Aerostructural Shape and Topology Optimization of Aircraft Wings by Kai James A thesis submitted
Transcript of Aerostructural Shape and Topology Optimization of Aircraft Wings by Kai James A thesis submitted
Aerostructural Shape and Topology Optimization ofAircraft Wings
by
Kai James
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of Institute for Aerospace StudiesUniversity of Toronto
Copyright c© 2012 by Kai James
Abstract
Aerostructural Shape and Topology Optimization of Aircraft Wings
Kai James
Doctor of Philosophy
Graduate Department of Institute for Aerospace Studies
University of Toronto
2012
A series of novel algorithms for performing aerostructural shape and topology opti-
mization are introduced and applied to the design of aircraft wings. An isoparametric
level set method is developed for performing topology optimization of wings and other
non-rectangular structures that must be modeled using a non-uniform, body-fitted mesh.
The shape sensitivities are mapped to computational space using the transformation de-
fined by the Jacobian of the isoparametric finite elements. The mapped sensitivities are
then passed to the Hamilton-Jacobi equation, which is solved on a uniform Cartesian
grid. The method is derived for several objective functions including mass, compliance,
and global von Mises stress. The results are compared with SIMP results for several
two-dimensional benchmark problems. The method is also demonstrated on a three-
dimensional wingbox structure subject to fixed loading. It is shown that the isoparamet-
ric level set method is competitive with the SIMP method in terms of the final objective
value as well as computation time.
In a separate problem, the SIMP formulation is used to optimize the structural topol-
ogy of a wingbox as part of a larger MDO framework. Here, topology optimization is
combined with aerodynamic shape optimization, using a monolithic MDO architecture
that includes aerostructural coupling. The aerodynamic loads are modeled using a three-
dimensional panel method, and the structural analysis makes use of linear, isoparametric,
hexahedral elements. The aerodynamic shape is parameterized via a set of twist vari-
ii
ables representing the jig twist angle at equally spaced locations along the span of the
wing. The sensitivities are determined analytically using a coupled adjoint method. The
wing is optimized for minimum drag subject to a compliance constraint taken from a 2g
maneuver condition.
The results from the MDO algorithm are compared with those of a sequential opti-
mization procedure in order to quantify the benefits of the MDO approach. While the
sequentially optimized wing exhibits a nearly-elliptical lift distribution, the MDO design
seeks to push a greater portion of the load toward the root, thus reducing the structural
deflection, and allowing for a lighter structure. By exploiting this trade-off, the MDO
design achieves a 42% lower drag than the sequential result.
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Dedication
To Carl and Rosita, for endowing me with wings so that I may know freedom and
opportunity beyond that for which I could have hoped.
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Acknowledgements
There are many people who have been instrumental in my being able to make it to this
point. This is especially true of my supervisor, Joaquim Martins, who I am extremely
fortunate to have had as my guide throughout journey. You always provided me with
everything I needed, whether it was technical expertise and advice, personal and profes-
sional mentorship, or simply some words of encouragement when it seemed nothing was
working. Your patience, diligence, and leadership made what can often be an arduous
and painful journey, into an enjoyable one. And for that, I thank you. I must also thank
the other professors on my doctoral examination committee. Prof. Jorn Hansen and
Prof. David Zingg both offered invaluable constructive criticism during my committee
meetings, and have been consistent allies in helping me pursue my academic aspirations.
A big thanks also goes out to my colleagues in the MDO lab. You all helped create a
trully enriching atmosphere for learning. I especially need to thank Graeme Kennedy for
sharing with me his vast wealth of knowledge on all things MDO. I would also like to
specifically thank Edmund Lee, Sandy Mader, and Gaetan Kenway, who, together with
Graeme, provided camaraderie and a crucial forum where we could discuss and exchange
ideas.
I would be remiss if I didn’t mention my close friends, Samuel Oduneye and Aman
Husbands. As fellow PhD students in the sciences, you two were like my brothers in
arms, offering an empathetic ear when I needed to voice my frustrations, and always
being there to share in the highs and lows of research.
Lastly, I have to thank my parents, Carl James and Rosita Thompson, who instilled
in me an appreciation for the power of education. Also, through his own academic
achievements, my father has been my inspiration to strive toward excellence, both in the
academy and in life. My parents never wavered in their encouragement and support for
all my academic endeavours, and their faith in me has been the engine that allowed me
to persist in this path even through my most trying times. Thank you.
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Contents
1 Introduction 5
1.1 The Flight of Icarus: A Brief History of the Wing (8AD–1903) . . . . . . 5
1.2 Numerical Methods in Aircraft Design . . . . . . . . . . . . . . . . . . . 7
1.3 The Importance of Aerostructural Optimization . . . . . . . . . . . . . . 8
1.4 Previous Work on Aerostructural Topology Optimization . . . . . . . . . 9
1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Dissertation Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Topology Optimization 12
2.1 Structural Optimization as a Material Distribution Problem . . . . . . . 12
2.2 The SIMP Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Handling Numerical Challenges and Instabilities . . . . . . . . . . . . . . 19
2.3.1 Checkerboarding . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Mesh-Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Local Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Manufacturing Considerations . . . . . . . . . . . . . . . . . . . . . . . . 37
3 The Level Set Method 39
3.1 An Alternative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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3.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 The Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 The Isoparametric Level Set Method . . . . . . . . . . . . . . . . . . . . 48
3.5.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . 48
3.5.2 Isoparametric Mapping . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.3 The Constitutive Relation . . . . . . . . . . . . . . . . . . . . . . 55
3.5.4 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5.5 Algorithm Overview . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Compliance Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6.1 Discretization and Finite Element Analysis . . . . . . . . . . . . . 61
3.6.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.3 Wingbox Optimization . . . . . . . . . . . . . . . . . . . . . . . . 69
3.7 Stress-Based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7.1 Global von Mises Stress Using an Isoparametric Formulation . . . 74
3.7.2 Sample Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Aerostructural Optimization 86
4.1 Multidisciplinary Optimization . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.1 MDO Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.2 Design Parameterization . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.3 Aerodynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.4 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.5 Load and Displacement Transfer . . . . . . . . . . . . . . . . . . . 97
4.1.6 The Newton–Krylov Method . . . . . . . . . . . . . . . . . . . . . 97
4.1.7 The Coupled Adjoint Method . . . . . . . . . . . . . . . . . . . . 98
4.2 Aeroelastic Tailoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3 The Aerostructural Problem . . . . . . . . . . . . . . . . . . . . . . . . . 107
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4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.2 Sizing Optimization Example . . . . . . . . . . . . . . . . . . . . 112
4.3.3 Sequential Optimization . . . . . . . . . . . . . . . . . . . . . . . 113
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Conclusions 126
5.1 Summary of Contributions and Findings . . . . . . . . . . . . . . . . . . 126
5.2 Significance of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Recommendations and Future Work . . . . . . . . . . . . . . . . . . . . . 132
A Compliant Mechanism Design 134
A.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B Aerostructural Problem Specifications 140
B.1 Initial Conditions & Constraint Values . . . . . . . . . . . . . . . . . . . 140
B.2 Material Properties & Finite Element Mesh Dimensions . . . . . . . . . . 141
B.3 Atmospheric & Flight Conditions . . . . . . . . . . . . . . . . . . . . . . 141
B.4 CRM Wing Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.5 Sizing Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
References 143
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List of Tables
2.1 Minimized compliance values for the cantilever beam problem . . . . . . 35
2.2 Minimized compliance values for the MBB-beam problem . . . . . . . . . 36
3.1 Comparison of SIMP and LSM compliance minimization results . . . . . 68
3.2 Comparison of L-bracket solutions optimized for various objectives . . . . 82
4.1 Sample adjoint sensitivity results for the aerostructural problem . . . . . 100
4.2 Drag results for the aeroelastic tailoring problem . . . . . . . . . . . . . . 107
4.3 Tabular comparison of sequential and MDO results (topology optimization)116
A.1 Initial and final state of the compliant gripper . . . . . . . . . . . . . . . 139
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List of Figures
2.1 The topology optimization problem . . . . . . . . . . . . . . . . . . . . . 14
2.2 Various interpolation functions for penalization of intermediate densities 16
2.3 The classic cantilever beam problem . . . . . . . . . . . . . . . . . . . . . 20
2.4 Node-Based solutions to the cantilever beam problem . . . . . . . . . . . 22
2.5 Mesh-dependent solutions to the cantilever beam problem . . . . . . . . 23
2.6 Filter coefficient function plot . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Schematic diagram of a density filter . . . . . . . . . . . . . . . . . . . . 25
2.8 Data flow for the density filtering algorithm . . . . . . . . . . . . . . . . 27
2.9 Mesh-independence due to filtering . . . . . . . . . . . . . . . . . . . . . 28
2.10 Optimization result found using a random starting point . . . . . . . . . 29
2.11 Penalization history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.12 Convergence history for the continuation method . . . . . . . . . . . . . 32
2.13 Optimized topology obtained using the continuation method . . . . . . . 33
2.14 Two-dimensional slice of the feasible design space . . . . . . . . . . . . . 34
2.15 The MBB-beam problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Generalized structural shape design problem . . . . . . . . . . . . . . . . 40
3.2 Sample level set function for a two dimensional problem . . . . . . . . . . 43
3.3 Mapping from a non-uniform mesh to a Cartesian grid . . . . . . . . . . 52
3.4 Mapping from local element coordinates to global coordinates . . . . . . 53
3.5 Level Set Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . 59
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3.6 The long L-bracket problem . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7 The short L-bracket problem . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.8 Minimum compliance L-bracket structures . . . . . . . . . . . . . . . . . 65
3.9 Convergence plot for the minimum compliance long-L-bracket . . . . . . 66
3.10 Comparison of SIMP and LSM solutions to the long L-bracket problem . 67
3.11 Comparison solutions to the short L-bracket problem . . . . . . . . . . . 67
3.12 Comparison convergence histories for the short L-bracket problem . . . . 68
3.13 The cantilevered ring problem . . . . . . . . . . . . . . . . . . . . . . . . 69
3.14 Optimized designs for the cantilevered ring problem . . . . . . . . . . . . 70
3.15 Finite-element mesh for the wingbox structure . . . . . . . . . . . . . . . 70
3.16 Wingbox loading conditions . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.17 Optimized wingbox structure . . . . . . . . . . . . . . . . . . . . . . . . 72
3.18 Minimum-stress L-bracket designs . . . . . . . . . . . . . . . . . . . . . . 79
3.19 Stress distribution in the minimum-stress L-bracket . . . . . . . . . . . . 80
3.20 Stress distribution in the minimum-compliance L-bracket . . . . . . . . . 81
3.21 The semi-circular cantilever beam problem . . . . . . . . . . . . . . . . . 81
3.22 The minimum-stress semi-circular cantilever beam topology . . . . . . . 83
3.23 The isoparametric arch bridge problem . . . . . . . . . . . . . . . . . . . 83
3.24 Finite element mesh used in the isoparametric arch bridge problem . . . 84
3.25 Initial shape and topology of the arch bridge structure . . . . . . . . . . 84
3.26 Optimized arch bridge topology . . . . . . . . . . . . . . . . . . . . . . . 84
4.1 MDF Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 The CRM Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 CRM wing with TriPan surface mesh . . . . . . . . . . . . . . . . . . . . 94
4.4 Structural wingbox for the CRM wing . . . . . . . . . . . . . . . . . . . 96
4.5 Flying configurations for the baseline and optimized CRM . . . . . . . . 104
4.6 Twist distribution for the aerodynamically optimized CRM wing . . . . . 105
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4.7 Lift distribution for the aerodynamically optimized CRM wing . . . . . . 106
4.8 Aerostructural algorithm architecture . . . . . . . . . . . . . . . . . . . . 111
4.9 The rib-spar structural model . . . . . . . . . . . . . . . . . . . . . . . . 113
4.10 Aerostructurally optimized lift distributions for the CRM wing with a fixed
rib-spar topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.11 Sequential algorithm architecture . . . . . . . . . . . . . . . . . . . . . . 115
4.12 Aerostructurally optimized lift distributions (maneuver condition) . . . . 117
4.13 Aerostructurally optimized lift distributions (cruise condition) . . . . . . 117
4.14 Convergence history for the MDO problem . . . . . . . . . . . . . . . . . 118
4.15 Convergence history of the constraint functions for the MDO problem . . 120
4.16 Uni-axial loading problem . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.17 Optimized material distribution (Sequential A) . . . . . . . . . . . . . . 123
4.18 Optimized material distribution (Sequential B) . . . . . . . . . . . . . . . 124
4.19 Optimized material distribution (MDO) . . . . . . . . . . . . . . . . . . 125
A.1 The electrostatic gripper problem . . . . . . . . . . . . . . . . . . . . . . 135
A.2 Optimized gripper mechanism . . . . . . . . . . . . . . . . . . . . . . . . 138
A.3 Gripper mechanism in various stages of actuation . . . . . . . . . . . . . 139
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1
Nomenclature
General Terms
B strain-displacement matrix
c vector of equality constraints
c vector of inequality constraints
D constitutive matrix
E0 material Young’s modulus
Ee effective Young’s modulus
F vector of applied forces
ke element stiffness matrix
K global stiffness matrix
u global displacement state vector
ε strain tensor
ρ relative material density
σ stress tensor
Chapter 2 Terms
rfilt filter radius
rij radial distance between elements i and j
s density filter diffusion parameter
w adjoint state vector
p SIMP penalization factor
β weight penalization function
θij density filter weight coefficient
Θ density filter weight coefficient matrix
φ general penalization function
2
Chapter 3 Terms
d global vector of nodal displacements
de element displacement vector
H mean curvature of material boundary
J Jacobian matrix
L Lagrangian function
M stress coefficient matrix
n local unit normal vector
N finite element shape function
r step size used in Lagrange multiplier update
v general advection velocity
vc advection velocity in computational space
vp advection velocity in physical space
vole element volume
w adjoint state vector
x, y, z global (physical) coordinates
ΓD Dirichlet boundary
ΓN Neumann boundary
θ reference vector field
λ Lagrange multiplier
ν Poisson’s ratio
ξ, η, ζ local (computational) coordinates
ψ level set function
Ω material domain
∂Ω material boundary
3
Chapter 4 Terms
A aerodynamic residual vector
c thrust specific fuel consumption
C compliance
Cp pressure coefficient
Cd drag coefficient
D drag
e span efficiency
g acceleration due to gravity (9.81m/s2)
L lift
m rate of fuel consumption
p∗ target value of SIMP penalization factor
r continuation parameter for SIMP penalization factor
~r vector specifying the magnitude and direction of a rigid link
S structural residual vector
T thrust
~u local displacement vector
~θ local rotation vector
U∞ free-stream velocity
v velocity field
w vector of doublet strengths (aerodynamic state vector)
W structural weight
Wfixed fixed weight
δW virtual work
x SIMP design variable
~y global state vector
4
α0 aircraft angle of attack
αi induced angle of attack
αj jig twist angle
αlocal local angle of attack
αs structural twist deflection angle
Φ potential function
ψ adjoint state vector
Appendix A Terms
B0 electrostatic coefficient
c speed of light
d structural deflection
Felec electrostatic force (magnitude)
q magnitude of electrostatic charge
r radial distance between two charges
µ0 magnetic constant (4πV·s/A·m)
Chapter 1
Introduction
1.1 The Flight of Icarus: A Brief History of the
Wing (8AD–1903)
According to Greek mythology, the catalyst for the advent of human flight, like so many
other revolutionary inventions, was dire necessity. Having fallen out of favour with King
Minos of Crete, Daedalus, the skilled Athenian craftsman, along with his son Icarus, was
imprisoned in the tower atop the Labyrinth once used to house the minotaur. Daedalus
himself had built the labyrinth a few years earlier and so he knew how to escape the tower
at any time. However, his primary challenge was in getting off the island, as he knew the
king patrolled all surrounding waterways and was careful to inspect all vessels entering
or leaving. So he decided to craft two sets of wings, which he fashioned out of feathers.
Using wax and thread, Daedalus set about fastening the feathers to one another. He
applied a mild camber to each surface mimicking that which he had observed in the
wings of the birds. When he was finished, he anxiously tried on his new creation. To
his delight, the wings worked, propelling him into the air with just a casual flapping of
his arms. Before equipping his son with the apparatus, Daedalus issued a stern warning.
Flying too high would take Icarus too close to the sun, causing the wax to melt. Flying
5
Chapter 1. Introduction 6
too low carried the risk of getting the feathers wet from the sea water, making them too
heavy to fly. And so, armed with his new wings, and the requisite instructions, Icarus
followed his father out of the labyrinth and into the sky, where freedom awaited them.
The story of Daedalus and Icarus, as told in Ovid’s Metamorphoses [64], offers a
window into humanity’s timeless fascination with flight. What is also telling about the
story is that, in this case, the gift of flight was procured through craftsmanship and not
divine provenance, suggesting an understanding that if humans ever did fly like the birds,
science and engineering would furnish the path that takes us there.
More than a century before the first winged aircraft took flight, humans successfully
harnessed the power of buoyancy to navigate the skies. As early as the late eighteenth
century engineers were launching manned flights in hot-air balloons. But ballooning
was slow and cumbersome. For humans to truly exert dominion over the skies, this
would require lift-based vehicles. Drawing upon the knowledge gleaned from observing
avian flight, it was understood that the key to generating sufficient lift was to use lifting
surfaces, or wings
Some people attribute the invention of the airplane to the Wright brothers. However,
this is inaccurate as the concept of winged, heavier-than-air flying vehicles had occupied
the human imagination for centuries prior to Orville and Wilbur Wright. One of the
earliest published scientific papers on the topic was authored by Emanuel Swedenborg
in 1716 and was titled “Sketch of a Machine for Flying in the Air” [22]. The eighteenth
century turned out to be a watershed period for aviation. A big reason for this is the
work of George Cayley, the man credited with inventing the concept of the modern
airplane [5], with a fuselage, wings, and a tail for controlling the aircraft. Prior to
Cayley’s work, the focus was on ornithopters, which use flapping wings to generate both
lift and thrust. Cayley was the first to separate the tasks of propulsion and lift generation
when he introduced the idea of a powered, fixed-wing airplane in 1799. This spawned a
renaissance characterized by successive groundbreaking developments, which culminated
Chapter 1. Introduction 7
in the first powered, heavier-than-air, manned flight by the Wright brothers in 1903.
Today fixed-wing aircraft continue to dominate aviation, and as foreshadowed by Ovid
in the tale of Daedalus and Icarus, the wing retains its central role in the landscape of
human flight.
1.2 Numerical Methods in Aircraft Design
Aircraft design is a highly complex, multidisciplinary field that places unique and aggres-
sive demands on the engineers and scientists involved. Aircraft structural design is no
exception. Historically, the need for light-weight, multifunctional aerospace structures
has pushed the limits of the available materials and technology. This has created a need
for accurate and efficient methods for analyzing and predicting the behaviour of struc-
tures, particularly for those working in the area of aeroelasticity [25, 26]. As a result,
aerospace scientists and engineers are responsible for a disproportionate number of semi-
nal contributions to numerical analysis of structures and structural design [30]. From the
first direct stiffness method in 1959 [90], to the isoparametric finite element method in
1966 [40], aerospace engineers have been at the forefront of numerical analysis methods,
many of which are now used in a variety of other disciplines.
Today, with the advent of high performance computing, much of the research on nu-
merical methods for structures focuses on design optimization. By combining numerical
analysis techniques with numerical optimization methods that systematically search the
space of possible designs, engineers can find the best design for a given objective. As
was the case in the past, the unique demand for light-weight structures in aircraft de-
sign makes this field well-suited to the application and development of new structural
optimization methods.
One of these new methods is topology optimization. Though the method itself was
introduced as early as the late 1980’s [10], its application to aircraft design only began
Chapter 1. Introduction 8
within the past decade [48]. Since then it has been used to create conceptual designs
for a number of aircraft-related problems. This thesis seeks to add to the existing body
of research by applying the method in new ways to a series of problems related to the
structural design of aircraft wings.
1.3 The Importance of Aerostructural Optimization
From the standpoint of fuel consumption, air travel is relatively inefficient when com-
pared with other modes of transportation. Air freight burns three times as much fuel per
kilometre per tonne of cargo than the next least efficient freight method, heavy trucks.
When compared with the most efficient freight methods, trains and marine vessels, to-
day’s aircraft are outperformed by more than an order of magnitude [24]. Consequently,
there is much to be gained by improving the fuel efficiency of aircraft. Improving fuel
efficiency not only reduces operating costs, but it also reduces greenhouse gas emissions.
Aircraft currently account for up to 4% of global greenhouse gas emissions [31], this
number is expected to rise due to the rapid growth in the number of air travellers each
year.
There are a variety of strategies with which the industry can mitigate the projected
environmental impact of the current trend. These strategies can be divided onto four
major categories: operations and trajectory optimization, design of fuel efficient en-
gines, use of alternative fuels, and design of light-weight aerodynamically efficient air-
frames. Aerostructural optimization is a powerful tool for implementing the last ap-
proach. Aerostructural optimization is the simultaneous optimization of the aerodynamic
and structural design features of a mechanism or component that is subject to aerody-
namic loads. Aerostructural optimization takes into account the coupled interaction
between the structural and aerodynamic responses of the mechanism in order to achieve
the best possible design. When applied to aircraft wings, this technique can drastically
Chapter 1. Introduction 9
improve the aerodynamic efficiency of the design, thereby reducing drag and improving
fuel efficiency.
1.4 Previous Work on Aerostructural Topology Op-
timization
Early efforts at topology optimization focused on producing maximally stiff structures
for some specified fixed load. However, the discipline has matured significantly and
is now routinely used to optimize a wide range of objectives including eigenfrequency,
maximum local stress, as well as various objectives relating to the performance of micro-
electromechanical mechanisms (MEMS). Because of its ability to generate efficient, light-
weight structures for a variety of objectives, several authors have applied the technique
to aircraft design [35, 58, 57, 82].
