Multi-objective optimization of transonic airfoils using variable-fidelity models, co-kriging surrogates

72 Pages·2017·2.41 MB·English

Iowa State University Capstones, Theses and
Graduate Theses and Dissertations
Dissertations
2016
Multi-objective optimization of transonic airfoils
using variable-fidelity models, co-kriging surrogates,
and design space reduction
Anand Amrit
Iowa State University
Follow this and additional works at:https://lib.dr.iastate.edu/etd
Part of theAerospace Engineering Commons
Recommended Citation
Amrit, Anand, "Multi-objective optimization of transonic airfoils using variable-fidelity models, co-kriging surrogates, and design
space reduction" (2016).Graduate Theses and Dissertations. 15148.
https://lib.dr.iastate.edu/etd/15148
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Multi-objective optimization of transonic airfoils using variable-fidelity models,
co-kriging surrogates, and design space reduction
by
Anand Amrit
A thesis submitted to the graduate faculty
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
Major: Aerospace Engineering
Program of Study Committee:
Leifur Leifsson, Major Professor
Christina Bloebaum
Jonathan Regele
Iowa State University
Ames, Iowa
2016
Copyright (cid:13)c Anand Amrit, 2016. All rights reserved.
ii
DEDICATION
I would like to dedicate this thesis to my father Ajaya Kumar Nayak and to my mother
Narayani Nayak who gave me emotional and moral strength to overcome all hurdles during my
research. I would like to thank my advisor Dr. Leifur Leifsson whose ideas and efforts to make
my research successful was instrumental. At the end, I would also like to thank my POSC
committee members Dr. Christina Bloebaum and Dr. Jonathan Regele.
iii
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
CHAPTER 2. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Definition and Formulation of Multi-Objective Optimization . . . . . . . . . . . 6
2.2 Multi-Objective Optimization Strategies and Algorithms . . . . . . . . . . . . . 7
2.2.1 Single-Objective Optimization using a Scalarized Objective Function . . 8
2.2.2 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.4 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Applications of Multi-Objective Optimization in Aerodynamic Design . . . . . 10
CHAPTER 3. MULTI-OBJECTIVE OPTIMIZATION METHODOLOGY . 12
3.1 Multi-Objective Aerodynamic Design Formulation . . . . . . . . . . . . . . . . 12
3.2 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Variable-Fidelity Surrogate Model . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Design Space Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
iv
3.5 Kriging Surrogate Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5.1 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5.2 Kriging Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6 Co-kriging Surrogate Construction . . . . . . . . . . . . . . . . . . . . . . . . . 23
CHAPTER 4. NUMERICAL APPLICATIONS . . . . . . . . . . . . . . . . . . 25
4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 Formulation of the MOO Problem . . . . . . . . . . . . . . . . . . . . . 25
4.1.2 Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.3 Training Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.4 Computational Fluid Dynamics Modeling . . . . . . . . . . . . . . . . . 28
4.1.5 Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.1 MOO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.3 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.3 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4.3 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Comparison of the Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.6 Comparison using Single-Objective Optimization and a Scalarized Objective . . 46
4.7 Parametric Study of Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.7.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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CHAPTER 5. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
vi
LIST OF TABLES
Table 4.1 Results of the grid convergence study at M = 0.734 and C = 0.824. 31
∞ l
Table 4.2 Comparison of the computatioal cost of the three strategies. . . . . . . 42
Table 4.3 Relative root mean square error (RMSE) of the initial kriging model
in Step 4 of the MOO algorithm in Strategy 3 for different number of
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Table 4.4 Relative root mean square error (RMSE) of the initial kriging model
in Step 4 of the MOO algorithm in Strategy 1 for different number of
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Table 4.5 Comparison of the computatioal cost of the original three strategies,
with the refined Strategy 3 (indicated by 3∗). . . . . . . . . . . . . . . 52
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LIST OF FIGURES
Figure 1.1 Representation of the design space and the corresponding feasible ob-
jective space. Designs A, B, and C are non-optimal solutions. Design
D lies on the Pareto-optimal front. . . . . . . . . . . . . . . . . . . . . 2
Figure 1.2 PDE simulations (such as computational fluid dynamics (CFD) simu-
lations) need dense computational grids that require long CPU times
(often on the order of days) on high performance computing clusters.
