One-Shot Methods for Aerodynamic Shape Optimization M.Sc. Emre¨Ozkaya

180 Pages·2014·10.2 MB·English

One-Shot Methods for Aerodynamic
Shape Optimization
Von der Fakult¨at fu¨r Mathematik, Informatik und Naturwissenschaften der
RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors
der Naturwissenschaften genehmigte Dissertation
vorgelegt von
¨
M.Sc. Emre Ozkaya
aus Bornova, Tu¨rkei
Berichter:
Univ.-Prof. Dr.rer.nat. Nicolas Ralph Gauger
Prof. Dr. Andreas Griewank
Tag der mu¨ndlichen Pru¨fung: 11.08.2014
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfu¨gbar
Abstract
This thesis is concerned with the aerodynamic shape optimization based on
the one-shot strategy, which aims at performing an optimization simultaneously
withthesimulationprocess. Combinedwiththeconsistentdiscreteadjointmethod
based on Automatic Differentiation (AD), one-shot methods enable performing a
shape optimization at a small multiple of the time required for a single flow sim-
ulation. In the present work, we first investigate the preconditioning techniques
and constructive conditions that are required to satisfy the contractivity of the
coupled one-shot iterations. Further, an analysis concerning the quantification of
theretardationrate, whichisanindicationofthecomputationalcostoftheoverall
optimization compared to a single flow simulation, is presented. We also discuss
the implementation aspects concerning the aerodynamic design and optimization
chains, and present a review of the shape parameterization techniques commonly
usedinaerodynamicshapeoptimization. Anassessmentofthemostcommonsensi-
tivityevaluationmethodswithrespecttoaccuracy,computationalcostandrobust-
ness is performed. Among all the methods, AD based consistent discrete adjoint
method is the most robust and flexible method, which enables computing accurate
sensitivity information at a fixed computation cost independent of the number of
design parameters used in the optimization. The application of this strategy as
well astheADtechniquestogenerate adjointsolversfrom thein-housestate equa-
tion solvers are discussed in detail. Furthermore, advanced techniques to improve
the computational performance are introduced. In the last section, first validation
results obtained from the developed adjoint Euler and Navier-Stokes solvers are
presented. The run-time and memory requirements of the adjoint solvers using
different grid levels are provided to demonstrate the efficiency of the chosen ad-
joint methodology. Then, optimization results that are obtained by applying the
one-shotmethodtothreedifferentairfoiloptimizationscenariosarepresented. Fur-
thermore,acomparisonbetweentheone-shotmethodandthenestedquasi-Newton
method is performed using one of the test cases. The run-time measurements are
presented to prove the efficiency of the one-shot method.
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Zusammenfassung
Die vorliegende Dissertation befasst sich mit der aerodynamischen Formopti-
mierunganhanddesOne-Shot-Verfahrens,welchesbeiderDurchfu¨hrungdesSimu-
lationsprozesses auf ein gleichzeitiges Erreichen der Optimalit¨atsbedingungen ab-
zielt.KombiniertmitderdiskretenAdjungiertenmethodeerm¨oglichtdasOne-Shot-
VerfahrendieDurchfu¨hrungeinerFormoptimierungmiteinemRechenaufwand,der
nur ein kleines Vielfaches der Zeit fu¨r die einzelnen Str¨omungssimulation ben¨otigt.
In der vorliegenden Arbeit werden zun¨achst Pr¨akonditionierungstechniken und Be-
dingungen untersucht, die die Kontraktivit¨at der gekoppelten One-Shot Iteration
sichern.EineAnalysezurQuantifizierungderVerz¨ogerungsrate,welchedieRechen-
aufwandsteigerung der Gesamtoptimierung im Hinblick auf einzelne Simulationen
misst,wirddargestellt.ImAnschlusswerdenimplementatorischeAspektederaero-
dynamischen Entwurfs- und Optimierungskette und ein U¨berblick u¨ber allgemeine
Parametrisierungstechniken geliefert. Verschiedene Methoden zur Berechnung der
Sensitivit¨aten in Bezug auf Genauigkeit, Rechenaufwand und Robustheit werden
untersuchtundbewertet.InsbesonderedieaufAutomatischemDifferenzieren(AD)
basierende konsistente diskrete Adjungiertenmethode zeichnet sich durch Robust-
heit und Flexibilit¨at aus und erm¨oglicht die exakte Berechnung der Sensitivit¨aten
zu einem festen Rechenaufwand unabh¨angig von der Anzahl der Entwurfsparame-
ter. Die Anwendung dieser Strategie sowie dafu¨r eingesetzte AD-Techniken werden
im Detail diskutiert. Daru¨ber hinaus werden Strategien eingefu¨hrt und diskutiert,
die die Laufzeit und den Speicherbedarf des adjungierten L¨osers reduzieren. Im
letzten Teil der Arbeit werden zun¨achst Validierungsergebnisse der entwickelten
adjungierten Euler und Navier-Stokes L¨oser vorgestellt. Laufzeit- und Speicherbe-
darfmessungen fu¨r unterschiedliche Gitterstufen unterstreichen dabei die Effizienz
der gew¨ahlten Methode. Anschließend werden Ergebnisse, die durch Anwendung
des One-Shot-Verfahrens auf drei unterschiedliche Profiloptimierungsszenarien er-
zielt wurden, vorgestellt. Eine Vergleichsstudie zwischen der One-Shot Methode
undeinemklassischenQuasi-Newton-Verfahrenwirddurchgefu¨hrt,wobeiLaufzeit-
messungen die Effizienz des One-Shot-Verfahrens nachweisen.