One of the earliest examples of topology optimization of aircraft is that of the Airbus
A380, where topology optimization was used to optimize inboard fixed leading edge ribs
as well as the fuselage door intercoastals [35]. It is estimated that the use of topology
optimization led to an overall weight savings of 1000kg per aircraft [48]. For the wingbox
ribs, engineers at Airbus performed a compliance minimization of the structure subject
to fixed aerodynamic loading. Maute and Allen [57] offered a more theoretical example,
in which they optimized the conceptual structural layout of a wing planform. Treating
the wing as a flat plate, topology optimization was used to determine the location of
a series of stiffeners. Here the mass was minimized subject to constraints on lift, drag,
and tip deflection. In this example, the coupled aerostructural system was solved using
a Newton-type method, with an adjoint method used to perform the sensitivity analysis.
A later paper, coauthored by Maute and Reich [58], would use a similar aerostructural
framework while optimizing the material distribution inside a two-dimensional morphing
airfoil in order to minimize drag on the deformed airfoil shape. More recently, Stanford
Chapter 1. Introduction 10
and Ifju [82] used topology optimization to design the layout of a two-material membrane-
skeleton structure in the wing of a micro air vehicle, for which they sought to maximize
the lift-to-drag ratio using an unconstrained formulation. This example also includes
aerostructural coupling with a vortex lattice method used to compute the aerodynamic
forces.
1.5 Objectives
The goal of this thesis is to develop and implement a series of original algorithms for
performing aerostructural topology optimization of aircraft wings. The first part of this
thesis focuses on the level set method. It extends the standard level set formulation to ac-
commodate non-uniform, structured finite-element meshes comprised of arbitrary quadri-
lateral and hexahedral elements. This is necessary in order to apply the level set method
to non-rectangular, contoured structures like a wingbox. The algorithm is demonstrated
on a series of two-dimensional benchmark problems involving compliance and stress min-
imization. It is also used for compliance minimization of a three-dimensional contoured
wingbox problem. In this example and those that follow it, the wing is treated as a
three-dimensional design domain, in which the optimizer is free to distribute material
anywhere. This represents a departure from previous approaches where the wing was
either treated as a two-dimensional plate, or it was assumed a priori that the structural
members would be arranged in the conventional rib-spar configuration [74]. By including
the full three-dimensional region interior to the wetted surface in the design domain,
these examples offer better insight into the full potential of topology optimization as a
tool for aerostructural design.
A separate SIMP-based framework was also developed for performing topology opti-
mization for a variety of aerostructural design. Here, several new and useful numerical
tools are introduced for performing aerostructural topology optimization. The topology
Chapter 1. Introduction 11
optimization is carried out as part of a larger MDO framework in which the aerodynamic
shape of the wing is also optimized. This differs from previous examples from the litera-
ture where the aerodynamic shape was treated as a fixed design feature. This approach
is then compared with a sequential optimization result in order to quantify the benefits
of the MDO approach, specifically when it is combined with topology optimization.
1.6 Dissertation Layout
The remainder of this dissertation begins by defining the topology optimization problem
in Chapter 2. This chapter also contains a brief history of the development of the topology
optimization method as well as a discussion of the numerical challenges associated with
it. This discussion includes some numerical experiments and an evaluation of the various
techniques used to overcome these challenges.
Chapter 3 focuses on the level set method and introduces a novel variation to the
method that makes use of isoparametric finite elements. This chapter discusses the math-
ematical basis for the isoparametric method along with some examples of its application
to the design of non-rectangular structures including a three-dimensional wingbox.
In Chapter 4 the aerostructural framework is introduced. Here the structural analysis
is coupled to an aerodynamic model used to generate the loads acting on the structure.
The topology is optimized along with the aerodynamic shape of the wing in a single MDO
framework. Finally, Chapter 5 contains the conclusions based on the results observed
throughout the dissertation.
Chapter 2
Topology Optimization
2.1 Structural Optimization as a Material Distribu-
tion Problem
Early efforts at structural topology optimization focused on optimal layouts of truss
structures. The first paper published on this topic was authored by the Australian
inventor Mitchell, in 1904 [69]. Seven decades later, several seminal papers were published
that extended Mitchell’s theory to beam systems [68, 67, 66]. Also during this period,
there was a significant amount of research devoted to sizing and shape optimization (i.e.
varying the thickness or cross-sectional area of the structural members [47], or moving
the boundary of the material domain [39]).
However, with the publication of a paper by applied mathematicians Bendsøe and
Kikuchi in 1988 [10], the paradigm shifted. This paper was the first to treat topology
optimization as a material distribution problem, and it modeled the design as a continuum
structure. This new approach sought to provide a means for achieving optimal structural
designs in which the topology, as well as the shape and sizing of the members, was
optimized.
The original work on topology optimization as a material distribution problem was
12
Chapter 2. Topology Optimization 13
rooted in homogenization theory, in which one obtains approximations of effective, macro-
level material properties for porous, or otherwise periodic composite materials [12, 71].
In this way, complex, composite materials could be modelled as being homogeneous. To
perform topology optimization, the domain of the structure was divided into a series of
cells, each containing one or a series of rectangular holes. Homogenization was then used
to obtain a set of isotropic material properties for the cell. By varying the size of the
holes in each cell, one could generate an optimized material distribution throughout the
domain of the structure. Cells where the holes were large in the optimal layout could
be interpreted as being void, whereas cells with very small holes could be interpreted as
being solid. By including hole orientation as a design variable, it was found that optimal
designs tended to be comprised almost exclusively of fully solid and fully void regions,
with very little intermediate density material, thus making the resulting structure viable
from a manufacturing standpoint [10].
The combination of solid and void cells forms a pixelized representation of the optimal
structure, whose topology could differ greatly from that of the initial guess. Because
changes in the material boundary were achieved by adding or removing material from
the cells, homogenization methods also offered the advantage of having a fixed mesh.
While many shape optimization algorithms require a re-meshing of the finite element
model to conform to the material boundary, homogenization methods do not require any
re-meshing, thereby greatly improving the computational efficiency of the optimization
algorithm.
2.2 The SIMP Method
2.2.1 Problem Formulation
The solid isotropic material penalization (SIMP) method borrows principles from the
homogenization method, but greatly simplifies the process. Whereas homogenization
Chapter 2. Topology Optimization 14
Figure 2.1: Sample topology optimization problem (left) with the optimized design (right)
containing a combination of solid and void cells.
methods include material density as a function of cell microstructure (i.e., size and quan-
tity of perforations) [8], the SIMP method introduces the concept of material density as
a non-physical, independent variable. The SIMP method also omits the rotation angle as
a design variable and therefore assumes isotropic material properties at the macro-scale.
Therefore, in two dimensions, the number of design variables per cell is reduced from 3
to 1 [70].
Under this assumption, the effective material stiffness, Ee, of a given cell or finite
element can be expressed as the product of the Young’s modulus of the solid material,
E0 and some interpolating function of the material density, ρe ∈ (0, 1],
Ee = φ(ρe)E0, (2.1)
where the function φ must be chosen such that during the optimization process each cell
is forced toward either the solid or void phase, by penalizing intermediate densities. Al-
though intermediate density material could, in theory, be manufactured by introducing
an infinite number of holes into the microstructure, this process would be impractical and
the cost would be prohibitive. Nonetheless Rozvany and Zhou [70] proposed an inter-
polation function based on the hypothetical cost of having to manufacture intermediate
density material. Based on the fact that a fully solid microstructure (i.e. no holes) and a
Chapter 2. Topology Optimization 15
fully void microstructure (i.e. no material ) would be cheapest to manufacture, they came
up with an approximate cost function, β that takes into account both manufacturing cost
as well as the raw material cost for the range of material densities. The resulting function
is shown in Figure 2.2(b). By imposing a constraint on the total material cost of the
structure, intermediate densities were effectively penalized and were therefore removed
during the optimization.
However, today the most commonly used penalization function is the one introduced
by Bendsøe [8], which is given by
φ(ρe) = ρpe (2.2)
⇒ Ee = ρpeE0, (2.3)
where the penalization parameter, p is some number number greater than 1 (usually
p = 3). When implemented in combination with a mass constraint, this function penalizes
intermediate density elements with reduced stiffness for a given mass. The plot of this
function is shown in Fig. 2.2(a). In more recent work, researchers have used alternative
penalization functions including the hyperbolic sine function [15], which is shown in
Fig. 2.2(c), and the rational approximation of material properties (RAMP) function [83],
which is shown in Fig. 2.2(d). Like the power-law function, both these alternatives
have the property that φ(0) = 0, φ(1) = 1, and they provide for reduced stiffness at
intermediate density values.
For a given set of element densities, ρk, finite element analysis is used to solve the
structural displacement state, d, corresponding to a given point, in the design space.
Each cell in the discretized domain contains one or more finite elements, so during each
optimization iteration, the global stiffness matrix of the structure is computed based on
the element densities as follows,
Chapter 2. Topology Optimization 16
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ
φ(ρ)
p = 1
p = 2
p = 3
p = 5
(a) Power-law function
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ
β(ρ)
p = 1
p = 2
p = 3
p = 5
(b) Weight penalization function
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ
φ(ρ) p = 1
p = 2
p = 3
p = 5
(c) Hyperbolic sine function
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ
φ(ρ)
p = 0
p = 4
p = 9
p = 1.5
(d) RAMP function
Figure 2.2: Various interpolation functions for penalization of intermediate densities:
(a) power-law: φ = ρp; (b) weight penalization function: β = ρ1p ; (c) hyperbolic sine
function: φ = sinh(pρ)sinh(p)
; (d) RAMP: φ = ρ1+p(1−ρ)
K =∑e
ke(ρe). (2.4)
Chapter 2. Topology Optimization 17
Based on the SIMP penalization function (2.2), the element stiffness matrix, ke, can also
be expressed as a function of ρe using
ke = ρpek0, (2.5)
where k0 is the stiffness matrix of the element in the solid phase (ρ = 1). Note that one
must enforce a bound such that ρ ≥ ρmin > 0, in order to avoid singularities in the global
stiffness matrix. Typically, ρmin is chosen as 10−3.
The generalized optimization problem can be expressed as follows,
minρ
J
subject to: c = 0
c ≤ 0 (2.6)
Ku− F = 0
0 < ρmin ≤ ρ ≤ 1,
where J is the objective function, which typically depends on the displacement state
u. The optimization may be subject to a set of equality and inequality constraints,
represented by c and c, respectively, The variable F denotes the global vector of externally
applied forces, which, along with K and u, forms the governing equilibrium equation for
the structure.
2.2.2 Sensitivity Analysis
Because of the unusually high dimensionality of the design space inherent in topology
optimization (approximately one design variable per element, meaning an order of magni-
tude of O(104) or greater), the vast majority of practitioners use gradient-based methods
for solving the optimization problem. The large number of design variables also demands
Chapter 2. Topology Optimization 18
an efficient analytical approach to computing the sensitivities. For this reason, in the
results presented, the adjoint method is used for evaluating all SIMP sensitivities. The
following is a derivation of the generalized formula for evaluating the sensitivities of an
arbitrary function J with respect to the SIMP design variables, ρ.
Given some function, J , one can define the equivalent function, G as the sum of
J , and the residual expression, Ku − F, which is known to be equal to zero from the
equilibrium equation,
G = J + wT (Ku− F), (2.7)
where the coefficient vector w is independent of the displacement state, u. Taking the
total derivative of G with respect to ρ one gets
dG
dρ=∂J
∂ρ+∂J
∂u
du
dρ+ wT
(∂K
∂ρu + K
du
dρ
), (2.8)
where the partial derivative term, ∂J/∂ρ, captures any explicit dependence of the func-
tion J with respect to the independent variable, which, in this case, is the vector of
element densities ρ. (Note that this value can be computed without solving the equilib-
rium equation.) One can find w such that all terms containing du/dρ vanish. This is
equivalent to solving the adjoint equation,
wTK = −∂J∂u
(2.9)
for the vector, w, which is referred to as the adjoint vector or adjoint state. Once the
adjoint vector is known, the total derivative can be computed using
dG
dρ=∂J
∂ρ+ wT ∂K
∂ρu. (2.10)
Chapter 2. Topology Optimization 19
Using the definition of K provided in Eqn. 2.4, along with the SIMP penalization
(2.5), one can deduce the following identity for ∂K/∂ρe,
∂K
∂ρe= pρp−1
e k0 (2.11)
Therefore the total sensitivity of J with respect to the material density of element e can
be written as
dJ
dρe=∂J
∂ρe+ pρp−1
e wTe k0ue (2.12)
where we and ue are the portions of the global adjoint and displacement vectors corre-
sponding to the nodes of element e. The above formula for the total sensitivity expressed
in terms of the adjoint vector is very useful since it provides the sensitivity of any function
with respect to all design variables, and, unlike finite difference methods, it requires no
additional solutions of the governing equations. In fact, once the state of the system has
been obtained by solving the governing equations, the only additional cost is to solve the
linear adjoint system (2.9) once for each function being differentiated. The low compu-
tational cost and the high degree of accuracy (all solutions are exact) make the adjoint
method an indispensable tool for handling topology optimization problems.
2.3 Handling Numerical Challenges and Instabilities
As several authors have noted the density-based formulation with penalization leads to
several numerical difficulties [79, 43]. Over the years, a number of strategies have been
developed to eliminate or suppress these phenomena. This section describes the three
main numerical challenges associated with the SIMP-related methods, and presents a
brief study including some discussion and evaluation of the various approaches taken to
address each problem. The results were obtained using a two-dimensional linear finite
Chapter 2. Topology Optimization 20
element model, with a uniform mesh comprised of four-node, linear elements. The opti-
mization was performed using the SIMP power law combined with the optimality criteria
method for compliance minimization subject to a volume constraint [9].
2.3.1 Checkerboarding
Checkerboarding refers to the formation of regions of alternating solid and void elements
in an optimized structure (Fig. 2.3(b)). These occur due to poor numerical modelling
of the stiffness of checkerboard patterns [43]. Checkerboards occur most commonly in
models that use 4-node linear, quadrilateral elements, where all forces acting on an
element can be transferred completely through point-connections at the corner nodes.
Therefore, when using these elements, the checkerboard pattern exhibits an artificially
high stiffness.
(a) Problem definition (b) Optimized solution with checkerboards
Figure 2.3: Geometry and loading conditions for the classic cantilever beam problem (a);
an optimized solution containing checkerboarding (b)
Various methods have been proposed to alleviate checkerboarding. The simplest,
from an implementation standpoint, is to use higher order, eight-node or nine-node ele-
ments. This effectively eliminates checkerboards; however it also significantly increases
the computational effort required to perform the structural analysis. An alternative is
to implement a patch method [11] or restriction methods [85]. These methods desig-
nate patches which consist of a group of 4 or more adjacent elements. Once patches
Chapter 2. Topology Optimization 21
are identified, constraints are enforced on the relationship between the element densities
in each patch in order to eliminate checkerboards or otherwise prevent undesirable so-
lutions. Checkerboarding can also be controlled with the use of perimeter constraints.
This approach, which was initially introduced by Ambrosio and Buttazzo [4], enforces a
constraint on the total perimeter of the material interface, thereby limiting the potential
for checkerboarding. The drawback to this approach is that it requires the designer to
arbitrarily select an acceptable perimeter limit. As a result, the solution obtained may
be suboptimal or may still retain some checkerboard regions.
Another approach that effectively eliminates checkerboarding is the node-based for-
mulation. Under this formulation, the design variables represent the material densities
and the structural nodes. These values are interpolated using the shape functions of the
finite elements to create a continuous density field. The variation in density within in-
dividual elements is also taken into account when finding the element stiffness matrices.
Figure 2.4(a) shows a node-based solution to the problem described in Fig. 2.3(a). As
the image shows, checkerboarding has been completely eliminated.
Although this method can produce converged solutions in some cases, it has been
shown to be unstable when used in the absence of some additional regularization tech-
nique such as higher-order elements [43]. When combined with linear elements, the
node-based formulation can lead to islanding (Fig. 2.4(b)), a phenomenon in which is-
lands of solid material are left unconnected to the remainder of the structure, thereby
yielding no structural advantage. Figure 2.4(b) shows a second solution to problem 2.3(a)
in which the node-based solution was combined with a continuation method (see section
2.3.3), resulting in several island regions. Due to its superior computational efficiency rel-
ative to the alternative methods mentioned above, the aerostructural results presented in
Chapter 4 use the node-based density formulation in combination with density filtering,
which eliminates islanding and is discussed in detail in the section that follows.
Chapter 2. Topology Optimization 22
(a) Node-based solution (b) Node-based solution with islanding
Figure 2.4: Node-based solutions to the cantilever beam problem. (a) a stable result in
which checkerboarding has been eliminated; (b) an second result in which islanding has
occurred
2.3.2 Mesh-Dependency
Mesh-dependency refers to the property through which the level of coarseness or fineness
of the finite element mesh affects the number of members and the overall complexity
of the optimized structure. Unless steps are taken to reduce this effect, a fine mesh
will yield a large number of excessively thin members, which reduces the viability and
manufacturability of the optimized structure. This occurs because the generalized phys-
ical description of the problem is ill-posed and therefore suffers from the nonexistence
of solutions [79]. An example of mesh-dependency is shown in Fig. 2.5, which contains
two different solutions to the problem shown in Fig. 2.3(a). Although the size of the
domain is the same for both solutions, the solution on the left is discretized using a mesh
comprised of 30 × 60 unit square elements, while the solution of the right is discretized
using a mesh of 60 × 120 elements of side length 0.5. Both solutions use a node-based
formulation in order to avoid checkerboarding. As shown in the figures, the two solutions
have different topologies, with the finer mesh producing a solution that contains more
members and, in some cases, more narrow members.
To some extent, one can control the number of structural members in the optimized
design — and therefore reduce mesh-dependency — using perimeter constraints. How-
Chapter 2. Topology Optimization 23
(a) 30× 60 elements (b) 60× 120 elements
Figure 2.5: Optimized solutions to the 2× 1 cantilever beam problem using two different
mesh densities
ever, typically mesh-dependency is addressed using filtering techniques. The results pre-
sented in this section, as well as in Chapter 4, use density filters, in which, the element
densities serve as intermediate variables that are dependent on a set of design variables
xe. Similar to the element densities, ρ, each element is assigned its own design variable,
xe, which varies from 0 to 1. This value is then projected onto the density variable of all
elements within some prescribed radius by defining those densities as a weighted sum of
the x-values for all elements with the radius of influence as follows.
ρi =
Ni∑j=1
θijxj, (2.13)
where Ni is the total number of elements whose centroids lie within the filter radius,
rfilt of element i. The weighting coefficients, θij, are generally some decreasing function
of the radial distance between the two elements. A common choice is to use a linearly
decreasing function [49, 16], however, in this study, a novel weighting function has been
developed. The function has the shape of a normal probability distribution and is given
by,
Chapter 2. Topology Optimization 24
θij = e−r2ij
2s2 , (2.14)
θij =1
Ni∑k=1
θik
θij, (2.15)
where rij is the radial distance between the centroids of elements i and j. Note that the
values θij are normalized so that∑
j θij = 1.
Figure 2.6 shows the form of the weighting function for different values of the diffusion
parameter s. This function is useful because it has de facto local support as θij effectively
vanishes at distances where θij(r) < ρmin, and, unlike the linear filtering function, it is
smooth. Although the smoothness of this function is not required for optimizations in
which the filter radius remains constant, it often desirable to reduce the radius of influence
over the course of the optimization. This technique can be used to eliminate the fuzzy
boundaries that appear along the material interface [79, 37]. Because the filter weights,
θij, are smooth with respect to both the radial distance, r, and the diffusion parameter,
s, one can reduce the filter radius over the course of the optimization by decreasing
s, without any loss of robustness in the overall algorithm. This approach is part of a
broader category of techniques referred to as continuation methods. The drawback of
implementing this type of continuation method on the filter coefficients is that it can
cause the design to suddenly become infeasible, and can lead to artificial perturbations
in the optimized topology [36]. However, the properties of the exponential θ-function
described above, specifically its smoothness and the fact that it is defined everywhere
throughout the domain, can facilitate filter-based continuation methods by avoiding these
numerical difficulties.
The relationship between the element densities and the design variables in a density
filtering scheme can be viewed as being similar to the relationship between successive
layers of neurons in a neural network as shown in in Fig. 2.7.
Chapter 2. Topology Optimization 25
0 0.5 1 1.5 2 2.5 3 3.5 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radial distance
θ
s = 1.0
s = 0.2
s = 0.5
Figure 2.6: The exponential function used to determine the weighting coefficients, θ in
the density filter
Figure 2.7: Schematic representation of a density filter
Chapter 2. Topology Optimization 26
This mapping can be represented mathematically by expressing Eqn. 2.13 in matrix form
as
ρ1
ρ2
...
ρn
=
θ11x1 + θ12x2 + . . .+ θ1nxn
θ21x1 + θ22x2 + . . .+ θ2nxn...
θn1x1 + θn2x2 + . . .+ θnnxn
=
θ11 θ12 · · · θ1n
θ21 θ22 · · · θ2n
......
. . ....
θn1 θn2 · · · θnn
x1
x2
...
xn
, (2.16)
or, more compactly, as
ρ = Θx. (2.17)
Because the coefficient matrix, Θ, is independent of the design variables, xi the ad-
ditional cost of performing sensitivity analysis in the presence of a density filter is very
low. Given a set of sensitivities, dJdρ
as computed in Section 2.2.2, one can obtain the
corresponding sensitivities dJdx
as follows.
dJ
dx
T
=dJ
dρ
T ∂ρ
∂x(2.18)
⇒ dJ
dx
T
=dJ
dρ
T
Θ (2.19)
⇒ dJ
dx= ΘT dJ
dρ(2.20)
Using the above equations, the data flow in the topology optimization algorithm with
density filtering proceeds as shown in Fig. 2.8.
Earlier filtering algorithms filtered the sensitivities directly [13]. Under this formula-
tion, rather than using the actual sensitivities to perform the optimization, the optimizer
would be passed a set of surrogate sensitivities dJdρ
that were given by
dJ
dρ= Θ
dJ
dρ. (2.21)
Chapter 2. Topology Optimization 27
Figure 2.8: Data flow in the topology optimization algorithm with density filtering
However, this approach lacks consistency as it solves a separate, albeit similar, problem
from the one defined by the optimization problem ( 3.1). For the sensitivity-filtered
problem the optimal solutions may differ from those of the original problem, and in some
case, solutions may not exist at all.
