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Efficiency improvement of evolutionary multiobjective optimization methods for CFD-based shape optimization [Elektronische Ressource] / von Hongtao Sun

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Efficiency Improvement of EvolutionaryMultiobjective OptimizationMethods for CFD-Based Shape OptimizationVom Fachbereich Maschinenbauan der Technischen Universität DarmstadtzurErlangung des Grades eines Doktor-Ingenieurs(Dr.-Ing.)genehmigte DissertationvonHongtao Sun, M. Sc.aus Liaoning, V. R. ChinaBerichterstatter: Prof. Dr. rer. nat. M. SchäferMitberichterstatter: Prof. Dr. rer. nat. S. UlbrichTag der Einreichung: 24. November 2009Tag der mündlichen Prüfung: 14. April 2010Darmstadt 2010D17PrefaceThis thesis contains the outcome of my research in the last five years at the institute ofNumerical Methods in Mechanical Engineering at TU Darmstadt.There are many people that helped, inspired and encouraged me to progress and completethis thesis, to whom I am deeply thankful. First and foremost, I would like to express mygratitude and appreciation towards Prof. Dr. rer. nat. Michael Schäfer for his great supervision,support and encouragement during the whole work. I also thank Prof. Dr. rer. nat. StefanUlbrich for kindly accepting to become the co-advisor of this thesis.My gratitude is extended to all the colleagues at our institute for the support and friendshipthat created a wonderful and motivating working environment. Particularly I would like tothank Dr.-Ing. Zerrin Harth for the pleasant collaboration, thank Michael Kornhass, PlamenPironkov, Gerrit Becker, Johannes Siegmann, Dr.-Ing. Markus Heck, Yu Du, Dr.-Ing.

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Published 01 January 2010
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Efficiency Improvement of Evolutionary
Multiobjective Optimization
Methods for CFD-Based Shape Optimization
Vom Fachbereich Maschinenbau
an der Technischen Universität Darmstadt
zur
Erlangung des Grades eines Doktor-Ingenieurs
(Dr.-Ing.)
genehmigte Dissertation
von
Hongtao Sun, M. Sc.
aus Liaoning, V. R. China
Berichterstatter: Prof. Dr. rer. nat. M. Schäfer
Mitberichterstatter: Prof. Dr. rer. nat. S. Ulbrich
Tag der Einreichung: 24. November 2009
Tag der mündlichen Prüfung: 14. April 2010
Darmstadt 2010
D17Preface
This thesis contains the outcome of my research in the last five years at the institute of
Numerical Methods in Mechanical Engineering at TU Darmstadt.
There are many people that helped, inspired and encouraged me to progress and complete
this thesis, to whom I am deeply thankful. First and foremost, I would like to express my
gratitude and appreciation towards Prof. Dr. rer. nat. Michael Schäfer for his great supervision,
support and encouragement during the whole work. I also thank Prof. Dr. rer. nat. Stefan
Ulbrich for kindly accepting to become the co-advisor of this thesis.
My gratitude is extended to all the colleagues at our institute for the support and friendship
that created a wonderful and motivating working environment. Particularly I would like to
thank Dr.-Ing. Zerrin Harth for the pleasant collaboration, thank Michael Kornhass, Plamen
Pironkov, Gerrit Becker, Johannes Siegmann, Dr.-Ing. Markus Heck, Yu Du, Dr.-Ing. Dörte
Sternel for the fruitful discussions. Special thanks go to our system administrator Michael
Fladerer for his availability and willingness to solve all kinds of software problems and our
secretary Monika Müller for her kind help on a lot of things. Furthermore I would like to thank
Dr. Andreas Schönfeld for his valuable suggestions on the efficient computing on HHLR, Gary
Hachadorian for the extensively grammatical and linguistic correction of this thesis.
I am also profoundly thankful to my parents who did all they could to support me. Last, but
not the least, I thank my husband, Yinghua Wang for all his enormous affection and incredible
patient during these five years. This dissertation is dedicated to my parents and my husband.
