSO-L4 TVIW, c1best = c2worst = 2 and c1worst = c2best = 0
The PSO-L1 test case showed the best improvement for the Rastrigin function compared to the
SPSO and CPSO. The dynamic approach showed a decrease in performance compared to the
static (with PSO-L4 as the exception). PSO-L3 and PSO-L4 showed poor performance compared
to the SPSO and showed a small improvement compared to the CPSO. For the Rosenbrock
function, the PSO-L2 for both the static and dynamic approaches showed a substantial
improvement compared to the SPSO and CPSO. The dynamic approach was unable to improve
the static case. PSO-L3 and PSO-L4 showed very poor performance for both static and dynamic.
For the Sphere function, PSO-L1 and PSO-L2 performed well for the static approach, but
achieved poor performance for the dynamic. The dynamic approach most likely encouraged too
much exploration and allowed the algorithm to converge too slowly. PSO-L3 and PSO-L4 again
had very poor performance. For the Griewank function, the static approach PSO-L2 showed the
101
best performance for the static approach and PSO-L1 had the best performance for the dynamic
approach.
Table A.2 - Linear test case results for unconstraint benchmark problems
Static Dynamic
Function Test Case Mean
Fitness
STD
Fitness
Mean
Fitness
STD
Fitness
Rastrigin
SPSO 56.90118 24.33535 55.06076 24.94498
CPSO 308.4377 47.85343 301.9652 47.02567
PSO-L1 40.19204 16.80198 41.30647 15.76697
PSO-L2 42.05146 18.95228 49.71422 16.6171
PSO-L3 252.2899 38.12768 281.7518 40.84652
PSO-L4 241.5799 41.00807 228.4859 39.55752
Rosenbrock
SPSO 5934.871 41186.66 7399.068 27373.19
CPSO 14327244 5527449 14687113 6658570
PSO-L1 179.4968 513.4332 314.4676 590.1053
PSO-L2 92.00987 110.5895 109.1248 308.7556
PSO-L3 1545192 1045339 3451285 2634375
PSO-L4 1297824 987565.8 2560736 1430502
Sphere
SPSO 62.20384 214.8213 132.9173 306.1166
CPSO 3414.737 841.5883 3306.931 825.3003
PSO-L1 1.07E-14 7.78E-14 0.000502 0.001621
PSO-L2 1.31E-15 1.01E-14 0.470508 3.020789
PSO-L3 1046.318 384.7626 1437.601 478.9931
102
PSO-L4 959.9526 392.4152 1369.438 409.9016
Griewank
SPSO 6.212958 15.61322 5.437311 16.05851
CPSO 122.8694 28.17922 127.2405 30.91161
PSO-L1 0.05742 0.058148 0.033803 0.037524
PSO-L2 0.050278 0.048272 0.199606 0.267707
PSO-L3 36.37878 15.66795 54.6751 18.26063
PSO-L4 31.34928 12.91106 48.86456 15.62182
A.2 Step Test Case Results for Static and Dynamic
This section will present the results for the step distribution cases. The parameters are listed in
Table A.3. U represents the upper group, M is the middle group, and L is the low group. With
U=0.2 means that the top 20% of the swarm belongs to the upper group, with M=0.6 means that
the particles that lie between 20% and 60% are part of the middle group, and L=1 means the
particles that lie between 100% and 60% are part of the low group.
Table A.3 - Step test case for unconstraint benchmark problems
Test Case Parameters
SPSO TVIW, c1=c2=2
CPSO w=0.72984, c1=c2=1.49618
PSO-S1 TVIW, U=0.2 c1=3 c2=1 and M=0.6 c1=2 c2=2 and L=1 c1=1 c2=3
PSO-S2 TVIW, U=0.2 c1=1 c2=3 and M=0.6 c1=2 c2=2 and L=1 c1=3 c2=1
PSO-S3 TVIW, U=0.4 c1=3 c2=1 and M=0.8 c1=2 c2=2 and L=1 c1=1 c2=3
PSO-S4 TVIW, U=0.4 c1=1 c2=3 and M=0.8 c1=2 c2=2 and L=1 c1=3 c2=1
For the Rastrigin function, the PSO-S4 performed the best for the static case and the PSO-S1
performed the best for the dynamic case. With the exception of PSO-S4, the dynamic approach
103
for the other test cases yielded better results. The static test cases for the Rosenbrock function,
PSO-S1 to PSO-S3 performed poorly compared to their respective dynamic test cases. The
dynamic technique introduced more exploration to the optimization process which allowed for a
better solution. For the Sphere function, the static approach performs better for all test cases
except for PSO-S3. For the Griewank Function, the PSO-S1 performed the best for both the
static and dynamic techniques. For all the test cases PSO-S1 to PSO-S4, the dynamic technique
performed much better than the static. Comparing the results, PSO-S1 is the best option for
multi-modal functions. The smaller upper group with greater cognitive learning proved to be
more explorative than the other test cases.
