Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments

Considering the uncertainties in mechanical engineering in order
to improve the performance of future products or systems is
becoming a competitive advantage, sometimes even a necessity,
when seeking to guarantee an increasingly high safety
requirement.

Mechanical Engineering in Uncertainties deals with modeling,
quantification and propagation of uncertainties. It also examines
how to take into account uncertainties through reliability analyses
and optimization under uncertainty. The spectrum of the methods
presented ranges from classical approaches to more recent
developments and advanced methods. The methodologies are
illustrated by concrete examples in various fields of mechanics
(civil engineering, mechanical engineering and fluid mechanics).
This book is intended for both (young) researchers and engineers
interested in the treatment of uncertainties in mechanical
engineering.

Autorentext

Christian Gogu is Associate Professor at the University Toulouse III-Paul Sabatier, France. His research, which he carries out at the Clement Ader Institute, focuses, in particular, on taking into account uncertainties in the design and optimization of aeronautical systems.

Klappentext
Considering the uncertainties in mechanical engineering in orderto improve the performance of future products or systems isbecoming a competitive advantage, sometimes even a necessity,when seeking to guarantee an increasingly high safetyrequirement.

Mechanical Engineering in Uncertainties deals with modeling,quantification and propagation of uncertainties. It also examineshow to take into account uncertainties through reliability analysesand optimization under uncertainty. The spectrum of the methodspresented ranges from classical approaches to more recentdevelopments and advanced methods. The methodologies areillustrated by concrete examples in various fields of mechanics(civil engineering, mechanical engineering and fluid mechanics).This book is intended for both (young) researchers and engineersinterested in the treatment of uncertainties in mechanicalengineering.

Zusammenfassung
Considering the uncertainties in mechanical engineering in order to improve the performance of future products or systems is becoming a competitive advantage, sometimes even a necessity, when seeking to guarantee an increasingly high safety requirement. Mechanical Engineering in Uncertainties deals with modeling, quantification and propagation of uncertainties. It also examines how to take into account uncertainties through reliability analyses and optimization under uncertainty. The spectrum of the methods presented ranges from classical approaches to more recent developments and advanced methods. The methodologies are illustrated by concrete examples in various fields of mechanics (civil engineering, mechanical engineering and fluid mechanics). This book is intended for both (young) researchers and engineers interested in the treatment of uncertainties in mechanical engineering.

Inhalt
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. JohnsonMehl 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 fieldbased 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 aeroelas...