Hongtao Sun
Darmstadt, Germany
November 2009Table of Contents
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Overview of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Foundations of Flow Shape Optimization . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Numerical Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Shape Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Free Form Deformation . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Optimization Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Automated Shape Optimization Loop . . . . . . . . . . . . . . . . . . . . . . 14
3 Multiobjective Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Multiobjective Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Pareto-optimal Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Classical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.3 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Modified NSGA-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 External Population and Final Selection . . . . . . . . . . . . . . . . . 24
3.2.3 Parallel Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.4 Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 RBFN-Based Approximation Model . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.1 Network Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
i4.1.2 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 Determination of Training Size . . . . . . . . . . . . . . . . . . . . . . 31
4.2.2 Determination of Output Coefficients . . . . . . . . . . . . . . . . . . 32
4.2.3 Determination of Network Centers . . . . . . . . . . . . . . . . . . . . 33
4.3 RBFN Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Hybrid Optimization Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1 Global Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.1 Global Search Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1.2 Control Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.1 Starting Points of Local Search . . . . . . . . . . . . . . . . . . . . . . 41
5.2.2 Multiobjective Problems . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.3 Deterministic Optimization Methods . . . . . . . . . . . . . . . . . . . 45
5.2.4 Local Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3.1 Analytical Test Case 1 - ZDT1 . . . . . . . . . . . . . . . . . . . . . . 50
5.3.2 Analytical Test Case 2 - FON . . . . . . . . . . . . . . . . . . . . . . 55
5.3.3 Numerical Test Case 1 - Pipe Junction . . . . . . . . . . . . . . . . . . 58
5.3.4 Numerical Test Case 2 - Heat Exchanger . . . . . . . . . . . . . . . . 65
6 Proper Orthogonal Decomposition (POD)-Based Reduced-Order Model . . . . 74
6.1 Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Combined Interpolation Approach . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4.1 Test Case 1 - Pipe Junction . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4.2 Test Case 2 - Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . 83
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
iiList of Tables
4.1 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.1 Global optimization parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Approximation control parameters (ZDT1) . . . . . . . . . . . . . . . . . . . . 52
5.3 Performance comparison of the optimal solutions after global search (ZDT1) . 53
5.4 Performance comparison of Pareto solutions with different p (ZDT1) . . . . . 540
5.5 Approximation control parameters (FON) . . . . . . . . . . . . . . . . . . . . 57
5.6 Performance comparison of the optimal solutions after global search (FON) . . 58
5.7 Approximation control parameters (pipe - 8 DVs) . . . . . . . . . . . . . . . . 62
5.8 Optimization performance comparison (pipe - 8 DVs) . . . . . . . . . . . . . . 64
5.9 Optimal solution obtained by NSGA-II+CONDOR (pipe - 8 DVs) . . . . . . . 65
5.10 Approximation control parameters (heat exchanger - 4 pipes) . . . . . . . . . . 68
5.11 Performance comparison after global search (heat exchanger - 4 pipes) . . . . . 71
5.12 Final Pareto-optimal solutions (heat exchanger - 4 pipes) . . . . . . . . . . . . 71
6.1 Comparison of CPU time (pipe - 4 DVs) . . . . . . . . . . . . . . . . . . . . . 82
6.2 Comparison of optimization results (pipe - 4 DVs) . . . . . . . . . . . . . . . . 83
6.3 Comparison of CPU time (fin-tube heat exchanger) . . . . . . . . . . . . . . . 87
6.4 Four exemplary optimal solutions (fin-tube heat exchanger) . . . . . . . . . . . 90
6.5 Performance comparison of two optimization runs (fin-tube heat exchanger) . . 90
iiiList of Figures
2.1 Methodology of numerical flow shape optimization . . . . . . . . . . . . . . . 6
2.2 Illustration of original (left) and deformed shape (right) using FFD . . . . . . . 11
2.3 A general flowchart of CFD-based shape optimization . . . . . . . . . . . . . . 15
3.1 Illustration of dominance concept, Pareto-optimal and reference vectors . . . . 17
