High-Fidelity Aerostructural Optimization of Nonplanar ... · High-Fidelity Aerostructural Optimization of Nonplanar Wings for Commercial Transport Aircraft Shahriar Khosravi Doctor

High-Fidelity Aerostructural Optimization of Nonplanar

Wings for Commercial Transport Aircraft

by

Shahriar Khosravi

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

c© Copyright 2016 by Shahriar Khosravi

Abstract

High-Fidelity Aerostructural Optimization of Nonplanar Wings for Commercial

Transport Aircraft

Shahriar Khosravi

Doctor of Philosophy

Graduate Department of Aerospace Science and Engineering

University of Toronto

2016

Although the aerospace sector is currently responsible for a relatively small portion of

global anthropogenic greenhouse gas emissions, the growth of the airline industry raises

serious concerns about the future of commercial aviation. As a result, the development

of new aircraft design concepts with the potential to improve fuel efficiency remains an

important priority. Numerical optimization based on high-fidelity physics has become an

increasingly attractive tool over the past fifteen years in the search for environmentally

friendly aircraft designs that reduce fuel consumption. This approach is able to discover

novel design concepts and features that may never be considered without optimization.

This can help reduce the economic costs and risks associated with developing new aircraft

concepts by providing a more realistic assessment early in the design process.

This thesis provides an assessment of the potential efficiency improvements obtained

from nonplanar wings through the application of fully coupled high-fidelity aerostruc-

tural optimization. In this work, we conduct aerostructural optimization using the Euler

equations to model the flow along with a viscous drag estimate based on the surface area.

A major focus of the thesis is on finding the optimal shape and performance benefits of

nonplanar wingtip devices. Two winglet configurations are considered: winglet-up and

winglet-down. These are compared to optimized planar wings of the same projected span

in order to quantify the possible drag reductions offered by winglets. In addition, the

drooped wing is studied in the context of exploratory optimization.

The main results show that the winglet-down configuration is the most efficient

ii

winglet shape, reducing the drag by approximately 2% at the same weight in comparison

to a planar wing. There are two reasons for the superior performance of this design.

First, this configuration moves the tip vortex further away from the wing. Second, the

winglet-down concept has a higher projected span at the deflected state due to the struc-

tural deflections. Finally, the exploratory optimization studies lead to a drooped wing

with the potential to increase range by 4.9% relative to a planar wing.

iii

Dedication

To my amazing parents, Sedigheh and Morteza.

iv

Acknowledgements

First and foremost, I would like to express my most sincere thanks to my advisor, Prof.

David W. Zingg, for giving me the opportunity to conduct this research investigation.

His scientific rigour, effective guidance, support, and patience have been essential to the

work that is presented in this thesis. It has truly been a privilege to learn from him over

the course of the past four years.

I am thankful to the members of my doctoral examination committee, Prof. Prasanth

B. Nair and Prof. Craig A. Steeves, for their helpful feedback during my committee

meetings. Their contributions have had a positive impact on the quality of the present

thesis. I also would like to express my gratitude to Prof. Case van Dam of the University

of California Davis for writing the appraisal letter, and to Prof. Markus Bussmann of the

University of Toronto for his contribution as the internal examiner.

There are a number of colleagues and friends who have had an instrumental role in

my studies here at the University of Toronto. I am especially grateful to Ms. Jenmy

Zhang for her great work in laying the foundation upon which this entire thesis is based.

In addition, I sincerely thank Dr. Tim Leung for his help at the beginning of my studies.

Special thanks go to Mr. Thomas Reist and the rest of my colleagues in the computational

aerodynamics and multidisciplinary design optimization groups for taking the time to

have numerous conversations with me about various aspects of my research. Finally,

I would not have been able to successfully complete this thesis without the incredible

support and encouragement I received from Sarah, Boone, Nick, Vishal, Nikhil, Jon, and

Roger.

I am fortunate to have had the opportunity to learn from Prof. Peter D. Washabaugh

during my undergraduate studies at The University of Michigan, Ann Arbor. His inspi-

rational teaching style, enthusiastic support, and passion for education motivated me to

become an aerospace engineer and further pursue my research ambitions at the graduate

level.

Finally, I am forever indebted to my wonderful parents and siblings for their ongoing

support throughout all of my difficult endeavours including this one.

Shahriar Khosravi

June 7, 2016

Toronto, Ontario

v

Contents

Abstract ii

List of Tables viii

List of Figures xiv

List of Symbols and Acronyms xvii

1 Introduction 1

1.1 Commercial Aviation and Climate Change . . . . . . . . . . . . . . . . . 1

1.2 High-Fidelity Numerical Optimization . . . . . . . . . . . . . . . . . . . 3

1.3 Nonplanar Wings for Improved Fuel Efficiency . . . . . . . . . . . . . . . 5

1.3.1 Wingletted Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 The Drooped Wing Concept . . . . . . . . . . . . . . . . . . . . . 8

1.4 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Numerical Optimization Methodology 13

2.1 Aerostructural Optimization Formulation . . . . . . . . . . . . . . . . . . 13

2.2 Geometry Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Mesh Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Aerodynamic Grid Movement . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Structural Grid Movement . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Coupled Aerostructural Analysis . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Aerodynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.3 Force and Displacement Transfer . . . . . . . . . . . . . . . . . . 23

2.4.4 Aerostructural Analysis . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Gradient Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

vi

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Validation and Verification of the Methodology 27

3.1 Effects of Fitting on Functional Convergence . . . . . . . . . . . . . . . 27

3.2 Validation Based on the HIRENASD Wing . . . . . . . . . . . . . . . . . 30

3.3 Inviscid Transonic Wing Sweep Optimization . . . . . . . . . . . . . . . . 32

3.4 Effects of Structural Topology on Optimization Trends . . . . . . . . . . 36

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Aerostructural Optimization of Nonplanar Wings 43

4.1 Winglet Shape Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Baseline Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.2 Aerodynamic Shape Optimization . . . . . . . . . . . . . . . . . . 45

4.1.3 Aerostructural Optimization with Variable Winglet Cant Angle . 49

4.1.4 Further Investigation Using Aerostructural Optimization . . . . . 59

4.1.5 Optimal Winglets for Cruise and Climb Conditions . . . . . . . . 67

4.2 The Drooped Wing Concept . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 The Drooped Wing Concept Optimization Results . . . . . . . . . 69

4.2.2 Aerodynamic Shape Optimization of the Drooped Wing Concept 78

4.2.3 Multimodality of the Drooped Wing Concept . . . . . . . . . . . 79

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Conclusions, Contributions, and Future Work 83

5.1 Primary Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Secondary Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Appendices 89

A Aerodynamic Shape Optimization of Winglets Using More Realistic

Constraints 90

B Aerodynamic-Structural Optimization of Winglets 93

References 96

vii

List of Tables

3.1 The optimization results obtained from the structural layout investigation 41

4.1 Nonlinear constraints used for optimization in all cases . . . . . . . . . . 52

4.2 Optimization design variables for all cases . . . . . . . . . . . . . . . . . 52

4.3 Summary of the cruise point performance of the planar and wingletted

wings from aerostructural optimization evaluated on the fine grid . . . . 57

4.4 Winglet-up and winglet-down optimization design variables . . . . . . . . 60

4.5 Summary of the aerostructural optimization results obtained from the fine

grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6 Summary of the multipoint aerostructural optimization results obtained

from the fine grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7 Optimization results for the drooped wing concept . . . . . . . . . . . . . 73

4.8 The final objective function comparison for all of the six optimized designs

evaluated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A.1 Differences in nonlinear constraints for the aerodynamic shape optimiza-

tion cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

viii

List of Figures

1.1 The total number of passengers carried worldwide by air transportation.

The exponential trendline closely fits the available data. . . . . . . . . . . 2

1.2 Carbon dioxide emission trends from international commercial aviation

from 2005 to 2050. Results were modeled for 2005, 2006, 2010, 2020, 2025,

2030, and 2040, then extrapolated to 2050. Dashed line in technology

contribution range represents an assumed 0.95% technology improvement

per year from 2010 to 2015, and 0.57% technology improvement per year

from 2015 to 2050. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 The B-spline control grid and the corresponding surface parameterization

of a wing geometry. The blue spheres represent the surface control points. 16

2.2 The computational grid described by the B-spline parameterization shown

on the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 The aerodynamic grid movement scheme is used to manually morph a

planar wing into a C-wing configuration. The initial control grids are

shown in (a) and (b). The corresponding initial and final computational

grids are shown in (c) and (d). In both cases, the blue spheres represent

the B-spline control points. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Cell aspect ratio distribution of the aerodynamic grids for the initial, jig,

and final shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Illustration of the ability of the structural mesh movement scheme to pre-

serve the shape of the structural components inside the wingbox over the

course of a large shape change. . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 The structural layout of the wingbox along with the outer mold line of the

wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 The planform of the wing along with the corresponding surface patches

and control points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ix

3.3 Convergence of lift coefficient for increasing grid density and control point

resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Convergence of drag coefficient for increasing grid density and control point

resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Convergence of lift coefficient for increasing grid density obtained from the

second framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Convergence of drag coefficient for increasing grid density obtained from

the second framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Geometric planform of the HIRENASD wing model. . . . . . . . . . . . . 30

3.8 Comparison of experimental and computational pressure coefficient results

for the HIRENASD wing geometry. The Mach number, angle of attack,

and Reynolds number are 0.80, 1.5◦, and 7.0 × 106, respectively. The

experimental (black), static aeroelastic (blue), and rigid-wing results (red)

are shown for each spanwise station. . . . . . . . . . . . . . . . . . . . . 31

3.9 Grid resolution of the surface and symmetry plane for the fine optimization

mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.10 The planforms for the three wings show that the optimal sweep angle, Λ,

increases with increasing β, i.e. increasing emphasis on drag. . . . . . . . 34

3.11 Cruise and 2.5g load distributions along the span of the wing for the

β = 1.0 case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.12 Cruise and 2.5g load distributions along the span of the wing for the

β = 0.5 case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.13 The optimized thickness distribution of skin elements for the β = 0.5 case. 35

3.14 The optimized thickness distribution of skin elements for the β = 1.0 case. 35

3.15 The four structural layouts considered. . . . . . . . . . . . . . . . . . . . 36

3.16 The geometric parameterization and deign variables for the purpose of the

structural layout optimizations. . . . . . . . . . . . . . . . . . . . . . . . 37

3.17 The merit function convergence history for the optimization using struc-

tural layout number 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.18 The optimality and feasibility convergence histories for the optimization

using structural layout number 3. . . . . . . . . . . . . . . . . . . . . . . 38

3.19 The optimized structural thickness distributions of the structures for all

four optimization cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.20 Optimal lift distributions for the cruise and 2.5g load conditions obtained

from optimization using structural layout number 1. . . . . . . . . . . . . 40

x







4.1 The planform of the baseline planar configuration is based on the Boeing

737NG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The geometric parameterization and design variables for the winglet-up

configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 The geometric parameterization and design variables for the winglet-down

configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 The convergence of optimality and feasibility measures for the winglet-up

aerodynamic optimization case. . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 The convergence of the Lagrangian merit function for the winglet-up aero-

dynamic optimization case. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6 The aerodynamically optimal lift distribution for the planar configuration. 47

4.7 The aerodynamically optimal lift distribution for the winglet-up configu-

ration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 The aerodynamically optimal lift distribution for the winglet-down config-

uration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.9 Comparison of total drag in cruise for the planar and wingletted wings

using the same lift constraint in all three cases. . . . . . . . . . . . . . . 48

4.10 Contours of x-vorticity behind the trailing edge of the wing for all three

cases in cruise condition (M = 0.785 at 35, 000 ft). The dashed line marks

the location of the maximum strength vortex at the tip of the wing for the

planar configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.11 The geometric parameterization and design variables for the exploratory

winglet optimization case. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.12 Three possible wing shapes permitted by the parameterization. (1) and

(2) are two variations of a winglet-down configuration, and (3) is planar. 50

4.13 Primary structural layout of the wing in relation to the outer mold line.

Skin elements are not shown for clarity. . . . . . . . . . . . . . . . . . . . 52

4.14 Each colored surface represents a structural component, the thickness of

which is a design variable. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xi

4.15 Convergence of optimality and feasibility measures for the winglet-down

aerostructural optimization case. The dashed line marks the beginning of

the optimization using the fine grid. . . . . . . . . . . . . . . . . . . . . . 53

4.16 Convergence of the Lagrangian merit function for the winglet-down aerostruc-

tural optimization case. The dashed line marks the beginning of the opti-

mization using the fine grid. . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.17 The initial and optimized wing shapes for the β = 0.5 case. Contours of

pressure coefficient in cruise condition (M = 0.785 at 35, 000 ft) are also

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.18 The initial and optimized wing shapes for the β = 1.0 case. Contours of

pressure coefficient in cruise condition (M = 0.785 at 35, 000 ft) are also

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.19 Plots of the pressure coefficient in cruise condition (M = 0.785 at 35, 000 ft)

for the optimal planar wing with β = 1.0 and the initial geometry. . . . . 54

4.20 Lift distributions for the optimized winglet-down configuration with β = 0.5. 55

4.21 Lift distributions for the optimized winglet-down configuration with β = 1.0. 55

4.22 Plots of the pressure coefficient in cruise condition (M = 0.785 at 35, 000 ft)

for the optimal winglet-down configurations with β = 0.5 and β = 1.0. . . 55

4.23 The wing twist distributions in cruise condition for the optimal winglet-

down configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.24 The initial and optimized planform shapes for the aerostructural optimiza-

tion cases with β = 0.5 and β = 1.0. . . . . . . . . . . . . . . . . . . . . . 57

4.25 Comparison of total drag for both values of β between the optimized pla-

nar wing with a fixed cant angle of zero and the optimized winglet-down

configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.26 View of the undeflected spans for the planar and winglet-down configura-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.27 View of the deflected spans for the planar and winglet-down configurations. 58

4.28 Geometric parameterization and design variables. . . . . . . . . . . . . . 61

4.29 The optimized winglet-down configuration with β = 0.5 along with the

initial wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.30 Merit function convergence history for the winglet-down configuration with

β = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.31 Feasibility and optimality convergence histories for the winglet-down con-

figuration with β = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

xii

4.32 Contours of pressure coefficient on the upper and lower surfaces in cruise

condition (M = 0.785 at 35, 000 ft) for the initial geometry. . . . . . . . . 61

4.33 Contours of pressure coefficient on the upper and lower surfaces in cruise

condition (M = 0.785 at 35, 000 ft) for the optimized winglet-down con-

figuration with β = 0.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.34 Skin thickness values in millimeters for the winglet-down configuration

with β = 0.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.35 Spanwise lift distributions at the cruise and 2.5g load conditions for the

winglet-down configuration with β = 0.5. . . . . . . . . . . . . . . . . . . 63

4.36 Spanwise lift distributions at the cruise and 2.5g load conditions for the

winglet-down configuration with β = 1.0. . . . . . . . . . . . . . . . . . . 63

4.37 Top-down view of the optimal winglet-down configurations. . . . . . . . . 64

4.38 Trade-off curves of optimal designs for all of the wingletted and planar

configurations considered in this investigation. . . . . . . . . . . . . . . . 65

4.39 Contours of x-vorticity behind the trailing edge of the wing for the planar

and winglet-down cases in cruise condition (M = 0.785 at 35, 000 ft). The

dashed line marks the location of the tip vortex for the planar configuration. 66

4.40 Primary structural layout of the ribs and spars used for the purpose of

this investigation. Skin elements are omitted for clarity. . . . . . . . . . . 68

4.41 The flight profile of a B737NG aircraft from Toronto to Vancouver. . . . 69

4.42 The flight profile of a B737NG aircraft from Toronto to Montreal. . . . . 69

4.43 Geometric parameterization and design variables for the drooped wing case. 71

4.44 Three possible wing shapes permitted by the parameterization. The di-

hedral angle for each region, φ can vary between −30◦ to +30◦. The red

circles represent the curved junctions between regions and their locations

along the span are allowed to vary. (1) is a planar wing, and (2) and (3)

are two possible nonplanar wing shapes. . . . . . . . . . . . . . . . . . . 71

4.45 Convergence of optimality and feasibility conditions for the drooped wing

optimization case. The dashed line marks the beginning of optimization

on the fine grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.46 Merit function convergence behavior for the drooped wing optimization

case. The dashed line marks the beginning of optimization on the fine grid. 72

4.47 The geometric evolution of the drooped wing over the course of optimiza-

tion. All shapes are at the deflected state of the wing and the correspond-

ing function evaluation number is also shown. . . . . . . . . . . . . . . . 73

xiii

4.48 The contours of x-vorticity behind the trailing edge of the wing for the

planar (top) and drooped (bottom) wings. The dashed line marks the

location of the tip vortex for the planar configuration. . . . . . . . . . . . 73

4.49 The additional freedom given to the optimizer to manipulate the nonlinear

shape of the leading and trailing edges of the wing. . . . . . . . . . . . . 74

4.50 Pressure coefficient contours in cruise condition (M = 0.785 at 35, 000 ft)

for the optimized drooped wing with curved edges. . . . . . . . . . . . . 74

4.51 Pressure coefficient contours in cruise condition (M = 0.785 at 35, 000 ft)

for the initial and optimized planar configurations when the shape of the

leading and trailing edges are free to vary. . . . . . . . . . . . . . . . . . 75

4.52 The optimal structural thickness distributions for the baseline planar con-

figuration and the planar wing with nonlinear leading and trailing edges. 76

4.53 Contours of normalized entropy in cruise condition (M = 0.785 at 35, 000 ft)

at 60% half-span for the optimized planar wing with straight edges. . . . 77

4.54 Contours of normalized entropy in cruise condition (M = 0.785 at 35, 000 ft)

at 60% half-span for the optimized planar wing with curved edges. . . . . 77

4.55 View of the aerodynamically optimized planar wing at M = 0.50 along

with the streamtraces that extend to the far-field. . . . . . . . . . . . . . 78

4.56 View of the aerodynamically optimized wing with curved leading and trail-

ing edges at M = 0.50 along with the streamtraces that extend to the

far-field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.57 The drooped wing configuration that results from purely aerodynamic

shape optimization in cruise condition (M = 0.785 at 35, 000 ft) using the

same geometric parameterization and design variables as the aerostruc-

tural optimization cases of the same concept. . . . . . . . . . . . . . . . . 79

4.58 Randomly generated initial geometries. . . . . . . . . . . . . . . . . . . . 80

4.59 Final optimized designs from the initial geometries shown in Figure 4.58. 80

A.1 Comparison of total drag in cruise for all wings while taking into account

differences in optimal lift and wing volume. . . . . . . . . . . . . . . . . . 91

B.1 Differences between the lift and drag performance of the planar and winglet-

ted wings obtained from purely aerodynamic and aerodynamic-structural

optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xiv

List of Symbols and Acronyms

A Aerostructural Jacobian matrix

Bi,j,k Coordinates of the de Boor control points

b Aerodynamic mesh state

bs Coordinates of the displaced B-spline control points

Ceq Equality constraints

Cin Inequality constraints

fM Implicit force vector

G Total gradient vector

J Objective function

KM Stiffness matrix for the mesh equations

N (4) Fourth-order B-spline basis function

q Aerodynamic state

RAS Aerostructural residual

RA Aerodynamic residual

RS Structural residual

RM Mesh residual

r Rigid link vector

u Structural state

uA Aerodynamic surface displacements

uS Structural surface displacements

V B-spline volume

v Design variables

b Wing projected span

b∗ Wing projected span at the deflected state

CD Inviscid drag coefficient of the wing

CL Inviscid Lift coefficient of the wing

Cp Inviscid pressure coefficient

xv

D Total drag of the aircraft

D0 Initial drag of the wing

Dinviscid Inviscid drag of the aircraft

e Span efficiency factor based on linear aerodynamic theory

g Gravitational acceleration constant on Earth

k Element number

L Inviscid lift

ℓ Distance between the wing-winglet junction and the wingtip

M Freestream Mach number

N Number of grid nodes

Ne Number of shell elements

Ni, Nj, Nk Number of B-spline control points in the x, y, z coordinate directions

R Range parameter

S Wetted surface area of the wing

W Weight of the wing

W0 Initial weight of the wing

Wf Final weight of the aircraft

Wfuel Fuel weight

Wi Initial weight of the aircraft

WMTO Aircraft maximum takeoff weight

x Streamwise coordinate direction

y Spanwise coordinate direction

z Vertical coordinate direction

ξ Parametric space

Ψ Lagrange multiplies or adjoint variables

α Angle of attack

β Parameter between zero and unity

ǫ Residual tolerance

ζ, η, ξ Parametric coordinate directions

θ Winglet cant angle

θS Structural rotations

Λ Wing sweep angle

λ Von Mises stress criterion

xvi

λmin Minimum von Mises stress criterion

ρ KS weighting parameter

σk Von Mises stress for the k-th element

σyield Yield stress of the material

φ Dihedral angle

CFD Computational Fluid Dynamics

FFD Free Form Deformation

GMF Global Market Forecast

HECS Hyper-Elliptic Cambered Span

HIRENASD HIgh REynolds Number AeroStructural Dynamics

ICAO International Civil Aviation Organization

KS Kreisselmeier-Steinhauser

MITC Mixed Interpolation of Tensorial Components

NASA National Aeronautics and Space Administration

pyOpt A Python-based package for optimization

RANS Reynolds-Averaged Navier-Stokes

SNOPT Sparse Nonlinear OPTimizer

TACS Toolkit for the Analysis of Composite Structures

xvii

Chapter 1

Introduction

1.1 Commercial Aviation and Climate Change

There is extensive and incontrovertible scientific evidence that anthropogenic contri-

butions are the dominant cause of the rapidly changing global climate here on Earth.

Without serious mitigation strategies, average air and ocean temperatures will continue

to rise, widespread melting of snow and ice will expand, and the rising of global aver-

age sea level will accelerate beyond its already alarming magnitude. These effects pose

significant interruptions to the natural functions of the ecological systems on our planet,

and will ultimately erode our ability as a species to maintain health and security. As

a result, the existential threat of climate change is undeniably real and requires urgent,

collective action to effectively reduce its harmful consequences [1, 2, 3, 4].

The most important contributing factor to climate change is the human-induced in-

creases in the amount of atmospheric greenhouse gases, including carbon dioxide, chlo-

rofluorocarbons, methane, and nitrous oxide. The relative contribution of commercial

aviation to the overall human-made greenhouse gas emissions is currently estimated to

be around 3% [5]. Although this relative percentage contribution is small, it translates

to a very large mass of greenhouse gas emissions. The amount of fuel burned by in-

ternational aviation alone in 2015 is estimated to exceed 200 million tonnes. Assuming

that 1 kg of fuel burned produces approximately 3.16 kg of carbon dioxide, international

aviation is responsible for emitting more than 630 million tonnes of carbon dioxide into

the atmosphere of Earth by the end of 2015 [6].

In order to gain a more practical perspective on the dangers that commercial aviation

faces due to climate change, one can examine the amount of carbon dioxide emitted

in the atmosphere during a typical passenger flight. According to the International

Civil Aviation Organization (ICAO), each passenger is responsible for emitting 263 kg

1

Chapter 1. Introduction 2

Year

1970 1980 1990 2000 2010 2020 2030 2040

Number

ofPassen

gersCarriedWorldwide

×109

0

1

2

3

4

5

6

7

8

9

ICAO Data

Exponential Trendline

Figure 1.1: The total number of passengers carried worldwide by air transportation. Theexponential trendline closely fits the available data.

of carbon dioxide into the atmosphere during a typical commercial flight from Toronto

to Vancouver over the approximately 3, 600 km distance.1 This amount of emissions is

undoubtedly large and reminds us that immediate action is necessary despite the fact

that the current greenhouse gas emissions from commercial aviation represent a small

percentage of the total human-induced emissions.

Although the relative contribution of the aviation sector to the overall human-induced

emissions is small, it is still a point of concern due to the exponential growth of the

airline industry, as indicated in Figure 1.1. In fact, the aircraft manufacturing company

Airbus projects that the industry will grow by an average annual rate of 4.6% in terms

of revenue passenger kilometer [7]. In the absence of any mitigation strategies, this

translates to a relative emissions contribution of 15% by 2050 [5]. As a result, the

current growth of the commercial aviation sector is clearly unsustainable. The challenges

that the airline industry is currently facing are going to be significantly more difficult to

address in the long term if serious research and development efforts focused on reducing

the environmental footprint of aviation are not pursued today.

