Final_MSc_Thesis_T.A.deJong.
Topology Optimization of 3D Linkages
with application to morphing winglets
Technische Universiteit Delft
T.A. de Jong
T OPOLOGY O PTIMIZATION
OF 3D L INKAGES
WITH APPLICATION TO MORPHING WINGLETS
by
T.A. de Jong
Faculty of Aerospace Engineering
Delft University of Technology
Supervisors:
dr. ir.
ir.
Copyright © Aerospace Structures and Materials (ASM)
All rights reserved.
Picture on the cover is taken from reference [26]
R. De Breuker
E. Gillebaart
D ELFT U NIVERSITY OF T ECHNOLOGY
D EPARTMENT A EROSPACE S TRUCTURES AND M ATERIALS (ASM)
The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering
for acceptance a thesis entitled:
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
WITH APPLICATION TO MORPHING WINGLETS
by
Twan de Jong
in partial fulfilment of the requirements for the degree of Master of Science
G RADUATION COMMITTEE
Committee chairman:
Dr. S. R. Turteltaub, MSc
Committee member:
Dr. Ir. R. Vos
Signature
Signature
Date
Date
Committee member:
Dr. Ir. R. De Breuker
Committee member:
Ir. E. Gillebaart
Signature
Signature
Date
Date
A CKNOWLEDGEMENTS
I would like to start by thanking both Roeland and Erik for their daily supervision. Their experience in the
field of morphing structures has been very valuable during the nine months of my thesis. I enjoyed the fruitful discussions during our biweekly meetings. In addition, I would like to thank the Sergio Turteltaub and
Roelof Vos for their willingness to participate in my graduation committee.
My thesis would have not have been the same without Ron, Johan, Adriaan, and Maarten. As fellow thesis
students they have proven both valuable in discussions as well as friends during the breaks.
Also, the support from my family, friends, and girlfriend was very useful. During the nine months they have
seen my progress, including its ups and downs, and could cheer me up when needed.
Finally, a special thanks to Ron and Johan for taking the time to read through my thesis. I appreciated and
incorporated the feedback to improve my thesis.
Thank you,
Twan de Jong
T.A. DE J ONG
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
A BSTRACT
Topology optimization is the process of optimizing both the material layout and the connectivity inside a design domain [56]. The first paper on topology optimization dates back to 1904, when the Australian inventor
Michell derived optimality criteria for minimum weight truss structures [48]. In 1988 Bendsøe and Kikuchi
[4] published the pioneering paper "Homogenization approach to topology optimization", laying the foundation of numerical optimization methods for topology optimization. Since then, extensive research has been
performed both in academia and industry trying to solve different topology optimization problems.
Due to its general applicability, topology optimization has been applied to the design of many morphing
aircraft structures including morphing leading edges [41, 50, 61, 66, 73], trailing edges [1, 19, 53, 69], or both
[44]. It has also been applied to complete morphing wings [14, 27, 28, 58, 59]. Morphing structures have
the ability to change their shape throughout the flight. This allows for possible weight savings and/or drag
reduction, resulting in a reduced fuel consumption. Despite the great interest in morphing winglets from
both Airbus and Boeing, topology optimization has not yet been used to design morphing winglets, except
for previous work done by E. Gillebaart and R. De Breuker [21]. This thesis continues with the research by
focusing on the following research objective:
"Developing a software tool to design a mechanism for morphing winglets, using ground-structure based
topology optimization, by improving, extending, and expanding the previous 2D inhouse tool."
The research in this thesis is based on previous work done by the faculty. The previous 2D tool is improved,
its capabilities are extended and the tool is expanded to 3D. The current tool effectively demonstrates how
topology optimization, based on the ground-structure approach, can be used to obtain mechanisms for morphing winglets. A two step optimization strategy is formulated, where the mechanism is designed in the first
step and sized to obtain minimum weight in the second step. Both optimizations are done using the globally convergent method of moving asymptotes (GCMMA) optimizer, combined with the adjoint sensitivity
technique. Due to the large rotations of the winglet, geometric non-linearity is taken into account using the
Green-Lagrange strain measure.
Various mechanisms for morphing winglets were successfully designed and sized both in 2D and in 3D. In 2D
mechanisms were found where the cant angle could be regulated, in 3D mechanisms were found where both
the cant angle and the toe angle could be regulated. An aerodynamic load case of 5 [kN] was defined. In 2D
half of this loading was assumed to act on the mechanism, resulting in a minimum weight of 15.0 [kg]. In 3D
the minimum weight was found to be 48.0 [kg].
T.A. DE J ONG
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
C ONTENTS
Acknowledgements
iii
Abstract
v
List of Figures
ix
List of Abbreviations
xi
List of Symbols
1 Introduction
1.1 State-of-the-art . . . . . . . . . . . . .
1.2 Beyond State-of-the-art . . . . . . . . .
1.3 Research Objective and Research Goals .
1.4 Outline . . . . . . . . . . . . . . . . .
xiii
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1
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. 8
. 9
. 10
2 Analysis
11
2.1 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Linear Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Optimization
3.1 Globally Convergent Method of Moving Asymptotes
3.2 Optimization Strategy. . . . . . . . . . . . . . . .
3.3 Mechanism Design . . . . . . . . . . . . . . . . .
3.4 Mechanism Sizing. . . . . . . . . . . . . . . . . .
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35
41
51
4 Design Cases
4.1 Two Dimensional Morphing Winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Three Dimensional Morphing Winglet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Trade-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
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66
75
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5 Conclusions and Recommendations
79
5.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Bibliography
83
A Abaqus Input File 3D Von Mises Truss
87
T.A. DE J ONG
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
L IST OF F IGURES
1.1
1.2
1.3
1.4
1.5
Illustration of the MBB beam based on the continuum approach [38] . . . . . . . . .
Illustration of the inverter mechanism based on the ground-structure approach [13]
Illustration of articulated mechanism (left) and compliant mechanism (right) [30] .
Picture of the model with articulated split wingtips used in the wind tunnel tests [8]
Flowchart of the design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
3
4
7
10
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Flowchart of the finite element analysis . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the 2D Von Mises truss . . . . . . . . . . . . . . . . . . . . . . . .
Verification of FEM code, 2D Von Mises truss with inclinations of 15◦ and 45◦
Illustration of the 3D Von Mises truss . . . . . . . . . . . . . . . . . . . . . . . .
Schematic views of the 3D Von Mises truss in Abaqus with inclination of 15◦ .
Verification of FEM code, 3D Von Mises truss with inclination of 15◦ . . . . . .
Illustration of the Von Mises truss, symmetry conditions applied . . . . . . . .
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11
18
20
21
22
23
25
3.1
3.2
3.3
3.4
3.5
Penalized Young’s Moduli for the SIMP approach (p = 2,3,10) . . . . . . . . . . . . . . . . . .
Scissors benchmark problem and solution by Kawamoto [30] . . . . . . . . . . . . . . . . . .
Topology of the scissors before post-processing (top) and after post-processing (bottom) .
Optimization history of the mechanism design for the scissors . . . . . . . . . . . . . . . . .
The five steps of the actuation of the scissors, before (left) and after (right) post-processing
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32
48
49
50
51
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
Picture of Airbus C295, military transport aircraft [26] . . . . . . . . . . . . . . . . . . . . . . . . .
Truss ground-structure (design space) for 2D winglets . . . . . . . . . . . . . . . . . . . . . . . . .
Converged 2D winglet mechanisms before post-processing . . . . . . . . . . . . . . . . . . . . . .
Optimization history of mechanism design for 2D winglets . . . . . . . . . . . . . . . . . . . . . .
Converged 2D winglet mechanisms after post-processing . . . . . . . . . . . . . . . . . . . . . . .
Trajectories of 2D winglets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Converged 2D winglet mechanisms after sizing. The colors give the cross-sectional areas in [cm2 ].
Optimization history of mechanism sizing for 2D winglets . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the toe angle and cant angle of a winglet . . . . . . . . . . . . . . . . . . . . . . . . .
Truss ground-structure (design space) for 3D winglets . . . . . . . . . . . . . . . . . . . . . . . . .
Views of design space used for the 3D winglets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Actuation of 3D winglet 1 with the input trajectory (blue) and the target output trajectory (pink).
The actual location of the output nodes is indicated in green if it lies within the bounding box
and red if it lies outside the bounding box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization history of mechanism design for 3D winglet, mechanism 1 . . . . . . . . . . . . . .
Converged 3D winglet mechanism 1 after sizing. The colors give the cross-sectional areas in
[cm2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization history of mechanism sizing for 3D winglet, mechanism 1 . . . . . . . . . . . . . .
Actuation of 3D winglet 2 with the input trajectory (blue) and the target output trajectory (pink).
The actual location of the output nodes is indicated in green if it lies within the bounding box
and red if it lies outside the bounding box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization history of mechanism design for 3D winglet, mechanism 2 . . . . . . . . . . . . . .
Converged 3D winglet mechanism 2 after sizing. The colors give the cross-sectional areas in
[cm2 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization history of mechanism sizing for 3D winglet, mechanism 2 . . . . . . . . . . . . . .
2D view of 3D winglet mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Winglet mechanisms trade-off, actuation efficiency versus mechanism weight. The weight of
2D mechanisms is multiplied with 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mechanism 3D2 before (black) and after (blue) aerodynamic loading, last actuation step . . . .
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
T.A. DE J ONG
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57
58
60
61
62
63
64
65
66
66
67
69
70
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71
72
73
74
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77
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
L IST OF A BBREVIATIONS
AaM
BWB
CCSA
DOF
ESO
FEA
FEM
GA
GCMMA
GL
HECS
KKT
LM
LSE
MBB
MDO
MIMO
MISO
MMA
MOPBIL
RAMP
SCP
SIMP
SISO
SRV
SSV
TO
TPE
UAV
T.A. DE J ONG
Aeroelasticity and Morphing
Blended Wing Body
Conservative Convex Separable Approximation(s)
Degree(s) Of Freedom
Evolutionary Structural Optimization
Finite Element Analysis
Finite Element Method
Genetic Algorithm
Globally Convergent Method of Moving Asymptotes
Green-Lagrange
Hyper-Elliptic Cambered Span
Karush-Kuhn-Tucker
Levenberg-Marquardt
Least Square Error
Messerschmitt-Bölkow-Blohm
Multidisciplinary Optimization
Multiple-Input-Multiple-Output
Multiple-Input-Single-Output
Method of Moving Asymptotes
Multi-Objective Population-Based Incremental Learning
Rational Approximation of Material Properties
Sequential Convex Programming
Solid Isotropic Material with Penalization
Single-Input-Single-Output
Sum of Reciprocal Variables
Sum of Squares of Variables
Topology Optimization
Total Potential Energy
Unmanned Air Vehicle
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
L IST OF S YMBOLS
Greek Symbols
αi n
αmax
γcr
γ
∆
∆
∆max
δ
²
²i
η
λcr
λi
λ, λ1 , λ2 , λ3 , λ4
µ
Π
ρi
ρ mi n
ρm
ρ
σi
σy
φcr
φi
Ω
Minimum connectivity correction factor
Minimum volume correction factor
Ith critical buckling load factor
Rotation vector
Downward deflection
Distance between actual and intended nodal location
Maximum allowable distance between actual and intended nodal location
Rotation vector
Strain
Strain of ith element
Vector with Lagrange multipliers
Ith critical buckling load factor
Ith buckling load factor
Vector with Lagrange multipliers
Levenberg-Marquardt scaling parameter
Total potential energy
Cross-sectional area design variable
Minimum cross-sectional area
Material density
Vector containing design variables
stress in ith element
Yield stress
Ith eigenvector describing the critical eigenmode
Eigenvector describing the ith eigenmode
Design space
Latin Symbols
A
A0
Ai
a
A
b
bi
B
B̂
B̃
Bi
B̂ i
B̃ i
C
T.A. DE J ONG
Cross-sectional area
Maximum/original cross-sectional area
Cross-sectional area of ith element
Volume vector
Vector with cross-sectional areas
B̂ x
B̂ i x
Strain rotation matrix
Strain rotation matrix with rows of fixed degrees of freedom removed
Strain rotation matrix with rows and columns of fixed degrees of freedom removed
Strain rotation matrix of ith element
Strain rotation matrix with rows of fixed degrees of freedom removed
of ith element
Strain rotation matrix with rows and columns of fixed degrees of freedom removed of ith element
Compliance
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
xiv
L IST OF F IGURES
C i nput
n
C imi
nput
nod e
nod e
Di
E
E0
Ei
F0
Connectivity at the input node
Minimum connectivity at the input node
Distance between actual and intended trajectory squared
Young’s modulus
Maximum/original cross-sectional Young’s Modulus
Young’s modulus of ith element
Objective function
i nput
Reaction force at the ith input node
Force vector
External force vector
Internal force vector
Levenberg-Marquardt gain factor
Jth constraint
Approximation of jth constraint
Gj
g l obal buckl i ng
Jth global buckling constraint
i nput nod e
Gj
t r a j ect or y
Gj
l ocal buckl i ng
Jth input node connectivity constraint
Fi
F
F ext
F i nt
g
Gj
G̃ j
G
G st r ess compr essi on
G st r ess t ensi on
G vol ume
h
I
Ii
Jf
k
Ks
KG
KH
KL
Kt
l
lˆ
li
lˆi
L
M
N
Nbuckl
Nc
Nd
Ne
Ni n
Nl b
Nn
Nout
Ns
Nt
p
P icr
p
Q
g l obal buckl i ng
Rj
i nput nod e
Rj
Jth trajectory constraint
Local buckling constraint
Compression stress constraint
Tension stress constraint
Volume constraint
Direction of descent vector
Second moment of area
Second moment of area of ith element
Jacobian
Spring stiffness of truss member
Scaling parameter for Kreisselmeier-Steinhauser function
Geometric stiffness matrix
Higher order stiffness matrix
Linear stiffness matrix
Tangent stiffness matrix
Undeformed length
Deformed length
Undeformed length of ith element
Deformed length of ith element
Lower moving asymptote
Total number of constraints
Total number of design variables
Total number of global buckling constraints per step
Total number of compression stress constraints
Total number of free degrees of freedom
Total number of elements
Total number of input nodes
Total number of local buckling constraints
Total number of nodes
Total number of output nodes
Total number of steps
Total number of tension stress constraints
Penalty factor
Critical column buckling load for the ith element
Nodal (external) force vector
Taylor series approximation of the potential energy
Critical buckling load factor of jth constraint
Ratio of C i nput
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
nod e
n
over C imi
nput
nod e
of jth constraint
T.A. DE J ONG
L IST OF F IGURES
xv
t r a j ec t or y
Ratio ∆ over ∆max of jth constraint
l oc al buckl i ng
u xac t
t
u ac
y
ac t
uz
Ratio of elemental force of ith element over P icr
Ratio of elemental stress of ith element over yield stress
Ratio of V over V max
Residual force vector
Residual force vector from the second (linear) step of the finite element analysis
Residual force vector from the first (non-linear) step of the finite element analysis
Actual x-displacement of the output node
Actual y-displacement of the output node
Actual z-displacement of the output node
ux
Target x-displacement of the output node
uy
Target y-displacement of the output node
Rj
Ri
R ist r ess
R vol ume
R
R (L)
R (N L)
tr a j
tr a j
tr a j
uz
U
U
u
∆u
u (L)
u (N L)
V
V
V max
wi
W
x max
x mi n
x
T.A. DE J ONG
Target x-displacement of the output node
Internal strain energy
Upper moving asymptote
Displacement vector
Displacement increment vector
Displacement vector from the second (linear) step of the finite element analysis
Displacement vector from the first (non-linear) step of the finite element analysis
Work done by external forces
Total volume
Maximum volume
Weighting factor of ith load case
Weight
Upper bound design variable
Lower bound design variable
Nodal location vector
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
1
I NTRODUCTION
Topology optimization is the process of optimizing both the material layout and the connectivity inside a design domain [56]. The first paper on topology optimization dates back to 1904, when the Australian inventor
Michell derived optimality criteria for minimum weight truss structures [48]. The author and his research
group extended the Michell’s theory to beam systems and presented the first general theory for topology optimization in the 1970s, see [48] for the contributing papers. In 1988 Bendsøe and Kikuchi [4] published the
pioneering paper "Homogenization approach to topology optimization", laying the foundation of numerical
optimization methods for topology optimization. Since then, extensive research has been performed both in
academia and industry trying to solve different topology optimization problems.
Due to its general applicability, topology optimization has been applied to many different disciplines, including civil engineering, material science, automotive, and aerospace engineering. Within aerospace engineering extensive research has been done on how topology optimization can be used to design aircraft structures.
Morphing aircraft structures are structures with the ability to change their shape, which potentially increases
the structural and aerodynamic efficiency. With the continuous drive to decrease the aircraft weight, extensive research is performed on morphing aircraft structures via topology optimization. Many papers have been
published where topology optimization has been used to design morphing leading edges [41, 50, 61, 66, 73],
trailing edges [1, 19, 53, 69], or both [44]. It has also been applied to entire morphing wings [14, 27, 28, 58, 59],
as well as to flexible skins [29].
Despite the fact that topology optimization is often applied to design morphing aircraft structures, it has
hardly been applied to design morphing winglets. To the author’s best knowledge only E. Gillebaart and R.
De Breuker have done research in this field [21], most of the research on the design of morphing winglets
does not involve topology optimization [5–8, 15–17, 22–24, 52, 60, 68, 72]. Also industry has shown great interest in morphing winglets due to the potential increase in structural and/or aerodynamic efficiency. Boeing
for example has filed two patents in 2010 and 2014, related to morphing winglets [49, 71]. In addition the EU
project Novel Air Vehicle Configurations: From Fluttering Wings to Morphing Flight (NOVEMOR) resulted in a
paper from DLR on morphing winglets [11]. Also the CleanSky 2 project from the EU has set goals to research
morphing winglets. However, the majority of these efforts focuses on 2D morphing winglets (including the
work done by E. Gillebaart and R. De Breuker), which need to be extended to 3D for practical applications. In
addition, only a few of these papers involve some sort of optimization [15, 16, 74]. Rather than optimizing the
mechanism or device that allows for the morphing of a winglet, these papers focus on maximizing range and
endurance or minimizing the drag of a wing with a morphing winglet.
The state-of-the-art of topology optimization as well as its application in the aerospace industry will be given
in Section 1.1. Section 1.2 will elaborate on the research gap identified and how this research will fill the gap.
Based on the gap the research objective for this thesis is formulated and stated in Section 1.3. Finally, Section
1.4 will provide the outline for this thesis.
T.A. DE J ONG
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
2
1. I NTRODUCTION
1.1. S TATE - OF - THE - ART
In this section a brief overview is given of topology optimization and how this has been used to design morphing aircraft structures. First a few important branches within topology optimization are discussed, including
the continuum and the ground-structure approach, gradient and non-gradient methods, as well as some
considerations regarding mechanism design. Then an overview is given of the various morphing aircraft
structures that have been designed using topology optimization.
1.1.1. T OPOLOGY O PTIMIZATION
As mentioned earlier, topology optimization can be defined as the process of optimizing both the material
layout and the connectivity inside a design domain [56], usually subjected to one or more constraints. Solving
topology optimization problems can be done by discretizing the design domain into many finite elements.
Then the density of each element, i.e. the material distribution, is determined such that the objective function is minimized while satisfying the given constraints. The two different ways of discretizing the design
domain are known as the continuum and the ground-structure approach, which will be discussed in the next
paragraph. Both approaches yield an optimization problem that can be solved with gradient or non-gradient
based methods. The different methods are covered here as well. Finally, mechanism design has implications for the problem formulation. The different approaches to design mechanisms are discussed in the last
paragraph.
C ONTINUUM AND G ROUND -S TRUCTURE A PPROACH
The two different types of topology optimization are the continuum approach and the ground-structure approach. The continuum approach is the most used both in academia and industry. The design space in this
approach is completely filled with material, allowing for more flexibility in the design. Figure 1.1 shows the
design space (top) and the optimized design (bottom) for the Messerschmitt-Bölkow-Blohm (MBB) beam
based on the continuum approach.
Figure 1.1: Illustration of the MBB beam based on the continuum approach [38]
When solving the continuum approach with gradient based methods like the density methods, the optimization process requires an additional step involving regularization techniques to prevent numerical issues such
as checker-boarding and mesh dependency. By checker-boarding is meant the formation of solid-void elements in a checkerboard pattern. Mesh dependency means that the topologies will change with different
discretization sizes, while the problem formulation remains unchanged [12]. The two primary regularization
methods are filtering and constraint techniques. Filtering techniques modify the density variables or the sensitivities directly, while the constraint techniques add local or global constraints to the optimization problem.
An overview of different density and sensitivity filtering techniques can be found in [12] and [55]. Density filters work by modifying the element density based on a function involving the surrounding density elements.
The sensitivity filter calculates the sensitivities based on a weighted average of surrounding sensitivities [55].
The original sensitivity filter was presented by Sigmund in 1997 [54]. An overview constraint techniques can
be found in [12]. As a consequence of the filtering techniques (both density and sensitivity) grey material will
be present between the solid and void/empty regions. To remove the grey regions, several schemes have been
developed including morphology-based operators [55, 70] and projections of filtered densities via the relaxed
Heaviside function [25, 34].
The second approach is the ground-structure approach. The design space is not completely filled with material, but consists of nodes interconnected via truss members. Figure 1.2 gives an example an inverter mechT OPOLOGY OPTIMIZATION OF 3D LINKAGES
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anism based on the ground-structure approach. The optimizer selects which combination of truss members
yields the optimal solution. The final design consists then only of a few truss members.
Figure 1.2: Illustration of the inverter mechanism based on the ground-structure approach [13]
The ground-structure approach is very suitable for optimizing truss structures. It has also several advantages
compared to the continuum approach. One is the limited post processing time, i.e. the ease of interpretation. Especially when designing mechanisms, the continuum approach might give difficulties in the post
processing step when thin regions are to be removed or function as a hinge. In addition, the density approach
based on the ground-structure does not require any regularization techniques. Kawamoto has done research
on how the ground structure approach can be used to design mechanisms. In his tool also the buckling is
included as a constraint [30].
G RADIENT AND N ON -G RADIENT O PTIMIZATION M ETHODS
Two different classes of gradient based methods include the density methods and the boundary variation
methods. Within the non-gradient based methods one distinguishes between the hard-kill methods and bioinspired cellular division methods. All four methods will be discussed here.
The first density-based method was proposed by Bendsøe in 1989 [3]. Since then several other methods have
been developed, all with the same working principle. These methods have in common that they make use
of the density variables to steer the optimizer towards a 0/1 solution. The most well known density method
is the SIMP approach as proposed 1989 [3] and can be categorized as a material interpolation scheme. This
implicit method penalizes intermediate densities to promote a 0/1 solution. Other approaches include the
RAMP, Hashin-Shtrikman, Reuss-Voigt and quadratic penalization. Why penalization is required for gradient
based optimization methods and how these different penalization schemes work, will be discussed in more
detail in Chapter 3.
Boundary variation methods are more recently developed and originate from shape optimization techniques.
In boundary variation methods implicit functions define the structural boundaries, in contrast to the density
methods where explicit parametrization of the design space is used [12]. The two most used boundary variation methods include the level set method and the phase-field method. These methods however are not
suitable for the ground-structure approach.
The first class of non-gradient based methods is the class of the hard-kill methods. These methods do not
involve any sensitivities [12]. Hard-kill methods work by gradual addition and removal of finite amount of
material from the design space. The addition and removal of material is based on heuristic rules. The most
well known hard-kill method is the Evolutionary Structural Optimization (ESO) method. This method works
with removal of material only.
The last class is the so-called bio-inspired cellular division method, proposed by Kobayashi in 2010 [37]. This
method is able to generate both discrete and continuum structures and is inspired on the cellular division
process. It develops the topology in stages based on a sequence of tasks. The set of rules that define the
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1. I NTRODUCTION
tasks are called the map-L system. When using a genetic algorithm (GA) for this method, the design variables
become the set of rules.
M ECHANISM D ESIGN
Two common types of mechanisms are the articulated mechanisms and the compliant mechanisms. Articulated mechanisms consist of several linkages connected via (hinged) joints. These mechanisms get their
mobility via their free joints. While compliant mechanisms gain their mobility via the elasticity of its components [30]. An example of both types of mechanisms can be found in Figure 1.3.
Figure 1.3: Illustration of articulated mechanism (left) and compliant mechanism (right) [30]
What is important in mechanism design (both for articulated as well as complaint mechanisms) is the trajectory of one or more nodes during the deformation, i.e. the kinematics from the mechanism. In general for
mechanisms it is required that one or more nodes follow a certain trajectory. One way to incorporate the trajectory is by formulating an objective function which minimizes the least squares error between the intended
trajectory and the actual trajectory of the mechanism [45].
A different way would be to include an additional constraint which specifies the allowed deviations from the
intended trajectory. This approach does not optimize the mechanism for the trajectory. In this formulation
the nodes follow the intended trajectory sufficiently well (as dictated by the constraint), and could then be
optimized for something else instead, like minimum compliance [35]. Kawamoto’s research was focused on
articulated mechanisms [30], where a least squares error was formulated as the objective function. The objective results in structures that are able to follow the intended trajectory, but that does not necessarily result
in mechanisms. It could very well be that the truss members are strained to be able to follow the intended trajectory. Therefore an additional constraint was included to regulate the number of degrees of freedom. Since
truss structures were considered the Maxwell’s rule could be adopted, which for the 2D discrete formulation
looks like [33]:
d m = 2n − b − r
(1.1)
In this equation d m represents the number of DOF, n represents the number of nodes in the structure, b
represents the number of independent non-redundant bars (i.e. no overlapping bars), and r represents the
number of reaction forces. This rule can be used since articulated mechanisms get their mobility via one or
more DOFs. The parameter b can be obtained by determining the rank of the equilibrium matrix. Kawamoto
also explains how this rule can be used in the relaxed formulation [32].
1.1.2. T OPOLOGY O PTIMIZATION AND M ORPHING A IRCRAFT S TRUCTURES
This section focuses on the research done in the field of topology optimization and morphing aircraft structures. Various morphing structures are covered, including the leading edge, trailing edge, wing, skin and
winglet. Since no research has been done on the design of morphing winglets via topology optimization, the
last section is focused on research done on morphing winglets in general.
M ORPHING L EADING E DGE
In 2003 Lu and Kota [39] have published a paper on the design of a compliant mechanism for a morphing
leading edge. They used the ground-structure approach to parametrize the design space. Their objective
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function involves the Least Square Error (LSE) measure as well as a modified Fourier Transform to rank the