3.2 Illustration of weighted sum method . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Illustration of weighted metric method . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Illustration of -constraint method . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Illustration of crowding distance . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 Generation of four sampling points in 2D design space using LHS . . . . . . . 25
3.7 Archive population A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26g
3.8 Master-slave model for parallel function evaluations . . . . . . . . . . . . . . . 26
3.9 Flowchart of modified NSGA-II . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 RBFN architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Generation of regression tree . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Structure of regression tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1 Evolutionary optimization procedure . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Working procedures in the control generation . . . . . . . . . . . . . . . . . . 40
5.3 Clustering method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 Local search using pseudo-weights . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Determination of nadir point, ideal point and scale region . . . . . . . . . . . . 45
5.6 Illustration of a local search case . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.7 Local search procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.8 Illustration of hypervolume of a bi-objective optimization problem . . . . . . . 49
5.9 Approximation error of the 1st objective against RBFN models (ZDT1) . . . . 52
5.10 Approximation error of the 2nd objective against RBFN models (ZDT1) . . . . 53
5.11 Approximation error and q in control generations (ZDT1) . . . . . . . . . . . . 54
5.12 Optimization results comparison after global search (ZDT1) . . . . . . . . . . 55
5.13 Comparison of approximation error and q with different p (ZDT1) . . . . . . 560
5.14 Comparison of final optimization results (ZDT1) . . . . . . . . . . . . . . . . 56
5.15 Approximation error and q in control generations (FON) . . . . . . . . . . . . 57
5.16 Optimization results comparison after global search (FON) . . . . . . . . . . . 58
5.17 Comparison of final optimization results (FON) . . . . . . . . . . . . . . . . . 59
5.18 Sketch of the initial geometry configuration (pipe - 8 DVs) . . . . . . . . . . . 60
iv
e5.19 Shape box discretization and the selected control points (pipe - 8 DVs) . . . . . 60
5.20 Deformation directions and the corresponding DVs (pipe - 8 DVs) . . . . . . . 60
5.21 Pressure contour of the initial configuration (pipe - 8 DVs) . . . . . . . . . . . 61
5.22 Approximation error and q in control generations (pipe - 8 DVs) . . . . . . . . 63
5.23 Convergence history of all optimization runs (pipe - 8 DVs) . . . . . . . . . . . 64
5.24 Pressure contour of initial and optimal configuration in xy-plane (pipe - 8 DVs) 65
5.25 Pressure contour of initial and optimal configuration in xz-plane (pipe - 8 DVs) 66
5.26 Recirculation of initial and optimal configuration in xz-plane (pipe - 8 DVs) . . 66
5.27 Sketch of the initial geometry (heat exchanger - 4 pipes) . . . . . . . . . . . . 67
5.28 Approximation error of pressure drop (heat exchanger - 4 pipes) . . . . . . . . 69
5.29 Approximation error of Nusselt number (heat exchanger - 4 pipes) . . . . . . . 70
5.30 Approximation error and q in control generations (heat exchanger - 4 pipes) . . 70
5.31 Optimization results comparison after global search (heat exchanger - 4 pipes) . 71
5.32 Temperature contour comparison (heat exchanger - 4 pipes) . . . . . . . . . . . 72
5.33 Pressure contour comparison (heat exchanger - 4 pipes) . . . . . . . . . . . . . 73
5.34 Comparison of final optimization results (heat exchanger - 4 pipes) . . . . . . . 73
6.1 Shape box discretization and the selected control points (pipe - 4 DVs) . . . . . 80
6.2 Deformation directions and the corresponding DVs (pipe - 4 DVs) . . . . . . . 80
6.3 Average reconstruction errors of 256 and 625 snapshots (pipe - 4 DVs) . . . . . 81
6.4 Pressure contour comparison using 256 snapshots (pipe - 4 DVs) . . . . . . . . 81
6.5 Pressure contour comparison using 625 snapshots (pipe - 4 DVs) . . . . . . . . 82
6.6 Comparison of optimization history (pipe - 4 DVs) . . . . . . . . . . . . . . . 83
6.7 Top view of a fin-tube heat exchanger . . . . . . . . . . . . . . . . . . . . . . 84
6.8 Selected optimization domain (fin-tube heat exchanger) . . . . . . . . . . . . . 84
6.9 Shape boxes and selected control points (fin-tube heat exchanger) . . . . . . . 85
6.10 Deformation directions and the corresponding DVs (fin-tube heat exchanger) . 85
6.11 Average reconstruction error of 200 snapshots (fin-tube heat exchanger) . . . . 87
6.12 Pressure contour comparison (fin-tube heat exchanger) . . . . . . . . . . . . . 88
