Variational methods for structural optimization:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Series: | Applied mathematical sciences
140 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVI, 545 S. graph. Darst. |
| ISBN: | 0387984623 |
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| 100 | 1 | |a Cherkaev, Andrej |d 1950- |e Verfasser |0 (DE-588)11560832X |4 aut | |
| 245 | 1 | 0 | |a Variational methods for structural optimization |c Andrej Cherkaev |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a XXVI, 545 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804128045909409792 |
|---|---|
| adam_text | Contents
List of Figures xi
Preface xv
I Preliminaries 1
1 Relaxation of One Dimensional Variational Problems 3
1.1 An Optimal Design by Means of Composites 3
1.2 Stability of Minimizers and the Weierstrass Test 7
1.2.1 Necessary and Sufficient Conditions 7
1.2.2 Variational Methods: Weierstrass Test 10
1.3 Relaxation 14
1.3.1 Nonconvex Variational Problems 14
1.3.2 Convex Envelope 16
1.3.3 Minimal Extension and Minimizing Sequences .... 19
1.3.4 Examples: Solutions to Nonconvex Problems .... 24
1.3.5 Null Lagrangians and Convexity 27
1.3.6 Duality 29
1.4 Conclusion and Problems 32
2 Conducting Composites 35
2.1 Conductivity of Inhomogeneous Media 35
2.1.1 Equations for Conductivity 35
2.1.2 Continuity Conditions in Inhomogeneous Materials . 39
2.1.3 Energy, Variational Principles 42
2.2 Composites 45
2.2.1 Homogenization and Effective Tensor 46
vi Contents
2.2.2 Effective Properties of Laminates 51
2.2.3 Effective Medium Theory: Coated Circles 55
2.3 Conclusion and Problems 57
3 Bounds and G Closures 59
3.1 Effective Tensors: Variational Approach 59
3.1.1 Calculation of Effective Tensors 59
3.1.2 Wiener Bounds 61
3.2 G Closure Problem 63
3.2.1 G convergence 63
3.2.2 G Closure: Definition and Properties 67
3.2.3 Example: The G Closure of Isotropic Materials ... 73
3.2.4 Weak G Closure (Range of Attainability) 75
3.3 Conclusion and Problems 76
II Optimization of Conducting Composites 79
4 Domains of Extremal Conductivity 81
4.1 Statement of the Problem 82
4.2 Relaxation Based on the G Closure 83
4.2.1 Relaxation 83
4.2.2 Sufficient Conditions 85
4.2.3 A Dual Problem 89
4.2.4 Convex Envelope and Compatibility Conditions ... 90
4.3 Weierstrass Test 92
4.3.1 Variation in a Strip 92
4.3.2 The Minimal Extension 99
4.3.3 Summary 101
4.4 Dual Problem with Nonsmooth Lagrangian 103
4.5 Example: The Annulus of Extremal Conductivity 108
4.6 Optimal Multiphase Composites 110
4.6.1 An Elastic Bar of Extremal Torsion Stiffness .... 110
4.6.2 Multimaterial Design Ill
4.7 Problems 115
5 Optimal Conducting Structures 117
5.1 Relaxation and G Convergence 117
5.1.1 Weak Continuity and Weak Lower Semicontinuity . 117
5.1.2 Relaxation of Constrained Problems by G Closure . 121
5.2 Solution to an Optimal Design Problem 123
5.2.1 Augmented Functional 123
5.2.2 The Local Problem 126
5.2.3 Solution in the Large Scale 129
5.3 Reducing to a Minimum Variational Problem 130
5.4 Examples 134
Contents vii
5.5 Conclusion and Problems 139
III Quasiconvexity and Relaxation 143
6 Quasiconvexity 145
6.1 Structural Optimization Problems 145
6.1.1 Statements of Problems of Optimal Design 145
6.1.2 Fields and Differential Constraints 148
6.2 Convexity of Lagrangians and Stability of Solutions 151
6.2.1 Necessary Conditions: Weierstrass Test 151
6.2.2 Attainability of the Convex Envelope 155
6.3 Quasiconvexity 158
6.3.1 Definition of Quasiconvexity 158
6.3.2 Quasiconvex Envelope 163
6.3.3 Bounds 165
6.4 Piecewise Quadratic Lagrangians 167
6.5 Problems 170
7 Optimal Structures and Laminates 171
7.1 Laminate Bounds 171
7.1.1 The Laminate Bound 172
7.1.2 Bounds of High Rank 174
7.2 Effective Properties of Simple Laminates 176
7.2.1 Laminates from Two Materials 177
7.2.2 Laminate from a Family of Materials 180
7.3 Laminates of Higher Rank 182
7.3.1 Differential Scheme 183
7.3.2 Matrix Laminates 189
7.3.3 ^ Transform 193
7.3.4 Calculation of the Fields Inside the Laminates . . . 195
7.4 Properties of Complicated Structures 198
7.4.1 Multicoated and Self Repeating Structures 198