Figure 1.2 shows the carbon dioxide emission contributions from international aviation

alone, as reported in the 2013 ICAO Environmental Report [6]. This figure includes

possible reductions in emissions from fleet renewal, technology improvements, and more

efficient air traffic management and infrastructure use. If the goal of the airline industry

is to cap the total amount of emissions to the 2020 levels by 2050, there will be at least a

billion metric tons of carbon dioxide to remove from the atmosphere by other mitigation

1The online emissions calculator is available at:

http://www.icao.int/environmental-protection/CarbonOffset/Pages/default.aspx


Figure 1.2: Carbon dioxide emission trends from international commercial aviation from2005 to 2050. Results were modeled for 2005, 2006, 2010, 2020, 2025, 2030, and 2040,then extrapolated to 2050. Dashed line in technology contribution range represents an as-sumed 0.95% technology improvement per year from 2010 to 2015, and 0.57% technologyimprovement per year from 2015 to 2050 [6]2.

strategies. Based on the existing targets established by the members of ICAO, it is

possible that alternative fuels can close 25% of this gap. However, there are significant

uncertainties in predicting the actual contribution of alternative fuels in reducing this gap.

As a result, it is necessary to pursue aircraft technologies that have a greater potential

for providing significant fuel efficiency gains than those currently under consideration.

1.2 High-Fidelity Numerical Optimization

It is clear that in order to sustain the current growth of the airline industry, commer-

cial aircraft must become highly efficient in terms of fuel consumption. Conventional

tube-and-wing designs are already highly optimized and offer little potential for further

significant efficiency improvements. Therefore, in order to achieve the necessary tech-

nology improvements, novel design concepts must be explored. However, the traditional

2Used with permission.


cut-and-try approach to aircraft design does not lend itself well to exploring new ideas

because it heavily relies on the past experience of aircraft designers. This kind of design

knowledge does not exist for novel concepts with significant potential.

Numerical design optimization of commercial aircraft is an effective approach for

discovering novel concepts with the ability to provide efficiency improvements. In par-

ticular, numerical optimization tools based on high-fidelity aerodynamic analysis using

the Euler and Navier-Stokes equations have become increasingly available due to the

advent of high-performance parallel computing and efficient numerical methods. These

tools have the capability to take into account nonlinear physics and provide a more accu-

rate assessment of the potential efficiency gains that various new concepts can offer. In

contrast, low- and medium-fidelity models such as Prandtl’s classical lifting-line theory

and the vortex lattice method ignore the nonlinear effects of compressibility [8]. High-

fidelity aerodynamic shape optimization can provide a more realistic assessment of new

concepts earlier in the design process and hence reduce the associated development costs

and economic risks.

Exploratory design optimization based on aerodynamic analysis has revealed many

novel wing concepts with the ability to reduce drag. However, it is essential to determine

how much of these drag benefits may be overshadowed by the effects of structural weight.

Some past applications of high-fidelity aerodynamic shape optimization with simplified

weight and structural models have shed light on the fundamental trade-off between weight

and drag [9, 10, 11, 12, 13, 14]. However, the effects of structures on the aerodynamic

performance of novel wings are more accurately captured if fully coupled stress analysis is

employed based on the aerodynamic loads. This highlights the importance of performing

high-fidelity aerostructural optimization in the context of exploratory design.

Recent applications of fully coupled high-fidelity aerostructural optimization to con-

ventional aircraft have shown the effectiveness of using numerical optimization tools to

recover important fundamental trade-offs between weight and drag in the design of com-

mercial aircraft wings [15, 16, 17, 18]. These have largely been facilitated by the advent of

the adjoint method for the purpose of conducting gradient-based aerostructural optimiza-

tion [19, 20]. This means that the application of high-fidelity aerostructural optimization

to novel wing concepts, especially those that have shown potential for considerable drag

reductions, may prove to be worthwhile.


1.3 Nonplanar Wings for Improved Fuel Efficiency

Lift-induced drag (or simply induced drag) is approximately 40% of the total drag of

a commercial aircraft in cruise [21]. It is therefore worthwhile to explore concepts that

reduce the induced drag. This has been the primary motivation for researchers to study

nonplanar wings. In this section, we present a literature survey on the potential efficiency

improvements offered by nonplanar wings for commercial aircraft. The literature survey

is divided into two sections. First, we examine studies related to nonplanar wingtip

devices. Second, we describe the previous studies that consider a more novel concept

called the drooped wing.

1.3.1 Wingletted Wings

There have been many studies in the past decades on the possible fuel efficiency im-

provements provided by winglets. We mention a few important examples in this section;

a more thorough review is provided by Kroo [21]. The term winglet was first used by

Whitcomb [22] at NASA. He showed that winglets can provide a significant improvement

in the lift-to-drag ratio over planar wingtip extensions at the same level of root bending

moment. While the winglets and wingtip extensions were designed by a combination of

available theory and wind tunnel tests, the wing itself was not redesigned. At the same

time, another study done by NASA considered the relative advantages of winglets and

wingtip extensions [23]. It concluded that at the same level of root bending moment,

winglets provide a greater induced drag reduction than a wingtip extension. Later, an-

other study by Jones and Lasinsky [24] concluded that when optimized wing shapes are

considered, similar reductions of induced drag may be achieved either by extending the

wingtip or by having a vertical winglet.

Asai [25] studied the relative advantages of planar and nonplanar wings and concluded

that the trade-off between the induced drag and wing root bending moment alone is not

enough to determine the effectiveness of winglets. The effects of the viscous drag penalty

incurred by winglets must also be taken into account. He further suggested that if both

the root bending moment and viscous drag are kept constant, it is possible to design a

planar wing with lower total drag than any nonplanar wing. Van Dam [26, 27] considered

planar wings that produce a nonplanar wake at a nonzero angle of attack and suggested

that these geometries can provide considerable induced drag reductions.

More recently, Takenaka and Hatanaka [28] performed a multidisciplinary design ex-

ploration for a winglet using high-fidelity computational fluid dynamics (CFD) and com-

putational structural mechanics. They considered a sample of 32 winglets having various


root chord lengths, taper ratios, sweep angles, spans, cant angles, and toe angles for mul-

tidisciplinary design optimization based on the Kriging model. Their optimized winglet

provides a reduction in total drag of approximately 22 drag counts, where one drag count

is equal to a drag coefficient of 0.0001, while increasing the wing root bending moment by

5.3%. Furthermore, they demonstrated that a conventional winglet, one that is a vertical

extension of the wingtip geometry, can only provide a reduction of 17 drag counts while

increasing the root bending moment by 3.5% in comparison to the baseline planar wing.

The total drag reductions provided by the wings with winglets were also validated using

wind tunnel tests. However, it is important to note that only the winglets were optimized

in this study. The shape of the wing was not optimized.

Another notable numerical study, conducted by Verstraeten and Slingerland [29],

focused on the drag characteristics of optimally loaded planar wings, wings with winglets,

and c-wings using a low-fidelity model of the aerodynamics and weight. They concluded

that when a span constraint exists, a wingletted wing with a height-to-span ratio of

28% provides a total drag reduction of 5.4% in comparison to a planar wing at identical

wing root bending moments. This study also demonstrated that winglets can be used

to provide induced drag reductions when there is a constraint on the aspect ratio of the

wing. Ning and Kroo [30] did a similar study, but included the area-dependent weight

in their calculations too. They also took into account the effects of a critical structural

load factor on the trade-offs in the design of wings with winglets. Another notable

difference in this study was the inclusion of a stall speed constraint. They demonstrated

that whether a winglet performs better than a wingtip extension depends on the ratio

of the maneuver lift coefficient to the cruise lift coefficient. This is due to the fact

that the area-dependent weight is a function of the ratio of the maneuver and cruise lift

coefficients in their proposed low-fidelity weight model. When this ratio is equal to unity,

a wingtip extension is slightly more advantageous while the winglet performs marginally

better when this ratio is equal to 2.5. This trend held true for both retrofits and new

wing designs. Using a medium-fidelity aerostructural optimization approach, Jansen et

al. [31] showed that a wing with a winglet is the globally optimal design when a span

constraint exists. Furthermore, when the span is unconstrained, the optimal design is a

raked wingtip. In a more recent study, Elham and van Tooren [32] integrated a quasi-

three-dimensional aerodynamic solver to a quasi-analytical weight estimation model in

order to investigate the benefits of retrofitted winglets for existing aircraft. An important

conclusion of this work is that a planar tip extension is a better choice than a winglet

for achieving the maximum drag reduction if there are no constraints on wing weight.

Despite the considerable research effort on this subject, there is still no clear consensus


on the ability of nonplanar wingtips to provide significant fuel efficiency improvements,

especially for a new wing design, and the conditions under which the potential benefits

are seen. Conclusions made in the past vary depending on the specific design problem

considered and the level of physical detail that the models used are able to capture.

This means that more work needs to be done in order to rigorously quantify the possible

efficiency gains from winglets. Furthermore, fully-coupled high-fidelity aerostructural

optimization is essential in order to capture the interdisciplinary and nonlinear effects

that are important in the study of winglets.

In the study of nonplanar wingtip devices, it is important to make a distinction

between retrofitted winglets and new wing designs. A retrofitted winglet is intended

as an after-market addition to an existing wing design in an attempt to improve the

aerodynamic performance of the wing. There are many examples of retrofitted winglets

on today’s modern transport aircraft. These include the B737NG, B747-400, B767-400,

MD-11, and KC-135. In all of these examples, the addition of a nonplanar wingtip device

increases the span of the wing [33]. This is an important consideration when discussing

the potential benefits of winglets. It is difficult to rigorously quantify the efficiency

improvements of winglets when the projected span of the wing is increased because

the induced drag is reduced quadratically with an increase in the span. Therefore, the

fact that these retrofitted winglets improve the performance of existing wings may not

necessarily mean that new wing designs with winglets will also outperform their planar

counterparts of the same projected span. Our conclusions apply in the context of new

wing designs, as opposed to retrofits.

The high-fidelity analysis in the present thesis uses the Euler equations to model

the flow along with a post-optimality viscous drag estimate based on the surface area.

This is sufficient for studying the main trends involved in the design of wingletted wings.

At the end of each optimization, the post-optimality viscous drag estimate will ensure

that the increase in the wetted surface area as a result of having a winglet is taken

into account in calculating the total drag of a wingletted wing. This approach does

not capture the effects of viscosity on the optimal design during optimization. However,

as the chord length is held fixed during the optimizations, the optimizer cannot alter

the chord Reynolds number. Despite neglecting viscous effects during optimization, the

current approach leads to similar geometries obtained from high-fidelity aerodynamic

shape optimization based on the Reynolds-averaged Navier-Stokes (RANS) equations

[34]. Thus, it is not necessary to model the flow based on the RANS equations for the

purpose of the current study.

In the present thesis, we size the structures based on the von Mises failure criterion at


a 2.5g load condition [35, 36, 37, 38]. We do not consider structural constraints such as

buckling and flutter. These are potentially important considerations in this context and

should be taken into account in future studies. Consequently, the results presented in

this thesis could overestimate the benefits of winglets, but are unlikely to underestimate

them. Furthermore, we do not consider additional maneuver and gust load cases in sizing

the structures. However, this does not adversely affect our main conclusions because our

approach leads to the same structural sizing trends as the ones obtained from studies

that include more load conditions [39, 40]. Furthermore, active maneuver and gust load

alleviation systems could potentially reduce the need to include many critical structural

load cases [41]. It is important to note that our objective in the present work is to study

the main trends, not to perform detailed wing design. The present structural sizing

strategy is sufficient for this purpose.

1.3.2 The Drooped Wing Concept

A notable numerical aerodynamic optimization study using both medium- and high-

fidelity tools from NASA [42] considered a large degree of geometric freedom in terms

of dihedral across the span using a fourth-order polynomial to allow for a continuously

varying spanwise camber. The optimal shape was found to be a drooped wing for a

Boeing 767 class commercial airplane. The optimal drooped wing has negative curvature

along the span with the wingtips lowered towards the ground continuously. The authors

attributed the aerodynamic benefit of the drooped wing (a 5.3% reduction in induced

drag in comparison to the baseline planar configuration) to its ability to move the core

of the tip vortex away from the wing.

NASA had experimentally investigated the drooped wing concept under the name

of Hyper-Elliptic Cambered Span (HECS) as early as 2006 [43]. This investigation was

inspired by the way a seagull shapes the spanwise camber of its wings in gliding flight.

The authors argued that since seagulls would naturally choose the most efficient spanwise

camber, it is possible that the drooped wing is able to provide higher efficiency than if the

wings were fully extended (leading to a higher span). In other words, the drooped wing

may be able to provide better aerodynamic performance than the fully extended wings

with a higher span. The experimental results of this investigation eventually revealed

that the drooped wing is the most beneficial configuration in comparison to planar and

wingletted wings of the same projected span. It was shown to improve the maximum

lift-to-drag ratio by more than 9%.

The two studies from NASA along with other investigations in the biomimetics com-


munity inspired another notable numerical optimization study on the seagull drooped

wings [44]. Specifically, the authors focused on whether the drooped wing configuration

might be an aerodynamic optimum or if it occurs due to the gull’s anatomical con-

straints. In other words, the authors were interested to see if the gulls voluntarily morph

into this particular configuration rather than having been forced into it by structural

motion constraints. The results of this numerical optimization study revealed that of

all the configurations that a gull is able to choose from for gliding flight as dictated by

its anatomical constraints, the drooped wing is the most optimal configuration from the

standpoint of maximum lift-to-drag ratio. Nevertheless, this does not necessarily mean

that the potential benefits of the drooped wing for gulls will scale the same way to the

size of a modern commercial aircraft.

The design implications of the drooped wing concept for today’s modern commercial

aircraft are largely unknown [44]. One reason for the lack of attention to this particular

configuration in the literature may be that there are uncertainties with regard to the

operational and manufacturing challenges that such a configuration may introduce to

the design process of an aircraft. Nonetheless, it is important to quantify the possible

performance gains from this concept as part of the effort to meet the fuel efficiency gains

demanded by the climate change challenge.

1.4 Thesis Objectives

The overall goal of the present thesis is to investigate the potential performance benefits

of nonplanar wings using fully coupled high-fidelity aerostructural optimization. We

consider two classes of nonplanar configurations: wings with winglets and a drooped wing

design. We will compare these nonplanar wings to their optimal planar counterparts of

the same projected span.

A major focus of this thesis is on wingletted wings with the potential to reduce the

induced drag. We aim to provide a comprehensive assessment of these nonplanar wings

using aerostructural optimization, where the effects of weight and structural deflections

are taken into account in addition to drag. Although the problem of winglet design has

been studied by many researchers in the past, this thesis presents a new (and somewhat

unexpected) perspective on important nonlinear and interdisciplinary effects that were

not previously considered.

There are two main reasons for studying the drooped wing. We would first like

to see if the fully coupled high-fidelity aerostructural optimizer is able to recover this

concept from an initially planar wing given the necessary geometric freedom. Further,


our investigation will hopefully serve as a next step towards assessing the design trade-

offs associated with the drooped wing in the context of commercial aircraft development

using high-fidelity tools. As a result, the reader is cautioned that our conclusions will be

limited in scope and preliminary in nature.

The fully coupled high-fidelity aerostructural optimization framework used for the

purpose of conducting our investigations has been under continuous development over

the course of the past six years. Therefore, it is necessary to rigorously establish that the

results obtained from the framework are credible. This involves validating the aerostruc-

tural analysis capability with static aeroelastic experimental results. Furthermore, it is

important to verify that the framework is able to recover the expected trends in the

design of commercial aircraft wings. Finally, certain aspects of the framework should be

tested specifically to ensure that they do not inhibit its ability to capture the correct

trends. We also aim to examine the effects of a few relevant limitations of the numerical

tools used on the main conclusions of our investigations.

The test cases presented in this thesis involve giving a large amount of geometric

freedom to the optimizer in order to produce nonplanar wings starting from an initially

planar wing. It is important to note that this type of optimization by nature is unable to

account for all the relevant operational constraints that must be taken into account. For

example, a winglet-down configuration may violate the current airport requirements in

terms of wingtip vertical clearance. However, it is still important to quantify the possible

efficiency gains provided by this particular configuration. This will help to determine

whether it is worthwhile to consider, for example, high-wing aircraft in order to enable

a winglet-down design. The present thesis does not address the possible feasibility and

manufacturing challenges associated with novel designs because we believe it is first im-

portant to determine the potential performance gain. Studies of this nature are essential

if the aviation industry is committed to removing the carbon dioxide emission gap that

we discussed in Section 1.1.

With the preceding remarks in mind, the main objectives of the present thesis are

summarized below.

• establish the credibility of the numerical tools used for the purpose of studying the

potential performance benefits of nonplanar wings;

• find the optimal shape and quantify the potential benefits of wingletted wings for

commercial aircraft;

• determine if the fully coupled high-fidelity aerostructural optimizer is able to recover

a drooped wing from an initially planar wing given the necessary geometric freedom;


• investigate the potential efficiency gains of the drooped wing as a next step towards

assessing the design trade-offs associated with this concept.

1.5 Thesis Outline

The high-fidelity aerostructural analysis and optimization methodology is described in

Chapter 2. This chapter presents a brief overview of the geometric parameterization and

mesh movement strategies, the aerodynamic and structural solvers, and the sensitivity

calculation method for the purpose of conducting gradient-based optimization. It is

important to note that the methods described in this chapter are not part of the direct

contributions to the field as a result of this work, as they are primarily the work of others.

The main purpose of Chapter 3 is to validate and verify the most important aspects of

the aerostructural framework. This involves validating and comparing the computational

analysis results with static aeroelastic experimental data. Furthermore, we aim to verify

in this chapter that the aerostructural analysis and optimization framework is consistent

in terms of the grid convergence behavior of the functionals of interest. We also con-

sider important numerical optimization cases that help to establish that the optimizer

is capable of recovering the expected trade-off between weight and drag in the context

of wing design. Finally, the potential implications of using a fixed structural topology

during optimization for the sizing of the structures are studied through a preliminary

investigation.

In Chapter 4, we present the results of applying the current aerostructural analysis

and optimization framework to novel wing concepts. We begin by performing a thorough

investigation on nonplanar wingtip devices that have shown promise for reducing drag in

the past. We adopt a comprehensive step-by-step approach where the potential benefits

of winglets are first characterized on the basis of purely aerodynamic shape optimization.

We then move towards fully coupled aerostructural optimization using an incremental

approach in order to highlight the most important differences between purely aerody-

namic and aerostructural optimization. This chapter continues by giving the optimizer

more geometric freedom to explore a larger design space. This leads to a novel drooped

wing concept that shows considerable promise for providing fuel efficiency gains. Finally,

we make a first attempt to characterize the possible effects of multimodality on the main

conclusions in the context of the drooped-wing design.

A brief summary of the main conclusions from the present thesis is provided in Chap-

ter 5. Furthermore, a review of the most significant contributions to the field is presented.

We also discuss possible future research directions that are worthwhile to pursue in the


context of our main findings.

Chapter 2

Numerical Optimization

Methodology

The main objective of this chapter is to provide an overview of the solution methods

used for the purpose of conducting our numerical investigations. All major components

of the current framework are briefly discussed, and a few illustrative examples are given.

Zhang et al. [45, 46] provide a more thorough description of the tools used.

2.1 Aerostructural Optimization Formulation

From a mathematical standpoint, an aerostructural optimization problem seeks to mini-

mize an objective function J with respect to a set of aerodynamic and structural design

variables v. The problem formulation can be summarized as follows:

minv

J (v,q,u,b) , (2.1)

subject to: RAS(v,q,u,b) = 0 ,

Cin(v,q,u,b) ≤ 0 , Ceq(v,q,u,b) = 0 .

The steady-state aerostructural state variables are given by [q,u,b], where q is the aero-

dynamic state, u is the structural state, and b is the state of the aerodynamic grid. The

aerostructural state depends on the design variables through the steady-state aerostruc-

tural residual equation RAS(v,q,u,b) = 0. The optimization is conducted subject to a

set of inequality and equality constraints, denoted by Cin and Ceq, respectively. In this

case, we use the third-party gradient-based optimizer SNOPT [47] to solve the aerostruc-

13

Chapter 2. Numerical Optimization Methodology 14

tural optimization problem. SNOPT is a sequential quadratic programming algorithm

that is well-suited for nonlinear optimization problems with thousands of design variables.

PyOpt [48] is used to provide a Python interface to SNOPT.

Gradient-based optimizers typically require fewer function evaluations than genetic

algorithms [49]. The number of function evaluations is a critical consideration in the

context of high-fidelity aerostructural optimization because the cost of calculating the

objective is much higher in comparison to low- and medium-fidelity tools. However, this

has important implications for exploratory optimization due to the fact that gradient-

based optimizers converge to a local minimum, which may not be the global optimum.

Chernukhin and Zingg [50] quantify the level of multi-modality for high-fidelity aerody-

namic shape optimization problems of interest and provide gradient-based multi-start

strategies that may be used to alleviate this particular challenge. Although we do not

make use of these strategies in the present thesis, we do make an attempt to investi-

gate whether multi-modality is affecting our main conclusions obtained from exploratory

optimization.

For the purpose of this thesis, we are primarily interested in multidisciplinary objec-

tive functions that involve both aerodynamic and structural functionals. These include

linear combinations of weight and drag with varied emphasis on their relative impor-

tance, range, and drag. Note that drag is not a purely aerodynamic objective function

in aerostructural optimization due to the implicit dependence of lift-induced drag on the

total weight of the aircraft. Linear combinations of wing drag and weight allow us to

construct trade-off curves, and hence evaluate the inherent trade-off that exists between

weight and drag in the design of commercial aircraft wings.

The nonlinear constraints for a typical aerostructural optimization problem include

aerodynamic, structural, and geometric constraints. A lift constraint, where the total

lift is required to be equal to a multiple of the aircraft weight depending on the load

condition considered, is an example of a nonlinear aerodynamic constraint. Structural

constraints include stress aggregation functions that ensure the integrity of the wing

under the aerodynamic loads. In the present thesis, the Kreisselmeier-Steinhauser (KS)

stress aggregation technique [51] is used to ensure that the von Mises stress criterion for

various components inside the wingbox does not reach its critical value. A wing volume

constraint is an example of a nonlinear geometric constraint. Another practical example

is constraining the projected span of a wingletted wing when the winglet cant angle is

a design variable for rigorous induced drag comparisons. In all of the cases presented in

this thesis, the projected span is constrained and remains unchanged.

Typical design variables for aerostructural optimization can be grouped into three


categories: aerodynamic, structural, and geometric. Aerodynamic design variables are

the angles of attack for all of the load conditions being considered. In our cases, there are

at least two load conditions: one is the design flying condition, the other a 2.5g critical

load condition. Structural design variables are the thickness values of the components

inside the wingbox (ribs, spars, and skin elements). Since the thickness of the structural

components is allowed to vary, the weight of the aircraft changes over the course of

a single optimization. Thus, the corresponding lift coefficient changes depending on

the structures. This represents the most important difference in the formulation of the

lift constraint between purely aerodynamic and aerostructural optimization. Geometric

design variables control the sectional shape, twist, sweep, and dihedral angles along the

span, as well as the shape of the leading and trailing edges from a three-dimensional

standpoint.

The aerostructural analysis and optimization framework does not currently model

unsteady aeroelastic phenomena (such as flutter) or buckling. These are potentially im-

portant considerations in the detailed design of commercial aircraft wings. However, the

present thesis does not aim to accurately capture all of the detailed design aspects. In-

stead, the primary focus is on exploratory optimization using high-fidelity aerostructural

analysis to shed light on important design trade-offs between weight and drag. There-

fore, the existing methodology discussed in this chapter is sufficient for the purpose of

the present thesis. It is also important to note that these dynamic aeroelasticity effects

in addition to buckling are most likely to inhibit the ability of nonplanar wing concepts

to provide further drag reductions. As a result, if such wing designs do not provide sig-

nificant efficiency gains in the absence of these constraints, then it is unlikely that their

performance will improve in the presence of these constraints. In other words, our results

may overestimate the performance the nonplanar wings considered, but are unlikely to

underestimate them.

2.2 Geometry Parameterization

The current framework makes use of B-spline volumes for geometric parameterization.

This approach involves mapping a point from the parametric space ξ = (ξ, η, ζ) to the

physical space, as defined by the following relationship [52]:

V(ξ) =

Ni∑

i=0

Nj∑

j=0

Nk∑

k=0

N (4)i (ξ)N (4)

j (η)N (4)k (ζ)Bi,j,k , (2.2)


Figure 2.1: The B-spline control grid andthe corresponding surface parameteriza-tion of a wing geometry. The blue spheresrepresent the surface control points.

Figure 2.2: The computational grid de-scribed by the B-spline parameterizationshown on the left.

where N (4)i (ξ) ,N (4)

j (η) , and N (4)k (ζ) are fourth-order B-spline basis functions in each

parametric coordinate direction, and Bi,j,k are the coordinates of the de Boor control

points. If the parametric coordinates of all aerodynamic grid nodes are known, the

B-spline control points constitute a discrete mesh that analytically describes the aero-

dynamic grid. Thus, the B-spline volume control points may be used to control the

computational grid. The B-spline control points at the surface of the geometry can be

used as a means to control and manipulate the shape.

