Stability-Constrained Aerodynamic Shape Optimization with Applications to Flying Wings by ...

199 Pages·2012·13.73 MB·English

Stability-Constrained Aerodynamic Shape Optimization with
Applications to Flying Wings
by
Charles Alexander Mader
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Institute for Aerospace Studies
University of Toronto
Copyright (cid:13)c 2012 by Charles Alexander Mader
Abstract
Stability-Constrained Aerodynamic Shape Optimization with Applications to Flying Wings
Charles Alexander Mader
Doctor of Philosophy
Graduate Department of Institute for Aerospace Studies
University of Toronto
2012
A set of techniques is developed that allows the incorporation of flight dynamics metrics
as an additional discipline in a high-fidelity aerodynamic optimization. Specifically, techniques
for including static stability constraints and handling qualities constraints in a high-fidelity
aerodynamic optimization are demonstrated. These constraints are developed from stability
derivative information calculated using high-fidelity computational fluid dynamics (CFD). Two
techniques are explored for computing the stability derivatives from CFD. One technique uses
anautomaticdifferentiationadjointtechnique(ADjoint)toefficientlyandaccuratelycomputea
fullsetofstaticanddynamicstabilityderivativesfromasinglesteadysolution. Theothertech-
niqueusesalinearregressionmethodtocomputethestabilityderivativesfromaquasi-unsteady
time-spectral CFD solution, allowing for the computation of static, dynamic and transient sta-
bility derivatives. Based on the characteristics of the two methods, the time-spectral technique
is selected for further development, incorporated into an optimization framework, and used to
conduct stability-constrained aerodynamic optimization. This stability-constrained optimiza-
tion framework is then used to conduct an optimization study of a flying wing configuration.
This study shows that stability constraints have a significant impact on the optimal design of
flying wings and that, while static stability constraints can often be satisfied by modifying the
airfoil profiles of the wing, dynamic stability constraints can require a significant change in the
planform of the aircraft in order for the constraints to be satisfied.
ii
Dedication
To my parents:
For nurturing my curiosity and giving me the tools to succeed
iii
Acknowledgements
The end product of a doctoral program is a thesis with a single name on it. However, the
creation of that thesis would not be possible without the support of many, many people. I
would like to take this opportunity to thank the various people who have supported me in this
endeavor.
First and foremost I would like to thank my supervisor, Professor Joaquim Martins. When
this project first started, it was, in many ways, out in left field. Professor Martins allowed me
the freedom to pursue the research where it led. While perhaps costly in the short term, I think
that in the end this proved very fruitful. Further, Professor Martins’ enthusiasm for research in
general and optimization research in particular provided a constant impetus pushing this work
along. Finally, Professor Martins’ advice and guidance has been invaluable over the course of
my degree. Thank you.
I would also like to thank Professor Zingg and Professor Damaren, my other committee
members, for their time and effort. Their insightful questions pushed me to better my under-
standing of various subjects and to fill in holes in certain areas of my research. This has led
to significantly better final product. I would also like to thank Professor Kyle Anderson, my
external examiner, for taking the time to review my thesis and provide constructive feedback
on my work.
I would like to express my gratitude to Dr. Ruben Perez, whose insights into aircraft design
and in particular, stability and control have been instrumental in my work.
Finally, I would like to thank all of my colleagues in the MDO lab. We have succeeded
in creating an excellent collaborative atmosphere that has been a pleasure to work in. The
impromptu brainstorming sessions and theoretical discussions are invigorating and the positive
energy in the lab helps keep things going on those occasional days when things don’t go exactly
as planned. Thanks Everyone!