Table A.4 - Step test case results for unconstraint benchmark problems
Static Dynamic
Function Test Case Mean
Fitness
STD
Fitness
Mean
Fitness
STD
Fitness
Rastrigin SPSO 56.90118 24.33535 55.06076 24.94498
CPSO 308.4377 47.85343 301.9652 47.02567
PSO-S1 41.7291 16.57188 38.12451 19.76411
PSO-S2 41.45962 16.3556 41.1666 13.6602
PSO-S3 43.06124 21.12839 42.63005 21.90669
PSO-S4 36.51926 12.15523 42.84084 29.20735
Rosenbrock SPSO 5934.871 41186.66 7399.068 27373.19
CPSO 14327244 5527449 14687113 6658570
PSO-S1 1970.379 12653.4 74.551 83.64479
PSO-S2 126.4138 311.8671 87.002 92.801
PSO-S3 199.2434 589.2289 91.0184 100.6304
PSO-S4 1035.156 8993.523 3721.739 17717.38
104
Sphere SPSO 62.20384 214.8213 132.9173 306.1166
CPSO 3414.737 841.5883 3306.931 825.3003
PSO-S1 5.15E-15 1.92E-14 5.92E-12 2.78E-11
PSO-S2 3.15E-11 3.12E-10 8.99E-08 8.99E-07
PSO-S3 8.95E-13 5.28E-12 9.53E-15 7.59E-14
PSO-S4 1.62E-14 7.71E-14 1.33E-11 5.84E-11
Griewank SPSO 6.212958 15.61322 5.437311 16.05851
CPSO 122.8694 28.17922 127.2405 30.91161
PSO-S1 0.052928 0.048512 0.029862 0.036609
PSO-S2 0.063826 0.059922 0.046133 0.045377
PSO-S3 0.07038 0.052941 0.047013 0.049769
PSO-S4 0.056182 0.040877 0.038422 0.035625
A.3 Sinusoidal Test Case Results for Static and Dynamic
This section will present the results for the sinusoidal distribution cases. The parameters are
listed in Table A.5.
Table A.5 - Sinusoidal test case for unconstraint benchmark problems
Test Case Parameters
SPSO TVIW, c1=c2=2
CPSO w=0.72984, c1=c2=1.49618
PSO-Sin1 TVIW, ω=2π and φ=π/2
PSO-Sin2 TVIW, ω=4π and φ=π/2
105
PSO-Sin3 TVIW, ω=6π and φ=π/2
PSO-Sin4 TVIW, ω=8π and φ=π/2
PSO-Sin5 TVIW, ω=2π and φ=0
PSO-Sin6 TVIW, ω=2π and φ=π
PSO-Sin3 provided the best solution for both the static and dynamic techniques for the Rastrigin
function; however their results are very similar. For each test case the static approach yielded
lower mean fitness values compare to the dynamic, but there is slight discernible difference in
performance for PSO-Sin1 to PSO-Sin4. PSO-Sin3 and PSO-Sin5for the Rosenbrock function;
the static and dynamic approach respectively have very similar results and are the best
performers. PSO-Sin1 for the dynamic approach had the best performance for the sphere
function. For all test cases of the Griewank function, except PS0-Sin5, the dynamic approach
was able to improve the mean fitness with PSO-Sin6 being the best performer.
