Biomechanics Optimization, Uncertainties and Reliability

258 Pages·2017·15.013 MB·English

by  Ghias Kharmanda, Abdelkhalak El Hami

Biomechanics

Reliability of Multiphysical Systems Set
coordinated by
Abdelkhalak El Hami
Volume 5
Biomechanics

Optimization, Uncertainties and Reliability

Ghias Kharmanda
Abdelkhalak El Hami
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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ISBN 978-1-78630-025-6
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter 1. Introduction to Structural Optimization . . . . . . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. History of structural optimization . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Sizing optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2. First works in sizing optimization . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3. Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4. Shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2. First works in shape optimization . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.3. Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5. Topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.2. First works in topology optimization . . . . . . . . . . . . . . . . . . . . . 17
1.5.3. Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 2. Integration of Structural Optimization
into Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2. Integration of structural optimization into
orthopedic prosthesis design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vi Biomechanics
2.2.1. Structural optimization of the hip prosthesis . . . . . . . . . . . . . . . . . 24
2.2.2. Sizing optimization of a 3D intervertebral
disk prosthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3. Integration of structural optimization into
orthodontic prosthesis design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.1. Sizing optimization of a dental implant . . . . . . . . . . . . . . . . . . . . 47
2.3.2. Shape optimization of a mini-plate . . . . . . . . . . . . . . . . . . . . . . 49
2.4. Advanced integration of structural optimization
into drilling surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.1. Case of treatment of a crack with a single hole . . . . . . . . . . . . . . . . 53
2.4.2. Case of treatment of a crack with two holes . . . . . . . . . . . . . . . . . 54
2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 3. Integration of Reliability into
Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2. Literature review of reliability-based optimization . . . . . . . . . . . . . . . . 58
3.3. Comparison between deterministic and
reliability-based optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.1. Deterministic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.2. Reliability-based optimization . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4. Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1. Description and modeling of the studied problem . . . . . . . . . . . . . . 64
3.4.2. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5. Approaches and strategies for
reliability-based optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.1. Mono-level approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.2. Double-level approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.3. Sequential decoupled approaches . . . . . . . . . . . . . . . . . . . . . . . 68
3.6. Two points of view for developments of
reliability-based optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6.1. Point of view of “Reliability” . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6.2. Point of view of “Optimization” . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.3. Method efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7. Philosophy of integration of the concept of
reliability into structural optimization groups . . . . . . . . . . . . . . . . . . . . . 72
3.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 4. Reliability-based Design
Optimization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2. Classic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Contents vii
4.2.1. Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2. Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.3. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.4. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3. Hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.2. Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.3. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.4. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4. Improved hybrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.1. Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.2. Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.3. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.4. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5. Optimum safety factor method . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5.1. Safety factor concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5.2. Developments and optimality conditions . . . . . . . . . . . . . . . . . . . 92
4.5.3. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.4. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6. Safest point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6.1. Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6.2. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.6.3. Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7. Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.7.1. RBDO of a hook: CM and HM . . . . . . . . . . . . . . . . . . . . . . . . 105
4.7.2. RBDO of a triangular plate: HM & IHM . . . . . . . . . . . . . . . . . . . 107
4.7.3. RBDO of a console beam
(sandwich beam): HM and OSF . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7.4. RBDO of an aircraft wing: HM & SP . . . . . . . . . . . . . . . . . . . . . 113
4.8. Classification of the methods developed . . . . . . . . . . . . . . . . . . . . . . 115
4.8.1. Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.8.2. Semi-numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.8.3. Comparison between the numerical- and
semi-numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter 5. Reliability-based Topology
Optimization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2. Formulation and algorithm for the RBTO model . . . . . . . . . . . . . . . . . 122
5.2.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
viii Biomechanics
5.2.2. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.3. Validation of the RBTO code developed . . . . . . . . . . . . . . . . . . . 125
5.3. Validation of the RBTO model . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3.1. Analytical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3.2. Numerical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4. Variability of the reliability index . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.1. Example 1: MBB beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4.2. Example 2: Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4.3. Example 3: Cantilever beam with double loads . . . . . . . . . . . . . . . 136
5.4.4. Example 4: Cantilever beam with a transversal hole . . . . . . . . . . . . . 136
5.5. Numerical applications for the RBTO model . . . . . . . . . . . . . . . . . . . 137
5.5.1. Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.2. Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5.3. Fatigue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.6. Two points of view for integration of reliability
into topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6.1. Point of view of “topology” . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6.2. Point of view of “reliability” . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6.3. Numerical applications for the two points of view . . . . . . . . . . . . . . 146
5.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Chapter 6. Integration of Reliability and
Structural Optimization into Prosthesis Design . . . . . . . . . . . . . . . . . 153
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2. Prosthesis design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3. Integration of topology optimization
into prosthesis design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.1. Importance of topology optimization
in prosthesis design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3.2. Place of topology optimization in
the prosthesis design chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4. Integration of reliability and structural
optimization into hip prosthesis design . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.4.1. Numerical application of the deterministic approach . . . . . . . . . . . . 158
6.4.2. Numerical application of the reliability-based approach . . . . . . . . . . 167
6.5. Integration of reliability and structural
optimization into the design of mini-plate systems
used to treat fractured mandibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.5.1. Numerical application of the
deterministic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.5.2. Numerical application of the
reliability-based approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181