The B-spline control grid that defines the aerodynamic mesh is typically two orders

of magnitude coarser than the aerodynamic mesh itself. Thus, the number of B-spline

control points that analytically define the surface of interest is also orders of magnitude

lower than the node density of the computational surface grid. As a result, the coordinates

of these B-spline control points can efficiently be used as geometric design variables.

This is further illustrated in Figures 2.1 and 2.2, where the surface parameterization

and the control grid along with the corresponding computational grid are shown for

a planar wing. The B-spline control points may also be grouped together in regions

to provide a more intuitive way of controlling planform variables such as wing sweep

and dihedral angles [53]. This particular geometry parameterization scheme has been

successfully applied to many high-fidelity aerodynamic shape optimization problems in

the past [52, 54, 50, 55, 12, 13].

In order the determine the initial coordinates of the B-spline control points (that


analytically describe the aerodynamic grid), a least-squares fitting is carried out prior

to the optimization. In this approach, each block of the multiblock structured grid is

described by a separate grid of B-spline control points. At the block interfaces, the

control points are coincident for continuity. In addition, the clustering of the control grid

mimics the mesh spacing distribution of the aerodynamic grid due to the nature of the

knot distribution [52]. This characteristic helps the aerodynamic mesh movement scheme

to maintain the quality of the grid in the presence of large shape changes.

2.3 Mesh Movement

When the optimizer makes changes to the geometry of interest, both the aerodynamic

and structural grids must be moved to reflect the new shape. The resultant grids must

have sufficiently high quality for robust and accurate predictions of functionals of interest.

Furthermore, the changes must be carried out in a computationally efficient manner, such

that the cost of performing the high-fidelity analysis remains manageable. The purpose

of this section is to provide a brief overview of the grid movement strategies used in the

current framework.

2.3.1 Aerodynamic Grid Movement

The optimizer makes changes to the B-spline control points that define the surface ge-

ometry of interest. The aerodynamic grid movement scheme must efficiently determine

how these changes should be propagated throughout the rest of the computational grid

to maintain a valid and high-quality mesh. To achieve this goal, the framework uses a

method based on the equations of linear elasticity [56]. In this approach, the computa-

tional grid is treated as a linearly elastic model, where the Young’s modulus is spatially

varying for each element of the control grid depending on the cell volume and skewness.

Elements in the grid that are at the risk of becoming more skewed after a shape change

have a higher stiffness. This ensures that the overall quality of the grid is maintained.

Although this scheme is computationally more expensive than algebraic methods, such

as the one used by Leung et al. [57], it is exceptionally robust and well-suited to our

exploratory applications.

Since the computational grid is treated as a linearly elastic solid, an underlying as-

sumption is that the grid movements are small. However, this is in contradiction to our

objective, which is to be able to perform exploratory optimization involving large shape

changes. Thus, in order to accommodate this, the mesh movement is carried out in linear


increments of equal size, i = 1, . . . ,m, in accordance with the following equation [52]:

R(i)M (b(i−1),b(i)) = K

(i)M (b(i−1))[b(i−1) − b(i)]− f

(i)M (b(i)

s ) = 0 , (2.3)

where b(i)s is the vector of the displaced B-spline surface control points, f

(i)M is the implicit

force vector defined by b(i)s , R

(i)M is the mesh residual, and K

(i)M is the stiffness matrix for

the ith increment.

In this work, the number of increments used for design changes may be as high as

ten. This still comes at a manageable computational cost thanks to the reduced size of

the B-spline control grid, which has up to two orders of magnitude fewer nodes than the

aerodynamic grid. The choice of how many increments to use greatly depends on the

nature of the design problem as well as the flying condition considered. For instance,

the grid movement necessary to accommodate the structural deflections in a 2.5g critical

structural load condition could require as high as fifteen increments due to the presence of

large structural deflections during aerostructural analysis. Furthermore, the intermediate

aerostructural states arising from the partitioned solution strategy used for the purpose

of this thesis may involve unphysically large deflections that must be accommodated in

order to maintain the robustness of the analysis.

The ability of the aerodynamic grid movement scheme to handle large shape changes

is illustrated in Figure 2.3. In this case, an initially planar wing is manually morphed into

a C-wing. Note that since the linear elasticity equations are solved for the control grid,

the size of the problem is considerably smaller than if we were to apply the equations

to the entire aerodynamic grid. Figure 2.4 shows the aspect ratio distribution from the

initially planar geometry to the jig shape, as well as the final deflected shape resulting

from a subsequent aerostructural analysis based on the Euler equations. While the aspect

ratio distribution is not the only relevant criterion for aerodynamic mesh quality, Figures

2.3 and 2.4 suggest that the deformed grid is of sufficiently high quality after the large

shape change.

2.3.2 Structural Grid Movement

When the outer mold line of the wing is changed by the optimizer, the structural mesh

needs to be moved consistently with the B-spline surfaces in order to reflect the new shape

of the geometry of interest. This is also necessary for maintaining the consistency of the

force and displacement transfer between the aerodynamic and structural grids throughout

the optimization. To accomplish this goal, a surface based free-form deformation (FFD)


Figure 2.3: The aerodynamic grid movement scheme is used to manually morph a planarwing into a C-wing configuration. The initial control grids are shown in (a) and (b). Thecorresponding initial and final computational grids are shown in (c) and (d). In bothcases, the blue spheres represent the B-spline control points.

approach is used to parametrize the space inside the wingbox in between the upper and

lower portions of the B-spline surfaces [45].

The structural mesh movement scheme is specifically designed to handle large shape

changes for the purpose of exploratory optimization. In particular, it is essential to

minimize the amount of distortion introduced in the components during shape changes.

To demonstrate the ability of the structural mesh movement scheme to handle large

shape changes, we examine the same C-wing case from Section 2.3.1. In this case, a

planar wing geometry is manually morphed into a C-wing configuration, and a subse-


Figure 2.4: Cell aspect ratio distribution of the aerodynamic grids for the initial, jig, andfinal shapes.

quent aerostructural analysis based on the Euler equations is performed on the resulting

geometry. Figure 2.5 illustrates that the structural mesh movement scheme is capable

of preserving the consistency of the structural components in relation to the outer mold

line after the mesh movement is carried out.

2.4 Coupled Aerostructural Analysis

The aerostructural analysis problem involves driving the coupled aerostructural residual

down to a specified value ǫ, according to the following equation [45]:

‖RAS‖ =

∥∥∥∥∥∥∥

RA(q,b)

RS(q,u,b)

RM(u,b)

∥∥∥∥∥∥∥

≤ ǫ , (2.4)

where RA is the aerodynamic residual, RS is the structural residual, and RM is the

flow grid residual from the linear elasticity mesh equations. The aerostructural analysis

framework used incorporates an interface for the aerodynamic, structural, and mesh

movement modules. Since the two solvers for aerodynamics and structures are written

in different programming languages, Python is used to construct the interface routines.

These are necessary to access data in each module, and to allow for the exchange of

information amongst the two discipline solvers.


Figure 2.5: Illustration of the ability of the structural mesh movement scheme to preservethe shape of the structural components inside the wingbox over the course of a large shapechange.

2.4.1 Aerodynamic Analysis

The flow solver is based on an efficient Newton-Krylov approach for multiblock structured

grids using both the Euler and Reynolds-averaged Navier Stokes (RANS) equations for

turbulent flow [58, 59]. The aerostructural analysis framework is capable of performing

simulations using the RANS equations, but it is currently able to carry out optimization

based on the Euler equations only. Thus, all optimization results presented in this thesis

are based on inviscid flow with post-optimality estimates for viscous drag based on the

wetted surface area of the geometry of interest. Second-order summation-by-parts finite-

difference operators are used for spatial discretization along with simultaneous approxi-

mation terms for the imposition of boundary and block interface conditions. This specific

block interface treatment makes the aerodynamic grid generation relatively straightfor-

ward due to the fact that it only requires C0 continuity along the grid lines across any

block interface.

The aerodynamic module makes use of a sophisticated parallel implicit Newton-

Krylov approach for efficient calculation of the flow state during aerostructural analysis.

Quadratic convergence can be achieved using Newton’s method, but it requires an initial

guess that is sufficiently close to the solution. For this reason, the solution strategy is

broken down into two stages: approximate-Newton and inexact-Newton. The first phase

seeks to find a suitable initial guess for the Newton’s method as efficiently as possible.

The second phase uses this initial iterate to achieve fast convergence to the steady-state

solution.

The approximate-Newton phase uses a pseudo-transient continuation approach that


is similar to the implicit Euler time-marching scheme with local time linearization. How-

ever, since we are seeking an initial guess for Newton’s method, time accuracy is not

required. Thus, important modifications can be made to accelerate convergence to the

steady-state solution. These include using a first-order Jacobian with a lagged update

in addition to a spatially varying time step. These speed-up strategies do not affect the

accuracy of the steady-state solution. Once a suitable initial guess for Newton’s method

is found, the algorithm switches to the inexact-Newton phase, where fast convergence

to the steady-state solution is achieved. The aerodynamic solver has been extensively

studied and validated in the past, and has been successfully applied to both analysis and

optimization problems involving high-fidelity calculations [60, 53, 61, 62, 63]

2.4.2 Structural Analysis

The structural solver is the Toolkit for the Analysis of Composite Structures (TACS)

[64, 65]. It is a parallel finite-element framework developed specifically for large-scale

gradient-based design optimization of thin-walled aircraft structures. TACS makes use

of efficient numerical methods that exhibit good parallel scalability for the purpose of

calculating the structural quantities of interest and the corresponding gradients with

respect to the design variables. Although TACS is capable of performing nonlinear

structural analysis, only the linear capability is used in the context of the present thesis.

The structural components inside the wingbox are modeled with second-order mixed

interpolation of tensorial components (MITC) shell elements, which are well-suited for the

analysis of thin-walled structures and have been shown to prevent shear- and membrane-

locking [65].

Structural sizing of the aircraft involves determining whether any of the components

have reached the critical level of stress as a result of the aerodynamic loads. From an

optimization standpoint using a finite-element framework, this means that every element

of the structures grid would have to be constrained to avoid reaching the critical stress

value. This involves tens of thousands of constraints, which would be prohibitive in

terms of the computational cost of optimization. Therefore, we use the Kreisselmeier-

Steinhauser (KS) stress aggregation technique [51, 66, 38] in order to reduce the number

of necessary structural integrity constraints. This is especially important in light of the

fact that the gradient calculation makes use of the coupled adjoint [20, 19]. The structural

integrity constraint requires that the von Mises stress σk in element k is lower than the

yield stress σyield of the material. Thus, we would like to ensure that λk = (σk/σyield) ≤ 1,

where λk represents the von Mises stress criterion for element k. The KS constraint


formulation can then be defined as [38]:

KS = λmin −1

ρln

{Ne∑

k=1

exp[−ρ(λk − λmin)]

}

, (2.5)

where ρ is the KS weighting parameter, and Ne is the number of shell elements in the

structural grid. It is important to note that using this strategy leads to a conservative

structural design depending on the choice of the weighting parameter. We choose a

weighting parameter of 30 to achieve sufficiently smooth constraint functions based on

prior experimentation.

We use a material based on the 7075 Aluminum with a Poisson’s ratio of 0.33, Young’s

modulus of 70GPa, and yield stress of 434MPa. These material properties are used in

all cases throughout the thesis. Furthermore, a safety factor of 2.0 is applied to the yield

stress. In practical design of aircraft, a safety factor of 1.5 is often used for the yield

stress. However, since we consider a single 2.5g structural load case to size the structures,

it is appropriate to apply a higher safety factor in order to better capture the correct

trends in structural sizing of the wing.

2.4.3 Force and Displacement Transfer

An integral step during an aerostructural analysis involves transferring the aerodynamic

loads from the flow solver to the structural solver, and transferring the corresponding

displacements from the structural solver to the aerodynamic surface grid for the flow

solver to evaluate the new forces. This step occurs at the interface level between the two

discipline solvers, and is central to capturing the coupling between aerodynamics and

structures. TACS provides the means for this exchange of information to take place [65].

This involves constructing rigid links between each aerodynamic surface grid node and

the closest point on the structural model prior to the beginning of optimization. The

forces and displacements can be extrapolated using these rigid links according to the

following mathematical expression [65]:

uA = uS + (θS × r) , (2.6)

where [uS, θS] is the vector of structural displacements and rotations, and r is a vector

for the rigid link.

Once the new deflected aerodynamic grid is determined using the rigid links approach,

we still need to find the physical coordinates of the B-spline control points that would

analytically define it. This is necessary for the mesh movement scheme because it relies


on the movement of the B-spline surface control points. Thus, the B-spline control points

that define the deflected surface are determined through a least-squares fitting process

[45]. This will introduce some level of error to the displacement transfer because the

fitted surface will slightly differ from the actual displaced surface determined from the

structural analysis. We will investigate the implications of this error on the convergence

of aerodynamic quantities of interest in Section 3.1. Whether or not a more sophisticated

transfer scheme should be developed to eliminate this error is outside the scope of the

present thesis.

2.4.4 Aerostructural Analysis

The present framework uses the nonlinear block Gauss-Seidel method to find the solution

to Equation 2.4 [45]. This involves iterating between the aerodynamic, structural, and

mesh movement modules until a solution that satisfies the residual criteria for all modules

is obtained. The following provides an outline of the basic solution procedure at every

iteration of the aerostructural analysis:

1. Perform a flow analysis to find the aerodynamic forces;

2. transfer the forces to the structural model;

3. perform a structural analysis to find the new displacements;

4. transfer the displacements calculated by the structural solver to the aerodynamic

surface grid;

5. find the coordinates of the B-spline control points that best define the displaced

surface using a least-squares fitting procedure;

6. solve the mesh equations to update the aerodynamic grid.

This procedure is repeated until convergence of the residual vector is achieved for all

three modules. It is important to note that during the intermediate aerodynamic and

structural analyses, the residual vector is only reduced by a few orders of magnitude.

Requiring deep convergence during these intermediate steps is not necessary and leads

to an inefficient analysis.

The nonlinear block Gauss-Seidel methodology used in the current framework is

straightforward and requires minimal intervention in the existing routines of each dis-

cipline solver. However, it does introduce some convergence difficulties particularly for


problems that exhibit strong coupling between the aerodynamics and structures. Exam-

ples of this scenario include cases where the structural deflections are large, as a result

of considering an extreme flying condition, having a large span, or using more advanced

materials. Addressing these challenges is outside the scope of the present thesis, but

more efficient solution methods including monolithic strategies are being pursued by

other researchers.

2.5 Gradient Calculation

The present aerostructural optimization framework uses the discrete coupled adjoint

method [19, 20] for sensitivity calculations, as outlined in [45]. We seek to determine the

gradient of the objective function J with respect to the design variables v [19, 20]:

G =∂J

∂v

∣∣∣∣q,u,b

+∂J

∂q

∣∣∣∣v,u,b

dq

dv+

∂J

∂u

∣∣∣∣v,q,b

du

dv+

∂J

∂b

∣∣∣∣v,q,u

db

dv. (2.7)

We can differentiate Equation 2.4 with respect to the design variables in order to rewrite

Equation 2.7 in terms of the aerostructural Jacobian A:

∂

∂v

RA(v,q,u,b)

RS(v,q,u,b)

RM(v,q,u,b)

=

∂RA

∂v∂RS

∂v∂RM

∂v

q,u,b

+

∂RA

∂q∂RA

∂u∂RA

∂b∂RS

∂q∂RS

∂u∂RS

∂b∂RM

∂q∂RM

∂u∂RM

∂b

︸︷︷︸

A

dqdvdudvdbdv

= 0 (2.8)

⇒

dqdvdudvdbdv

= −A−1

∂RA

∂v∂RS

∂v∂RM

∂v

q,u,b

.

The two entries (∂RA/∂u) and (∂RM/∂q) in A are equal to zero because the flow and

aerodynamic grid residuals do not explicitly depend on the structural and flow states,

respectively. Equation 2.7 then becomes:

G =∂J

∂v

∣∣∣∣q,u,b

−[∂J∂q

∂J∂u

∂J∂b

]

A−1

︸︷︷︸

Ψ

∂RA

∂v∂RS

∂v∂RM

∂v

q,u,b

, (2.9)

where Ψ = [Ψq Ψu Ψb] are the Lagrange multipliers or the adjoint variables. These can


be evaluated numerically via the following expression using a linear block Gauss-Seidel

solution strategy:

AT

Ψq

Ψu

Ψb

=

∂J∂q∂J∂u∂J∂b

. (2.10)

Equation 2.10 is solved for each objective and constraint in the optimization problem

formulation. The cost of calculating the gradient remains constant with increasing num-

ber of design variables. This makes the adjoint method more efficient than the direct

method when the number of constraints plus the objective function is much smaller than

the number of design variables. This is an important consideration because the number

of design variables for optimization problems of interest in the context of the present

thesis is typically an order of magnitude larger than the number of constraints plus the

objective function. The total gradient can then be calculated as follows:

G =∂J

∂v

∣∣∣∣q,u,b

−[

Ψq Ψu Ψb

]

∂RA

∂v∂RS

∂v∂RM

∂v

. (2.11)

Zhang et al. [45] provide a thorough description of how each individual term in the

gradient expression is calculated.

2.6 Summary

The aerostructural optimization framework used in this work consists of six main com-

ponents: 1) a multiblock Newton-Krylov-Schur flow solver for the Euler and Reynolds-

averaged-Navier-Stokes equations [58, 59], 2) a finite-element structural solver for the

analysis and optimization of the structure [64], 3) a mesh movement technique based

on the linear elasticity equations for moving the aerodynamic grid during aerostructural

analysis and optimization [52], 4) a surface-based free-form deformation (FFD) technique

for moving the structures mesh during optimization [45], 5) a B-spline parameterization

method for geometry control which is coupled with the linear elasticity mesh movement

technique [52], and 6) the gradient-based optimizer SNOPT [47] with gradients calcu-

lated using the discrete-adjoint method for the coupled aerostructural system. Since the

two discipline solvers are written in different programming languages, we use Python to

provide an interface for the solvers [48]. Zhang et al. [45] provide a detailed description

of the framework.

Chapter 3

Validation and Verification of the

Methodology

The main purpose of this chapter is to validate and verify the numerical optimization

framework. We aim to demonstrate that our approach is suitable for performing high-

fidelity aerostructural optimization with large shape changes. This involves comparing

computational results to experimental data as well as performing numerical analysis and

optimization studies that test specific aspects of the tools.

3.1 Effects of Fitting on Functional Convergence

In practical application of an aerostructural optimization framework, it is important to

perform a post-optimality analysis to ensure that the main conclusions are independent of

the aerodynamic grid resolution. This involves analyzing the final optimized geometries

using a finer aerodynamic grid than the one used for performing the optimization. We

focus on the aerodynamic grid because our experience suggests that the aerodynamic

functionals of interest are substantially less sensitive to structural mesh refinement than

aerodynamic grid refinement. The main objective of this section is to investigate whether

or not the fitting errors introduced during the displacement transfer, as described in

Section 2.4.3, affect the grid convergence during the post-optimality analysis. The fitting

error is unacceptable if it leads to inconsistent conclusions on the coarse optimization

and fine analysis grids. This section compares the post-optimality analysis results of the

current framework with another one that uses the same transfer scheme, but captures

the displacements without fitting [57]. For clarity, this methodology is referred to as the

second framework. The main goal is to establish that the grid convergence behavior of

the aerodynamic functionals of interest does not change in the presence of the fitting

27

Chapter 3. Validation and Verification of the Methodology 28

Figure 3.1: The structural layout of thewingbox along with the outer mold line ofthe wing.

Figure 3.2: The planform of the wing alongwith the corresponding surface patches andcontrol points.

error by demonstrating that they become independent of the grid resolution within a

specific tolerance.

The planform of the optimized geometry used in this study is shown in Figures 3.1

and 3.2 and is based on the Boeing 737-900 wing. The optimization minimizes the drag

under cruise conditions while maintaining the structural integrity of the wing at a 2.5g

load condition. The cruise Mach number is 0.74 at an altitude of 30, 000 feet, while the

Mach number for the 2.5g load condition is equal to 0.85 at an altitude of 20, 000 feet.

The optimizer is free to change the sectional shape and twist of the wing along the span.

However, the thickness-to-chord ratio of the wing is not allowed to reduce by more than

10% of its initial value anywhere along the chord and span. The optimization is launched

with the RAE 2822 airfoil as the initial section. The optimized wing is then used for the

purpose of this study using the current framework.

The wing geometry used to perform the mesh refinement studies with the second

framework is obtained by performing a similar aerostructural optimization case using the

second framework. This optimized wing differs slightly in shape and structural thickness

distribution from the one described above due to the different optimization methodology

used. Therefore, there are some differences in the grid converged lift and drag values.

However, discussing these differences is beyond the scope of the present thesis because

the focus here is on the convergence trends.

Figure 3.3 shows the convergence of lift coefficient with increasing flow grid density

and control point resolution obtained from the present framework. The number of grid

nodes in the flow grid is given by N . Figure 3.4 shows the drag coefficient in drag counts

(10−4). The legends indicate the number of control points used to fit every patch on the

surface of the wing. The upper and lower surfaces of the wing each have 10 patches,

as illustrated in Figure 3.2. Only the number of chordwise control points is increased


2 4 6 8 10 12 14 16

x 106

0.500

0.502

0.504

0.506

0.508

0.510

0.512

0.514

N

CL

6 × 5 Control Points7 × 5 Control Points8 × 5 Control Points

Figure 3.3: Convergence of lift coefficientfor increasing grid density and controlpoint resolution.

2 4 6 8 10 12 14 16

x 106

117

118

119

120

121

122

123

124

N

DragCounts

6 × 5 Control Points7 × 5 Control Points8 × 5 Control Points

Figure 3.4: Convergence of drag coeffi-cient for increasing grid density and con-trol point resolution.

2 4 6 8 10 12 14 16

x 106

0.491

0.493

0.495

0.497

0.499

0.501

0.503

0.505

N

CL

Figure 3.5: Convergence of lift coefficientfor increasing grid density obtained fromthe second framework.

2 4 6 8 10 12 14 16

x 106

113

114

115

116

117

118

119

120

N

DragCounts

Figure 3.6: Convergence of drag coefficientfor increasing grid density obtained fromthe second framework.

because it has the greatest impact on whether or not the airfoil shape is captured correctly

by the fitting process. The sequence of aerodynamic grids is obtained by doubling the

total number of grid nodes at every level using the B-spline parameterization technique

to maintain a consistent distribution of grid nodes. These graphs demonstrate that lift

and drag change by less than 1% from the second finest to the finest grid level and are

largely unaffected by the number of control points.

Figure 3.5 shows the convergence of lift coefficient with increasing grid density and

control point resolution obtained from the second framework, which is capable of cap-

turing the displaced surface without fitting error. Figure 3.6 shows the drag coefficient.


Figure 3.7: Geometric planform of the HIRENASD wing model.

Figures 3.5 and 3.6 show the same convergence trends observed from the results obtained

by the present framework. This demonstrates that the fitting error introduced during the

transfer process does not significantly affect the convergence of aerodynamic functionals

of interest.

3.2 Validation Based on the HIRENASD Wing

Although the individual components of the aerostructural analysis capability have been

separately validated with experimental results, it is also important to compare the static

aeroelastic analysis results with experiment. However, it is quite difficult to find a suitable

experimental study for validation. Most of the test articles used in relevant experimental

studies are structurally too stiff to provide a meaningful way of assessing the deflections.

Furthermore, it is a challenge to replicate the exact experimental conditions and test

setup in many cases. Nonetheless, the HIgh REynolds Number Aero-Structural Dynam-

ics (HIRENASD) Project does provide some useful static aeroelastic data along with

the relevant geometries for validating the framework. Figure 3.7 shows the geometric

planform of the wing model used in the experiment.

The HIRENASD Project was initiated to provide experimental aeroelastic data for a

large transport wing-body configuration [67, 68, 69]. This section compares static aeroe-

lastic computational results obtained using the present framework with the HIRENASD


X

Z

Y

Cp: -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

ExperimentCFD+FEARigid CFD

y/b=0.95

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

ExperimentCFD+FEARigid CFD

y/b=0.80

Figure 3.8: Comparison of experimental and computational pressure coefficient resultsfor the HIRENASD wing geometry. The Mach number, angle of attack, and Reynoldsnumber are 0.80, 1.5◦, and 7.0 × 106, respectively. The experimental (black), staticaeroelastic (blue), and rigid-wing results (red) are shown for each spanwise station.

experimental data. In order to model the test conditions accurately, the Reynolds-

Averaged-Navier-Stokes (RANS) capability of the flow solver has been used here for

the purpose of the aerostructural analysis. The main objective is to demonstrate that

the correct physics are captured even in the presence of the fitting errors. Furthermore,

the results of this section motivate the future extension of the current framework to

aerostructural optimization based on the RANS equations.

The test condition Mach number, angle of attack, and Reynolds number based on the

mean aerodynamic chord are 0.80, 1.5◦, and 7.0 × 106, respectively. An aerostructural

analysis is performed to obtain the computational results. The one-equation Spalart-

Allmaras turbulence model is used. Osusky and Zingg [59, 63] provide comprehensive

details and validation of the RANS flow solver.