design shapes. A GA algorithm is used to solve the optimization problem. In 2005 they published a different
paper [40], which could be seen as an improvement of their previous work. In this paper a modified groundstructure approach is used, by them described as the load-path method. This method analyses a limited
number of distinct load paths, rather than every truss element separately. By combining different load paths,
the optimal solution is sought. This new approach should overcome the issues of disconnected structures.
In addition, this approach does not require an a-priori specified mesh. In their paper they find the desired
shapes effectively with their new method. The load path method is further improved from 2007 and onwards
[50, 51]. This method also prevented intersecting members to guarantee manufacturability.
Also the faculty of aerospace engineering of Delft University of Technology has done research in this field.
In this research a design method, using topology optimization, was presented for an actuation system of a
morphing leading edge [66]. This work was done as part of the Leading Edge Actuation Topology Design and
Demonstration (LeaTop) project. The Solid Isotropic Material with Penalization (SIMP) interpolation scheme
was used. One year later additional research was done by the faculty and a paper on the design and demonstration of the leading edge actuation system was presented [41]. Recently, in 2015, research was presented
on the experimental evaluation of the morphing leading edge. They demonstrated a rotation of the leading
edge of at least 5 degrees. An extensive outline of the experimental set-up used can be found in their paper
[61].
Other work has been done by the Northwestern Polytechnical University [73]. In their research, topology
optimization was used to design a compliant morphing leading edge involving composite materials. The objective function was to minimize the LSE of the actual shape and the target shape. The SIMP method was
adopted to penalize intermediate densities, while the sensitivities were calculated using the adjoint method.
The optimized solution was also experimentally tested.
Finally, Kintscher from DLR has done research on morphing leading edges as well [36]. They have developed
several leading edge devices for morphing. Experimental testing was done on a full-scale leading edge. A
compliant skin has been proposed, and the internal structure is designed via topology optimization. This
structure is used to morph the leading edge.
M ORPHING T RAILING E DGE
Amongst others, Friswell and Baker are two well-known researchers for their work in morphing aircraft structures. They have published two papers [1, 19] on the optimization of morphing trailing edges. The paper
from 2008 focused on the optimization of hierarchical structures. Their model has been applied to two examples, being the optimization of structure’s anisotropy and the optimization of truss and skin elements of
compliant trailing edge mechanism. Their optimization process involves both gradient based optimization
algorithms and GA algorithms. In their second work they investigated truss structures for a variable camber
trailing edge. In the first step a statically and kinematically stable structure is proposed, where in the second
step truss members are replaced by actuators. To reduce the complexity of the design, Baker and Friswell
propose for further research to consider compliant members rather than articulated truss members.
In 2008 another paper [53] was published on the design of a compliant mechanism for a morphing trailing edge. The proposed tool was written in MATLAB, which called the commercial finite element software
package ANSYS to conduct the finite element analysis (FEA). The topology optimization was based on the
ground-structure approach to design the compliant mechanism. Due to the large deflections geometric nonlinearity was accounted for as well. Although the GA algorithm used could be able to optimize the topology
and dimension simultaneously, a second optimization run (direct search) was used to optimize the dimensions.
Also Delft University of Technology has done research in this field. Vos et al. [69] used topology optimization
techniques to optimize a pressure adaptive honeycomb structure for a morphing flap. The trailing edge was
designed to morph and act as a flap. In further research the computational efficiency should be improved to
reduce the time to find the best topology.
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M ORPHING L EADING AND T RAILING E DGE
Trease and Kota [67] have developed a tool to optimize a compliant mechanism with embedded actuators and
sensors. The tool optimizes simultaneously the structural topology as well as the location of the actuators and
sensors for maximum efficiency and adaptive performance. The efficiency is specified as the work done by
the output node(s) divided by the work done by the input node(s). Their research is also compatible with one
or more input and output nodes. They classify the problems as single-input-single-output (SISO) problems,
multiple-input-single-output (MISO), and multiple-input-multiple-output (MIMO) problems. Their work is
based on the general ground-structure approach and was applied to morph an airfoil. Further improvements
would be to include non-linear FEA and the load-path approach, as previously proposed in 2005 [40].
In 2004 Maute and Allen presented a paper on 3D aeroelastic structures [43]. In their work they consider
fluid-structure interaction as well. Central in the work is a Sequential Augmented Lagrangian method for
solving the optimization problem. The analytical adjoint method is used to determine the sensitivities. In
2006 Maute and Reich [44] came up with an integrated design tool, where both an aerodynamic and structural module are combined. Their approach is compared to the traditional two step process, where first
the aerodynamic shape is optimized and then the structure that could maintain the shape. The problem is
solved via the sequential convex programming (SCP) algorithm method of moving asymptotes (MMA). The
sensitivities required for MMA were calculated using the adjoint method and the SIMP material interpolation
scheme was adopted. It was found that the design performed better compared to the two step process, but
the computational time increased by a factor ten.
M ORPHING W INGS
Topology optimization has also been applied to design complete wings. Eves et al. for example have published a paper in 2009 [14] on the wing design of non-conventional aircraft. Topology optimization turned
out to be a very effective tool to use in the design for unconventional aircraft, where little knowledge in advance is available for the optimal structure. In their research they optimized the wing of a blended wing body
(BWB) unmanned air vehicle (UAV), using the SIMP material interpolation scheme. Also, special attention
was given to wing skin buckling and two different approaches were suggested to incorporate this into the
design. They suggested to include the aerodynamic performance in further research.
Topology optimization has also been applied to design an in-plane morphing wing [27]. In this paper the
ground-structure approach is adopted. The optimization problem is relaxed and solved by using the globally
convergent method of moving asymptotes (GCMMA) with adjoint sensitivities. In addition to the topology,
also the location and the relative force intensities of distributed actuators are optimized. In addition, both
single and multi-objective formulations were used in the optimizations. In 2008 [28] the authors published
a second paper, with the aim of improving and extending their previous work. In the first paper the wing
possessed only two stable configurations. In their second paper multiple configurations of the wing were
possible. For further research the authors propose to include out-of-plane deflection of the wing as well as
aeroelastic effects.
Sleesongsom, Bureerat and Tai published a paper on the design of a morphing wing in 2013 [59]. Their work
is based on the ground-structure approach, split up into two different strategies. The first strategy uses topology and sizing optimization for the design variables, the second strategy includes the nodal locations as well.
They used a multi-objective formulation, to minimize the mass, prevent aeroelastic effects such as divergence
and flutter, as well as stress limits. The optimization problem was solved with a multi-objective populationbased incremental learning (MOPBIL) algorithm. Both strategies give Pareto fronts with unconventional results.
M ORPHING S KINS
The majority of the research on morphing structures is focused on the internal structure. This structure is
designed to morph and should be able to carry the external loads. As a consequence of these morphing motions, the requirements on the morphing skin are challenging. Not only does the skin experience very large
in-plane deformations, usually there are also requirements on the out-of-plane behavior of the skin. As a
result, often regular materials are not able to cope with these requirements and man-made materials have
to be designed. The importance of morphing skins has also being stressed in the paper of Friswell et al. that
focused on the design of morphing trailing edges [18].
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Work that deals with the design of morphing skins is performed by Joo, Reich, and Westfall [29, 46]. Their
work focused on the development of an engineered composite skin for morphing aircraft applications by a
two-step process. The first step determines the macroscopic bulk properties across the three layer skin, while
the second step optimizes the micro-structure of the skin. The authors mentioned that the manufacturing
process would be easier for monolithic skins with equivalent properties.
Thuwis et al. [65] have also published a paper on the optimization of skins. This work proposes a tool to
optimize a variable-stiffness composite skin suitable for morphing applications. The variable-stiffness is
achieved by changing the skin thickness and the spatial fibre angle. A two-dimensional aero-solvo-elastic
framework has been developed. The skin optimization is performed by a gradient-based algorithm, making
use of adjoint sensitivities. The authors suggest to include the aerodynamic behavior in the early design stage
due to its large effect on the skin.
M ORPHING W INGLETS
This section is concerned with the research field of morphing wingtips/winglets. Many different papers have
been published about research in this field. Important papers will be discussed here, starting with those from
P. Bourdin, A. Gatto, and M.I. Friswell.
Over the years Bourdin, Gatto, and Friswell have published several papers on morphing winglets. They proposed variable cant angle winglets for morphing aircraft control [5, 7]. These winglets are proposed as an
alternative to the conventional control surfaces such as the aileron and rudder. Both experimental and numerical tests were performed and it was found that these winglets result in a highly coupled flight control
system. In addition, they found that not the complete flight envelope could be realized with only one morphing winglet per wing, additional control surfaces were necessary to support the manoeuvres. A different
suggestion was made to split the winglets, such that on each wingtip two winglets are present. The suggestion
of articulated split wingtips was further researched and two paper were published [6, 8]. Again numerical and
experimental methods were used to analyze the model. It was found that these split wingtips were capable of
providing the required control moments. In fact multiple different wingtip configurations existed for a particular control moment, meaning that a second objective (such as drag) could be minimize. The proposed
model worked well at moderate and high lift coefficients, making it suitable for low speed aircraft. In Figure
1.4 the prototype used in the wind tunnel tests of Friswell can be found.
Figure 1.4: Picture of the model with articulated split wingtips used in the wind tunnel tests [8]
Friswell has also contributed to other research entitled "Morphing Winglets for Aircraft Multi-phase Improvement" [68]. This paper focuses on the hierarchical methodology towards the final morphing application
(MORPHLET). The paper starts with the selection process of smart materials, followed by an optimization
of non-planar wing schedules. This optimization tool includes an aerodynamic module, wing weight module, and a Breguet range module. The optimization is performed by a GA algorithm.
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In addition, DLR presented their work on the 4th EASN Association International Workshop on Flight Physics
& Aircraft Design Conference [11] in Germany. The research, as part of the NOVEMOR project, focused on
a droop-nose morphing wingtip. In the design tool first the composite skin was optimized, followed by the
internal compliant structure. The full-scale wind tunnel model demonstrated the working principle of the
morphing wingtip.
Three other researchers (J. Wittmann, H. Baier, and M. Hornung) have also presented their work [72] on the
German Aerospace Congress in 2010. Friswell has done research with them as well, but that was focused on
an hierarchical approach for the conceptual design of morphing devices. The research done by Wittmann,
Baier, and Hornung was focused on the mission performance optimization by making use of morphing wing
tip devices. More on this work can be found in reference [72].
Also Gomes, Falcão and Suleman have done research in this field. Their first work dates back to 2009 [22]. In
this work a multidisciplinary optimization (MDO) strategy is proposed to design multi stable composite plate
configurations for morphing wingtips. One year later additional work has been presented on MDO of a morphing wingtip [15]. In this work the wingtip is able to rotate over the vertical axis (toe angle) and the aircraft’s
longitudinal axis (cant angle). Their work focuses on optimizing a performance metric (such as lift, drag,
lift-to-drag-ratio, etc.), subjected to stress an displacement constraints. The structural analysis was done in
ANSYS and the computational fluid dynamics in CFX. In 2011 additional work was presented [16]. This work
included experimental tests of the morphing wingtip. A major improvement of 20% reduction in ground roll
was found. Also, a 2% increase in fuel efficiency in cruise was realized. Also in 2013, research concerned
with morphing winglets was presented [23]. This work involved significant experimental work and a detailed
prototype was manufactured. Two separate servos were used to change the cant angle and the toe angle respectively. The experimental work included tip displacement measurements for different cant and toe angles.
Other work has been done by the Georgia Institute of Technology [52, 60]. Their first work dates back to
2001, focusing on active multiple winglets to improve the performance of UAVs. By applying numerous active
winglets it was found that these could replace ailerons. In addition, these winglets provide gust alleviation,
which improved the handling quality as well as the sensor performance. Two different methods were used
to predict the performance of the winglets. The first method adopted a panel method and CAD geometry
for the analysis. The second method used the characteristics of the baseline model combined with results
from winglets from a generic wing. It was concluded that up to 40% increase in range and endurance could
be achieved. Also, due to the active winglets a higher lift coefficient was found for cruise conditions, which
could be used to increase the payload. Their second paper was in collaboration with Star Technology and
Research, Inc. on the performance analysis of a wing with multiple winglets. In this work a NACA 0012 airfoil
was used with morphing winglets and tested in the wind tunnel. A reduction of induced drag and an increase
in lift-over-drag (L/D) of 15-50% was experimentally found.
1.2. B EYOND S TATE - OF - THE - ART
In this section the research gap derived from the state-of-the-art will be addressed. In addition, the relevance
will be covered as well. Hence this section covers the progress beyond state-of-the art, which forms the basis
for the research goals mentioned in the next section. From the previous section two things become apparent.
To the best of the author’s knowledge, the following statements can be made:
1. Topology optimization has not been applied to design the mechanism for morphing winglets
2. 3D ground-structure based topology optimization has not been used to design mechanisms in the
aerospace industry
Topology optimization techniques are common in the design of morphing aircraft structures. However, the
research done was mainly focused on 2D. In addition, it was observed that topology optimization has not
yet been applied to design the mechanism of morphing winglets. Optimizations have been performed where
morphing winglets were included, but the objective was to maximize range and endurance or to minimize
drag. Here the ability of the winglet to deflect was used to increase the range, rather than optimizing the material distribution to design a morphing winglet.
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It has been observed that topology optimization had not been applied to design a mechanism for morphing
winglets. This is interesting given the great interest shown in the design of morphing winglets and the use of
topology optimization for morphing aircraft structures. Many project including those from the EU (NOVEMOR and Clean Sky) and Airbus (MORPHLET) have shown interest in morphing winglets. Given the gap and
its relevance it has been decided to focus the research on the design of morphing winglets using topology
optimization. As mentioned earlier, most research was limited to 2D, whereas 3D is required for practical
applications. Therefore, the research will also focus on 3D morphing winglets using topology optimization .
Since this research area is very broad, a specific research objective and corresponding goals are defined in the
next section.
1.3. R ESEARCH O BJECTIVE AND R ESEARCH G OALS
In the previous section it was mentioned that only limited research has been done on 3D topology optimization. It was also found that no research has been done on the design of morphing winglets using topology
optimization techniques. Given this research gap and the significant interest the industry has shown in morphing winglets, it would be interesting to explore the field of topology optimization to design a 3D morphing
winglet. In this section the specific objective and goals are formulated, which will be researched for this thesis.
1.3.1. R ESEARCH O BJECTIVE
As mentioned in Section 1.1.1 there exist two ways of discretizing the design space. The first approach is
the continuum approach, the second is the discrete or ground-structure approach [30]. Advantages of the
ground-structure approach include the limited post processing, the absence of additional filtering techniques, and the limited problem size due to the less dense design space compared to the continuum approach
[30]. Based on these advantages it has been decided to develop a tool to design 3D morphing winglets using
the ground-structure topology optimization approach based on the previous work done by E. Gillebaart and
R. De Breuker [21]. The research objective can be more formally stated as:
"Developing a software tool to design a mechanism for morphing winglets, using ground-structure based
topology optimization, by improving, extending, and expanding the previous 2D inhouse tool."
1.3.2. R ESEARCH G OALS
The research objective can be split into three sub-goals, i.e. improving, extending and expanding of the previous tool. Improving the tool can be achieved by improving the robustness of the tool, the computational time
of the tool and the separation of the design vector. The tool is not always stable and could converge towards
non-feasible solutions. In addition, the computational efficiency of the previous implementation could be
improved, which is highly favorable once the tool will be expanded to 3D. Also, in some design cases rather
poor separation was observed, where several elements had densities unequal to 0 or 1. These results could
not be interpreted and therefore not used. In addition, the tool can be extended in terms of its capabilities.
Buckling criteria can be included to guarantee stability of the mechanism. Also, the previous tool is only compatible with one input node and one output node. Multiple input and output nodes increases the flexibility
of the tool. Finally, as mentioned before, the tool designs 2D mechanisms. Once the tool is improved and
extended in terms of capabilities, the tool will be expanded to 3D. These goals can be summarized as follows.
1. Improve previous tool
(a) Increase robustness
(b) Reduce computational time
(c) Increase separation
2. Extend previous tool
(a) Include multiple input nodes
(b) Include multiple output nodes
(c) Include buckling
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3. Expand previous tool
(a) Expand to 3D
1.4. O UTLINE
To meet the research goals defined in the previous section, a tool has been developed. The tool is schematically represented in Figure 1.5. The tool consists of four steps, the input, mechanism design, mechanism
sizing, and output step. First the steps are briefly covered, then the structure of the thesis will be explained.
The structure of the thesis can be better understood by going through the steps of the tool first.
Figure 1.5: Flowchart of the design procedure
In the input step the design space is defined. This means that the size of the design space has to be chosen,
where the nodes are located, how many elements are included, how the mechanism is supported and which
nodes are used for actuation (input nodes) of which nodes are traced (output nodes). Also, the initial design
vector (the cross-sectional areas of the truss members) has to be chosen.
In the second step the mechanism is designed. This is an optimization where the optimizer determines which
elements are needed to be able to follow the trajectory. The objective and constraints used, will be discussed
in Chapter 3. The (initial) design vector/geometry is evaluated in the analysis section. How the structure deforms, what the internal forces are, what the reaction forces are, etc. are determined here. Then the optimizer
determines how the design vector/geometry should change to minimize the objective while satisfying the
constraints. The output is a converged design vector which tells which elements are needed for the mechanism.
The next step then removes all elements except for those that are needed according to the previous step. Then
a second optimization is performed to minimize the weight of the mechanism while satisfying several failure
constraints. This step finds the minimum weight of the mechanism, while preventing failure.
In the final step, the output step, the sizing optimization is converged a final geometry will result. This mechanism is able to follow the given trajectories, will have minimum weight and will not fail under the given
loading.
In Chapter 2 will cover the different analysis tools from both optimizations. The finite element tool and the
linear buckling analysis will be covered. The tools determine its structural behavior, which functions as input for the objective functions and constraints. Chapter 3 elaborates on the optimization process. It will
cover which optimizer is used, what strategy is used to design morphing winglets, and covers in detail the
objective, constraints, and their sensitivities of both the mechanism design as well as the sizing optimization.
Then Chapter 4 contains the results obtained from the optimizations. Both a 2D winglet and a 3D winglet
design case has been included. Finally, Chapter 5 contains the conclusions and recommendations for future
research.
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A NALYSIS
This chapter describes the analysis tools used in the optimization procedure to design the morphing winglet.
Two different analyses can be identified, the finite element analysis (Section 2.1) and the linear buckling
analysis (Section 2.2). The FEA calculates the nodal displacements that result in equilibrium. The FEA also
calculates the elemental strains and reaction forces that result from this equilibrium. In the linear buckling
analysis, an eigenvalue analysis is performed to calculate the buckling load factors. When load factors are
found between 0 and 1, the structure is unable to withstand the applied loading and will buckle. Finally,
Section 2.3 covers how the output from the analysis tools are post-processed. Here the structural behavior is
compared to predefined limits to assess how the structure performs.
2.1. F INITE E LEMENT A NALYSIS
As discussed previously, the FEA is used in the optimization procedure. The module solves for the displacement vector u which satisfies equilibrium. These equilibrium equations are derived from the total potential
energy (TPE). The tool calculates the structural deformation as a result of prescribed displacements and/or
applied loading. The tool also calculates the elemental strains as well as the nodal reaction forces. These
structural responses are used as in input for the optimizer. Objective functions and constraint functions are
formulated based on these outputs, which will be explained in the next chapter.
The developed finite element tool consists of two steps. In the first step the structure is only actuated, i.e.
only the prescribed displacements are enforced. Then, the corresponding displacement vector for equilibrium is calculated. This displacement field corresponds to the movement of the nodes when no loading is
applied. In the second step the loading is applied and the deformations due to the loading are obtained. In
this finite element tool it is assumed that the large displacements and rotations result from the actuation step.
The deformations of the elements due to the loading are assumed to be small. As a result, in the first step it
is assumed that the stiffness matrix is a function of the displacements. This requires an iterative procedure
to calculate the displacement vector u (N L) . Here the superscript (N L) refers to the non-linear/iterative solver
used in the first step of the finite element analysis. Once that equilibrium is found, the loading is applied.
Here it is assumed that the stiffness matrix remains constant and equal to the converged stiffness matrix obtained in the first step. The displacement vector obtained in second (linear) step is denoted as u (L) . The finite
element tool is schematically represented in Figure 2.1.
Figure 2.1: Flowchart of the finite element analysis
T.A. DE J ONG
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12
2. A NALYSIS
The decoupling of the prescribed displacements and applied loading is done to be able to calculate the displacements only caused by the loading. These displacements, or deviations from the equilibrium state due
to loading, are used in the objective function for the mechanism design, covered in the next chapter.
First the TPE will be formulated, followed by how the equilibrium equations are obtained from it. Then, the
strain measure used will be covered, followed by which iterative algorithm is used to solve the equilibrium
equations (in step I). The displacement vector obtained in this step is denoted as u (N L) . The method used
here is based on the PhD work of Kawamoto [30]. Then briefly the linear step is covered to calculate u (L) . In
addition, the FEA is able to solve the equilibrium equations both for 2D as well as 3D structures, hence the
changes required to extend the FEA from 2D to 3D will be covered, followed by the verification for both 2D
and 3D.
2.1.1. D ESCRIPTION
The TPE (Π) is defined as the sum of the internal strain energy (U ) and the work done by the external conservative forces (V ).
Π =U +V
(2.1)
The structures considered here solely consist of truss members, which can only carry axial loading. As a
consequence the strain energy of the structure can be written as the sum of the elemental axial strain energies.
In addition, the forces are assumed to be constant. This allows the external work to be written as a constant
external force vector times the displacement vector. Rewriting the TPE gives:
Π(A, x, u) =
Ne
1X
A i E i l i (x)²i (u, x)2 − p T u
2 i =1
(2.2)
Here A ∈ RNe represents the vector containing the cross-sectional areas of all truss members, x ∈ R2Nn is the
vector containing all nodal locations, u is the nodal displacement vector, and p is the nodal force vector.
Furthermore, Ne represents the total number of elements (truss members) of the mechanism, Nn is the total
number of nodes of the structure, and Nd is the total number of free degree of freedoms (DOF). This number
is (in 2D) equal to twice the number of nodes minus the fixed DOF. The fixed DOF are the ones constrained
by the supports. Finally, A i represents the cross-sectional area, E i the Young’s modulus, l i the undeformed
length, and ²i the strain of the ith element.
Minimizing the TPE will yield equilibrium. Therefore Equation 2.3 can be solved to find the displacement
vector that yields equilibrium. However, solving Equation 2.3 will yield stationary values for the TPE, which
could also maximize the TPE. Therefore Equation 2.4 has to be satisfied as well to guarantee that the solution
found minimizes the TPE.
∇u Π(A, x, u) = 0
(2.3)
∇2uu Π(A, x, u) ≥ 0
(2.4)
S TRAIN M EASURE
As was already indicated in Equation 2.2, the strain is a function of both the nodal location as well as the nodal
displacement. The dependence on the displacement vector will be non-linear, which is required to model the
large displacements and rotations that occur during actuation. Even though the displacements and rotations
are large, the strains are assumed to be small and the material is assumed to behave linear elastic. The large
nodal displacements are a result of the kinematics of the mechanism, the deformation of the members of
the structure will be small. This would mean that the functionality for mechanisms is not changed when the
members were replaced by rigid elements. A suitable strain measure for modelling large displacements and
rotations, while the strains remain small, is the Green-Lagrange (GL) strain measure. This strain measure is
therefore used. Consider a single element, which is under uni-axial loading, then the GL strain can be written
as:
²=
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
lˆ2 − l 2
2l 2
(2.5)
T.A. DE J ONG
2.1. F INITE E LEMENT A NALYSIS
13
Here lˆ is the deformed length of the element. In 2D the strain can also be written as:
∂u 1 1
²=
+
∂x
2
·µ
∂u 1
∂x
¶2
∂u 2
+
∂x
µ
¶2 ¸
(2.6)
In Equation 2.6 the partial derivatives are in local coordinates. These partial derivatives can be expressed in
terms of the global displacement coordinates and the corresponding rotations, given by Equation 2.7.
∂u 1
∂x
1
= γu
l
∂u 1
∂x
1
= δu
l
γ
= [−cos(α)
δ
= [si n(α)
(2.7)
− si n(α) cos(α) si n(α)]
− cos(α)
− si n(α) cos(α)]
Substituting Equation 2.7 into Equation 2.6 gives:
i
1
1 h¡ ¢2
² = γu + 2 γu + (δu)2
l
2l
(2.8)
Carrying out the multiplication gives the following for the second term:
1