Figure 2.9 illustrates the effects of density filtering. The solutions shown in the figure
were obtained using procedures identical to those used in Fig. 2.5, with the density of
the mesh for the solution (b) exactly twice the density of the mesh in solution (a). The
difference is that in this example, a density filter was added. In order to achieve an
equivalent length scale in both results, a diffusion parameter of s = 0.65 was used for
both solutions. As shown in the figure, the optimized designs are identical in spite of the
significant difference in mesh density.
In the three-dimensional aerostructural problems presented in Chapter 4, the node-
based formulation is combined with a density filter. The node-based formulation provides
a more appropriate representation of the density field, which must ultimately be inter-
polated in order to accurately visualize the optimized structure.
Chapter 2. Topology Optimization 28
(a) 30× 60 elements (b) 60× 120 elements
Figure 2.9: Optimized solutions to the 2× 1 cantilever beam problem using two different
mesh densities with density filtering.
2.3.3 Local Minima
The topology optimization problem described above ( 3.1) contains many local optima,
as evidenced by the large variety of solutions found in the literature for a given problem.
The inclusion of a penalty parameter (i.e. p > 1) has the effect of making the problem
non-convex, with most local minima appearing at those locations within the design space
where nearly all, variables take on the extreme values ρmin, 1. When performing opti-
mization on non-convex problems, it is common to run the optimization multiple times
using different starting points and to select the best optimum of the results obtained.
However this approach offers no significant advantage in topology optimization prob-
lems. There are several reasons for this. Because the dimensionality of the design space
is so high, one would have to test a very large number of starting points in order that
these points would be sufficiently distributed throughout the design space so that they
could confer a noticeable advantage. Secondly, the nature of the power-law interpolation
function already provides an implicit guideline for a suitable starting point that produces
good solutions.
To understand this, it is useful to define the concept of variable neutrality within the
context of a penalization method. Penalization biases the optimizer so that it favours
extreme values of the design variables. The more pronounced this bias is at a given value
Chapter 2. Topology Optimization 29
between 0 and 1, the less neutral the value is. So for example, for ρ = 0.8, there is
a strong incentive to move toward the extreme value of 1, since, based on the slope of
the function φ at this point, the optimizer can achieve a large increase in stiffness, for
a relatively small increase in mass. Similarly, for ρ = 0.1, there is a strong incentive
to move toward ρmin since decreasing the variable yields a large reduction mass with a
relatively small cost in terms of stiffness. These biases may cause a variable to move away
from its ideal value as it would appear in the globally optimal solution. This hypothesis
can be tested using the following experiment, which has been crafted specifically for this
study. Returning to the cantilever beam problem shown in Fig. 2.3(a), the initial design
is chosen at random so that it contains a large number of design variables at or near the
variable bounds as shown in Fig. 2.10(a). Running the optimization using this design as
the starting point produces the structure shown in Fig. 2.10(b). Based on the symmetry
of the boundary conditions, it is clear that the optimal should be symmetric, which
suggests that the asymmetric structure in Fig. 2.10(b) corresponds to a local optimum.
In this case, non-neutrality of the starting point biased the optimization process, causing
it to converge to a local optimum
(a) Randomly generated initial design (b) Optimized design
Figure 2.10: Initial design (a) and optimized design (b) when using a random generator
to select the starting point of the optimization search
In order to prevent these biases from overwhelming the search for the optimal solution,
it is useful to begin the optimization with all variables taking on the most neutral value
Chapter 2. Topology Optimization 30
possible. In the case where there is a constraint on the total mass, one typically chooses
the initial value φ0 that causes the constraint to be satisfied with all design variables set
to an equal value. By choosing a uniform intermediate density field as the starting point
for the optimization (i.e., ρe = 0.4 ∀ e [80]), one ensures that all design variables begin
at a safe distance from the locations the where penalization bias is the strongest.
Another way designers handle the problem of local minima is by using a continua-
tion method. Generally speaking, a continuation method is any technique in which the
parameters of the optimization problem (i.e. penalty parameter, filter radius, etc.) vary
incrementally over the course of the optimization, in order to address certain numerical
challenges. In the context of local minima, one applies the continuation technique to
the SIMP penalization parameter, p, increasing it gradually toward some target value
(usually p = 3). Using this technique, the optimization starts out with no penaliza-
tion and proceeds this way for several iterations, and then the penalty parameter is
increased incrementally at pre-defined intervals, or when the norm of the gradients satis-
fies some convergence criterion. This method prevents the situation where the design is
prematurely pushed toward a 0− 1 solution, thereby hindering the search for the global
optimum. As will be shown later, for aerostructural optimization, this approach has an
added advantage. In the early stages of the optimization, when all or nearly all of the
element densities are intermediate, the overall structure can become overly penalized.
For example, consider an initial starting point of ρi = 0.4∀i. Without penalization, this
structure will have an initial Young’s modulus that is 40% that of the bulk material. How-
ever with full penalization (i.e. p = 3), this structure will have a Young’s modulus that
is 6.4% (approximately 1/16) that of the bulk material. This can lead to unreasonably
high deflections that may hinder the aerodynamic analysis and forestall convergence.
To evaluate the merits of the continuation method, the 2×1 cantilever beam problem is
solved using a uniform initial design with a continuation method. The result is compared
with the standard solution shown in Fig. 2.9(a) as well as with the solution obtained using
Chapter 2. Topology Optimization 31
0 50 100 150 200 2500
1
2
3
4
5
Iteration Number
Pen
aliz
atio
n P
aram
eter
(p)
Figure 2.11: Chart showing the change in the SIMP penalty parameter over the course
of the optimization
a randomly-generated initial design shown in Fig. 2.10. All results use a density filter
with a diffusion parameter of s = 0.6. For the continuation method, Fig. 2.11 shows the
value of the SIMP penalization parameter as a function of the iteration number.
Figure 2.12 shows the convergence history of the compliance objective function for
all three solutions. From the plot, it is clear that the continuation method increases the
number of iterations required to reach convergence.
From the convergence plot, it appears as though the continuation method actually pro-
duces a worse objective value than the two competing approaches. However this is some-
what misleading. As indicated by Fig. 2.13, the continuation method tends to produce
more complex structures with a larger number of members. Therefore these structures
have a longer material boundary and, consequently, a larger proportion of intermediate
density elements, which always appear along the material interface. In order to provide
a more accurate depiction of how all three solutions compare to one another, Table 2.1
shows the compliance of all three optimized structures calculated using three different
metrics. In the first column, the optimized compliance values are shown with full pe-
Chapter 2. Topology Optimization 32
0 50 100 150 200 250 30010
1
102
103
Iteration Number
Com
plia
nce
standardrandom x
0
continuation
Figure 2.12: Convergence histories for the results obtained using the continuation,
random- initial-design, and standard methods
Chapter 2. Topology Optimization 33
(a) Continuation solution (b) Locally optimal solution
Figure 2.13: The optimized topology of the cantilever beam obtained using the contin-
uation method (a), shown with an analogous solution to the same problem produced
without the use of a continuation method (b)
nalization (i.e. p = 3). In the second column, the compliance is evaluated without
penalization using p = 1. In the third column, the element densities are passed through
a Heaviside filter, so that those above the designated threshold are treated as being fully
solid (ρ = 1) and those below the threshold are treated as being fully void (ρ = ρmin).
The threshold value is chosen so that the constraint is satisfied in each case.
The table shows that, based on both measurements in which penalization is not a
significant factor, the results obtained using the continuation method are superior. The
table also shows that, even though the three solutions vary significantly in their topology,
all three exhibit fairly similar compliance. This suggests that, in the vicinity of local
minima, the objective function is relatively flat, thus making the algorithm especially
vulnerable to converging to one of the many local minima that are known to populate
the design space. Figure 2.14 depicts a two-dimensional slice of the design space that
passes through all three solutions, with contour lines indicating the actual value of the
objective function in this region. The white region outside the contour plot is infeasible
due to the bounds on the design variables. As one might predict, the three solutions
are local minima, and as one moves away from the vertices of the triangle, the objective
function increases.
Chapter 2. Topology Optimization 34
Figure 2.14: Two-dimensional slice of the feasible design space with the three solutions
located at the vertices of the triangle
Chapter 2. Topology Optimization 35
Cp=3 Cp=1 Cthreshold Topology
Standard 86.27 73.99 76.66
Random 88.07 73.34 76.93
Continuation 90.64 72.35 76.63
Table 2.1: Minimized compliance values for the three solutions to the cantilever beam
problem (the best result for each metric appears in bold)
The same set of experiments was performed using the classic MBB-beam problem [62].
Figure 2.15 shows the optimized topologies obtained using the three different approaches
(note that symmetry was enforced in all three cases).
Comparing the numbers in Table 2.2, one sees that the trend is virtually identical to
that of the cantilever beam experiments, with the continuation method outperforming
the other two in both categories where penalization is not a factor. From both the MBB-
beam and cantilever beam results, it is also apparent that the numerical islanding effect
shown in Fig. 2.4(b) that occurred when combining the node-based formulation with the
continuation method, has been eliminated through the addition of a density filter. Due to
these findings, in the three dimensional aerostructural problems presented in Chapter 4,
the node-based formulation, along with density filtering and the continuation method are
Chapter 2. Topology Optimization 36
(a) Geometry, starting point, and loading (b) Standard solution
(c) Random solution (d) Continuation solution
Figure 2.15: Geometry and loading conditions for the MBB-beam problem, along with
optimized designs obtained using each of the three methods discussed
Cp=3 Cp=1 Cthreshold Topology
Standard 256.66 225.22 253.24
Random 263.90 221.20 233.37
Continuation 259.48 219.79 226.80
Table 2.2: Minimized compliance values for the three solutions to the MBB-beam problem
used.
Chapter 2. Topology Optimization 37
2.4 Manufacturing Considerations
Over the past two decades, structural optimization has experienced a significant growth
in popularity among designers in industry. Topology optimization is one of the major
factors behind this trend due to its ability to generate highly original design concepts
that deviate dramatically from conventional designs [104]. However, in industrial appli-
cations, this desire for new, leading-edge designs must be balanced with the numerous
practical considerations and constraints that must be taken into account before a design
is approved for production. When designing a wing structure, for example, designers
must consider whether a design is possible to manufacture, the cost of manufacturing,
and also maintenance and access requirements [74].
When incorporated into the industrial design process, topology optimization is typi-
cally used for generating conceptual designs [104, 35]. Manufacturing, as well as buckling
and other failure constraints are handled during the later stages of the design cycle, at
which time the size and cross-sectional shape of the structural members are determined
[48]. Still, there are several strategies one can use in order to improve the manufactura-
bility of the conceptual designs generated during the topology optimization phase.
A simple way to lower manufacturing costs is to impose geometric constraints so
that a design pattern is repeated over multiple sections of the structure. As a result
one can now purchase commercial topology optimization software that comes equipped
with the ability to enforce such constraints [104]. An example of how this would be
beneficial from a manufacturing standpoint is the use of a single topology design for all
ribs in an aircraft wing. Similarly, one can enforce symmetry constraints, either planar or
cyclical, which can also bring down manufacturing costs. It may also be the case that a
manufacturer requires an extrudable structure. This can be achieved either by enforcing
equality of the element densities in the direction of extrusion, or simply by redefining the
three-dimensional problem as a two-dimensional one.
Another feature that can be found in some commercial codes is the enforcement of a
Chapter 2. Topology Optimization 38
minimum member size. To some extent this can be achieved with density filtering, but
a more deliberate approach is to enforce constraints on the density change across adja-
cent elements [104]. When taking this approach, the constraint is relaxed on boundary
elements in the final stage of the optimization in order to minimize grey regions. Similar
constraints on individual element densities have been proposed to account for the re-
strictions imposed by specific manufacturing techniques, including casting and stamping
[104].
The results presented in the following chapters do not enforce the manufacturing
constraints discussed here. However, it should be noted that the algorithms developed
in this study are fully compatible with these constraints, and can be augmented to
incorporate these manufacturing considerations with a reasonable amount of effort.
Chapter 3
The Level Set Method
3.1 An Alternative Approach
While SIMP and homogenization methods remain the most prevalent methods in the lit-
erature, there are many other approaches to performing topology optimization including
genetic algorithms [7], integer programming [86], and evolutionary structural optimiza-
tion (ESO)[98]. The most popular and most effective of the alternatives is the level
set method. The level set method was first introduced in the late 1980’s by Osher and
Sethian [63] as a method for tracking front propagation. It has been used in a variety of
fields including computer vision and fluid dynamics, and in the past decade it has been
widely used by researchers in structural topology optimization. The method shares some
important similarities with SIMP. However the main difference lies in the way the level
set problem is parameterized.
In both the level set and SIMP methods, the design domain is discretized into a mesh
of finite elements. When using the ersatz material approach, the level set method, like
SIMP, seeks to determine which elements are solid and which are void in the optimal
structure. However, while the SIMP approach optimizes these material densities directly,
the level set method instead optimizes the location of the material boundary, as shown
39
Chapter 3. The Level Set Method 40
Figure 3.1: Generalized design problem using level set parametrization. Here ΓD denotes
the fixed boundary and ΓN denotes the boundary to which surface tractions are applied
in Fig. 3.1. From the location of this interface, ∂Ω, one extracts the relative material
densities of the elements based on whether they lie inside or outside the boundary. By
formulating the problem in this way, the level set method avoids checkerboarding and
mesh-dependence [3, 41]. Another advantage of the level set method as compared with
the SIMP method, is that the level set method precludes the existence of gray regions
due to its boundary-based parameterization. However, this feature also comes with the
drawback that the designer must decide a priori what the material boundary will look
like in the initial design. As a result, optimized structures obtained using the level set
method are sensitive to the choice of initial design [3]. Therefore the designer must
carefully select the number of holes in the initial design in order to control the length
scale and the level of detail found in the optimized structure.
3.2 Problem Definition
Given the parameterization described above, the level set optimization problem can be
defined as follows.
Chapter 3. The Level Set Method 41
min∂Ω
J
subject to: c = 0
c ≤ 0 (3.1)
K(ρ)u− F = 0
where J can be any objective functional that is dependent on the displacement state,
u. The function J is optimized with respect to the material boundary, ∂Ω. As in the
previous chapter, c and c represent the equality and inequality constraints respectively.
The structure must also satisfy the governing equation, which in this case is the linear
equilibrium equation Ku − F = 0. The global stiffness matrix, K, is dependent on the
density field, ρ, which determines the local Young’s modulus, E, at a given location, x,
within the domain according to the equation
E(x) = ρ(x)E0. (3.2)
The density field is discretized so that it is piecewise constant within each element.
Elements located inside the solid region, Ω, are assigned a material density of 1, while
those lying outside the boundary are given a small non-zero density, ρmin = 0.001, so that
this region effectively mimics the behaviour of a void space, while avoiding singularities
in the global stiffness matrix.
ρ(xi) =
1, xi ∈ Ω
10−3, xi ∈ Ω, i = 1, 2, ...n , (3.3)
where n is the number of the elements in the finite element mesh. When an element is
bisected by the material boundary, its density is interpolated based on the fraction of
the element’s volume that lies inside Ω. Unlike the SIMP method, there is no need to
Chapter 3. The Level Set Method 42
penalize intermediate densities since grey regions are precluded due to the presence of
an enforced material boundary.
The use of interpolated intermediate densities, sometimes refered to as the ersatz
material approach [3], allows for continuity in the movement of the material boundary.
If one is not careful, this approach can lead to numerical instabilities in both the SIMP
and level set approaches. Bruns and Tortorelli [17] and Yoon and Kim [101] observed
that when performing non-linear analysis, elements within the void regions modeled
using low-density material, experience large deformations that can cause the tangent
stiffness matrix to lose its positive-definiteness. To address this problem, van Dijk et al.
[91] employ an element connectivity parameterization method in which void elements are
allowed to slide over each other. Prior to this work, Ha and Cho [38] avoided this problem
by only modeling the solid material domain and remeshing the finite element model after
each optimization iteration so that the mesh boundary would conform to the updated
material boundary. Alternatively, Duysinx et al. presented a third option in which
they used a fixed mesh combined with the eXtended Finite Element Method, which can
model intra-element discontinities using enriched shape functions [29]. However, since
the problems being solved in this chapter involve relatively small displacements, all finite
element analysis assumes linear elasticity, which is compatible with the ersatz material
approach.
The level set algorithm begins by defining the working domain, or the region within
which material will be distributed. Mathematically, this is represented as a bounded
domain D ⊂ <d, d = 2, 3, of which all admissible structural shapes Ω are a subset. The
material boundary, ∂Ω, is represented implicitly as the zero contour of a higher order
function, ψ. This function is known as the level set function, and it is from this function
that the method gets its name. The level set function is chosen to satisfy the following
conditions.
Chapter 3. The Level Set Method 43
010
2030
4050
60
0
10
20
30
40−10
−5
0
5
10
15
XY
ψ
(a) Surface plot
5 10 15 20 25 30 35 40 45 50 55 60
5
10
15
20
25
30
35
40
(b) Contour plot
Figure 3.2: Two-dimensional example of the level set function corresponding to a can-
tilever beam structure (the material boundary is given by the thick black line in the
contour plot on the right)
ψ(x) = 0; x ∈ ∂Ω ∩D,
ψ(x) < 0; x ∈ Ω,
ψ(x) > 0; x ∈ (D ∩ Ω).
This function is an indispensable component of the level set method, as it allows the
algorithm to achieve changes in the structural topology. Topological changes such as
the merging or elimination of holes and structural members can occur when the value
of the level set function rises above or dips below the zero threshold. The value of the
level set function at any given time during the optimization is determined entirely by
the optimization process and the location of the material boundary, therefore it has
no closed-form representation. Rather, its value is stored discretely at the nodes of a
structured Cartesian mesh and this data is periodically interpolated in order to ascertain
the precise location of the zero contour.
The topology optimization problem (3.1) is generally non-convex. Therefore, although
Chapter 3. The Level Set Method 44
the method is mesh-independent, the final solution does depend on the initial topology
[3, 41]. One must carefully select the number of holes in the initial design to avoid
excessively limiting the number of possible designs that can be obtained. The choice of
initial topology can also be used to control the length-scale and number of holes in the
final solution. Once an initial topology is chosen and the material boundary is defined,
the level set function is initially set to equal the distance of each point in the domain
from the nearest point on the boundary. Negative distances indicate points inside the
solid region, Ω, and positive distances are used for points outside Ω. Optimization is
then performed to advance the boundary toward its optimal location.
3.3 The Hamilton–Jacobi Equation
The Hamilton–Jacobi equation relates the movement of the material boundary to changes
in the value of the level set function. It is used to advance the level set function and,
by extension, the material boundary, based on optimality criteria. In order to derive the
equation, it is necessary to return to the definition of the level set function. For some
point X on the material boundary at some time, t, from the definition of ψ one obtains
the following identity,
ψ(X(t), t) = 0, (3.4)
where the time parameter, t, does not represent time in the physical sense but rather it
tracks the progression of the optimization process, and can be thought of as a continuous
representation of the iteration number. Taking the total derivative of this equation with
respect to t, one gets
Chapter 3. The Level Set Method 45
∂ψ
∂t+∂ψ
∂X
∂X
∂t= 0 (3.5)
⇒∂ψ
∂t+∇ψ · X = 0 (3.6)
⇒∂ψ
∂t+∇ψ · vn = 0 (3.7)
where n is the unit vector point in the direction normal to the material surface, ∂Ω,
and the advection velocity, v, is the speed with which X travels the normal direction.
Noting that n = ∇ψ/|∇ψ|, the third equation can be further simplified to yield the
Hamilton–Jacobi equation, which is given by
∂ψ
∂t+ v|∇ψ| = 0. (3.8)
One can rearrange this equation to obtain the following scalar formula for updating the
level set function at a given point in the domain.
ψt+dt = ψt − v|∇ψt|dt (3.9)
The spatial derivative term ∇ψ is computed numerically at each point in the compu-
tational mesh by taking finite differences of the values of the level set function at adjacent
nodes. The advection velocity, v, used in this formula is given by the shape sensitivity
of the objective function at the point X. Performing this update moves the material
boundary by a distance of vdt in the outward normal direction. The time step dt is
chosen so that the Courant–Friedrichs–Lewy (CFL) condition is satisfied [95]. Conver-
gence is reached once the advection velocities are within a small tolerance of zero at all
points along the boundary. This procedure is equivalent to using an explicit Euler time
marching method to solve the partial differential equation (3.8).
After several Hamilton–Jacobi updates, the level set function can become very steep,
which negatively impacts the stability of the algorithm. In order to mitigate this effect
it is necessary to periodically restore the level set function to its signed distance form
Chapter 3. The Level Set Method 46
while retaining the current location of the zero contour [3]. In order to achieve this, the
current study relies on an implementation of the fast sweeping method [60, 75, 89], for
solving the Eikonal equation, (3.10), at each point in the computational mesh.
|∇ψ| = 1. (3.10)
Recently, some researchers have begun to use the reaction diffusion equation as an
alternative to the Hamilton–Jacobi equation [99]. This approach has the advantage that
it does not require the recovery of the signed distance function, which can be challenging
to implement, especially for three-dimensional problems. The reaction diffusion approach
also has the additional advantage that it allows for the creation of new holes even in two-
dimensional problems. When using the Hamilton-Jacobi equation with shape sensitivities
alone, as is done in the examples presented later, it is impossible to create new holes in
two-dimensional structures because changes in shape and topology can only be achieved
through movement of the existing boundary. Therefore, while the removal of holes is
possible, the creation of new holes would require the introduction of additional boundary
surfaces that are not connected to the original boundary. Allaire et al. [3] get around this
problem by using a bubble method in which the topological derivative is used in addition
to shape derivatives in order to create new holes during the optimization. However, the
Hamilton–Jacobi method performs quite differently in three-dimensions. In this case, new
holes are introduced when two separate parts of the material boundary intersect with one
another. Because this study is primarily concerned with three-dimensional optimization
of wing strucures, Hamilton–Jacobi equation with shape sensitivities is sufficient for these
purposes.
