Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments

Foreword xi
Maurice LEMAIRE

Preface xv
Christian GOGU

Part 1. Modeling, Propagation and Quantification of Uncertainties 1

Chapter 1. Uncertainty Modeling 3
Christian GOGU

1.1. Introduction 3

1.2. The usefulness of separating epistemic uncertainty from aleatory uncertainty 6

1.3. Probability theory 10

1.3.1. Theoretical context 10

1.3.2. Probabilistic approach for modeling aleatory uncertainties 13

1.3.3. Probabilistic approach for modeling epistemic uncertainties 16

1.4. Probability box theory (p-boxes) 21

1.5. Interval analysis 24

1.6. Fuzzy set theory 25

1.7. Possibility theory 27

1.7.1. Theoretical context 27

1.7.2. Comparison between probability theory and possibility theory 30

1.7.3. Rules for combining possibility distributions 34

1.8. Evidence theory 35

1.8.1. Theoretical context 35

1.8.2. Rules for combining belief mass functions 38

1.9. Evaluation of epistemic uncertainty modeling 40

1.10. References 40

Chapter 2. Microstructure Modeling and Characterization 43
François WILLOT

2.1. Introduction 43

2.2. Probabilistic characterization of microstructures 45

2.2.1. Random sets 45

2.2.2. Covariance 47

2.2.3. Granulometry 50

2.2.4. Minkowski functionals 51

2.2.5. Stereology 53

2.2.6. Linear erosion 53

2.2.7. Representative volume element 54

2.3. Point processes 55

2.3.1. Homogeneous Poisson point processes 56

2.3.2. Inhomogeneous Poisson point processes 58

2.4. Boolean models 59

2.4.1. Definition and Choquet capacity 59

2.4.2. Properties 61

2.4.3. Covariance 63

2.4.4. Other characteristics 63

2.5. RSA models 66

2.6. Random tessellations 67

2.6.1. Voronoi tessellation 68

2.6.2. Johnson-Mehl tessellation 69

2.6.3. Laguerre tessellation 69

2.6.4. Random Poisson tessellation 70

2.6.5. The dead-leaves model 71

2.6.6. Generalized random partition models 72

2.7. Gaussian fields 73

2.8. Conclusion 76

2.9. Acknowledgments 77

2.10. References 77

Chapter 3. Uncertainty Propagation at the Scale of Aging Civil Engineering Structures 83
David BOUHJITI, Julien BAROTH and Frédéric DUFOUR

3.1. Introduction 83

3.2. Problem positioning 85

3.2.1. Probabilistic formulation 85

3.2.2. Thermo-hydro-mechanical-leakage transfer function 86

3.2.3. Resulting probabilistic THM-F problem 87

3.3. Random field-based modeling of material properties 88

3.3.1. Random fields 88

3.3.2. Generation methods for discretized random fields 88

3.3.3. Random fields and autocorrelations 91

3.3.4. Application: contribution to modeling the cracking of reinforced concrete works by self-correlated r.f 92

3.4. Modeling uncertainty propagation using response surface methods 98

3.4.1. Probabilistic coupling strategies 98

3.4.2. Polynomial chaos method 101

3.5. Conclusion 108

3.6. References 108

Chapter 4. Reduction of Uncertainties in Multidisciplinary Analysis Based on a Polynomial Chaos Sensitivity Study 113
Sylvain DUBREUIL, Nathalie BARTOLI, Christian GOGU and Thierry LEFEBVRE

4.1. Introduction 113

4.2. MDA with model uncertainty 115

4.2.1. Formalism 115

4.2.2. Solving the random MDA 119

4.2.3. Approximation of the quantity of interest using sparse polynomial chaos 122

4.3. Sensitivity analysis and uncertainty reduction 124

4.3.1. Introduction 124

4.3.2. Sobol' indices approximated by polynomial chaos 126

4.4. Application to an aeroelastic test case 128

4.4.1. Presentation 128

4.4.2. Construction of disciplinary metamodels 131

4.4.3. Sensitivity analysis and uncertainty reduction 133

4.5. Conclusion 140

4.6. References 140

Part 2. Taking Uncertainties into Account: Reliability Analysis and Optimization under Uncertainties 143