¡ ¢2
0
γu + (δu)2 = 
−1
0
0
1
0
−1

−1
0
1
0

0
−1

0
(2.9)
1
Calling this matrix B allows one to rewrite Equation 2.8 as a function of this matrix. Also the matrices B̂ and
B̃ are introduced. Matrix B̂ is defined as matrix B after removing the rows corresponding to the fixed DOF.
Matrix B̃ is defined as matrix B after removing both the rows and columns equal to the fixed DOF. Then for
the single element the strain can be written as:
²=
1
1
B̂ xu + 2 u T B̃ u
2
l
2l
(2.10)
²=
1 T
1
b u + 2 u T B̃ u
l2
2l
(2.11)
or
Here b divided by l will yield the direction cosines of the element. Translating this concept to a mechanism
consisting of several elements, will result in a strain for the ith element given by:
²i =
1
l i2
B̂ i xu +
1
2l i2
u T B̃ i u
(2.12)
Where B̂ i ∈ Rd ×2Nn is equal to B with the rows corresponding to the fixed DOF removed. It can also be written
as:
²i =
1
l i2
b Ti u +
1
2l i2
u T B̃ i u
(2.13)
Note that b i and B̃ i are determined by the connectivity of the element. Here b i ∈ RNd is equal to B̂ i x and
B̃ i ∈ RNd ×Nd is equal to B i with the rows and columns corresponding to the fixed DOF removed. Also note
that the linear strain measure is recovered if the non-linear contribution is neglected. Finally, due to the nonlinear GL strain measure, solving the equilibrium equations has to be done iteratively. This will explained in
the next section.
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14
2. A NALYSIS
I TERATIVE ROOT FINDING
Due to the non-linear strain formulation, the minimization of the TPE will has to be done iteratively. Over
the years many different iterative algorithms have been developed for iterative root finding. The standard
algorithm looks as follows:
x = x0
k =0
repeat
find h
x = x + αh
k = k +1
(2.14)
until stop
Here x is updated iteratively until a certain convergence criteria is met. The variable h is the direction of
descent and a is the step size. A popular algorithm is the steepest descent approach, where the direction is
calculated from:
h = −F 0 (x)
(2.15)
The algorithm is robust, even when the initial guess x 0 lies far from the converged solution. The convergence,
however, is rather poor close to the optimum. A different algorithm with faster convergence is Newton’s
method, where h is calculated from:
F 00 (x)h = −F 0 (x)
(2.16)
The downside of this algorithm is its lack of robustness and the computational time required to calculate the
00
Hessian F . If the higher order derivatives are neglected, the Hessian can be written as the product of the
Jacobian (J f ) transposed times the Jacobian. Then h is calculated from:
J f (x)T J f (x)h = −F 0 (x)
(2.17)
This method is known as the Gauss-Newton’s method. Another type of algorithm combines the fast convergence from the Gauss-Newton’s method and the robustness of the steepest descent. This algorithm is known
as the Levenberg-Marquardt, and h is calculated from:
(J f (x)T J f (x) + µI )h = −F 0 (x)
(2.18)
The LM parameter µ influences the direction h, for small values of µ the search direction will be close to
the Gauss-Newton’s direction. For large values of µ the search direction will be close to the steepest descent
direction. The algorithm was improved by Marquardt to avoid slow convergence in the direction of small
gradients. Here h is calculated from:
¡
£
¤¢
J f (x)T J f (x) + µ d i ag J f (x)T J f (x) h = −F 0 (x)
(2.19)
When the LM algorithm is applied to minimize the TPE, then F = Π and x = u (N L) . For clarity the superscript
is left out and the displacement vector is represented as u.
u = u0
k =0
µ = µ0
repeat
solve
u = u + ∆u
k = k +1
(K t + µ d i ag [K t ])h = −R
(2.20)
until stop
The LM parameter both influences the direction as well as the size of h. Now the LM parameter µ will be
updated based on a gain factor g , given by:
g=
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
Π(u) − Π(u + ∆u)
Q(0) −Q(∆u)
(2.21)
T.A. DE J ONG
2.1. F INITE E LEMENT A NALYSIS
15
Π(u + ∆u)
≈ Q(∆u)
= Π(u) +
∂Π
1
∂2 Π
∆u
∆u + ∆u T
∂u
2
∂u 2
(2.22)
1
= Π(u) + R∆u + ∆u T K t ∆u
2
Therefore the algorithm can be written as:
u = u0
k =0
µ = µ0
repeat
solve
Compute g
if g > δ
(K t + µ d i ag [K t ])h = −R
(2.23)
u = u + ∆u ©
ª
µ = µ max 1/γ, 1 − (β − 1)(2g − 1)p
else
µ = 2µ
k = k +1
until stop
For a converged displacement vector the incremental change ∆u should be (almost) zero. The vector should
also give equilibrium, which results in the residual vector R being (close to) zero. Therefore both the norm of
the vector ∆u or the norm of the vector R could be used as a stopping criteria. In the current implementation
the norm of ∆u has to be below a certain threshold for equilibrium.
The next step is to derive the residual force vector and tangent stiffness matrix. The residual force factor is
defined as the gradient of the TPE and the tangent stiffness matrix is defined as the Hessian of the TPE.
R ESIDUAL FORCE VECTOR
The residual force vector can be calculated as shown in Equation 2.24. Note that the partial derivative of a
summation is the same as the summation of the partial derivatives. The order can be interchanged.
R
=
∂Π
∂u
Ã
!
Ne
∂ 1X
∂ ¡ T ¢
2
=
A i E i l i ²i −
p u
∂u 2 i =1
∂u
=
Ne
X
A i E i l i ²i
i =1
(2.24)
∂²i
− pT
∂u
∂²i
The next step is to calculate
, which is shown below.
∂u
Ã
!
∂²i
∂ 1 T
1 T
=
b u + 2 u B̃ i u
∂u
∂u l i2 i
2l i
Ã
!
Ã
!
∂ 1 T
∂
1 T
=
b u +
u B̃ i u
∂u l i2 i
∂u 2l i2
=
=
T.A. DE J ONG
1
l i2
1
l i2
b Ti +
b Ti +
1
T
2l i2
1
l i2
(2.25)
u T (B̃ i + B̃ i )
u T B̃ i
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
16
2. A NALYSIS
Realizing that the matrix B̃ i is symmetric, which means that the transpose of the matrix is equal to the matrix
itself, allows for further simplifications as done in the last step. Substituting this expression into Equation
2.24 gives the final expression for the residual force factor. This can also be written as the internal force
vector minus the external force vector. Once the internal force vector balances the external force vector, the
structure is in equilibrium.
!
¶Ã
µ
Ne
X
1 T
1 T 1 T
1 T
A i E i l i 2 b i u + 2 u B̃ u
b + 2 u B̃ i − p T
R =
2 i
(2.26)
l
2l
l
li
i =1
i
= F i nt − F ext
TANGENT STIFFNESS MATRIX
Similarly, the tangent stiffness matrix can be calculated.
Kt
=
∂2 Π
∂u 2
=
∂R
∂u
Ã
!
Ne
∂ X
∂²i
∂ ¡ T¢
=
A i E i l i ²i
−
p
∂u i =1
∂u
∂u
=
Ne
X
Ã
Ai Ei li
i =1
∂²i T ∂²i
∂2 ²i
+ ²i
∂u ∂u
∂u 2
(2.27)
!
∂²i
has already been determined. Only an expression for the second derivative of the strains has
∂u
not yet been derived. This is done in Equation 2.28.
The term
∂2 ²i
∂u 2
=
µ
¶
∂ ∂²i
∂u ∂u
Ã
!
∂ 1 T 1 T
b + 2 u B̃ i
=
∂u l i2 i
li
=
1
l i2
(2.28)
B̃ i
Substituting gives the final expression for the tangent stiffness matrix:
Kt =
Ne
X
i =1
"Ã
Ai Ei li
1
bT
l i2 i
+
1
l i2
!T Ã
T
u B̃ i
1
bT
l i2 i
! µ
!#
¶Ã
1 T
1 T
1
+ 2 u B̃ i + 2 b i u + 2 u B̃ u
B̃ i
l
2l
li
l i2
1
T
(2.29)
As a consequence of the GL strain measure, the stiffness matrix is a function of the nodal displacements. The
stiffness matrix is no longer constant, as is the case for a linear analysis. This dependence requires therefore
an iterative procedure to solve for the nodal displacements. In the linear case the stiffness matrix is assumed
to be independent of the displacement vector and will therefore remain constant, then the displacement
vector could be obtained in a single step. The linear step will be covered next.
L INEAR STEP
Once the actuation step has converged the second step will be performed, where the loading is applied. The
equation to be solved is given by Equation 2.30. Here K t is the converged tangent stiffness matrix from first
step.
K t u (L) = F
(2.30)
As mentioned in the beginning of this chapter in the second step the stiffness matrix is assumed to remain
constant. Hence solving for u (L) can be done in a single step by pre-multiplying both sides of Equation 2.30
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
2.1. F INITE E LEMENT A NALYSIS
17
with the inverse of K t . The displacement field obtained in this step can be interpreted as the deviation from
the intended trajectory, resulting from the applied loading. This deviation will be minimized in the mechanism design optimization process, as will be explained in the next chapter. If the loading was applied simultaneously with the prescribed displacements, then only one displacement field would be found and the
deviation from the intended trajectory would remain unknown.
T HREE D IMENSIONAL
In this section the changes in the FEA required for 3D are discussed. In 2D the nodal displacements were
arranged as follows in the vector u:

u 1(1)
 u (1) 
 2 
 (2) 
 u1 
 (2) 


u =  u2 
 . 
 . 
 . 
 (Nn ) 
u 1 
(N )
u2 n

(2.31)
On the first entry the displacement in the 1-direction of node 1 is displayed, on the second entry the displacement in the 2-direction of node 1 is displayed. Similarly, on the third and fourth entry the displacements in
the 1- and 2-direction of the second node are located. This resulted in the following B matrix for the formulation of the strains, as was given by Equation 2.9 and repeated here:
1
0
B =
−1
0

0
1
0
−1
−1
0
1
0

0
−1

0
(2.32)
1
For 3D the nodal displacements will be arranged as follows:

u 1(1)
 u (1) 
 2 
 (1) 
 u3 
 (2) 
u 
 1 
 u (2) 
 2 

u =
 u 3(2) 


 .. 
 . 


 (Nn ) 
u 1 
 (Nn ) 
u 2 

(2.33)
(Nn )
u3
In analogy to the 2D strain formulation, the following B matrix can be used to calculate the strains.

1
0


0
B =
−1

0
0
0
1
0
0
−1
0
0
0
1
0
0
−1
−1
0
0
1
0
0
0
−1
0
0
1
0

0
0


−1

0

0
1
(2.34)
The expression for the strain therefore remains unchanged and the strain can in 3D be written as was already
given by Equation 2.11, and repeated here:
²=
1
1 T
b u + 2 u T B̃ u
l2
2l
(2.35)
Carrying out the multiplications yields as expected the strain measure for 3D, which is given by Equation 2.36.
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
18
2. A NALYSIS
²=
∂u 1 1
+
∂x
2
·µ
∂u 1
∂x
¶2
µ
+
∂u 2
∂x
¶2
µ
+
∂u 3
∂x
¶2 ¸
(2.36)
The entire approach previously described remains the same. Only the B matrix is slightly changed and as a
consequence the dimensions of the problem, accounting for the third dimension. With this newly defined
matrix the total potential energy can be minimized.
2.1.2. V ERIFICATION
An important step in developing a software tool is verification. Verification is done by answering the following
question:
"Are we building the system right?"
Answering the question means that the different components of the tool developed for this thesis should be
checked with other (numerical) tools, if available, to see whether the tool developed behaves the way it is
supposed to behave. For the finite element tool both 2D and 3D will be verified.
In 2D the tool will be verified with the Von Mises truss structure, since the analytical solution is known. In 3D
no analytical solution is known, therefore a simple geometry (a 3D version of the Von Mises truss) has been
sketched in Abaqus to see how the forces and displacements compare. The input file for the Abaqus model is
included in Appendix A.
T WO D IMENSIONAL
The Von Mises truss is a relatively simple structure of which the analytical solution is known. Which makes
this structure very suitable for verification. The force-displacement curve will be compared for this verification. An illustration of the structure can be found in Figure 2.2. The symmetric structure consists of two
connected truss members, each at one end pinned. In the tool the deformation was displacement controlled
to prevent snap-through.
Figure 2.2: Illustration of the 2D Von Mises truss
Two verification cases of the Von Mises truss will be used to verify the finite element tool in 2D. The first
verification case has an initial inclination of 15 degrees and the second one has 45 degrees. For the first
verification case the analytical solution presented by Bathe [2] will be used. The analytical solution for the
structure gives the relation between the applied force and the nodal displacement. The assumption, however,
is that the axial force of the truss member could be modeled as F = ku. Hence it is assumed that the axial
stiffness of the member remains constant, which is only valid for small strains. The following solution was
derived by Bathe:



µ
¶


1
∆
◦


F = 2kl −1 + ·
µ ¶2 ¸1/2  si n(15 ) − l


∆
∆
◦
1 − 2 si n(15 ) +
l
l
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
(2.37)
T.A. DE J ONG
2.1. F INITE E LEMENT A NALYSIS
19
Here l represents the length of each truss, k is the axial stiffness, equal to E A/l . E here is the Young’s Modulus
and A is the cross-sectional area. Finally, ∆ represents the downward deflection.
Since the first verification case is only able to verify small strains accurately, another verification case has
been used. This case has been obtained from the book of Crisfield [10]. The inclination angle is 45 degrees,
resulting in larger strains. Furthermore in the book is Crisfield the following parameters were used:
x
h
A
E
= 2500
= 2500
= 100
= 5x105
p
The length of the truss member, l , can be calculated by x 2 + h 2 . It has been decided that the same values
for the cross-sectional area, Young’s modulus and length of the truss members were used in both verification
cases. Therefore the only difference is the initial inclination.
As can be clearly seen from Figure 2.3a, the results of the tool developed match very well the analytical solution from Bathe. Only a very small deviation can be seen at the peak and the valley of the force-displacement
curve. This minor difference can be explained by the fact that only for small strains the analytical solution
can be used for verification, while at the peak and valley the strains are the largest. It can be concluded that
the finite element tool matches the results from Bathe and the tool has been verified for small strains.
Even for large strains the FEM code is able to calculate the force displacement relation. This, however, can
no longer be compared to the analytical solution. Therefore a different numerical implementation, shown in
the book of Crisfield, has been used. Figure 2.3b shows the force displacement curves of both tools. One can
clearly see that both implementations give the same result. It can be concluded that the finite element tool
is able to correctly calculate the force-displacement curve, even for large deformations. Hence, the tool has
been successfully verified.
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
20
2. A NALYSIS
(a) Force-displacement curves of FEM code and analytical solution by Bathe with initial inclination 15◦
(b) Force-displacement curves of FEM code and numerical solution by Crisfield with initial inclination 45◦
Figure 2.3: Verification of FEM code, 2D Von Mises truss with inclinations of 15◦ and 45◦
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
2.1. F INITE E LEMENT A NALYSIS
21
T HREE D IMENSIONAL
For the 3D verification of the finite element tool no analytical solution is known, which could be used. Therefore it has been decided to draw a structure in Abaqus, which is then compared to the results obtained from
the finite element tool. The structure chosen for the verification is shown in Figure 2.4.
Figure 2.4: Illustration of the 3D Von Mises truss
The structure behaves similar to the 2D Von Mises truss. The structure will experience snap through as well.
However the deformation is displacement controlled, thereby capturing also the unstable parts of the force
displacement path. Similar to the 2D verification case, the initial inclination of 15 degrees is used, and the
same length and cross-sectional area (for all three elements) were used. Only the angles between the elements
are not yet defined, this can best be seen in Figure 2.5, which represents the model used in Abaqus. The angles
between the three elements are defined as α, β, and γ. For this structure the angles are chosen such that no
symmetry plane exists. As a consequence displacements in all three dimensions will occur, hereby verifying
all three dimensions. The angles are defined in the xz-plane and set equal to:
α
β
γ
= 90◦
= 112.5◦
= 157.5◦
The geometry as described above was modeled in Abaqus, the three lower nodes were all pinned, thereby
constraining the three translational degrees of freedom. On the top node a displacement was enforced in the
y- or U2-direction. The last step was to apply a mesh with elements that have the same strain measure. Different strain measures will solve a different set of equilibrium equations and as a result a different displacement
field is obtained. However, the truss element in Abaqus uses logarithmic strain, rather than the GL strain
used in the current finite element model. Therefore the truss members had to be meshed with 2 node cubic
beam elements (B33 elements), which use the GL strain. The mesh had to be chosen such that each member
had a single element. To reproduce the pinned connection at the top node, only the translational degrees
of freedom were coupled. However, having a coupling constraint at the top node does not allow any other
boundary conditions, such as the prescribed displacement in the y-direction. To overcome this, a reference
node was added. This node was via a MPC constraint coupled with the top nodes of each element. Again,
only the translational degrees of freedom were coupled to reproduce the pinned joint. Finally, the prescribed
displacement could be applied to the reference node.
Of the four nodes three are pinned, hence only three degrees of freedom remain. The top node has three
displacements in the x-, y-, and z-direction, called U1, U2, and U3 respectively. In addition, there will be a reaction force at the top node in the y-direction, as a result of the prescribed displacements. Hence, as an output
the force displacement curve of U2 and R2 are plotted, shown in Figure 2.6. As can be seen a similar structural
behavior is observed as was done in the 2D case. The structure will experience snap-through. In addition, the
displacements U1 and U3 will be plotted as a function of U2, also shown in Figure 2.6. Comparing the results
of Abaqus with the finite element code, the result match perfectly. Both for the force displacement curve as
well as the displacements in U1 and U3. Hereby the tool is also verified in 3D and it works as intended.
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T OPOLOGY OPTIMIZATION OF 3D LINKAGES
22
2. A NALYSIS
(a) top view
(b) 3D view
Figure 2.5: Schematic views of the 3D Von Mises truss in Abaqus with inclination of 15◦
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
2.1. F INITE E LEMENT A NALYSIS
23
(a) Force-displacement curves of FEM code and Abaqus with initial inclination 15◦
(b) Displacement curves of FEM code and Abaqus with initial inclination 45◦
Figure 2.6: Verification of FEM code, 3D Von Mises truss with inclination of 15◦
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
24
2. A NALYSIS
2.2. L INEAR B UCKLING A NALYSIS
Buckling is a critical state of the structure at certain load levels at which the structure exhibits large displacements or collapses. Global buckling, buckling of the structure as a whole, is often a critical failure mode and
is always associated with compressive forces. When the mechanism is sized, i.e. when the cross-sectional
areas of all elements are determined, they should be sufficient thick to prevent buckling failure.
The tool explained in this section performs a linear buckling analysis which calculates the buckling load factors and the corresponding buckling modes. When these factors are multiplied with the applied loading, the
critical buckling loads are obtained. Hence as long as the factors are not between 0 and 1, the applied loading will not cause buckling. First will be explained how these buckling load factors and buckling modes are
obtained, followed by the verification of the linear buckling analysis.
2.2.1. D ESCRIPTION
The first step would be to rewrite the TPE as a function of the linear stiffness matrix (K L ), the geometric stiffness matrix (K G ), and the stiffness matrix containing the higher order terms (K H ). Here the linear stiffness
matrix is the initial stiffness of the structure, which contains the terms that are not a function of the displacement. Hence, this matrix would describe the stiffness of the structure when there are no displacements. The
geometric stiffness matrix contains all the terms that are linear in displacement. Finally, the higher order
terms (in displacement) are collected in K H , this term only becomes significant for large displacements. The
TPE can be written as [30]:
Π(A, x, u) =
Ne
1X
u T (K L (A, x) + K G (A, u, x) + K H (A, u, x)) − p T u
2 i =1
(2.38)
With:
KL
=
Ne
X
Ai Ei
i =1
KG
=
Ne
X
Ai Ei
i =1
KH
=
l i3
l i3
b i b Ti
b Ti u B̃ i
(2.39)
Ne
X
¢T
¢¡
Ai Ei ¡
B̃ i u B̃ i u
3
i =1 l i
In a linear buckling analysis the following generalized eigenvalue problem is solved:
(K L + λi K G ) φi = 0
(2.40)
Here λi is the ith buckling load factor and φi describes the corresponding ith buckling mode. The buckling
load factors are the solutions for which the geometric stiffness matrix results in a loss of stability. The smallest
positive eigenvalue (λcr ) gives the critical buckling load. To prevent buckling, λcr should be larger than one.
As long as that is the case, the load is allowed to increase and no buckling will occur for the applied load.
To solve the generalized eigenvalue problem, both sides of the equations can be left-multiplied with the
eigenvector transposed, giving:
φTcr (K L + λcr K G ) φcr = 0
(2.41)
Solving this equation for the λcr gives:
λcr = −
φTcr K L φcr
φTcr K G φcr
(2.42)
However, instead of solving the generalized eigenvalue problem given by equation 2.40, a slightly modified
eigenvalue problem is solved as proposed by Bathe [2]. The reason is the possible indefiniteness of the geometric stiffness matrix. The linear stiffness matrix is real, symmetric and positive definite, whereas the geometric stiffness matrix is real, symmetric, but could be indefinite. Numerical difficulties could occur due to
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
2.2. L INEAR B UCKLING A NALYSIS
25
the indefiniteness of the geometric stiffness matrix. The modified problem solved is a generalized eigenvalue
problem where both matrices are definite:
¡
¢
φTcr (K L + K G ) + γcr K L φcr = 0
Now λcr can be obtained from:
λcr =
1
1 − γcr
(2.43)
(2.44)
The relation given by Equation 2.44 can be verified by calculating λcr from Equation 2.42. Both equations
give indeed the same eigenvalue.
2.2.2. V ERIFICATION
Also the linear buckling analysis tool will be verified. This will be done by using the same structure as used
for the 2D finite element analysis. The Von Mises truss with an inclination of 45 degrees is used here. Also,
since the geometry is symmetric, the structure could be reduced to a single truss member when applying the
symmetry boundary conditions. Due to the symmetry, the upper node will only move up and down, it will
not move left or right. Hence, the structure can be reduced to the geometry given by Figure 2.7. Since the
structure only a a single free degree of freedom, the verification will be done analytically.
Figure 2.7: Illustration of the Von Mises truss, symmetry conditions applied
The exact same geometry is used as was done for the verification case with 45 degrees inclination. Hence the
parameters used are:
x
h
A
E
= 2500
= 2500
= 100
= 5x105
In addition, a unit force F pointing downward is assumed. This yields a buckling factor which is equal to the
critical buckling load. In the reduced structure the applied loading will therefore be equal to a half.
Since this structure consists of a single element, the linear stiffness matrix and geometric stiffness matrix
reduce to a scalars and no summation over the elements drops out as well. Hence, dropping the subscript i ,
the linear and geometric stiffness can be written as:
KL =
KG =
AE
bb T
l3
(2.45)
AE T
b u B̃
l3
(2.46)
Recall that b is defined as B̂ x. Hence b is given by:
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T OPOLOGY OPTIMIZATION OF 3D LINKAGES
26
2. A NALYSIS
£
b = b = B̂ x = 0
−1
0
 
