Global Optimization Algorithms for Aerodynamic Design by Oleg Chernukhin A thesis submitted in ...

109 Pages·2011·12.1 MB·English

Global Optimization Algorithms for Aerodynamic Design
by
Oleg Chernukhin
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Aerospace Engineering
University of Toronto
Copyright (cid:13)c 2011 by Oleg Chernukhin
Abstract
Global Optimization Algorithms for Aerodynamic Design
Oleg Chernukhin
Master of Applied Science
Graduate Department of Aerospace Engineering
University of Toronto
2011
This work focuses on an investigation of multi-modality in typical aerodynamic shape
optimization problems and development of optimization algorithms that can find a global
optimum. First, a classification of problems based on the degree of multi-modality is
introduced. Then, two optimization algorithms are described that can find a global
optimum in a computationally efficient manner: a gradient-based multi-start Sobol algo-
rithm, and a hybrid optimization algorithm. Two additional algorithms are considered
as well: a gradient-based optimizer and a genetic algorithm. Finally, we consider a set
of typical aerodynamic shape optimization problems. In each problem, the primary ob-
jectives are to classify the problem according to the degree of multi-modality, and to
select the preferred optimization algorithm for the problem. We find that typical two-
dimensional airfoil shape optimization problems are unimodal. Three-dimensional shape
optimization problems may contain local optima. In these problems, the gradient-based
multi-start Sobol algorithm is the most efficient algorithm.
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Acknowledgements
I would like to thank Prof. David Zingg for his guidance. He is an excellent teacher
and mentor. His leadership and support were instrumental in helping me to achieve the
results in this work. I am very honoured to have had the opportunity to work with Prof.
Zingg.
I would like to thank all my colleagues in the CFD lab for making my time at UTIAS
both fun and fulfilling. In particular, many thanks to Jason Hicken for his help with
Jetstream, and Howard Buckley for helping me understand Optima2D.
I would like to thank my family for their support and encouragement. I am fortunate
to be able to always count on their support and to help keep things in proper perspective.
FinancialsupportfromtheUniversityofToronto,MITACS,andBombardierAerospace
is gratefully acknowledged.
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Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Aerodynamic Shape Optimization Methods 8
2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Flow Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Geometry Parameterization and Mesh Movement . . . . . . . . . . . . . 10
2.4 Discrete Adjoint Method for Gradient Calculation . . . . . . . . . . . . . 10
3 Optimization Algorithms 13
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Gradient-based Optimizer: SNOPT . . . . . . . . . . . . . . . . . . . . . 16
3.3 Gradient-free Optimizer: Evolver . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Gradient-based Multi-start Sobol Optimization Algorithm . . . . . . . . 18
3.5 Hybrid Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6 Linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6.1 Two-Patch Wing Topology . . . . . . . . . . . . . . . . . . . . . . 22
3.6.2 Blended-Wing-Body Topology . . . . . . . . . . . . . . . . . . . . 32
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3.7 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Results 40
4.1 Algorithm Assessment: Finding a Global Minimum of an Analytical Test
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Highly Multi-modal Problem . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Moderately Multi-modal Problem . . . . . . . . . . . . . . . . . . 44
4.2 2-D Airfoil Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 3-D Wing Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Transonic Wing Section Optimization . . . . . . . . . . . . . . . . 50
4.3.2 Subsonic Wing Optimization . . . . . . . . . . . . . . . . . . . . . 52
4.3.3 Transonic Wing Optimization . . . . . . . . . . . . . . . . . . . . 57
4.3.4 Transonic Planar Wing Optimization . . . . . . . . . . . . . . . . 59
4.4 Blended-Wing-Body Aircraft Optimization . . . . . . . . . . . . . . . . . 62
5 Conclusions and Future Work 66
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.1 Multi-modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.2 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.1 More Practical Optimization Problems . . . . . . . . . . . . . . . 70
5.2.2 Linear Constraint Systems . . . . . . . . . . . . . . . . . . . . . . 70
5.2.3 Optimization Algorithm Improvements . . . . . . . . . . . . . . . 71