A multiblock structured grid is used for the aerodynamic solver with 3, 548, 095 nodes.

The average y+ value is equal to 0.24. The finite-element model provided by the HIRE-

NASD project contained solid elements. However, the structural solver, TACS, accepts

MITC shell elements only. Furthermore, the current structural model does not include

the leading and trailing edges. Thus, an effort has been made to ensure that the struc-

tural finite-element model used in this analysis represents the original structure of the

HIRENASD wing as closely as possible within these constraints. This finite-element

model for the structures has approximately 38, 000 second-order MITC shell elements.

Figure 3.8 provides a comparison of the computational static aerostructural results


with the experimental data. The rigid-body results (where there are no structural deflec-

tions) are also provided for reference. Figure 3.8 demonstrates that the static aerostruc-

tural results obtained from the present framework consistently show much better agree-

ment with the experimental data than the rigid CFD computations, especially towards

the wingtip. Moreover, the computed tip deflection of 12.6 mm is in excellent agreement

with the experimental value of 12.5 mm [70].

3.3 Inviscid Transonic Wing Sweep Optimization

There is a fundamental trade-off between weight and drag in the design of aircraft wings.

For instance, at transonic speeds, increasing the quarter-chord sweep angle of a wing

reduces the wave drag, but the corresponding increase in the weight may overshadow the

drag benefit in such a way that the resulting range is reduced. The main objective of this

section is to investigate whether the current framework is able to capture this important

trade-off correctly in the context of an aerostructural optimization of a conventional

planar wing.

The choice of the objective function in optimization influences the final optimized

design. In the practical design of aircraft wings, the objective is carefully chosen based

on the design requirements for a particular aircraft. However, for the purpose of this

study, only the trade-off between weight and drag is of interest. For this reason, the

objective function has the form

J = βD

D0

+ (1− β)W

W0

, (3.1)

where β is a parameter between zero and unity, D is the inviscid drag of the wing in

cruise, W is the calculated weight of the wing satisfying the structural failure constraints

at a 2.5g load condition, and D0 and W0 are the respective initial values. As β is varied

from zero to unity, the emphasis on drag in the objective function is increased while

reducing the emphasis on weight. Three values for β have been chosen: 0.50, 0.75, and

1.00.

There are two lift constraints; one corresponds to the cruise load condition, the other

to the 2.5g load condition. The cruise Mach number is 0.785 at an altitude of 35, 000 ft,

while the Mach number for the 2.5g load condition is 0.798 at an altitude of 12, 000 ft.

Since the weight of the wing is a function of the structural thickness values, it changes

over the course of the optimization. The total weight of the aircraft is assumed to be

equal to the computed weight of the wing plus a fixed weight of 785, 000N for the whole


aircraft. This fixed weight is estimated based on the maximum takeoff weight of a Boeing

737-900 discounted by the approximate wing weight. The approximate wing weight is

equal to 7% of the maximum takeoff weight. It is important to note that we are ignoring

the lift produced by the fuselage. Furthermore, we are ignoring the variations in the

weight of the fuselage including the center wingbox. These are potentially important

nonlinear effects that should be taken into account in the future. However, the majority

of the lift is produced by the wing. In addition, based on empirical relations between the

wing weight and the fuselage weight, variations in the wing weight lead to small changes

in the weight of the fuselage [71]. Therefore, the impact of neglecting these effects on our

main conclusions is most likely small.

The stresses on the wing due to the aerodynamic loads at the 2.5g load condition are

aggregated using three KS functions with an aggregation parameter of 30.0. There is one

KS function for the ribs and spars, one for the top skin, and one for the bottom skin of

the wing. These KS functions are constrained to ensure structural integrity of the wing.

The reduction in the thickness-to-chord ratio of the wing is limited to 10% of the initial

value.

The aerostructural optimizations are initiated with a planar wing geometry based on

the Boeing 737-900 planform. Figure 3.1 shows the layout of the wing and the structures

inside the wingbox. Initially, a coarse CFD grid is used with 193, 536 nodes and 112

blocks. Once the optimizer satisfies the nonlinear constraints on this coarse mesh, the

optimization is restarted using a finer mesh with 653, 184 nodes and 112 blocks. While the

lift and drag values may not be accurate on this grid, their dependence on the geometry

will be accurately captured [54]. Figure 3.9 shows the grid resolution of the surface and

symmetry plane for the fine mesh. Each block is parameterized with 6 × 6 × 6 control

points. The upper and lower surfaces of the wing are parameterized with 10 B-spline

surface patches. The structures mesh has 30, 473 second-order MITC shell elements.

The initial airfoil is the RAE 2822. The optimizer is free to change the tip twist

and section shape at 16 spanwise stations in addition to the quarter-chord sweep angle.

Each spanwise station is parameterized by 24 control points, 14 of which are design

variables. The remaining 10 control points are fixed to ensure curvature continuity on

the surface of the wing. The sweep angle is varied in such a way that the initial span

of the wing is maintained. The total number of geometric design variables is equal to

226. Furthermore, there are a total of 156 structural design variables which determine

the thickness of structural components inside the wingbox. Finally, there are two angle

of attack design variables; one for cruise, the other for the 2.5g load condition.

As β is varied from 0.5 to 1.0, i.e. as more emphasis is placed on drag and less on


Figure 3.9: Grid resolution of the surfaceand symmetry plane for the fine optimiza-tion mesh.

Figure 3.10: The planforms for the threewings show that the optimal sweep angle,Λ, increases with increasing β, i.e. increas-ing emphasis on drag.

0 2 4 6 8 10 12 14 16−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

y [m]

Lift

CruiseCruise Elliptical2.5g Load2.5g Elliptical

Figure 3.11: Cruise and 2.5g load distri-butions along the span of the wing for theβ = 1.0 case.

0 2 4 6 8 10 12 14 16−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

y [m]

Lift


Figure 3.12: Cruise and 2.5g load distri-butions along the span of the wing for theβ = 0.5 case.

weight, the optimizer should take advantage of the available freedom to increase the

sweep angle of the wing in order to reduce drag. As a result, the sweep angles of the

optimized designs should increase with increasing β. Figure 3.10 shows the planform of

the three optimized wings. The wing with β = 1.0 has a 16% lower drag and a 49%

higher weight than the wing with β = 0.5. It is clear that the optimizer has produced

the expected trend. This demonstrates that the present aerostructural optimization

framework is capable of correctly capturing the trade-off between weight and drag.

Figures 3.11 and 3.12 show the spanwise lift distributions at the cruise and 2.5g load

conditions for the β = 1.0 and β = 0.5 cases, respectively. All lift values have been

normalized by the elliptical lift at the root of the wing for cruise. For the β = 1.0


Figure 3.13: The optimized thickness distribution of skin elements for the β = 0.5 case.

Figure 3.14: The optimized thickness distribution of skin elements for the β = 1.0 case.

case, the cruise lift distribution closely follows the elliptical load, while the 2.5g spanwise

lift distribution is much more triangular in comparison to cruise. This means that the

optimizer is taking advantage of aeroelastic tailoring to minimize inviscid drag in cruise

both by maintaining an optimal lift distribution and increasing the quarter-chord sweep

angle. This is done while maintaining the structural integrity of the wing at the 2.5g

load condition by reducing the tip loading. It is also insightful to examine the β = 0.5

case. With β = 0.5, the lift distributions for both the cruise and 2.5g load conditions

are triangular because the objective function is more heavily biased towards the weight

of the wing.

Figures 3.13 and 3.14 show the optimized skin thickness distribution for the β = 0.5

and β = 1.0 cases, respectively. The optimizer has increased the thickness inboard in both

cases. Furthermore, it is clear that the β = 0.5 has lighter components in comparison

to the β = 1.0 case. Although only a single critical structural load condition has been

considered, these results show that at least some of the correct trends in the structural

sizing of a wing have been captured.


Figure 3.15: The four structural layouts considered.

3.4 Effects of Structural Topology on Optimization

Trends

One important limitation of the current methodology is that the structural layout inside

the wingbox remains constant throughout the course of an optimization. This means

that the number of structural components is fixed in addition to their topology relative

to the outer mold line of the wing. However, we can experiment with different structural

layouts. The main objective of this section is to conduct a preliminary investigation on

whether or not the choice of the structural layout has a significant influence on the main

sizing trends of the wingbox and the resultant aerostructural interactions.

In order to investigate the implications of using different structural layouts on the final

sizing trends from our aerostructural optimizations, we choose four different topologies

that are inspired by publicly available documentation on today’s modern wing structures.

These layouts are shown in Figure 3.15. Skin elements are omitted here for clarity. An

effort has been made to ensure that the rib spacings remain approximately constant in

all cases. One critical factor in choosing the rib spacing is the imposition of skin buckling


Figure 3.16: The geometric parameterization and deign variables for the purpose of thestructural layout optimizations.

constraints in practical design of wings to ensure the integrity of the skin under critical

buckling loads. However, our current methodology does not provide the capability to

impose buckling constraints. Thus, we maintain a rib spacing that is similar to that of

the Boeing 737NG wing structure in all cases. We discuss the implications of this and

other limitations on our final optimization results in Section 4.1.3.

Since the wing thickness-to-chord ratio of a well-designed wing generally decreases

towards the wingtip, the structures finite element mesh that is made of regular quadri-

lateral elements may become more distorted at the wingtip than elsewhere. This may

lead to a poor quality mesh and influence our final conclusions. For this reason, layouts

number 3 and 4 include fewer components toward the tip to examine the possible effects.

In theory, the force and displacement transfer scheme should still be able to perform its

intended function without compromise in either case [45].

The four aerostructural optimizations are initiated with a starting geometry based

on the Boeing 737-900 planform. We initially use a coarse CFD grid with 193, 536 nodes

and 112 blocks. The optimizations are continued on a finer grid with 653, 184 nodes

and 112 blocks once the merit function plateaus on the coarse grid. This is done in

part to minimize the challenges associated with converging these difficult and expensive

numerical optimization cases. Each block of both grids is parameterized with 6 × 6 × 6

control points. The upper and lower surfaces of the wing are parameterized with 10 B-

spline surface patches. The finite element grids for the structures all have approximately

30, 000 second-order shell elements.

The objective function is the inviscid drag of the wing in transonic cruise conditions.

Figure 3.16 shows the geometric parameterization and design variables used in all cases.

The optimizer can only vary the sectional shape and twist distribution of the wing along

the span. There are two lift constraints: cruise and 2.5g maneuver. The cruise and 2.5g


Iteration Number

0 50 100 150 200 250

Merit

Function

0

2

4

6

8

10

12

14

Figure 3.17: The merit function conver-gence history for the optimization usingstructural layout number 3.

Iteration Number

0 50 100 150 200 25010−8

10−6

10−4

10−2

100

Feasibility

Optimality

Figure 3.18: The optimality and feasibilityconvergence histories for the optimizationusing structural layout number 3.

load conditions are the same as the ones used in Section 3.3. Furthermore, the same

structural sizing strategy and material are used for the purpose of this investigation.

The reader is reminded that the inviscid drag is not a purely aerodynamic objective

function in the context of aerostructural optimization. This is due to the fact that the

inviscid drag of the aircraft does indeed depend on the total weight by the means of the

imposed cruise lift constraint. In other words, the optimizer will not increase the weight

of the wing to gain a potential aerodynamic advantage if it means that the lift required

will lead to a higher lift-induced drag.

Figures 3.17 and 3.18 provide the optimization convergence history for the layout

number 3 on the coarse grid. The merit function is equal to the objective function once the

optimizer satisfies all nonlinear constraints. Optimality is a measure of the Lagrangian

gradient, and feasibility is the highest nonlinear constraint violation. We would consider

an optimization fully converged when both optimality and feasibility measures have been

reduced to machine-zero. However, in the vast majority of cases in practice, this may

not even be possible because it requires a large amount of computational resources. As

a result, we typically consider an optimization converged when the merit function has

plateaued and all nonlinear constraints are satisfied to a tolerance of 10−6. It is clear from

Figures 3.17 and 3.18 that this particular optimization case has converged sufficiently.

The same convergence trends hold true for the rest of the cases.

Figure 3.19 shows the optimal skin thickness distribution along the span for all four

cases. In each case, regardless of the layout used, the aerostructural optimizer thickens

the skin inboard especially near the wing crank. This is consistent with our expectation


Figure 3.19: The optimized structural thickness distributions of the structures for allfour optimization cases.

from the 2.5g critical load condition. The same trends hold for the ribs and spars as well.

This means that although the individual component thickness values do vary depending

on the structural layout used, the main sizing trends remain the same. These trends

have also been observed in other numerical studies in the literature that include more

critical structural load conditions in addition to buckling [40], and studies that make

use of multipoint structural topology optimization techniques [72]. However, it should

be noted that some structural layouts are more practical than others. For instance,

the sudden variations in the thickness values along the chordwise direction observed in

layout number 3 makes it a less feasible structure from a manufacturing standpoint.

These considerations, however, are largely outside the scope of the present thesis.

An important consideration in the context of fully coupled high-fidelity aerostructural

optimization is whether or not the optimizer is able to take advantage of the bend-

twist coupling of the wing in order to achieve aeroelastic tailoring. In our specific case,

aeroelastic tailoring refers to the ability of the optimizer to maintain an optimal lift

distribution in cruise while reducing the tip loading as much as possible at the 2.5g load

condition to reduce weight. This can be achieved by introducing tip washout (twisting the

tip downward at the deflected state under the aerodynamic loads to reduce the effective

angle of attack). We would like to ensure that our choice of the structural layout does not

inhibit the ability of the optimizer to accomplish this goal. For this reason, we examine

the optimal lift distributions at both load conditions for all cases and confirm that the

trends are the same. Figures 3.20 through 3.23 show the optimal lift distributions for


y (m)0 5 10 15

Normalized

Lift

0.0

0.5

1.0

1.5

2.0

2.5

3.0Cruise

Elliptical

Maneuver

Figure 3.20: Optimal lift distributions forthe cruise and 2.5g load conditions ob-tained from optimization using structurallayout number 1.

y (m)0 5 10 15

Normalized

Lift

0.0

0.5

1.0

1.5

2.0

2.5

3.0Cruise

Elliptical

Maneuver


y (m)0 5 10 15

Normalized

Lift

0.0

0.5

1.0

1.5

2.0

2.5

3.0Cruise

Elliptical

Maneuver


y (m)0 5 10 15

Normalized

Lift

0.0

0.5

1.0

1.5

2.0

2.5

3.0Cruise

Elliptical

Maneuver


cruise and maneuver for all structural layouts used. These figures demonstrate that

the aeroelastic tailoring trends have remained consistent in all cases regardless of the

individual layout used.

Table 3.1 provides a few important optimization results for each case. It is clear that

the differences between all four cases are small. The results and conclusions should be

viewed as a first attempt to ensure that keeping the structural topology fixed relative

to the outer mold line of the wing does not inhibit the optimizer from capturing impor-


Table 3.1: The optimization results obtained from the structural layout investigation

Parameter Layout 1 Layout 2 Layout 3 Layout 4Aircraft Weight (kN) 732 732 729 727Aircraft Drag (kN) 47.2 47.2 47.1 47.0

tant interdisciplinary interactions. We emphasize that our aim is to take a step further

from purely aerodynamic shape optimization towards including the effects of weight and

structural deflections in addition to drag. Our goal is not to perform detailed design

studies that are typically done in later stages of the development of commercial aircraft.

Instead, we would like to examine design trends from aerodynamic shape optimization

that are shown to have potential for significant efficiency improvements using aerostruc-

tural optimization tools.

The present study shows that keeping the structural layout fixed does not have a

significant effect on the main trends. However, it does not address the possibility that a

given structural layout may become unsuitable due to large shape changes as a result of

optimization. This can be addressed by conducting aerostructural topology optimization

[73], where the structural topology is optimized concurrently with the shape. However,

this is outside the scope of the present thesis. In our optimization cases, we have made an

attempt to ensure that the structural layout is reasonable and does not become infeasible

due to shape changes.

3.5 Summary

The main objective of this chapter is to verify and validate various aspects of the

aerostructural analysis and optimization methodology used for the purpose of our in-

vestigations. We perform a systematic aerodynamic mesh refinement study in order to

examine the effect of the fitting step during the force and displacement process on the

convergence of functionals of interest including lift and drag. Our investigation shows

that the convergence behavior of functionals of interest is largely unaffected by the fitting

process. Results obtained from the framework are also compared with experimental data

from the HIRENASD project. This investigation demonstrates that the aerostructural

analysis methodology is capable of capturing the physics accurately. Furthermore, there

is good agreement between the computed tip deflection and the experimental value. Our

wing sweep study shows that the framework is able to capture the trade-off between

weight and drag correctly. Finally, we have examined the effect of changing the layout

of the ribs and spars on the main trends. This preliminary study shows that the choice


of the layout does not have a significant effect on the behavior of the aerostructural op-

timizer. However, this assumes that the layout is reasonable, including the constraint

that the spacing between ribs is not significantly increased. In the cases considered in

this thesis, the rib spacings remain reasonable throughout the optimizations because the

structural spans do not increase significantly.

Chapter 4

Aerostructural Optimization of

Nonplanar Wings

In this chapter, we investigate the potential efficiency gains provided by nonplanar wing

configurations from an aerostructural perspective. We consider two classes of nonplanar

wings, wings with winglets and the drooped wing configuration. We compare the possible

efficiency gains from nonplanar designs in relation to their planar counterparts of the same

projected span.

4.1 Winglet Shape Optimization

The present study uses a step-by-step approach to studying winglets where purely aero-

dynamic shape optimization is conducted first before performing fully-coupled aerostruc-

tural optimization. This approach is central to understanding the arguments presented

in this section. Therefore, it is worthwhile to provide a brief description of what each

individual step is intended to achieve. In Section 4.1.2, we conduct purely aerodynamic

shape optimization in order to establish that the numerical tools work as intended and

recover the expected trends. Moreover, this section demonstrates that a winglet oriented

downward is more effective than a winglet oriented upward, even for purely aerodynamic

shape optimization with no deflections. Section 4.1.3 presents results from fully-coupled

aerostructural optimization for planar and wingletted wings of the same projected span.

The main objective is to see what the aerostructural optimizer does given the freedom to

choose between a winglet-up and winglet-down configuration starting from an initially

planar wing. In Section 4.1.4, we aim to determine what the optimizer chooses to do

given the freedom to either keep the wing planar or create a winglet for each config-

uration. Finally, Section 4.1.5 will explore the possibility that a wingletted wing may

43

Chapter 4. Aerostructural Optimization of Nonplanar Wings 44

Figure 4.1: The planform of the baseline planar configuration is based on the Boeing737NG.

provide a higher efficiency improvement in high-lift, low-speed conditions where the ratio

of the induced to total drag is higher than in cruise. Sections 4.1.2 and 4.1.3 use the

Boeing 737-600 wing as the baseline configuration, and Sections 4.1.4 and 4.1.5 consider

the larger Boeing 737-900 as the baseline design. This will help to gain a better under-

standing of the sensitivity of the conclusions with respect to different baseline aircraft.

The B737-600 and B737-900 aircraft are part of the same family known as the B737NG

and have identical wing planforms. However, they differ in terms of the maximum takeoff

weight, range, and length of the fuselage.

4.1.1 Baseline Geometry

The baseline wing geometry for this study is based on the planform of the Boeing 737NG

wing shown in Figure 4.1 with the RAE 2822 supercritical airfoil. All of the wingletted

and planar configurations have the same projected span. There are two main reasons

for constraining the projected span. First, we are assuming that there is an airport gate

constraint that prohibits any increase in the span relative to the baseline wing. Second,

if the wingletted wings grow in span, then it will not be clear whether any aerodynamic

benefit is due to the increased span or the nonplanar feature. This is due to the fact that,

based on linear aerodynamic theory, the induced drag is reduced in a quadratic fashion

with any increase in wing span. The relationship between induced drag and wing span

is given by Dinduced = L2/(πb2q∞e), where Dinduced is the induced drag, L is the lift, b is

the span, q∞ is the dynamic pressure, and e is the span efficiency factor [8].


The Boeing 737NG aircraft with winglets have a span that is 4.5% larger than the

span of the Boeing 737NG without winglets [74]. According to linear aerodynamic theory,

the total drag reduction in cruise as a result of the same span increase is approximately

4%. The Boeing Commercial Airplanes company claims a 3% to 4% total cruise drag

reduction for the wingletted wings in comparison to the baseline planar wing [33]. As a

result, it is not clear how much of this improvement is due to the winglet alone. This

is one of the main reasons that we do not allow our wingletted wings to grow in span

relative to the planar configuration. Thus, we should be cautious in comparing the

potential aerodynamic performance improvement from this investigation with the values

published for the Boeing 737NG blended winglets, since the latter involve an increase in

span.

4.1.2 Aerodynamic Shape Optimization

Although the primary focus of the present study is on using high-fidelity aerostructural

analysis and optimization, it is insightful to begin this investigation on winglets by con-

ducting purely aerodynamic shape optimization. This step-by-step approach, where the

optimization studies are first performed using high-fidelity aerodynamic analysis based

on the Euler equations, will lead to a solid understanding of the best-case scenario for

wingletted wings. It also helps to establish the methodology used by reproducing sim-

ilar trends observed in previous studies [54, 75, 34]. In this section, we will show that

the benefits of winglets are smaller for transonic wings of fixed span than might appear

from studies of wings at lower speeds. Furthermore, we will provide evidence that the

winglet-down configuration offers a larger drag reduction than the winglet-up design in

the context of purely aerodynamic shape optimization.

There are three configurations considered in this study: winglet-up, winglet-down,

and planar. All three configurations have identical projected spans. Figures 4.2 and

4.3 show the geometric parameterization and design variables for the winglet-up and

winglet-down cases, respectively. Each surface patch on the geometry is parameterized

by 6 × 6 B-spline control points. There are a total of 252 section control points of

which 224 are design variables. The coordinates of the remaining control points are a

function of the neighboring design variables such that slope continuity is maintained over

the entire surface of the wing. The optimizer has the freedom to change the sectional

shape of the wing by manipulating the z-coordinates of the control points at all spanwise

stations across the wing and winglet. The winglet cant angle is constrained to +71◦

for the winglet-up, and −71◦ for the winglet-down configuration. This replicates the


Figure 4.2: The geometric parameteriza-tion and design variables for the winglet-up configuration.

Figure 4.3: The geometric parameteriza-tion and design variables for the winglet-down configuration.

baseline configuration since the cant angle of the Boeing 737NG blended winglets is

equal to +71◦. The planar configuration has a cant angle of zero. In each case, the wing

sweep and winglet cant angles are fixed. As a result, only the sectional shape and twist

distribution of the wings and winglets are free to vary over the course of the optimization.

The thickness-to-chord ratio of the wing and winglet cannot reduce by more than 30%

to avoid unrealistic geometries. Furthermore, the volume of the wing is constrained to

the initial value.

The objective function is the inviscid drag of the wing in cruise. The cruise Mach

number is M = 0.785 at an altitude of 35, 000 ft. In all three cases, the lift constraint is

set to CL = 0.486. This lift constraint is based on performing a fully coupled aerostruc-

tural optimization for a planar wing. The aerodynamic optimizations are performed on

a grid with 653, 184 nodes. This grid does not have sufficiently fine grid spacings for

accurate prediction of drag, but is able to capture the dependence of drag on the ge-

ometry. The final optimal geometry is analyzed using a grid with 36, 793, 008 nodes to

provide a more accurate estimate of the final objective function values during the post-

optimality analysis. We have verified that any relative performance benefit obtained on

the optimization grid translates in the same way to the fine mesh. The parameter refine-

ment capability of the B-spline volume grid is used in order to confirm that drag values

obtained from optimized planar and nonplanar wings exhibit similar grid convergence

characteristics. This indicates that the numerical errors on the distorted grids resulting

from the optimization produce comparable numerical errors to the original grid around

the baseline geometry. Figures 4.4 and 4.5 show the optimization convergence for the

winglet-up case. It is clear from Figures 4.4 and 4.5 that the optimization problem has

converged to an optimal design. An important secondary conclusion is that the optimizer


Iteration Number

0 200 400 60010−8

10−6

10−4

10−2

100

Feasibility

Optimality

Figure 4.4: The convergence of op-timality and feasibility measures forthe winglet-up aerodynamic optimizationcase.

Iteration Number

0 200 400 600

Merit

Function

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Figure 4.5: The convergence of the La-grangian merit function for the winglet-upaerodynamic optimization case.

y (m)0 5 10 15 20

Norm

alizedLift

0.0

0.2

0.4

0.6

0.8

1.0

1.2

CruiseElliptical

Figure 4.6: The aerodynamically optimallift distribution for the planar configura-tion.

y (m)0 5 10 15 20

Norm

alizedLift

0.0

0.2

0.4

0.6

0.8

1.0

1.2

CruiseElliptical

Figure 4.7: The aerodynamically optimallift distribution for the winglet-up config-uration.

achieved the majority of the reduction in the merit function over the first 200 iterations.