Chapter 3. The Level Set Method 47
3.4 Sensitivity Analysis
Following the framework introduced by Allaire et al. [3], the advection velocities used in
the Hamilton–Jacobi update are given by the shape sensitivities. In their study, Allaire et
al. proved that the use of these sensitivities to propagate the level set function guaranteed
a decrease in the level set function during each time step for some small value of dt > 0.
Prior to this work, Sethian and Weigmann [77] used an intuitive approach in which the
advection velocities were determined by the values of the von Mises stress field. However,
this approach is ad hoc and cannot guarantee convergence to a local minima. The use of
shape sensitivities can be augmented through the introduction of topological derivatives
[6], however, because this study is primarily focused on three-dimensional structures,
shape sensitivities are sufficient to achieve all necessary topological changes.
The shape sensitivities are computed using the Frechet functional derivative [81],
which can be written as,
J ′(Ω)(θ) =
∫∂Ω
vθ · nds, (3.11)
where θ is an arbitrary small vector field, and n is the unit normal vector as before. For
an arbitrary objective of the form
J(Ω) =
∫Ω
j(x)dx+
∫∂Ω
l(x)ds, (3.12)
the generalized shape sensitivity is given by
J ′(Ω)(θ) =
∫Ω
div(θ(x)j(x))dx+
∫∂Ω
θ(x) · n(x)
(∂l(x)
∂n(x)+H(x)l(x)
)ds (3.13)
⇒ J ′(Ω)(θ) =
∫∂Ω
θ(x) · n(x)j(x)ds+
∫∂Ω
θ(x) · n(x)
(∂l(x)
∂n(x)+H(x)l(x)
)ds. (3.14)
where H is the mean curvature of ∂Ω. For objective and constraint functions that depend
on the displacement state u, the shape sensitivity is calculated using the adjoint method.
Chapter 3. The Level Set Method 48
3.5 The Isoparametric Level Set Method
3.5.1 Background and Motivation
Compared to the element-based approaches such as SIMP and ESO, the level set method
for topology optimization was introduced relatively recently. Due to this fact, as well as
the increased complexity of implementing the algorithm, the level set method remains
primarily an area for academic and theoretical innovation, with most engineers turning
to the SIMP method to handle the more complex problems found in industry. As such,
the majority of the research devoted to the level set method focuses on a small class of
problems, seeking only to improve the numerical performance of the method [2]. As a
result, there remain large classes of problems to which the level set method has yet to be
applied.
One such group of problems is involves the optimization of structures confined to
irregularly-shaped physical domains. This class of problems is important because it
encompasses the vast majority of problems encountered in real-world engineering appli-
cations. This is especially true in aerostructural optimization where the structure is often
contoured according to the shape of the lifting surface. The aerostructural examples pre-
sented in this study involve wingbox structures which are shaped so that they roughly
conform to the outer surface of the wing. Structures like the wingbox are modelled us-
ing body-fitted finite-element meshes which tend to be non-uniform and non-rectilinear.
However, because the Hamilton–Jacobi equation must be solved on a Cartesian grid, the
structures found in the level set literature are often limited to those that can be modelled
with a uniform rectangular mesh. This way the structural shape sensitivities, computed
at the Gauss points of the finite elements [96], coincide with the nodes of the compu-
tational grid. By contrast, the SIMP method can easily be used to optimize structures
with non-uniform finite-element meshes, and numerous examples of these problems can
be found throughout the SIMP literature [48, 50, 42, 94].
Chapter 3. The Level Set Method 49
A few authors have proposed techniques for solving the Hamilton–Jacobi equation in
the presence of an unstructured or non-uniform finite element mesh[38, 52]. One option
is to compute the shape sensitivities at the nodes or Gauss points of the finite element
mesh and then interpolate these values to construct a continuous representation of the
sensitivity field. This sensitivity field can then be sampled at the nodes of the computa-
tional grid which is superimposed onto the finite element mesh. The interpolation of the
sensitivity values can be achieved in a variety of ways including the use of radial basis
functions [38], least squares fitting [27], and boundary elements [1].
Although these strategies make it possible to solve the Hamilton–Jacobi equation
using any arbitrary structural mesh, they do not specifically address the issue of having
a working domain that is non-rectangular. The isoparametric level set method is a
novel approach developed over the course of this research in order to be able to apply
the level set method to this large and indispensable class of problems. When using
the isoparametric method, as in previous methods, the Hamilton–Jacobi equation is
solved on a uniform, Cartesian computational grid. The structural analysis is performed
on a fixed, non-uniform structured mesh whose cells represent units in actual physical
space, and are modelled using linear, isoparametric, quadrilateral or hexahedral elements.
The shape sensitivities are calculated using this finite element mesh and they are then
mapped to computational space using the isoparametric transformation taking place
within each cell. Because each element in the finite element mesh has its own unique
node in the rectilinear computational grid, a one-to-one mapping is maintained between
the structural sensitivities and advection velocities used to solve the Hamilton–Jacobi
equation, thereby avoiding any significant loss of robustness 1. This mapping of the
1Note that here as well as throughout the thesis, the term “robust” refers to the robustness of thealgorithm itself and not the robustness of the structures produced by the algorithm. The “robustness”of the algorithm refers to its ability to consistently converge to a local optimum regardless of the initialdesign or the boundary conditions of the structural optimization problem. In order to generate robuststructures, the algorithms would have to take into account uncertainty and multiple load cases. Robustdesign is outside the scope of this thesis. Therefore, the optimized structures are designed only with thespecified load cases in mind, and are not generally robust with respect to off-design load cases.
Chapter 3. The Level Set Method 50
shape sensitivities is a unique and original feature of the isoparametric level set method,
and it is the key feature that drives the method.
In general, Hamilton-Jacobi equations have the following form,
∂ψ
∂t+ f(x,
∂ψ
∂x, t), (3.15)
where f is some arbitrary functional that is dependent on the spatial coordinates, x, the
spatial derivatives, ∂ψ∂x
, and the time parameter, t. This class of equations includes both
Eqn. 3.8 and the Eikonal equation (3.10).
There exist several techniques for solving generalized Hamilton-Jacobi equations on
unstructured meshes [76, 102]. However, most of these techniques assume there exists a
closed for expression for the function f , which can be solved exactly given the values of
the function ψ. By contrast, when solving the structural optimization problem described
in (3.1), no such closed form expression exists. Rather, the function f is dependent on
the velocity field, v, which is evaluated as a discrete approximation of the continuous
shape sensitivity field. However, the algorithm benefits from the fact that, when the
level set function is equal to signed distance function, the finite difference calculation of
the spacial derivatives is nearly exact because the signed distance function is linear with
respect to spatial location.
However, as noted by Sethian and Vladimirsky [77], when the computational mesh
is unstructured, the spatial derivatives must be approximated as linear combinations of
derectional derivatives, which are computed by taking finite differences of the value of
ψ at adjacent grid points. There is also an additional challenge when performing the
signed-distance recovery task on an unstructured mesh. Before one can begin to solve
the Eikonal equation, one must obtain accurate values for the signed distance of grid
points immediately adjacent to the material boundary. In the case of three-dimensional
structures, this is a fairly difficult geometry problem that admits only approximate solu-
tions, even when the mesh cells are perfect cubes. In an unstructured mesh, those cubes
Chapter 3. The Level Set Method 51
are replaced by arbitraty hexahedrons and tetrahedrons, vastly complicating the geome-
try problem and potentially increasing the inaccuracy of the signed distance calculation.
This effectively adds an additional layer of approximation on top of an already inexact
solution scheme, and reduces the likelihood that the algorithm will converge. Therefore,
the advantage of the isoparametric level set method is that it allows for the use of an
arbitrary non-uniform, non-rectangular finite element mesh while maintaining uniformity
and orthogonality in the solution of the Hamilton–Jacobi equation.
3.5.2 Isoparametric Mapping
The shape sensitivities discussed in Section 3.4 are approximated using isoparametric fi-
nite elements. Using the element shape functions, it is possible to generate a continuous
representation of the displacement state, from which one can construct the strain field
necessary for computing the sensitivity values. This strain field is evaluated at the Gauss
quadrature points of each element to compute the shape sensitivities. The resulting val-
ues are then averaged over the volume of the element to obtain a discretized sensitivity
field that is piecewise constant across each element. The choice to use a volume-averaged
sensitivity field ensures that the advection velocities used are representative of the shape
sensitivity throughout the entire element, since the boundary may occupy multiple lo-
cations within the element domain during a major optimization iteration in which the
structural analysis problem is solved only once. However, other authors have chosen
to sample sensitivity field at discrete points such as the element centroid, or the Gauss
points [27]. These techiniques have also produced statisfactory results.
After performing the element-wise sensitivity analysis, the sensitivities are converted
to their equivalent value in computational space using an isoparametric mapping. The
use of Isoparametric elements allows the method to be applied to working domains of
any shape, provided the finite element mesh is structured. Furthermore, the use of
isoparametric mapping for converting the shape sensitivities, means that each element
Chapter 3. The Level Set Method 52
in the finite element mesh provides its own unique sensitivity value to be used in the
solution of the Hamilton–Jacobi equation. This one-to-one mapping between the finite
element mesh and the computational grid eliminates the need to interpolate the element
sensitivities, thus removing an extraneous layer of approximation and maintaining the
stability and robustness of the algorithm.
Figure 3.3 illustrates the mapping between a non-uniform, finite element mesh and
the corresponding computational mesh used to perform the level set optimization.
Figure 3.3: Two-dimensional mapping of an arbitrary structured mesh to the correspond-
ing uniform rectilinear computational mesh.
The transformation from physical to computational space takes place at the element
level and is based on the isoparametric transformation required to morph a unit square
(or cube) into the quadrilateral (or hexahedral) shape of the element in question. In order
to describe this transformation in mathematical terms, within each element one defines
a local coordinate system ξ, η, ζ, where the orthogonal basis vectors ~ξ, ~η, ~ζ represent the
directions in computational space. Figure 3.4 shows the local coordinate system for a
two-dimensional quadrilateral element whose nodes are located at the points xi, yi.
The image on the left shows how these coordinates appear in physical space, while the
image on the right depicts the local coordinates in computational space. Note that in the
figure, the computational coordinates ξ, η range from −1 to 1. This choice of domain is
used to facilitate Gauss quadrature integration, however a further mapping is performed
Chapter 3. The Level Set Method 53
Figure 3.4: Two-dimensional illustration of the mapping from local to global coordinates
for an arbitrary quadrilateral element
so that each cell in the computational mesh has a unit area or volume.
The physical location x, y, z of any point within the element can be expressed in
terms of the node locations and the linear basis functions Ni.
x =8∑i=1
Ni(ξ, η, ζ)xi (3.16)
The basis functions are dependent on the local coordinates, ξ, η, ζ, and they have the
following form,
Ni(ξ, η, ζ) =(1 + ξξi)(1 + ηηi)(1 + ζζi)
8, ξi, ηi, ζi = ±1. (3.17)
Using this relationship between the physical coordinates and the local or computa-
tional coordinates, one can write the Jacobian matrix of the coordinate transformation
within each element,
J(ξ, η, ζ) =
∂x∂ξ
∂x∂η
∂x∂ζ
∂y∂ξ
∂y∂η
∂y∂ζ
∂z∂ξ
∂z∂η
∂z∂ζ
. (3.18)
From here, one can derive a relationship between the sensitivities calculated in physical
Chapter 3. The Level Set Method 54
space and their equivalent values in computational space.
Consider the infinitesimal patch located at point p on the material interface. The
orientation of the local coordinate axes is arbitrary, therefore one can choose them so
that the ζ axis points in the direction normal to the patch. When measured in compu-
tational space, the patch has an area of dξdη. Suppose that during one iteration of the
optimization process, the patch moves a distance dζ in the outward normal direction. In
doing so, it will trace a prism whose volume is given by dξdηdζ in computational space.
The same prism occupies a total volume of dξdηdζ|J| in physical space. Here it is useful
to introduce a new quantity that will be referred to as the relative impact. Given the
shape sensitivity, v, at some point, p, on the material surface, the relative impact of a
shape perturbation at p is defined as the product of the incremental volume change, Q,
due to that perturbation, and the shape sensitivity at that point.
If the shape sensitivity at p has a value of vp, computed in physical space, one can
find a corresponding value, vc, such that the relative impact of a shape perturbation at
p, is consistent in both physical and computational space. Therefore,
Qcvc = Qpvp (3.19)
⇒ dξdηdζvc = dξdηdζ|J|vp (3.20)
⇒ vc = |J|vp. (3.21)
Because the shape sensitivities are computed at discrete locations (one per element), the
conversion factor |J| must be averaged over the domain of each element. In order to
do this, one integrates the determinant of the Jacobian matrix over the volume of the
element. The result of this integration is simply the element volume as follows,
vole =
∫Ωe
det(J(ξ, η, ζ))dξdηdζ. (3.22)
Therefore given some shape sensitivity, vp, computed in physical space, the corre-
Chapter 3. The Level Set Method 55
sponding advection velocity in computational space is simply vc = volevp. By performing
a Hamilton–Jacobi update using the set of transformed computational sensitivities, one
is taking a step in the direction of steepest descent.
3.5.3 The Constitutive Relation
The constitutive matrix, D, describes the relationship between the stress, σ, and the
strain, ε, i.e.,
σ = Dε (3.23)
For the two-dimensional case, this equation can be expressed in matrix form as,
σxx
σyy
σxy
=E
(1− ν2)
1 ν 0
ν 1 0
0 0 1−ν2
εxx
εyy
εxy
, (3.24)
where ν is Poisson’s ratio of the isotropic material. The isoparametric method uses
an ersatz material approach [3, 54], in which the Young’s Modulus is interpolated to
obtain a appropriate stiffness for elements that are bisected by the material boundary.
For elements that fall only partially within the material domain, The Young’s modulus
is scaled according the portion of the element’s volume that is inside the boundary.
Therefore the effective Young’s modulus, Ee, and, by extension, the effective constitutive
matrix are given by
Ee = ρeE0 (3.25)
⇒ De = ρeD0, (3.26)
where E0 is the Young’s modulus of the solid material. The relative material density, ρe,
is a measure of the fraction of the element’s volume that lies inside the material boundary
Chapter 3. The Level Set Method 56
and makes the algorithm continuous as the shape sensitivities can be computed for any
location of the material boundary. Under the isoparametric formulation, the relative
material density is obtained using the following integral.
ρe =1
vole
∫Ωe
h(−ψ(ξ, η))det(J(ξ, η))dξdη (3.27)
ρe ≈1
npvole
np∑1
h(−ψp)|Jp|. (3.28)
This essentially divides the element into a series of pixels and performs a weighted sum
of those pixels that lie inside the material interface. The Heaviside function h is used
to identify the pixels that are inside the material domain. The terms in the summation
are weighted by the determinant of the Jacobian matrix, J, in accordance with spatial
mapping from computational to physical space. The number or pixels, np, must be large
enough to achieve a sufficiently accurate measure of the volume fraction, but small enough
to maintain computational efficiency. Experience indicates that np should satisfy np ≥ 50
for two dimensional problems and np ≥ 500 for three-dimensional problems. This process
can be computationally expensive. However, the cost is significantly reduced by dividing
the domain of the structure into blocks, and performing multiple integrals in parallel on
separate processors. This parallelization strategy is especially beneficial when solving
three-dimensional problems, and was used to solve the wingbox optimization problem
presented in Section 3.6.3.
3.5.4 Constraint Handling
Most optimization algorithms use a Lagrange multiplier approach for handling equality
and inequality constraints. In many level set schemes, however, the Lagrange multiplier is
replaced by a constant coefficient. Therefore, the optimization is equivalent to minimizing
a weighted sum of the objective and constraint functions [3]. However, since the coefficient
is fixed and arbitrary, the designer has no control over the final value of the constraint
Chapter 3. The Level Set Method 57
function. Consequently, the optimized design could be infeasible (if the desired constraint
value is breached) or sub-optimal if the constraint is inactive. Therefore this procedure
will always fail to yield the true optimum of the constrained problem.
Other authors have used a line search in order to find the value of the Lagrange
multiplier that causes the constraint to be satisfied with each optimization step [96]. This
approach is also inadequate since it requires the repeated evaluation of the constraint
function within each major optimization iteration. When using isoparametric elements
this can can be computationally expensive, even in the case of a single volume constraint.
Furthermore, the non-gradient-based updating of the Lagrange multiplier precludes the
use of a reliable convergence criterion that is based on the optimality conditions [59].
Instead, the isoparametric level set method uses an adaptive Lagrangian approach
that is designed to address both these issues. Using this approach, one can strictly
enforce a single equality constraint, while performing only one evaluation of the constraint
function within each iteration. The Lagrangian L is defined as a weighted sum of the
objective function, J , and the constraint function, c. The Lagrange multiplier λ serves
as the weight coefficient in the summation expression.
L = J(Ω) + λc(Ω) (3.29)
One then performs unconstrained optimization of the Lagrangian with respect to the
design variable, Ω. At the beginning of each iteration, the Lagrange multiplier is updated
using the heuristic
λk+1 = λk + rc(Ω), (3.30)
where, r, is a step size that is chosen to obtain an acceptable trade-off between conver-
gence time and stability. This update corresponds to a descent step, whose length is
proportional to the derivative of the Lagrangian with respect to λ. Experience suggests
Chapter 3. The Level Set Method 58
that a good choice of r should be in the range of [0.01λ0, 0.1λ0], where λ0 is the initial
value of the Lagrange multiplier. In the numerical experiments conducted in this study,
this method yielded converged solutions in which the final constraint value was within
0.01% of the target value. This level of precision is comparable to that found using SIMP
parameterization in combination with established numerical optimization methods such
as the frequently-used method of moving asymptotes (MMA) [84].
3.5.5 Algorithm Overview
The level set method and isoparametric level set method are considerably more involved
than the SIMP method, from an implementation standpoint. This can be attributed,
in large part, to the fact that the level set parametrization and the optimization algo-
rithm are highly interdependent and must be implemented accordingly. Indeed, one of
the major drawbacks of the level set method is the inability to combine it with any arbi-
trary optimization algorithm. In spite of the additional computational tasks required to
perform level set optimization, the structural analysis is still the most computationally
expensive step in the process, and it accounts for the majority of the computation time.
Figure 3.5 contains a flow chart of the general algorithm.
Due to the finite element discretization, the shape sensitivities never converge to the
tolerance one might expect from a typical optimization algorithm [3]. Still, it is possible
to define a reliable convergence criterion based on the optimality conditions. Based on
the CFL condition [20], the time step is selected so that
maxvcdt ≤ 1, (3.31)
noting that the spatial discretization in the computational grid always gives a cell size
of 1. The gradient vector can be expected to be reduced by two orders of magnitude,
therefore the algorithm has converged once the following condition is satisfied.
Chapter 3. The Level Set Method 59
Initialize
Boundary/Compute
Signed Distance (ψ)
Interpolate ψ
Structural Analysis
Hamilton-Jacobi
Update
Signed
Distance
Recovery
Convergence?
Map Sensitivities
ψ(0)
k)
ρ
vc
vp
ψ
k)
ψ
k)
Figure 3.5: Flow diagram for the isoparametric level set algorithm
Chapter 3. The Level Set Method 60
maxvcdt ≤ 0.01 (3.32)
This value for the convergence tolerance (10−2) is higher than that of most numerical
optimization algorithms. This is not due to the isoparametric formulation but rather,
it is a feature of the level set method in general. The relative lack of precision occurs
because there is an upper bound on the maximum achievable accuracy of the shape
sensitivities due to the finite element mesh used to discretize the continuum structural
analysis problem, and the intra-element discretization used to approximate the element
volume fraction. Whereas the SIMP approach takes a physical problem and converts
it into a mathematical problem that can be solved exactly, the level set method adds
a second layer of approximation since the mathematical problem itself (3.1) must be
approximated by implicitly representing the design variable, Ω, using a series of discrete
level set function values.
3.6 Compliance Minimization
Compliance minimization is the minimization of the work done by externally applied
loads. Alternatively it can be viewed as minimizing the total strain energy within a loaded
structure, or maximizing stiffness for a given load. Compliance is a popular objective
among researchers of the level set method [3, 27, 95] partly due to the relative ease with
which it is implemented. However it has also been shown to be a useful tool in the design
of practical engineering structures including aircraft wings [48]. The compliance function
is defined mathematically as
Jcomp(Ω) =
∫Ω
g(x) · u(x)dx+
∫ΓN
f(x) · u(x)ds
=
∫Ω
(εT (x)DεT (x)
)dx, (3.33)
Chapter 3. The Level Set Method 61
which is self-adjoint, and therefore its shape sensitivity can be written as
J ′comp(Ω)(θ) =
∫Γ0
θ(x) · n(x)(εT (x)DεT (x)
)ds, (3.34)
where, g and f denote body forces and surface tractions respectively, with u representing
the continuous displacement field. The variable Γ0 refers to the traction-free surface,
which is the only portion of the surface domain that varies during the optimization.
From the Frechet derivative shown above, the formula for the advection velocity is given
by
v = εT (x)DεT (x), (3.35)
which is equal to twice the strain energy density at x. This velocity value corresponds
to the shape sensitivity of the compliance function computed in physical space. Once
calculated, it is converted into computational space using Eqn. 3.19 and then passed to
the Hamilton–Jacobi equation.
3.6.1 Discretization and Finite Element Analysis
In practice, the shape sensitivity shown above must be approximated using finite ele-
ments. In order to do this, the expression εT (x)DεT (x) is averaged over the volume of
the element to obtain the physical advection velocity vp. This calculation begins by ap-
proximating the strain energy density expression using the discretized vector of element
nodal displacements, de.
εT (x)DεT (x) ≈ dTe BT (ξ, η)D0B(ξ, η)de, (3.36)
where the strain tensor has been replaced using
Chapter 3. The Level Set Method 62
ε(ξ, η) = BT (ξ, η)de. (3.37)
The strain-displacement matrix, B, is comprised of the set of first derivatives of the
element shape functions (Eqn. (3.17)), and it describes the relationship between the
strain tensor and the element nodal displacements, de.
