Modern Trends in Structural and Solid Mechanics 3: Non-deterministic Mechanics

This book – comprised of three separate volumes – presents the recent developments and research discoveries in structural and solid mechanics; it is dedicated to Professor Isaac Elishakoff. This third volume is devoted to non-deterministic mechanics. Modern Trends in Structural and Solid Mechanics 3 has broad scope, covering topics such: design optimization under uncertainty, interval field approaches, convex analysis, quantum inspired topology optimization and stochastic dynamics. The book is illustrated by many applications in the field of aerospace engineering, mechanical engineering, civil engineering, biomedical engineering and automotive engineering. This book is intended for graduate students and researchers in the field of theoretical and applied mechanics.

Table of Contents:
Preface xi Noël CHALLAMEL, Julius KAPLUNOV and Izuru TAKEWAKI Chapter 1. Optimization in Mitochondrial Energetic Pathways 1 Haym BENAROYA 1.1. Optimization in neural and cell biology 1 1.2. Mitochondria 3 1.3. General morphology; fission and fusion 5 1.4. Mechanical aspects 9 1.5. Mitochondrial motility 13 1.6. Cristae, ultrastructure and supercomplexes 14 1.7. Mitochondrial diseases and neurodegenerative disorders 15 1.8. Modeling 16 1.9. Concluding summary 17 1.10. Acknowledgments 18 1.11. Appendix 18 1.12. References 19 Chapter 2. The Concept of Local and Non-Local Randomness for Some Mechanical Problems 23 Giovanni FALSONE and Rossella LAUDANI 2.1. Introduction 23 2.2. Preliminary concepts 24 2.2.1. Statically determinate stochastic beams 24 2.2.2. Statically indeterminate stochastic beams 26 2.3. Local and non-local randomness 29 2.3.1. Statically determinate stochastic beams 31 2.3.2. Statically indeterminate stochastic beams 32 2.3.3. Comments on the results 36 2.4. Conclusion 36 2.5. References 37 Chapter 3. On the Applicability of First-Order Approximations for Design Optimization under Uncertainty 39 Benedikt KRIEGESMANN 3.1. Introduction 39 3.2. Summary of first- and second-order Taylor series approximations for uncertainty quantification 41 3.2.1. Approximations of stochastic moments 42 3.2.2. Probabilistic lower bound approximation 43 3.2.3. Convex anti-optimization 44 3.2.4. Correlation of probabilistic approaches and convex anti-optimization 45 3.3. Design optimization under uncertainty 46 3.3.1. Robust design optimization 46 3.3.2. Reliability-based design optimization 47 3.3.3. Optimization with convex anti-optimization 48 3.4. Numerical examples 48 3.4.1. Imperfect von Mises truss analysis 48 3.4.2. Three-bar truss optimization 50 3.4.3. Topology optimization 52 3.5. Conclusion and outlook 56 3.6. References 57 Chapter 4. Understanding Uncertainty 61 Maurice LEMAIRE 4.1. Introduction 61 4.2. Uncertainty and uncertainties 61 4.3. Design and uncertainty 63 4.3.1. Decision modules 63 4.3.2. Designing in uncertain 66 4.4. Knowledge entity 67 4.4.1. Structure of a knowledge entity 67 4.5. Robust and reliable engineering 70 4.5.1. Definitions 70 4.5.2. Robustness 71 4.5.3. Reliability 72 4.5.4. Optimization 72 4.5.5. Reliable and robust optimization 73 4.6. Conclusion 74 4.7. References 75 Chapter 5. New Approach to the Reliability Verification of Aerospace Structures 77 Giora MAYMON 5.1. Introduction 77 5.2. Factor of safety and probability of failure 78 5.3. Reliability verification of aerospace structural systems 84 5.3.1. Reliability demonstration is integrated into the design process 86 5.3.2. Analysis of failure mechanism and failure modes 87 5.3.3. Modeling the structural behavior, verifying the model by tests 87 5.3.4. Design of structural development tests to surface failure modes 88 5.3.5. Design of development tests to find unpredicted failure modes 88 5.3.6. “Cleaning” failure mechanism and failure modes 88 5.3.7. Determination of required safety and confidence in models 89 5.3.8. Determination of the reliability by “orders of magnitude” 89 5.4. Summary 92 5.5. References 93 Chapter 6. A Review of Interval Field