Table A.6 – Sinusoidal test case results for unconstraint benchmark problems
Static Dynamic
Function Test Case Mean
Fitness
STD
Fitness
Mean
Fitness
STD
Fitness
Rastrigin SPSO 56.90118 24.33535 55.06076 24.94498
CPSO 308.4377 47.85343 301.9652 47.02567
PSO-Sin1 40.26834 15.29541 40.7902 17.28688
PSO-Sin2 37.88888 17.27999 38.4533 19.66409
PSO-Sin3 35.76533 14.66962 35.92581 13.06116
PSO-Sin4 37.68982 15.62986 38.84474 17.90841
PSO-Sin5 39.45006 17.18446 42.39678 14.60561
106
PSO-Sin6 36.92318 13.62176 41.57725 15.47511
Rosenbrock SPSO 5934.871 41186.66 7399.068 27373.19
CPSO 14327244 5527449 14687113 6658570
PSO-Sin1 122.27 141.3482 124.19 315.0992
PSO-Sin2 86.95107 108.5391 164.8982 514.0281
PSO-Sin3 80.87965 111.1392 127.6445 317.5367
PSO-Sin4 1017.51 8995.2 96.64395 123.9145
PSO-Sin5 109.3905 314.6066 80.86337 113.5386
PSO-Sin6 94.72582 109.1958 2000.426 12648.13
Sphere SPSO 62.20384 214.8213 132.9173 306.1166
CPSO 3414.737 841.5883 3306.931 825.3003
PSO-Sin1 2.26E-14 1.02E-13 3.67E-18 1.85E-17
PSO-Sin2 1.81E-14 7.28E-14 2.17E-15 7.94E-15
PSO-Sin3 2.54E-14 9.62E-14 7.83E-14 3.74E-13
PSO-Sin4 3.57E-14 1.52E-13 1.76E-13 9.17E-13
PSO-Sin5 8.21E-15 2.87E-14 1.12E-13 1.11E-12
PSO-Sin6 1.85E-14 8.04E-14 9.72E-06 2.94E-05
Griewank SPSO 6.212958 15.61322 5.437311 16.05851
CPSO 122.8694 28.17922 127.2405 30.91161
PSO-Sin1 0.055881 0.045171 0.051112 0.053198
PSO-Sin2 0.045247 0.041617 0.034147 0.044257
PSO-Sin3 0.059125 0.051338 0.032451 0.032623
107
PSO-Sin4 0.051446 0.045137 0.031386 0.029491
PSO-Sin5 0.047068 0.04089 0.06631 0.059792
PSO-Sin6 0.044868 0.045726 0.029877 0.035616
In summary, Table A.7 presents the results for the best social and cognitive distributions. For
each function the dynamic approach provided the best solutions. The sinusoidal and step
distributions achieved the best performance for all the benchmark functions.
Table A.7 – Best overall social and cognitive distributions
Function Static Dynamic
Rastrigin PSO-Sin3 PSO-Sin3
Rosenbrock PSO-Sin3 PSO-S1
Sphere PSO-L2 PSO-Sin1
Griewank PSO-Sin6 PSO-S1/PSO-Sin6
108
Appendix B – Beam Finite Element Model Validation
Figure B.1 - Problem 5.2
Table B.1 - Material and geometric properties for problem 5.2
A 100 in2
Iz 1000 in4
E 30,000 ksi
Table B.2 - Nodal displacement and nodal reaction comparison
[68] FE Model
dx 0.0033 in 0.00329 in
dy -0.0097 in -0.00974 in
rotz -0.0033 rad -0.00329 rad
F1x 26.6 kip 26.8 kip
F1y -2.268kip -2.261 kip
109
M1y -389 k-in -381.5 k-in
F3x 20.63 kip 20.6 kip
F3y 17.42 kip 17.4 kip
M3y 767.4 k-in 769.4 k-in
Figure B.2 - Problem 5.8
Table B.3 - Material and geometric properties for problem 5.8
A 10 in2
Iz and Iy 100 in4
J 50 in4
E 30,000 ksi
G 10,000 ksi
110
Table B.4 - Nodal displacement comparisons
[68] FE Model
dx 7.098e-5 in 7.09e-5 in
dy -0.014 in -0.014 in
dz -2.35e-5 in -2.35e-5 in
rotx -4e-3 rad -4e-3 rad
roty 1.78e-5 rad 1.78e-5 rad
rotz -1.03e-4 rad -1.03e-4 rad
111
Appendix C – MDO Result Extras
C.1 Optimal Reference Wing Mass
Figure C.1- Main wing and wing-module junction cross-sectional areas
Figure C.2 - Modules and wingtip cross-sectional area
112
C.2 Two Morphing Stages
Table C.1 - Optimal module design for 2 morphing stage case study
Design Variable Module 1 Module 2 Module 3
Stage 1
Sweep 9.6 -19.8 18.9
Dihedral 10.5 13.3 19.3
Twist -1.8 -1.28 3.1