This provides some practical justification for ending the optimization once the majority

of the merit function reduction is achieved, especially in high-fidelity optimization where

the cost of a function evaluation and the corresponding gradient calculation is high.

Figures 4.6, 4.7, and 4.8 show the optimal lift distributions obtained for the planar,

winglet-up, and winglet-down configurations, respectively. The lift distribution for the

planar wing closely follows the elliptical distribution, suggesting that the optimizer has

minimized the induced drag. The optimal spanwise lift distributions for the winglet-up

and winglet-down configurations have a higher tip loading in comparison to the pla-


y (m)0 5 10 15 20

Norm

alizedLift

0.0

0.2

0.4

0.6

0.8

1.0

1.2

CruiseElliptical

Figure 4.8: The aerodynamically optimallift distribution for the winglet-down con-figuration.

Planar Up Down

Norm

alizedDrag

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Figure 4.9: Comparison of total drag incruise for the planar and wingletted wingsusing the same lift constraint in all threecases.

nar case. This is consistent with previous studies that show aerodynamically optimal

wingtip devices tend to increase the tip loading [54, 75, 33]. This increase in tip loading

has important implications in terms of wing weight that are ignored by the aerodynamic

shape optimizer. A higher tip loading tends to increase the weight of the wing by requir-

ing a heavier root structure to support the greater moment. The implications of these

differences in weight are further explored and discussed in Appendices A and B.

Figure 4.9 provides a comparison of the total drag in cruise between the wingletted and

planar wings. A post-optimality viscous drag estimate of the entire aircraft is included in

each case based on the wetted surface area in order to account for the increased viscous

drag of the wingletted configurations as a result of the larger surface area in comparison

to the planar wing. For the baseline aircraft, we use the Vehicle Sketch Pad modeling

tool developed at NASA Langley Research Center [76] to estimate that the viscous drag

at cruise is equal to 192 drag counts. The larger viscous drag for the wingletted wings is

taken into account based on the increase in the wetted surface area relative to the baseline

case [77]. Figure 4.9 indicates that the winglet-up configuration provides a 1.1% total

drag reduction in comparison to the planar wing of the same projected span. The reader

is reminded that, unlike the B737NG wings with blended winglets, these wingletted wings

have the same projected span as the planar wing. Hence the benefit of the winglet is

smaller than that achieved when the winglet leads to a span increase. It is also noteworthy

that the optimal winglet oriented downward produces a drag reduction of 2.6% relative to

the planar wing, more than twice that of the optimal winglet oriented upward. In order

to examine why the winglet-down configuration produces a larger benefit, it is insightful


Figure 4.10: Contours of x-vorticity behind the trailing edge of the wing for all threecases in cruise condition (M = 0.785 at 35, 000 ft). The dashed line marks the locationof the maximum strength vortex at the tip of the wing for the planar configuration.

to plot contours of x-vorticity behind the trailing edge of the wing in each case. These

provide some insight into the physical mechanism by which induced drag is generated.

Figure 4.10 shows that the winglet-down configuration is able to push the tip vortex

further away from the wake of the wing in the positive y-direction. This means that the

induced velocity (downwash) on the wing is reduced because it has an inverse relation

with the distance from the center of the vortex core [8]. This phenomenon has also

been observed in other optimization studies using the Euler [78] and Reynolds-averaged

Navier-Stokes equations [34], where the winglet-down configuration is shown to provide

a larger drag reduction in comparison to an optimal winglet-up design.

4.1.3 Aerostructural Optimization with Variable Winglet Cant

Angle

This section presents results obtained from fully coupled high-fidelity aerostructural op-

timization, where the effects of weight and structural deflections are taken into account

in addition to drag. The baseline geometry is still based on the Boeing 737NG planform

shown in Figure 4.1 with the RAE 2822 airfoil section. We consider the same three

configurations as before: winglet-up, winglet-down, and planar. The projected span is

constrained in all cases. The optimizer is free to vary the wing sweep and winglet cant

angles in addition to the section shape and twist distribution of the wing along the span,

as shown in Figure 4.11. Note that in this case, the optimizer is free to choose the

optimal wingtip configuration by varying the cant angle design variable. For a winglet

configuration to form from the initially planar wing, the cant angle is allowed to vary be-


Figure 4.11: The geometric parameter-ization and design variables for the ex-ploratory winglet optimization case.

Figure 4.12: Three possible wing shapespermitted by the parameterization. (1)and (2) are two variations of a winglet-down configuration, and (3) is planar.

tween −90◦ and +90◦. A positive cant angle corresponds to a winglet-up and a negative

cant angle to a winglet-down configuration. The optimizer is also free to keep the wing

planar by forcing the cant angle to be equal to zero. The vertical height of the wingletted

feature can increase by up to 6% of the wing span. To further clarify this, Figure 4.12

shows a simplified front view of three possible wing shapes permitted by this particular

parameterization. The parameter θ represents the cant angle. In all cases, the projected

span of the wing at the undeflected state remains constant even though the optimizer is

free to move the location of wing-winglet junction along the span. In Figure 4.12, the

curved wing-winglet junction is depicted by a red circle, and its physical location along

the span is measured by the parameter ℓ.

The choice of the objective function significantly influences the final optimized design.

For practical design of wings, the objective is carefully chosen based on the operating

requirements for a particular aircraft. However, our main goal in this study is not to

discover the best wing for a particular commercial airplane. Instead, we are interested

in the fundamental trade-offs between drag and weight that are involved in the design of

wings with winglets. For this reason, we choose an objective function of the form

J = βDinviscid

D0

+ (1− β)W

W0

(4.1)

where β is a parameter between zero and unity, Dinviscid is the inviscid drag of the wing in

cruise, W is the calculated weight of the wing satisfying the structural failure constraints

at the 2.5g load condition, and D0 and W0 are the respective initial values. As we vary

β from zero to unity, we place more emphasis on drag and less on weight. This allows

us to focus on the most relevant aspects of aerostructural design for various geometries

and gain insight into the fundamental trade-off between weight and drag. Two values


for β have been chosen: 0.5 and 1.0. The same viscous drag estimate as the one used in

Section 4.1.2 is included in the post-optimality calculations [77, 76].

Table 4.1 lists the constraints for each optimization test case considered in this study.

There are two lift constraints; one corresponds to the cruise load condition, the other to a

2.5g load condition. The cruise condition is M = 0.785 at an altitude of 35, 000 ft, while

the 2.5g load condition is M = 0.798 at an altitude of 12, 000 ft. Since the weight of the

wing is a function of the structural thickness values, it changes over the course of the

optimization. The total weight of the aircraft is assumed to be equal to the computed

weight of the wing plus a fixed weight of approximately 650, 000N. This fixed weight

is estimated based on the maximum takeoff weight of a Boeing 737-600 discounted by

the approximate wing weight. The wing weight is approximately equal to 8.5% of the

maximum takeoff weight.

In practical wing design, the structures are sized based on many critical structural

load conditions in order to ensure the structural integrity of the wing. The structural

sizing has a profound effect on the aerodynamic performance of the wing. By considering

a single 2.5g load condition, we aim to capture some of the effects of structural sizing on

the trade-off between drag and weight [31, 38, 37]. This means that we must constrain

the calculated stresses on the structures at the 2.5g load condition to prevent structural

failure. Three KS constraints with a weighting parameter of 30 are used at the maneuver

condition: one for the ribs and spars, one for the top skin, and one for the bottom skin.

These will ensure that none of the 30, 473 finite elements in the structural model exceed

the yield stress.

Table 4.2 provides a list of the design variables used in each case. These cases have a

total of 389 design variables that control the angle of attack, geometric shape, and struc-

tural thickness distribution of the wing. There are two angle of attack design variables:

one for cruise, the other for the 2.5g load condition.

Figure 4.13 shows the layout of the primary structural components in relation to

the outer mold line of the wing. The structural layout does not include the leading

and trailing edges because the current model cannot accurately represent them. This

does not adversely affect our conclusions because these secondary wing structures do not

carry a significant amount of load in comparison to the primary structural components.

Furthermore, their weight is mostly dependent on the wing projected area [30], which

remains constant in our study. The primary components consist of 30 ribs, 3 spars,

and 60 skin patches. The structural design variables are the thickness values of these

components. Figure 4.14 shows every component for which there is a thickness design

variable. The thickness of each component can vary between 5mm and 50mm.


Table 4.1: Nonlinear constraints usedfor optimization in all cases

Constraint DescriptionCruise L−WMTO = 0.0

Maneuver L− 2.5WMTO = 0.0

Top Skin KS ≤ 1.0

Bottom Skin KS ≤ 1.0

Rib/Spar KS ≤ 1.0

Wing Span b = 103 ft

Total 6

Table 4.2: Optimization design vari-ables for all cases

Design Variable QuantityTwist Angle 3Sweep Angle 3Cant Angle 1Section Shape 224Angle of Attack 2Skin Thickness 60Spar Thickness 66Rib Thickness 30Total 389

X

Y

Z

Figure 4.13: Primary structural layout ofthe wing in relation to the outer mold line.Skin elements are not shown for clarity.

Figure 4.14: Each colored surface repre-sents a structural component, the thick-ness of which is a design variable.

These large aerostructural optimization cases are challenging to converge due to the

presence of hundreds of design variables and many nonlinear constraints. In order to

reduce the difficulties associated with convergence, we first perform the optimizations on

a coarse CFD mesh with 193, 536 nodes. After 240 optimization iterations on the coarse

grid, the merit function plateaus and the optimization is continued on a finer mesh with

653, 184 nodes in 112 blocks. We use the same post-optimality analysis strategy as the

one in Section 4.1.2 in order to provide more accurate estimates of lift and drag for the

trade-off curves.

Figures 4.15 and 4.16 show the aerostructural optimization convergence trends for the

winglet-down configuration with β = 1.0 using the coarse and fine optimization grids.

The dashed line marks the beginning of the optimization using the fine grid. At the end


Iteration Number

0 50 100 150 200 250 300 35010−10

10−8

10−6

10−4

10−2

100

Feasibility

Optimality

Figure 4.15: Convergence of optimalityand feasibility measures for the winglet-down aerostructural optimization case.The dashed line marks the beginning ofthe optimization using the fine grid.

Iteration Number

0 50 100 150 200 250 300 350

Merit

Function

0

2

4

6

8

10

12

14

Figure 4.16: Convergence of the La-grangian merit function for the winglet-down aerostructural optimization case.The dashed line marks the beginning ofthe optimization using the fine grid.

Figure 4.17: The initial and optimizedwing shapes for the β = 0.5 case. Con-tours of pressure coefficient in cruise con-dition (M = 0.785 at 35, 000 ft) are alsoshown.

Figure 4.18: The initial and optimizedwing shapes for the β = 1.0 case. Con-tours of pressure coefficient in cruise con-dition (M = 0.785 at 35, 000 ft) are alsoshown.

of the optimization using the coarse grid, the optimized geometry is used as the initial

design for the subsequent fine optimization. On the finer aerodynamic grid, the optimizer

achieves a further merit function reduction of around 2.2% in 103 function evaluations

and satisfies the nonlinear constraints to a tolerance of 10−6. The reduction in optimality

on the fine grid is around an order of magnitude, but the fact that the merit function

has plateaued means that the majority of the reduction in the merit function is already

achieved. These trends hold true for the rest of the cases as well.

Figures 4.17 and 4.18 show the optimal winglet configurations for the β = 0.5 and


Figure 4.19: Plots of the pressure coefficient in cruise condition (M = 0.785 at 35, 000 ft)for the optimal planar wing with β = 1.0 and the initial geometry.

β = 1.0 cases, respectively. In both cases, the optimization leads to a winglet-down

configuration with a cant angle of −90◦. It is important to note that the optimizer does

have the freedom to vary the cant angle and choose between a winglet-up, winglet-down,

or planar configuration. Thus, the winglet-down configuration has better performance

in comparison with planar and winglet-up configurations of the same projected span

regardless of the value of β.

Figure 4.19 shows plots of the pressure coefficient along the span for the initial and

optimized wings on the fine optimization grid. The initial geometry is not shock-free as

indicated by the rapid pressure drop at half-chord over a large region of the upper surface

of the wing. The initial pressure distribution is produced by the starting geometry at

an angle of attack of 2◦. Note that the initial design simply serves as a starting point

for the optimization and is not meant to yield good lift and drag performance. The

optimizer has eliminated the associated wave drag by eliminating the shocks. The rapid

pressure recovery near the trailing edge is typical for inviscid optimizations that ignore

separation [53]. Ignoring viscous effects does not affect our main conclusions because if

wingletted wings do not provide significant drag reductions in inviscid flow, then it is

unlikely that their potential improvements will increase in the presence of viscous effects.

We acknowledge that if the optimized wings obtained here are analyzed using a RANS-

based flow solver, they will not perform well. However, the trade-offs are still captured

correctly.

Figures 4.20 and 4.21 show the optimal lift distributions for the winglet-down con-

figuration for both values of β. With β = 0.5, i.e. when there is equal emphasis on both

weight and drag of the wing in the objective function, the optimal cruise lift distribution

is more triangular than the elliptical load. At the 2.5g load condition, the spanwise lift


Span (m)0 5 10 15

Norm

alizedLift

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Cruise2.5g LoadElliptical

Figure 4.20: Lift distributions for the op-timized winglet-down configuration withβ = 0.5.

Span (m)0 5 10 15

Norm

alizedLift

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Cruise2.5g LoadElliptical

Figure 4.21: Lift distributions for the op-timized winglet-down configuration withβ = 1.0.

Figure 4.22: Plots of the pressure coefficient in cruise condition (M = 0.785 at 35, 000 ft)for the optimal winglet-down configurations with β = 0.5 and β = 1.0.

distribution is much more triangular with a significant amount of tip load relief. In fact,

the tip loading at the maneuver condition is lower than at cruise. When the objective

function is equal to the inviscid drag, i.e. β = 1.0, the optimal cruise lift distribution

closely follows the elliptical load, which is consistent with the expected trend. The op-

timizer still reduces the tip loading at the 2.5g load condition down to the cruise level

by introducing washout near the tip at the deflected state. These plots indicate that the

optimizer is actively taking advantage of aeroelastic tailoring. Furthermore, it establishes

the fact that the methodology used is able to capture the expected trends.

It is insightful to see how the optimizer achieves differences in optimal span loading

through aeroelastic tailoring for different values of β. Figure 4.22 shows plots of the

pressure coefficient along the span for both cases. With β = 0.5, the pressure coefficient


y (m)0 5 10 15 20

Twist(deg)

-6

-5

-4

-3

-2

-1

0

β = 0.5β = 1.0

Figure 4.23: The wing twist distributions in cruise condition for the optimal winglet-downconfigurations.

curves for the upper and lower surfaces near the wing-winglet junction cross over. This

indicates that the force near the leading edge is downward, but it switches direction after

the cross-over point. This results in more washout near the wingtip than the β = 1.0 case.

We can see evidence of the differences in the corresponding twist distributions along the

span as shown in Figure 4.23. It is clear that the optimizer is introducing greater washout

towards the tip for the β = 0.5 case. This is consistent with our expectation because

with β = 0.5, there is equal emphasis in the objective function on weight and drag. As

a result, the optimizer should reduce the tip loading and shift the load inboard in order

to maintain a lighter structure than the β = 1.0 case. Note that the twist angle at the

tip of the winglet in Figure 4.23 is equivalent to the toe angle and partly controls the

loading on the winglet, which in turn determines the structural deflection (bending) of

the winglet. We will later on demonstrate that the deflection of the winglet is important

to consider for induced drag comparisons because it influences the tip vortex location.

It is clear from Figure 4.24 that the optimizer modifies the planform of the wing from

the baseline geometry in both cases. With β = 0.5, the overall sweep angle is lower than

the case with β = 1.0. As the emphasis on drag in the objective function is increased,

the optimizer increases the sweep angle in order to remove the shocks that are present

in the β = 0.5 case.

Figure 4.25 provides a comparison of drag between optimized planar wings with a

fixed cant angle of zero and the optimized winglet-down configurations with a variable

winglet cant angle for both values of β. The drag improvement relative to the planar wing

is up to 1.7%. It is clear that in both cases, the winglet-down configuration has lower

drag than the optimized planar counterparts. Table 4.3 provides a summary of the most


Figure 4.24: The initial and optimizedplanform shapes for the aerostructural op-timization cases with β = 0.5 and β = 1.0.

β = 0.5 β = 1.0

Norm

alizedTotalDrag

0.90

0.92

0.94

0.96

0.98

1.00

1.02 Planar

Winglet-Down

Figure 4.25: Comparison of total drag forboth values of β between the optimizedplanar wing with a fixed cant angle of zeroand the optimized winglet-down configura-tion.

Table 4.3: Summary of the cruise point performance of the planar and wingletted wingsfrom aerostructural optimization evaluated on the fine grid

Planar Winglet-Down

Parameter β = 0.5 β = 1.0 β = 0.5 β = 1.0CL 0.472 0.478 0.473 0.488Total drag (counts) 321 301 317 296∆D 0.00% 0.00% -1.3% -1.7%L/D 14.7 15.9 14.9 16.5e 0.814 0.984 0.881 1.138

important results of this section. The drag difference relative to the planar wing for each

value of β is indicated by ∆D. The inviscid span efficiency factor e = L2/(Dinviscidb2q∞π)

is also provided, where L is the lift, Dinviscid is the inviscid drag, b is the wingspan, and q∞

is the freestream dynamic pressure. The inviscid drag does not include the viscous drag

estimate. It is noteworthy that although the span efficiency factor for the wingletted wing

with β = 1.0 is significantly higher than the planar counterpart, it does not translate

to a total drag reduction of the same magnitude. The is partly due to the fact that

the induced drag is only part of the total drag. Furthermore, the wingletted wing has

a higher viscous drag. This leads to a smaller total drag reduction. The lift coefficient

is also higher for the winglet-down configuration than the planar counterpart due to the

higher weight.

We saw in Section 4.1.2 that, from a purely aerodynamic standpoint, the winglet-down

configuration is able to push the tip vortex further away from the wing than the winglet-

up counterpart. When aerostructural optimization is considered, the winglet-down design


Figure 4.26: View of the undeflected spansfor the planar and winglet-down configura-tions.

Figure 4.27: View of the deflected spansfor the planar and winglet-down configu-rations.

pushes the tip vortex even further away from the wing due to the outboard deflection

of the winglet-down feature. Figures 4.26 and 4.27 demonstrate how the winglet-down

configuration has a slightly higher span at the deflected state under the aerodynamic

loads. This effect is subtle and is only captured if the structural deflections are taken

into account. It is important to note that although the increase in span as a result of

the deflections is small, its effect is larger due to the quadratic relation between span

and induced drag. This along with the ability of this design to push the tip vortex

further away from the wing from a purely aerodynamic standpoint explains why the

aerostructural optimizer produces a winglet-down given the freedom to do so.

There is an important point about the aerostructural optimization results that we

must mention. Although the fully coupled aerostructural optimization results include

more physics than the purely aerodynamic optimization counterparts, we are still largely

ignoring the implication of additional critical structural load cases, buckling, and flutter.

Consequently, the performance improvements obtained from these results still represent

a best-case scenario in the context of practical aircraft design. For instance, adding a

winglet may lead to a reduction of the flutter speed by reducing the torsional rigidity of

the wing [79, 80]. Similarly, a buckling constraint at the wing-winglet junction of a winglet

configuration may overshadow the potential aerodynamic benefit. As a result, if the

wingletted wings do not produce a considerable benefit in the absence of such prohibitive

constraints, then one can argue that they would not provide greater improvements if

we were to take these additional considerations into account. Thus, these results could

overestimate the benefits of winglets, but are unlikely to underestimate them.

We have so far limited our discussion of the aerostructural optimization cases to the

winglet-down configuration due to the fact that the optimizer seems to favor it over the


planar and winglet-up designs. However, there is an important yet subtle point about

the winglet-up configuration. The inboard deflection of the winglet-up would bring the

tip vortex even closer to the wing relative to the aerodynamically optimal counterpart.

This can potentially make the winglet-up configuration inferior from an aerostructural

standpoint. For this reason, it will be interesting to see what the optimizer chooses to

do given the freedom to either keep the wing planar or create a winglet-up configuration.

The next section explores this in more detail.

4.1.4 Further Investigation Using Aerostructural Optimization

Results from Section 4.1.3 indicate that when the aerostructural optimizer is given the

freedom to choose the optimal winglet shape at the tip of the wing, it produces a winglet-

down configuration. An alternative optimization strategy is to limit the choice of the cant

angle in such a way that the optimizer is forced to either produce a nonplanar feature

or keep the wing planar for both winglet-up and winglet-down configurations. If the

optimizer does not create a winglet in either case, then the planar wing is optimal. In

this section, we are particularly interested to see if the aerostructural optimizer creates

a winglet-up configuration from an initially planar wing for any value of β, especially in

light of the fact that this configuration may bring the tip vortex even closer to the wing

at the deflected state relative to the purely aerodynamic optimization case.

For the purpose of this investigation we choose the Boeing 737-900 as the baseline

aircraft, which is heavier and has a longer fuselage than the one used in the previous

sections. This is done in part to gain a better understanding of whether our conclusions

are sensitive to the choice of the baseline design. The planform of the wing and the corre-

sponding structural wingbox are shown in Figure 4.13. All of the winglet configurations

that are considered have the same undeflected span as the baseline wing and, as a result,

have the same planform area as the baseline wing. The projected span of the wing,

however, may change due to the structural deflection of the wing. We consider the same

cruise and 2.5g load conditions as before. The objective function is the same as the one

used in Section 4.1.3 and is shown in Equation 4.1. Three values of β have been chosen:

0.5, 0.75, and 1.0. The optimizations are conducted in a single stage using the grid with

653, 184 nodes in 112 blocks. Based on our investigations in the previous sections, there

is practical justification for ending the optimization cases after approximately 150 design

iteration as the majority of the reduction in the merit function will be achieved.

The nonlinear constraints used in this investigation are the same as the ones listed in

Table 4.1. However, the total weight of the aircraft is assumed to be equal to the com-


Table 4.4: Winglet-up and winglet-down optimization design variables

Design Variable QuantitySweep Angle 3Twist Angle 2Cant Angle 1Section Shape 84Angle of Attack 2Skin Thickness 60Spar Thickness 66Rib Thickness 30Total 248

puted weight of the wing plus a fixed weight of 785, 000N. This fixed weight is estimated

based on the maximum takeoff weight of a Boeing 737-900 excluding the approximate

wing weight. The structural material and sizing methodology are the same as before.

Table 4.4 provides a list of the design variables used for winglet-up and winglet-down

optimization cases. These cases have a total of 248 design variables. The geometric pa-

rameterization is slightly different from the previous sections in that the sectional shape

is controlled at fewer spanwise stations to reduce the number of design variables. Figure

4.28 shows the geometric design variables that control the shape of the wing and winglet.

The optimizer has the freedom to change the sectional shape of the wing by manipulating

the z-coordinates of the control points at 6 spanwise stations. The airfoil shapes are in-

terpolated between these stations using the remaining control points such that curvature

continuity is maintained. For a winglet-up configuration to form from an initially planar

wing, the cant angle at the wingtip is allowed to vary between 0 and +90◦. Similarly, for

a winglet-down configuration, the optimizer is free to vary the cant angle between 0 and

−90◦. In both cases, the optimizer is free to keep the wing planar. The maximum height

of the nonplanar feature at the wingtip is constrained to 6% of the wing span to replicate

the Boeing 737NG blended winglets. For the planar wing optimization case, the wingtip

cant angle design variable is inactive and hence the optimizer does not have the freedom

to create a nonplanar feature at the tip of the wing. As an example, Figure 4.29 shows

the optimized winglet-down configuration with β = 0.5 along with the initial wing.

Figure 4.30 shows the merit function history for the winglet-down configuration with

β = 1.0. Figure 4.31 shows the corresponding convergence histories for optimality and

feasibility. Such aerostructural optimization cases tend to converge rather slowly [16].

Nevertheless, Figure 4.30 indicates that we have obtained the majority of the merit func-

tion improvement. The optimizer completed 174 design iterations in 6 days of walltime


Figure 4.28: Geometric parameterizationand design variables.

Figure 4.29: The optimized winglet-downconfiguration with β = 0.5 along with theinitial wing.

0 20 40 60 80 100 120 140 1600.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Design Iterations

MeritFunction

Figure 4.30: Merit function convergencehistory for the winglet-down configura-tion with β = 1.0.

0 20 40 60 80 100 120 140 16010

−8

10−6

10−4

10−2

100

Design Iterations

FeasibilityOptimality

Figure 4.31: Feasibility and optimal-ity convergence histories for the winglet-down configuration with β = 1.0.