Approaches for Uncertainty Quantification in Numerical Models 95 Matthias FAES, Maurice IMHOLZ, Dirk VANDEPITTE and David MOENS 6.1. Introduction 95 6.2. Interval finite element analysis 97 6.3. Convex-set analysis 99 6.4. Interval field analysis 100 6.4.1. Explicit interval field formulation 101 6.4.2. Interval fields based on KL expansion 103 6.4.3. Interval fields based on convex descriptors 105 6.5. Conclusion 105 6.6. Acknowledgments 106 6.7. References 106 Chapter 7. Convex Polytopic Models for the Static Response of Structures with Uncertain-but-bounded Parameters 111 Zhiping QIU and Nan JIANG 7.1. Introduction 111 7.2. Problem statements 114 7.3. Analysis and solution of the convex polytopic model for the static response of structures 116 7.4. Vertex solution theorem of the convex polytopic model for the static response of structures 119 7.5. Review of the vertex solution theorem of the interval model for the static response of structures 122 7.6. Numerical examples 127 7.6.1. Two-step bar 127 7.6.2. Ten-bar truss 130 7.6.3. Plane frame 135 7.7. Conclusion 141 7.8. Acknowledgments 141 7.9. References 141 Chapter 8. On the Interval Frequency Response of Cracked Beams with Uncertain Damage 145 Roberta SANTORO 8.1. Introduction 146 8.2. Crack modeling for damaged beams 148 8.2.1. Finite element crack model 148 8.2.2. Continuous crack model 149 8.3. Statement of the problem 150 8.3.1. Interval model for the uncertain crack depth 151 8.3.2. Governing equations of damaged beams 152 8.3.3. Finite element model versus continuous model 154 8.4. Interval frequency response of multi-cracked beams 162 8.4.1. Interval deflection function in the FE model 162 8.4.2. Interval deflection function in the continuous model 165 8.5. Numerical applications 167 8.6. Concluding remarks 173 8.7. Acknowledgments 173 8.8. References 173 Chapter 9. Quantum-Inspired Topology Optimization 177 Xiaojun WANG, Bowen NI and Lei WANG 9.1. Introduction 177 9.2. General statements 180 9.2.1. Density-based continuum structural topology optimization formulation 180 9.2.2. Characteristics of quantum computing 181 9.3. Topology optimization design model based on quantum-inspired evolutionary algorithms 183 9.3.1. Classic procedure of topology optimization based on the SIMP method and optimality criteria 183 9.3.2. The fundamental theory of a quantum-inspired evolutionary algorithm – DCQGA 186 9.3.3. Implementation of the integral topology optimization framework 189 9.4. A quantum annealing operator to accelerate the calculation and jump out  of local extremum 191 9.5. Numerical examples 195 9.5.1. Example of a short cantilever 195 9.5.2. Example of a wing rib 196 9.6. Conclusion 198 9.7. Acknowledgments 198 9.8. References 199 Chapter 10. Time Delay Vibrations and Almost Sure Stability in Vehicle Dynamics 203 Walter V. WEDIG 10.1. Introduction to road vehicle dynamics 203 10.2. Delay resonances of half-car models on road 205 10.3. Extensions to multi-body vehicles on a random road 209 10.4. Non-stationary road excitations applying sinusoidal models 212 10.5. Resonance reduction or induction by means of colored noise 215 10.6. Lyapunov exponents and rotation numbers in vehicle dynamics 218 10.7. Concluding remarks and main new results 221 10.8. References 222 Chapter 11. Order Statistics Approach to Structural Optimization Considering Robustness and Confidence of Responses 225 Makoto YAMAKAWA and Makoto OHSAKI 11.1. Introduction 225 11.2. Overview of order statistics 226 11.2.1. Definition of order statistics 226 11.2.2. Tolerance intervals and confidence intervals of quantiles 227 11.3. Robust design 229 11.3.1. Overview of the robust design problem 229 11.3.2. Worst-case-based method 230 11.3.3. Order statistics-based method 230 11.4. Numerical examples 231 11.4.1. Design response spectrum 231 11.4.2. Optimization of the building frame considering seismic responses 232 11.4.3. Multi-objective optimization considering robustness 236 11.5. Conclusion 239 11.6. References 240 List of Authors 243 Index 245 Summaries of Volumes 1 and 2 249