Robust Design Optimization of Structures
under Uncertainties
Von der Fakulta¨t Luft- und Raumfahrttechnik und Geod¨asie
der Universitat Stuttgart¨
zur Erlangung der Wu¨rde eines Doktor-Ingenieurs (Dr. -Ing.)
genehmigte Abhandlung
vorgelegt von
Zhan Kang
aus Liaoning, V.R. China
Hauptberichter: Priv.Doz. Dr. -Ing. Ioannis Doltsinis
Mitberichter: Prof. Dr. -Ing. habil. Bernd Kr¨oplin
Mitberichter: em. Prof. Dr. -Ing. Werner Schiehlen
Tag der mu¨ndlichen Pru¨fung: 27. Juni 2005
Institut fur Statik und Dynamik der Luft- und Raumfahrkonstruktionen¨
Universitat Stuttgart¨
2005Sich zu ¨andern und sich zu verbessern sind zwei verschiedene Dinge.
-Deutsches Sprichwort
To change and to change for the better are two different things.
- a German proverbAcknowledgements
This thesis has been accomplished during my three years’ stay at the Faculty of
Aerospace Engineering and Geodesy of the University of Stuttgart. This work was
financially supported by the Deutscher Akademischer Austausch Dienst (DAAD) in
theframeworkofthescholarshipfordoctoralstudyandbytheFacultyofAerospace
Engineering and Geodesy, University of Stuttgart.
First and foremost, I wish to express my deepest gratitude to my advisor PD. Dr.
-Ing. Ioannis Doltsinis for giving me the opportunity to do my doctoral study at
the Institute of Statics and Dynamics of Aerospace Structures (ISD) and for his
invaluable guidance and support throughout this research. I have benefited greatly
from frequent and stimulating discussions with him. Moreover, his critical review
also contributes significantly to this thesis. I feel extremely fortunate to have had
an excellent supervision from him during the past years.
I would like to express my sincerest appreciation to my co-advisor Prof. Dr. Cheng
Gengdong from Dalian University of Technology (China), who has encouraged me
to do my doctoral research at the University of Stuttgart and has given me valuable
advice on defining the subject of this thesis. I have also benefited a lot from his
constructive suggestions regarding the research work throughout the years.
It is my great pleasure to thank Prof. Dr. -Ing. Bernd Kroplin and em Prof.¨
Dr. -Ing. Werner Schiehlen for taking the responsibility to read and to evaluate
my dissertation. I am also very grateful to them for their interests and valuable
comments on my work.
My special gratitude goes to Prof. Dr. Gu Yuanxian from Dalian University of
Technology (China). Without his constant encouragement and continual support
in every aspects of my research work, the fulfilment of my doctoral study would be
impossible.
IamindebtedtomycolleaguesattheFacultyofAerospaceEngineeringandGeodesy,
in particular Dr. -Ing. Kurt A. Braun, Gerhard Frik, Helmut Schmid, PeterGelpke,
SibylleFuhrmann, MarionHackenberg, IngeBiberger, VlastaReber-Hangi, fortheirVI Acknowledgements
excellent work and kindly help during my stay at the faculty.
Moreover, I would like to express my thanks to all the friends that made my stay
in Stuttgart a memorable part of my life, specially Dr. -Ing. Haupo Mok, Friedlich
Rau, Dr. -Ing. Yan Shuiping, Chen Zhidong, Xiao Li and the DAAD Coordinator
Ursula Habel.
Lastbutnottheleast,Iwouldliketothankmywife,SuiChanghongandmyparents
for their patience and encouragement throughout my doctoral study.VII
Contents
Acknowledgements V
Zusammenfassung XI
Abstract XIII
Nomenclature XV
List of Figures XVII
List of Tables XIX
1 Introduction 1
1.1 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Fundamentals of structural optimization 7
2.1 Conventional structural optimization . . . . . . . . . . . . . . . . . . 7
2.2 Multi-criteria optimization and Pareto optimum . . . . . . . . . . . . 11
2.3 Approximation concepts and sensitivity analysis . . . . . . . . . . . 12
3 Structural optimization under uncertainty 19
3.1 Mathematical models of uncertainty . . . . . . . . . . . . . . . . . . . 19
3.2 Overview of problem formulations . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Reliability based design optimization (RBDO) . . . . . . . . . 22
3.2.2 Worst case scenario-based design optimization . . . . . . . . . 24
3.2.3 Fuzzy set based design optimization . . . . . . . . . . . . . . . 25
3.2.4 Robust design . . . . . . . . . . . . . . . . . . . . . . . . . . . 26VIII Contents
3.3 Structural robust design . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Concept of structural robust design . . . . . . . . . . . . . . . 26
3.3.2 Differences between structural robust design and RBDO . . . 30
3.3.3 Current state of research on structural robust design . . . . . 33
4 Perturbation based stochastic finite element method (SFEM) 41
4.1 Overview of stochastic structural analysis . . . . . . . . . . . . . . . . 41
4.1.1 Statistical methods . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Non-statistical methods . . . . . . . . . . . . . . . . . . . . . 42
4.1.3 A comparison between Monte Carlo simulation and Perturba-
tion based method . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Perturbation based SFEM for linear structures . . . . . . . . . . . . . 45
4.2.1 Perturbation equations for static problems . . . . . . . . . . . 45
4.2.2 Perturbation based stochastic analysis for transient problems . 50
5 Formulation of structural robust design 53
5.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.1 Uncertainty and design variables in the problem . . . . . . . . 53
5.1.2 Numerical representation of structural robustness . . . . . . . 55
5.1.3 Employment of the perturbation based stochastic finite ele-
ment analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . 56
5.2.2 Computational aspects . . . . . . . . . . . . . . . . . . . . . . 59
5.2.3 Comparison with formulations based on Taguchi’s methodology 62
6 Robust design of linear structures 67
6.1 Response moments sensitivity analysis . . . . . . . . . . . . . . . . . 67
6.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7 Robust design of nonlinear structures with path dependence 83
7.1 Stochastic finite element analysis for nonlinear structures with path
dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Response moment sensitivity analysis . . . . . . . . . . . . . . . . . 89Contents IX
7.3 Numerical examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.4 Discussions and remarks . . . . . . . . . . . . . . . . . . . . . . . . . 100
8 Robust design of inelastic deformation processes 101
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Quasi-static deformation of inelastic solids . . . . . . . . . . . . . . . 103
8.2.1 Constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2.2 The equilibrium equations and computational aspects . . . . . 105
8.3 Stochastic finite element analysis of inelastic deformation processes . 107
8.3.1 Steady-state problems . . . . . . . . . . . . . . . . . . . . . . 107
8.3.2 Non-stationary problems . . . . . . . . . . . . . . . . . . . . . 110
8.4 Robust design of deformation processes . . . . . . . . . . . . . . . . . 114
8.4.1 Optimization for process robust design . . . . . . . . . . . . . 114
8.4.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 115
8.5 Discussions and remarks . . . . . . . . . . . . . . . . . . . . . . . . . 124
9 Summary and outlook 127
9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Bibliography 131
Curriculum vitae 141