Gradient-Based Optimum Aerodynamic Design Using Adjoint Methods

139 Pages·2012·0.81 MB·English

Gradient-Based Optimum Aerodynamic Design Using Adjoint
Methods
Lei Xie
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulflllment of the requirements for the degree of
Doctor of Philosophy
in
Aerospace Engineering
Dr. Eugene M. Clifi, Chairman
Dr. Bernard Grossman
Dr. William Mason
Dr. Jefi Borggaard
Dr. William Devenport
April 12, 2002
Blacksburg, Virginia
Keywords: Adjoint Method, Boundary Conditions, Optimum Aerodynamic Design,
Transonic Airfoil Design
Copyright 2002, Lei Xie
Gradient-Based Optimum Aerodynamic Design Using Adjoint Methods
Lei Xie
(ABSTRACT)
Continuous adjoint methods and optimal control theory are applied to a pressure-matching
inverse design problem of quasi 1-D nozzle (cid:176)ows. Pontryagin’s Minimum Principle is used
to derive the adjoint system and the reduced gradient of the cost functional. The properties
of adjoint variables at the sonic throat and the shock location are studied, revealing a log-
arithmic singularity at the sonic throat and continuity at the shock location. A numerical
method, based on the Steger-Warming (cid:176)ux-vector-splitting scheme, is proposed to solve the
adjoint equations. This scheme can flnely resolve the singularity at the sonic throat. A
non-uniform grid, with points clustered near the throat region, can resolve it even better.
The analytical solutions to the adjoint equations are also constructed via Green’s function
approach for the purpose of comparing the numerical results. The pressure-matching inverse
design is then conducted for a nozzle parameterized by a single geometric parameter.
In the second part, the adjoint methods are applied to the problem of minimizing drag
coe–cient, at flxed lift coe–cient, for 2-D transonic airfoil (cid:176)ows. Reduced gradients of
several functionals are derived through application of a Lagrange Multiplier Theorem. The
adjoint system is carefully studied including the adjoint characteristic boundary conditions
at the far-fleld boundary. A super-reduced design formulation is also explored by treating
the angle of attack as an additional state; super-reduced gradients can be constructed either
by solving adjoint equations with non-local boundary conditions or by a direct Lagrange
multiplier method. In this way, the constrained optimization reduces to an unconstrained
design problem. Numerical methods based on Jameson’s flnite volume scheme are employed
to solve the adjoint equations. The same grid system generated from an e–cient hyperbolic
grid generator are adopted in both the Euler (cid:176)ow solver and the adjoint solver. Several
computational tests on transonic airfoil design are presented to show the reliability and
e–ciency of adjoint methods in calculating the reduced (super-reduced) gradients.
Dedication
To Yan Yang.
iii
Acknowledgments
I would like to thank Dr. Eugene M. Clifi, my advisor, for his invaluable guidance, constant
support, and constructive criticism during the research. His vast knowledge of mathemat-
ics and sharp skills for the optimization methods have proved irreplaceable in this work. I
must thank committee members for their help to my research and comments to my disser-
tation. Thanks also go to Dr. Chao Yan, under whom I began the research in the optimum
aerodynamic design.
Of course, I am grateful to my wife, my parents, my sister for love and encouragement.
Without them this work would never have come into existence.
Finally, I wish to thank the following: A. Shenoy (for providing code of the Euler (cid:176)ow
solver); Lihai Qin, Xiaofeng Wu, Yanping Wang (for their friendships); Rulong Ma and
Donghui Chen (for useful discussions in research).