Figure 4.32: Contours of pressure coefficient on the upper and lower surfaces in cruisecondition (M = 0.785 at 35, 000 ft) for the initial geometry.

on 240 processors. These general convergence trends hold true for the rest of the opti-

mization cases as well.

Figure 4.32 shows the pressure coefficient on the top and bottom surfaces of the


Figure 4.33: Contours of pressure coefficient on the upper and lower surfaces in cruisecondition (M = 0.785 at 35, 000 ft) for the optimized winglet-down configuration withβ = 0.75.

Figure 4.34: Skin thickness values in millimeters for the winglet-down configuration withβ = 0.75.

initial geometry in cruise condition with an angle of attack of 2◦ evaluated on the fine

mesh. Contours of pressure coefficient for the optimized winglet-down configuration with

β = 0.75 are shown in Figure 4.33. The initial geometry does not satisfy the nonlinear

lift constraint using the initial value of the angle of attack and leads to a shock that is

present over a large portion of the upper surface. The optimizer has eliminated this shock

on the upper surface of the optimized design while satisfying all nonlinear constraints to

a tolerance of 10−6.

The optimized skin thickness values for the winglet-down configuration are shown in

Figure 4.34. Note that the aerostructural optimization has thickened the skin inboard.

This is somewhat expected because the failure criterion at the 2.5g load condition tends

to be closer to the critical value at the root of the wing and near the crank. The optimized

thickness distributions for the ribs and spars follow a similar pattern. Although we have

not considered all of the necessary critical load conditions that are required in the context

of wing design, this result shows that we have captured at least some of the correct trends

dictated by the structural sizing of the wing.

The maximum takeoff weight of the baseline aircraft considered in this section is


0 2 4 6 8 10 12 14 16−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

y [m]

Lift


Figure 4.35: Spanwise lift distributionsat the cruise and 2.5g load conditions forthe winglet-down configuration with β =0.5.

0 2 4 6 8 10 12 14 160.0

0.5

1.0

1.5

2.0

2.5

3.0

y [m]

Lift


Figure 4.36: Spanwise lift distributionsat the cruise and 2.5g load conditions forthe winglet-down configuration with β =1.0.

significantly larger than the one in Section 4.1.3. In order to show that the optimizer

can recover the fundamental tradeoff between weight and drag correctly regardless of

the baseline maximum takeoff weight, we examine the optimized lift distributions as

well as the planform shapes. Figure 4.35 shows the spanwise lift distributions for the

cruise and 2.5g load conditions corresponding to the winglet-down configuration with

β = 0.5. These have been normalized with respect to the lift at the root for an elliptical

lift distribution at cruise. The optimizer is aeroelastically tailoring the wing in order to

significantly reduce the tip loading at the 2.5g load condition while maintaining a lift

distribution closer to elliptical for the wing at the cruise condition. Figure 4.36 shows

the spanwise lift distributions for the winglet-down configuration with β = 1.0. In this

case, the cruise lift distribution more closely follows the elliptical lift distribution than

the β = 0.5 case. Furthermore, the tip loading at the 2.5g load condition is greater in

comparison to the β = 0.5 result. In other words, the optimizer is further reducing the

cruise drag at the cost of increasing the wing weight. Thus, the optimizer has captured

the expected trade-off between weight and drag.

Figure 4.37 shows the top-down view of optimal designs for the winglet-down con-

figuration. The values of β are also provided. As the value of β increases from 0.5 to

1.0, i.e. as we place increasing emphasis on the drag in the objective function, the sweep

angles of the wings also increase. This is the expected trend because wing sweep reduces

wave drag. The same trend exists for the planar configurations.

Figure 4.38 shows the trade-off curves of the optimal solutions obtained for all of the

configurations considered. The optimized planar wing with β = 0.5 has a normalized drag


Figure 4.37: Top-down view of the optimal winglet-down configurations.

value equal to unity. Similarly, the optimized planar wing with β = 1.0 has a normalized

weight value equal to unity. Note that for the purpose of this plot, the weight values

are obtained by adding the calculated wing weight to a fixed weight of 785, 000N to get

an approximate total aircraft weight. Similarly, the drag values are obtained by adding

the calculated inviscid drag based on the Euler equations to a viscous drag estimate of

the entire aircraft based on the same methodology used in Section 4.1.2. In this case,

we estimate that the viscous drag at cruise is equal to 200 drag counts for the baseline

planar configuration due to the longer fuselage length [77, 76]. In all cases, as we place

increasing emphasis on inviscid drag in the objective function, i.e. as we vary β from 0.5

to 1.0, the drag decreases at the cost of increasing weight.

Figure 4.38 shows that the winglet-down configuration can provide a total drag re-

duction of up to 2% for the same weight in comparison to the planar wings. Although

the reduction in inviscid drag is as high as 8% at the same weight, the inviscid drag is

only part of the total drag. Furthermore, the increase in the surface area due to the

nonplanar feature at the tip increases the viscous drag. This means that the total drag

reduction is lower than the inviscid drag reduction. Figure 4.38 does not include the

winglet-up results1 because the optimizer did not create a winglet-up feature for any of

the β values even though it did have the geometric freedom to do so. In other words,

when the optimizer has the freedom to create a winglet-up feature, it chooses to keep

1The forced winglet-up case will be discussed in the following paragraph.


Figure 4.38: Trade-off curves of optimal designs for all of the wingletted and planarconfigurations considered in this investigation.

Table 4.5: Summary of the aerostructural optimization results obtained from the finegrid

Planar Winglet-Down

Parameter β = 0.50 β = 0.75 β = 1.00 β = 0.50 β = 0.75 β = 1.00CL 0.576 0.587 0.596 0.581 0.594 0.603Total drag (counts) 412 400 374 399 383 364∆D 0.0% 0.0% 0.0% -3.2% -4.3% -2.7%L/D 14.0 14.7 15.6 14.6 15.1 16.6e 0.782 0.863 0.981 0.870 0.991 1.143

the wing planar regardless of the value of β. On the other hand, the optimizer produces

a winglet-down feature for each value of β when it is given the freedom to do so. This

reaffirms our original findings presented in Section 4.1.3. Table 4.5 summarizes the re-

sults obtained from the fine mesh analysis. It is important to note that although the

inviscid span efficiency of the winglet-down configuration for β = 1.0 is relatively high,

it does not translate to a drag reduction of the same magnitude because inviscid drag is

only part of the total drag and the winglet adds viscous drag. In addition, the wingletted

wing has a higher lift coefficient than the planar counterpart.

Although the aerostructural optimizer does not create a winglet-up configuration

given the freedom to do so, it is important to eliminate the possibility that the winglet-

up is a local optimum in a multimodal design space. In order to address this, we have


Figure 4.39: Contours of x-vorticity behind the trailing edge of the wing for the planarand winglet-down cases in cruise condition (M = 0.785 at 35, 000 ft). The dashed linemarks the location of the tip vortex for the planar configuration.

run an optimization case where the cant angle is forced to be equal to +90◦. This case

is depicted by the red circle in Figure 4.38. It is clear that this design is dominated by

the planar and winglet-down configurations. The reader is reminded that even from a

purely aerodynamic perspective, the winglet-up configuration reduces the total drag only

by 1.1% in comparison to an optimal planar wing of the same span, as shown in Section

4.1.2. From an aerostructural perspective, this configuration reduces the projected span

at the deflected state and brings the tip vortex closer to the wing. Furthermore, the

addition of the winglet adds weight both by increasing the structural span and modifying

the tip loading. The combination of these effects makes the winglet-up configuration have

inferior performance relative to its planar counterpart.

The winglet-down configuration performs better than the planar and winglet-up coun-

terparts for two reasons. First, this design moves the tip vortex further away from the

wing even when the deflections are not taken into account. Second, the projected span

of the winglet-down configuration at the deflected state is in fact larger than its planar

and winglet-up counterparts. The tip vortex moves even further away from the wing

in comparison to the purely aerodynamic optimization case. This highlights the impor-

tance of capturing the deflected shape of the wing under the aerodynamic loads. We have

reached the same conclusion even when using different geometry parameterization and

mesh movement schemes for the purpose of optimization [81]. In the case of a winglet-up

configuration, the structural deflection reduces the projected span and brings the tip

vortex closer to the wing. From an aerostructural perspective, this leads to a higher

objective function for the winglet-up configuration relative to the planar counterpart.

Figure 4.39 shows that the winglet-down configuration pushes the tip vortex further

away from the wing in the positive y-direction in comparison to the optimized planar


wing of the same projected span. This means that the induced downwash is reduced in

the winglet-down case. As a result, the induced drag is lower.

The winglet-down concept will have practical implications for the design of the aircraft

that we have presently ignored. For example, it may violate current wingtip clearance

regulations for commercial aircraft. As a result, it may require a high-wing configuration

to overcome such practical limitations. This may even be more beneficial for the aero-

dynamic performance of the aircraft in comparison to today’s modern low-wing designs,

as shown by Hashimoto et al. [82].

Although the winglet-down configuration appears to be the most competitive design,

its relative drag benefit in comparison to the planar wing is smaller than those reported

in past studies based on purely aerodynamic shape optimization that considered low-

speed flying conditions [54]. Furthermore, our results indicate that at least some of the

induced drag advantage of the winglet-down configuration is due to the increased span

of the wing at the deflected state. This suggests that perhaps we should include other

phases of a commercial flight profile such as the steady climb, where the ratio of induced

to total drag is higher than in cruise. We will explore this in the next section.

4.1.5 Optimal Winglets for Cruise and Climb Conditions

The main objective of this section is to explore the possibility that wingletted wings may

provide a larger benefit in high-lift, low-speed conditions. The induced drag constitutes

up to 80% of the total drag of the aircraft in high-lift, low-speed conditions such as

takeoff [21]. This is largely due to the fact that the coefficient of lift is larger than in

cruise. The aerostructural analysis and optimization framework used in this work does

not have the capability to analyze a takeoff condition, but we are indeed able to study

other low-speed, high-lift conditions, including steady climb.

The optimization problem formulation in terms of the initial geometry, geometric

design variables, and parameterization is the same as the one in Section 4.1.3, but the

structural layout is slightly simpler, as shown in Figure 4.40. There is now an additional

representative steady climb condition where the Mach number is equal to 0.40 at an

altitude of 10, 000 ft. The objective function is the sum of the total drag in cruise and

climb conditions. This choice of objective function may not be practical, but is effective

in studying the most important trends that we are interested in.

Table 4.6 shows the comparison of the total drag reduction from the optimal winglet-

ted wing in comparison to the planar configuration with the climb condition included

in the optimization as a design point. The nonplanar optimization produces a winglet-


Figure 4.40: Primary structural layout of the ribs and spars used for the purpose of thisinvestigation. Skin elements are omitted for clarity.

Table 4.6: Summary of the multipoint aerostructural optimization results obtained fromthe fine grid

Cruise Climb

Parameter Planar Winglet-Down Planar Winglet-DownCL 0.477 0.482 0.619 0.623Total drag (counts) 313 305 376 362Dinduced/Dtotal 0.38 0.35 0.49 0.45∆D 0.0% -2.5% 0.0% -3.7%L/D 15.2 15.8 16.5 17.2

down configuration when the optimizer is given the freedom to do so. It is clear that

the nonplanar wing is providing a higher drag reduction in climb than in cruise. This

is due to the fact that the ratio of the induced to total drag is higher in climb than in

cruise. Therefore, the viscous drag penalty associated with the increased surface area of

the wingletted design is lower in comparison to cruise. In other words, any induced drag

reduction provided by the winglet has a greater influence on the total drag value.

The most important conclusion from our climb optimization study is that a wingletted

wing may provide a larger total drag reduction in high-lift, low-speed conditions such as

steady climb. Thus, the fuel savings from a particular winglet configuration will depend

on its intended mission profile. In order to further illustrate this point, Figure 4.41 shows

the flight profile of a B737NG aircraft from Toronto to Vancouver, where the distance

flown is approximately 3, 600 km. It is clear that the time spent during the climb and

descent segments are significantly shorter in comparison to cruise. However, the same


Time (min)0 100 200 300

Altitude(ft)

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mach

Figure 4.41: The flight profile of a B737NGaircraft from Toronto to Vancouver.

Time (min)

0 10 20 30 40 50

Altitude(ft)

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mach

Figure 4.42: The flight profile of a B737NGaircraft from Toronto to Montreal.

aircraft is also used on shorter routes such as the one shown in Figure 4.42, where the

distance flown is just under 600 km. In this case, the aircraft spends a significant amount

of time in low-speed, high-lift conditions relative to cruise and as a result, a wingletted

wing may be more useful.

4.2 The Drooped Wing Concept

Although we have thus far given the optimizer a reasonable degree of geometric freedom

to perform exploratory optimization for winglets, much more geometric freedom can be

given to the optimizer to explore a larger design space in the context of nonplanar wings.

For instance, in Section 4.1.3, we limited the nonplanar freedom to the wingtip region

of the wing only. In theory, it is possible to allow the same type of variation at more

stations along the span. This will allow the optimizer to create highly nonplanar wings

with the potential to reduce the total drag in comparison to optimal planar wings of the

same span.

4.2.1 The Drooped Wing Concept Optimization Results

The objective function for the drooped wing case is based on the Breguet range equation

and is of the form

J = −L

Dinviscid

logWMTO

WMTO −Wfuel

. (4.2)

The fuel weight Wfuel is estimated to be around 206, 000N, and WMTO is the maximum


takeoff weight. The cruise lift L is constrained to be equal to WMTO. The cruise condition

is M = 0.785 at an altitude of 35, 0000 ft. We choose a different objective function here

than previously because producing trade-off curves of optimal designs for weight and

drag is expensive in terms of computational resources. We emphasize that this particular

choice of objective function may not be appropriate in a practical aircraft design context.

However, it is sufficient for our preliminary investigation of these nonplanar wings in the

context of exploratory aerostructural optimization.

Equation 4.2 differs from the Breguet range formulation because it does not include

the viscous drag. As a result, minimizing this objective function does not necessarily

maximize range due to the possible increase in the wetted surface area and hence viscous

drag. Increasing the height-to-span ratio leads to a reduction in the induced drag [21].

However, from a purely aerodynamic standpoint, this reduction in induced drag will

eventually be overshadowed by the increase in the viscous drag at a certain threshold

if the surface area is free to increase. In our optimization studies of the drooped wing,

where we include a viscous drag estimate in the post-optimality calculations, there are

two possible scenarios: the height-to-span ratio of the optimized drooped wing is either

below or above this threshold. In the former case, including the viscous drag estimate

in the objective function will not change the optimized geometry. In the latter case, the

potential performance improvement of the drooped wing from the present study will be

conservative. Therefore, our results could underestimate the potential benefits of the

drooped wing, but are unlikely to overestimate them.

The structural material and sizing strategy based on a 2.5g critical load condition are

the same as before. The weight of the aircraft is assumed to be equal to the weight of

the wing plus a fixed weight of 785, 000N to capture the engines, fuselage, and payload

weight. The weight of the wing is calculated by multiplying the weight obtained from

the finite-element model by a factor of 1.5 to account for the weight of the load-bearing

members that are not included in the structural finite-element model of the wing [65].

Similarly, in calculating the final drag values, we include the same estimate of the viscous

drag for the entire aircraft as the one used in Section 4.1.2. These are determined for

the baseline planar configuration which is based on the planform of the Boeing 737-900

aircraft.

The same structural layout of the ribs and spars shown in Figure 4.40 is used for

the drooped wing optimization cases. The finite element model of the structure has ap-

proximately 30, 000 second-order shell elements. The thickness values of the structural

components are allowed to vary between 5mm and 50mm. Figure 4.43 shows the corre-

sponding geometric parameterization and design variables. The upper and lower surfaces


Figure 4.43: Geometric parameterization and design variables for the drooped wing case.

Figure 4.44: Three possible wing shapes permitted by the parameterization. The dihedralangle for each region, φ can vary between −30◦ to +30◦. The red circles represent thecurved junctions between regions and their locations along the span are allowed to vary.(1) is a planar wing, and (2) and (3) are two possible nonplanar wing shapes.

are each broken into 5 regions. The twist and dihedral of each region are geometric design

variables. The optimizer is free to manipulate the wing section at 10 spanwise stations.

The airfoil shapes are interpolated between every pair of spanwise stations such that cur-

vature continuity is maintained. To further illustrate the freedom given to the optimizer

to develop a nonplanar wing, Figure 4.44 shows simplified views of three possible wing

shapes permitted by the geometric parameterization. In essence, the wings are made of

five straight wing segments joined with curved junctions to maintain curvature continu-

ity. In all cases, the projected span, leading edge sweep angle, and chord length remain

constant.

We apply the same nonlinear constraints that we used for the winglet cases. The

optimization is initiated with an initially planar geometry. Due to the broad range of


Design Iterations

0 100 200 300 40010−8

10−6

10−4

10−2

100

Feasibility

Optimality

Figure 4.45: Convergence of optimalityand feasibility conditions for the droopedwing optimization case. The dashed linemarks the beginning of optimization onthe fine grid.

Design Iterations

0 100 200 300 400

MeritFunction

-10

0

10

20

30

40

50

60

Figure 4.46: Merit function convergencebehavior for the drooped wing optimiza-tion case. The dashed line marks the be-ginning of optimization on the fine grid.

geometric freedom given to the optimizer, this case is particularly challenging in terms

of optimization convergence. To mitigate some of these challenges, we use the same

two-stage optimization strategy used in Section 4.1.3. The first stage uses a coarse grid

with approximately 140, 000 nodes. Once the merit function plateaus, we continue the

optimization on a finer mesh with 650, 000 nodes.

Figures 4.45 and 4.46 show the optimization convergence history for the drooped wing

case. After approximately 200 design iterations on the coarse grid, the merit function

plateaus and the optimization is continued on the finer grid. The dashed line marks

the beginning of the optimization on the finer grid. These figures indicate that the

optimization reached an acceptable level of convergence. Similar convergence trends

exist for the other optimization cases in this section as well.

Figure 4.47 shows the geometric evolution of the design during optimization. It is

evident that the optimizer has the freedom to analyze highly nonplanar wings over the

course of the optimization. Figure 4.47 also shows that the optimizer eventually converges

to a drooped wing. This confirms that our aerostructural analysis and optimization

methodology is able to recover a drooped wing concept starting from an initially planar

wing.

Table 4.7 shows that the drooped wing is more efficient than a planar wing of the

same projected span. Figure 4.48 sheds some light on the reason for this improvement.

As Lazos and Visser [43] and Nguyen et al. [42] point out, the drooped wing moves the

core of the tip vortex further away from the wing in comparison to the planar counter-


Figure 4.47: The geometric evolution ofthe drooped wing over the course of op-timization. All shapes are at the deflectedstate of the wing and the correspondingfunction evaluation number is also shown.

Figure 4.48: The contours of x-vorticitybehind the trailing edge of the wing for theplanar (top) and drooped (bottom) wings.The dashed line marks the location of thetip vortex for the planar configuration.

Table 4.7: Optimization results for the drooped wing concept

Planar Drooped Drooped Planar Winglet-DownParameter (Straight Edges) (Straight Edges) (Curved Edges) (Curved Edges) (Straight Edges)CL 0.624 0.632 0.612 0.599 0.632D (Counts) 410 402 384 380 404L/D 15.2 15.7 15.9 15.8 15.6W (×105 N) 1.32 1.37 1.30 1.28 1.40b∗ (m) 32.27 32.80 32.89 32.27 32.60Swetted (m2) 211.8 218.4 222.5 217.1 230.4R 3.92 4.02 4.12 4.08 3.99∆R 0.0% 2.6% 4.9% 4.0% 1.8%

part. However, our results include the additional effect of increased span at the deflected

state under the aerodynamic loads, which we observed in the case of the winglet-down

configuration as well. Nonetheless, the drooped wing achieves this without increasing the

wetted surface area and weight as much as the winglet-down configuration does and leads

to a higher range parameter improvement, as shown in Table 4.7. The range parameter

is given by R = (L/D) log(Wi/Wf ), where L is the lift, D is the total drag including the

same viscous drag estimate based on the wetted surface area used in the winglet study,

Wi is the initial weight, and Wf is the initial weight discounted by the fuel weight. The

differences in range are indicated by ∆R. The deflected span and surface area of the wing

are denoted by b∗ and Swetted, respectively. Note that the winglet-down configuration in

Table 4.7 is optimized for the same objective function as the one shown in Equation 4.2.

This is done in order to gain a better understanding of the relative performance benefits


Figure 4.49: The additional freedom given to the optimizer to manipulate the nonlinearshape of the leading and trailing edges of the wing.

Figure 4.50: Pressure coefficient contours in cruise condition (M = 0.785 at 35, 000 ft)for the optimized drooped wing with curved edges.

of the winglet-down and drooped concepts.

The drooped wing concepts presented by NASA [43, 42] have nonlinear leading and

trailing edges. We initially did not allow the optimizer to change the linear nature of

the leading and trailing edges. We next increase the amount of geometric freedom that

the optimizer has to manipulate the shape of the leading and trailing edges of the wing.

Figure 4.49 demonstrates this additional freedom for the leading and trailing edges of the

wing. Note that the streamwise location of the root and tip of the wing as well as the chord

length remain fixed in space. In this case, there is an additional geometric constraint to

keep the projected area unchanged. This is done because the required projected area of

the wing is strongly dependent on the takeoff performance requirements of the aircraft,

which we do not include in our problem formulation. For the purpose of this case, we

initiate the optimization using the optimal drooped wing that resulted from the previous

case.

Figure 4.50 shows the pressure coefficient contours for the optimized drooped wing


Figure 4.51: Pressure coefficient contours in cruise condition (M = 0.785 at 35, 000 ft)for the initial and optimized planar configurations when the shape of the leading andtrailing edges are free to vary.

with curved edges. The optimizer has taken advantage of the additional geometric free-

dom for the leading and trailing edges in order to further improve the initial drooped

wing design. The fine-mesh analysis indicates that this drooped wing concept with curved

edges provides a 4.9% range improvement in comparison to an optimized planar wing of

the same span, as shown in Table 4.7.

An interesting question to ask is why the optimizer is able to deliver an improved

performance when the leading and trailing edges can be curved. In order to find an

answer, we should try to characterize the potential benefits of having curved leading and

trailing edges separately from the drooped effect. Therefore, we try an optimization case

where we start from an initial geometry with linear leading and trailing edges and allow

the optimizer to move towards curved edges. The problem formulation is the same as

before except the geometric freedom to create a nonplanar wing is removed. Figure 4.51

shows the initial and optimized designs that result from this optimization. It is clear

that the optimizer has modified the leading and trailing edge shapes. In fact, Table 4.7

indicates that the wing with nonlinear leading and trailing edges improves the objective

function by 4% on the fine mesh relative to the optimal baseline planar wing with straight

edges.

The optimized planar wing with curved leading and trailing edges has a lower wing

weight in comparison to the optimal planar wing with linear leading and trailing edges,

as shown in Table 4.7. Evidence for this reduction in weight can be seen in the optimal

thickness distributions for the two wings. Figure 4.52 shows the optimal thickness values

of the top and bottom skins for both designs. The additional freedom to change the


Figure 4.52: The optimal structural thickness distributions for the baseline planar con-figuration and the planar wing with nonlinear leading and trailing edges.

shape of the edges has allowed the optimizer to add curvature to the main spars around

the wing crank. This lowers the stress concentration by creating a fillet near the planform

break and reduces the weight of the wing [83]. Although the structural topology is fixed

in the present study, the aerostructural optimizer is still able to curve the spars by making

changes to the leading and trailing edges of the wing. The reduction in weight leads to an

improved objective function value. Furthermore, a lower weight leads to a lower induced

drag due to the quadratic relationship between the lift required and the induced drag.

The reader is reminded that the overall sweep angle of the geometry remains un-

changed due to the fact that the physical locations of the wing root and tip are fixed.

Having the ability to create curved leading and trailing edges allows the optimizer to

vary the wing sweep locally along the span and reduce the wave drag well below its value

for the optimized wing with straight edges. This may explain why the optimizer adds

curvature to the wing away from the planform break. In order to examine this effect,

Figures 4.53 and 4.54 show contours of normalized entropy taken at 60% of the half-span

for the baseline design and the planar wing with curved leading and trailing edges, re-

spectively. The weak shock near the trailing edge of the baseline wing is diminished on

the wing with curved edges. As a result, some of the drag reduction provided by this

design can be attributed to its ability to reduce the wave drag.


Figure 4.53: Contours of normalized en-tropy in cruise condition (M = 0.785 at35, 000 ft) at 60% half-span for the opti-mized planar wing with straight edges.

Figure 4.54: Contours of normalized en-tropy in cruise condition (M = 0.785 at35, 000 ft) at 60% half-span for the opti-mized planar wing with curved edges.