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Contents
Abstract 1
Zusammenfassung 3
Acknowledgements 7
Chapter 1. Introduction and outline 1
Chapter 2. Theoretical framework of the one-shot method 5
2.1. The generic optimization problem 6
2.2. The Lagrangian and the optimality conditions 7
2.3. Adjoint procedures 7
2.4. Transition from simulation to optimization 11
2.5. Contractivity of the single-step one-shot method 15
2.6. Contractivity of the multi-step one-shot method 24
2.7. Characterization of the bounded retardation rate 25
Chapter 3. Basic ingredients for aerodynamic shape optimization 37
3.1. Aerodynamic design and shape optimization chains 37
3.2. Shape parameterization techniques 40
3.3. Shape and grid deformation 46
3.4. Simulation tools 48
Chapter 4. Solution of Flow Equations 51
4.1. Density-based schemes 54
4.2. Pressure-based schemes 65
4.3. Turbulence modeling 73
Chapter 5. Sensitivity evaluation methods for gradient-based optimization 83
5.1. Finite difference method 84
5.2. Complex Taylor series expansion method 85
5.3. Linearization of the state equation 86
5.4. Adjoint methods 87
Chapter 6. Automatic differentiation 107
6.1. Forward mode 108
6.2. Computational complexities of the forward and reverse modes 120
6.3. AD Tools and implementation 125
6.4. Performance improvement techniques for the adjoint solvers 128
Chapter 7. Numerical results and discussion 139
7.1. The run-time and the memory requirements of the adjoint codes 139
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6 CONTENTS
7.2. Airfoil optimization in inviscid transonic flow 141
7.3. Airfoil optimization in viscous laminar supersonic flow 152
7.4. Airfoil optimization in incompressible turbulent flow 156
Chapter 8. Conclusion and outlook 163
Bibliography 165
Acknowledgements
I started this Ph.D. thesis in the Department of Mathematics in Humboldt
University Berlin and finalized it in MathCCES, RWTH Aachen University. First
of all I would like to thank Prof. Dr. Nicolas Gauger who supervised me during
the thesis and enabled me to enjoy the great research atmosphere in these two
distinguished institutes. It has been for me always a pleasant time in his research
group.
I am a lot indebted to Prof. Dr. Andreas Griewank for his invaluable contri-
butions to my thesis work and for many other fruitful discussions. It was a great
pleasure and honor for me to work with him.
Special thanks to Dr. Anil Nemili and Dr. Adel Hamdi for their numerous
suggestions for the thesis work, and most importantly for their company during all
the years, which I enjoyed a lot.
Also many thanks to Prof. Dr. Andrea Walther and Dr. Laurent Hasco¨et for
theirfriendlysupport, especiallyforAutomaticDifferentiationtools. Theirhelpto
me was of great value.
I wish to thank Dr. Angelo Carnarius for his great support on the flow solver
and interesting discussions about CFD. I really enjoyed the nice interdisciplinary
collaboration with him.
In addition, I am thankful to Stefanie Gu¨nther and Torsten Bosse for their
help in correcting the thesis. I also thank all my ex-colleagues at the Humboldt
UniversityaswellascolleaguesattheRWTHAachenUniversityforthegreattime.
Finally, I would like to thank my family for their support and patience during
all the years. Without them, completing this thesis would not have been possible.
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CHAPTER 1
Introduction and outline
In the last decades, Computational Fluid Dynamics (CFD) methods have be-
come a standard tool in aerodynamic shape optimization. CFD tools, which had
been initially used only for the performance evaluation of the final design, have
now found a wide usage in the design process. In order to take advantage of the
large amount of information provided by the high-fidelity physical models of fluid
flow, it is desirable to give a high degree of freedom to the shape parameteriza-
tion, which leads to a large number of design parameters. In fact, aerodynamic
shape optimization has two clear trends: increasing number of design parameters
and increasing complexity of the simulation tools. Therefore, one requires more
sophisticated optimization and sensitivity evaluation methods to tackle with the
large number of design parameters and high computational cost. Among the de-
terministic optimization methods, gradient-based search methods are widely used
as they require much less number of flow simulations compared to other methods.
Forlarge-scaleaerodynamicshapeoptimizationusinggradient-basedmethods,
efficient evaluation of the shape sensitivities is an important aspect. Among the
varioussensitivityevaluationmethods,theadjointmethodisthemostefficientway
of evaluating the gradient vector of a scalar objective function with respect to a
large number of shape parameters. In contrast to finite difference or linearization
methods, computational cost of the adjoint methods remains bounded irrespective
of the number of shape parameters involved in the optimization.
As the adjoint methods became rapidly popular, they have been utilized for
diverse applications of shape optimization and flow control by various research
groups. In general, these works were mainly based on the nested gradient-based
searchalgorithmsforconstraintproblemssuchassequentialquadraticprogramming
(SQP) or interior point methods, in which adjoint and state equations are solved
with a good accuracy in each optimization cycle. As an alternative to the nested
methods,theone-shotmethodwaspresentedinaworkbyTa’asan[Ta’91],inwhich
he suggested updating design parameters in a multi-grid framework. His basic
idea was to decouple the smooth low-frequency and local high-frequency shape
deformations from each other, and apply these on the most suitable grid level.
Kuruvila et al. [KTS95] applied this method successfully for the optimization
of a NACA 0012 airfoil using potential flow equations. Later, Ta’asan [Ta’95]
suggestedanimprovedmethod,inwhichthestateandadjointequationsaresolved
inapseudo-timeembeddingwiththedesignequationbeingtreatedasanadditional
boundarycondition. Incontrasttothepreviousmethod, the newmethoddoesnot
require the multi-grid capability of flow solver. Iollo et al. [IKT96] have applied
this method to the inverse design problems, in which a tracking type of objective
functionsforpressuredistributionareoptimized. Hazraetal. [HSBG05]extended
Ta’asan’s approach further and used simultaneous pseudo time-stepping, which is
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