Integrating Eqn. 3.36 over the domain of element Ωe and dividing by the element
volume yields the average strain energy density, which provides the advection velocity
for element e as calculated in physical space.
vp =
∫Ωe
(dTe B
T (ξ, η)D0B(ξ, η)dTe |J|)dξdη
vole(3.38)
This formula can be written more compactly by making use of the element stiffness
matrix, k, which is defined as
k0 =
∫Ωe
BT (ξ, η)D0B(ξ, η)|J|dξdη (3.39)
ke =
∫Ωe
BT (ξ, η)DeB(ξ, η)|J|dξdη (3.40)
= ρek0, (3.41)
Therefore, the physical advection velocity becomes
vp =ρed
Te k0devole
. (3.42)
As discussed earlier, in order to convert this value to its computational analog, one
simply multiplies by the element volume. And so the computational advection velocity
is given by
vc = ρedTe k0de. (3.43)
Chapter 3. The Level Set Method 63
If there is a volume constraint, one must add an extra term to the above formula to
account for the shape sensitivity of the volume function. From Eqn. (3.13), the shape
sensitivity of the volume function is just unity. The when converted into computational
space the sensitivity in equal to the element volume (i.e. vc = vole). Combining the two
velocity expressions, one obtains the shape sensitivity of the Lagrangian for compliance
minimization subject to a volume constraint.
vc = ρedTe k0de − λvole. (3.44)
This value can now be is passed directly to the Hamilton–Jacobi equation in order to
update the level set function and advance the material boundary.
3.6.2 Numerical Examples
In the following section, the isoparametric formulation derived above is demonstrated on
several benchmark problems. The first two examples are based on the classic L-bracket
problem, which appears frequently in topology optimization studies [2, 49]. Here the
problem is modified so that the finite-element mesh is comprised of trapezoidal elements
as shown in Fig. 3.6(b). This mesh is mapped to a single uniform rectangular mesh,
upon which the Hamilton–Jacobi equation is solved. Both the finite element mesh and the
computational mesh measure 32× 128 cells so the mapping is one-to-one. Figure 3.6(a)
shows the geometry and loading conditions for the problem. In this and all subsequent
two-dimensional examples, the applied load has a magnitude of 1.
The L-bracket structure is optimized for minimum compliance subject to a constraint
on the structural volume so that the total volume of the optimized structure is 40% of
the volume of the working domain. In addition to the problem described in Fig. 3.6, a
similar problem is solved in which the aspect ratio of the vertical and horizontal sections
of the structure is half that used in the original problem as shown in Fig. 3.7. The finite
element mesh contains 36 × 72 elements. This truncated geometry causes the elements
Chapter 3. The Level Set Method 64
F
(a) Initial topology and loading conditions
0 20 40 60 800
10
20
30
40
50
60
70
80
90
(b) Finite element mesh
Figure 3.6: The long L-bracket problem
along the outer edge of the L-bracket to have higher aspect ratios, which would present
a problem for most level set optimization algorithms.
F
(a) Initial topology and loading conditions
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
(b) Finite element mesh
Figure 3.7: The short L-bracket problem
Chapter 3. The Level Set Method 65
Figure 3.8 shows the optimized topologies for the long and short L-bracket problems.
The images show the optimized location of the zero contour of the level set function for
each problem. Both solutions are fully converged and feasible. Also, the smoothness of
the material boundary indicates that the solutions are unaffected by the orientation of
the mesh lines.
(a) Long L-bracket (b) Short L-bracket
Figure 3.8: Optimized topologies for long and short L-bracket problems
From the convergence plot for the long L-bracket problem (Fig. 3.9), one sees that the
convergence is smooth and stable. Although the constraint function does oscillate slightly
during the optimization, the objective is reduced monotonically, and the oscillations do
not appear to have hindered convergences. By carefully selecting the coefficient, r, in
Eqn. (3.30), one can be reduce the amplitude and number of these oscillations. In the
examples shown here, r = 0.01λ0.
The above results have been compared with solutions to the same problems found
using the SIMP method. In both problems, the geometry and finite element mesh are
unchanged. In order to avoid checkerboarding in the SIMP solutions, the element-based
density approach is combined with density filtering. The SIMP optimization is carried
out using the optimality criteria method [9], which is similar to the level set optimizer in
that it only makes use of sensitivity information at the current design point and performs
Chapter 3. The Level Set Method 66
0 20 40 60 80 100 120 140 160 1800
400
800
1200
1600
2000
Iteration Number
Com
plia
nce
0 20 40 60 80 100 120 140 160 1800
0.2
0.4
0.6
0.8
1
Vol
ume
Fra
ctio
n
Figure 3.9: Convergence history for compliance minimization of the long L-bracket
no line search. In order to achieve a more accurate comparison, the convergence criterion
for the SIMP solution is chosen as
max|ρ(k+1)e − ρ(k)
e | ≤ 0.01, (3.45)
where ρ(k)e is the relative material density of element e during iteration k. This represents
a decrease or two orders of magnitude in the maximum design variable change and is
analogous to the convergence criterion used in the level set method as shown in Eqn. 3.32.
In the case of the short L-bracket problem, the isoparametric level set method and the
SIMP method produce nearly identical solutions, as illustrated in Fig. 3.11. By contrast,
the two methods generate significantly disparate solutions to the long L-bracket problem.
In spite of this, both solutions have similar compliance values as shown in Table 3.1. This
is further evidence of the flatness of the compliance objective function as discussed in
Chapter 2.
Table 3.1 shows the optimized compliance value, function calls, and computation time
Chapter 3. The Level Set Method 67
(a) Level set solution (b) SIMP solution
Figure 3.10: Optimized density distribution for the long L-bracket
(a) Level set solution (b) SIMP solution
Figure 3.11: Optimized density distribution for the short L-bracket problem
for each case when run on a dual core 2.0 GHz AMD Turion processor. The table reveals
that, not only do the two methods perform similarly in terms of the final objective
function, but the number of function calls (i.e. structural analyses) required to reach
convergence is also close for both methods. However, the level set method requires a
greater computation time since it comes with the added task of interpolating the level
set function to find the element densities. Figure 3.12 shows the convergence history of
Chapter 3. The Level Set Method 68
the two methods for the short L-bracket problem. Note that, the final compliance value
for the SIMP solution is higher than in the table. This is because the convergence plot
displays the compliance value with penalization, whereas the table shows the unpenalized
compliance, (i.e. p = 1).
0 50 100 150 200 2500
100
200
300
400
500
600
700
800
Iteration Number
Com
plia
nce
level set methodSIMP
Figure 3.12: Comparison of the convergence histories of the SIMP and level set methods
for the short L-bracket problem
Compliance Function calls Wall time (hh:mm)
Geometry long short long short long short
LSM 364.44 83.84 174 237 28188 2:57
SIMP 366.18 84.31 159 231 23532 2:44
Table 3.1: Comparison SIMP and LSM results for the minimum compliance L-Bracket
problem
Chapter 3. The Level Set Method 69
3.6.3 Wingbox Optimization
The main advantage of the isoparametric level set method is that it allows users to
optimize contoured structures confined to non-rectangular domains, such as the wing
box structure. In order to illustrate the usefulness of this feature, consider the problem
shown in Fig. 3.13. The image on the left depicts a ring-like structure subject to cantilever
boundary conditions and applied loading. Material must be distributed throughout the
circular meshed region to generate the minimum compliance structure.
(a) Initial topology
−30 −20 −10 0 10 20 30−30
−20
−10
0
10
20
30
(b) Finite element mesh
Figure 3.13: The two-dimensional cantilevered ring problem
Using the isoparametric level set method, this mesh is mapped to a uniform Cartesian
grid, upon which level set optimization can be performed. Figure 3.14 shows the opti-
mized material distribution for the problem, alongside a SIMP-generated solution. The
resemblance of the two solutions is significant. This result further validates the isopara-
metric formulation as it is able to nearly reproduce the design found by SIMP, which
takes a different path to the solution, and is already known to be effective at handling
Chapter 3. The Level Set Method 70
non-uniform meshes.
(a) Level set solution (b) SIMP solution
Figure 3.14: Optimized designs for the cantilevered ring problem
This same strategy can be applied to the optimization of a wingbox structure. In the
following example, the isoparametric level set formulation is used to generate a minimum-
compliance wingbox subject to fixed distributed loading. The contours of the working
domain are roughly determined by the shape of the wing’s outer surface, the optimizer
can place distribute material anywhere within this three-dimensional region.
Figure 3.15: Working domain and finite element mesh for the wingbox optimization
problem
Chapter 3. The Level Set Method 71
Figure 3.16: Loading conditions for the wingbox optimization problem
The cross-section of the domain is taken from a symmetric airfoil with the leading
and trailing edges removed in order to replicate the shape of a wingbox. The resulting
cross-section is then extruded to produce an aspect ratio of 3.02. A taper ratio of 0.91,
and a leading-edge sweep angle of 9.4 are also added. The working domain and finite
element mesh are shown in Fig. 3.15. Figure 3.16 shows the loading conditions. A uniform
distributed load is applied to the top surface. The load on the bottom surface is constant
in the chordwise direction (x), and decreases elliptically in the spanwise direction (z).
A fixed boundary condition is applied along the entire face at the root of the wing,
where it connects with the fuselage of the aircraft. The finite element mesh is comprised
of 32 × 16 × 96 linear, eight-node hexahedral elements in the x, y, and z directions
respectively. A minimum skin thickness is enforced on the top and bottom surfaces, so
that at any given location along the span, the local skin thickness is no less than 1/2
the thickness of the local surface element measured in the y-direction. This is enforced
by setting the minimum relative material densities of all surface elements to ρmin = 0.5.
The structure is optimized for minimum compliance, with a 25% volume fraction.
Figure 3.17 shows the initial and optimized internal topologies for the wingbox struc-
ture. These figures do not include the wing skin, which covers the top and bottom surface
of the structure and is independent of the level set function. The optimized structure is
comprised of two main types of component. Much of the material is devoted to reinforc-
ing the top and bottom skin of the structure, thereby providing added bending stiffness.
Chapter 3. The Level Set Method 72
This is especially true near the root, where the internal bending moment of the structure
is the highest. As illustrated in Fig. 3.17(c), the remainder of the material forms a shear
web comprised of spar-like members. These members are aligned at roughly 45 from the
horizontal xz-plane, and offer resistance to shear in the z direction. From Fig. 3.17(d)
one sees that these members are also slightly slanted in the chordwise direction. This
is consistent with the asymmetry of the wing caused by the sweep angle. For a swept
wing, any upward bending produces a torsional moment about the z-axis, causing shear
to occur in both the spanwise and chordwise directions. Therefore, the chordwise slant of
the spar members allows the structure to resist the twisting that occurs due to bend-twist
coupling.
(a) Initial wing topology (b) Optimized wing topology
(c) Span-wise view (d) Chord-wise view
Figure 3.17: Optimized wingbox structure
The wingbox structure shown in Fig. 3.17 differs significantly from the structures
found in actual wingboxes. Most noticeably, the structure shown here does not con-
tain a recognizable spar or ribs, which are the primary structural components used in
Chapter 3. The Level Set Method 73
most wings [74]. In order to achieve a viable structure, one would need to implement
additional constraints to prevent buckling and material failure, as well as implementing
multiple load cases that capture the entire flight envelope of the aircraft. However, as
discussed in Chapter 1, this still would not ensure convergence to anything resembling a
conventional structural design since the conventional configurations also take into account
manufacturing and maintenance considerations, which are difficult to represent mathe-
matically in an optimization framework. Furthermore, because these considerations often
take precedence, conventional structural designs are generally not optimal.
Chapter 3. The Level Set Method 74
3.7 Stress-Based Design
Stress constraints are frequently used in aircraft design optimization as a means of produc-
ing feasible designs that are not vulnerable to material failure [65, 55]. However, because
of the unusually large number of dimensions in a typical topology optimization problem
and also the high number of constraints needed to be enforced, only a small minority
of topology optimization papers include any discussion or handling of stress constraints.
These papers are almost exclusively limited to SIMP-based frameworks. The literature
on stress-constraints for SIMP optimization dates back more than a decade [18], and
several seminal papers on the topic have been published in the intervening years [28, 49].
In spite of this, few authors have sought to incorporate stress constraints into a level
set framework. One notable exception can be found in the work of Allaire et al. [2].
This paper was the first to introduce a framework for producing minimum stress designs
using the level set method. Furthermore, it was shown that the level set method offers
an advantage over the SIMP method in that it is free from the stress singularity problem
[28], which holds that as an element’s material density approaches zero, its local stress
grows intractably, causing the optimizer to get stuck in undesirable local optima. In the
section that follows, the framework introduced by Allaire et al. [2] is extended to the
isoparametric formulation.
3.7.1 Global von Mises Stress Using an Isoparametric Formu-
lation
The von Mises stress function [44] establishes a failure criterion based on the yield
strength of an elastic material. It describes an elliptical envelope within which the com-
ponents of the stress tensor must lie. For the two-dimensional case, the von Mises stress
is given by
Chapter 3. The Level Set Method 75
σVM =√σ2xx + σxxσyy + σ2
yy + 3σ2xy. (3.46)
This function is particularly useful because it provides scalar quantity with which to
analyze the stress tensor. During stress-based topology optimization, the von Mises
stress is evaluated for each element. These scalar stress values form the bases of the
stress constraint functions, which are either treated separately (i.e. one constraint for
each element [28]) or combined to form one or more aggregated constraint functions [49]
In the examples presented, the local von Mises stresses are used to form a global
objective function. The stress values are aggregated using a variation of the p-norm
function to yield the following objective
G =
∫Ω
σbV M(x)dx, (3.47)
where σVM is the local von Mises stress, and b is the aggregation parameter, which is
some integer value greater than 1. Using the adjoint method derived in Eqns. 2.7-2.12,
along with the general formula for the partial derivative given in Eqn. 3.13, one obtains
the following shape derivative for G.
G′ =
∫∂Ω
(σbV M(x) + εT (w(x))Dε(u(x))
)dx, (3.48)
where εT (w(x)) is the strain tensor evaluated using the adjoint state w(x). Both the
displacement state, u, the adjoint state, w, are solved for discretely at the nodes of the
finite element mesh. The adjoint state is given by the solution of the adjoint equation:
Kw = −∂G∂d
. (3.49)
The element shape functions are then used to interpolate these nodal values and produce
a continuous adjoint field, from which a set of adjoint elements strains, εT (w(x)), can be
calculated.
Chapter 3. The Level Set Method 76
As was the case with compliance, the stress sensitivity field is approximated as being
piecewise constant across each element. Therefore, to obtain the element sensitivities one
takes the integrand from Eqn. 3.48 and finds its average value over the domain of each
element. The following is a derivation of the discretized formula for the global von Mises
stress function (3.47) and the corresponding shape sensitivity using isoparametric finite
elements.
The von Mises stress can be written in terms of the strain tensor as follows:
σ2VM(ξ, η) = εT (ξ, η)DMDε(ξ, η), (3.50)
where the coefficient matrix M has the form
M =
1 −1
20
−12
1 0
0 0 3
, (3.51)
as introduced by Svanberg and Werme [87]. From Eqn. 3.37 one can replace the stain
tensor to obtain the following equation for the squared von Mises stress at any point,
ξ, η.
σ2VM(ξ, η) = dTe B
T (ξ, η)DMDB(ξ, η)de (3.52)
Integrating over the domain of the element and dividing by the element volume, one
arrives at the average squared von Mises stress within the element.
σ2VM =
1
vole
∫Ωe
dTe BT (ξ, η)DMDB(ξ, η)de|J(ξ, η)|dξdη (3.53)
⇒ σbV M =1
vole
∫Ωe
(dTe B
T (ξ, η)DMDB(ξ, η)de) b
2 |J(ξ, η)|dξdη. (3.54)
Chapter 3. The Level Set Method 77
Using Gauss quadrature, this integral can be re-written as
σbV M =1
vole
∑i
ωi(dTe B
T (ξi, ηi)DMDB(ξi, ηi)de) b
2 |J(ξi, ηi)|, (3.55)
where ωi are the Gauss weights, and ξi, ηi are the coordinates of the Gauss points.
For compactness, it is useful to introduce the matrix S, which is defined as
Si = BT (ξi, ηi)DMDB(ξi, ηi) (3.56)
⇒ σbV M =1
vole
∑i
ωi(dTe Sei
de) b
2 |J(ξi, ηi)| (3.57)
Since the S matrix is dependent on the constitutive matrix, D, it scales quadratically
with the relative material density ρ.
Also, making use of the element stiffness matrix defined in Eqn. 3.39, one can now
write the global stress function, G, in terms of the element nodal displacement vectors,
de, the element adjoint vectors, we, the element stiffness matrices, ke, and the matrices,
Sei:
G =∑e
(voleσbV Me
)(3.58)
=∑e
∑i
(ωi(dTe Sei
de) b
2 |Je(ξi, ηi)|)
(3.59)
In order to solve for the adjoint vector, one must first evaluate the partial derivative
on the right hand side of the adjoint equation (3.49). Using the Gauss quadrature form
of the global von Mises stress function shown above, the partial derivative is given by
∂G
∂de=∑i
|Je(ξi, ηi)|(ωib(dTe Sei
de) b
2−1
Si
)de. (3.60)
Chapter 3. The Level Set Method 78
Given the adjoint vector, w, one can write the expression for the advection velocity
vp of each element as
vp =1
vole
(∑i
ωi(dTe Sei
de) b
2 |Je(ξi, ηi)|+ wTe kede
), (3.61)
Multiplying by the element volume, one obtains the corresponding computational advec-
tion velocity, vc.
vc =∑i
ωi(dTe Sei
de) b
2 |Je(ξi, ηi)|+ wTe kede (3.62)
Typically, when performing stress-based design, one wishes to control the maximum
local von Mises stress in the structure [28]. This can be accomplished using an optimizer-
based approach, such as the bound formulation [61], or by assigning a high value to the
aggregation parameter, b. However, because with the level set method one is somewhat
restricted in terms of the choice of optimizer, it is advantageous to avoid multiple con-
straints, as would arise with a bound formulation. Also, if b is too large, the elements
with high stress will dominate the sensitivity field, and the contribution from low-stress
elements will become negligibly small, potentially rendering the algorithm unstable. In
order to maintain the stability of the algorithm, in the examples presented, the aggrega-
tion parameter is limited to small values. Therefore, the examples presented below use
values of b = 2 and b = 6.
3.7.2 Sample Problems
The isoparametric stress formulation is demonstrated on the L-bracket problem described
in Section 3.6.2. In these examples the finite element mesh contains 28×84 elements. The
objective is to minimize the global von Mises stress subject to a 40% volume constraint.
The problem is solved for two different values of the aggregation parameter b. Figure 3.18
Chapter 3. The Level Set Method 79
(a) b = 2 (b) b = 6
Figure 3.18: Optimized structures for the L-bracket with minimum global von Mises
stress
shows the optimized density distributions for the two objective functions. The difference
in the optimized topologies for the two problems underscores the significance of the
aggregation parameter.
Figure 3.19 shows the von Mises stress distribution in the optimized structures for
the two problems. In the b = 6 case, the optimizer has rounded out the entrant corner
at the inner elbow of the L-bracket, where the stress is highest. This has the effect of
distributing the stress concentration over a wider area, thus reducing the maximum local
stress.
Table 3.2 contains a breakdown of the numerical performance of each of the minimum-
stress designs, as well as for the minimum-compliance solution. Each row in the table
corresponds to a different design optimized for a specific objective. The columns represent
the different performance criteria used to evaluate each design. As expected, the two
stress-based designs perform well with respect to their specific objectives. It is also shown
that the b = 6 design has a significantly lower maximum stress than its counterpart. It
is also interesting to note that the structure design for minimum compliance has stress
Chapter 3. The Level Set Method 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
(a) b = 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(b) b = 6
Figure 3.19: Local von Mises stress distribution in the minimum-stress L-bracket
values similar to the stress-based design for the b = 2 case, which, in turn, exhibits good
compliance.
Two additional examples are used to demonstrate the effectiveness of the algorithm
for solving problems with non-rectangular domains, which the isoparametric formulation
is particularly adept at handling. The first of these examples is short cantilever beam
problem whose working domain is given by a semi-circle. Figure 3.21(a) shows the initial
design and boundary conditions for the problem, while Fig. 3.21(b) shows the finite
element mesh. The aggregation parameter is set to b = 6.
Figure 3.22 shows the solution to the semi-circular cantilever beam problem, along
with the design at various stages of optimization. As in previous cases, the algorithm
produces straight, well-defined members that are unaffected by the variations in the size
and orientation of the finite element grid.
The final example is that of an arch bridge with a semicircular underpass. Figure 3.23
shows the geometry and boundary conditions for the problem. The structure is modeled
using the finite element mesh shown in Fig. 3.24, and the initial topology is given in
Fig. 3.25. The bridge is optimized for minimum global von Mises stress subject to
Chapter 3. The Level Set Method 81
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Figure 3.20: Local von Mises stress distribution for the minimum-compliance L-bracket
(a) Loads and constraints (b) Finite element mesh
Figure 3.21: The semi-circular cantilever beam problem
Chapter 3. The Level Set Method 82
Optimization Result
∑σ2
∑σ6 maxσ Compliance
Optim
ized
for
∑σ2 184.75 33.621 1.146 197.11
∑σ6 194.38 7.951 0.715 218.90
Comp. 182.23 29.594 1.104 192.96
Table 3.2: Comparison of L-bracket solutions optimized for various objectives
a volume constraint requiring that the optimized structure occupy 20% the working
domain. The aggregation parameter for the problem is set to b = 6.
Figure 3.26 shows the evolution of structure over the course of the optimization by
depicting the material boundary at various stages of the process. In the pictures, the
dashed line represents the domain boundary. The structure initially converges rapidly
toward a two-member truss configuration. However, after this initial phase, the evolution
slows down considerably as the two members begin to migrate inward toward the edge of
the semi-circular void. This apparent deceleration of the process is due to the reduction
in the magnitude of the advection velocities, which converge toward zero during the
optimization.