We can use purely aerodynamic shape optimization based on the Euler equations to

confirm that the planar wing with curved leading and trailing edges can reduce the wave

drag by locally varying wing sweep along the span. We perform a lift-constrained drag

minimization with CL = 0.62 at Mach 0.785 using the same geometric parameterization

as the previous case. The inviscid lift-to-drag ratio of the wing with curved edges is

approximately equal to 34.0, which is 6.7% higher than the planar wing with straight

edges. However, we still need to address the possibility that some of this drag reduction

may be due to induced drag savings from a nonplanar wake.

In order to examine the possibility that a wing with curved leading and trailing

edges can reduce the induced drag by producing a nonplanar wake, we consider a purely

aerodynamic shape optimization case at a cruise Mach number of 0.50. Since the flow is

inviscid, the low Mach number will eliminate wave drag as a potential source of drag in the

calculations and hence simplify the optimization problem. We conduct a lift-constrained

drag minimization with CL = 0.62 using the same geometric parameterization as before.

Figures 4.55 and 4.56 show the streamlines of the two wings behind the trailing edge.

The curved edges have not altered the wake significantly. Furthermore, the inviscid lift-

to-drag ratio of the wing with curved leading and trailing edges is 36.6, which is 0.5%

higher than 36.4 for the baseline wing. This suggests that curved leading and trailing

edges do not significantly enhance the induced drag performance of swept-back wings,

which already produce a nonplanar wake at a nonzero angle of attack [26, 27].

Our preliminary investigation of the drooped wing concept has shown that it can im-

prove the range of the baseline aircraft. However, as with the winglet-down configuration,

this nonplanar geometry will have important practical implications on the design of the

aircraft that we have ignored in the present study. The negative dihedral of the drooped

wing may necessitate a high-wing configuration in order to maintain a reasonable lateral


Figure 4.55: View of the aerodynamicallyoptimized planar wing at M = 0.50 alongwith the streamtraces that extend to thefar-field.

Figure 4.56: View of the aerodynamicallyoptimized wing with curved leading andtrailing edges at M = 0.50 along with thestreamtraces that extend to the far-field.

static stability margin [84]. A high-wing configuration may also be required to accommo-

date wingtip vertical clearance requirements. These aspects are beyond the scope of the

current study because it is important to first quantify the possible efficiency benefits of

nonplanar wings. Furthermore, a high-wing configuration may be aerodynamically more

efficient than today’s modern low-wing aircraft [82].

4.2.2 Aerodynamic Shape Optimization of the Drooped Wing

Concept

It is insightful to examine the result of purely aerodynamic shape optimization, ignoring

the effects of weight and structural deflections, using the same geometric parameteri-

zation and design variables defined in Section 4.2.1. Focusing on important differences

between aerodynamically and aerostructurally optimal designs helps to gain a deeper

understanding of when aerostructural optimization is essential in the design of uncon-

ventional wings.

The problem formulation in terms of geometric design variables is the same as be-

fore. However, the objective function is the inviscid drag of the wing. There is a lift

constraint at the cruise condition which requires that CL = 0.62. The cruise condition is

M = 0.785 at an altitude of 35, 000 ft. There is no maneuver constraint since structural

considerations are ignored in purely aerodynamic optimization.

Figure 4.57 shows a front-view of the optimal wing that results from the aerodynamic


Figure 4.57: The drooped wing configuration that results from purely aerodynamic shapeoptimization in cruise condition (M = 0.785 at 35, 000 ft) using the same geometricparameterization and design variables as the aerostructural optimization cases of thesame concept.

shape optimization. The optimization leads to a drooped wing with a height-to-span ratio

of 19%. This is significantly larger than the height-to-span ratio achieved from aerostruc-

tural optimization, which is approximately equal to 6%. The fact that the aerostructural

optimizer did not create the same optimal geometry means that this particular shape is

most likely too heavy from a structural standpoint. As a result, fully coupled high-fidelity

aerostructural optimization is necessary for the drooped wing.

4.2.3 Multimodality of the Drooped Wing Concept

Past research efforts have found that, in the context of purely aerodynamic shape op-

timization based on the Euler equations, wing design problems with significant amount

of geometric freedom can be somewhat multimodal [50]. Furthermore, the number of

local optima increases with increasing geometric freedom. This presents a well-known

and significant challenge for gradient-based optimization methodologies that do not have

a mechanism to ensure convergence to the global optimum, especially in cases where the

number of design variables is on the order of hundreds.

Our drooped wing optimization cases involve a large degree of geometric freedom.

Since we use a gradient-based approach for the purpose of numerical optimization, it

is possible that this particular concept represents a local optimum. To the best of our

knowledge, the degree of multimodality for exploratory cases of this nature is largely

unknown in the context of aerostructural optimization. Therefore, we aim to make a

preliminary effort to determine whether the current optimization problem is multimodal.

A formal investigation of multimodality involves generating many samples to serve

as initial designs in order to ensure that all regions of the design space are sufficiently

represented. Sophisticated sampling strategies, including Latin hypercube and Sobol

sequences, already exist for efficient exploration of the design space. However, using


Figure 4.58: Randomly generated initialgeometries.

Figure 4.59: Final optimized designs fromthe initial geometries shown in Figure4.58.

these strategies requires optimizing many different initial geometries and assessing the

shape and performance of each optimized sample. Such a study is outside the scope of

the present study and would require an excessive amount of computational resources due

to the fact that performing high-fidelity aerostructural optimization is computationally

expensive. As a result, we limit our preliminary investigation to using a few randomly

generated initial designs using the same geometric parameterization presented in Section

4.2.1. The randomly perturbed design variables include the five dihedral and twist angle

values at the specified spanwise stations. The problem formulation in terms of objective

function and nonlinear constraints is the same as before.

Figure 4.58 shows the six initial geometries used for the purpose of this investigation.

Although the number of initial samples is not large enough to explore the design space

thoroughly, it still represents a significant step away from using a single initial design.

Figure 4.59 shows all final geometries. It is clear that the optimizations lead to similar

drooped wing concepts regardless of the initial design used. Table 4.8 shows the final

objective function values of all designs. Differences in the objective function are indicated

by ∆J . The performance of all six optimal designs is also similar. These results show

that the drooped wing concept is unlikely to have a multimodal design space from an

aerostructural optimization standpoint.

Although this brief investigation is not sufficiently comprehensive to be conclusive,

it is a first step towards determining whether taking into account the effects of weight

and structural deflections in addition to drag reduces the multimodality of the design

space for this type of wing optimization problem. For example, it is possible that the


Table 4.8: The final objective function comparison for all of the six optimized designsevaluated

Parameter Design 1 Design 2 Design 3 Design 4 Design 5 Design 6J −8.73 −8.76 −8.78 −8.77 −8.75 −8.75∆J 0.0% 0.3% 0.6% 0.5% 0.2% 0.2%

fundamental trade-off between weight and drag prevents the optimizer from exploring

designs that would have been an aerodynamically local optimum.

4.3 Summary

The main objective of this chapter is to quantify the possible efficiency gains from nonpla-

nar wings using fully coupled high-fidelity aerostructural optimization, where the effects

of weight and structural deflection are taken into account in addition to drag. We con-

sider two classes of nonplanar wings: wings with winglets and the drooped wing. These

are compared with optimized planar wings of the same projected span.

The comprehensive investigation on wingletted wings presented in this chapter pro-

vides a new perspective on the potential efficiency gains from winglets. The purely

aerodynamic shape optimization studies demonstrate that it is possible to obtain total

drag reductions from winglets in comparison to planar wings of the same span. This

indicates that the numerical methodology works as intended. A second important con-

clusion is that the winglet-down configuration performs better than a winglet-up even

from a purely aerodynamic standpoint. This is due to the fact that the winglet-down

configuration pushes the tip vortex further away from the wing. Furthermore, our results

confirm that the aerodynamic benefits of winglets for transonic wings are smaller than

those reported in other studies that consider low-speed wings. Fully coupled aerostruc-

tural optimization shows that the winglet-down configuration reduces the drag at the

same total weight by 2% in comparison to its optimized planar counterpart of the same

projected span. This is due to the fact that when the structural deflections are included,

the projected span of the wing increases due to the outboard deflection of the winglet-

down feature. Therefore, the winglet-down configuration is able to push the tip vortex

even further away relative to the purely aerodynamic case. Finally, it is shown that

the total drag reduction possible from wingletted wings is higher in low-speed, high-lift

conditions because the ratio of induced to total drag is higher than in cruise.

The drooped wing results from giving the optimizer as much geometric freedom as

possible in terms of the dihedral angle at various stations along the wing. We empha-


size that this study is not the first to suggest the drooped wing as a novel concept with

potential for fuel efficiency gains. A few previous studies have used biomimicry as an

inspiration for this concept since it mimics the wing configuration of seagulls in gliding

flight. However, this investigation serves as a next step towards characterizing the poten-

tial efficiency gains of the drooped wing for commercial aircraft in transonic flight. There

are two main conclusions from our studies of the drooped wing. First, the aerostructural

optimizer produces a drooped wing from an initially planar wing given the freedom to

do so. Second, this design has the potential to improve the range by 2.6% relative to an

optimized planar wing of the same projected span. The drooped wing is more efficient

than the winglet-down design, which improves the range by 1.8%. The reason is that

the drooped wing pushes the tip vortex further away than a planar wing and increases

the projected span at the deflected state, as with the winglet-down configuration, but

with a lower wetted surface area and weight. Furthermore, our preliminary investigation

on the multimodality of the drooped wing indicates that this design is most likely not a

local optimum. Finally, the drooped wing with curved leading and trailing edges leads to

a 4.9% range improvement in comparison to its planar counterpart by further reducing

wing weight and wave drag.

Chapter 5

Conclusions, Contributions, and

Future Work

The purpose of this chapter is to review the main contributions, conclusions, and future

work from the present thesis. The primary and secondary conclusions are discussed first.

The most important contributions are then highlighted by bullet points. Finally, a list

of possible future research directions is presented.

5.1 Primary Conclusions

The primary conclusions of this thesis on the potential performance gains from nonplanar

wings are listed below.

• From a purely aerodynamic perspective, winglets oriented downward produce a

larger drag reduction than winglets oriented upward by moving the tip vortex

further away from the wing in the spanwise direction.

• When fully coupled aerostructural optimization is considered, the results indicate

that the winglet-down configuration is able to push the tip vortex even further away

from the wing relative to the purely aerodynamic case by increasing the projected

span at the deflected state. This configuration reduces the drag at the same weight

by 2% while keeping the (undeflected) projected span unchanged in comparison to

the planar counterpart.

• Purely aerodynamic shape optimization results show that the winglet-up configu-

ration reduces the total drag by 1.1% in comparison to an optimized planar wing

83

Chapter 5. Conclusions, Contributions, and Future Work 84

of the same projected span. From an aerostructural perspective, however, this con-

figuration brings the tip vortex closer to the wing due to the inboard deflection

of the winglet-up feature. Furthermore, the addition of the winglet adds weight

to the wing by changing the spanwise loading and increasing the structural span.

The combination of these effects lead to an inferior performance for the winglet-up

design in comparison to an optimized planar wing of the same span.

• The potential benefits of wingletted wings are larger in low-speed, high-lift con-

ditions such as steady climb because the ratio of induced to total drag is higher

than in cruise. Thus, the total drag reduction associated with a wingletted design

depends on the flight profile of the mission that an aircraft is intended to fly.

• The aerostructural optimizer produces a drooped wing design when it is given the

freedom to do so. This is significant because it indicates that the high-fidelity

numerical optimization framework is able to recover design trends that have shown

potential for improving fuel efficiency. This is especially relevant in exploratory

design where one cannot solely rely on the designer’s experience.

• The drooped wing can improve the range by 2.6% in comparison to an optimal

planar wing of the same span. This configuration increases the projected span to a

larger extent compared to the winglet-down configuration, which provides a 1.8%

improvement in range. Furthermore, the drooped wing achieves this at a lower

wetted surface area and weight than the winglet-down design, as shown in Table

4.7.

• Curved leading and trailing edges may be beneficial for commercial aircraft wings.

Curved edges allow the optimizer to create local sweep variations in the wing and

hence minimize the wave drag at a constant overall sweep angle. Furthermore,

creating a smooth transition near the planform break can reduce the wing weight.

Heavier wings require more lift and as a result lead to higher lift-induced drag.

While there are practical reasons and constraints for having straight leading and

trailing edges, our purpose in exploratory optimization studies of this nature is

to quantify the relative efficiency improvement that is possible by relaxing such

constraints.

• The drooped wing with curved leading and trailing edges can improve the range

by 4.9% relative to its planar counterpart of the same projected span.


• Our preliminary multimodality investigation on the drooped wing concept suggests

that the aerostructural optimization is not sensitive to the choice of the initial

design.

• The optimal drooped wing produced by fully coupled aerostructural optimization

is a more feasible design in terms of the tip clearance constraint as compared

to an aerodynamically optimal counterpart. Thus, including more physics in the

analysis reduces the need for arbitrary constraints on the geometry of the wing. This

highlights some of the subtle but important differences between purely aerodynamic

and fully coupled aerostructural optimization.

5.2 Secondary Conclusions

The secondary conclusions of the present thesis are mostly related to establishing the

credibility of the main results by validating and verifying the numerical analysis and

optimization framework. These conclusions are listed below.

• The results of our systematic aerodynamic mesh refinement study indicate that

the errors associated with the fitting process during displacement transfer do not

significantly affect the convergence behavior of the flow solver.

• Comparisons made between aerostructural analysis results and static aeroelastic

experimental data from the HIRENASD project indicate that the computational

results are in good agreement with the experiment. Furthermore, the computed

wingtip deflection agrees well with the experimental result. Therefore, the fitting

errors during the displacement transfer process are not affecting the accuracy of

the computational results.

• The planar wing sweep angle optimization case indicates that the aerostructural

optimization framework is able to recover the expected trade-off between weight

and drag for wings flying at transonic speed.

• We performed a preliminary investigation to assess the potential impact of keep-

ing the structural topology fixed on the final trends. The results indicate that

the aerostructural optimizer is still able to produce the same aeroelastic behavior

for different structural layouts. Furthermore, the variation in the final results is

minimal.


• We chose to create trade-off curves of optimal designs for various configurations in

order to gain a deeper insight into the fundamental trade-off between wing weight

and drag and how it may be affected by the presence of certain features. However,

generating high-resolution fronts (with many points on the front itself to resolve

the trade-off more accurately) is computationally expensive in the context of high-

fidelity calculations. We found that a tremendous amount of insight could still be

acquired by limiting the number of points to two or three. Furthermore, insightful

comparisons can be made by simply choosing drag as the objective function even

in an aerostructural context due to the fact that drag is not simply an aerodynamic

objective. The total drag of the aircraft implicitly depends on the weight through

lift-induced drag. As a result, the aerostructurally optimized designs still vary

significantly from their aerodynamically optimized counterparts even when drag is

used as the objective function.

5.3 Main Contributions

A list of the most important contributions is provided below.

• studied the potential benefits of wingletted wings by adopting a comprehensive

step-by-step approach using purely aerodynamic and fully coupled high-fidelity

aerostructural optimization;

• conducted a preliminary investigation on the possible efficiency gains of a novel

drooped wing concept for commercial aircraft;

• validated the aerostructural analysis capability by comparing computational results

to experimental data;

• formulated test cases in order to verify specific aspects of the numerical optimization

framework and establish its suitability for performing aerostructural optimization

with large shape changes.

5.4 Future Work

The numerical optimization studies conducted over the course of this project attempted

to include the most important aspects of wing design in the context of aerostructural

optimization. However, there are several interesting research directions that may be


pursued based on the results presented in this thesis. A list of these possible extensions

is provided below.

• All of the optimization results presented in this thesis were based on inviscid flow

analysis. Although an attempt was made to account for the impact of viscous drag

using post-optimality estimates, a natural extension is to include viscous effects in

the optimization. This can be accomplished by extending the current framework

to incorporate the RANS-based flow solver.

• The effects of flutter and buckling on the sizing of the wings were ignored in the

present thesis. Although neglecting these effects does not change the main conclu-

sions made based on our results, it will be necessary to take them into account in

the context of detailed wing design. This is an essential next step for concepts that

shown promise in the present thesis.

• The structural sizing approach used in this thesis for the purpose of aerostructural

optimization relied on a single critical structural load condition. This is due to

the fact that incorporating more load cases would increase the cost of the opti-

mizations significantly due to the high-fidelity nature of the calculations. As the

computational resources grow and more efficient numerical methods emerge, it will

be worthwhile to include more load conditions in order to capture the correct sizing

trends more accurately.

• In this work, only the wing surfaces were modeled for the flow analysis. Viscous

drag estimates of the aircraft including the fuselage were considered in the post-

optimality analyses. However, including the fuselage, especially in the case of the

drooped wing where a high-wing configuration may be necessary, will provide a

better representation of the physical flow field particularly in the context of viscous

flow. This is important because a high-wing configuration may provide further

viscous drag reductions [82].

• The present aerostructural analysis and optimization framework lacks the capability

to perform structural topology optimization. Hence the layout of the ribs and spars

remained unchanged in every optimization case presented in this thesis. Although

we established to some extent that this did not affect our final results significantly,

the ability to find the optimal structural layout remains an important avenue of

research in the context of exploratory optimization.


• Gradient-based optimization is ultimately limited by the fact that the final solu-

tion may not be a global optimum. Gradient-based multistart algorithms address

this shortcoming [50]. However, there are reasons to believe that incorporating

more disciplines may reduce the occurrence of local optima in the design space.

For instance, different local optima resulting from purely aerodynamic shape opti-

mization may not be structurally feasible, and as a result, will not be present in

the aerostructural design space as optimal designs. It will be interesting to explore

this possibility by performing purely aerodynamic and aerostructural optimizations

using the same geometric parameterization methodology in order to quantify the

multimodality of the design space in each case.

• Performing a fine-mesh analysis on the optimized geometries was often challenging

due to convergence difficulties for the nonlinear block Gauss-Seidel method used in

aerostructural analysis. While some of these challenges may be mitigated by using

more robust solution strategies such as monolithic methods, creating a nonlinear

model of the aerodynamic functionals of interest on the fine mesh may be helpful.

Lunia et al. [85] provide a successful application of using neural networks for the

purpose of building such a nonlinear model.

• The current aerostructural analysis and optimization framework can be extended

to incorporate the FFD geometry control technique of Gagnon and Zingg [75]. This

geometry control strategy has been shown to be effective for the design optimization

of some unconventional aircraft configurations such as the box-wing and the truss-

braced wing [86, 61].

Appendices

89

Appendix A

Aerodynamic Shape Optimization of

Winglets Using More Realistic

Constraints

In Section 4.1.2, we demonstrate that from an aerodynamic perspective the winglet-up

configuration can reduce the total drag by 1.1%, while the winglet-down design can

provide a total drag reduction of 2.6% in comparison with a planar wing of the same

projected span. These purely aerodynamic shape optimization problems are formulated

using the same lift, volume, and thickness-to-chord ratio constraints. The primary goal

of the present section is to investigate the implications of using these constraints as they

relate to the main trends observed at the end of the optimizations.

There are two main issues related to the conventional approach for formulating the

winglet aerodynamic shape optimization problem. First, we use the same lift constraint

in all three cases, as seen in other similar studies [54, 14, 34]. This choice implies that the

weight of the aircraft is exactly the same in all three cases. The addition of a nonplanar

feature at the tip increases the weight of the wing in two ways. A winglet simply increases

the structural span, which requires more material. It also leads to a heavier structure

as a result of modifying the span loading near the tip of the wing. A higher tip loading

produced by the winglet requires structural modifications elsewhere along the span in

order to ensure the structural integrity of the wing [33]. Even a small increase in the

required lift will lead to a considerable increase in the induced drag due to the quadratic

relation between lift and induced drag [8]. Second, we allow the optimizer to reduce

the thickness-to-chord ratio by a certain amount while constraining the wing volume

to the initial value. However, there is a weight penalty associated with decreasing the

thickness-to-chord ratio of the wing that is not captured in purely aerodynamic shape

90

Appendix A. Further Aerodynamic Optimization of Winglets 91

Table A.1: Differences in nonlinear constraints for the aerodynamic shape optimizationcases

Original Constraints More Realistic Constraints

Constraint Planar Winglet-Up Winglet-Down Planar Winglet-Up Winglet-Down

Lift CL = 0.486 CL = 0.486 CL = 0.486 CL = 0.486 CL = 0.487 CL = 0.490

Volume (m3) V ≥ 13.2 V ≥ 13.3 V ≥ 13.3 V = 14.9 V = 15.3 V = 15.0

Planar Up Down

Norm

alizedDrag

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Figure A.1: Comparison of total drag in cruise for all wings while taking into accountdifferences in optimal lift and wing volume.

optimization. As a result, we are ignoring the structural implications of reducing the

thickness-to-chord ratio along the span.

In formulating the aerodynamic shape optimization problem for winglets, it is im-

possible to know in advance what the correct lift, volume, and thickness-to-chord ratio

constraints should be in order to account for the associated structural effects. Such non-

linear constraints are typically set in an arbitrary fashion. However, suppose for the

purpose of this discussion that we have access to the necessary information about the

optimal aircraft weight and wing volume to be used as more realistic constraints in the

formulation of our aerodynamic shape optimization problem, as listed in Table A.1. (We

obtained this information by performing fully coupled aerostructural optimization for

each individual configuration.) What happens to our aerodynamic shape optimization

results if we do take into account the differences between optimal lift and volume? Fig-

ure A.1 provides the same total drag comparison as before while taking into account

these differences. It is clear that our main conclusions have changed. The wingletted

wings no longer have a significant aerodynamic advantage in comparison to the optimal

planar wing. This means that the arbitrary constraints imposed on our aerodynamic

shape optimization problems have an influence on the final results. This highlights the

Appendix A. Further Aerodynamic Optimization of Winglets 92

importance of performing high-fidelity aerostructural optimization in the context of the

design of wings with winglets.

Appendix B

Aerodynamic-Structural

Optimization of Winglets

We have seen in Section 4.1.2 that the aerodynamic shape optimizer is able to recover a

total drag reduction in cruise for wingletted wings in comparison to an optimal planar

wing of the same span. However, as shown in Appendix A, when we consider differences

between optimal lift and volume, this aerodynamic advantage is diminished. Another

interesting question to consider is what happens if we take an aerodynamically opti-

mal design, freeze the geometry, and optimize the structures only using high-fidelity

aerostructural optimization? This slightly differs from sequential optimization in that

the optimizer can still manipulate the lift distribution in addition to the weight of the

wing by controlling the bend-twist coupling (through aeroelastic tailoring). In sequential

optimization, however, the structure is optimized for a fixed aerodynamically optimal

lift distribution. The answer to this question will shed some light on whether or not the

final geometries obtained from purely aerodynamic shape optimization can satisfy the

structural failure constraints at the weight implied by the lift constraint.

It is worthwhile to review the steps involved in conducting this investigation to provide

further clarity. We performed purely aerodynamic shape optimization for wingletted and

planar wings using identical CL constraints in Section 4.1.2. The lift constraint was

obtained from performing a fully coupled aerostructural optimization for a planar wing.

Thus, we know that the aerostructural optimizer is able to find a feasible structure for the

weight that is implied by this lift constraint. However, it is possible that the aerodynamic

optimizer has converged to a final design that requires a heavier structure because it does

not have the mechanism to take the weight into account. For instance, the aerodynamic

shape optimizer may have increased the tip loading beyond a feasible level, or reduced the

thickness-to-chord ratio of the wing to an infeasible extent. Thus, in order to determine

93

Appendix B. Aerodynamic-Structural Optimization of Winglets 94

CDinviscid

0.010 0.011 0.012 0.013 0.014 0.015

CL

0.484

0.486

0.488

0.490

0.492

0.494

0.496

0.498

Aerodynamic

Aerodynamic-StructuralDown

Up

Planar

Down Up Planar

Figure B.1: Differences between the lift and drag performance of the planar and winglet-ted wings obtained from purely aerodynamic and aerodynamic-structural optimization.

if this is indeed the case, we will fix the aerodynamically optimized geometries and try

to find a feasible structure for each case using fully coupled aerostructural optimization.

The aerodynamic grid used for the optimization is the same as the one used in Section

4.1.2. Since the geometries in each case are fixed to those obtained from aerodynamic

shape optimization, there are no geometric design variables, but the angle of attack is

free to vary. Comprehensive details about the approach used for structural sizing are

provided in Section 4.1.3. Here, we limit our discussion to a brief overview because the

main focus of the current study is still on the aerodynamic aspects. The optimizer is

allowed to vary the thickness values of the ribs, spars, and skin elements of the internal

structure of the wing. Thus, the optimizer is able to minimize the weight as long as

the structural integrity of the wing at a 2.5g critical load condition is maintained. We

consider the same three configurations as before: winglet-up, winglet-down, and planar.

The aerostructural optimizer has the freedom to modify the structure in such a way as

to maintain the same lift constraint originally specified in the aerodynamic optimization

problem formulation. Furthermore, it can stiffen the wing in order to minimize the struc-

tural deflections, so that the same optimal geometry is produced as the one obtained from

purely aerodynamic shape optimization. However, this may require a heavier structure

than the original aerodynamic shape optimization lift constraint of CL = 0.486 implies.