0
¤ 0

1 
x  = h
(2.47)
h
The displacement vector is the solution of the equilibrium equation after the force has been applied. This
can be obtained by solving the following:
KL u = −
F
2
(2.48)
Since the structure has only a single degree of freedom, the equation reduces to a scalar equation and the
downward displacement is simply equal to the applied loading (-1/2) divided by the stiffness. This gives:
u=−
1
2K L
(2.49)
The matrix B̃ is equal to B , where the rows and columns equal to the fixed degrees of freedom are removed.
Since this structure has only a single degree of freedom (the 4th one), the first three rows and columns will be
removed. This yield simply the value 1 for B̃ .
Substituting all this gives the following expressions for the linear and geometric stiffness:
KL =
KG = −
AE h 2
l3
(2.50)
AE h
AE hl
1
=− 3
=−
2l 3 K L
2l AE h 2
2h
(2.51)
Substituting Equations 2.50 and 2.51 into Equation 2.40 gives:
µ
µ
¶¶
1
AE h 2
+
λ
−
φ=0
l3
2h
(2.52)
Since the equation is a scalar equation, the critical buckling factor can be calculated as:
λ=−
K L 2AE h 3
=
KG
l3
(2.53)
And the normalized eigen mode would then be equal to 1 or -1, hence:
φ = 1 or − 1
(2.54)
Substituting the values yields a critical buckling factor of 3.5355e+07 for the analytical solution. Running the
linear buckling analysis yields also 3.5355e+07 for λ. Also, the normalized buckling mode is the same. The
results are shown in Table 2.1. Based on this it can be concluded that the linear buckling analysis performs as
intended. Hence the tool has been verified.
Table 2.1: Verification of linear buckling analysis
Implementation
Eigenvalue
Eigenmode
Analytically
Numerically
3.5355e+07
3.5355e+07
1 or -1
1 or -1
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2.3. P OST- PROCESSING
27
2.3. P OST- PROCESSING
The research objective defined in the previous chapter is to develop a tool to design a mechanism for morphing winglets, using ground-structure based topology optimization. Hence the primary objective is to design a suitable mechanism for a morphing winglet. This will be done using ground-structure based topology
optimization techniques. Now a mechanism is characterized by its kinematics, its ability to perform a certain motion. The actuation of the mechanism is done by applying prescribed displacements at one or more
nodes, defined as the input node(s). The desired motion of the mechanism is checked at different nodes,
the output nodes. The post-processing is therefore tailored to calculate parameters which are needed in the
optimizations to design and size a mechanism which is able to perform a specified motion without failure.
This section elaborates on how these different parameters are calculated. How these parameters are used in
the optimizations, will be covered in the next chapter.
2.3.1. F INITE E LEMENT A NALYSIS
In the previous sections the finite element analysis and linear buckling analysis were explained. The FEA
solves the equilibrium equations, calculates the corresponding nodal displacements and reaction forces, as
well as the elemental strains. The tool performs these calculations twice, the first time after the prescribed
displacements were enforced (to obtain u (N L) ), the second time in a single step after the loading is applied
(to obtain u (L) ). The first step is done incrementally, since the displacements and rotations as a result of the
actuation could be large. In the second step the nodal displacements as a result of the loading are assumed
to be small, hence this step is a linear one. How the displacements, reaction forces and strains are postprocessed, will be explained in this section.
Compliance
The first parameter calculated is the compliance, which is defined as [62]:
C = F T u (L)
(2.55)
Here F is the external force vector and u (L) is the displacement vector that is calculated in the linear step
of the finite element analysis. A slightly different definition is 1/2F T u (L) , but the definition given by [62] is
often used in topology optimization and is therefore used here. Furthermore, the problems considered in this
thesis assume a constant force vector, hence the loading does neither change over time nor change with the
deformations. Compliance can be interpreted as the ability to deform. The smaller the compliance is, the
greater the resistance to deform. Since the force vector is constant, the smaller the compliance, the smaller
the displacement vector u (L) must be. This results in a greater resistance against the deformations caused by
the loading.
Actuation Force
The internal force vector calculates the nodal forces as a result of the prescribed displacements and/or applied loading. In the first step of the FEA only the prescribed displacements were enforced, hence this internal
force vector represents their resistance against the prescribed displacements. The force required to enforce
the prescribed displacement, i.e. the actuation force, is given by the entry of the internal force vector corresponding to the prescribed displacement. Defining a row vector I i nput which contains all zeros except for
the entry of the prescribed displacement, where it contains a one. Then the actuation force is given by:
F i nput = I i nput F i nt
(2.56)
i nput
How F i nt is calculated, is given by Equation 2.26. Note that the variable F
is zero for mechanisms,
but will be nonzero otherwise. Mechanisms, which posses a degree of freedom, can be actuated without
deforming the elements.
Trajectory
The desired motion is checked at one or more the output nodes. Assuming a single output node, then the distance ∆ between the actual location of the output node and the intended location is in 2D given by Equation
2.57 and in 3D given by 2.58.
∆=
T.A. DE J ONG
r
³
´ ³
´
tr a j 2
tr a j 2
t
u xac t − u x
+ u ac
y − uy
(2.57)
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
28
2. A NALYSIS
∆=
r
³
´
tr a j 2
u xac t − u x
³
´ ³
´
tr a j 2
tr a j 2
t
ac t
+ u ac
−
u
+
u
−
u
y
y
z
z
(2.58)
t
ac t
Here u xac t , u ac
are the actual x-, y-, and z-displacements of the output node, respectively. Similarly,
y , and u z
tr a j
tr a j
tr a j
u x , u y , and u z
are the displacements corresponding to the intended trajectory. Defining a distance
∆max , then the following ratio can be calculated:
∆
(2.59)
∆max
If this ratio is smaller than 1, the distance between the specified location and the actual location is smaller
than ∆max . For values larger than 1, this distance is larger than ∆max . How this will be used in the optimizations, will be covered in the next chapter.
R t r a j ec t or y =
Local Buckling
For a pinned truss the critical column buckling load is given by 2.60:
P icr =
π2 E i I i
l i2
(2.60)
Here E i is the Young’s modulus, I i is the second moment of area (the moment of inertia), and l i is the length
of the ith member. Once the compressive force in element i exceeds P icr , column/local buckling will occur.
Assuming a solid cylinder as he shape for the truss members, the second moment of area is given by:
2 2
π 4 π 4 A0ρi
r = r =
4
4
4π
A similar ratio as was done for the trajectory can be introduced.
Ii =
l ocal buckl i ng
Ri
=
i nt er nal
4l i2 ²i
ρ i A 0 E i ²i F i
=
=
−
P icr
P icr
πA 0 ρ i
(2.61)
(2.62)
As long as the ratio is smaller than 1, the compressive force in the ith truss member is smaller than the critical
load and no buckling will occur. As soon as the ratio exceeds 1 for an element, it will fail due to column
buckling.
Stress
In the beginning of this chapter it was assumed that the material remains linear elastic and that Hooke’s law
applies. Therefore the stress in the ith element is simply given by:
σi = E ²i
(2.63)
Dividing the stress by the yield stress σ y , the following ratio can be introduced.
R ist r ess =
σi
σy
(2.64)
As soon as the ratio exceeds one, the element stress exceeds the yield stress. This ratio can be defined for
material failure in tension as well as in compression.
2.3.2. L INEAR B UCKLING A NALYSIS
The second analysis tool, the linear buckling analysis, solves a generalized eigenvalue problem to obtain the
buckling load factors and the buckling modes. The post-processing of these results will also be covered here,
which only applies to global buckling.
Global Buckling
It was previously covered that values smaller than 1 for λcr mean global buckling. To keep the consistency
within the post-processing, λcr is renamed R g l obal buckl i ng .
R g l obal
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
buckl i ng
= λcr
(2.65)
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2.3. P OST- PROCESSING
29
2.3.3. G EOMETRY
The last section of the post-processing covers the geometry. For a given geometry, its weight, number of
elements and input node connectivity can be determined.
Weight
The weight of the mechanism is calculated by multiplying the density of the material ρ m with the volume of
the structure. The total volume is simply the sum of the volume of each element. The weight calculation is
given by Equation 2.66.
Ne
X
W = ρm
Ai li
(2.66)
i =1
Volume
Topology optimization was defined as the process of optimizing both the material layout and the connectivity inside a design domain [56]. For the ground-structure approach this means that the optimizer determines
which elements are needed and which ones can be removed from the design space. The presence of the elements are determined by the vector ρ. The entries of this vector range from (almost) 0 to 1, thereby expressing
the required presence (or absence). If for the ith element this value is 1, the element should be maintained,
if it is 0 then it should be removed. How this vector is updated (and thereby changing the geometry) will be
covered in the next chapter. Now, summing over all the entries of this vector, the number of element in the
geometry can be determined. This is mathematically shown in Equation 2.67.
V=
Ne
X
ρi
(2.67)
i =1
Now introducing a maximum number of elements V max , the following ratio can be determined.
R vol ume =
V
V max
(2.68)
Below 1 the number of elements is smaller than Vmax , above 1 the maximum number of elements is exceeded.
Input node connectivity
The input node connectivity describes the number of elements attached to the input node. It was already
explained that the vector ρ contains the informations which elements are present in the geometry. Now
define a set i np.nod e which contains the elements connected to the input node, then summing over those
elements only will give the input node connectivity.
C i nput
nod e
=
ρi
X
(2.69)
i ∈i np.nod e
n
Now introducing a lower limit C imi
nput
nod e
, the ratio R i nput nod e can be determined.
R i nput
nod e
=
C i nput
nod e
(2.70)
n
C imi
nput nod e
n
For values greater than 1, the number of elements is larger than the C imi
nput
nod e
. And for values smaller than
n
C imi
nput nod e
1
is larger than the input node connectivity number. How this will be incorporated in the optimizations, will be covered in the next chapter.
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T OPOLOGY OPTIMIZATION OF 3D LINKAGES
3
O PTIMIZATION
As mentioned in the introduction, topology optimization can be defined as the process of optimizing both the
material layout and the connectivity inside a design domain [56]. Within this context topology optimization
can be described in general as the optimization problem of finding the material distribution that minimizes
an objective function F 0 , usually subjected to (1) a volume constraint in the form of G 1 ≤ 0, and (2) possibly
M − 1 other constraints G j ≤ 0, j = 2, ..., M . In mathematical form the problem can be written as [56]:
Z
f (u(ρ), ρ)dV
mi n F 0 = F 0 (u(ρ), ρ) =
ρ
Ω
R

 G 1 (ρ) = Ω ρ(x)dV − V0 ≤ 0
G (u(ρ), ρ)
≤0
s.t . =
 j
ρ(x)
= {0, 1}
(3.1)
j = 2, ..., M
∀x ∈ Ω
In this formulation the material distribution is described by the density variable ρ(x), which takes the values
0 (no material present) or 1 (material present) throughout the design space Ω. The state field u satisfies the
(non-)linear state equations. As is commonly done to solve Equation 3.1, the design space Ω is discretized
into many finite elements. Each element i (i = 1, ..., N ) is then described by the density variable ρ i , being
either 0 or 1. Defining the density variables as the design variables, then the optimizer determines which
elements remain in the design space (ρ i = 1) and which elements need to be removed (ρ i = 0) in order to
minimize the objective function, while satisfying the constraints. The problem can be rewritten as shown in
Equation 3.2 [56].
mi n F 0 = F 0 (u(ρ), ρ) =
XZ
ρρ
i
Ωi
f (u(ρ i ), ρ i )dV

P
 G 1 (ρ) = i v i ρ i − V0
G (u(ρ), ρ)
s.t . =
 j
ρi
≤0
≤0
= {0, 1}
(3.2)
j = 2, ..., M
i = 1, ..., N
However, it is well-known that problems described by Equations 3.1 and 3.2 may lack solutions [57], closedness of the design space, and/or exhibit mesh-dependent solutions [56]. Attempts have been made to solve
these issues by relaxation of the integer constraints. By relaxing is meant that the density variables are no
longer discrete, but can take all values between 0 and 1. Mathematically, the optimization problem can be
represented as [56]:
mi n F 0 = F 0 (u(ρ), ρ) =
ρρ
i
s.t . =



T.A. DE J ONG
XZ
0≤
Ωi
f (u(ρ i ), ρ i )dV
P
G 1 (ρ) = i v i ρ i − V0
G j (u(ρ), ρ)
ρi
≤0
≤0
≤1
(3.3)
j = 2, ..., M
i = 1, ..., N
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
32
3. O PTIMIZATION
This problem formulation allows for gradient-based solvers. These solvers are in general faster, because they
make use of derivatives (sensitivities) to determine in which direction to proceed the search. In this way not
the entire design space has to be searched, thereby reducing the computational time. A downside for nonconvex problems is that gradient based algorithms tend to find only a local minimum rather than the global
minimum.
By relaxing the problem formulation, the design variables ρ i are also relaxed. This means that the design
variables are no longer discrete, i.e. either 0 or 1, but are continuous and therefore range from 0 to 1. However, interpreting intermediate values is difficult, since the design variable ρ i reflects the presence of element
i. The element cannot be partially present, therefore in the optimization the design vector should be stimulated to yield either 0 or 1 for the design variables. This is done by introducing a penalization method for
the intermediate densities. Within the different penalization techniques the implicit and explicit methods
can be distinguished. The explicit methods are characterized by a constraint (or additional term in the objective function) which promotes a 0/1 design. Examples of these constraints include the quadratic penalization method, the sum of reciprocal variables (SRV) and the squared sum of variables (SSV). The implicit
penalization methods are characterized by a material interpolation scheme where intermediate densities are
penalized to make these elements less efficient in carrying the load. In combination with a volume constraint
and a loading the optimizer will be steered towards using only efficient members. This also promotes a 0/1
design. Examples of different implicit penalization schemes include the Rational Approximation of Material Properties (RAMP), Hashin-Shtrikman, and the Reuss-Voigt models. However, the most popular implicit
penalization is the Solid Isotropic Material with Penalization (SIMP) method, as proposed by Bendøe and
originally introduced for continuum topology optimization in 1989 [3]. The penalization scheme is given by
equation 3.4.
E = ρp E0
p ≥2
(3.4)
The penalized Young’s modulus is plotted as a function of ρ for three different values of the penalization factor
p in Figure 3.1.
Figure 3.1: Penalized Young’s Moduli for the SIMP approach (p = 2,3,10)
The continuous line represents the actual stiffness, the dashed lines represent the penalized stiffnesses. At
ρ equal to 0 or 1, the actual stiffness is recovered but for all intermediates densities a reduced stiffness will
result. As a consequence the members with an intermediate value for ρ encounter a penalized stiffness and
are therefore less efficient in carrying load. If the objective contains a term that tries to maximize the stiffness
and problem is subjected to a volume constraint, then the optimizer will steer the design variables to 0 and 1
to maximize stiffness. As an example, due to the penalization the optimizer prefers one element on a certain
location with the density equal to 1 than two (overlapping) elements with densities equal to a half.
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3.1. G LOBALLY C ONVERGENT M ETHOD OF M OVING A SYMPTOTES
33
A very similar approach has been used by Kawamoto [30] for the ground structure approach. Instead of
penalizing the Young’s modulus, the cross-sectional areas of the elements are penalized:
p
Ai = ρi A0
p ≥2
(3.5)
Common practice is to introduce a lower bound on the density variables (ρ mi n ). This prevents numerical
issues such as a singular stiffness matrix and/or insensitivity of the objective function to changes in the design
variables below a lower bound [32]. The lower bound is a small number, slightly larger than zero:
0 < ρ mi n ≤ ρ i ≤ 1
(3.6)
Due to the limited computational time, a gradient-based method has been selected to solve the optimization
problems. The optimizer used is GCMMA as developed by professor Svanberg [64]. SIMP has been selected as
the penalization method. The penalty factors was set to 3, which is common for the SIMP method. In Section
3.1 the optimizer GCMMA will be discussed in more detail. The optimizer is used for the optimizations. In
the next section the optimization strategy will be covered. Here will be explained why the tool developed
consists of two optimizations (i.e. mechanism design and mechanism sizing) and how these optimization
problems are formulated. Finally, the Sections 3.3 and 3.4 will cover each optimization problem in more
detail, including the derivation of the sensitivities. These sensitivities are used by the GCMMA optimizer to
determine the search direction.
3.1. G LOBALLY C ONVERGENT M ETHOD OF M OVING A SYMPTOTES
This section starts with the basics of the optimizer. The globally convergent method of moving asymptotes is a
convex sequential programming technique. The optimizer replaces the objective function and constraints by
convex approximations, which are then solved. How these approximations are formulated, will be discussed
here. Once the approximations are solved, the sensitivities will be calculated to determine how the design
vector is updated to solve the optimization problem. The method selected for obtaining the sensitivities is
the adjoint method. Why this method is selected and how this method works, will be covered here as well.
3.1.1. D ESCRIPTION
A powerful approach to solve non-linear structural optimization problems is to replace the (usually implicit)
objective and constraint functions with convex explicit separable functions. This technique is known as convex sequential programming. There are several ways to construct a separable explicit approximations via
Taylor expansions, including convex expansion, reciprocal expansion, direct (linear) expansion, diagonal
quadratic expansion, shifted convex expansion (incl. MMA and GCMMA), and power expansion [47]. The
basic steps in the algorithms are [63]:
• In each iteration, start with the current design vector x (k)
• Formulate a sub-problem by replacing the objective and constraints with convex approximations
• The unique and optimal solution of the sub-problem will be the next design vector x (k+1)
For the MMA it is possible not to converge, based on the chosen initial vector. If an infeasible solution is used
as the initial vector, in might not find a local minimum. Svanberg has improved the MMA algorithm to make
sure it finds a local minimum even for an infeasible starting point. The new updated algorithm is known as
GCMMA. It stands for globally convergent method of moving asymptotes. Globally convergent refers to its
ability to find a local minimum even for infeasible starting points. It does not mean the algorithm will find
the global minimum. The GCMMA makes use of an inner and an outer loop. The inner loop checks, based on
the function and constraint values, whether the approximation is conservative or not. If that is the case, then
the conservative convex separable approximations (CCSA) are solved [64], according to the steps described
previously. The solution to these approximations will be the updated design vector.
The choice of these approximations are based on the first derivatives of the function at the current iteration
and on the parameters U (k) and and L (k) . These parameters are known as the moving asymptotes. These
moving asymptotes increase both the stability and convergence rate of the optimization process. The approximations are constructed as follows [63]:
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
34
3. O PTIMIZATION
G̃ j (x
(k)
)) = G j (x
(k)
)+
N
X
i =1
Ã
p i(k)
j
1
Ui(k) − x i
−
!
1
Ui(k) − x i(k)
+
N
X
i =1
Ã
q i(k)
j
1
x i − L (k)
i
−
!
1
x i(k) − L (k)
i
(3.7)
With
#
¶+
¶
µ
δG j (k) −
10−5
1.001
(u ) + 0.001
(u ) + max
δx i
δx i
xi
− x imi n
"
#
µ
¶
¶
µ
δG j (k) +
δG j (k) −
10−5
(k) 2
(k)
(k)
(u ) + 1.001
(u ) + max
q i j = (x i − L i ) 0.001
δx i
δx i
xi
− x imi n
"
p i(k)
j
µ
= (Ui(k) − x i(k) )2
δG j
(k)
(3.8)
Where
³ δG
j
δx i
³ δG
j
δx i
(u (k) )
´+
(u (k) )
´−
= max
n δG
j
(u (k) ), 0
o
i
o
n δxδG
j
= max − δx (u (k) ), 0
(3.9)
i
In these equations the index i indicates the ith entry of the design vector, i.e. the ith design variable. It runs
from 1 to N , where N is the total number of variables in the design vector. The index j describes the jth constraint function. The exact formulation applies also to the objective function. The index j runs from 1 to M ,
where M is the total number of constraints. Finally, the superscript k indicates the kth sub-problem, which is
being solved. This will continue until all sub-problems are solved.
As one can see from Equation 3.7 to 3.9, the chosen convex approximation depends on the sign of the sensitivities. For positive values for the sensitivities, the first summation of Equation 3.7 is dominant. This term
contains the upper bound Ui of the variable x i . If the sensitivity becomes negative, the second summation
will be dominant. In that case the second term will be used in the summation, which involves the lower
bound L i . The lower bound will always between x mi n and x, the upper bound will always be between x and
x max .
The GCMMA algorithm updates the design vector based on the objective, constraints and their sensitivities.
Hence, each iteration will have a different design vector (i.e. a different geometry) until the optimizer is
converged. The optimizer is said to be converged when the Karush-Kuhn-Tucker (KKT) convergence criteria
are met.
3.1.2. A DJOINT M ETHOD
Gradient-based optimizers make use of derivatives/sensitivities to search for the optimum. The derivatives
can be determined both numerically and analytically. For very complicated derivatives doing this analytically could be hard if not impossible, in such cases numerical derivatives will be used. The disadvantage,
however, is the increased computational effort required to calculate the derivatives compared to the analytical derivatives. Calculating the derivatives numerically can be done via the finite difference scheme, such as
the forward difference, backward difference, and the central difference schemes [20]. The forward difference
scheme is defined as:
f 0 (x) =
f (x + h) − f (x) f 00 (ξ)
−
h
h
2!
(3.10)
The scheme makes use of the next function value f (x + h) to calculate the derivative. Similarly, the backward
difference scheme makes use of the previous function value f (x − h), as shown in Equation 3.11.
f 0 (x) =
f (x) − f (x − h) f 00 (ξ)
−
h
h
2!
(3.11)
Both these methods have a first order accuracy, which might not be sufficient for some applications. The
central difference scheme on the other hand has a second order accuracy, as shown in Equation 3.12.
f 0 (x) =
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
f (x + h) − f (x − h) f 000 (ξ) 2
−
h
2h
3!
(3.12)
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Calculating the sensitivities analytically can be done directly or via the adjoint method [9]. Consider a function of the form:
G j = f (u(ρ), ρ)
(3.13)
Then the sensitivity will be:
dG j
d ρi
=
δG j (u(ρ), ρ)
δρ i
δG j (u(ρ), ρ) d u(ρ)
+
δu
(3.14)
d ρi
In the direct method the term d u(ρ)/d ρ i is obtained by differentiating the equilibrium equations. This requires an inverted stiffness matrix for every design variable. Inverting large matrices is computationally expensive, therefore it is preferred to reduce the required number of inverted matrices.
A different approach which prevents the calculation of d u(ρ)/d ρ i and could therefore save computational
time, is the adjoint method. In the adjoint method the residual force vector, multiplied with Lagrange multipliers, is added. Since the residual force vector is zero, the problem is not altered and the Lagrange multipliers
can be chosen freely. The function can now be written as:
G j = G j (u(ρ), ρ) + λTj R(u(ρ), ρ)
(3.15)
The sensitivity will be:
dG j
d ρi
=
δG j (u(ρ), ρ)
δρ i
+
δG j (u(ρ), ρ) d u(ρ)
δu
d ρi
+ λTj
µ
∂R(u(ρ), ρ) ∂R(u(ρ), ρ) d u(ρ)
+
∂ρ i
∂u
d ρi
¶
(3.16)
Rewriting gives:
dG j
d ρi
µ
=
δG j (u(ρ), ρ)
δu
+ λTj
¶
∂R(u(ρ), ρ) d u(ρ) δG j (u(ρ), ρ)
∂R(u(ρ), ρ)
+
+ λTj
∂u
d ρi
δρ i
∂ρ i
(3.17)
Since the Lagrange multipliers could be chosen freely, they can be chosen such that the first term vanishes.
As a result the term d u(ρ)/d ρ i does not need to be calculated. Then the expression reduces to:
dG j
d ρi
=
δG j (u(ρ), ρ)
δρ i
+ λTj
∂R(u(ρ), ρ)
∂ρ i
(3.18)
Where λ j is obtained from:
λTj
δG j (u(ρ), ρ)
∂R(u(ρ), ρ)
=−
∂u
δu
(3.19)
In the direct method one inverts the equilibrium equations N times, where N is the total number of variables. In the adjoint method the first step is to calculate λ j M +1 times (including the objective), where M is
the number of constraints. The calculation of the Lagrange multipliers requires also inverted matrices, as a
consequence it is computationally less expensive to use the adjoint method if the design problem has more
design variables than constraints. If the number of constraints is greater or equal to the number of variables,
then the direct method is computationally less expense. The optimization problems solved for the morphing
winglet there are much more elements (i.e. design variables) compared to the number of constraints defined.
Therefore the adjoint method has been selected.
3.2. O PTIMIZATION S TRATEGY
The optimization strategy is focused on designing and sizing a mechanism which is able to perform a specified motion without failure. This will be done by improving, extending, and expanding the previous tool. To
understand where these research goals come from and how the current model meets these research goals,
first the previous model will be covered briefly, then the areas of improvement will be identified, which result
in the research goals. Finally, two optimization problems will be formulated. Here will be explained how the
problem formulations meet the research goals.
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3. O PTIMIZATION
3.2.1. P REVIOUS M ODEL
The approach in the previous tool for the design of mechanisms is largely based on the approach Kawamoto
took to design truss mechanisms. Kawamoto used in the non-linear finite element tool the LM-algorithm
to solve the equilibrium equations. The objective function formulated was the least squares error between
the intended trajectory and the actual trajectory of the mechanism. Hence the optimizer will try to find the
mechanism that is best able to follow the trajectory, i.e. to minimize the error between the target and actual
trajectory. To make sure no additional degrees of freedom are present (and the mechanism is able to carry a
load), two additional load cases were formulated, each with a small disturbance force. By defining the objective function as a weighted sum of the three load cases, the mechanism that will follow the intended trajectory
best, while not deviating from the trajectory due to the disturbance forces, will be the optimal solution. If additional degrees of freedom were present, the disturbance forces would result in a large deviation from the
intended trajectory. Hence by including these two additional load cases no additional degrees of freedom,
other than those required to perform the trajectory, will be present in the solution. The objective function
can therefore be written as:
F0 =
3
X
wi Di
(3.20)
i =1
The subscript i indicates the ith load case, w i are the weighting factors for each load case, and D i is given by:
³
´ ³
´
tr a j 2
tr a j 2
t
D i = u xac t − u x
+ u ac
−
u
y
y
(3.21)
tr a j
tr a j
t
Here u xac t and u ac
represent the actual x- and y-displacement of the output node and u x
and u y
the
y
intended displacements in x and y, respectively. In addition, multiple constraints were specified, including a
volume constraint on the number of elements.
As mentioned at the beginning of this section, the previous tool is largely based on the on the approach
described by Kawamoto. The tool uses the LM-algorithm as well. Similarly, the objective function is defined
as the least squares error, with one small extension. The lease squares error is calculated at different instances
of the trajectory and then summed. In this way the trajectory is checked throughout the entire path. The
objective function used in the previous tool can be written as:
F0 =
3
X
i =1
wi
Ns ³
X
´
tr a j 2
u xac t − u x
³
´
tr a j 2
t
+ u ac
−
u
y
y
(3.22)
j =1
Here N s represents the number of steps that the trajectory consists of. In addition, several constraints were
formulated. One constraints was a volume constraints on the number of elements, the other constraints were
related to the connectivity at the supports, input node, and output node. At the supports at least one member
must be present, to make sure all supports are included in the final design. At the input and output nodes
at least two elements must be present to be load carrying. The optimization problem was solved with the
GCMMA algorithm, also using the adjoint method for the sensitivities. The optimizer often yielded a design
vector with intermediate densities, where the elastic deformation of the members influenced the trajectory.
Now, the 0/1 design depends on a couple of parameters, which include a penalization scheme, volume constraint and an applied force. Therefore, to improve the separation, the applied loading was incrementally
increased, where for each load increment an additional optimization has to be performed.
Once a mechanism was found, a linear finite element analysis was performed to size the structure. For each
step j in the trajectory a finite element analysis was performed to calculate the required cross-sectional areas of the elements to prevent material failure. Then for each member the largest cross-sectional area was
selected to be conservative.
3.2.2. A REAS OF I MPROVEMENT
With the previous implementation of the tool, several improvements could be made, which could be categorized in robustness, computational time and separation (or 0/1 solution). Improving the robustness of the
tool can be achieved by removing the incremental load. For a given load the optimizer will search in a particular direction for the minimum, however when the load is increased, the minimum could lie elsewhere.
As a consequence the optimizer searches in a direction which later during the optimization no longer is a
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37
minimum. Changing its search direction is sometimes not possible since that would require a large increase
in the objective. As a result the optimizer was unable to find a feasible solution. Hence the robustness could
be improved by removing the incremental load.
An additional disadvantage of the incremental load is the increased computational time. For each load increment an optimization is performed. Another costly aspect of the previous tool is the formulation of the
objective function. For each step j three analyses have to be performed, one for the trajectory and two additional ones to regulate the degrees of freedom. If this could be reduces to a single load case, the computational
is reduced to approximately a third of the original computational time. Finally, more vector operations and
additional preallocation of variables could save some time as well.
Another aspect area of improvement is limited separation that occurred. In the previous tool the majority of
the runs resulted in intermediate densities, which are difficult to interpret and its elastic deformation might
influence the motion of the mechanism. Improving the separation is therefore preferred.
Once these aspects are improved, the functionality of the tool could be extended. The previous tool for example is only compatible with a single input and single output node. The tool could be extended to handle
multiple input and output nodes. Also, in the sizing step, only material failure was considered as a failure
mode. An important failure mode is buckling, or the loss of stability. Both column buckling as well as global
buckling can be included in the sizing step as well. These aspects could be implemented to extend the previous capabilities.
Finally, the previous tool is only compatible for 2D structures. Expanding the tool to 3D is another improvement, necessary for practical applications. All these improvements are considered the research goals for this
thesis, which were already introduced in Chapter 1. The goals are summarized below.
1. Improve previous tool
(a) Increase robustness
(b) Reduce computational time
(c) Increase separation
2. Extend previous tool
(a) Include multiple input nodes
(b) Include multiple output nodes
(c) Include buckling
3. Expand previous tool
(a) Expand to 3D
3.2.3. C URRENT M ODEL
Designing and sizing a mechanism would ideally be done in a single optimization, but that is not possible for
two reasons:
1. For the mechanism design, penalization was introduced to steer the design variables to 0 or 1. Sizing,
on the other hand, tries to minimize the cross-sectional areas to be as light as possible, just to prevent
failure. This would result in a design vector that has intermediate values for which the failure constraints are not accurate. Members with intermediate densities are penalized and will be able to carry
less load that it actually could. Therefore once a failure constraint becomes active, this might not yet
be active when the penalization was not applied. Due to the penalization the structure would be able
to carry less load that it actually could.
2. The elements that need to be removed will have a density variable equal to ρ mi n , as determined by the
optimizer. These very small and slender members are only there in the model, but would be removed
after post-processing. However, these members can hardly carry any load and would violate a stress or
local buckling constraint, while in practice they would not exist.
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3. O PTIMIZATION
Therefore, it has been decided to decouple the mechanism design and mechanism sizing. For the mechanism
design an optimization will be performed, taking into account the research objectives as discussed in the
previous section. The approach to design a mechanism is partially based on the approach Kawamoto took to
design truss mechanisms [31] and partially based on the work done by Kim [35]. The mechanism sizing step
has also been formulated as an optimization problem. In the previous tool the largest cross-sectional areas
were selected from each step to be conservative. But by minimizing the weight with different failure modes
as constraints, possible weight reductions could be achieved. The selected formulation which complies with
the research goals for both the mechanism design and sizing will be covered here.
Mechanism Design
In the previous tool the least squares error will be minimized for the mechanism design. As a result the optimizer steers towards a solution which is able to perform the trajectory as good as possible. This objective
function does not steer the solution towards a 0/1 design. A proper applied load and volume constraint in
combination with penalization could overcome this, as was shown by Kawamoto [31]. However, Kawamoto
had applied other optimizations to find out what the solution must be. Hence with this knowledge the optimization problem could be tailored. In general this information is not known a priori, therefore it was
observed that with the previous objective function it was very difficult to obtain a 0/1 design.
The objective function selected is based on the objective function used by Kim [35]. The objective function
was equivalent to the compliance. The minimization of the compliance yields the stiffest structure possible,
given the constraints. Then with the trajectory specified as a constraint, the optimizer will search for the
stiffest possible structure. Hence the optimizer actively steers towards a 0/1 design, thereby improving the
separation. Another very important aspect of the compliance minimization is the regulation of the degrees of
freedom. Having a mechanism which is able to follow the intended trajectory, then minimizing the compliance means a minimization of the displacements (u (L) ) at the node(s) where the load is applied. If additional
degrees of freedom would be present, then the displacements would be large. This, however, is prevented by
the minimization of the displacements as a result of the compliance [35]. The objective function could then
be written as:
F 0 = C = F T u (L)
(3.23)
However, it was observed that with the minimization of the compliance not always mechanisms were obtained, sometimes the elements were stretched to be able to perform the required motion. The compliance
of those geometries was indeed smaller than mechanisms that were able to perform the trajectory without
stretching elements. Therefore the reaction force at the input node (after step 1 of the finite element) was
included as an additional term in the objective function. In the first step of the finite element analysis no
loading was applied, hence the reaction force should be zero for mechanisms, and will be non-zero when
there is no degree of freedom. Including the reaction force as an additional term yields the following objective function:
³
´2
F 0 = C + F i nput
(3.24)
The term is squared to also minimize negative reaction forces in magnitude. Otherwise the optimizer would
be able to find negative objective function values, due to the reaction force. This formulation is able to find a
mechanism due to the reaction force term. The mechanism will also be as stiff as possible (i.e. 0/1 solution)
due to compliance term F T u (L) . By having an objective function which promotes 0/1 designs and regulates
the degrees of freedom automatically, there incremental load and additional load cases could be removed.
This improves the robustness, separation and computational time. To include multiple input nodes, the
objective function can be written as:
F0 = C +
N
in ³
X
i =1
´
i nput 2
Fi
(3.25)
Here Ni n is the total number of actuations at the various input nodes. Assuming that at each actuation/input
node only one degree of freedom has a prescribed displacement, then this number is also equal to the number
of input nodes.
As mentioned earlier, the trajectory must be included as a constraint. In the post-processing section of the
previous chapter the ratio R t r a j ect or y was introduced. For values smaller than 1, the mechanism is able to
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39
follow the specified trajectory. Note that for the ability to follow the specified trajectory the displacement
vector u (N L) is used. Using this displacement vector the applied loading will not influence the structure’s
ability to follow the specified trajectory. If the displacements after the applied loading were taken, then a
different loading would mean a different trajectory the mechanism follows. This could mean a violation of
the trajectory, which should be prevented. The trajectory constraint can be written as:
R t r a j ec t or y < 1
(3.26)
G t r a j ect or y = R t r a j ec t or y − 1 < 0
(3.27)
Rewriting this in standard form gives:
To include multiple output nodes, for each node a trajectory constraint can be formulated. This gives:
t r a j ect or y
Gj
t r a j ec t or y
= Rj
j = 1, ..., Nout
−1 < 0
(3.28)
Here, Nout is the total number of output nodes. Similarly, for the volume constraint the ratio should be
smaller than 1 which yields:
G vol ume = R vol ume − 1 < 0
(3.29)
However, due to the lower limit of ρ mi n for the design variables, the constraint needs to be modified slightly.
The volume will at least be equal to the total number of elements times ρ mi n . Therefore, defining αmax as
(Ne − Vmax )ρ mi n , the constraint will be redefined as:
G vol ume =
V vol ume
− 1 = R vol ume − 1 < 0
Vmax + αmax
(3.30)
To prevent the optimizer to yield mechanisms without any element at the input node (thereby having no
reaction force). The input node connectivity ratio should be larger than 1 to have at least the number of
elements specified by Vmax Rewriting this in standard form gives:
G i nput
nod e
= 1 − R i nput
nod e
<0
(3.31)
Similar to the volume constraint, the introduction of a lower bound of ρ mi n has to be incorporated. Therefore
αi n will be introduced. This will be equal to the number of elements at the input node minus the minimum
n
), then multiplied with ρ mi n . The constraint will be redefined as:
number of elements (C imi
nput nod e
G i nput
nod e
= 1−
C i nput
n
C imi
nput
nod e
+ αi n
nod e
= 1 − R i nput
nod e
<0
(3.32)
Finally, multiple input nodes can be included. Similar to the trajectory constraint, for each input node a
constraint can be defined.
i nput nod e
Gj
i nput nod e
= 1−Rj
j = 1, ..., Ni n
<0
(3.33)
Combining all this gives the following optimization problem for mechanism design:
mi n F 0 = C +
ρ
N
in ³
X
i =1
´
i nput 2
Fi