Blacksburg, Virginia Lei Xie
April 12, 2002
iv
Contents
1 Introduction 1
1.1 General Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sensitivity Method vs. Adjoint Method . . . . . . . . . . . . . . . . . . . . . 5
1.3 Development of Adjoint Methods . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Optimization Methods 10
2.1 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Quasi-Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Rank-one update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Rank-two updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Trust-Region Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Constrained Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Augmented Lagrangian method . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 Projected Lagrangian method . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Constrained Optimum Aerodynamic Design . . . . . . . . . . . . . . . . . . 19
v
3 Adjoint Methods for Quasi 1-D Nozzle Flows 22
3.1 Adjoint Equations and Boundary Conditions . . . . . . . . . . . . . . . . . . 23
3.1.1 Isentropic subsonic (cid:176)ow . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Isentropic transonic (cid:176)ow . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.3 Shocked (cid:176)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Analytical Adjoint Solutions for Isentropic Transonic Flow . . . . . . . . . . 31
3.3 Properties of Adjoints at Sonic Throat . . . . . . . . . . . . . . . . . . . . . 35
3.4 Steger-Warming Flux-Vector-Splitting Scheme . . . . . . . . . . . . . . . . . 36
3.5 Computational Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Transonic Airfoil Optimal Design Using Adjoint Methods 47
4.1 2-D Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Flow boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Optimality System on Computational Domain . . . . . . . . . . . . . . . . . 52
4.2.1 Boundary-value problem for adjoints . . . . . . . . . . . . . . . . . . 53
4.2.2 Adjoint boundary conditions on the airfoil . . . . . . . . . . . . . . . 56
4.2.3 Adjoint boundary conditions on the far-fleld . . . . . . . . . . . . . . 58
4.2.4 Adjoint boundary conditions on the surfaces › ;› . . . . . . . . . . 60
1 2
4.2.5 Reduced gradients of functionals . . . . . . . . . . . . . . . . . . . . . 60
4.3 Super-Reduced Design Structure . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 Adjoint system for super-reduced problem . . . . . . . . . . . . . . . 65
4.3.2 Direct Lagrange multiplier approach . . . . . . . . . . . . . . . . . . 68
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4.4 Numerical Solutions of Adjoint Systems . . . . . . . . . . . . . . . . . . . . . 70
4.5 Hyperbolic Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Geometric Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6.1 Hicks & Henne shape perturbations . . . . . . . . . . . . . . . . . . . 83
4.6.2 A NACA 4-digit airfoil family . . . . . . . . . . . . . . . . . . . . . . 85
4.7 Optimal Design for Transonic Airfoils . . . . . . . . . . . . . . . . . . . . . . 87
4.7.1 Reduced problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.7.2 Super-reduced problem . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Concluding Remarks 99
A Mathematical Basis for Adjoint Method 102
A.1 Deflnition of Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.2 An Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.3 A Lagrange Multiplier Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.4 Pontryagin’s Minimum Principle for Parameter Optimization . . . . . . . . . 108
B Recursive Projection Method 110
B.1 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
vii
List of Figures
2.1 The double dogleg curve, x C:P: N xN . . . . . . . . . . . . . . . . 17
c ! ! ! +
3.1 Relationships among d , – and dx . . . . . . . . . . . . . . . . . . . . . . 29
F F
3.2 Comparison of adjoints from numerical and analytical solutions on uniform
grid system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Comparisonofadjointsfromnumericalandanalyticalsolutionsonnonuniform
grid system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Target pressure distribution, fi = 0:6 . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Initial pressure distribution, fi = 0:8 . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Adjoints at M = 0:5533;fi = 0:8 . . . . . . . . . . . . . . . . . . . . . . . . 44
i
3.7 Convergence history of adjoint solver on uniform grid system . . . . . . . . . 45
3.8 Optimizationprocedureofpressure-matchinginversedesignforquasi1-Dnoz-
zle (cid:176)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Physical domain of O-type grid . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Computational domain (from O-type grid) . . . . . . . . . . . . . . . . . . . 49
4.3 Close-up view of 97 49 O-type grid around NACA 0012 airfoil shown in
£
every fourth line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
viii
4.4 Overall view of 65 33 O-type grid around airfoil m = 0:3;p = 0:5;t = 0:06 . 80
£
4.5 Close-up view of trailing edge vicinity . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Close-up view of the leading edge vicinity . . . . . . . . . . . . . . . . . . . . 81
4.7 Grid convergence study on O-type grid system: C . . . . . . . . . . . . . . 81
L
4.8 Grid convergence study on O-type grid system: C . . . . . . . . . . . . . . 82
D
4.9 Shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.10 Gradient-based optimization procedures using adjoint method . . . . . . . . 88
4.11 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil in Test Case 0, C = 0:5 . . . . . . . . . . . . . . . . . . . . 89
L0
4.12 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil in Test Case 1, C = 0:5 . . . . . . . . . . . . . . . . . . . . 91
L0
4.13 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil in Test Case 2, C = 0:5 . . . . . . . . . . . . . . . . . . . . 92
L0
4.14 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil, C = 0:3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
L0
4.15 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil, C = 0:4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
L0
4.16 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil, C = 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
L0
4.17 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil, C = 0:6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
L0
4.18 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil, C = 0:7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
L0
ix
4.19 C C comparisons for NACA0012 and optimal airfoils . . . . . . . . . . 97
L D
¡
4.20 Comparisons of airfoil shapes and pressure distributions of NACA 0012 and
optimal airfoil, super-reduced problem . . . . . . . . . . . . . . . . . . . . . 98
B.1 Convergence history at CFL = 0:95 . . . . . . . . . . . . . . . . . . . . . . . 115
B.2 Convergence history at CFL = 1:15 . . . . . . . . . . . . . . . . . . . . . . . 115
x