Otimiza§£o Robusta

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    Robust Design Optimization of Structures under Uncertainties

    Von der Fakult ät Luft- und Raumfahrttechnik und Geod äsie der Universit ät Stuttgart

    zur Erlangung der W ürde eines Doktor-Ingenieurs (Dr. -Ing.) genehmigte Abhandlung

    vorgelegt von

    Zhan Kang aus Liaoning, V.R. China

    Hauptberichter: Mitberichter: Mitberichter:

    Tag der m ündlichen Pr üfung:

    Priv.Doz. Dr. -Ing. Ioannis Doltsinis Prof. Dr. -Ing. habil. Bernd Kr öplin em. Prof. Dr. -Ing. Werner Schiehlen

    27. Juni 2005

    Institut f ür Statik und Dynamik der Luft- und Raumfahrkonstruktionen Universit ät Stuttgart

    2005

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    Sich 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 proverb

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    Acknowledgements

    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

    nancially supported by the Deutscher Akademischer Austausch Dienst (DAAD) in the framework of the scholarship for doctoral study and by the Faculty of Aerospace 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 beneted greatly from frequent and stimulating discussions with him. Moreover, his critical review also contributes signicantly 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 dening the subject of this thesis. I have also beneted a lot from his constructive suggestions regarding the research work throughout the years.

    It is my great pleasure to thank Prof. Dr. -Ing. Bernd Kr öplin 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 fullment of my doctoral study would be impossible.

    I am indebted to my colleagues at the Faculty of Aerospace Engineering and Geodesy, in particular Dr. -Ing. Kurt A. Braun, Gerhard Frik, Helmut Schmid, Peter Gelpke, Sibylle Fuhrmann, Marion Hackenberg, Inge Biberger, Vlasta Reber-Hangi, for their

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    VI 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.

    Last but not the least, I would like to thank my wife, Sui Changhong and my parents for their patience and encouragement throughout my doctoral study.

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    VIII 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 nite 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 nite 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 nite element analysis for nonlinear structures with path dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    7.2 Response moment sensitivity analysis . . . . . . . . . . . . . . . . . 89

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    Contents 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 nite 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

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    Zusammenfassung

    Die vorliegende Arbeit befaßt sich mit der Formulierung und den numerischen Meth- oden f ̈ur das robuste Design von Strukturen mit stochastischen Parametern. Die

    Theorie und die numerischen Methoden der Strukturoptimierung haben sich in den letzten zwei Jahrzehnten stark entwickelt. Ausserdem erm öglichen die schnell wach- senden Berechnungsm öglichkeiten die Berücksichtigung der Ungewissheiten im opti- malen Strukturdesign. Die vorliegende Arbeit soll zu einem besseren Verst ändnis der Strukturoptimierung beitragen, indem man den Einuß der stochastischen Streuung auf die Designrobustheit unter realistischen Bedingungen betrachted.

    Robustes Strukturdesign bietet zuverl ässige, quantitativ bestimmbare und leistungs- f ̈ahige Methoden an, Produkte und Prozesse zu entwerfen, die gegen über System- schwankungen unempndlich sind. Robustes Design kann in verschiedenen Phasen des Strukturdesigns, wie im Konzeptdesign, Parameterdesign und Toleranzdesign, erreicht werden. In dieser Arbeit wird das