t r a j ect or y

Gj



 vol ume
G
s.t . =
i nput nod e

Gj




0 ≤ ρ mi n
t r a j ec t or y
= Rj
=R
vol ume
−1
−1
j = 1, ..., Nout
(3.34)
<0
i nput nod e
= 1−Rj
≤ ρi
<0
<0
≤1
j = 1, ..., Ni n
However, the actuation can be applied in a number of steps (N s ). This changes the optimization problem
slightly, where now the average of these steps is taken as the objective function and the average of the trajectory steps. The constraints for the volume and input node connectivity do not change during the actuation
steps, hence no average was taken. Then the optimization problem can finally be written as:
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3. O PTIMIZATION
#
"
N
Ns
´
in ³
X
1 X
i nput 2
F i ,k
Ck +
mi n F 0 =
ρ
N s k=1
i =1

i
1 PNs h t r a j ec t or y
t r a j ect or y


G
=
R
−
1

j


N s k=1 j ,k
 vol ume
vol ume
G
=
R
−1
s.t . =
i
nput
nod
e
i nput nod e


= 1−Rj
 Gj



0 ≤ ρ mi n
≤ ρi
<0
j = 1, ..., Nout
(3.35)
<0
<0
≤1
j = 1, ..., Ni n
Mechanism Sizing
The mechanism design yields a mechanism which is able to follow the specified trajectory. The elements,
however, are not optimized for weight. So far only their required presence was determined. Hence in the sizing step the mechanism will be optimized for weight, while preventing failure as a result of the aerodynamic
loading. The objective to be minimized is weight and can be written as:
F0 = W = ρ m
Ne
X
ρi A0li
(3.36)
i =1
Here ρ m is the density of the material.
The failure modes taken into account are global buckling, material failure in tension, material failure in compression, and local buckling. Of these different failure modes only global buckling is specified for the entire
structure, whereas the other three are specified for each element individually. Starting with the global buckling constraint, if the critical eigenvalue is greater than 1, no global buckling will occur. Hence the constraint
can be written as:
G g l obal
buckl i ng
= 1 − R g l obal
buckl i ng
(3.37)
Taking only the lowest buckling mode into account might result in mode jumping. The optimizer will prevent
the lowest buckling mode to occur, but the second buckling mode could become critical. Therefore it has
been decided to take the lowest three buckling modes into account. In addition, in every step the geometry has changed (due to the actuation). As a result, different buckling loads and possibly different buckling
modes are encountered for each step. Therefore, for every step the lowest three buckling modes are taken
into account during the sizing by specifying a constraint for each buckling mode:
g l obal buckl i ng
Gj
g l obal buckl i ng
= 1−Rj
<0
j = 1, ..., Nbuckl × N s
(3.38)
Here Nbuckl is the number of buckling modes taken into account per step. Hence the total number of constraints for global buckling are the product of Nbuckl and the number of steps N s .
The next constraint would be material failure in tension, which would occur if the stress in the element exceeds the yield stress. For the ith element in the kth step the constraint would be:
G ist,kr ess t ensi on =
σi ,k
σy
−1 < 0
(3.39)
In contrast to the global buckling constraint, all stress constraints can be combined in a single KreisselmeierSteinhauser function [12]. This function will always be more strict than any of the constraints put into this
function. Hence when this single function is satisfied, none of the constraints will be violated. The function
is given by:
Ã
!
N
X
1
KsG j
ln
G=
e
Ks
j =1
(3.40)
Here K s is a scaling parameter, the larger the value the closer the approximation of the actual constraints. Also,
N represents the total number of constraints combined in the single constraint and G j is the jth constraint.
Applying this function of the stress constraints would give the following constraint for tension:
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41
G
st r ess t ensi on
Ã
!
Nt
X
r ess t ensi on
1
K s G st
j
=
ln
e
<0
Ks
j =1
(3.41)
Note that j represents a single combination of i and k. And N t would be the product of the number of steps
times the number of elements in the mechanism. For example, if the actuation is done in five steps and the
mechanism consists of 4 elements, N t would be equal to twenty.
Similarly, the compression constraint can be formulated, with the small difference that the stress is now negative. Here is assumed that the yield stress in compression is equal but opposite in sign to the yield stress in
tension. It is assumed that the material yields at the same stress level in tension as it does in compression.
st r ess compr essi on
G i ,k
=−
σi ,k
σy
−1 < 0
(3.42)
And the final stress constraint for compression would be:
G
st r ess compr essi on
Ã
!
Nc
st r ess compr essi on
X
1
KsG j
ln
e
=
<0
Ks
j =1
(3.43)
Here Nc is equal to N t . The last constraint is local buckling, which is again specified for each element in each
step. For the ith element and the kth step, the constraint would be written as:
l ocal buckl i ng
G i ,k
l ocal buckl i ng
= R i ,k
−1 =
ρ i A 0 E i ²i ,k
P icr,k
−1 < 0
(3.44)
Or
l ocal buckl i ng
G i ,k
=−
4²i ,k l i2
ρ i πA 0
−1 < 0
(3.45)
Again, combining all these constraints into a single local buckling constraint would give:
G
l ocal buckl i ng
!
Ã
N
l ocal buckl i ng
lb
X
1
KsG j
<0
=
ln
e
Ks
j =1
(3.46)
Here the subscripts i and k are replaced by j . Also, Nl b is the number total number of local buckling constraints. This is also equal to N t .
Now combining all this into a single optimization would give:
mi n F 0 =ρ m
ρ
Ne
X
ρi A0li
i =1
 g l obal buckl i ng

Gj




 G st r ess t ensi on
s.t . =
G st r ess compr essi on




G l ocal buckl i ng


0 ≤ ρ mi n
<0
≤ ρi
<0
<0
<0
≤1
j = 1, ..., Nbuckl × N s
(3.47)
3.3. M ECHANISM D ESIGN
In the previous section the problem formulation for mechanism design was introduced. This section will
cover the problem formulation in more detail, elaborates on how the functions depend on the design variables, and derives the sensitivities of the objective function and constraints. First the objective function will
be discussed, then the constraints, followed by the sensitivities, and finally the verification of the mechanism
design is done. For simplicity a single actuation step is assumed here. If the actuation was done in multiple
steps, then the average of the function values and sensitivities has to be taken.
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
42
3. O PTIMIZATION
3.3.1. O BJECTIVE
The objective function chosen for the mechanism design, assuming that the actuation is done in a single step,
was:
F0 = C +
N
in ³
X
´
i nput 2
Fi
i =1
(3.48)
or
F 0 = F T u (L) +
N
in ³
X
i =1
i nput
Ii
F i nt
´2
(3.49)
In this equation u (L) and F i nt are a function of the design variables.
3.3.2. C ONSTRAINTS
Several constraints are included in the optimization problem, these include the trajectory constraint, volume
constraint, and input node connectivity constraint. Each of these constraints will be covered here.
T RAJECTORY C ONSTRAINT
The trajectory constraint(s) can be written as:
t r a j ect or y
Gj
t r a j ec t or y
= Rj
j = 1, ..., Nout
−1 < 0
(3.50)
or
t r a j ect or y
Gj
=
∆j
∆max
j
−1 < 0
j = 1, ..., Nout
(3.51)
Where ∆ j (in 3D) was given by:
r
∆j =
´
´ ³
³
´ ³
tr a j 2
tr a j 2
tr a j 2
t
+ u zac t − u z
+ u ac
u xac t − u x
y − uy
(3.52)
t
The actual nodal locations (u xac t , u ac
and u zac t ) are entries of the displacement vector u (N L) . Hence these
y
variables are a function of the design variables.
V OLUME CONSTRAINT
The volume constraint was defined as:
G vol ume = R vol ume − 1 < 0
(3.53)
or
G vol ume =
V
−1 < 0
Vmax + αmax
(3.54)
Here the volume of the structure is calculated by summing over the design variables, hence Vmax is a function
of ρ i .
I NPUT N ODE C ONNECTIVITY C ONSTRAINT
The last constraint is the input node connectivity. This is a constraint on the number of elements attached to
the input node(s). For each input node a constraint is specified of the form:
G i nput
nod e
= 1 − R i nput
nod e
C i nput
nod e
<0
(3.55)
or
G i nput
nod e
= 1−
n
C imi
nput nod e
+ αi n
<0
(3.56)
Each input node should have at least 2 elements attached to it to be able to transfer any load. Therefore, for
n
each input node a value of 2 is used for the minimum number of elements C imi
.
nput nod e
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
3.3. M ECHANISM D ESIGN
43
G i nput
Here C i nput
nod e
nod e
= 1−
C i nput
nod e
2 + αi n
<0
(3.57)
is a function of the design variables.
3.3.3. S ENSITIVITIES
In this section the sensitivities, using the adjoint method, are determined for the objective function as well as
the constraints. First the objective will be covered, followed by the trajectory constraint(s), volume constraint,
and input node connectivity constraint(s).
O BJECTIVE FUNCTION
The objective function consisted of two parts, one involving the compliance, the other involving the reaction
forces. For a single-step actuation, the sensitivity can be written as:
Ã
!
´
in ³
d ¡ T (L) ¢
d NX
d F0
i nput 2
=
F u (ρ) +
F
(3.58)
d ρi
d ρi
d ρ i i =1 i
If the actuation is done in multiple steps, simply the average of these steps is taken as the objective function
and its sensitivities. The two terms (compliance and reaction force) will be determined separately and added
in the end. First, the sensitivity of the compliance term will be calculated.
Compliance
The adjoint method adds an additional term to the sensitivity, which is the product of unknown Lagrange
multipliers and the residual force vector. This would not change the sensitivities, since the residual force vector is zero. However, by rearranging the terms, the computationally intensive term of the total derivative of the
displacement with respect to the design variables is not necessary. Since the compliance calculation involves
u (L) , the residual added should eliminate the sensitivities d u (L) (ρ)/d ρ i . This will be achieved when adding
the residual force vector from the linear (second) step of the finite element analysis. Hence the residual R (L)
will be added. Important to note is that this residual is a function of both ρ and u (N L) (ρ).
d ¡ T (L) ¢
F u (ρ)
d ρi
=
£
¤¢
d ¡ T (L)
F u (ρ) + λT1 R (L) (ρ, u (N L) (ρ), u (L) (ρ))
d ρi
=
£
¤¢
d ¡ T (L)
F u (ρ) + λT1 K t (ρ, u (N L) (ρ))u (L) (ρ) − F
d ρi
·
¸
¢
d ¡
d ¡ T (L) ¢
d
T
(N L)
(L)
F
=
F u (ρ) + λ1
K t (ρ, u
(ρ))u (ρ) −
d ρi
d ρi
d ρi
·
¸
(N L)
d u (L) (ρ)
d ¡ (L) ¢
(ρ)) (L)
T d K t (ρ, u
(N L)
u (ρ) + λ1
u (ρ) + K t (ρ, u
(ρ))
=F
d ρi
d ρi
d ρi
(3.59)
T
·
¸
(N L)
d u (L) (ρ)
(ρ)) (L)
d u (L) (ρ)
T d K t (ρ, u
(N L)
=F
+ λ1
u (ρ) + K t (ρ, u
(ρ))
d ρi
d ρi
d ρi
T
¸
·
¡
£
¤¢ d u (L) (ρ)
d K t (ρ, u (N L) (ρ)) (L)
+ λT1
u (ρ)
= F T + λT1 K t (ρ, u (N L) (ρ))
d ρi
d ρi
Note that the tangent stiffness matrix evaluated for the non-linear displacement vector is called K t . This is
done since the tangent
£ stiffness matrix
¤ for both steps in the finite element analysis are the same. Now choose
λT1 such that F T +λT1 K t (ρ, u (N L) (ρ)) becomes zero, thereby eliminating the need to calculate d u (L) (ρ)/d ρ i .
This yields the following expression for the sensitivity, taking the displacement vector out of the brackets.
d ¡ T (L) ¢
F u (ρ)
d ρi
= λT1
·
¸
d K t (ρ, u (N L) (ρ)) (L)
u (ρ)
d ρi
= λT1
·
¸
∂K t (ρ, u (N L) (ρ)) ∂K t (ρ, u (N L) (ρ)) d u (N L) (ρ) (L)
+
u (ρ)
∂ρ i
d ρi
∂u (N L) (ρ))
(3.60)
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44
3. O PTIMIZATION
To avoid evaluating d u (N L) (ρ)/d ρ i , the adjoint technique will be applied again. This time the non-linear
residual force vector will be added, the vector obtained from the first step of the finite element analysis.
d u (N L) (ρ)
d ρi
=
d u (N L) (ρ)
d R (N L) (ρ, u (N L) (ρ))
+ λT2
d ρi
d ρi
· (N L)
¸
d u (N L) (ρ)
(ρ, u (N L) (ρ)) ∂R (N L) (ρ, u (N L) (ρ)) d u (N L) (ρ)
T ∂R
=
+ λ2
+
d ρi
∂ρ i
d ρi
∂u (N L) (ρ)
(3.61)
· (N L)
¸
(ρ, u (N L) (ρ))
d u (N L) (ρ)
d u (N L) (ρ)
T ∂R
(N L)
=
+ λ2
+ K t (ρ, u
(ρ))
d ρi
∂ρ i
d ρi
· (N L)
¸
¡
¢ d u (N L) (ρ)
∂R
(ρ, u (N L) (ρ))
+ λT2
= I + λT2 K t (ρ, u (N L) (ρ))
d ρi
∂ρ i
Now choose λT2 such that I + λT2 K t (ρ, u (N L) (ρ)) becomes zero, thereby eliminating the need to calculate
d u (N L) (ρ)/d ρ i . This yields the following expression for the sensitivity.
d u (N L) (ρ)
d ρi
= λT2
·
∂R (N L) (ρ, u (N L) (ρ))
∂ρ i
¸
(3.62)
Recall from Chapter 2 the expressions for the residual force vector and tangent stiffness matrix, which are
repeated below and evaluated for u (N L) .
R
(N L)
=
Ne
X
µ
Ai Ei li
i =1
Kt =
¢T
1 T (N L)
1 ¡
bi u
+ 2 u (N L) B̃ u (N L)
2
l
2l
Ne
X
¶Ã
1
bT
l i2 i
!
1 ¡ (N L) ¢T
+ 2 u
B̃ i − p T
li
(3.63)
Ai Ei li
i =1
·Ã
bT
l i2 i
1 ¡
!T Ã
¢T
1 ¡
+ 2 u
B̃ i
+ 2 u (N L) B̃ i
×
li
li
!¸
µ
¶Ã
1
1 T (N L)
1 ¡ (N L) ¢T
(N L)
+ 2 bi u
B̃ u
B̃ i
+ 2 u
l
2l
l i2
1
¢
(N L) T
1
!
(3.64)
bT
l i2 i
As mentioned at the beginning of this chapter, all cross-sectional areas are penalized with the SIMP interpolation scheme. Substituting this into the expressions for the residual force vector and tangent stiffness matrix
and simplifying the expressions give:
R
(N L)
=
Ne p
X
ρ A max E i
l i3
i =1
Kt =
·
Ne p
X
ρ A max E i ³
i =1
l i3
µ
¶³
¡
¢T ´
1 ¡ (N L) ¢T
T (N L)
(N L)
bi u
+ u
B̃ u
b Ti + u (N L) B̃ i − p T
2
(3.65)
µ
¶
¸
¡
¢T ´T ³ T ¡ (N L) ¢T ´
¢T
¡ ¢
1¡
bi + u
b Ti + u (N L) B̃ i
B̃ i + b Ti u (N L) + u (N L) B̃ u (N L) B̃ i
2
(3.66)
Calculating the remaining derivatives:
p−1
∂R (N L) pρ i
=
∂ρ i
A max E i
l i3
¶³
µ
¡
¢T ´
¢T
1¡
b Ti u (N L) + u (N L) B̃ u (N L) b Ti + u (N L) B̃ i − p T
2
(3.67)
µ
¶
¡ ¢
∂K t
pρ p−1 A max E i ³ T ¡ (N L) ¢T ´T ³ T ¡ (N L) ¢T ´
1 ¡ (N L) ¢T
T (N L)
(N L)
B̃
+
b
u
+
B̃ i
=
b
+
u
B̃
b
+
u
u
B̃
u
i
i
i
i
i
3
∂ρ i
2
li
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
(3.68)
T.A. DE J ONG
3.3. M ECHANISM D ESIGN
45
(N L)
Nd
∂K t d u (N L) X
∂K t d u j
=
(N L) d ρ
∂u (N L) d ρ i
i
j =1 ∂u
j
=
Nd
X
∂K t d u (N L) ( j )
(N L)
d ρi
j =1 ∂u
j
=
(3.69)
Nd X
Ne p
X
ρ A max E i
l i3
j =1 i =1
×
·³
I Tj B̃ i
´T ³
¡
¢T ´ ³
¡
¢T ´T ³ T ´
b Ti + u (N L) B̃ i + b Ti + u (N L) B̃ i
I j B̃ i
µ
´¶ ¡ ¢¸ d u (N L) ( j )
¡
¢T
1³
+ b Ti I j + I Tj B̃ u (N L) + u (N L) B̃ I j
B̃ i
2
d ρi
The partial derivative of the tangent stiffness matrix with respect to the displacement vector would be a 3D
matrix, multiplying this with the second term reduces it to a matrix. Note that j runs from 1 to the number of
free DOF. To summarize, the final sensitivity of the compliance term is given by:
¸
·
∂K t (ρ, u (N L) (ρ)) ∂K t (ρ, u (N L) (ρ)) d u (N L) (ρ) (L)
d ¡ T (L) ¢
+
u (ρ)
F u (ρ) = λT1
d ρi
∂ρ i
d ρi
∂u (N L) (ρ))
(3.70)
Where the three derivative terms are given by the Equations 3.62, 3.68, and 3.69.
Reaction force
As mentioned earlier the internal force vector in the first step is used in the formulation of the reaction force
term. The internal force vector F i nt , defined in the previous chapter, is a function of ρ and u (N L) (ρ). Hence
the non-linear residual force vector will be added for the adjoint approach. Also, there has been made use of
the following rule, where the order (summation versus derivative) has been interchanged.
´2 NX
´2
in d ³
in ³
d NX
i nput
i nput
Ii
F i nt (ρ, u (N L) (ρ)) =
Ii
F i nt (ρ, u (N L) (ρ))
d ρ i i =1
i =1 d ρ i
(3.71)
Now for a single reaction force i the sensitivity will be determined. Then summing those individual sensitivities will yield the sensitivity of all the reaction forces.
´2
d ³ i nput
Ii
F i nt (ρ, u (N L) (ρ))
d ρi
=
´2
¢
d ³ i nput
d ¡ (N L)
Ii
F i nt + λ3 T
R
(ρ, u (N L) (ρ))
d ρi
d ρi
³
´
(N L)
(ρ))
i nput
i nput d F i nt (ρ, u
= 2 Ii
F i nt I i
d ρi
+λ3 T
(3.72)
¢
d ¡ (N L)
R
(ρ, u (N L) (ρ))
d ρi
Note that the residual force vector added R (N L) has been used in the sensitivity calculation for the compliance as well. Hence the derivatives of the residual force vector do not have to be recalculated. Also, the
dependencies will be removed for clarity.
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
46
3. O PTIMIZATION
´2
d ³ i nput
Ii
F i nt
d ρi
´
³
d ¡ (N L) ¢
i nput d F i nt
i nput
= 2 Ii
F i nt I i
+ λ3 T
R
d ρi
d ρi
i nput i nput
= 2F i
Ii
+λ3
T
µ
µ
∂F i nt
∂F i nt d u (N L)
+
∂ρ i
∂u (N L) d ρ i
∂R (N L) ∂R (N L) d u (N L)
+
∂ρ i
∂u (N L) d ρ i
¶
¶
(3.73)
µ
(N L) ¶
d u (N L)
i nput i nput ∂F i nt
T ∂R
= 2F i
Ii
+
λ
3
d ρi
∂u (N L)
∂u (N L)
i nput i nput
Ii
+2F i
∂F i nt
∂R (N L)
+ λ3 T
∂ρ i
∂ρ i
Similarly, choose λ3 such that the first term will disappear. This results in solving the following equation for
λ3 :
¶
µ
i nput i nput ∂F i nt
T
+
λ
K
(3.74)
2F i
Ii
3
t =0
∂u (N L)
Here is made use of the fact that the partial derivative of the residual force vector with respect to the displacements gives the tangent stiffness matrix. The sensitivity can be written as:
´2
d ³ i nput
Ii
F i nt
d ρi
i nput i nput
Ii
= 2F i
∂F i nt
∂R (N L)
+ λ3 T
∂ρ i
∂ρ i
(3.75)
The partial derivative of the residual force vector has been determined previously. Hence the term
remains to be determined. Recall from Chapter 2 the expression for the internal force vector F i nt :
!
µ
¶Ã
Ne
X
1 T
1 T
1 T 1 T
A i E i l i 2 b i u + 2 u B̃ u
F i nt =
b + 2 u B̃ i
l
2l
l i2 i
li
i =1
∂F i nt
∂ρ i
(3.76)
p
Realizing that A i = ρ i A 0 , gives the following expression for the sensitivity:
!
¶Ã
µ
1 T 1 T
1 T
∂F i nt
1 T
p−1
= pρ i A 0 E i l i 2 b i u + 2 u B̃ u
b + 2 u B̃ i
∂ρ i
l
2l
l i2 i
li
(3.77)
Substituting this expression yields the final expression for the sensitivity:
´2
d ³ i nput
Ii
F i nt
d ρi
i nput i nput
p−1
= 2F i
Ii
pρ i A 0 E i l i
+λ3 T
³
1 T
b u + 2l12 u T B̃ u
l2 i
Ã
´ 1
bT
l i2 i
+
1
l i2
!
T
u B̃ i
(3.78)
∂R (N L)
∂ρ i
Finally, the sensitivities of the entire objective function F 0 can be calculated. This can be obtained by summing the expression for the compliance sensitivity (equation 3.70) and the sensitivity of the reaction force for
each prescribed displacement (equation 3.78).
T RAJECTORY C ONSTRAINT
For the calculation of the trajectory sensitivities, the subscript j has been left out for clarity. As mentioned
earlier for every output node j a trajectory constraint will be specified. Hence the displacements shown here,
are those for the jth output node. Also, the trajectory constraint was specified before the loading was applied.
Hence the displacements in the constraint are those obtained from u (N L) . As a result, the residual force vector
R (N L) is used in the adjoint sensitivity analysis. One final note, the Lagrange multipliers are collected in the
vector λ4 .
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
3.3. M ECHANISM D ESIGN
47
dG t r a j ect or y
d ρi
1
=
∆max
1
∆max
∆max
+λ4
T
tr a j
2((u (N L) )T I x − u x )I x + 2((u (N L) )T I y − u y )I y d u (N L)
q
tr a j
tr a j
d ρi
2 ((u (N L) )T I x − u x )2 + ((u (N L) )T I y − u y )2
µ
∂R (N L) ∂R (N L) d u (N L)
+
∂ρ i
∂u (N L) d ρ i
¶
(3.79)
tr a j
1
∆max
+λ4 T
µ
1
∆max
+λ4 T
= λ4 T
µ
tr a j
t
(u xac t − u x )I x + (u ac
)I y d u (N L)
y − uy
q
tr a j
t r a j 2 d ρi
t
(u xac t − u x )2 + (u ac
)
y − uy