In addition to extending to level set method to be able to handle problems with
contoured domains, the isoparametric formulation can offer an additional advantage when
Chapter 3. The Level Set Method 83
iteration 0 Iteration 15 Iteration 35 Iteration 60 Iteration 120 Iteration 250
Figure 3.22: The semi-circular cantilever beam at various stages of optimization
Figure 3.23: Geometry and boundary conditions for the isoparametric arch bridge prob-
lem
solving stress-based problems. In locations where the structure experiences low stress, or
where material is not present, the isoparametric formulation allows a designer to coarsen
the mesh in this region. Conversely, one can refine the mesh in regions of high stress,
thereby improving accuracy and computational efficiency. The algorithms presented in
this chapter contribute to the closing of the gap between SIMP and the level set method
when it comes to solving complex engineering design problems such as structural wingbox
optimization. However, its inflexibility in terms of handling large numbers of constraints,
Chapter 3. The Level Set Method 84
Figure 3.24: Finite element mesh used in the isoparametric arch bridge problem
Figure 3.25: Initial shape and topology of the arch bridge structure
iteration 0 iteration 10 iteration 25
iteration 50 iteration 200 iteration 590
Figure 3.26: Material boundary of the bridge structure at various stages of optimization
and its inefficient optimization strategy continue to make it a less favourable option than
the SIMP method. Therefore it is necessary to return to the SIMP method in the
Chapter 3. The Level Set Method 85
remaining chapters where aerodynamic loads and aerostructural coupling are added to
the analysis.
Chapter 4
Aerostructural Optimization
4.1 Multidisciplinary Optimization
Aircraft design is an inherently interdisciplinary endeavour as it draws upon knowl-
edge from a wide range of science and engineering fields, such as dynamics and control,
propulsion, aerodynamics, and structural mechanics. Therefore, when performing design
optimization it is necessary to use an approach that accounts for the multidisciplinary
nature of the design problem [55]. Also, because the various disciplines governing the
aircraft’s performance are coupled, one must take this into account when modeling the
physics of the aircraft. Performing this multidisciplinary analysis can greatly improve
the accuracy of the computational model, thereby causing the optimization process to
yield larger gains. This is especially true of aerostructural design optimization, where the
aerodynamic and structural analyses are tightly coupled, since the structural deflection
influences the shape of the aerodynamic surface, which, in turn, determines the aerody-
namic loads acting on the structure. In addition to performing multidisciplinary analysis,
it is also beneficial to optimize the structural design concurrently with the other design
parameters, such as aerodynamic shape, using a multidisciplinary optimization (MDO)
approach. In cases where the disciplines associated with the various design parameters
86
Chapter 4. Aerostructural Optimization 87
are coupled, MDO tends to produce superior designs when compared with sequential ap-
proaches, where the design variables are grouped according to discipline, and each group
of variables is optimized separately and in sequence [19, 55]
As mentioned in Chapter 1, previous studies on topology optimization of aeroelastic
structures assumed a fixed aerodynamic shape [35, 58, 57, 82]. This is analogous to se-
quential optimization in the sense that the aerodynamic design was treated separately
(usually before the structural design was determined), and the structural optimization
took place independently at a later stage of the design process. The approach does not
fully take into account the interdependence between variables from different disciplines,
since it fails to consider all possible combinations of variables. Therefore this method gen-
erally leads to suboptimal solutions. By contrast, in this chapter topology optimization is
performed as part of a larger MDO framework. All objective and constraint functions, as
well as their sensitivities, are computed in a way that takes aerostructural coupling into
account. It is demonstrated that when structural topology optimization of a structural
wingbox is performed using MDO, the resulting designs outperform those obtained using
a sequential approach. Although there are many other disciplines that are also coupled
to the structural design, this study is restricted to looking only at aerodynamics as it
bears the strongest dependence on the structural performance.
4.1.1 MDO Architecture
There exist several algorithm architectures for performing MDO. By strategically dis-
tributing the various tasks associated with the overall optimization problem among the
optimizer and the analysis modules, one can tailor the algorithm to suit the nature of
the optimization problem in order to improve computational efficiency. The choice of
architecture is dependent on many factors. These include the level of coupling between
the different disciplines, the level of accuracy required for each analysis, and the relative
computational load demanded by the various disciplines.
Chapter 4. Aerostructural Optimization 88
Optimizer
Aerostructural Solver
Aerodynamic
Analysis
Structural
Analysis
u
F
x
Figure 4.1: MDF architecture for a generalized aerostructural problem
The multidisciplinary feasible (MDF) architecture [21] is used for the aerostructural
problems solved in this chapter. As shown in Fig. 4.1, this approach solves the coupled
aerostructural system to full convergence in each optimization iteration, with only the
coupled total sensitivities being passed to the optimizer. As indicated in the figure,
for aerostructural problems, aerodynamic and structural analyses are coupled via the
aerodynamic forces, F, and the structural displacements, u. In the diagram, the design
variable is represented by x, the objective function by, f , and the equality and inequality
constraints by c and c respectively.
Due to the large number of coupling variables shared by the two disciplines (i.e.,
all surface forces and displacements), the MDF architecture is the most appropriate for
the topology optimization problems being dealt with in this chapter. Furthermore, the
MDF architecture has been shown to be effective for solving aerostructural problems as it
Chapter 4. Aerostructural Optimization 89
offers advantages, such as greater simplicity and accuracy, over more complex, optimizer-
driven architectures [88]. All results were obtained using SNOPT, an optimizer that
uses sequential quadratic programming for solving the numerical optimization problem.
This algorithm has been shown to be effective for large-scale, constrained optimization
problems of the sort dealt with in this chapter [34].
4.1.2 Design Parameterization
For the aerostructural optimization problem, the structural topology is parameterized
using SIMP density variables. This return to the SIMP method, as opposed to the level
set method, is motivated by the large number of nonlinear constraint functions required
to solve the aerostructural problem, and the consequent need to use a flexible, robust,
and efficient optimizer, which is not readily compatible with the level set method. In the
results presented, the density filter discussed in Chapter 2 is used in combination with a
node-based H8/H8 density formulation [43].
An Optimizer-Based Continuation Method
As shown in Chapter 1, continuation methods are useful for avoiding convergence to
local minima when solving the non-convex topology optimization problem. In aerostruc-
tural topology optimization, continuation methods offer an added benefit as the baseline
structural design (generally a block of uniform, intermediate density) can have extremely
high deflections, which may lead numerical difficulties, including unrealistic modeling of
the aerodynamic loads, or aeroelastic divergence. Also, when topology optimization is
combined with aerodynamic shape optimization, these large deflections may cause the
optimizer to compensate by altering the aerodynamic shape in extreme and impracti-
cal ways. Continuation methods mitigate these issues since there is no penalization on
intermediate densities in the beginning of the optimization.
However, continuation methods also generate their own set of numerical challenges.
Chapter 4. Aerostructural Optimization 90
Because the penalization parameter is increased arbitrarily during the optimization, the
objective and constraint functions undergo discontinuous jumps. This may cause the
design to suddenly become infeasible, which may trigger premature termination of the
algorithm. This approach is also computationally inefficient as it slows down the algo-
rithm’s convergence [36]. Furthermore, when using an SQP method such as SNOPT, this
discontinuous change in the problem definition can invalidate the Hessian approximation,
which may also forestall convergence.
In order to avoid these numerical instabilities, the aerostructural problem uses an
novel optimizer-based continuation method, developed and introduced specifically for
the problems dealt with in this study. The method is fully continuous and requires no
adjustments to the definitions of the optimization problem once the optimization has
started. The method requires that the SIMP penalization parameter, p, be treated as a
design variable. One then enforces the following equality constraint:
(p− p∗)r = 0, r > 1. (4.1)
Here, p∗ is the target final value of the penalization parameter, which is usually chosen
as p∗ = 3. Experimental results indicate that this value is sufficiently high so that it
eliminates most intermediate density elements, yet it is low enough that the optimization
and multidisciplinary analysis avoid divergence. By initializing p to unity, the initial
design is unpenalized as required. One could also set p0 such that p0 < 1, in order to
ensure that the baseline design exhibits satisfactory stiffness. As the optimizer attempts
to satisfy this constraint, the penalization parameter will gradually approach its target
value. One can increase the speed with which p approaches p∗, by decreasing the exponent
r. In the examples presented, the constant r is set to r = 8. In order to maximize stability
and reduce the impact of local minima, the value of r should be set as high as possible.
However, in addition to slowing down convergence, if r is higher than 8, the parameter p
may converge to a value much lower than the target, p∗, since the high exponent means
Chapter 4. Aerostructural Optimization 91
(a) planform view (b) airfoil view
Figure 4.2: Outer mold line of the baseline CRM wing model (semispan = 29.38m)
that parity between the two variables is not required to satisfy (4.1) within the specified
tolerance of the optimization. Alternatively, one could achieve a similar effect by treating
Eqn. (4.1) as a penalty term, and adding it to the objective.
Aerodynamic Shape Variables
The baseline wing shape is based on the Common Research Model (CRM), a wing-body-
tail configuration developed by NASA [92]. The CRM model uses a transonic supercritical
wing designed for aerodynamic performance at high subsonic speeds. The swept planform
of the wing causes it to twist considerably when subject to aerodynamic loads, due to
bend-twist coupling. For this reason the wing provides a good case study for examining
the effects of aerostructural coupling and evaluating the benefits of the MDO approach
being used.
For the aerodynamic portion of the design problem, the planform and airfoils are
fixed, while the spanwise twist distribution is allowed to vary. The twist distribution is
parameterized via eight design variables representing the jig twist angle at equally spaced
Chapter 4. Aerostructural Optimization 92
locations along the span. These angles are then interpolated linearly in order to obtain
a continuous twist profile along the span of the wing. The local angle of attack, αlocal,
at a given location along the span is a superposition of the aircraft angle of attack, α0;
the jig twist angle, αj; the twist due to structural deflection, αs; and the induced angle
of attack, αi, due to downwash.
αlocal = α0 + αj + αs − αi (4.2)
During each optimization iteration, the structural finite element mesh is transformed
to conform to the jig shape of the current outer mold line. Shape changes on the surface
of the wingbox are propagated through the finite element mesh using a free-form defor-
mation technique [73, 46]. By optimizing the jig twist, the goal is to exploit the interplay
between the structural and aerodynamic responses of the wing, and ultimately generate
significant improvements over the baseline design.
4.1.3 Aerodynamic Analysis
The aerodynamic loads are computed using the TriPan software package [45], an unstruc-
tured three-dimensional panel code. TriPan models inviscid, incompressible, external lift-
ing flows using constant-source, doublet singularity elements distributed over the lifting
surface. The computation begins with the continuity equation, which, for incompressible
flows, reduces to
∇ · v = 0, (4.3)
where v is the velocity field. By defining the potential function, Φ, such that
v = ∇Φ, (4.4)
Chapter 4. Aerostructural Optimization 93
one arrives at the following Laplace equation,
∇2Φ = 0, (4.5)
which can be solved by applying boundary conditions and assuming the solution can
be expressed as a linear combination of doublet singularities. For a discretized problem
containing N panels, one obtains an N -dimensional vector of aerodynamic residuals,
A, which are linear functions of the doublet strengths, w, associated with each panel.
Therefore, the governing equation of the aerodynamic system can be represented in vector
form as
A(w) = 0, A,w ∈ RN . (4.6)
Once the doublet strengths are found, one can compute the components, vi, of the
velocity field using,
vl =∂w
∂l, vm =
∂w
∂n, vn =
∂w
∂n, (4.7)
where l,m, n represent the local coordinates. From the velocity field, one can evaluate
the local pressure coefficient using the following relationship
Cp = 1− q2
U2∞
(4.8)
Here, U∞, is the free-stream velocity of the steady flow condition. The aerodynamic
forces and moments are then calculated by integrating the local pressure values over the
surface of the wing. Figure 4.3 shows the CRM wing with the TriPan mesh used to solve
for all aerodynamic forces in this study.
Chapter 4. Aerostructural Optimization 94
Figure 4.3: The CRM wing with the TriPan surface mesh
Chapter 4. Aerostructural Optimization 95
The use of a panel method allows for the rapid solution of the aerodynamic forces,
thus making the coupled aerostructural analysis, and the optimization procedure converge
more quickly. The trade-off for these time savings is the loss of accuracy. In particular,
the panel method is unable to predict stall, viscous effects, and wave drag. However, this
is an acceptable trade-off, as the focus here is on the aerodynamic benefits of a lighter
structure and optimized twist distribution, which mainly affect the induced drag. The
alternative to a panel method would be to use CFD methods, which are generally more
accurate, and are capable of capturing viscous effects. However, for problems of this
size, the TriPan solution is several orders of magnitude less expensive than CFD, both
in terms of memory requirements and computation time [100].
4.1.4 Structural Analysis
The structural model is identical to that described in Chapter 3. The finite element
mesh is comprised of eight-node, linear, hexahedral isoparametric elements. The only
difference in this chapter is that the consistent force vector is no longer constant, but
rather, is a function of the structural and aerodynamic state variables, denoted as w and
u, respectively. The resulting structural governing equation is expressed in vector form
as
S(u) = K(ρ)u− F(w,u) = 0, (4.9)
where S is the structural residual, K is the global stiffness matrix, and ρ is the vector
of relative material densities. The assembly of the stiffness matrix and the solution of
the linear algebraic system are performed using the TACS software tool, a parallel finite
element code [45].
Chapter 4. Aerostructural Optimization 96
Figure 4.4: The structural wingbox and finite element mesh for the CRM wing
Chapter 4. Aerostructural Optimization 97
4.1.5 Load and Displacement Transfer
The forces and displacements are transferred using the technique that was originally
introduced by Brown [14], and has since been used for high-fidelity aerostructural opti-
mization [45, 56] .Aerodynamic loads and structural displacements are transmitted via a
series of rigid links that connect each node in the panel grid to its nearest point on the fi-
nite element mesh. Therefore, the displacement, ~uA, at a given node on the aerodynamic
surface can be expressed as
~uA = ~us + ~θs × ~r, (4.10)
where ~us and ~θs are the displacement and rotation at the nearest point on the finite
element mesh, and the vector ~r represents the length and direction of the rigid link con-
necting the two points. Conversely, given the pressure distribution along the aerodynamic
surface, the corresponding nodal forces acting on the finite-element mesh are computed
using the principle of virtual work. The virtual work, δW , done by the aerodynamic
forces is given by
δW =
∫SA
pn · δ~uAdS =
∫SA
(pn · δ~us − pn · (~rs × δ~θ)
)dS, (4.11)
where p is the surface pressure, and n is the unit vector normal to the aerodynamic
surface.
4.1.6 The Newton–Krylov Method
The aerodynamic and structural residual equations combine to form a nonlinear global
system of residual equations given by
Chapter 4. Aerostructural Optimization 98
R =
A(u,w)
S(u,w)
= 0. (4.12)
Under the current scheme, this equation is solved iteratively using Newton’s method, in
which the Newton update of the global state vector, y = [wTuT ]T , is obtained by solving
the following linear system.
∂R
∂y∆y(n) = −R(y) (4.13)
This equation is solved using the GMRES Krylov subspace method. A block diagonal
preconditioner is applied to the linear system, with one block for each discipline. An
approximate Schur preconditioner is used for the structural block, and an ILU precondi-
tioner is used for the aerodynamic block.
4.1.7 The Coupled Adjoint Method
As in previous chapters, due to the large number of design variables, an adjoint analytical
method is used to perform the sensitivity analysis. In this case, it is useful to use a coupled
adjoint method that reflects the multidisciplinary nature of the aerostructural problem.
From the derivation provided in Chapter 2 (beginning with Eqn.2.7), the total sensitivity
of some function f , with respect to the vector of design variables, x, is given by
df
dx=∂f
∂x− ψT
∂R
∂x, (4.14)
where the adjoint vector, ψ, is found by solving the adjoint equation,
∂RT
∂qψ =
df
dq. (4.15)
Chapter 4. Aerostructural Optimization 99
This equation is solved iteratively using a staggered approach in which the contributions
to the adjoint vector from each discipline are computed sequentially, following the proce-
dure introduced by Martins et al. [56] for sensitivity analysis of aerostructural systems.
The resulting procedure is equivalent to a block Gauss–Seidel method in which the up-
dates for the aerodynamic and structural portions of the global adjoint vector are given
by
∂AT
∂w∆ψ
(n)A =
∂f
∂w− ∂AT
∂wψ
(n)A − ∂ST
∂wψ
(n)S
∂ST
∂u∆ψ
(n)S =
∂f
∂u− ∂ST
∂uψ
(n)S − ∂AT
∂uψ
(n+1)A , (4.16)
Once this process has converged, the adjoint vector is updated using ψ(n+1)A = ψ
(n)A +∆ψ
(n)A
and ψ(n+1)S = ψ
(n)S + ∆ψ
(n)S .
In both (4.16) and (4.13), the exact values of the partial derivatives in the Jacobian
matrix, ∂R/∂y, are evaluated analytically. The values for the diagonal blocks ∂S/∂u
and ∂A/∂w are obtained using the chain rule, with each being expressed as the product
of two simpler partial derivative terms,
∂AT
∂u=∂AT
∂XA
∂XA
∂u, (4.17)
∂ST
∂w=∂ST
∂FA
∂FA
∂w, (4.18)
where the intermediate variable XA represents the locations of the nodes in the panel
mesh, and the variable FA represents the forces on the aerodynamic surface. With the
exception of ∂AT/∂XA, each of the terms on the right-hand-side is sparse and can be
solved efficiently using matrix-free methods [45].
The sensitivities computed using the adjoint method are verified by comparing them
with finite difference results. By taking the directional derivative of the function, f , with
respect to a vector of design variables, x, along a randomly generated vector b, one can
Chapter 4. Aerostructural Optimization 100
check the sensitivities for all variables with a single finite difference calculation. Equation
(4.19) gives the relationship between the exact directional derivative taken along b, and
the second-order central difference approximation.
∂f(x)
∂x· b ≈ f(x + hb)− f(x− hb)
2h(4.19)
Reducing the step size, h, improves accuracy, however if h is too small, subtractive
cancellation error reduces the accuracy of the approximation. The maximum achievable
accuracy is also limited by the tolerance to which the Newton–Krylov method is converged
in the evaluation of f(x + hb) and f(x− hb). Furthermore, the accuracy of the adjoint
sensitivity is limited by the tolerance used in solving for the adjoint vector. In the results
presented, both the adjoint and residual vectors are solved to a tolerance of ε = 10−7.
Under these conditions, the best agreement between the sensitivity analysis methods
was achieved with a step size of h = 10−7. Table 4.1 shows the adjoint and central
difference sensitivity results for a sample aerostructural analysis that included 19159
density variables, ρ; 8 twist variables, αj; one angle of attack variable, α0; and a SIMP
penalization variable, p, for a total of n = 19169 variables.
Drag Lift Compliance
Central difference method 55.95073679 1003.612219 954479.4238
Adjoint method 55.95075349 1003.616505 954475.8612
Adjoint wall time (s) 13.30 11.49 9.88
Table 4.1: Sample adjoint sensitivity results for the aerostructural problem
The results verify the adjoint sensitivity analysis as there is strong agreement with
the finite difference method for all three test functions. The table also illustrates the
Chapter 4. Aerostructural Optimization 101
computational efficiency of the adjoint method. Once the aerodynamic and structural
state vectors have been solved for, all n sensitivities for a given function can be calculated
with just one solution of the adjoint equation, which in this case required an average
of 11.56 seconds per function. Therefore, the total time required to solve all adjoint
sensitivities is on par with the time required to solve the aerostructural system, which
took 33.52 seconds to converge. This is in stark contrast to finite-difference methods,
which require O(n) full aerostructural analyses for each function being differentiated.
4.2 Aeroelastic Tailoring
Structures designed to support aerodynamic surfaces present a unique challenge since the
natural deflection of the structure causes the aerodynamic surface to change shape. If
this coupling is not taken into consideration during the design process, the deflection is
likely to negatively impact the efficiency of the aerostructural system, as it will deviate
from its optimized configuration when subjected to loading. Recognizing this fact, many
engineers have sought to harness this effect by tailoring the design so that the reduction
in performance due to aeroelastic effects is minimized or, if possible, reversed.
One of the most prominent examples of successful aeroelastic tailoring is the Grum-
man X-29 aircraft [23]. This plane is unique in that its wings are swept forward, as
opposed to being swept back as is the case for most jet aircraft. Also, the wings are
mounted to the back portion of the fuselage, while canard stabilizers for pitch control
are mounted just behind the cockpit. This configuration provides increased maneuver-
ability. However, it is susceptible to aeroelastic divergence at high speeds. Due to the
forward-swept design, the deflection of the wing causes upward twisting and increased
loading at the tips As a result, the composite materials that comprise wing skin are
tailored through strategic arrangement of the ply angles, so that the upward twisting at
the tips is minimal. This creates the effect of having an extremely stiff wing, without the
Chapter 4. Aerostructural Optimization 102
increased material and weight penalty that would otherwise be required to achieve that
stiffness
Academic research on the tailoring of composite structures for forward-swept wings
dates back several decades [97, 51]. Authors have also applied the same principle to
a variety aerostructural problems including the design of turbine blades [93] and heli-
copter rotors [33]. In both examples, light-weight, flexible blades are designed for a small
amount of desirable twist using composite materials. In this section, aeroelastic tailoring
is performed on the CRM wing in order to achieve an aerodynamically optimal flying
shape. This experiment offers some useful insight into the interdependence between the
aerodynamic and structural behaviour of the wing, and underscores the importance of
including aeroelastic effects in the analysis. This insight is later used when implementing
and analyzing the aerostructural problem in Section 4.3, where the structural topology
is added to the set of design variables.
In this problem the aeroelastic response is tailored by varying only the eight jig
twist angles along with the angle of attack, which is used to enforce the lift constraint,
L = W , where W is the total weight of the plane. Coupled aerostructural analysis and
coupled adjoint sensitivity analysis are used to optimize the deflected shape of the wing
under aerodynamic loading. The aerodynamic efficiency of the wing is determined by
the magnitude of the drag force, which serves as the objective function in this problem.
For a wing in which the airfoil, planform and mass are all fixed, drag is minimized when
the spanwise lift distribution is elliptical. The lift at a given location along the span is
proportional to the local angle of attack, αlocal. Therefore, the optimizer will search for a
combination of jig twist angles that produces a deflected twist distribution, which results
in an elliptical spanwise lift profile.