6.13 x-velocity contour comparison (fin-tube heat exchanger) . . . . . . . . . . . . 88
6.14 Temperature contour comparison (fin-tube heat exchanger) . . . . . . . . . . . 88
6.15 Pareto solutions achieved by POD evaluations (fin-tube heat exchanger) . . . . 89
6.16 Four exemplary optimal shapes (fin-tube heat exchanger) . . . . . . . . . . . . 89
6.17 Pareto front comparison (fin-tube heat exchanger) . . . . . . . . . . . . . . . . 90
vList of Symbols and Acronyms
Latin Symbols
A archive population
A area of the temperature surfaces
A autocorrelation matrix used in POD
B splitting boundary of the regression tree
c specific heatp
c constraints of the optimization problemi
c RBFN center
C cost function for RBFN training
C cluster
d crowding distance
d normalized Euclidean distance between two objective solutions
d minimum distance of the i-th Pareto solution to all the other solu-i,min
tions in the Pareto front
d average d of all solutions in the Pareto frontave,min i,min
d distance between two clustersC
ini defd d , coordinate vector of the initial and deformed grid point in the
physical domain
D distribution function of the design variable in the design space
D hydraulic diameterh
e internal energy
e average percentage approximation errorave
e maximum allowed approximation errormax
f , f maximum and minimum values of the m-th objectivem,max m,minef interpolation model used in the trust-region method
f, f volume force per mass uniti
f objective functions of the optimization problemi
iF, fff snapshots matrix
g deformation vector
g POD basis vector
h heat transfer coefficient
h,h heat fluxi
hhh,h hidden layer of RBFNm
HV hypervolume
viIII sorted indices vector
I identity matrix of size KK
K size of the RBFN training set
K ,K maximum and minimum size of the RBFN training setmax min
K ,K number of data points in the subsets S and SL R L R
M truncation degree
l l metrics with p varying from 1 top p
N number of constraintscon
N number of design variablesdv
N number of binary design variablesdvb
N number of real design variablesdvr
N number of recalculated solutions in the control generatione
N total number of required exactly evaluated functionsfun,e
N total number of required exactly evaluated generationsgen,e
N number of solutions on Pareto frontFJ J
N number of starting points for the local searchlocal
N maximum allowed times for design vector regenerationsmax
N number of optimization objectivesobj
N number of parallel runsp
N population sizepop
N number of solutions in the Pareto frontP
Nu Nusselt number
p pressure
p generation control frequency
p recombination probabilityc
p mutation probabilitym
p number of intial exactly evaluated generations0
p¯ , p¯ mean pressure of the inlet or outlet cross-sectionin out
P parent population
Pr Prandtl number
P projection matrix
q scalar heat source
q intial exactly evaluated generations in p generationsini
q general source terms
q minimum number of exactly evaluated generations in p genera-min
tions
q average number of exactly evaluated generations calculated in allave
control generations
Q child population
Q total heat transfer
mr RBFN radius corresponding to the n-th dimension of input vectorn
and m-th network center
R combined parent-child population
Re Reynolds number
s coordinate vector of all grid points
vii
¥ini defsss ,sss initial and deformed coordinate vector of all grid points
S size of the database
S Pareto solution
SCM set coverage metric
S , S subsets obtained by splitting the node of the regression treeL R
SP spacing
t time
t ,t time required by serial and parallel computingserial parallel
it shape basis vector casued by a initial displacement of i-th control
point
T temperature
T inlet temperaturein
¯ ¯T ,T mean temperature of the inlet or outlet cross-sectionin out
T,T Cauchy stress tensori j
uu x-component of inlet velocityx,in
u,u velocity vectori
x hydrodynamic entrance regione,h
x thermal entrance regione,t
L Ux ,x low and upper bound of design variable xii i
L Ub bx ,x low and upper bound of binary design variable xii i
L Ur rx ,x low and upper bound of real design variable xii i
x,x cartesian coordinatei
x,x input layer of RBFNi
x,x design vectori
y output layer of RBFN
y¯ ,y¯ mean value of subset S and SL R L R
y output vector of a set of training points
z weighted sum objective.
z distance between input x to the RBFN center c
xz average distance between input and the point in the databaseave
Izzz ideal vector
Uzzz utopian vector
Nz nadir vector
Greek Symbols
sharpness coefficientm
, , total number of control points in three directionsn n n
Kronecker delta operatori j
, constraint vector defined in -constraint methodi
efficiency of pressure drop reductionp
coordinate vector of the grid point in the logical domain
ini def, coordinate vector of the initial and deformed grid point in the
logical domain
heat conductivity
viii
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