=
µq
¶
tr a j 2
tr a j 2
(N
L)
T
(N
L)
T
((u
) I x − u x ) + ((u
) I y − uy )
tr a j
1
=
d
d ρi
d R (N L)
d ρi
+λT4
=
µq
¶
tr a j
tr a j
((u (N L) )T I x − u x )2 + ((u (N L) )T I y − u y )2 − 1
d R (N L)
d ρi
+λT4
=
d
d ρi
µ
∂R (N L) ∂R (N L) d u (N L)
+
∂ρ i
∂u (N L) d ρ i
¶

tr a j
tr a j
t
(N L)
(N L)
−
u
)I
(u xac t − u x )I x + (u ac
y
∂R
y
y
 du
+ λ4 T
q

(N
L)
tr a j
tr a j 2
d ρi
∂u
t
(u xac t − u x )2 + (u ac
)
y − uy
∂R (N L)
∂ρ i
∂R (N L)
∂ρ i
¶
¶
In the derivation I x and I y are vectors that contain zeros at every entry except for one, which corresponds to
the entry of the output node. This entry contains the value 1. Also, for simplicity the sensitivity was shown
only in 2D, but the exact same approach holds for 3D. Only a third term of the z-direction will be included as
well.
The partial derivative of the non-linear residual force vector is given by equation 3.67. The Lagrange multipliers λ4 are obtained from equation 3.80.
1
∆max
tr a j
tr a j
t
(u xac t − u x )I x + (u ac
)I y
∂R (N L)
y − uy
+ λ4 T
=0
q
tr a j
tr a j 2
∂u (N L)
t
(u xac t − u x )2 + (u ac
)
y − uy
(3.80)
V OLUME C ONSTRAINT
The sensitivity of the volume constraint can directly be determined and is equal to:
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
48
3. O PTIMIZATION
dG vol ume
d ρi
=
µ
¶
d
V
−1
d ρ i Vmax + αmax
Ne
X


ρi




i


−
1
V

 max + αmax

=
d
d ρi
=
1
Vmax + αmax
I NPUT N ODE C ONNECTIVITY C ONSTRAINT
n
As mentioned earlier the minimum number of elements C imi
nput
tivity is given by:
i nput nod e
dG j
d ρi
=
−1
2 + αi n

0


nod e
(3.81)
was set to 2. Therefore, the final sensi-
if i ∈ i nput
(3.82)
otherwise
Here the set i nput contains the elements which are attached to the input node j . The sensitivity is only nonzero if the element is attached to the input node. Then it is equal to −1/(2 + αi n ).
Finally, all the analytical sensitivities calculated were compared to the sensitivities obtained from central
differences. The sensitivities were found to be the same. Hence the sensitivities were verified.
3.3.4. V ERIFICATION
The problem formulation for the mechanism design should be able to find black-and-white mechanisms
which follow a specific trajectory. Whether the tool is indeed able to find black-and-white mechanisms with
a specified trajectory will be verified with a mechanism found by Kawamoto [30]. In his PhD Kawamoto also
considered the design of articulated truss mechanisms. He considered a problem formulation illustrated in
Figure 3.2a. On two nodes a displacement was prescribed and the objective was to find the mechanism where
the output node (node 11 in the figure) displaces as much as possible to the right. Via brute force the optimal
solution for this 2D problem was found to be the mechanism shown in Figure 3.2b.
(a) Truss ground-structure for benchmark
problem (initial geometry)
(b) Optimal solution for the benckmark
problem (final geometry)
Figure 3.2: Scissors benchmark problem and solution by Kawamoto [30]
The kinematics is a result of the relative cross-sectional areas. The actual values of the areas are not relevant. The optimizer will steer towards either ρ mi n or the upper bound, whether this is set to 0.1, 1 or 2
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
3.3. M ECHANISM D ESIGN
49
does not change the mechanisms found. Members with an area equal to ρ mi n can be removed, whereas elements with significant cross-sectional areas (ideally all equal to the upper bound) cannot be removed without
changing its kinematics. The upper bound has conveniently set to 1, just like Kawamoto did. Also the value
of the Young’s modulus does not influence the mechanisms found, therefore it has been set to 10 (just like
Kawamoto did). Also the same size for the ground-structure was used, with dimensions 1-by-1.
On the input nodes a prescribed displacement of 0.5 was enforced, for node 1 an upward displacement and
for node 3 a downwards displacement. In addition, on the output node a trajectory was specified which coincided with the displacement of the node for mechanism shown in Figure 3.2b. This means that the geometry
has two input nodes and a single output node. In addition, the actuation has been done in 5 steps, hence the
optimization problem can be written as:
#
"
´
2 ³
5
X
1X
i nput 2
F i ,k
Ck +
mi n F 0 =
ρ
5 k=1
i =1

i
1 P5 h t r a j ec t or y


G t r a j ect or y =
Rk
−1

k=1


5


 G vol ume
= R vol ume − 1
i nput nod e
i nput nod e
s.t . =
G1
= 1 − R1


 i nput nod e
i nput nod e


G
= 1 − R2


 2
0 ≤ ρ mi n
≤ ρi
<0
<0
(3.83)
<0
<0
≤1
Running the simulation, for a strict trajectory constraint with ∆max chosen to be 0.01. The mechanism found
in the optimization was exactly the same mechanism (scissors) Kawamoto found, as can be seen in Figure 3.3.
Op top the converged solution is shown, where all elements are included, below the post-processed geometry
is shown. Here the elements with ρ mi n are removed.
Figure 3.3: Topology of the scissors before post-processing (top) and after post-processing (bottom)
The corresponding optimization history is shown in Figure 3.4 for a random input vector. In blue the objective
function is shown, which was converged after approximately 140 iterations. In red the trajectory constraint
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50
3. O PTIMIZATION
was shown. The original geometry was indeed not able to follow the specified trajectory and the constraint
was violated at the start. Then, in green the volume constraint is shown. Finally, in pink the connectivity
for first input node (the lower input node) and in yellow the connectivity for second input node 2 (the upper
input node) was shown. In the optimized result all constraints are satisfied. It must be noted that all function
values were normalized with the maximum value for the objective function (which for this optimization was
the starting geometry). Hence the objective starts at the value 1.
Figure 3.4: Optimization history of the mechanism design for the scissors
Finally, in Figure 3.5 the actuation of the mechanism is illustrated. This is done for 5 steps in the actuation
process, both before and after the post-processing.
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3.4. M ECHANISM S IZING
51
Figure 3.5: The five steps of the actuation of the scissors, before (left) and after (right) post-processing
3.4. M ECHANISM S IZING
The second optimization sizes the individual truss members to obtain the minimum weight, while preventing
failure. In contrast to the first optimization, penalization is no longer required. After post-processing the
geometry obtained from the first optimization, only the necessary elements will remain. By removing the
penalization the structural response from the finite element now accurately reflects the structure’s ability to
carry the aerodynamic loading. The stresses and strains from this finite element analysis can be used in the
formulation of various failure modes. In addition, a linear buckling analysis is also performed to take global
buckling into account.
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52
3. O PTIMIZATION
3.4.1. O BJECTIVE
After the mechanism design, the sizing optimization is performed. As mentioned earlier the objective is
minimum weight, repeated below.
F0 = W = ρ m
Ne
X
ρi A0li
(3.84)
i =1
The weight is calculated by multiplying the density of the material with the volume. The volume depends
on the design variables ρ i . Hence by formulating minimum weight, the optimizer tries to reduce the crosssectional areas, until one or more failure constraints become active.
3.4.2. C ONSTRAINTS
Several constraints are included in the optimization problem, these include the global buckling constraints,
stress constraint in tension and compression, and local buckling constraint. Each of these constraints will be
covered here.
G LOBAL B UCKLING C ONSTRAINT
For global buckling several constraints were defined, a single constraint was given by:
g l obal buckl i ng
Gi
g l obal buckl i ng
= 1 − Ri
<0
i = 1, ..., Nbuckl × N s
(3.85)
Ideally, all these constraints are combined into a single Kreisselmeier-Steinhauser function. However the
use of this function has numerical limitations. This functions is the natural logarithm of a sum. So when
the sum is equal to zero, the natural logarithm will be minus infinity. The sum would (numerically) be zero
when all the buckling constraints are very negative. In practice this would mean than the structure would
not buckle, however the optimizer will be stuck once this phenomenon occurs at a certain iteration. For
various optimizations this was encountered. Therefore it was decided not to combine the various buckling
constraints into a single Kreisselmeier-Steinhauser function, but several separate buckling constraints. In
these constraints the buckling load factors are a function of the design variables.
T ENSION S TRESS C ONSTRAINT
The final stress constraint for tension is given by:
G
st r ess t ensi on
!
Ã
Nt
X
r ess t ensi on
1
K s G st
j
e
<0
ln
=
Ks
j =1
(3.86)
r ess t ensi on
Where G st
is a stress constraint for a particular element in a step, given by:
j
r ess t ensi on
G st
= G ist,kr ess t ensi on =
j
σi ,k
σy
=
²i ,k E i
σy
(3.87)
In this constraint the strain of the ith element depends on the design variables ρ i . If one assumes the same
material for each element, then the Young’s modulus and yield stress would be the same for all elements.
C OMPRESSION S TRESS C ONSTRAINT
This constraint is very similar to the tension constraint, only with a minus in front of the actual stress ratio.
In this way the constraint is violated if the stress exceeds the negative yield stress.
L OCAL B UCKLING C ONSTRAINT
The last constraint is local buckling. The constraint was defined as:
Ã
!
N
l ocal buckl i ng
lb
X
1
KsG j
l ocal buckl i ng
ln
e
<0
G
=
Ks
j =1
(3.88)
With:
l ocal buckl i ng
Gj
l oc al buckl i ng
= G i ,k
=−
4²i ,k l i2
ρ i πA 0
−1
(3.89)
In the local buckling constraint is depends in ρ i directly, as well as via the strain indirectly.
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3.4. M ECHANISM S IZING
53
3.4.3. S ENSITIVITIES
In this section the sensitivities of the functions are derived. Similar to the previous optimization, the adjoint
technique will be used.
W EIGHT
The sensitivities of the objective function can be derived as shown below:
!
Ã
Ne
X
d
d F0
m
ρi A0li
=
ρ
d ρi
d ρi
i =1
(3.90)
The density of the material, the length of the element and the maximum cross-sectional area are independent
of the design variables. Hence, the sensitivities simply become:
d F0
= ρm A0li
d ρi
(3.91)
G LOBAL B UCKLING C ONSTRAINT
The global buckling constraint is slightly more complicated. For convenience the subscript j has been left
out. But for every buckling constraint the sensitivity is calculated as shown here. Similar to the first optimization (mechanism design), the residual force vector will be added. Since the eigenvalues of this problem
are a function of u (L) , the linear residual force vector will be added. Also note that the vector containing
the Lagrange multipliers is here denoted as η, to avoid confusion with the λ indicating the eigenvalue. The
derivation is shown for the critical eigenvalue, but also applies to any other eigenvalue.
dG g l obal buckl i ng
d ρi
=
¢
¢
d ¡
d ¡ (L)
1 − λcr ((ρ, u (L) (ρ)) + ηT
R (ρ, u (N L) (ρ), u (L) (ρ))
d ρi
d ρi
=−
¢
d λcr ((ρ, u (L) (ρ))
d ¡
+ ηT
K t (ρ, u (N L) (ρ))u (L) (ρ) − F
d ρi
d ρi
∂λcr ((ρ, u (L) (ρ)) ∂λcr (ρ, u (L) (ρ)) d u (L) (ρ)
+
=−
∂ρ i
d ρi
∂u (L) (ρ)
µ
+ηT
µ
¶
d u (L) (ρ)
d K t (ρ, u (N L) (ρ)) (L)
u (ρ) + K t (ρ, u (N L) (ρ))
d ρi
d ρi
¶
(3.92)
µ
¶
∂λcr (ρ, u (L) (ρ))
d u (L) (ρ)
T
(N L)
= −
+
η
K
(ρ,
u
(ρ))
t
d ρi
∂u (L) (ρ)
−
d K t (ρ, u (N L) (ρ)) (L)
∂λcr (ρ, u (L) (ρ))
+ ηT
u (ρ)
∂ρ i
d ρi
µ
¶
∂λcr
d K t (L)
d u (L) ∂λcr
= − (L) + ηT K t
−
+ ηT
u
d ρi
∂ρ i
d ρi
∂u
The partial derivatives of λcr with respect to the displacement vector and design variables have to be determined. Important to note is that both stiffness matrices are a function of the design variables ρ i , where only
the geometric stiffness matrix is a function of the displacement vector. Starting with ∂λcr /∂u (L) gives:
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54
3. O PTIMIZATION
∂λcr
=
∂u (L)
=
=
=
The jth entry of the vector φTcr
φTcr
∂K G
∂u (L)
−
∂u (L)
φTcr K L φcr
!
φTcr K G φcr
φTcr K L φcr
¶
µ
T ∂K G
φ
φ
cr
cr
∂u (L)
(φTcr K G φcr )2
−λcr
φTcr K G φcr
−λcr
φTcr K G φcr
(3.93)
µ
¶
T ∂K G
φcr
φ
∂u (L) cr
µ
¶
T ∂K G
φcr
φ
∂u (L) cr
φcr is given by:
Nd
X
ρ p A max E i
j =1
Ã
∂
l i3
∂u (L)
B̃ i φcr
∂u j
b Ti
Nd
X
ρ p A max E i
= φTcr
j =1
l i3
b Ti I j B̃ i φcr
(3.94)
Here, I j is a vector will containing zeros, except for the jth entry, which is equal to one. With this the Lagrange
multipliers can be calculated. Finally, the partial derivative of λcr with respect to the design variables is given
by:
Ã
!
φTcr K L φcr
∂
∂λcr
=
− T
∂ρ i
∂ρ i
φcr K G φcr

T
 φcr
∂K L
φ
∂ρ i cr

µ
¶
φT K L φcr
∂K G


φTcr
φcr 
− Tcr
= − T

 φcr K G φcr
∂ρ i
(φcr K G φcr )2

T ∂K L
φ
φ
µ
¶
cr
 cr ∂ρ

λcr
∂K G


i
= − T
+ T
φTcr
φcr 
 φcr K G φcr

∂ρ i
φcr K G φcr

φTcr
=−
=−
∂K L
φ
∂ρ i cr
φTcr K G φcr
1
φTcr K G φcr
−
λcr
φTcr K G φcr
(3.95)
¶
µ
T ∂K G
φ
φcr
∂ρ i cr
µ
µ
¶¶
T ∂K L
T ∂K G
φcr
φ + λcr φcr
φ
∂ρ i cr
∂ρ i cr
The other terms are simply given by:
∂K L
∂ρ i
p−1
=
pρ i
A max E i
l i3
b i b Ti
(3.96)
∂K G
∂ρ i
p−1
=
pρ i
A max E i
l i3
b Ti u (L) B̃ i
With all the derivatives determined, the final sensitivity can be calculated. The sensitivity is given by Equation
3.92, where the partial derivatives of λcr are given by Equations 3.93 and 3.94.
T ENSION S TRESS C ONSTRAINT
The next constraint is the tension stress constraint. First the total derivative of the Kreisselmeier-Steinhauser
function will be given. This will be derived in this section, but is also used for the compression stress constraint and local buckling constraint.
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3.4. M ECHANISM S IZING
55
!
Ã
Nt
X
r ess t ensi on
d 1
K s G st
j
=
e
ln
d ρi K s
j
dG st r ess t ensi on
d ρi
Nt
X
=
Ã
e
r ess
K s G st
j
t ensi on
r ess t ensi on
dG st
j
e
(3.97)
d ρi
j
Nt
X
!
r ess
K s G st
j
t ensi on
j
Now the total derivatives of the individual tension constraints have to be derived. Important to note is that
or the derivation of the sensitivities a mechanism is assumed. In other words, the total strain would normally
be a function of both u (N L) and u (L) . However, for mechanisms the prescribed displacements and the corresponding displacement vector do not result in strains. Therefore the strain is only a function of the u (L) . This
simplifies the derivation of the sensitivities. Hence the sensitivities can be written as:
r ess t ensi on
dG st
j
d ρi
=
E d ² j (u (L) (ρ))
σy
d ρi
(3.98)
E d ² j (u (L) (ρ))
d R (L) (ρ, u (L) (ρ))
=
+ λT
σy
d ρi
d ρi
The Lagrange vector is denoted as λ. For clarity the dependencies are not shown, now the sensitivities become:
r ess t ensi on
dG st
j
d ρi
µ (L)
¶
E ∂² j d u (L)
∂R (L) d u (L)
T ∂R
=
+λ
+
σ y ∂u (L) d ρ i
∂ρ i
∂u (L) d ρ i
(3.99)
(L) ¶
(L)
E ∂² j
d u (L)
T ∂R
T ∂R
+
λ
+
λ
=
σ y ∂u (L)
∂ρ i
∂u (L) d ρ i
µ
The partial derivatives of the linear residual force vector have already been determined in the sensitivities for
the mechanism design, hence will not be repeated here. Also, the partial derivative of the strain with respect
to the displacement vector has already been derived previously, and was given by Equation 2.25. With all the
terms known, the sensitivities can be computed.
C OMPRESSION S TRESS C ONSTRAINT
The sensitivities of the compression stress constraint are equal in magnitude but opposite in sign compared
to the tension constraint.
L OCAL B UCKLING C ONSTRAINT
Finally, the sensitivities of the local buckling constraint are needed. Again only the sensitivities for a single
constraint will be derived. This will have to be repeated for each element in each step of the actuation. The
sensitivities have to be combined as given by equation 3.97. Now the sensitivities for a single constraint is
given by:
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56
3. O PTIMIZATION
l ocal buckl i ng
dG j
d ρi
d
=
d ρi
Ã
−4l 2j ² j (u (L) (ρ))
!
πA 0 ρ j
=
−4l 2j d ² j (u (L) (ρ))
πA 0 d ρ i
ρj
=
µ (L)
¶
−4l 2j d ² j (u (L) (ρ))
∂R
∂R (L) d u (L)
+ λTlb
+
πA 0 d ρ i
ρj
∂ρ i
∂u (L) d ρ i
=
¶
µ (L)
¶
−4l 2j µ ∂² j d u (L)
∂ 1
∂R (L) d u (L)
T ∂R
+
+
λ
+
lb
πA 0 ∂u (L) d ρ j
∂ρ i ρ j
∂ρ i
∂u (L) d ρ i
Ã
=
−4l 2j ∂² j
πA 0
∂R (L)
+ λTlb
(L)
∂u
∂u (L)
!
(3.100)
4l 2j ∂ 1
d u (L)
∂R (L)
−
+ λTlb
dρj
πA 0 ∂ρ i ρ j
∂ρ i
The Lagrange multipliers are collected in the vector λl b . This vector can be obtained as was done for the
stress constraints. The partial derivative of the residual force vector with respect to the design variables is
also previously determined. Only the term ∂/∂ρ i (1/ρ j ) has to be determined, which is given below:


if i 6= j
0
∂ 1
−1
=
(3.101)
∂ρ i ρ j 
 ρ 2 if i = j
j
Finally, all the sensitivities derived in this chapter have been verified with finite differences. The sensitivities
were found to be the same.
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4
D ESIGN C ASES
In the previous two chapters the tool has been discussed in detail. In Chapter 2 the finite element analysis
and the linear buckling analysis were covered. In Chapter 3 the optimization strategy, including the mechanism design and mechanism sizing, was explained. With these two optimizations, mechanisms for morphing
winglets could be designed and sized.
Over the years there has been a significant growth in the regional air transport. These trajectories are characterized by relative short distances. As a result, the time associated with take-off and landing becomes significant compared to the entire duration, this in contrast to long-haul flights. Where for long-haul aircraft the
winglet can be designed for cruise only, for regional aircraft the take-off and landing should be taken into account. Therefore in this thesis the design cases are focused on regional aircrafts, specifically the C295 military
transport aircraft from airbus. A picture of this aircraft can be found in Figure 4.1.
Figure 4.1: Picture of Airbus C295, military transport aircraft [26]
Based on the C295, the dimensions of the winglet and the aerodynamic loading could be defined. For he
design cases the length of the winglet is set to 2 meters. The width starts at 1 meter and has a taper ratio of a
half, resulting in a width of 50 centimeters at the tip. The aerodynamic loading is simplified to a point load,
acting in the middle of the winglet length (and in 3D at a quarter of the chord). The lifting force was estimated
using Proteus, an aeroelastic model developed in house by the Aeroelasticity and Morphing Wings (AaM)
research group of my faculty aerospace engineering of the Delft University of Technology. The aerodynamic
force was estimated to be 5 [kN] in the maneuver load case of 2.5g. It is assumed that the loading remains
constaint at 5 [kN] and remains perpendicular to the winglet. With these specifications the mechanisms
could be designed and sized. In Section 4.1 the 2D results can be found and in Section 4.2 the 3D results.
4.1. T WO D IMENSIONAL M ORPHING W INGLET
In this section the 2D mechanisms are covered. As a result of the geometric non-linearity the design space is
non-convex, resulting in several (local) minima. To overcome this several optimizations are performed, each
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58
4. D ESIGN C ASES
with a random design vector. From these optimizations 4 different mechanisms were obtained, which are
presented in this section. These mechanisms are also sized to obtain the minimum weight, while satisfying
several failure constraints. First the optimization problems are defined, followed by the results obtained from
it.
4.1.1. P ROBLEM F ORMULATION
As mentioned in the beginning of this chapter, the winglet was set to 2 meters long. The width of the design
space was set to 0.4 meters and the height to 0.3 meters. These dimensions were chosen to be the same
as used in the previous work done by the AaM research group of my faculty aerospace engineering of the
Delft University of Technology [21]. This design space is discretized by defining 3 nodes along the width
and 3 nodes along the height, resulting in a total of 9 nodes. Connecting these 9 nodes with all possible
combinations, results in truss members in the design space. The cross-sectional area of these 36 elements
are used as design variables. An illustration of the design space could be found in Figure 4.2. To prevent rigid
body motions, the upper and lower left nodes are pinned. This is illustrated by the roller supports. Also, the
middle left node (indicated with the blue circle) is chosen to be the input node, whereas the most right node
(the tip node of the winglet, indicated with the green circle) is selected for the output node. The actuation
will be done in 5 steps. The horizontal displacement is prescribed, with a maximum actuation of 0.11 meters.
The vertical displacement of the input node is free. For the output node a circular trajectory of 45 degrees is
prescribed. The radius of this circular motion is set to 2.2 meters, which is equal to the winglet length plus
half of the design space. This again is chosen to be the same as was done in the previous work of the faculty.
The layout of the winglet itself is chosen such that in the middle of a winglet a node is present, where the
aerodynamic loading will be acting on. This is illustrated by the red arrow.
Figure 4.2: Truss ground-structure (design space) for 2D winglets
With this description, the following optimization problem could be defined for the mechanism design in 2D.
As mentioned before only a single input node and a single output node are used, which together with the
volume constraint, results in 3 constraints. For the output node the average of the trajectory constraint in
each step is taken. The design variables ρ i are the cross-sectional area scaling parameters. where A i is given
by:
A i = ρ 3i A 0
(4.1)
The penalization factor is set to 3, which is commonly done in this field. The lower bound for the design
variables was chosen to be 0.001. This is very small, which would not affect the kinematics of the mechanism.
mi n F 0 =
ρ
³
´ ¸
5 ·
1X
i nput 2
C k + Fk
5 k=1

i
1 P5 h t r a j ec t or y


Rk
−1
G t r a j ec t or y =

k=1

5

= R vol ume − 1
G vol ume
s.t . =

i nput nod e

= 1 − R i nput nod e
 G


0 ≤ ρ mi n
≤ ρi
<0
(4.2)
<0
<0
≤1
The maximum allowable deviation for the output trajectory (∆max ) was set to 0.1. The maximum number of
elements in the design space (V max ) was set 4. And, as already mentioned earlier at least 2 elements should
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59
be attached to the input node.
Finally, for the mechanism design not the actual material properties are used, because only the relative crosssectional areas determine which elements need to be removed and which ones are needed. Therefore the
same material properties for the mechanism design were used as Kawamoto used for his optimizations [30].
The Young’s modulus E was set to 10 and the cross-sectional area A 0 to 1. Finally, the aerodynamic loading
has been set to 1 for convenience.
Once a mechanism is found, the individual elements of the mechanism are sized to obtain the minimum
weight, under the condition that no failure occurs. In 2D it is assumed that two mechanisms would carry the
total aerodynamic loading, which would yield a load of 2.5 [kN] for the 2D mechanisms. Also, as discussed in
the previous chapter for stress in tension, stress in compression and local buckling a single constraint could
be formulated. For the global buckling a set of constraints is added. Again the actuation will be done in 5
steps, where in each step the lowest three buckling modes are taken into account. This result in 15 global
buckling constraints. This gives the following optimization problem:
mi n F 0 =ρ m
ρ
Ne
X
ρi A0li
i =1
 g l obal buckl i ng