The optimization problem and aerostructural analysis are performed at a cruise Mach
number of 0.74. This choice of Mach number is near the transonic flow regime, for which
swept wings are primarily designed. However, this speed is low enough to justify the use
Chapter 4. Aerostructural Optimization 103
of the panel method, which is not equipped to model shocks. The structural box has
uniform intermediate density ρ = 0.35 and no SIMP penalization. The cruise constraint is
enforced so that L = Wbaseline = 1964 kN. Drag is minimized with respect to the jig twist
angles, while the structural design is held constant. This is essentially an aerodynamic
shape optimization in which aeroelastic coupling is enforced. The resulting optimization
problem can be expressed mathematically as follows.
minαj
D
subject to: L = Wbaseline (4.20)
− 6 ≤ αji ≤ 6, i = 1, ...8
Ku− F(w) = 0
Figure 4.5 shows the rigid and flying configurations of the outer mold lines for the
baseline and optimized CRM wings. The corresponding twist distributions are shown
in Fig. 4.6. One can obtain a hypothetical measure of aerodynamic performance of the
jig shapes by performing a pure aerodynamic analysis and treating the wing as a rigid
body. By comparing the result of this analysis with that of the aerostructural analy-
sis, it is possible to quantify the effect of the aeroelastic deformations on aerodynamic
performance.
Figure 4.7 shows the lift distributions corresponding to the four twist distributions
shown in Fig. 4.6. These results represent two different wing designs (baseline and opti-
mized), each of which is analyzed and evaluated using two different approaches, a pure
aerodynamic analysis in which the structure is assumed to be rigid, and an aeroelastic
analysis, in which the structure deflects and is subject to aeroelastic coupling. The op-
timization uses aeroelastic analysis, therefore both the optimized and baseline wings are
suboptimal in their rigid state.
For both the baseline and optimized wings, the upward bending of the structure
Chapter 4. Aerostructural Optimization 104
X
Y
Z
rigid configuration
flying configurationX
Y
Z
X
Y
Z
(b) Baseline wing
X
Y
Z
(b) Aerodynamically optimized Wing
Figure 4.5: Baseline and optimized outer mold lines for the CRM; The flying configuration
is superimposed onto the rigid configuration
causes a negative (leading edge down) structural twist angle, αs, that increases in mag-
nitude toward the tip. This reduces the local angle of attack in the outer sections of
the wing, thus pushing the load rootward. For the baseline CRM wing, this bend-twist
coupling causes the lift distribution to deviate further away from the optimal elliptical
lift distribution. As shown in Fig. 4.6, optimization increases the jig twist angle in the
outer sections of the wing in order to compensate for the structural deflection. As a
result, when the aerodynamic loads are applied, the wing deflects into a configuration
that produces a nearly elliptical lift distribution. Because the load is pushed out toward
the tip in the optimized case, this wing exhibits greater tip deflection under aerodynamic
loading as shown in Fig. 4.5
Table 4.2 contains the drag values of for each of the four cases analysed. The table
shows that the aerodynamic shape optimization yields an 8.5% drag reduction when
compared with the baseline design. It is also interesting to note that this optimized
structure is tailored so that the aerodynamic load applied under the specified flight
condition causes the wing to deflect into a more efficient flying shape. This is inferred
from the fact that the drag on the optimized wing is higher when the wing is assumed to
be rigid than when structural deflection is included in the analysis. This illustrates the
Chapter 4. Aerostructural Optimization 105
0 5 10 15 20 25 30−2
0
2
4
6
8
10
12
z[m]
α [°
]
baseline jig twistbaseline deflected twistoptimized jig twistoptimized deflected twist
Figure 4.6: Twist distribution for the aerodynamically optimized CRM wing
Chapter 4. Aerostructural Optimization 106
0 5 10 15 20 25 300
10
20
30
40
50
60
70
80
90
100
110
z [m]
Lift
[kN
/m]
baseline wing; aerodynamic analysisbaseline wing; aeroelastic analysisoptimized wing; aerodynamic analysisoptimized wing; aeroelastic analysiselliptical distribution
Figure 4.7: Lift distribution for the aerodynamically optimized CRM wing
Chapter 4. Aerostructural Optimization 107
Aerodynamic Analysis Aeroelastic analysis
(rigid structure) (flexible structure)
Baseline wing 0.014868 (−4.9%) 0.015642 (0.0%)
Optimized wing 0.014412 (−7.9%) 0.014303 (−8.5%)
Table 4.2: Drag coefficient results (CD) for the baseline and optimized wing (The quantity
in parentheses represents the percentage of improvement over the baseline wing)
impact of aeroelastic coupling on the optimization process and the eventual performance
of the optimized design. However, as demonstrated in the section that follows, this
approach is still insufficient to achieve an aerostructurally optimal design, which can
only be achieved by optimizing both the structure and the aerodynamic shape together.
4.3 The Aerostructural Problem
The aerostructural optimization procedure expands upon the aeroelastic optimization
framework described above by including the structural design as part of the optimization
problem. Previous studies on this subject have performed topology optimization of struc-
tures subject to aeroelastic coupling with a fixed outer moldline [35, 58, 82], the method
and subsequent investigation presented here are unique in that the topology optimization
is carried out as part of an MDO algorithm in which the structural topology is optimized
together with the aerodynamic shape of the wing.
In order to quantify the benefit of the MDO method, the results are compared with
those produced using a sequential optimization procedure that is analogous to that which
is followed in the studies mentioned above. Other studies have investigated the merits
of MDO as compared with sequential optimization [19, 55], however these studies pa-
rameterized the structural design using only sizing variables with the topology remaining
Chapter 4. Aerostructural Optimization 108
fixed. The current study seeks to validate the hypothesis that performing MDO in order
to obtain an optimized topology can offer significant advantages that are in addition of
those which have been observed in MDO problems with sizing variables.
4.3.1 Problem Formulation
The choice of objective function should be one that captures the efficiency of both the
aerodynamic and structural aspects of the design. The function should also have practical
significance in that its minimization should yield tangible benefits for the user of the
optimized product. The drag function, D, meets both these criteria. The amount of
drag an aircraft experiences is highly dependent on the aerodynamic shape, which, in
this case, refers the twist distribution. For the cruise condition in which L = W , the total
drag on the wing is also dependent on the structural weight, since a heavier structure
necessitates an increase in lift, which leads to higher induced drag. Furthermore, for an
aircraft in cruise, drag is equal, in magnitude, to the thrust force, T , provided by the
engines.
T = D (4.21)
For a jet engine, the amount of thrust generated is directly proportional to the rate of
fuel consumption, m, with the proportionality constant, c, given by the engine’s thrust-
specific fuel consumption (TSFC).
m = cT (4.22)
Using these two equations, one can derive the following expression for the amount of fuel
burned per unit of distance travelled.
Chapter 4. Aerostructural Optimization 109
m
V=cD
V, (4.23)
where V is the velocity of the aircraft. Therefore, for a constant velocity and constant
TSFC, minimizing drag is equivalent to minimizing fuel consumption per unit distance.
This translates directly into reduced emissions, and cost savings for the operator of the
aircraft.
In the results that follow, drag is minimized with respect to the jig twist angles, αj,
as well as a set of structural design variables, xs, which get passed through a density
filter to obtain the nodal material densities used in the SIMP formulation. The resulting
optimization problem can be written as follows.
minxs,αj ,α0,p
D
subject to: Lcruise = W
Lmaneuver = 2W
Cmaneuver < Cmax (4.24)
(p− p∗)r = 0
0 < ρmin ≤ xs ≤ 1
The constraint L = W is enforced, with W accounting for the total mass of the aircraft
including the structural weight of the wing, Wwing, plus some fixed weight, Wfixed, used
to account for the fuselage and all other parts of the plane that are not subject to
optimization. In the results presented below, the fixed weight has been set to Wfixed =
7 × 105 N. This choice is based on the design of Boeing 777-300 aircraft, whose wing
size and geometry are similar to that of the CRM, and whose operating empty weight
(OEW) is 1572900 N. To find the fixed weight, an additional 400000 N is added to the
OEW to account for payload. This number is then divided by two (since only one wing
Chapter 4. Aerostructural Optimization 110
is being analyzed), which results in a total weight of approximately 106 N for half the
aircraft. It is estimated that the fuselage with payload accounts for 70% of this total
weight, which leads to a fixed weight of 7 × 105 N. The remaining 3 × 105 N are due
to the wings and engines. Neither the engines nor the fuel weight are included in the
analysis. Because the fuel is housed inside the wings and the engines are mounted to the
bottom of the wings, the weight of the fuel and the engines opposes the lift force, causing
inertial relief. Therefore, although the fuel and the engines increase the overall weight of
the plane resulting in increased drag, neither of these will cause a significant net increase
in the total force acting on the wing itself.
The constraint on the structural compliance, Cmaneuver < Cmax, is used to ensure some
requisite stiffness. When aerostructural analysis is performed on the baseline wing struc-
ture with no stiffness penalization, this compliance corresponds to a vertical tip deflection
of approximately 2 m, which is equal to 6.7% of wingspan. In topology optimization prob-
lems, this constraint often takes the place of material failure and buckling constraints, due
to the inherent difficulty of enforcing local constraints in topology optimization schemes,
as demonstrated in Chapter 3. Once topology optimization has been used to generate a
conceptual structural layout, stress constraints and buckling constraints can later be used
to help determine the optimal shape and sizing of the structural members, as described
by Grihon et al. [35].
The compliance constraint is enforced for a 2g maneuver condition since typically a
higher g maneuver is required to cause structural failure. as this is a circumstance under
which structural failure is more likely to occur than during cruise. During each opti-
mization iteration, two separate aerostructural analyses must be performed, one for the
cruise condition, where the optimizer enforces L = W , and one for the maneuver condi-
tion where the constraint L = 2W is enforced. The optimizer satisfies these constraints
by varying the angle of attack α0 associated with each flight condition. Therefore both
angles of attack are included as design variables. Because the two cases are independent
Chapter 4. Aerostructural Optimization 111
Optimizer
Aerostructural Solver
Aerodynamic
Analysis
Structural
Analysis
u F
xs, p
Structural
Analysis
Aerodynamic
Analysis
F u
SIMP Module
Cruise Case Maneuver Case
Figure 4.8: MDO architecture for the aerostructural optimization problem
of one another, they are computed in parallel. The third and final nonlinear constraint
is used to implement the continuation method so that the SIMP penalization parameter,
p, approaches its target value, p∗. Figure 4.8 shows a flow chart detailing the flow of
information and the distribution of tasks between the various computational modules.
In order to achieve the effect of having a minimum skin thickness at the top and
bottom surfaces of the wingbox, the material densities of the elements along these faces is
constrained to remain above ρmin = 0.15 (compared to ρmin = 10−3 for interior elements).
Chapter 4. Aerostructural Optimization 112
Additionally, the penalization penalization applied to these elements is half that of the
interior elements. Therefore, the minimum penalized density for the top and bottom
skin elements is 0.058. This ensures that the stiffness at the structural nodes to which
the external forces are applied, remains above a minimum threshold. This guarantees
that material is always present to support the surface load and transfer that load to the
primary structure, which is attached to the fuselage. This also prevents divergence of
the aerostructural analysis.
4.3.2 Sizing Optimization Example
The aerostructural optimization problem is unique and distinct from the aerodynamic
optimization problem (4.20). Although the objective function is the same, the inclusion
of the structural constraint causes the optimizer to seek out a design whose aerodynamic
shape differs significantly from that of the aerodynamically optimized wing. Here, this
concept is demonstrated with an example in which the aerostructural problem is solved
using a sizing optimization approach. In this example, the topology of the structure is
fixed and the design is that of a conventional rib-spar lay-up, which is typical of transport
jet aircraft [74]. The ribs and spars are modeled using a structured mesh of linear shell
elements. In this case, the structural design variables are the thicknesses of the shell
elements. The aerodynamic shape is parameterized in using the method described in
Section 4.1.2. Figure 4.9 shows the rib-spar structural model and the finite element
discretization.
The optimization was carried out using the problem specifications given in Ap-
pendix B, which contains the values the parameters (i.e., atmospheric conditions, flight
data, initial conditions, and material properties) used in this and all other aerostruc-
tural problems. Figure 4.10 shows the optimized lift distributions for the aerostructural
problem. The graph contains plots for both the cruise and maneuver conditions. Both
lift distributions are normalized with respect to the total lift during cruise so that areas
Chapter 4. Aerostructural Optimization 113
Figure 4.9: The rib-spar structural model with finite element mesh
underneath the cruise and maneuver plots are 1 and 2 respectively. Under both sets of
flight conditions, the wing is more root-loaded than would be the case for aerodynami-
cally optimal wing. It is assumed during the optimization that the compliance constraint
is inactive in the during cruise flight and therefore, the constraint is only enforced for
the maneuver condition. This structural constraint encourages a load distribution that
places a greater portion of the load near the root of the wing. This reduces the structural
demand placed on the wing, thus allowing for a lighter structures, which, in turn, results
in reduced drag.
4.3.3 Sequential Optimization
In order to analyze and quantify the impact of the MDO framework, the problem de-
scribed in 4.24 is also solved using a sequential optimization procedure. This method is
carried out in two stages. In the first stage, drag is minimized with respect to the jig
twist only, while the material density remains constant and uniform at ρ = ρ0. Because
Chapter 4. Aerostructural Optimization 114
0 5 10 15 20 25 300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
z [m]
L loca
l/Lto
tal
CruiseElliptical (1g)ManeuverElliptical (2g)
Figure 4.10: Aerostructurally optimized lift distributions for the CRM wing with a fixed
rib-spar topology
the structure is not allowed to change, only the cruise constraint L = W is enforced.
Once a minimum is achieved, this optimized jig twist distribution is used in a second
optimization where these angles are kept constant and the nodal densities are allowed to
vary. Aerostructural coupling is included in the analysis during both stages. Figure 4.11
shows the data flow for the sequential algorithm.
The algorithm described in Fig. 4.11 is implemented using two approaches. In the
first approach, which is hereafter referred to as approach A, the shape optimization is
performed using a pure aerodynamic framework, which is identical to the procedure
implemented in Section 4.2. In approach B, the compliance constraint is enforced during
the shape optimization. Therefore, the optimizer must tailor the twist distribution,
without varying the topology, to ensure the aerodynamic load do not cause a violation
of the compliance constraint.
Chapter 4. Aerostructural Optimization 115
Optimizer
Aerostructural Solver
Structural
Analysis
Aerodynamic
Analysis
u
F
Aerodynamic Shape Optimization
Optimizer
Aerostructural Solver
Aerodynamic
Analysis
Aerodynamic
Analysis
Structural
Analysis
Structural
Analysis
u
u
F
F
SIMP Module xs, p
,
Cruise
Case:
Maneuver
Case:
Topology Optimization
Figure 4.11: Algorithm architecture for the sequential optimization procedure
4.4 Results and Discussion
Table 4.3 shows the optimized drag values for the sequential and MDO methods. For the
aerostructural topology optimization problem, the MDO design achieved a 42% lower
drag than the design produced by the standard sequential optimization algorithm. In
the case where the compliance constraint was included in the first phase of the sequential
optimization procedure (sequential B), the resulting sequential design experienced 21%
more drag than the MDO result. The numbers indicate that this is largely due to the
MDO result has the lightest structure, followed by the sequential B result. (It should be
noted that the weight values listed in Table 4.3 refer only to the weight of the wingbox.
The total weight of the plane is the sum of this value plus the fixed weight.)
The normalized lift distributions for each design under the maneuver load are shown
in Figure 4.12. In each case, the lift distribution is normalized with respect to its own to-
tal lift value so that the area under all curves is equal to 1. When the load near the tip of
the wing is reduced, this causes a reduction in the bending moment throughout the span.
By exploiting this aerostructural trade-off, the MDO and sequential B algorithms allow
for a lighter structure, which reduces the amount of lift-induced drag acting on the wing.
Chapter 4. Aerostructural Optimization 116
CD Weight [kN] CL Span efficiency Wall time (hh:mm)
Sequential optimization A 0.00588 572 0.404 0.984 9 : 33
Sequential optimization B 0.00414 312 0.322 0.884 10 : 24
MDO 0.00340 253 0.303 0.955 13 : 12
Table 4.3: Comparison of the optimized sequential and MDO results for the aerostruc-
tural topology optimization problem
By contrast, the sequential A lift distribution is approximately elliptical during both the
maneuver and cruise condition, which is shown in Fig. 4.13. From Fig. 4.7, it is known
that in the absence of a compliance constraint, the shape optimization performed dur-
ing the sequential algorithm produces an elliptical lift distribution. Therefore, Fig. 4.13
demonstrates that the shape of the lift distribution plot remains largely unchanged by
the structural optimization phase of the sequential algorithm. When the compliance con-
straint is enforced during the shape optimization, this mitigates the problem by pushing
the load rootward in order to satisfy the constraints. Therefore, in the case of sequen-
tial algorithm A, the shape optimization turns out to be counterproductive as it pushes
the load in the outward direction as opposed to moving it inward which leads to an
aerostructural optimum.
The convergence history for the MDO problem (shown in Fig. 4.14) further supports
the claim that most of the reduction in drag is due to the reduction in structural mass.
The figure illustrates how closely the aerodynamic drag is tied to the structural mass, as
the plots for the two quantities follow very similar paths.
Returning to Table 4.3, the numbers indicate that the reduction in weight alone does
not account for the superior performance of the MDO design. Based on the CL values for
Chapter 4. Aerostructural Optimization 117
0 5 10 15 20 25 300
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
L loca
l/Lto
tal
z [m]
Sequential ASequential BMDOElliptical
Figure 4.12: Normalized lift distributions for the aerostructurally optimized wings under
the maneuver load
0 5 10 15 20 25 300
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
z [m]
L loca
l/Lto
tal
Sequential ASequential BMDOElliptical
Figure 4.13: Normalized lift distributions for the aerostructurally optimized wings under
the cruise load
Chapter 4. Aerostructural Optimization 118
0 20 40 60 80 100 120 140 160 180 200
0.004
0.008
0.012
0.016
0.02
Iteration Number
Dra
g C
oeffi
cien
t
0 20 40 60 80 100 120 140 160 180 200
2
4
6
8
10
x 104
Mas
s [k
g]
Figure 4.14: Convergence history of the drag and mass functions for the MDO case
each design, it is evident that, when compared with the sequential B design, some of the
reduction in drag is due to improved aerodynamic efficiency. The span efficiency factor
is a measure of how much induced drag a wing produces for a given lift and a given span.
For a wing with an aspect ratio AR, the induced drag co-efficient, CDi, can be expressed
in terms of the lift coefficient, CL, the aspect ratio, AR, and the span efficiency, e, as
follows.
CDi=
C2L
πeAR(4.25)
As shown in Table 4.3, the sequential A design has the highest span efficiency. This is
expected since this design has a nearly elliptically lift distribution. However, the sequen-
tial B design has a much lower span efficiency since its aerodynamic performance has
been sacrificed in the interest of having a lighter structure. The MDO design, although
not as aerodynamically efficient as the sequential A design, has a higher span efficiency
than the sequential B design. This can be seen in the cruise lift distribution plots for
Chapter 4. Aerostructural Optimization 119
each solution. The MDO lift distribution is closer to the elliptical plot than the sequen-
tial B lift distribution. This is in spite of the fact that, for the maneuver condition, the
lift distributions for these two designs are much closer to one another. This suggests an
additional advantage to the MDO approach. MDO yields an aeroelastic tailoring effect
in which the displacement of the structure is tailored to produce a more efficient shape.
Ideally, one would like a root-heavy load distribution during maneuver, and an elliptical
load distribution during cruise. However, because the jig shape remains fixed as the
plane transitions from one condition to the other, achieving this separation between the
two load distributions is difficult. From the lift distribution plots, it is clear that only
the MDO algorithm was able to generate any useful separation between its cruise and
lift distributions by tailoring the structural response to each load case.
Table 4.3 also shows the computation time required for each algorithm. In all three
cases, the algorithms were run on a 96-processor, distributed-memory cluster with each
core operating at 2.53GHz. All three algorithms had similar convergence times, how-
ever, the MDO algorithm was slightly slower than the sequential algorithms due to the
added computational cost of having addtional design variables (i.e. jig shape plus SIMP
densities), and having to compute sensitivities with respect to each of them.
The convergence histories for compliance and the SIMP penalization factor are shown
in Fig. 4.15, which illustrates the interdependence between these two values. During
the early stages of the optimization process, the penalization factor increases smoothly,
while the compliance remains constant at its upper bound. This plot is a testament to
the effectiveness of the optimizer-based continuation method. Using the optimizer, one
can achieve a steady, monotonic increase in the penalization factor while maintaining
stability in the objective and constraint functions.
Figures 4.17 to 4.19 show the optimized material distribution inside the wingbox for
each of the three algorithms tested. The figures showing the contour plot ρp∗
provide
a graphic representation of the relative stiffness at the various locations throughout the
Chapter 4. Aerostructural Optimization 120
0 50 100 150 20050
100
150
200
250
Iteration Number
Com
plia
nce
[kJ]
0 50 100 150 2000
1
2
3
4
Pen
aliz
atio
n fa
ctor
Figure 4.15: Convergence history of the compliance function and the SIMP penalization
factor for the MDO case
domain, since the finite element analysis uses this quantity (the relative material density
raised to the power of p∗) to determine the effective Young’s modulus of the material at
a given location. Sub-figures 4.17(c), 4.18(c) and 4.19(c) show the contour of ρp∗
with
the coloring mapped to a logarithmic scale. These plots are designed to reveal areas in
the structure where the relative material density is non-void (i.e., ρ > 0.01), but is low
enough (i.e., ρ < 0.1) that it is not visible when plotted on the linear scale. Material
densities in this range are sufficient to transfer loads between structural components, and,
as in the case of the skin elements, these regions can be integral to the overall viability
of the structure. Sub-figures 4.17(d), 4.18(d) and 4.19(d) show the distribution of the
design variable x subject to filtering, but with no penalization.
The optimized structures are each dominated by a single spar that extends from the
root toward the tip of the wing. Furthermore, in all three cases, the spar starts out in
the trailing half of the chord and extends toward the leading edge as it moves outward
Chapter 4. Aerostructural Optimization 121
along the span. This means that the sweep angle of the spar is smaller than that of the
overall wingbox, which lessens the bend-twist coupling and reduces compliance.
