Optimization and Anti-optimization of Structures Under Uncertainty

Contents......Page 18
Preface......Page 8
1.1 Probabilistic Analysis: Bad News......Page 24
1.2 Probabilistic Analysis: Good News......Page 33
1.3 Convergence of Probability and Anti-Optimization......Page 36
2.1 Introduction......Page 40
2.2 What Can We Get from Structural Optimization?......Page 42
2.3 Definition of the Structural Optimization Problem......Page 43
2.4.1 Overview of optimization problems......Page 46
2.4.2 Classification of optimization problems......Page 47
2.4.3 Parametric programming......Page 49
2.4.4 Multiobjective programming......Page 51
2.5 Approximation by Metamodels......Page 53
2.6.1 Overview of heuristics......Page 54
2.6.2 Basic approaches of single-point search heuristics......Page 55
2.6.2.2 Basic algorithm of single-point search heuristics......Page 56
2.6.2.3 Greedy method......Page 57
2.6.3 Simulated annealing......Page 58
2.7 Classification of Structural Optimization Problems......Page 59
2.8 Probabilistic Optimization......Page 62
2.9 Fuzzy Optimization......Page 64
3.1 Introduction......Page 70
3.2 Models of Uncertainty......Page 72
3.3.1 Introduction......Page 73
3.3.2 A simple example......Page 74
3.3.3 General procedure......Page 75
3.4.1 Definition of the ellipsoidal model......Page 80
3.4.2 Properties of the ellipsoidal model......Page 81
3.5 Anti-Optimization Problem......Page 84
3.6.1 Roles of sensitivity analysis in anti-optimization......Page 86
3.6.2 Sensitivity analysis of static responses......Page 87
3.6.3 Sensitivity analysis of free vibration......Page 90
3.6.4 Shape sensitivity analysis of trusses......Page 91
3.7.1 Overview of exact reanalysis......Page 92
3.7.2 Mathematical formulation based on the inverse of the modi ed matrix......Page 93
3.7.3 Mechanical formulation based on virtual load......Page 97
4.1 A Simple Example......Page 100
4.2 Boley\'s Pioneering Problem......Page 103
4.3 Anti-Optimization Problem for Static Responses......Page 108
4.4 Matrix Perturbation Methods for Static Problems......Page 109
4.5.1 Introduction......Page 113
4.5.2 An asymptotic solution......Page 114
4.5.3 A worst-case investigation......Page 120
4.6.1 Introduction......Page 125
4.6.2.1 Equilibrium equations......Page 126
4.6.2.2 Self-equilibrium forces......Page 127
4.6.2.3 Tangent stiffness matrix......Page 128
4.6.2.5 Compliance against external load......Page 129
4.6.3 Anti-optimization problem......Page 130
4.6.4 Numerical examples......Page 132
5.1 Introduction......Page 136
5.2 A Simple Example......Page 137
5.3 Buckling Analysis......Page 139
5.4 Anti-Optimization Problem......Page 140
5.5.1 Definition of frame model......Page 141
5.5.2 Worst imperfection of optimized frame......Page 143
5.5.3 Mode interaction......Page 146
5.5.4 Worst-case design and worst imperfection under stress constraints......Page 149
5.6 Anti-Optimization Based on Convexity of Stability Region......Page 151
5.7.1 Introduction......Page 156
5.7.2 Hilltop branching point of perfect system......Page 157
5.7.3 Imperfection sensitivity of hilltop branching point......Page 158
5.7.4 Worst imperfection......Page 159
5.7.5 Worst imperfection of an arch-type truss......Page 162
5.8 Some Further References......Page 167
6.1 Introduction......Page 168
6.2 A Simple Example of Anti-Optimization for Eigenvalue of Vibration......Page 169
6.2.1 Anti-optimization for forced vibration......Page 171
6.3 Bulgakov\'s Problem......Page 172
6.4.1 Introduction......Page 173
6.4.2 Basic equations for vibrating viscoelastic beam......Page 174
6.4.3 Application to a simply supported beam......Page 178
6.4.4 Least and most favorable responses......Page 180
6.4.5 Numerical examples and discussion......Page 183
6.5.1 Introduction......Page 186
6.5.2 Formulation of the problem......Page 187
6.5.4.1 Basic ideas......Page 189
6.5.4.2 Preliminary statements......Page 190
6.5.4.3 Search of the smallest ellipsoid containing all points......Page 195