Gj




 G st r ess t ensi on
s.t . =
G st r ess compr essi on




G l ocal buckl i ng


0 ≤ ρ mi n
<0
≤ ρi
<0
<0
<0
≤1
j = 1, ..., 15
(4.3)
The material selected is aluminum 2024-T3. The Young’s modulus (E ) is 73.1 [GPa], the yield stress (σ y ) for
tension and compression is 310 [MPa], and the material density (ρ m ) is 2780 [kg/m3 ]. The results obtained
from the optimizations are presented in the next section.
4.1.2. R ESULTS
The resulted obtained from the mechanism design optimizations can be found in Figure 4.3. Four different
mechanisms were found, each of which was able of follow the specified trajectory. This will be covered in
more detail later on. The first mechanism consists of four elements, hence satisfies the volume constraint. In
addition, two elements are attached to the input node.
The second mechanism has five thick elements. This is very interesting, since the volume constraint specified
only four elements. However, looking closely at the mechanism, two of the elements have only a thickness
which is half the maximum thickness. Therefore, mechanism 2 also complies with the volume constraint of
four elements in total, which is this case results in two elements of a half. These two elements are a factor
2 smaller compared to the other three elements, but are several order of magnitude larger than the thin elements which need to be removed. These elements all have a cross-sectional area equal to the lower bound.
this means that these two elements do influence the kinematics and should be maintained in the solution.
This, in contrast to the other elements which have cross-sectional areas (almost) equal to the lower bound.
This mechanism has a total of three elements attached to he input node.
The third mechanism is very similar to the second one. Both mechanisms follow the same trajectory. Where
in the second mechanisms five elements turned out to be needed, here only four elements are needed. All
other, very thin, elements need to be removed. Note that also three elements are attached to the input node.
The fourth and final mechanism also consists of four elements. Two of those elements are attached to the
input node.
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(a) Mechanism 1
(b) Mechanism 2
(c) Mechanism 3
(d) Mechanism 4
Figure 4.3: Converged 2D winglet mechanisms before post-processing
The optimization histories of the four mechanisms are shown in Figure 4.4. Since the objective function for
these optimizations does not represent a physical quantity, the results are normalized with the converged
value of the objective function. This will give for all four mechanisms a value 1 for the converged objective
function. Interesting to note is that for mechanism 1 the trajectory constraint is not active. This means that
the mechanism is able to satisfy (on average) the trajectory constraint very well. The bounding box ∆max
could have been smaller at this mechanism could still be found. For the other mechanisms the trajectory
constraint is active, which means that these mechanisms just satisfy the trajectory. For the mechanisms 2
and 3 it is interesting to note that the input node connectivity constraint is not active. This means that more
than 2 elements are attached to the input node. This is confirmed by the topology, where indeed 3 elements
are attached to the input node. Finally, for all mechanisms the volume constraint is active, which would
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mean that each mechanism consists of four elements, this is indeed the case. Also for mechanism two the
total number of full elements would be four.
(a) Mechanism 1
(b) Mechanism 2
(c) Mechanism 3
(d) Mechanism 4
Figure 4.4: Optimization history of mechanism design for 2D winglets
The results obtained from the mechanism design are post-processed before the mechanism sizing could be
done. With post-processing the very thin elements are removed, while the significant elements are maintained. This would yield the final mechanisms as shown in Figure 4.5.
The volume constraint and input node connectivity could be visualized in the topology. The trajectory, however, does not become clear form the geometry. Therefore the trajectory constraint is visualized in Figure
4.6. The target trajectory is plotted with the circular bounding box. Also mechanism 4 is included to have
an illustrate how the trajectory is specified with respect to the mechanisms. As can be seen from the figure,
mechanism one satisfies the trajectory constraint in every step. In each of the five steps the output node lies
within the bounding box. For the other mechanisms this is not the case, in some instances the output node
does lie within the bounding box, in other instances it lies beyond the bounding box. This can be explained
by the fact that on average the nodes should lie within the bounding boxes, which allows for some flexibility in the designs. This was done on purpose, since this approach yields mechanisms like 2, 3 and 4, which
would not have been found in the trajectory should be satisfied in each step. Then only mechanism 1 would
be found, whereas the other mechanisms are also very worth considering. One final note, mechanism 1 lies
in each step within the bounding box. This means that in each step the trajectory is satisfied, but not active.
Hence the average of these steps would yield a constraint which is also not active. This explains why the
trajectory constraint or mechanism 1 is not active.
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(a) Mechanism 1
(b) Mechanism 2
(c) Mechanism 3
(d) Mechanism 4
Figure 4.5: Converged 2D winglet mechanisms after post-processing
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Figure 4.6: Trajectories of 2D winglets
The next optimizations are the sizing optimizations. As mentioned before, 15 global buckling constraints
were specified and 3 other failure constraints. Hence a total of 18 constraints will have to be satisfied. The
optimized results are shown in Figure 4.7. The mechanisms have weights ranging from 15.0 [kg] to 20.0 [kg].
Inspecting the geometries it is interesting to see that the upper element in the mechanism will always be
the thickest. Clearly the mechanism consist of a load carrying part and an actuation part. In the figure the
colors of the members represent the cross-sectional areas in square centimeters. Mechanisms 1 and 2 are the
lightest with a weight of 15.2 [kg] and 15.0 [kg] respectively. Interesting to note is that mechanism 2, which
consists of 5 elements, turns out to be lighter than mechanism 3, which consists of 4 elements. Therefore the
minimum number of elements does not necessarily yield the lightest designs.
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(a) Mechanism 1, weight 15.2 [kg]
(b) Mechanism 2, weight 15.0 [kg]
(c) Mechanism 3, weight 16.9 [kg]
(d) Mechanism 4, weight 20.0 [kg]
Figure 4.7: Converged 2D winglet mechanisms after sizing. The colors give the cross-sectional areas in [cm2 ].
The optimization histories for sizing are shown in Figure 4.8. In the left figures the optimization history is
given, where also the weight (objective function) is visualized. The right figures zoom in on the constraints, to
see which constraint or constraints become active. From these figures it becomes clear that global buckling
is the critical failure mode for these geometries. The other failure modes (local buckling, stress in tension
and stress in compression) are not driving the design. This is somewhat intuitive, since the structures are
relatively large. During the optimizations the weight continuously decreases, and the constraints tend to go
to zero, meaning that they become active. As mentioned earlier, 15 buckling modes are included. The lowest
three buckling modes of each step are taken into account, therefore not all buckling modes could become
active. Either 1 or 2 buckling modes become active, limiting the geometry to reduce further in weight.
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(a) Mechanism 1
(b) Mechanism 1 zoom
(c) Mechanism 2
(d) Mechanism 2 zoom
(e) Mechanism 3
(f) Mechanism 3 zoom
(g) Mechanism 4
(h) Mechanism 4 zoom
Figure 4.8: Optimization history of mechanism sizing for 2D winglets
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One of the research goals was to extend the tool to 3D. This has been done and the results are presented in
this section. First the problem formulations are covered, which slightly deviate from the 2D optimizations,
followed by the mechanisms found. In 3D two different mechanisms are shown, one mechanism can only
change the cant angle, whereas the other mechanism can both change the cant angle and the toe angle. An
illustration of the cant angle and the toe angle can be found in Figure 4.9 [42]. When the cant angle is zero,
the winglet becomes a wing extension. The toe angle is the angle the winglet makes with the aircraft’s vertical
axis.
Figure 4.9: Illustration of the toe angle and cant angle of a winglet
4.2.1. P ROBLEM F ORMULATION
The dimensions of the winglet were already covered. The length was set to 2 meters, the width started at 1
meter, and was 50 centimeters at the tip. The width and the weight were kept the same and equal to 0.4 meters and 0.3 meters respectively. The depth is equal to the width of the winglet and therefore set to 1 meter.
The total number of nodes in the design space was set to 18, again 3 by 3, but only 2 in the third direction.
This is illustrated in Figure 4.10. Three different views of the design space are given in Figure 4.11. For clarity
the nodes were amplified, the better visualize the design space.
Similar to the 2D design space, the corner nodes of the design space are pinned. This is illustrated with the
triangles at the four corner nodes. Also, the middle two nodes were used as input nodes (illustrated with the
blue circles). The actuation is only prescribed in the horizontal direction, where the other dimensions were
free. the actuation will be done in 5 steps, up to 0.11 meters. The two most right nodes, the winglet tip nodes,
were selected for the output nodes. This is illustrated with the green circles. For the first mechanism for both
output nodes the same circular motion was prescribed as done in 2D. Hence for both nodes a trajectory of
45 degrees was specified. This trajectory would yield mechanisms which can change the cant angle of the
winglet. For the second mechanism, however, a different trajectory was specified for the nodes. For one node
45 degrees was specified, where for the other node 55 degrees was specified. This was done to find mechanisms which both regulate the cant angle as well as the toe angle. The layout of the winglet was chosen such
that in the middle of the winglet length and at a quarter chord a node is present for the aerodynamic loading.
The loading is illustrated by the red arrow.
Figure 4.10: Truss ground-structure (design space) for 3D winglets
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(a) Top view
(b) Side view
(c) Front view
Figure 4.11: Views of design space used for the 3D winglets
With this geometry, the optimization problem in 3D could be written as shown in Equation 4.4. The reaction
force term in the objective function becomes a summation of the two input nodes. Also, given the two input
nodes, two constraints are specified for the input node connectivity. In addition, two trajectory constraints
are specified for the output nodes. Hence, a total of 5 constraints are defined. The variable ∆max was kept
the same as for 2D, where Vmax was set to 10 elements. or the mechanism design the material properties are
again set to 10 for the Young’s modulus and 1 for the cross-sectional area.
"
#
´
5
2 ³
X
1X
i nput 2
mi n F 0 =
Ck +
F i ,k
ρ
5 k=1
i =1

i
1 P5 h t r a j ec t or y
t r a j ect or y

R
−
1
=
 G1


5 k=1 h 1,k

i

1 P5

t r a j ect or y
t r a j ec t or y


G2
=
R
−
1


5 k=1 2,k
G vol ume
= R vol ume − 1
s.t . =


i nput nod e
i nput nod e

G1
= 1 − R1




i
nput
nod
e
i nput nod e

 G2
= 1 − R2


0 ≤ ρ mi n
≤ ρi
<0
<0
(4.4)
<0
<0
<0
≤1
Given the two input nodes, two input node constraints were specified. The first constraint refers to the first
input node (indicated with the blue 1 in Figure 4.10) and the second constraint refers to the second input
node (indicated with the blue 2 in Figure 4.10). Similarly, two output node constraints were specified. The
first output node is indicated with the green number 1, the second output node with the green number 2, also
shown in Figure 4.10.
The optimization problem for sizing remains unchanged compared to the 2D formulation. The same material
properties are used, the actuation is done in 5 steps and in each step the three lowest buckling modes are
taken into account. The problem formulation for the sizing can therefore be written as:
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mi n F 0 =ρ m
ρ
Ne
X
ρi A0li
i =1
 g l obal buckl i ng

