Uncertainty and Optimization in Structural Mechanics

Optimization is generally a reduction operation of a definite quantity. This process naturally takes place in our environment and through our activities. For example, many natural systems evolve, in order to minimize their potential energy. Modeling these phenomena then largely relies on our capacity to artificially reproduce these processes. In parallel, optimization problems have quickly emerged from human activities, notably from economic concerns. This book includes the most recent ideas coming from research and industry in the field of optimization, reliability and the recognition of accompanying uncertainties. It is made up of eight chapters which look at the reviewing of uncertainty tools, system reliability, optimal design of structures and their optimization (of sizing, form, topology and multi-objectives) ? along with their robustness and issues on optimal safety factors. Optimization reliability coupling will also be tackled in order to take into account the uncertainties in the modeling and resolution of the problems encountered. The book is aimed at students, lecturers, engineers, PhD students and researchers.

Contents

1. Uncertainty.
2. Reliability in Mechanical Systems.
3. Optimal Structural Design.
4. Multi-object Optimization with Uncertainty.
5. Robust Optimization.
6. Reliability Optimization.
7. Optimal Security Factors Approach.
8. Reliability-based Topology Optimization.

About the Authors

Abdelkhalak El Hami is Professor at the Institut National des Sciences Appliquées, Rouen, France. He is the author of many articles and books on optimization and uncertainty.
Bouchaib Radi is Professor in the Faculty of Sciences and Technology at the University of Hassan Premier, Settat, Morocco. His research interests are in such areas as structural optimization, parallel computation, contact problem and metal forming. He is the author of many scientific articles and books.

Preface ix

Chapter 1 Uncertainty 1

1.1. Introduction 1

1.2. The optimization problem 3

1.3. Sources of uncertainty 4

1.4. Dealing with uncertainty 6

1.4.1. Reliability optimization 11

1.4.2. Robust optimization 12

1.4.3. Multi-object optimization 13

1.4.4. Stochastic optimization 14

1.4.5. Worst-case scenario based optimization 14

1.4.6. Non-probabilistic optimization 15

1.4.7. Interval modeling 15

1.4.8. Fuzzy sets 15

1.5. Analyzing sensitivity 16

1.5.1. Local sensitivity analysis 16

1.5.2. Global sensitivity analysis 16

Chapter 2 Reliability in Mechanical Systems 17

2.1. Introduction 17

2.2. A structure reliability problem 18

2.3. Modeling a structure reliability problem 18

2.3.1. A deterministic mechanical model 18

2.3.2. Risks and probabilistic modeling 18

2.3.3. Types of failure in a structure 19

2.3.4. Probability of failure in a structure 19

2.4. Calculating the probability of failure in a structure 20

2.4.1. Calculating the probability of failure using the Monte Carlo method 20

2.4.2. Calculating the probability of failure using a reliability index 21

2.5. Reliability indices 21

2.5.1. The Rjanitzyne–Cornell index 21

2.5.2. The Hasofer–Lind index 22

2.5.3. The FORM method 23

2.5.4. The SORM method 25

2.6. Overview of the resistance–sollicitation problem 26

2.6.1. Probability of failure 27

2.6.2. Reliability indices 28

2.7. System reliability in mechanics 33

2.7.1. Combinations of types of failure 34

2.7.2. Assessment of the failure probability of a system 35

2.8. The finite element method and structural reliability 36

2.8.1. Context and objectives of the problem 36

2.8.2. Discretization and modeling random fields 36

2.8.3. Mechano-reliability coupling 37

2.8.4. Surface response coupling 41

Chapter 3 Optimal Structural Design 43

3.1. Introduction 43

3.2. Historical development of structural optimization 44

3.3. Classifying structural optimization problems 44

3.3.1. Dimensional optimization 45

3.3.2. Topological optimization 45

3.3.3. Shape optimization 47

Chapter 4. Multi-object Optimization with Uncertainty .. 51

4.1. Introduction 51

4.1.1. Choice of an optimization method 52

4.1.2. Classifying optimization methods 52

4.2. User classification 53

4.3. Design classification 54

4.4. Multi-objective genetic algorithms 54

4.5. Robust multi-objective optimization 56

4.5.1. Robustness criteria in multi-objective optimization 56

4.6. Normal boundary intersection method 57

4.6.1. Description of the NBI method 58

4.7. Multi-objective structural optimization problem 66

Chapter 5 Robust Optimization 69

5.1. Introduction 69

Table of Contents vii

5.2. Modeling uncertainty 69

5.2.1. Parametric methods 70

5.2.2. Non-parametric methods 71

5.3. Accounting for robustness in optimum research 73

5.4. Robustness criteria 74

5.4.1. Defining uncertainty in design parameters 74

5.4.2. Robustness criteria in multi-objective optimization 75

5.5. Resolution method 76

5.6. Examples of mono-objective optimization 77

Chapter 6 Reliability Optimization 79

6.1. Introduction 79

6.2. Overview of reliability optimization 80

6.3. Reliability optimization methods 81

6.4. The reliability indicator approach 81

6.5. The single-loop approach 82

6.6. The sequential optimization and reliability assessment approach 87

Chapter 7 Optimal Security Factors Approach 93

7.1. Introduction 93

7.2. Standard method 93

7.3. The optimal security factors (OSFs) method 95

7.4. Extension of the OSF method to multiple failure scenarios 99

Chapter 8 Reliability-based Topology Optimization 113

8.1. Introduction 113

8.2. Definitions in topology optimization 114

8.3. Topology optimization methods 115

8.4. Reliability coupling and topology optimization 118

8.5. Illustration and validation of the RBTO model 120

8.6. Application of the RBTO model to mechanics 122

8.6.1. Static analysis 122

8.6.2. Modal analysis 123

Bibliography 125

Index 131 

Abdelkhala El Hami is Professor at INSA Rouen, France.

Bouchaïb Radi is Professor at the Faculty of Sciences and Technology Settat, Morocco.