Charles Alexander Mader
University of Toronto Institute for Aerospace Studies
February 2012
iv
Contents
1 Introduction 1
1.1 The Case for Unconventional Configurations . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Need for Stability Constraints in High-Fidelity Optimization . . . . . . . . . 2
1.3 Overview of the Stability Constraint Approach . . . . . . . . . . . . . . . . . . . 3
1.4 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Linear Flight Dynamics 7
2.1 Decoupled Longitudinal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Short-Period Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 CFD-Based Stability Derivatives 12
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Sensitivity Analysis Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 ADjoint Stability Derivative Method . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2 ADjoint Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.3 Verification of the ADjoint for Stability Derivatives . . . . . . . . . . . . . 25
3.4 Time-Spectral Stability Derivative Method . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Time-spectral CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 Linearized Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.3 Verification and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Time-Spectral ADjoint Method 46
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
v
4.3.1 Single-Cell Routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.2 ∂R/∂ζ and ∂R/∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.3 ∂I/∂ζ and ∂I/∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.4 Solution of the Adjoint System . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 Test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Auxilliary Analyses 64
5.1 Center of Gravity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Moment of Inertia Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Root Bending Moment Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Stability Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4.1 Static Longitudinal Stability Definition . . . . . . . . . . . . . . . . . . . 70
5.4.2 Static Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.3 Control Anticipation Parameter . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Geometry Handling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Design Optimization Results 76
6.1 Multidisciplinary Optimization Overview . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Optimization Study Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2.1 Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2.2 Additional Constraints for Shape Variables . . . . . . . . . . . . . . . . . 84
6.3 Optimization Problem Statements and Qualitative Results . . . . . . . . . . . . . 85
6.3.1 Reference Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.2 Static Stability Constrained Problems . . . . . . . . . . . . . . . . . . . . 95
6.3.3 Dynamic Stability Constrained Problems . . . . . . . . . . . . . . . . . . 104
6.4 Results Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.1 Mach = 0.5 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.2 Mach = 0.7 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.3 Mach = 0.85 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 Conclusions 116
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8 Future Work 118
References 120
Appendix 129
A Linear Flight Dynamics 130
B Results Data Tables 134
C Relevant Optimization Parameters 139
D FFD Coordinate Tables 145
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List of Tables
3.1 Comparison of lift and moment coefficients for the NACA 0012 at α = 0 for
various values of qˆ(results from Limache [55] in parentheses) . . . . . . . . . . . 22
3.2 Sensitivity verification: NACA 0012 test case, Mach = 0.5, 131,072 cells: 10−12
relative convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 ADjoint stability derivatives for a NACA 0012 airfoil at α = 0.0 degrees . . . . . 25
3.4 Effect of point location on time-spectral derivative location . . . . . . . . . . . . 36
3.5 SACCON test case conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Sensitivity verification: ONERA M6, pitching motion, 917,000 cells: 10−12 rela-
tive convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Sensitivity verification: ONERA M6, plunging motion, 917,000 cells: 10−12 rel-
ative convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Time-spectral ADjoint computational cost breakdown for ONERA M6 (normal-
ized with respect to a total flow solution cost of 160.3 seconds) . . . . . . . . . . 61
5.1 Stability bounds for damping ratio and CAP parameter . . . . . . . . . . . . . . 73