Gj




 G st r ess t ensi on
s.t . =
G st r ess compr essi on




G l oc al buckl i ng


0 ≤ ρ mi n
<0
≤ ρi
<0
<0
<0
≤1
j = 1, ..., 15
(4.5)
4.2.2. R ESULTS
As mentioned earlier, the first mechanism is obtained when a trajectory of 45 degrees is specified for both
nodes. The mechanism found can be seen in Figure 4.12. In blue the input trajectory is specified and in pink
the target output trajectory. Also, only the post-processed geometry is shown, since the mechanism is very
difficult to recognize given the many elements in the design space. It can be seen that the mechanism is able
to follow the trajectory well. The actual location of the output nodes are given in green (when it lies within
the bounding box) or red (when it lies outside the bounding box). Plotting the bounding boxes, as was done
in 2D, was found to be unclear. The bounding boxes in 3D are spheres, which were difficult to represent in a
clear way. Therefore it was decided to plot the actual nodal locations for the output nodes in green and red.
For this mechanism the output nodes lie in each step within the bounding box, hence all the locations are
given in green. In fact, the mechanism is similar to the first mechanism found in 2D. The mechanism indeed
follows a 2D trajectory. In 3D however, a few more elements are needed to prevent additional degrees of freedom. The final mechanism consists of 11 elements. As can be seen 8 elements actually carry the load and
determine its kinematics, the other 3 are connected to the input nodes to restrain the out of plane displacements.
The optimization history for the 3D mechanism design is shown in Figure 4.13. Similar to the 2D cases, the
objective does not represent a physical quantity. Hence, the values are normalized with respect to the converged objective function value. This can be seen in the optimization history, where the objective function
converges to 1. Also, as can be seen, the volume constraint is active. This is expected, since the more material
in the design space, the smaller the compliance. Hence for optimizations with compliance in the objective,
the volume constraint is expected to be active. Also 1 input node constraint is active, while the other is not.
This can be explained by the fact that before post-processing a bit more material is attached to the first input
node. A bit more that a total of 2 was connected at the input node. As a result the constraint is not active.
Similar behavior is observed when the two trajectory constraints are compared. Both constraints were not
active, but one is more negative than the other. This can be explained by the fact that 10 elements were specified for the volume constraint. The actual mechanisms consists of 11 elements, which means that the total
volume has to be distributed among these elements. This, in combination with the loading not acting in the
middle of the chord, yields an unsymmetrical material distribution, explaining the different paths the two
output nodes travel.
The second optimization would be the sizing. The optimized geometry can be found in Figure 4.14. The
geometry is shown for two different color bars. The left sub-figure has he same color range as the 2D mechanisms, allowing for a good comparison. The right sub-figure has a slightly different scale, which amplifies the
differences between the cross-sectional areas. The first thing to notice is that the left and right side are not
sized the same. This could be explained by the fact that the loading is not acting in the middle of the chord.
Also, similar to the 2D cases, the upper members are relatively large. Finally, the 3 elements connecting the
left side of the mechanism with the right side have a small cross-sectional area. This confirms the primary
role of these elements, i.e. preventing the additional degrees of freedom, rather than carrying the aerodynamic load. This could also explain why the 3D mechanism weights 48.0 [kg], which is more than twice the
weight of a 2D mechanism. For 2D the aerodynamic loading was divided by 2 and two mechanisms would be
needed to carry the aerodynamic load. The weight of a single mechanism in 2D ranged from 15.0 to 20.0 [kg].
Hence for the full aerodynamic load the total weight for the mechanisms would range from 30.0 to 40.0 [kg].
Comparing this to the 48.0 [kg] found for the 3D mechanism, the 3D mechanism is slightly heavier.
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(a) Step 1
(b) Step 2
(c) Step 3
(d) Step 4
(e) Step 5
Figure 4.12: Actuation of 3D winglet 1 with the input trajectory (blue) and the target output trajectory (pink). The actual location of the
output nodes is indicated in green if it lies within the bounding box and red if it lies outside the bounding box.
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Figure 4.13: Optimization history of mechanism design for 3D winglet, mechanism 1
(a) Color bar from 0 to 160 cm2
(b) Color bar from 0 to 100 cm2
Figure 4.14: Converged 3D winglet mechanism 1 after sizing. The colors give the cross-sectional areas in [cm2 ].
The optimization history of the sizing is given in Figure 4.15. Similar to the 2D, global buckling is the critical
buckling mode. All the buckling constraints become less negative with decreasing weight, which is expected.
In the first sub-figure the objective function can be seen. The weight decreases and reaches a minimum
weight of 48.0 [kg]. In the second and third sub-figures the critical buckling mode can be seen. The other
failure modes are not critical.
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(a) Mechanism 1
71
(b) Mechanism 1 zoom 1
(c) Mechanism 1 zoom 2
Figure 4.15: Optimization history of mechanism sizing for 3D winglet, mechanism 1
For the second mechanism in 3D different trajectories were specified for the output nodes. For the first output
node a 45 degrees rotation was specified. This is the same as done for the previous mechanisms. However,
for the second output node 55 degrees rotation is specified. The result of these trajectories would yield a
mechanism which both changes its cant angle as well as its toe angle during the actuation. The result obtain
is given in Figure 4.16. The mechanism consists of 13 elements. How the mechanism deflects is given in 5
steps. As can be seen, the mechanism indeed changes the toe angle during the actuation. It was found that
the maximum toe angle was 15 degrees. This is more than the difference between the 45 and 55 degrees, but
that can be explained by the presence of the bounding box.
In blue the trajectories of the input nodes are given, in pink the target output node trajectories. The actual
location of the output nodes are plotted in green when it falls within the bounding box, or in red when it lies
outside of it. As can be seen in steps 3, 4, and 5 only the second output node lies outside the bounding box.
This is possible, since only on average the output nodes should lie within these boxes.
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(a) Step 1
(b) Step 2
(c) Step 3
(d) Step 4
(e) Step 5
Figure 4.16: Actuation of 3D winglet 2 with the input trajectory (blue) and the target output trajectory (pink). The actual location of the
output nodes is indicated in green if it lies within the bounding box and red if it lies outside the bounding box.
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The optimization history is given in Figure 4.17. As done for every mechanism design, the objective function
converges to 1. Interesting to note is that trajectory constraint for output node 1 is not active, which it is
active for output node 2. This could also be seen from Figure 4.16. Output node 1 lies in each step within the
bounding box. As a result the constraint cannot be active. The volume constraint (10 elements) is active. This
volume is distributed among the 13 members of the mechanism, as a result not every member has ρ i equal
to 1. This explains why an input node constraint is active, while after post-processing more than 2 elements
are present.
Figure 4.17: Optimization history of mechanism design for 3D winglet, mechanism 2
Now that the topology is known, the minimum weight can be determined. The sizing optimization has been
performed and the results are given in Figure 4.18. The weight was found to be 102.7 [kg]. This is significantly
higher than the weight calculated for the first 3D mechanism. This can be understood when inspecting the
geometry in more detail. The mechanism is much more complicated than the mechanism which only regulates the cant angle. Due to the toe angle, the aerodynamic loading rotates out of plane, this results in an out
of plane force, which can not be carried efficiently by the mechanism. In addition, the force also induces a
torque, complicating the load case even further. As a result the geometry turned out to be quite sensitive to
buckling. An interesting observation is how the different elements are sized. Most of the material is concentrated at one side of the mechanism, this in contrast to the previous mechanism. Hence the twist in the mechanism greatly influences what the final geometry looks like. Also, the optimization history provides insight in
why this mechanism is heavier than the previous one found. In figure 4.18 the optimization history can be
found. By zooming in on the constraints, it can be observed that the local buckling constraint becomes the
critical buckling mode. Hence a different failure mode drives this geometry. By combining all local buckling
constraints into a single Kreisselmeier-Steinhauser function, this constraint will always be more conservative.
This means that the new constraint will always be more strict than the individual local buckling constraints.
Once this constraint becomes active and thereby prevents additional weight reduction, the actual structure
will not fail in local buckling. By using the Kreisselmeier-Steinhauser function and this constraint is active,
a more conservative (i.e. higher) weight is obtained for the mechanism. Another possible explanation could
be the non-convexity in the design space. This means that several local minima exist, of which one is found.
Multiple runs have been performed, but did not yield a lower weight that given here. However, it could be
that the lowest weight is not yet found. Also for 2D multiple runs were performed, but the vast majority of
the runs did find the same weight. The increased size of the design space for 3D mechanisms makes it more
difficult to find the global minimum.
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Figure 4.18: Converged 3D winglet mechanism 2 after sizing. The colors give the cross-sectional areas in [cm2 ].
(a) Mechanism 2
(b) Mechanism 2 zoom
Figure 4.19: Optimization history of mechanism sizing for 3D winglet, mechanism 2
Finally, the difference in trajectory between mechanism 1 and 2 are illustrated in Figure 4.20. A 2D view of
the mechanisms is shown. In blue the input trajectories are plotted and in pink the target output trajectories.
In Figure 4.20a mechanism 1 is shown. As was already mentioned earlier in this chapter, mechanism 1 could
be interpreted as 2 mechanisms, each one acting on the sides of the winglet. These two mechanisms are
identical, resulting in the same input trajectory (which coincide in this figure) for both input nodes. Also,
the same output trajectory was specified for both output nodes. When looking at the output nodes, these
coincide in the figure as well, which confirms that mechanism 1 only changes the cant angle. When looking
at mechanism 2 (Figure 4.20b), a different behavior is observed. The trajectories for the input nodes are
different. Also the output trajectories differed, as previously explained. Interesting to note is that the output
nodes do not coincide for mechanism 2, this clearly illustrates the toe angle. Hence, mechanism 2 influences
both the cant angle as well as the toe angle.
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(a) Mechanism 1 with cant angle
(b) Mechanism 2 with cant angle and toe angle
Figure 4.20: 2D view of 3D winglet mechanisms
4.3. T RADE - OFF
In the previous two sections the mechanisms for the 2D and 3D winglet were presented. In 2D 4 mechanisms
were found, in 3D 2 mechanisms were given. In this section a trade-off is made between these various mechanisms. This will be done by comparing both the weight of the mechanisms and the efficiency. The efficiency
is defined as the output work divided by the input work. At the input node(s) a displacement is prescribed,
which requires a certain actuation force for equilibrium. The total input work is the average actuation force
multiplied with the maximum actuation displacement. For all these mechanisms the input nodes were displaced over a distance of 0.11 meters. The actuation has been done in 100 steps to obtain an accurate average
reaction force. The actuation forces are different for the various mechanisms. Part of the input work is used
to carry the aerodynamic loading and to deflect the winglet, some energy is ’lost’ by straining the members
and is stored as strain energy. This energy will be released when unloaded, but it can not be used to deflect
the winglet. Hence the ratio given by Equation 4.6 gives the efficiency for a mechanism:
η=
Wout Wi n −U
U
=
= 1−
Wi n
Wi n
Wi n
(4.6)
Here it is assumed that no fiction is present in the hinges. The only efficiency losses occur due the strain
energy of the truss members. To calculate the output work is challenging, since the aerodynamic force is not
always perfectly perpendicular to winglet’s surface. The aerodynamic force rotates with the same angle as
the target output trajectory. The actual angle of the winglet could be slightly different, due to the bounding
boxes specified for the trajectory. Therefore it has been decided to calculated the aerodynamic efficiency by
calculating the strain energy (i.e. the energy loss) and calculate the mechanism efficiency using this energy
loss.
Ideally, the mechanism is as light as possible and has the highest efficiency. Both factors are favorable and
therefore preferred. The lighter the mechanism itself, the lighter the aircraft will be. This will result in a lower
fuel consumption. For a given aerodynamic loading, the higher the efficiency the lower the input work required. The lower the input work, the smaller the actuator can be, which on its turn also results in a lower
weight. For the different mechanisms the weight of the mechanisms (which excludes the actuator weight)
is plotted versus the efficiency. To compare the 2D mechanisms with the 3D mechanisms, the weight of the
2D mechanisms is multiplied with a factor 2, since half the aerodynamic load was assumed to act on a 2D
mechanism. The results can be found in Figure 4.21. The mechanisms 1, 2, 3, and 4 in 2D are denoted 2D1,
2D2, 2D3, and 2D4 respectively. In 3D the mechanisms are denoted as 3D1 and 3D2.
Interesting to note is that the 2D mechanisms are lower in weight than the 3D mechanisms. This can be
explained by the fact that some elements in 3D are only present to prevent additional degrees of freedom
in the mechanisms, while they carry hardly any load. Also, in 3D the aerodynamic loading acts at a quarter
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chord, resulting in a torsional moment. This gives a more complicated load case, which could also explain
the increased weight. It must be noted that the weight optimization uses a simplified aerodynamic loading.
A more advanced model of the loading could change the weight calculations for the various mechanisms.
Further research would be required to obtain detailed weight estimations. Also, the actuator weight should
then be taken into account.
Figure 4.21: Winglet mechanisms trade-off, actuation efficiency versus mechanism weight. The weight of 2D mechanisms is multiplied
with 2.
The efficiencies of the 2D mechanisms are very high, ranging between 97.3% and 98.3%. For these mechanisms only 1-2% is lost in the deformation of the mechanisms. This means that most of the input work is
used to deflect/actuate the mechanism rather than to strain the elements. These high efficiencies are expected, since only energy is lost in the axial deformation of the truss members. In contrast to compliant
mechanisms, where thin regions of the mechanisms are strained during actuation, no strain energy is stored
in the hinges of the articulated mechanisms. When looking at 3D, the first mechanism also has a high efficiency of 88.2%. However, for the second mechanism an efficiency of 72.9% was found. This means that
more than 25% of the energy that is put into the mechanism is stored as strain energy. Comparing the two
mechanisms in 3D, the more complicated the trajectory of the winglet, the lower the efficiency. Additional
3D mechanisms will have to be examined to see whether this is true in general. To better understand why the
efficiency of mechanism 3D2 is lower, a wireframe of the mechanism is shown before and after aerodynamic
loading in Figure 4.22. As can be seen in the left sub-figure, the aerodynamic loading deflects the geometry.
When zooming in on the mechanism, the elements experience a significant strain. As a consequence of these
strains, the efficiency reduces.
Based on the results from Figure 4.21, the mechanisms 1 and 2 in 2D perform the best. These have the lowest
weight and the highest efficiency. These mechanisms have an additional advantage when considering the
morphing skin. The skin could fit between the upper and lower nodes of the mechanism, without any contact
with the mechanism. This contact would increase the friction and results in additional wear of the skin. This
should be taken into account for mechanism 4 for example. In 3D the mechanisms were found to be heavier,
but there is an opportunity to design mechanisms with more complex trajectories.
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(a) Entire geometry
(b) Zoomed in on mechanism
Figure 4.22: Mechanism 3D2 before (black) and after (blue) aerodynamic loading, last actuation step
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C ONCLUSIONS AND R ECOMMENDATIONS
In Chapter 1 the following research objective was specified:
"Developing a software tool to design a mechanism for morphing winglets, using ground-structure based
topology optimization, by improving, extending, and expanding the previous 2D inhouse tool."
Based on this objective the following research goals were defined:
1. Improve previous tool
(a) Increase robustness
(b) Reduce computational time
(c) Increase separation
2. Extend previous tool
(a) Include multiple input nodes
(b) Include multiple output nodes
(c) Include buckling
3. Expand previous tool
(a) Expand to 3D
In the Chapters 2 and 3 the tool developed in this thesis was discussed in detail. In Chapter 4 the results
obtained from this tool were included. In this chapter will be discussed to what extend the tool meets the
research goals. The conclusions are discussed in Section 5.1, the recommendations for future research will
be covered in Section 5.2.
5.1. C ONCLUSIONS
A tool for the design of articulated mechanisms was developed. It consists of two optimization steps, which
was illustrated in Figure 1.5. The first optimization, mechanism design, focused on finding the topology for
the mechanism to perform the specified trajectory. Once the topology was found, the second optimization
sized the individual elements to obtain the minimum weight, while preventing failure. For this thesis mechanisms for morphing winglets were designed, both in 2D and in 3D.
The conclusions can be categorized into three. The first category contains the conclusions related to the
improvements of the tool, the second covers the extensions and the third covers the conclusions related to
the expansion.
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5. C ONCLUSIONS AND R ECOMMENDATIONS
5.1.1. I MPROVEMENTS
In the mechanism design the objective function was changed to one including compliance and the sum of
the reaction forces at the input nodes squared. By selecting this objective, the optimizer tries to minimize
the reaction forces at the input nodes and the compliance. The reaction forces will therefore be steered toward zero, yielding a mechanism. As a result, the trajectory will be performed without straining the elements
needed for the topology. If elements would be strained, then the reaction force at the input nodes would be
non-zero.
The compliance term was included to improve the separation, once a mechanism was found. By minimizing the compliance, for constant forces, the stiffness of the mechanism is maximized. Hence the compliance
term promotes 0/1 solutions. Therefore, the current tool improves the separation.
Also, by minimizing the compliance, the degrees of freedom are automatically regulated. The mechanism will
have the minimum number of degrees of freedom needed to perform the trajectory, due to the term containing the reaction forces. By including the compliance term, the optimizer will prevent any additional degrees
of freedom in the mechanism. If an additional degree of freedom would be present, then one or more nodes
would experience large displacements. With the minimum compliance term these displacements are minimized.
Another advantage of the compliance term is the opportunity to remove the incremental load increase in the
previous tool. The incremental load increase was included to improve the separation, which is already promoted via the compliance term. By removing the incremental load the robustness of tool is improved. The
previous tool experienced optimization which did not converge as a result of the incremental load increase,
which is now improved. Another advantage of the removal of the incremental load is the improved computational time. In the previous tool for each load increment an optimization is performed, which considerably
impacts the computational time.
Finally, in the previous tool the degrees of freedom were regulated with two additional load cases, each with
a disturbance force. By minimizing the error between the target and actual trajectory for all three load cases,
the best solution is the one where the disturbance forces do not affect the output trajectory. Hence, these
additional load cases prevented more degrees of freedom that needed for the trajectory. The current tool has
an objective which automatically regulates the degrees of freedom, thereby eliminating the need of two additional load cases. This resulted in a significant reduction in computational time.
With the current objective function the tool’s robustness was increased, the computational time was reduced,
and the separation was improved. Therefore it could be concluded that the tool was successfully improved.
5.1.2. E XTENSIONS
The next step was to extend the tool’s capabilities. The previous tool was only able to include a single input
node and a single output node. The current tool is able to have multiple input and output nodes. The number
of input nodes and output nodes can be chosen freely. For each input node and output node a constraint is
specified for the mechanism design. For the input node at least two elements should be attached, while for
every output node a trajectory is specified. In the previous chapter different examples can be found where
two input nodes and two output nodes were included in the design.
The previous tool sized the elements with a linear finite element analysis, only taken into account yielding
as a failure mode. In each step an analysis was performed to find the cross-sectional area’s, then for each
element the largest cross-sectional area from all steps is selected. This does not necessarily give the lightest
design, and does not account or buckling. Hence, the current tool included a second optimization, where
the minimum weight was defined as the objective function, and various failure modes were included as constraints. The constraints prevent material failure in tension and compression, local buckling of the individual
truss members, and global buckling.
The current tool is now compatible with multiple input and output nodes. Also, local and global buckling
were included during the sizing. Therefore, it can be concluded that the tool’s capabilities are successfully
extended.
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5.1.3. E XPANSION
Also, the tool should be expanded to 3D for practical applications. The changes required for 3D were limited
to the analysis part. These changes were implemented and verified. Therefore it can be concluded that the
tool was successfully expanded to 3D. Various 3D mechanisms for the winglet were found. By expanding the
tool to 3D, the winglet deflection is no longer limited to 2D. Where the mechanism for the winglet in 2D is
only able to is able to regulate the cant angle, the mechanism for the 3D winglet is able to regulate both the
cant angle and the toe angle.
It can be concluded that the tool is successfully improved, extended, and expanded. The tool meets all the
specified research goals, thereby meeting the overall research objective. The current tool successfully designed and sized 2D and 3D mechanisms. In 2D 4 different mechanisms were found, where the mechanisms
1 (2D1) and 2 (2D2) were found to perform the best. A trade-off was made on the mechanism efficiency and
weight. For mechanisms 2D1 a weight of 15.2 [kg] and an efficiency of 98.3% was found. For 2D2 a weight of
15.0 [kg] and an efficiency of 98.2% was found. In 3D the mechanisms were found to be heavier (minimum
weight was found to be 48.0 [kg]), but were able to perform more complex trajectories. Mechanism 3D2 was
able to both regulate the cant angle as well as the toe angle. A variation in toe angle of up to 15 degrees was
found.
5.2. R ECOMMENDATIONS
In the future several aspects can be researched in more detail. These aspects can be categorized into two, the
first would cover the recommendations within the scope defined for this thesis (internal), the second would
be beyond this scope (external).
5.2.1. I NTERNAL
In the current tool only the cross-sectional areas of the elements were used in the optimizer as design variables. The nodal locations could also be included to increase the flexibility of the tool. In that case the design
vector would consist of both cross-sectional areas as well as nodal locations. This also means that the sensitivities of objective functions and constraints with respect to the nodal locations have to be determined.
Currently, only aluminum 2024-T3 is considered in the sizing optimizations. It would be interesting to see
how different materials (with different yield stress, Young’s modulus and density) would impact the weight
of the mechanisms. Also combining different materials for different elements would be interesting to look
at, since the current tool assumes the same material for all elements. Related to this, composites could also
be considered in the design of these mechanisms. Since the truss members are only axially loaded, possible
weight savings could be obtained when using composites.
Also, the design space of the test cases in this thesis were in 2D limited to 9 nodes and in 3D limited to 18
nodes. It is worth considering larger design spaces, where mechanisms might be found with a more continuous transition between the wing and the winglet. An example would be the hyper-elliptic cambered span
(HECS) wing from NASA, where a very continuous transition can be observed. In this thesis the shape of the
winglet does not change, for future research additional actuators could be considered within the winglet, allowing for a more advanced actuation.
In addition, in the current tool no fatigue is considered. The deflection of the winglet would be very repetitive. Assuming that for each flight the winglet will deflect once up and down, at least one cycle is performed
for each flight. This will result in several thousands of cycles, which might become critical. An additional
constraints for fatigue could be included.
Finally, in 3D it might be possible for elements to intersect. In 2D this would not be a problem, since an
offset could be given in the third dimension. However in 3D this is not possible, which might result in mechanisms which are non-feasible. In the test cases this behavior was not observed, but this might be possible
for more complex mechanisms. To solve this, the research done by Santer et. al. [50] might be useful. They
incorporated this constraint to guarantee manufacturability.
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5. C ONCLUSIONS AND R ECOMMENDATIONS
5.2.2. E XTERNAL
Beyond the scope of the current research objective, one could consider a more accurate representation of the
loading. Currently, this is simplified to a constant point load, acting in the middle of the winglet. Improvements could be a distributed loading or a loading that changes over time. Also, the aeroelasticity could be
considered. Currently, the stiffness is not affected by he aerodynamic loading. When a more accurate loading is considered for the winglet, the structural representation of the winglet could also be improved. This is
currently modeled as a structure, consisting of truss members. By modeling the winglet with aerodynamic
profiles, the aeroelastic behavior will be more accurate. By incorporating this, the load introduction into the
mechanism will also be more accurate. As a result, a more accurate weight optimization can be performed.
Also, the current weight optimization only takes the weight of the mechanism into account. The weight of the
actuator is not included. An improved weight optimization also includes the weight of the actuator, which
could be estimated based on the force-displacement requirements on the input nodes. In the current tool the
weight of the mechanism is minimized, which also impacts the efficiency. Therefore, it might be possible that
the minimum weight of the mechanism does not yield the minimum weight of the entire system, including
the actuator.
In addition, the current tool is only verified. It would be interesting to validate the results. This can be done
by manufacturing the mechanism and perform experiments. Other experiments in the wind tunnel, with a
more accurate representation of the winglet, are also possible.
Given the general applicability of the tool, other applications can be considered as well. In this thesis only
mechanisms for morphing winglets are designed and sized. The tool can also be used to design mechanisms
for morphing leading edges, trailing edges, or even entire wing structures.
Also, the research objective was focused on the design of mechanisms for morphing winglets. The skin between the wing and the winglet, where the mechanism will be positioned, will experience large deformations.
Additional research is needed on how the skin should be designed. Topology optimization can also be used
to design morphing skins [29, 46], which could be used to design this part of the wing skin.
Finally, some remarks considering the manufacturability. During manufacturing some imperfections in the
members could be introduced. This could decrease the performance of the geometry. Hence buckling could
be considered with members which have initial imperfections. In addition, additional research would be
needed on the design of the joints. Especially in 3D these joints could be challenging. Some sort of ball and
socket joint might be needed.
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B IBLIOGRAPHY
[1] D. Baker and M. I. Friswell. Determinate structures for wing camber control. Smart Materials and Structures, 18(3):1–13, 2009.
[2] K.-J. Bathe. Finite Element Procedures in Engineering Analysis. Prentice Hall, Upper Saddle River, New
Jersey, 1982.
[3] M. P. Bendsøe. Optimal shape design as a material distribution problem. Structural Optimization,
202:193–202, 1989.
[4] M. P. Bendsøe and N. Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71(2):197–224, 1988.
[5] P. Bourdin, A. Gatto, and M. I. Friswell. The Application of Variable Cant Angle Winglets for Morphing
Aircraft Control. In 24th Applied Aerodynamics Conference, San Francisco, California, 2006.
[6] P. Bourdin, A. Gatto, and M. I. Friswell. Potential of Articulated Split Wingtips for Morphing-Based Control of a Flying Wing. In 25th AIAA Applied Aerodynamics Conference, Miami, FL, 2007. American Institute
of Aeronautics and Astronautics, Inc.
[7] P. Bourdin, A. Gatto, and M. I. Friswell. Aircraft Control via Variable Cant-Angle Winglets. Journal of
Aircraft, 45(2):414–423, 2008.
[8] P. Bourdin, A. Gatto, and M. I. Friswell. Performing co-ordinated turns with articulated wing-tips as
multi-axis control effectors. Aeronautical Journal, 114(1151):35–47, 2010.
[9] P. W. Christensen and A. Klarbring. An Introduction to Structural Optimization. Springer, 2009.
[10] M. Crisfield. Non-Linear Finite Element Analysis. John Wiley and Sons, New York, vol 1: ess edition, 1997.
[11] A. De Gaspari, S. Ricci, S. Vasista, and H. Monner. Morphing Devices for a Wing and Wingtip Based
on Compliant Structures. In 4th EASN Association International Workshop on Flight Physics & Aircraft
Design, Aachen, Germany, 2014.
[12] J. D. Deaton and R. V. Grandhi. A survey of structural and multidisciplinary continuum topology optimization : post 2000. Structural and Multidisciplinary Optimization, 49:1–38, 2014.
[13] P. Eberhard and K. Sedlaczek. Grid-Based Topology Optimization of Rigid Body Mechanisms. In III
European Conference on Computational Mechanics, 2006.
[14] J. Eves, V. V. Toropov, H. M. Thompson, P. H. Gaskell, J. J. Doherty, and J. Harris. Topology Optimization
of aircraft structures with non-conventional configurations. In 8th World Congress on Structural and
Multidisciplinary Optimization,, Lisbon, Portugal, 2009.
[15] L. Falcão, A. A. Gomes, and A. Suleman. Multidisciplinary Design Optimisation of a Morphing Wingtip.
In 2nd International Conference on Engineering Optimization, Lisbon, Portugal, 2010.
[16] L. Falcão, A. A. Gomes, and A. Suleman. Aero-structural Design Optimization of a Morphing Wingtip.
Journal of Intelligent Material Systems and Structures, 22(July):1113–1124, 2011.
[17] L. Falcão, A. A. Gomes, and A. Suleman. Design and Analysis of an Adaptive Wingtip.
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2011.
In 52nd
[18] M. I. Friswell. The prospects for morphing aircraft. In IV ECCOMAS Thematic Conference on Smart
Structures and Materials, Porto, Portugal, 2009.
T.A. DE J ONG
T OPOLOGY O PTIMIZATION OF 3D L INKAGES
84
B IBLIOGRAPHY
[19] M. I. Friswell, J. E. Herencia, D. Baker, and P. M. Weaver. The optimisation of hierarchical structures
with applications to morphing aircraft. In Second International Conference on Multidisciplinary Design
Optimization and Applications, Gijon, Spain, 2008.
[20] C. F. Gerald and P. O. Wheatley. Applied Numerical Analysis (7th edition). Pearson, 2003.
[21] E. Gillebaart and R. De Breuker. Optimisation of a mechanical linkage for a morphing winglet. In Proceedings of the DeMEASS VI Conference, Ede, The Netherlands, 2014.
[22] A. A. Gomes, L. Falcão, and A. Suleman. Optimisation of multistable composites for morphing wingtips.
In - 8th World Congress on Structural and Multidisciplinary Optimization, Lisbon, Portugal, 2009.
[23] A. A. Gomes, L. Falcão, and A. Suleman. Study of an Articulated Winglet Mechanism. In 54th
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference AIAA, Boston, MA,
2013. American Institute of Aeronautics and Astronautics, Inc.
[24] J. C. Gomez and E. Garcia. Morphing unmanned aerial vehicles. Smart Materials and Structures,
20(10):16, 2011.
[25] J. K. Guest, J. H. Prévost, and T. Belytschko. Achieving minimum length scale in topology optimization
using nodal design variables and projection functions. International Journal for Numerical Methods in
Engineering, 61:238–254, 2004.
[26] IANS. Airbus Military launches improved C295W transport for India, 2013 (accessed February 11, 2016).
[27] D. Inoyama, B. P. Sanders, and J. J. Joo. Conceptual design and multidisciplinary optimization of in-plane
morphing wing structures. Proceedings of SPIE, 01(1):1–11, 2006.
[28] D. Inoyama, B. P. Sanders, and J. J. Joo. Topology Optimization Approach for the Determination of the
Multiple-Configuration Morphing Wing Structure. Journal of Aircraft, 45(6):1853–1862, 2008.
[29] J. J. Joo, G. W. Reich, and J. T. Westfall. Flexible Skin Development for Morphing Aircraft Applications
via Topology Optimization. Journal of Intelligent Material Systems and Structures, 20(November):1969–
1985, 2009.
[30] A. Kawamoto. Generation of Articulated Mechanisms by Optimization Techniques. PhD thesis, Technical
University of Denmark, 2004.
[31] A. Kawamoto. Path-generation of articulated mechanisms by shape and topology variations in nonlinear truss representation. International Journal for Numerical Methods in Engineering, 64(12):1557–
1574, 2005.
[32] A. Kawamoto, M. P. Bendsøe, and O. Sigmund. Articulated mechanism design with a degree of freedom
constraint. International Journal for Numerical Methods in Engineering, 61(9):1520–1545, 2004.
[33] A. Kawamoto, M. P. Bendsøe, and O. Sigmund. Planar articulated mechanism design by graph theoretical
enumeration. Structural and Multidisciplinary Optimization, 27(4):295–299, 2004.
[34] A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki. Heaviside projection based topology optimization by a PDE-filtered scalar function. Structural and Multidisciplinary
Optimization, 44:19–24, 2011.
[35] S. I. Kim and Y. Y. Kim. Topology optimization of planar linkage mechanisms. International Journal for
Numerical Methods in Engineering, 98(February):265–286, 2014.
[36] M. Kintscher, H. P. Monner, and O. Heintze. Experimental Testing of a Smart Leading Edge High Lift Device for Commercial Transportation Aircrafts. In 27th International Icongress of the aeronautical sciences,
Nice, France, 2010.
[37] M. H. Kobayashi. On a biologically inspired topology optimization method. Communications in Nonlinear Science and Numerical Simulation, 15(3):787–802, 2010.
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
T.A. DE J ONG
B IBLIOGRAPHY
85
[38] G. Kotucha and K. Hackl. Density gradient based regularization of topology optimization problems.
Proceedings in Applied Mathematics and Mechanics, 5(1):423–424, 2005.
[39] K. J. Lu and S. Kota. Design of Compliant Mechanisms for Morphing Structural Shapes. Journal of
Intelligent Material Systems and Structures, 14(June):379–390, 2003.
[40] K. J. Lu and S. Kota. An Effective Method of Synthesizing Compliant Adaptive Structures using Load Path
Representation. Journal of Computational Science and Technology, 16(April):307–317, 2005.
[41] A. Mauchle, R. De Breuker, and J. Simpson. Design and demonstration of a leading edge actuation
system through topology optimisation. In 23rd International Conference on Adaptive Structures and
Technologies (ICAST), pages 418–429, Nanjing, China, 2012, 2012.
[42] M. D. Maughmer, T. S. Swan, and S. M. Willits. Design and Testing of a Winglet Airfoil for Low-Speed
Aircraft. Journal of Aircraft, 39(4):654–661, 2002.
[43] K. Maute and M. Allen. Conceptual design of aeroelastic structures by topology optimization. Structural
and Multidisciplinary Optimization, 27(1-2):27–42, 2004.
[44] K. Maute and G. W. Reich. Integrated Multidisciplinary Topology Optimization Approach to Adaptive
Wing Design. Journal of Aircraft, 43(1):253–263, 2006.
[45] C. B. W. Pedersen, T. Buhl, and O. Sigmund. Topology synthesis of large-displacement compliant mechanisms. International Journal for Numerical Methods in Engineering, 50:2683–2705, 2001.
[46] G. W. Reich, B. Sanders, and J. J. Joo. Development of Skins for Morphing Aircraft Applications via Topology Optimization. In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials
Conference, Honolulu, Hawaii, 2007.
[47] G. I. N. Rozvany, editor. Optimization of Large Structural Systems. Kluwer Academic Publisher, volume
ii edition, 1993.
[48] G. I. N. Rozvany. A critical review of established methods of structural topology optimization. Structural
and Multidisciplinary Optimization, 37(3):217–237, 2008.
[49] M. Sankrithi and J. Frommer. Controllable winglets, 2010.
[50] M. Santer and S. Pellegrino. Topology Optimization of Adaptive Compliant Aircraft Wing Leading Edge.
In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii, 2007.
[51] M. Santer and S. Pellegrino. Topological Optimization of Compliant Adaptive Wing Structure. AIAA
journal, 47(3):523–534, 2009.
[52] A. Shelton, A. Tomar, J. Prasad, M. J. Smith, and N. Komerath. Active Multiple Winglets for Improved
Unmanned-Aerial-Vehicle Performance. Journal of Aircraft, 43(1):110–116, 2006.
[53] L. Shili, G. Wenjie, and L. Shujun. Optimal Design of Compliant Trailing Edge for Shape Changing. Chinese Journal of Aeronautics, 21:187–192, 2008.
[54] O. Sigmund. On the Design of Compliant Mechanisms Using Topology Optimization. Mechanics of
Structures and Machines: An International Journal, 25(4):493–524, 1997.
[55] O. Sigmund. Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 33(4-5):401–424, 2007.
[56] O. Sigmund and K. Maute. Topology optimization approaches - A comparative review. Structural and
Multidisciplinary Optimization, 48:1031–1055, 2013.
[57] O. Sigmund and J. Petersson. Numerical instabilities in topology optimization: A survey on procedures
dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, 16(1):68–
75, 1998.
T.A. DE J ONG
T OPOLOGY OPTIMIZATION OF 3D LINKAGES
86
B IBLIOGRAPHY
[58] S. Sleesongsom and S. Bureerat. Using Opposite-Based Population-Based Incremental Learning and
Multigrid Ground Elements. Mathematical Problems in Engineering, 2015:16, 2015.
[59] S. Sleesongsom, S. Bureerat, and K. Tai. Aircraft morphing wing design by using partial topology optimization. Structural and Multidisciplinary Optimization, 48:1109–1128, 2013.
[60] M. J. Smith, N. Komerath, R. Ames, and O. Wong. Performance analysis of a wing with multiple winglets.
In 19th Applied Aerodynamics Conference. American Institute of Aeronautics and Astronautics, Inc.,
2001.
[61] J. Sodja, M. J. Martinez, J. C. Simpson, and R. De Breuker. Experimental Evaluation of the Morphing
Leading Edge. In 23nd AIAA/AHS Adaptive Structures Conference, Kissimmee, Fl, 2015. American Institute of Aeronautics and Astronautics, Inc.
[62] M. Stolpe and K. Svanberg. An alternative interpolation scheme for minimum compliance topology
optimization. Structural and Multidisciplinary Optimization, 22:116–124, 2001.
[63] K. Svanberg. The method of moving asymptotes a new method for structural optimization. International
Journal for Numerical Methods in Engineering, 24:359–373, 1987.
[64] K. Svanberg. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization, 12(2):555–573, 2002.
[65] G. A. A. Thuwis, M. M. Abdalla, and Z. Gürdal. Optimization of a variable-stiffness skin for morphing
high-lift devices. Smart Materials and Structures, 19(12):1–10, 2010.
[66] G. A. A. Thuwis, R. De Breuker, and J. Simpson. Conceptual design of an actuation system for a morphing
leading edge high-lift device using topology optimisation. In 22nd International Conference on Adaptive
Structures and Technologies (ICAST), Corfu, Greece, 2011.
[67] B. Trease and S. Kota. Adaptive and controllable compliant systems with embedded actuators and sensors. Proceedings of SPIE, 6525:13, 2007.
[68] N. Ursache, T. Melin, A. Isikveren, and M. I. Friswell. Morphing Winglets for Aircraft Multi-phase Improvement. In 7th AIAA Aviation Technology Integration and Operations Conference (ATIO), Belfast,
Northern Ireland, 2007. American Institute of Aeronautics and Astronautics, Inc.
[69] R. Vos, J. Scheepstra, and R. Barrett. Topology optimization of pressure adaptive honeycomb for a morphing flap. In Active and Passive Smart Structures and Integrated Systems 2011. SPIE, 2011.
[70] F. Wang, B. S. Lazarov, and O. Sigmund. On projection methods, convergence and robust formulations
in topology optimization. Structural and Multidisciplinary Optimization, 43(6):767–784, 2011.
[71] E. White, B. Rawdon, Z. Hoisington, and C. Droney. Aircraft with movable winglets and method of control, 2014.
[72] J. Wittmann, M. Hornung, and H. Baier. Mission performance optimization via morphing wing tip devices. In Proc. DGLR - 2010, Hamburg, Germany, 2010.
[73] T. Xinxing, G. Wenjie, S. Chao, and L. Xiaoyong. Topology optimization of compliant adaptive wing
leading edge with composite materials. Chinese Journal of Aeronautics, 27(6):1488–1494, 2014.
[74] M. Zhang, R. Nangia, S. Ricci, and A. Rizzi. Design and Shape Optimization of Morphing Winglet for
Regional Jetliner. In 2013 Aviation Technology, Integration, and Operations Conference, Los Angeles, CA,
2013. American Institute of Aeronautics and Astronautics, Inc.
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T.A. DE J ONG
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