Chapter 5. Rare-event Probability Estimation 145
Jean-Marc BOURINET

5.1. Introduction 145

5.1.1. Mapping to the multivariate standard normal space 147

5.1.2. Copulas and correlation 149

5.1.3. Isoprobabilistic transformations 152

5.2. MPFP-based methods 159

5.2.1. First-order reliability method 159

5.2.2. Second-order reliability method 163

5.3. Simulation methods 166

5.3.1. Crude MC simulation 167

5.3.2. Subset simulation 168

5.3.3. IS and CE methods 182

5.4. Sensitivity measures 189

5.4.1. Introduction 189

5.4.2. FORM 191

5.4.3. Crude MC simulation and subset simulation 195

5.5. References 198

Chapter 6. Adaptive Kriging-based Methods for Failure Probability Evaluation: Focus on AK Methods 205
Cécile MATTRAND, Pierre BEAUREPAIRE and Nicolas GAYTON

6.1. Introduction 205

6.2. Presentation of Kriging 208

6.2.1. Principle 208

6.2.2. Identification of Kriging hyperparameters 209

6.2.3. Kriging-based prediction 210

6.2.4. Illustration of Kriging-based prediction 210

6.3. Employing Kriging to calculate failure probabilities 211

6.3.1. The EFF function 212

6.3.2. The U function 212

6.3.3. The IMSE_T function 213

6.3.4. The SUR function 213

6.3.5. The H function 214

6.3.6. The OBJ function 214

6.3.7. The L function 214

6.3.8. Discussion 214

6.4. The AK-MCS method: presentation and generic principle 215

6.4.1. Presentation of the AK-MCS method 215

6.4.2. Illustration of the AK-MCS method 217

6.4.3. Discussion 219

6.5. The AK-IS method for estimating probabilities of rare events 219

6.5.1. Presentation of the AK-IS method 219

6.5.2. Illustration of the AK-IS method 220

6.5.3. Discussion 220

6.6. The AK-SYS method for system reliability problems 222

6.6.1. Some generalities about system reliability analysis 222

6.6.2. Presentation of the AK-SYS method 223

6.6.3. Illustration of the AK-SYS method 225

6.6.4. Alternatives to the AK-SYS method 226

6.6.5. Application to problems indexed by a subset 227

6.7. The AK-HDMR1 method for high-dimensional problems 229

6.7.1. HDMR functional decomposition 230

6.7.2. Presentation of the AK-HDMR1 method 231

6.8. Conclusion 233

6.9. References 234

Chapter 7. Global Reliability-oriented Sensitivity Analysis under Distribution Parameter Uncertainty 237
Vincent CHABRIDON, Mathieu BALESDENT, Guillaume PERRIN, Jérôme MORIO, Jean-Marc BOURINET and Nicolas GAYTON

7.1. Introduction 237

7.2. Theoretical framework and notations 242

7.3. Global variance-based reliability-oriented sensitivity indices 244

7.3.1. Introducing the Sobol' indices on the indicator function 244

7.3.2. Rewriting Sobol' indices on the indicator function using Bayes' Theorem 245

7.4. Sobol' indices on the indicator function adapted to the bi-level input uncertainty 247

7.4.1. Reliability analysis under distribution parameter uncertainty 247

7.4.2. Bi-level input uncertainty: aggregated versus disaggregated types of uncertainty 249

7.4.3. Disaggregated random variables 250

7.4.4. Extension to the bi-level input uncertainty and pick-freeze estimators 251

7.5. Efficient estimation using subset sampling and KDE 253

7.5.1. The problem of estimating the optimal distribution at failure 253

7.5.2. Data-driven tensorized KDE 257

7.5.3. Methodology based on subset sampling and data-driven tensorized G-KDE 258

7.6. Application examples 258

7.6.1. Example #1: a polynomial function toy-case 261

7.6.2. Example #2: a truss structure 264

7.6.3. Example #3: application to a launch vehicle stage fallback zone estimation 267

7.6.4. Summary about numerical results and discussion 274

7.7. Conclusion 274

7.8. Acknowledgments 275

7.9. References 275

Chapter 8. Stochastic Multiobjective Optimization: A Descent Algorithm 279
Quentin MERCIER and Fabrice POIRION

8.1. Introduction 279

8.2. Mathematical refresher 281

8.2.1. Stochastic processes 281

8.2.2. Convex analysis 282

8.3. Multiobjective optimization and common descent vector 288

8.3.1. Binary relations 288

8.3.2. Multiobjective optimization, Pareto preorder 290

8.3.3. Common descent vector 296

8.4. Descent algorithm for multiobjective optimization and its extension to the stochastic framework 298

8.4.1. Multiple gradient descent algorithm 298

8.4.2. Stochastic multiple gradient descent algorithm 300

8.5. Illustrations 305

8.5.1. Performance of the SMGDA algorithm 305

8.5.2. Multiobjective approach to RBDO problems 309

8.5.3. Rewriting the probabilistic constraint 310

8.6. References 316

List of Authors 319

Index 321