Aerostructural Shape and Topology Optimization of … · Aerostructural Shape and Topology...

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.

iii

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.

iv

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.

v

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

vi

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

vii

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

viii

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

ix

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

x

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

xi

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

xii

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


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


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


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


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


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


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


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,


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)


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


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)


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


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


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.


(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-


(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,


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.


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


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)


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.


(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


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


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


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-


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


(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.


Figure 2.14: Two-dimensional slice of the feasible design space with the three solutions

located at the vertices of the triangle


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


(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.


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


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.


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


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.


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


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


∂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


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.


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.


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].


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.


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


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


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


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


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-


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


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


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


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.


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


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)


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


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)


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


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


Figure 3.7: The short L-bracket problem


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


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


(a) Level set solution (b) SIMP solution

Figure 3.10: Optimized density distribution for the long L-bracket


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


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


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


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


non-uniform meshes.


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


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.


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


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.


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


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.


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)


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)


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


(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


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


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


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


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,


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


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.


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


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.


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


(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


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)


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.


Figure 4.3: The CRM wing with the TriPan surface mesh


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].


Figure 4.4: The structural wingbox and finite element mesh for the CRM wing


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


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)


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


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


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


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


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


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


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


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


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


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.


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


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


Optimizer


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).


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


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


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.


Optimizer


Structural

Analysis

Aerodynamic

Analysis

u

F

Aerodynamic Shape Optimization

Optimizer


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.


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


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


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


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


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


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


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.


(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




Figure 4.18: Material distribution for the CRM wing optimized using sequential algorithm

B




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-


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


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,


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


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.


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


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


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


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


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.


(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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