The graph shows how the time and the objective function values of a
2D CFD simulation of transonic airfoil flow vary with the grid density. 3
Figure 3.1 An illustration of the design space reduction technique using single-
objective optimization runs (Koziel et al. [1]). The illustration assumes
a three-dimensional design space and two design objectives. . . . . . . 19
Figure 3.2 Flowchart describing the data-driven surrogate modelling process . . . 20
Figure 3.3 Latin Hypercube Sampling Illustration . . . . . . . . . . . . . . . . . . 21
Figure 4.1 Example B-spline parameterization of an airfoil. The designable control
points are restricted to vertical movements only. . . . . . . . . . . . . . 26
Figure 4.2 The baseline airfoil (RAE 2822) and sample airfoils from the base train-
ing set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Figure 4.3 Example training points sampled using Latin Hypercube Sampling. . . 28
Figure 4.4 Visualization of the high-fidelity model computational mesh. . . . . . . 30
Figure 4.5 Visualization of the low-fidelity model computational mesh. . . . . . . 30
Figure 4.6 A close-up view of the airfoil surface mesh for the high-fidelity model.. 31
viii
Figure 4.7 Evolution of lift, drag and pitching moment coefficients obtained by the
low fidelity model at M = 0.734. . . . . . . . . . . . . . . . . . . . . 32
∞
Figure 4.8 A comparison of the pressure distribution obtained by the high and
low-fidelity models at M = 0.734. . . . . . . . . . . . . . . . . . . . . 32
∞
Figure 4.9 Variation of the simulation time with respect to the grid size for the
grid study in Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Figure 4.10 Results of Strategy 1 showing the Pareto fronts obtained in at several
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Figure 4.11 The final Pareto front of Strategy 1 with high-fidelity valiation samples. 35
Figure 4.12 Results of Strategy 2 showing the Pareto fronts obtained in at several
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 4.13 The final Pareto front of Strategy 2 with high-fidelity valiation samples. 38
Figure 4.14 Results of Strategy 3 showing the Pareto fronts obtained in at several
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 4.15 The final Pareto front of Strategy 3 with high-fidelity valiation samples. 41
Figure 4.16 Comparision of the final Pareto fronts obtained by the three strategies. 42
Figure 4.17 DesignsselectedalongthefinalParetofrontofstrategy3forvisualization. 43
Figure 4.18 Airfoil shape of the designs selected from the final Pareto front of Strat-
egy 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 4.19 Pressure coefficient of designs selected from the final Pareto front of
Strategy 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 4.20 Pressure coefficient contours for design 1. . . . . . . . . . . . . . . . . . 44
Figure 4.21 Pressure coefficient contours for design 2. . . . . . . . . . . . . . . . . . 45
Figure 4.22 Pressure coefficient contours for design 3. . . . . . . . . . . . . . . . . . 45
Figure 4.23 Acomparisonoftheproposedmulti-objectivealgorithmwiththesingle-
objective optimization using a scalarized objective function. . . . . . . 47
Figure 4.24 ResultsofStrategy3showingtheParetofrontsobtainedwith500initial
sampling points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
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Figure 4.25 ResultsofStrategy3showingtheParetofrontsobtainedwith300initial
sampling points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 4.26 ResultsofStrategy3showingtheParetofrontsobtainedwith100initial
sampling points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Figure 4.27 ResultsofStrategy3showingtheParetofrontsobtainedwith100initial
sampling points and 3 refinement points. . . . . . . . . . . . . . . . . . 51
Figure 4.28 ResultsofStrategy3showingtheParetofrontsobtainedwith1,600and
100 initial sampling points. . . . . . . . . . . . . . . . . . . . . . . . . . 52