The lift necessary to sustain this heavier structure may lead to a higher induced drag.

Therefore, if the aerostructural optimization is unable to produce the same lift and drag

performance as the one obtained from purely aerodynamic shape optimization, then the

original lift constraint specification for the aerodynamic shape optimization is infeasible.

We are interested to see if this is the case for any of the configurations considered.

Appendix B. Aerodynamic-Structural Optimization of Winglets 95

Figure B.1 plots the lift and inviscid drag performance of all designs from purely aero-

dynamic and aerodynamic-structural optimization evaluated on the optimization grid.

The single-discipline designs have the same lift due to the same lift constraint specified

in the original aerodynamic shape optimization problem formulation. It is clear that the

aerostructural optimizer is not able to reduce both lift and drag for any of the configu-

rations to the level obtained from purely aerodynamic shape optimization. This means

that our original lift constraint specification for the optimal aerodynamic geometry is not

feasible as far as structural design is concerned (even though it was originally obtained

from performing fully coupled aerostructural optimization). In other words, it is impossi-

ble to find a structure at the specified weight for any of the configurations obtained from

purely aerodynamic shape optimization. This result provides stronger motivation for

conducting fully coupled high-fidelity aerostructural optimization in this context, where

the fundamentally inherent tradeoff between drag and weight is automatically taken into

account throughout the course of the optimization. Furthermore, it suggests that we

should be cautious in comparing conclusions made based on purely aerodynamic shape

optimization to those obtained by aerostructural optimization.

Our main conclusions here should not be mistaken for an attempt to suggest that

high-fidelity aerodynamic shape optimization is simply insufficient. Instead, we hope to

clarify subtle (but undoubtedly important) differences between purely aerodynamic and

aerostructural optimization that influence our final conclusions. We emphasize that these

points should be viewed in the context of our step-by-step approach. This approach aims

to highlight the reasons that the present results may appear to differ from conclusions

made in the past based on purely aerodynamic shape optimization and low- and medium-

fidelity aerostructural optimization.

References

[1] “Climate Change 2001: Synthesis Report. A Contribution of Working Groups I, II,

and III to the Third Assessment Report of the Integovernmental Panel on Climate

Change [Watson, R.T. and the Core Writing Team (eds.)],” Tech. rep., Intergovern-

mental Panel on Climate Change, Cambridge, United Kingdom, 2008.

[2] “Climate Change 2007: Synthesis Report. Contribution of Working Groups I, II and

III to the Fourth Assessment Report of the Intergovernmental Panel on Climate

Change [Core Writing Team, Pachauri, R.K and Reisinger, A. (eds.)],” Tech. rep.,

Intergovernmental Panel on Climate Change, Geneva, Switzerland, 2008.

[3] Olivie, D. J. L., Cariolle, D., Teyssedre, H., Salas, D., Voldoire, A., Clark, H.,

Saint-Martin, D., Michou, M., Karcher, F., Balkanski, Y., Gauss, M., Dessens, O.,

Koffi, B., and Sausen, R., “Modeling the climate impact of road transport, maritime

shipping and aviation over the period 18602100 with an AOGCM,” Atmospheric

Chemistry and Physics , Vol. 12, No. 3, 2012, pp. 1449–1480.

[4] “Climate Change 2014: Synthesis Report. Contribution of Working Groups I, II

and III to the Fifth Assessment Report of the Intergovernmental Panel on Climate

Change [Core Writing Team, R.K. Pachauri and L.A. Meyer (eds.)],” Tech. rep.,

Intergovernmental Panel on Climate Change, Geneva, Switzerland, 2014.

[5] Sgouridis, S., Bonnefoy, P. A., and Hansman, R. J., “Air transportation in a car-

bon constrained world: Long-term dynamics of policies and strategies for mitigat-

ing the carbon footprint of commercial aviation,” Transportation Research Part A:

Policy and Practice, Vol. 45, No. 10, 2011, pp. 1077 – 1091, A Collection of Pa-

pers:Transportation in a World of Climate Change.

[6] “ICAO Environmental Report 2013,” Tech. rep., International Civil Aviation Orga-

nization, Montreal, Quebec, Canada, October 2013.

[7] “Global Market Forecast 2015,” Tech. rep., Airbus S.A.S., August 2015.

96

References 97

[8] Anderson, J. D., Fundamentals of Aerodynamics , McGraw-Hill, 1221 Avenue of the

Americas, New York, NY 10020, United States of America, 4th ed., 2007.

[9] Leoviriyakit, K., Kim, S., and Jameson, A., “Viscous Aerodynamic Shape Opti-

mization of Wings Including Planform Variables,” 21st AIAA Applied Aerodynamics

Conference, No. 2003-3498, Orlando, Florida, June 2003.

[10] Leoviriyakit, K., Kim, S., and Jameson, A., “Aero-Structural Wing Planform Opti-

mization Using the Navier-Stokes Equations,” 10th AIAA/ISSMO Multidisciplinary

Analysis and Optimization Conference, No. 2004-4479, Albany, New York, Septem-

ber 2004.

[11] Jameson, A., Leoviriyakit, K., and Shankaran, S., “Multi-point Aero-Structural Op-

timization of Wings Including Planform Variables,” 45th AIAA Aerospace Sciences

Meeting and Exhibit , No. 2007-764, Reno, Nevada, January 2007.

[12] Reist, T. A., , and Zingg, D. W., “Aerodynamic Shape Optimization of a Blended-

Wing-Body Regional Transport for a Short Range Mission,” 31st AIAA Applied

Aerodynamics Conference, No. 2013-2414, San Diego, California, June 2013.

[13] Reist, T. A., , and Zingg, D. W., “Optimization of the Aerodynamic Performance of

Regional and Wide-Body-Class Blended Wing-Body Aircraft,” 31rd AIAA Applied

Aerodynamics Conference, No. 2015-3292, Dallas, Texas, June 2015.

[14] Gagnon, H. and Zingg, D. W., “High-fidelity Aerodynamic Shape Optimiza-

tion of Unconventional Aircraft through Axial Deformation,” AIAA SciTech, 52nd

Aerospace Sciences Meeting , No. 2014-0908, National Harbor, Maryland, January

2014.

[15] Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., “High-Fidelity Aerostructural

Design Optimization of a Supersonic Business Jet,” Journal of Aircraft , Vol. 41,

No. 3, 2004, pp. 523–530.

[16] Kenway, G. K. W., Kennedy, G. J., and Martins, J. R. R. A., “Multipoint High-

Fidelity Aerostructural Optimization of a Transport Aircraft Configuration,” Jour-

nal of Aircraft , Vol. 51, 2014, pp. 144–160.

[17] Kenway, G. K. W., Kennedy, G. J., and Martins, J. R. R. A., “Scalable Parallel Ap-

proach for High-Fidelity Steady-State Aeroelastic Analysis and Adjoint Derivative

Computations,” AIAA Journal , Vol. 52, 2014, pp. 935–951.

References 98

[18] Liem, R. P., Kenway, G. K. W., and Martins, J. R. R. A., “Multimission Aircraft Fuel

Burn Minimization via Multipoint Aerostructural Optimization,” AIAA Journal ,

Vol. 53, 2015, pp. 104–122.

[19] Maute, K., Nikbay, M., and Farhat, C., “Coupled Analytical Sensitivity Analysis and

Optimization of Three-Dimensional Nonlinear Aeroelastic Systems,” AIAA Journal ,

Vol. 39, No. 11, 2001, pp. 2051–2061.

[20] Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., “A Coupled-Adjoint Sensitiv-

ity Analysis Method for High-Fidelity Aero-Structural Design,” Optimization and

Engineering , Vol. 6, 2005, pp. 33–62.

[21] Kroo, I., “Drag Due to Lift: Concepts for Prediction and Reduction,” Annual Review

of Fluid Mechanics , Vol. 33, No. 1, 2001, pp. 587–617.

[22] Whitcomb, R., “A Design Approach and Selected Wind Tunnel Results at High

Subsonic Speeds for Wing-Tip Mounted Winglets,” Tech. Rep. TN-D-8260, NASA,

1976.

[23] Heyson, H. H., Riebe, G. D., and Fulton, C. L., “Theoretical Parametric Study of the

Relative Advantages of Winglets and Wing-Tip Extensions,” Tech. Rep. TP-1020,

NASA, 1977.

[24] Jones, R. T. and Lasinsky, T. A., “Effects of Winglets on the Induced Drag of Ideal

Wing Shapes,” Tech. Rep. TM-81230, NASA, 1980.

[25] Asai, K., “Theoretical Considerations in in Aerodynamic Effectiveness of Winglets,”

Journal of Aircraft , Vol. 22, No. 7, 1985, pp. 635–637.

[26] van Dam, C. P., “Induced-Drag Characteristics of Crescent-Moon-Shaped Wings,”


[27] van Dam, C. P., “Aerodynamic Characteristics of Crescent and Elliptic Wings at

High Angles of Attack,” Journal of Aircraft , Vol. 28, No. 4, 1991, pp. 253–260.

[28] Takenaka, K. and Hatanaka, K., “Multidisciplinary Design Exploration for a

Winglet,” Journal of Aircraft , Vol. 45, No. 5, 2008, pp. 1601–1611.

[29] Verstraeten, J. G. and Slingerland, R., “Drag Characteristics for Optimally Span-

Loaded Planar, Wingletted, and C Wings,” Journal of Aircraft , Vol. 46, No. 3, 2009,

pp. 962–971.

References 99

[30] Ning, S. A. and Kroo, I., “Multidisciplinary Considerations in the Design of Wings

and Wing Tip Devices,” Journal of Aircraft , Vol. 47, No. 2, 2010, pp. 534–543.

[31] Jansen, P., Perez, R. E., and Martins, J. R. R. A., “Aerostructural Optimization

of Nonplanar Lifting Surfaces,” Journal of Aircraft , Vol. 47, No. 5, 2010, pp. 1491–

1503.

[32] Elham, A. and van Tooren, M. J., “Winglet multi-objective shape optimization,”

Aerospace Science and Technology , Vol. 37, 2014, pp. 93 – 109.

[33] Faye, R., Laprete, R., and Winter, M., “Blended Winglets for Improved Airplance

Performance,” Tech. rep., Boeing Commercial Airplanes, http://www.boeing.com/

commercial/aeromagazine/aero_17/winglets.pdf.

[34] Koo, D. and Zingg, D. W., “Progress in Aerodynamic Shape Optimization Based on

the Reynolds-Averaged Navier-Stokes Equations,” The AIAA Science and Technol-

ogy Forum and Exposition (AIAA SciTech), No. 2015-1292, San Diego, California,

January 2016.

[35] Ciampa, P. D. and Nagel, B., “Aerostructural Interaction in a Collaborative MDO

Environment,” International Conference of Computational Methods in Sciences and

Engineering 2014 (ICCMSE), Athens, Greece, April 2014.

[36] Viti, A., Druot, T., and Dumont, A., “Aero-structural approach coupled with direct

operative cost optimization for new aircraft concept in preliminary design,” 17th

AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, No. AIAA

2016-3512, Washington, District of Columbia, June 2016.

[37] Kenway, G. K. W., Kennedy, G. J., and Martins, J. R. R. A., “A CAD-

Free Approach to High-Fidelity Aerostructural Optimization,” Proceedings of

the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort

Worth,Texas, September 2010, AIAA 2010-9231.

[38] Kennedy, G. J. and Martins, J. R. R. A., “Parallel Solution Methods for

Aerostructural Analysis and Design Optimization,” 13th AIAA/ISSMO Multidisci-

plinary Analysis Optimization Conference, No. AIAA-2010-9308, Fort Worth, Texas,

September 2010.

[39] Kennedy, G. J. and Martins, J. R. R. A., “A Comparison of Metallic and Compos-

ite Aircraft Wings Using Aerostructural Design Optimization,” 14th AIAA/ISSMO

http://www.boeing.com/commercial/aeromagazine/aero_17/winglets.pdf

http://www.boeing.com/commercial/aeromagazine/aero_17/winglets.pdf

References 100

Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN, Septem-

ber 2012, AIAA-2012-5475.

[40] Kenway, G. K. W., Kennedy, G. J., and Martins, J. R. R. A., “Aerostructural

optimization of the Common Research Model configuration,” 15th AIAA/ISSMO

Multidisciplinary Analysis and Optimization Conference, No. 2014-3274, Atlanta,

Georgia, June 2014.

[41] Fonte, F., Ricci, S., and Mantegazza, P., “Gust Load Alleviation for a Regional

Aircraft Through a Static Output Feedback,” Journal of Aircraft , Vol. 52, No. 5,

2015, pp. 1559–1574.

[42] Nguyen, N., Trinh, K., Reynolds, K., Kless, J., Aftosmis, M., Urnes, J., and Ip-

polito, C., “Elastically Shaped Wing Optimization and Aircraft Concept for Im-

proved Cruise Efficiency,” 51st AIAA Aerospace Sciences Meeting including the New

Horizons Forum and Aerospace Exposition, No. AIAA 2013-0141, Grapevine (Dal-

las/Ft. Worth Region), Texas, January 2013.

[43] Lazos, B. and Visser, K., “Aerodynamic Comparison of Hyper-Elliptic Cambered

Span (HECS) Wings with Conventional Configurations,” 24th AIAA Applied Aero-

dynamics Conference, No. AIAA 2006-3469, San Francisco, California, June 2006.

[44] Andrews, S. A., Perez, R. E., and Allan, W. D. E., “Aerodynamic implications

of gull’s drooped wing-tips,” Bioinspiration and Biomimetics , Vol. 8, No. 4, 2013,

pp. 046003.

[45] Zhang, Z. J., Khosravi, S., and Zingg, D. W., “High-Fidelity Aerostructural Opti-

mization with Integrated Geometry Parameterization and Mesh Movement,” 56th

AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confer-

ence, No. AIAA 2015-1132, Kissimmee, Florida, January 2015.

[46] Zhang, Z. J., Khosravi, S., and Zingg, D. W., “High-Fidelity Aerostructural Opti-

mization with Integrated Geometry Parameterization and Mesh Movement,” Struc-

tural and Multidisciplinary Optimization, 2016, doi: 10.1007/s00158-016-1562-7.

[47] Gill, P. E., Murray, W., Michael, and Saunders, M. A., “SNOPT: An SQP Algo-

rithm For Large-Scale Constrained Optimization,” SIAM Journal on Optimization,

Vol. 12, 1997, pp. 979–1006.

References 101

[48] Perez, R. E., Jansen, P. W., and Martins, J. R. R. A., “pyOpt: A Python-Based

Object-Oriented Framework for Nonlinear Constrained Optimization,” Structures

and Multidisciplinary Optimization, Vol. 45, No. 1, 2012, pp. 101–118.

[49] Zingg, D. W., Nemec, M., and Pulliam, T. H., “A comparative evaluation of ge-

netic and gradient-based algorithms applied to aerodynamic optimization,” Revue

europeenne de mecanique numerique, Vol. 17, No. 1-2, March 2008, pp. 103–126.

[50] Chernukhin, O. and Zingg, D. W., “Multimodality and Global Optimization in

Aerodynamic Design,” AIAA Journal , Vol. 51, No. 6, 2013, pp. 1342–1354.

[51] Wrenn, G. A., “An Indirect Method for Numerical Optimization Using the Kreis-

selmeier–Steinhauser Function,” Tech. Rep. CR-4220, NASA, 1989.

[52] Hicken, J. E. and Zingg, D. W., “Aerodynamic Optimization Algorithm with Inte-

grated Geometry Parameterization and Mesh Movement,” AIAA Journal , Vol. 48,

No. 2, 2010, pp. 400–413.

[53] Osusky, L., Buckley, H., Reist, T., and Zingg, D. W., “Drag Minimization Based

on the Navier-Stokes Equations Using a Newton-Krylov Approach,” AIAA Journal ,

Vol. 53, No. 6, June 2015, pp. 1555–1577.

[54] Hicken, J. E. and Zingg, D. W., “Induced-Drag Minimization of Nonplanar Ge-

ometries Based on the Euler Equations,” AIAA Journal , Vol. 48, No. 11, 2010,

pp. 2564–2575.

[55] Reist, T. A. and Zingg, D. W., “Aerodynamically Optimal Regional Aircraft Con-

cepts: Conventional and Blended Wing-Body Designs,” 52nd AIAA Aerospace Sci-

ences Meeting , No. AIAA 2014-0905, National Harbor, Maryland, January 2014.

[56] Truong, A. H., Oldeld, C. A., and Zingg, D. W., “Mesh Movement for a Discrete-

Adjoint Newton-Krylov Algorithm for Aerodynamic Optimization,” AIAA Journal ,

Vol. 46, No. 7, 2008, pp. 1695–1704.

[57] Leung, T. M., Zhang, J., Kenway, G. K. W., Kennedy, G. J., Wang, X., Zingg,

D. W., and Martins, J. R. R. A., “Aerostructural Optimization Using a Coupled-

Adjoint Approach,” Canadian Aeronautics and Space Institute 60th Aeronautics and

Annual General Meeting , Toronto, Ontario, May 2013.

[58] Hicken, J. E. and Zingg, D. W., “Parallel Newton-Krylov Solver for the Euler

Equations Discretized Using Simultaneous-Approximation Terms,” AIAA Journal ,

Vol. 46, No. 11, 2008, pp. 2773–2786.

References 102

[59] Osusky, M. and Zingg, D. W., “Parallel Newton-Krylov-Schur Solver for the Navier-

Stokes Equations Discretized Using Summation-By-Parts Operators,” AIAA Jour-

nal , Vol. 51, No. 12, 2013, pp. 2833–2851.

[60] Brown, D. A. and Zingg, D. W., “Performance of a Newton-Krylov-Schur Algorithm

for Solving Steady Turbulent Flows,” AIAA Journal , Vol. 54, No. 9, 2016, pp. 2645–

2658.

[61] Gagnon, H. and Zingg, D. W., “Aerodynamic Optimization Trade Study of a Box-

Wing Aircraft Configuration,” Journal of Aircraft , Vol. 53, No. 4, 2016, pp. 971–981.

[62] Lee, C., Koo, D., Telidetzki, K., Buckley, H., Gagnon, H., and Zingg, D. W., “Aero-

dynamic Shape Optimization of Benchmark Problems Using Jetstream,” 53rd AIAA

Aerospace Sciences Meeting , No. 2015-0262, Kissimmee, Florida, January 2015.

[63] Osusky, M. and Zingg, D. W., “Results from the Fifth AIAA Drag Prediction

Workshop obtained with a parallel Newton-Krylov-Schur ow solver discretized using

summation-by-parts operators,” 31st AIAA Applied Aerodynamics Conference, No.

AIAA 2013-2511, San Diego, California, June 2013.

[64] Kennedy, G. J. and Martins, J. R. R. A., “A parallel finite-element framework

for large-scale gradient-based design optimization of high-performance structures,”

Finite Elements in Analysis and Design, Vol. 87, September 2014, pp. 56–73.

[65] Kennedy, G. and Martins, J., “A parallel aerostructural optimization framework

for aircraft design studies,” Structural and Multidisciplinary Optimization, Vol. 50,

No. 6, 2014, pp. 1079–1101.

[66] Akgun, M. A., Haftka, R. T., Wu, K. C., Walsh, J. L., and Garcelon, J. H., “Efficient

Structural Optimization for Multiple Load Cases Using Adjoint Sensitivities,” AIAA

Journal , Vol. 39, No. 3, 2001, pp. 511–516.

[67] Ballmann, J., Dafnis, A., Braun, C., Korsch, H., Reimerdes, H. G., and Olivier,

H., “The HIRENASD Project: High Reynolds Number Aerostructural Dynamics

Experiments in the European Transonic Windtunnel (ETW),” 25th International

Congress of the Aeronautical Sciences (ICAS) 2006 , No. ICAS 2006-10.3.3, Ham-

burg, Germany, 2006.

[68] Ballmann, J., Dafnis, A., Korsch, H., Buxel, C., Reimerdes, H. G., Brakhage, K. H.,

Olivier, H., Braun, C., Baars, A., and Boucke, A., “Experimental Analysis of High

References 103

Reynolds Number Aero-Structural Dynamics in ETW,” 46th AIAA Aerospace Sci-

ences Meeting and Exhibit , No. AIAA 2008-841, Reno, Nevada, January 2008.

[69] Ballmann, J., Boucke, A., Dickopp, C., and Reimer, L., “Results of Dynamic Exper-

iments in the HIRENASD Project and Analysis of Observed Unsteady Processes,”

International Forum on Aeroelasticity and Structural Dynamics (IFASD) 2009 , No.

IFASD 2009-103, Seattle, Washington, January 2009.

[70] Chwalowski, P., Florance, J. P., Heeg, J., and Wieseman, C. D., “Preliminary Com-

putational Analysis of the HIRENASD Configuration in Preparation for the Aeroe-

lastic Prediction Workshop,” International Forum of Aeroelasticity and Structural

Dynamics (IFASD) 2011 , No. IFASD-2011-108, Paris, France, June 2011.

[71] Raymer, D. P., Aircraft Design : A Conceptual Approach, American Institute of

Aeronautics and Astronautics, 12700 Sunrise Valley Drive, Suite 200 Reston, VA

20191-5807, United States of America, 5th ed., 2012.

[72] Dunning, P., Stanford, B., and Kim, H., “Coupled aerostructural topology optimiza-

tion using a level set method for 3D aircraft wings,” Structural and Multidisciplinary

Optimization, Vol. 51, No. 5, 2015, pp. 1113–1132.

[73] James, K. A., Kennedy, G. J., and Martins, J. R., “Concurrent aerostructural topol-

ogy optimization of a wing box,” Computers and Structures , Vol. 134, 2014, pp. 1–17.

[74] “737 Airplane Characteristics for Airport Planning,” Tech. rep., Boeing Commercial

Airplanes, September 2013, http://www.boeing.com/assets/pdf/commercial/

airports/acaps/737.pdf.

[75] Gagnon, H. and Zingg, D. W., “Two-Level Free-Form and Axial Deformation for

Exploratory Aerodynamic Shape Optimization,” AIAA Journal , Vol. 53, No. 7,

2015, pp. 2015–2026.

[76] Fredericks, W., Costa, K. A. G., Deshpande, N., Moore, M., Miguel, E. S., and

Snyder, A., “Aircraft Conceptual Design Using Vehicle Sketch Pad,” 48th AIAA

Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Ex-

position, No. AIAA 2010-658, Orlando, Florida, January 2010.

[77] Gur, O., Mason, W. H., and Schetz, J. A., “Full-Configuration Drag Estimation,”


http://www.boeing.com/assets/pdf/commercial/airports/acaps/737.pdf

http://www.boeing.com/assets/pdf/commercial/airports/acaps/737.pdf

References 104

[78] Bourdin, P., “Numerical Predictions of Wing-Tip Effects on Lift-Induced Drag,”

International Council of Aeronautical Sciences (ICAS) Congress , No. 2002-2.2.3,

Toronto, Ontario, September 2002.

[79] Cui, P. and Han, J., “Prediction of flutter characteristics for a transport wing with

wingtip devices,” Aerospace Science and Technology , Vol. 23, No. 1, 2012, pp. 461–

468.

[80] Snyder, M. P. and Weisshaar, T. A., “Flutter and Directional Stability of Aircraft

with Wing-Tip Fins: Conflicts and Compromises,” Journal of Aircraft , Vol. 50,

No. 2, 2013, pp. 615–625.

[81] Khosravi, S. and Zingg, D. W., “A Numerical Optimization Study on Winglets,”

15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, No.

AIAA 2014-2173, Atlanta, Georgia, June 2014.

[82] Hashimoto, A., Jeong, S., and Obayashi, S., “Aerodynamic Optimization of Near-

future High-wing Aircraft,” Transactions of the Japan Society for Aeronautical and

Space Sciences , Vol. 58, No. 2, 2015, pp. 73–82.

[83] Gregg, R. D. and McDowell, D. L., “Blended Leading and Trailing Edge Wing

Planform,” June 2005, U.S. Patent 10/717,366.

[84] Anderson, J. D., Introduction to Flight , McGraw-Hill, 1221 Avenue of the Americas,

New York, NY 10020, United States of America, 4th ed., 2000.

[85] Lunia, A., Isaac, K. M., Chandrashekhara, K., and Watkins, S. E., “Aerodynamic

testing of a smart composite wing using fiber-optic strain sensing and neural net-

works,” Smart Materials and Structures , Vol. 9, No. 6, 2000, pp. 767.

[86] Gagnon, H. and Zingg, D. W., “Euler-Equation-Based Drag Minimization

of Unconventional Aircraft Configurations,” Journal of Aircraft , 2016, doi:

10.2514/1.C033591.

Documents

High-Fidelity Aerostructural Optimization of Nonplanar ... · High-Fidelity Aerostructural Optimization of Nonplanar Wings for Commercial Transport Aircraft Shahriar Khosravi Doctor