6.5.4.4 Numerical application......Page 196
6.6.1 Preliminary comments......Page 198
6.6.2 Credible accelerograms......Page 203
6.6.3 Application......Page 207
6.6.4 Discussion and conclusion......Page 211
6.7.1 Introduction......Page 213
6.7.2 Deterministic theoretical analysis......Page 214
6.7.3 Stability analysis within two-term approximation......Page 216
6.7.4 Convex modeling of uncertain moduli......Page 218
6.7.5 Anti-optimization problem: polygonal region of uncertainty......Page 220
6.7.6 Anti-optimization problem: ellipsoidal region of uncertainty......Page 223
6.7.7 Minimum weight design......Page 224
6.7.8 A numerical example......Page 229
6.8 Some Further References......Page 231
7.1 Introduction......Page 234
7.2 Interval Analysis of MDOF Systems......Page 235
7.3 Interval Finite Element Analysis for Linear Static Problem......Page 237
7.4.1 Basic equations......Page 240
7.4.2 A numerical example......Page 243
7.5 Interval Analysis for Pattern Loading......Page 244
7.6 Some Further References......Page 248
8.1 Introduction......Page 250
8.2.1 Problem formulation......Page 252
8.2.2 Probabilistic analysis......Page 254
8.2.3 Uniformly distributed random initial imperfections......Page 255
8.2.4 Random initial imperfections with truncated exponential distribution......Page 263
8.2.5 Random initial imperfection with generic truncated distribution......Page 267
8.2.7 Conclusion......Page 269
8.3.1 Introduction......Page 270
8.3.2 Deterministic analysis......Page 271
8.3.3 Probabilistic analysis......Page 273
8.3.4 Initial imperfections with uniform probability density: Rectangular domain .......Page 274
8.3.5 Initial imperfections with general probability density: Rectangular domain......Page 285
8.3.6 Initial imperfection with uniform probability density function: Circular domain......Page 287
8.3.8 Initial imperfections as convex variables: Circular domain......Page 292
8.3.9 Concluding remarks......Page 295
9.1 Introduction......Page 296
9.2 A Simple Example......Page 297
9.3 Formulation of the Two-Level Optimization–Anti-Optimization Problem......Page 299
9.4.1 Cycle-based method......Page 300
9.4.2 Methods based on monotonicity or convexity/concavity......Page 301
9.4.3.2 Parametric programming approach......Page 302
9.5.1 Introduction......Page 303
9.5.2 Problem formulation......Page 304
9.5.3 Numerical examples......Page 308
9.6 Stress and Displacement Constraints......Page 311
9.7.1 Introduction......Page 314
9.7.2 Optimization problem and optimization algorithm......Page 317
9.7.3 Numerical examples......Page 318
9.8.1 Introduction......Page 321
9.8.2 Deterministic loading......Page 323
9.8.3 A numerical example......Page 325
9.8.4 Convex model of uncertain loading......Page 326
9.8.5 Worst-case estimation......Page 327
9.8.6 A numerical example of worst-case estimation of homology design......Page 328
9.8.7 Concluding remarks......Page 329
9.9.1 Introduction......Page 330
9.9.2 Definition of asymptotic stability......Page 331
9.9.3 Optimization problem......Page 332
9.9.4 Numerical examples......Page 333
9.9.4.2 A 24-bar truss......Page 334
9.10.1 Introduction......Page 339
9.10.2 Equations for self-equilibrium state......Page 340
9.10.3.1 Force errors......Page 341
9.10.4 Sensitivity analysis with respect to nodal coordinates......Page 343
9.10.5.2 Solution process......Page 345
9.10.6 Numerical examples......Page 346
9.11 Some Further References......Page 348
10.1 Why Were Practical Engineers Reluctant to Adopt Structural Optimization?......Page 350
10.2 Why Didn\'t Practical Engineers Totally Embrace Probabilistic Methods?......Page 354
10.3 Why Don\'t the Probabilistic Methods Find Appreciation among Theoreticians and Practitioners Alike?......Page 356
10.4 Is the Suggested Methodology a New One?......Page 357
10.5 Finally, Why Did We Write This Book?......Page 362
Bibliography......Page 366
Index......Page 410
Author Index......Page 414