6.1 Baseline wing: geometry specifications . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Primary design variables and their bounds . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Compatibility design variables and their bounds. . . . . . . . . . . . . . . . . . . 81
6.4 NACA 0012 wing: planform only optimization results: 1107k cells, M = 0.5 . . . 111
6.5 NACA 0012 wing: shape optimization results: 1107k cells, M = 0.5 . . . . . . . . 112
6.6 NACA 0012 wing: planform only optimization results: 1107k cells, M = 0.7 . . . 113
6.7 NACA 0012 wing: shape optimization results: 1107k cells, M = 0.7 . . . . . . . . 114
6.8 NACA 0012 wing: shape optimization results: 1107k cells, M = 0.85 . . . . . . . 114
B.1 NACA 0012 wing: planform only optimization results summary: 1107k cells,
M = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B.2 NACA 0012 wing: shape optimization results summary: 1107k cells, M = 0.5 . . 135
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B.3 NACA 0012 wing: planform only optimization results summary: 1107k cells,
M = 0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
B.4 NACA 0012 wing: shape optimization results summary: 1107k cells, M = 0.7 . . 137
B.5 NACA 0012 wing: shape optimization results summary: 1107k cells, M = 0.85 . 138
C.1 Relevant parameters: M=0.5 cases, planform only . . . . . . . . . . . . . . . . . 140
C.2 Relevant parameters: M=0.5 cases, with shape variables . . . . . . . . . . . . . . 141
C.3 Relevant parameters: M=0.7 cases, planform only . . . . . . . . . . . . . . . . . 142
C.4 Relevant parameters: M=0.7 cases, with shape variables . . . . . . . . . . . . . . 143
C.5 Relevant parameters: M=0.85 cases, with shape variables . . . . . . . . . . . . . 144
D.1 FFD coordinates for baseline case, M = 0.5 . . . . . . . . . . . . . . . . . . . . . 146
D.2 FFD coordinates for twist-only case, M = 0.5 . . . . . . . . . . . . . . . . . . . . 147
D.3 FFD coordinates for bending moment constrained case, M = 0.5 . . . . . . . . . 148
D.4 FFD coordinates for C constrained case, M = 0.5 . . . . . . . . . . . . . . . . 149
mα
D.5 FFD coordinates for K constrained case, M = 0.5 . . . . . . . . . . . . . . . . . 150
n
D.6 FFD coordinates for CAP constrained case, M = 0.5 . . . . . . . . . . . . . . . . 151
D.7 FFD coordinates for baseline case with shape variables, M = 0.5, part 1 . . . . . 152
D.8 FFD coordinates for baseline case with shape variables, M = 0.5, part 2 . . . . . 153
D.9 FFD coordinates for C constrained case with shape variables, M = 0.5, part 1 154
mα
D.10FFD coordinates for C constrained case with shape variables, M = 0.5, part 2 155
mα
D.11FFD coordinates for K constrained case with shape variables, M = 0.5, part 1 . 156
n
D.12FFD coordinates for K constrained case with shape variables, M = 0.5, part 2 . 157
n
D.13FFD coordinates for CAP constrained case with shape variables, M = 0.5, part 1 158
D.14FFD coordinates for CAP constrained case with shape variables, M = 0.5, part 2 159
D.15FFD coordinates for bending moment constrained case, M = 0.7 . . . . . . . . . 160
D.16FFD coordinates for C constrained case, M = 0.7 . . . . . . . . . . . . . . . . 161
mα
D.17FFD coordinates for K constrained case, M = 0.7 . . . . . . . . . . . . . . . . . 162
n
D.18FFD coordinates for CAP constrained case, M = 0.7 . . . . . . . . . . . . . . . . 163
D.19FFD coordinates for bending moment constrained case with shape variables,
M = 0.7, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
D.20FFD coordinates for bending moment constrained case with shape variables,
M = 0.7, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
D.21FFD coordinates for C constrained case with shape variables, M = 0.7, part 1 166
mα
D.22FFD coordinates for C constrained case with shape variables, M = 0.7, part 2 167
mα
D.23FFD coordinates for K constrained case with shape variables, M = 0.7, part 1 . 168
n
D.24FFD coordinates for K constrained case with shape variables, M = 0.7, part 2 . 169
n
ix
D.25FFD coordinates for CAP constrained case with shape variables, M = 0.7, part 1 170
D.26FFD coordinates for CAP constrained case with shape variables, M = 0.7, part 2 171
D.27FFD coordinates for bending moment constrained case with shape variables,
M = 0.85, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
D.28FFD coordinates for bending moment constrained case with shape variables,
M = 0.85, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
D.29FFD coordinates for C constrained case with shape variables, M = 0.85, part 1174
mα
D.30FFD coordinates for C constrained case with shape variables, M = 0.85, part 2175
mα
D.31FFD coordinates for K constrained case with shape variables, M = 0.85, part 1 176
n
D.32FFD coordinates for K constrained case with shape variables, M = 0.85, part 2 177
n
D.33FFD coordinates for CAP constrained case with shape variables, M = 0.85, part 1178
D.34FFD coordinates for CAP constrained case with shape variables, M = 0.85, part 2179
x