From the contour slice plots, one sees that in both the MDO result and the sequential
B result, the optimizer has hollowed out the interior of the wing, which is fully void. Low
density material has been distributed along the faces at the leading edge and trailing
edge of the wingbox. This material serves as a shear web, transferring load between the
top and bottom skin. By contrast, the sequential A algorithm retains some intermediate
density material in the interior of the structure. This is especially true near the tip,
where the shear strain is the highest. These three-dimensional regions of intermediate
density are undesirable, since they are structurally inefficient and provide no insight into
how the eventual structural members should be configured. This distinct feature of the
sequential A structure is due to its higher mass, which results in a greater load being
placed on the structure.
These figures further demonstrate that the inclusion of a compliance constraint during
the shape optimization phase, improves the sequential algorithm significantly. Once this
improvement has been made, the optimized design is much closer to that of the MDO
result. Nonetheless, all three structures deviate significantly from the traditional rib-spar
configuration typically used in the design of wings. The topologies obtained are consistent
with previous efforts to optimize structures subject to distributed loads [32]. When loads
of this nature are applied, it is common that the optimized structures contain some
intermediate density material that acts as a secondary structure transferring the applied
load to a primary structure, which is connected to the supports [32]. This phenomenon
is related to the mesh-dependency problem discussed in Chapter 2. In this case the
mesh-dependency is caused by the non-uniqueness of the solution [79]. This problem
often arises in cases where the structure is subject to distributed uni-axial loads (shown
in Fig. 4.16), which permit an infinite number of equally desirable solutions [79]. In this
situation, the optimizer selects for intermediate density regions, which are analogous to
Chapter 4. Aerostructural Optimization 122
having an infinitely fine discretization. This problem cannot be prevented by using a
finer mesh, however one strategy is to have the designer select a discrete design based on
manufacturing preferences.
(a) Geometry and loading conditions (b) Coarse solution (c) Fine solution
Figure 4.16: Example of non-unique solutions due to uni-axial distributed loading
Another reason for the disparity between the results presented here, and conventional
configurations is that no buckling constraints were enforced. If buckling were to be in-
cluded in the optimization, it is likely that rib-like structures would appear in order to
prevent buckling of the skin. As has been the case with previous aerostructurally opti-
mized topologies [35], any optimized structures produced by this new algorithm would
require some post-processing and interpretation in order to generate a feasible, manufac-
turable wing.
Chapter 4. Aerostructural Optimization 123
(a) Block contour plot with finite element mesh; ρp∗ (b) Slice contour plot; ρp∗
(c) Logarithmic contour plot; ρp∗ (d) Density field; ρ
Figure 4.17: Material distribution for the CRM wing optimized using sequential algorithm
A
Chapter 4. Aerostructural Optimization 124
(a) Block contour plot with finite element mesh; ρp∗ (b) Slice contour plot; ρp∗
(c) Logarithmic contour plot; ρp∗ (d) Density field; ρ
Figure 4.18: Material distribution for the CRM wing optimized using sequential algorithm
B
Chapter 4. Aerostructural Optimization 125
(a) Block contour plot with finite element mesh; ρp∗ (b) Slice contour plot; ρp∗
(c) Logarithmic contour plot; ρp∗ (d) Density field; ρ
Figure 4.19: Material distribution for the CRM wing optimized using MDO
Chapter 5
Conclusions
5.1 Summary of Contributions and Findings
This thesis presents several novel algorithms and mathematical tools for performing topol-
ogy optimization of aircraft wings. Although topology optimization is fairly mature as
a discipline, the vast majority of studies involving topology optimization are limited to
benchmark test cases that bear little resemblance to real-world engineering problems.
This study was partially aimed at expanding the capability of topology optimization
by introducing robust techniques for designing complex, three-dimensional aeroelastic
structures. Several strategies were also introduced for overcoming the various numerical
challenges associated with the two most popular topology optimization approaches: the
SIMP method and the level set method.
Chapter 3 focused on the level set method, which offers the advantage that its solu-
tions are generally independent of the finite element mesh used to model the structure.
However, there is a trade-off in that the level set method is much less flexible than its
element-based counterparts, including SIMP. Consequently, previous studies on level set
methods have dealt almost exclusively with rectangular structural domains and uniform
finite element meshes. Due to the contoured profile of the standard wingbox, this ap-
126
Chapter 5. Conclusions 127
proach is insufficient for optimizing wing structures.
In order to address this issue, an isoparametric formulation was developed, which
allows users to apply the level set method to problems involving irregularly shaped do-
mains and non-uniform finite element meshes. Isoparametric quadrilateral or hexahedral
elements were used to compute the structural response and evaluate the shape sensi-
tivities along the material boundary. Using the Jacobian transformation corresponding
to each element’s shape functions, these sensitivities are then mapped to computational
space, where the Hamilton–Jacobi equation is solved on a uniform, Cartesian grid. The
one-to-one mapping between the shape sensitivities computed in physical space and the
corresponding sensitivities expressed in computational space maintains the robustness of
the algorithm as there is no need for interpolation or smoothing of the sensitivities.
The isoparametric formulation was derived for several objective and constraint func-
tions including mass, compliance, and global von Mises stress. The method was tested
on a series of two-dimensional benchmark problems and the results were compared with
the SIMP method, which is readily applicable to problems involving non-uniform finite
element meshes with no need for modification or adaptation of the method. Results
showed that the isoparametric level set method is competitive with the SIMP method
both in terms of the final value of the objective function, and the computational time
required to reach convergence. Finally, the method was used to optimized a structural
wingbox subject to fixed pseudo-aerodynamic loads. The three-dimensional structure
was optimized for minimum compliance subject to a weight constraint. Results con-
firmed that the method is also effective when applied to three-dimensional problems, and
the optimization converged to produce a feasible structure with well-defined members
that formed a spar-like sheer web, providing the structure with torsional and bending
stiffness.
Chapter 4 looked at the aerostructural problem in which topology optimization was
performed as part of a larger MDO framework. Here the aerodynamic loads were com-
Chapter 5. Conclusions 128
puted using a three-dimensional panel method which was coupled to the structural model
in order to account for aeroelastic effects. Whereas previous studies on aerostructural
topology optimization limited the design domain to two-dimensional plates, this study
treated the wingbox as a three-dimensional domain, thereby allowing the optimizer to
distribute material throughout the planform as well as through the thickness of the wing.
The other major contribution of this section was to perform full MDO in which aerody-
namic shape of the wing was optimized concurrently with the structural topology.
The baseline design of the wing was taken from NASA’s common research model,
which uses a swept supercritical wing typical of long-haul transport aircraft. The aero-
dynamic shape was parameterized using a series of jig twist variables defining the local
jig twist angle at equally spaced locations along the span. The structural topology
was parameterized using a node-based SIMP formulation, with 8-node hexahedral finite
elements. The aerodynamic loads and structural displacements were coupled using a
consistent and conservative approach.
The wing was optimized for minimum drag during cruise flight with a constraint on
the maximum compliance due to a 2g maneuver condition. In addition to the full multi-
disciplinary optimization approach, two sequential optimization algorithms were tested.
In the first algorithm, the aerodynamic shape was optimized purely for minimum drag,
which the resulting jig shape being used as a fixed design feature in the subsequent struc-
tural optimization. This approach is analogous to the method implemented in previous
aerostructural topology optimization studies where the structural topology is optimized
using a predetermined, fixed jig shape. An enhanced sequential optimization proce-
dure was also implemented. In this second algorithm, the compliance constraint from
the MDO problem was enforced during both the shape optimization and the structural
optimization. All three results were compared in order to quantify the advantages of
combining aerodynamic shape optimization with topology optimization.
The results were consistent with previous studies on MDO for aerostructural design
Chapter 5. Conclusions 129
where it was shown that the MDO design outperforms the design obtained using sequen-
tial optimization. In this case, the MDO design had a minimum drag value that was 42%
lower than that of the design obtained using the standard sequential procedure. While
the sequential result exhibited an aerodynamically optimal lift distribution, the struc-
tural optimization was forced to use extra material in order to satisfy the compliance
constraint, thereby increasing induced drag. The enhanced sequential algorithm offered
a significant improvement over the standard algorithm, yielding a final design that was
closer to the MDO result, but this design still had 25% higher drag than that which
was produced by MDO. The enhanced sequential result and the MDO result deviated
significantly from the elliptical lift distribution in order to place a greater portion of the
load near the root of the wing which reduces bending and compliance. This allows for
a lighter structure, which reduces the amount of lift-induced drag. However, the MDO
result also had a significantly higher span efficiency than the enhanced sequential result
due to MDO’s ability to achieve aeroelastic tailoring of the structural deflection under
the two load cases considered. This illustrates MDO’s unique ability to exploit the inter-
dependence of the aerodynamic and structural behaviour of the wing in order to produce
an optimal aerostructural trade-off.
5.2 Significance of Findings
The level set method is a useful and elegant tool for solving topology optimization prob-
lems. However, the method remains limited in terms of the range of problems for which
it can be used. Because the conventional level set method uses a Cartesian grid to solve
the Hamilton-Jacobi equation, the vast majority of examples from the literature include
only structures that are confined to rectangular domains and modeled using uniform fi-
nite element meshes. This excludes a large and important class of structures that, like
the wingbox, are not rectangular, and therefore, must be modeled using a non-uniform,
Chapter 5. Conclusions 130
body-fitted mesh.
In the past this issue was addressed using one of two approaches. One could use
a fixed, non-uniform mesh and interpolate the shape sensitivities in order to produce
an approximate continuous sensitivity field, which was then sampled at the points in
the Cartesian grip upon which the Hamilton-Jacobi equation was solved. However, this
greatly reduces the accuracy of the sensitivity values that get passed to the Hamilton
Jacobi equation, which could slow down convergence or prevent it altogether. Alterna-
tively, one could re-mesh the structure after each optimization step so that the element
boundaries always coincide with the material boundary. However, this approach sig-
nificantly increases the computational cost of the algorithm, especially when solving
three-dimensional problems.
The isoparametric level set method addresses both these challenges by providing an
accurate and robust means of mapping the sensitivities from physical to computational
space, while avoiding the increased computational cost associated with re-meshing. The
ability to apply the method to stress-based design problems is also useful as it allows a
designer to consider failure constraints during the conceptual design process. This impor-
tant feature is omitted from most topology optimization schemes. These contributions
are significant as they provide designers with a convenient method for solving a class
of problems that more closely resemble those encountered in a real-world engineering
context, and that, until now, have been largely ignored by researchers.
Among this group of unexplored problems is the design of aeroelastic structures. With
only a handful of previous examples present in the topology optimization literature, these
constitute a tiny fraction of the topology optimization research that has taken place.
The existing examples all share several shortcomings. Most treat the design domain
as a two-dimensional region, with no effort made the optimize the through-thickness
material distribution. Furthermore, although most previous examples include some form
of aerostructural coupling, the structural topology was optimized separately from the
Chapter 5. Conclusions 131
aerodynamic shape in accordance with the sequential optimization paradigm.
As the results demonstrate, the combination of structural topology and aerodynamic
shape design yields designs that are far superior to those produced using sequential
optimization. By performing aerostructural optimization using the MDO framework im-
plemented in Chapter 4, one can achieve an optimal trade-off between structural and
aerodynamic performance, and exploit the interplay between the two disciplines. More-
over, the use of a fully three-dimensional wingbox represents a significant step forward
in the area of topology optimization of wing structures. The unique three-dimensional
design parameterization of the structure implemented in this study, is useful as it allows
for the optimization of the through-thickness design. This is not possible with the two-
dimensional models used in previous studies [82]. Because the through-thickness design
determines that determines the second moment of area and the torsional moment of the
structure, the two-dimensional approaches are limited in the amount of insight they can
provide regarding the optimal design of the three-dimensional structure.
Examples from the aircraft industry have already shown that including topology
optimization in the design cycle, along with sizing optimization, can generate significant
weight savings, even when using relatively rudimentary topology optimization methods
[35]. Advancements in the optimization technique, such as those presented in this study,
provide the optimizer with a more detailed picture of the feasible design space that more
accurately reflects the physics of the aircraft. This can lead to additional weight savings
as well as improvements in aerodynamic performance, which entails greater fuel efficiency.
When combined with improvements in propulsion, aerodynamic shape, operations, and
non-conventional airframe configurations, these advancements will ultimately contribute
to lower emissions and more environmentally-friendly aircraft.
Chapter 5. Conclusions 132
5.3 Recommendations and Future Work
There are a number of features that could be added to the current optimization frame-
work in order to improve the viability of the optimized designs from an industry stand-
point. One of the most important of these is the inclusion of failure constraints based
on yield stress and buckling. Although these are typically not considered until later in
the design process after an optimized topology has been generated, including these con-
straints during the topology optimization phase could lead to lighter structures. Careful
consideration would be needed to find a way to effectively enforce the thousands of lo-
cal constraints in a way that does not overwhelm one’s computational resources. This
could likely be achieved though the use of an aggregation method similar to the one
implemented in Chapter 3. Another addition that would be complementary to the incor-
poration of failure constraints is the use of more load cases. By identifying and simulating
the cases in the flight envelope that are most likely to cause a structural failure one can
improve the reliability and the viability of the optimized topology.
The method should also be expanded to include optimization of entire airframes.
Since the aerostructural behaviour of the tail and fuselage are naturally coupled to the
wing design, including these other components in the design problem is consistent with
the MDO philosophy and should yield all the usual benefits associated with the MDO
approach. The guiding principle is to include in the optimization procedure as many
aspects of the design as possible. Provided one has an effective strategy for handling local
minima, this approach allows the optimizer the opportunity to find a global optimum
that is often not reachable through experience and intuition alone.
Moreover, the more freedom afforded to the optimizer, the more likely it will be to
produce novel design concepts that deviate drastically from conventional designs. As a
potential future project it would be useful to spend some time interpreting and analyzing
the optimized topologies using discrete finite element models to see what feasible designs
can be gleaned. Using the three-dimensional approach introduced in this study, one
Chapter 5. Conclusions 133
could potentially uncover new concepts for structural configurations that outperform the
traditional rib-spar layout.
Appendix A
Compliant Mechanism Design
A.1 Problem Formulation
A compliant mechanism is a structure or device that achieves changes in configuration
through the bending of flexural hinges. The absence of moment-free hinges confers several
advantages from an engineering standpoint. Because they are often comprised of a single
material with no ’moving parts’ (in the conventional sense of the term) compliant mech-
anisms can be manufactured cheaply. Also, these devices require no lubrication, thus
eliminating the possibility of outgassing, which can be a liability for space applications
[72].
Compliant mechanisms are generally designed to transfer input forces or displace-
ments from one location to another. This can be accomplished by maximizing the dis-
placement of the structure at some output location. In this way, the compliant mechanism
design problem is analogous to the aeroelastic tailoring problem described in Chapter 4.
In both problems, one seeks to minimize or maximize an explicit function of the de-
flected shape of the structure, by strategically modifying the design of the unloaded
structure. Topology optimization is well-suited to handling this particular design prob-
lem, as several authors have demonstrated [53, 78]. This section illustrates how topology
134
Appendix A. Compliant Mechanism Design 135
optimization can be used to perform compliant mechanism design through the use of
a classic example. The gripper mechanism was among the first problems solved using
topology optimization [78], and it remains a popular benchmark problem for validating
new topology optimization techniques [53, 103]. However, in this example, an added
feature has been incorporated to further enhance the analogy with aerostructural design.
In the example presented, the applied loads are due to an electrostatic force. Therefore
the magnitude and direction of the applied loads are dependent upon the design and its
displacement due to electro-structural coupling.
The optimization problem is illustrated is Fig. A.1. The working domain is given
by a rectangle with a 16mm × 16mm square in removed from the centre of the right
edge, where the device will grip objects. A positive electrode is placed at the centre
of the left side of the domain. When turned on, this electrode will attract the two
negatively charged electrodes that are affixed to the top left and bottom left corners of
the domain, thus exerting an inward force at both points. The dimensionality of both
the optimization and the structural analysis problem can be cut in half using symmetry,
as shown in Fig. A.1(b).
(a) Problem domain (b) Finite element model with symmetry boundary conditions
Figure A.1: The electrostatic gripper problem
Appendix A. Compliant Mechanism Design 136
The magnitude of the electrostatic force Felec, is determined by Coulomb’s law, which
is given by
Felec =c2µ0
4π
q1q2r2
, (A.1)
where c is the speed of light, µ0 is the magnetic constant, q1 and q2 are the respective
magnitudes of the interacting charges measured in Coulombs, and r is the radial distance
between the two charges. Note that any electrostatic interaction between the two nega-
tively charged electrodes is assumed to be negligible. For compactness, Coulomb’s law is
hereafter expressed as
Felec =B0
r2(A.2)
where
B0 =c2µ0q1q2
4π. (A.3)
The electrostatic force acts in the radial direction defined by the relative locations of the
two charges. Because these locations change as the structure displaces under the load,
both the magnitude and direction of the electrostatic force are dependent on the design.
The above formula provides the governing electrostatic equation used in the multi-
disciplinary analysis of the electro-structural system. The equation can be written as
Felecx
Felecy
=
dxB((h/2+dy)2+d2x)1.5
(h/2+dy)B
((h/2+dy)2+d2x)1.5
, (A.4)
where dx and dy are the components of the structural displacement at the input location
measured in the reference frame fixed to the positive charge. Therefore, dx is equal to the
Appendix A. Compliant Mechanism Design 137
difference between the horizontal displacement at the input location, and the horizontal
displacement of the positive charge due to structural deflection. The components of the
electrostatic force, Felecx and Felecy , are treated as state variables. The above equation is
coupled to the structural governing equation K(ρ)u−F, as the input displacements are
taken directly form the displacement vector u. The coupled non-linear system is solved
using a block Gauss-Seidel method.
In order to achieve a gripping effect at the output location, the output displacement
is maximized. Although some flexibility is necessary for the mechanism to function, the
structure must also have sufficient stiffness so that it can effectively transfer the input
force to the output location. This is accomplished by enforcing a constraint on the
maximum displacement at the input location. Therefore, the optimization problem can
be expressed as follows.
minρ
uout
subject to: dy < dinmax
K(ρ)u− F = 0 (A.5)
Felec = Felec(dx, dy)∑e
ρe = 0.4 ∗ ne
(A.6)
Note that in the above problem, the output displacement is minimized since the down-
ward direction is defined as negative.
A.2 Numerical Results
The optimization problem was solved using the optimality criteria method introduced in
Chapter 2. The non-linear constraint on the input displacement was was enforced using
Appendix A. Compliant Mechanism Design 138
Figure A.2: Optimized material distribution for the gripper mechanism with electrostatic
actuation
the adaptive Lagrangian method described in section 3.5.4. The electrostatic constant
was chosen as B0 = 0.125N ·m2, and the sprint constant for the output spring was chosen
as ks = 10kN/m. The structure has material properties E = 5×104Pa and ν = 0.3, with
thickness in the z-dimension measuring 2mm. The input displacement was constrained to
be less than 1cm. The structure was parameterized using element-based SIMP densities,
with a density filter for eliminating checkerboarding. The domain was discretized using
a uniform finite element mesh containing 60× 84 square, bilinear elements.
Figure A.2 shows the material distribution of the optimized gripper in pixel form,
while Fig. A.3 shows the configuration of the mechanism in the ’off’ and ’on’ positions.
This figure demonstrates that topology optimization can be used to tailor the deflected
shape of a structure without altering the shape of the working domain. In fact, from
Table A.1, it is clear that, not only has the optimization procedure produced an increase
in the magnitude of the output displacement, but in doing so, it has actually reversed
direction of the output displacement in the baseline (initial) structure.
Appendix A. Compliant Mechanism Design 139
(a) off (b) on
Figure A.3: ’off’ and ’on’ configurations for the electrostatic gripper mechanism
Fx Fy dx dy uout
Initial value −0.249N −35.568N 0.415mm −0.720mm 2.345µm
Optimized value −2.204N −49.854N 2.21mm −10.00mm −1.72mm
Table A.1: A comparison of the state variable values for the baseline and optimized
compliant gripper mechanisms
Appendix B
Aerostructural Problem
Specifications
B.1 Initial Conditions & Constraint Values
Quantity Symbol Value
Initial SIMP design variable value x0 0.25
Maximum SIMP design variable value xmax 1.0
Minimum SIMP design variable value (skin elements) xminskin0.15
Minimum SIMP design variable value (interior elements) xminint0.001
Initial value of SIMP penalization factor p0 0.2
Target value of SIMP penalization factor p∗ 3.0
Initial cruise angle of attack α0cruise2.0
Initial maneuver angle of attack α0maneuver 8.0
Maximum jig twist angle αjmax 6.0
Minimum jig twist angle αjmin−6.0
Maximum allowable compliance Cmax 20kJ
Fixed weight Wfixed 7× 105N
140
Appendix B. Aerostructural Problem Specifications 141
B.2 Material Properties & Finite Element Mesh Di-
mensions
Young’s modulus E 350× 109 N/m2
Poisson’s ratio ν 0.3
Density ρstruct 2.7× 103 kg/m3
Number of elements in the chordwise direction nex 16
Number of elements through the thickness ney 6
Number of elements in the spanwise direction ney 160
B.3 Atmospheric & Flight Conditions
Temperature T 245.0 K
Pressure p 42795.0 N/m2
Density ρatm 0.6096 kg/m3
Speed of sound a 313.5 m/s
Mach number M 0.74
B.4 CRM Wing Geometry
Quarter-chord sweep angle ΛC/4 35
Taper ratio λ 0.275
Semispan Span 29.38m
Aspect ratio AR 9.0
Wing reference area Sref 191.85m2
Appendix B. Aerostructural Problem Specifications 142
B.5 Sizing Optimization
Number of spars nspars 3
Number of ribs nribs 20
Minimum element thickness tmin 0.01m
Maximum allowable compliance Cmax 20 kJ
Fixed weight Wfixed 7× 105N
*Note: All problem specifications not listed in the above table are the same as for
the topology optimization problem.
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