Span 0.37 0.25 0.4
X-scale 0.865 0.865 0.865
Y-scale 0.82 0.82 0.82
Taper 0.7 0.917 0.7
Stage 2
Sweep -0.2 7.4 0.1
Dihedral 0 0 8.5
Twist -0.72 -1 2.45
Span 0.25 0.5 0.4
X-scale 0.865 0.865 0.865
Y-scale 0.82 0.82 0.82
Taper 0.7 0.917 0.7
Table C.2 – Optimal main wing and wingtip design for 2 morphing stage case study
Design Variable Main Wing Wingtip
Sweep 20 -3.6
113
Dihedral 0 1.7
Twist 0 -5
Span 6.85 0.723
X-scale 0.99 0.99
Y-scale 0.9 0.7
Taper 0.3 0.82
Chord 3.08 0.3443
Leading Edge 0.13 0.13
Trailing Edge 0.6 0.6
Figure C.3 - First morphing stage for 2 morphing stage case
114
Figure C.4 - Second morphing stage for 2 morphing stage case
C.3 Four Morphing Stages
Table C.3 - Optimal module design for 4 morphing stage case study
Design Variable Module 1 Module 2 Module 3
Stage 1
Sweep 1.85 20 -13.4
Dihedral 0 0 12.2
Twist -1.1 1.66 -2.8
Span 0.3 0.5 0.31
X-scale 0.99 0.99 0.99
Y-scale 0.882 0.882 0.882
Taper 0.89 1 0.78
Stage 2
Sweep -0.24 -0.81 20
Dihedral 0 20 18.5
115
Twist -3.3 5 -5
Span 0.5 0.4 0.5
X-scale 0.99 0.99 0.99
Y-scale 0.882 0.882 0.882
Taper 0.89 1 0.78
Stage 3
Sweep -20 20 -20
Dihedral 0 7 2.7
Twist 2.5 -1.7 -1.34
Span 0.36 0.2 0.31
X-scale 0.99 0.99 0.99
Y-scale 0.882 0.882 0.882
Taper 0.89 1 0.78
Stage 4
Sweep 18.5 -20 -20
Dihedral 8.5 0.96 0
Twist 1.14 -4.1 3.3
Span 0.34 0.39 0.25
X-scale 0.99 0.99 0.99
Y-scale 0.882 0.882 0.882
Taper 0.89 1 0.78
Table C.4 - Optimal main wing and wingtip design for 4 morphing stage case study
Main Wing Wingtip
116
Sweep 20 -11
Dihedral 0 0
Twist 0 2.5
Span 6 1
X-scale 0.99 0.75
Y-scale 0.932 0.813
Taper 0.3 0.61
Chord 3 0.38
Leading Edge 0.1 0.1
Trailing Edge 0.74 0.74
Figure C.5 - First morphing stage for 4 morphing stage case
117
Figure C.6 – Second morphing stage for 4 morphing stage case
Figure C.7 - Third morphing stage for 4 morphing stage case
118
Figure C.8 - Fourth morphing stage for 4 morphing stage case
C.4 High Fidelity Results
Table C.5 - High fidelity first and second stage module results
Design Variable Module 1 Module 2 Module 3
Stage 1
Sweep 13.04 -0.6 7.28
Dihedral 7.4 5.4 6.55
Twist 1.07 2.06 2.35
Span 0.43 0.325 0.464
Rib Thickness 0.006 0.006 0.006
Skin Thickness 0.0061 0.0061 0.0061
Taper 0.814 0.863 1
Passive Radius 0.0031 0.0031 0.0031
Active Radius 0.0038 0.0038 0.0038
Stage 2 Sweep -0.05 -4.7 6.85
119
Dihedral 1.9 4.95 3.24
Twist 2.67 0.67 -0.1
Span 0.29 0.35 0.49
Rib Thickness 0.006 0.006 0.006
Skin Thickness 0.0061 0.0061 0.0061
Taper 0.814 0.863 1
Passive Radius 0.0031 0.0031 0.0031
Active Radius 0.0038 0.0038 0.0038
Table C.6 - High fidelity main wing and wingtip results
Design Variable Main Wing Wingtip
Sweep 34.8 -3.6
Dihedral 0 1.7
Twist 6.2 -5
Span 7.85 0.757
Taper 0.19 0.82
Chord 3.9 0.3443
Ribs 19 0
Stringers 6 0
Leading Edge 0.25 0.25
Trailing Edge 0.72 0.721
Skin Thickness 0.0166 0.006
120
Rib Thickness 0.0052 0.0061
Spar Thickness 0.0086 0.0022
Stringer Radius 0.0049 0
Structural Taper 0.38 1
Figure C.9 - Pressure distribution of first morphing stage wing showing shockwave at
trailing edge
121
Figure C.10 - Pressure distribution of second morphing stage wing showing shockwave at
trailing edge
Figure C.11 - Comparing deformed to undeformed wing for the first morphing stage
Figure C.12 - Comparing deformed and undeformed for second morphing stage