5.2.4 Multi-Modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References 72
Appendices 78
A BWB Linear Constraints 79
v
B HYBROPT Input Parameters 91
C Multi-Modality as a Result of Constraining the Design Space 97
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List of Tables
3.1 Classification of Optimization Problems by Multi-Modality . . . . . . . . 15
3.2 Typical Values for the Input Parameters . . . . . . . . . . . . . . . . . . 26
3.3 Values for the Sampled Design Variables . . . . . . . . . . . . . . . . . . 39
4.1 Linear Constraint System Inputs for Subsonic Wing Optimization Problem 53
4.2 C Values for Optimal Geometries in Subsonic Wing Optimization Problem 54
D
4.3 Grid Refinement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 C ValuesforOptimalGeometriesinTransonicWingOptimizationProblem 59
D
4.5 C Values for Optimal Geometries in BWB Problem . . . . . . . . . . . 65
D
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List of Figures
1.1 Griewank Test Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Block Diagram for Hybrid Optimization Algorithm . . . . . . . . . . . . 19
3.2 Wing Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Main Regions of the Wing . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Baseline Blended-Wing-Body Geometry . . . . . . . . . . . . . . . . . . . 33
3.5 Comparison of Sampling Methods . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Sampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 R Sampling 2-Patch Wing . . . . . . . . . . . . . . . . . . . . . . . . . 38
L
4.1 Convergence Plots for Test Problems . . . . . . . . . . . . . . . . . . . . 42
4.2 Explanation of Convergence Properties of the Hybrid Algorithm . . . . . 43
4.3 Baseline Grid Used for 2-D Airfoil Optimization Problem . . . . . . . . . 45
4.4 Comparison of the Original and Optimized Airfoils . . . . . . . . . . . . 46
4.5 Optimization Results for Airfoil Optimization Problem . . . . . . . . . . 47
4.6 Baseline Grid Used for 3-D Wing Optimization Problems . . . . . . . . . 49
4.7 Results for Wing Section Optimization Problem in Transonic Flow . . . . 51
4.8 Convergence Plots for Wing Section Optimization Problem in Transonic
Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.9 32 Initial Guesses for Subsonic Wing Optimization Problem . . . . . . . 53
4.10 Convergence Plots for Subsonic Wing Optimization Problem . . . . . . . 54
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4.11 Local Optima for Subsonic Wing Optimization Problem . . . . . . . . . . 55
4.12 Convergence Plots for Transonic Wing Optimization Problem . . . . . . 57
4.13 Local Optimum 1 for Transonic Wing Optimization Problem . . . . . . . 58
4.14 Local Optimum 2 for Transonic Wing Optimization Problem . . . . . . . 58
4.15 Local Optimum 3 for Transonic Wing Optimization Problem . . . . . . . 59
4.16 Mach Number Contours for Geometries in Transonic Planar Wing Opti-
mization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.17 Effect of Side-Edge Separation in Transonic Planar Wing Optimization
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.18 Baseline Geometry for BWB Problem . . . . . . . . . . . . . . . . . . . . 64
4.19 Mach Contour Plots for BWB Problem . . . . . . . . . . . . . . . . . . . 64
4.20 Local Optima for BWB Problem . . . . . . . . . . . . . . . . . . . . . . 65
C.1 Multi-Modality as a Result of Constraining the Design Space . . . . . . . 100
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Chapter 1
Introduction
1.1 Background
Rising fuel prices and concerns about climate change present the main challenges for the
aviation industry in the 21st century. These considerations will increase the pressure on
the aircraft industry to design more fuel-efficient aircraft, which in turn will increase the
demand for more advanced design tools available to the engineers. Over the past decades,
gainsinfuelefficiencyhavemainlycomefromimprovementsinenginetechnology,reduced
weight due to the use of composite materials, and reduction of overall drag through more
aerodynamically efficient airframe designs.
The use of computational techniques has enabled aircraft designers to improve the
process of designing new aircraft. For example, Computational Fluid Dynamics (CFD)
has been a major breakthrough, allowing the designers to reduce their reliance on more
expensive and time-consuming wind tunnel tests. CFD is now a mature technology, used
by engineers in many industries every day.
Aerodynamic shape optimization (ASO) is an active and promising area of research
that has the potential to become the next major breakthrough in aircraft design. This
approach involves the use of CFD in combination with an optimization algorithm to
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