7.4.2 Structures of Contrast Properties 201
7.5 Optimization in the Class of Matrix Composites 206
7.6 Discussion and Problems 211
8 Lower Bound: Translation Method 213
8.1 Translation Bound 213
8.2 Quadratic Translators 220
8.2.1 Compensated Compactness 220
8.2.2 Determination of Quadratic Translators 224
8.3 Translation Bounds for Two Well Lagrangians 228
8.3.1 Basic Formulas 228
8.3.2 Extremal Translations 229
8.3.3 Example: Lower Bound for the Sum of Energies . . . 232
viii Contents
8.3.4 Translation Bounds and Laminate Structures .... 235
8.4 Problems 237
9 Necessary Conditions and Minimal Extensions 239
9.1 Variational Methods for Nonquasiconvex Lagrangians . . . 239
9.2 Variations 241
9.2.1 Variation of Properties 241
9.2.2 Increment 242
9.2.3 Minimal Extension 246
9.3 Necessary Conditions for Two Phase Composites 248
9.3.1 Regions of Stable Solutions 248
9.3.2 Minimal Extension 249
9.3.3 Necessary Conditions and Compatibility 251
9.3.4 Necessary Conditions and Optimal Structures .... 253
9.4 Discussion and Problems 257
IV G Closures 259
10 Obtaining G Closures 261
10.1 Variational Formulation 261
10.1.1 Variational Problem for GTO Closure 262
10.1.2 G Closures 269
10.2 The Bounds from Inside by Laminations 270
10.2.1 The/^Closure m Two Dimensions 274
11 Examples of G Closures 279
11.1 The Gm Closure of Two Conducting Materials 279
11.1.1 The Variational Problem 279
11.1.2 The GTO Closure in Two Dimensions 280
11.1.3 Three Dimensional Problem 284
11.2 G Closures 289
11.2.1 Two Isotropic Materials 289
11.2.2 Polycrystals 291
11.2.3 Two Dimensional Polycrystal 292
11.2.4 Three Dimensional Isotropic Polycrystal 293
11.3 Coupled Bounds 296
11.3.1 Statement of the Problem 296
11.3.2 Translation Bounds of Gm Closure 299
11.3.3 The Use of Coupled Bounds 305
11.4 Problems 308
12 Multimaterial Composites 309
12.1 Special Features of Multicomponent Composites 311
12.1.1 Attainability of the Wiener Bound 311
12.1.2 Attainability of the Translation Bounds 316
Contents ix
12.1.3 The Compatibility of Incompatible Phases 321
12.2 Necessary Conditions 325
12.2.1 Single Variations 326
12.2.2 Composite Variations 328
12.3 Optimal Structures for Three Component Composites . . . 334
12.3.1 Range of Values of the Lagrange Multiplier 334
12.3.2 Examples of Optimal Microstructures 338
12.4 Discussion 341
13 Supplement: Variational Principles for
Dissipative Media 343
13.1 Equations of Complex Conductivity 344
13.1.1 The Constitutive Relations 344
13.1.2 Real Second Order Equations 347
13.2 Variational Principles 348
13.2.1 Minimax Variational Principles 349
13.2.2 Minimal Variational Principles 350
13.3 Legendre Transform 352
13.4 Application to G Closure 353
V Optimization of Elastic Structures 357
14 Elasticity of Inhomogeneous Media 359
14.1 The Plane Problem 359
14.1.1 Basic Equations 359
14.1.2 Rotation of Fourth Rank Tensors 363
14.1.3 Classes of Equivalency of Elasticity Tensors 371
14.2 Three Dimensional Elasticity 373
14.2.1 Equations 373
14.2.2 Inhomogeneous Medium. Continuity Conditions . . . 377
14.2.3 Energy, Variational Principles 378
14.3 Elastic Structures 379
14.3.1 Elastic Composites 379
14.3.2 Effective Properties of Elastic Laminates 380
14.3.3 Matrix Laminates, Plane Problem 382
14.3.4 Three Dimensional Matrix Laminates 385
14.3.5 Ideal Rigid Soft Structures 387
14.4 Problems 391
15 Elastic Composites of Extremal Energy 393
15.1 Composites of Minimal Compliance 393
15.1.1 The Problem 393
15.1.2 Translation Bounds 395
15.1.3 Structures 398
15.1.4 The Quasiconvex Envelope 402
x Contents
15.1.5 Three Dimensional Problem 403
15.2 Composites of Minimal Stiffness 405
15.2.1 Translation Bounds 406
15.2.2 The Attainability of the Convex Envelope 407
15.3 Optimal Structures Different from Laminates 410
15.3.1 Optimal Structures by Vigdergauz 410
15.3.2 Optimal Shapes under Shear Loading 413
15.4 Problems 417
16 Bounds on Effective Properties 419
16.1 Gm Closures of Special Sets of Materials 419
16.2 Coupled Bounds for Isotropic Moduli 422
16.2.1 The Hashin Shtrikman Bounds 423
16.2.2 The Translation Bounds 425
16.2.3 Functionals 429
16.2.4 Translators 431
16.2.5 Modification of the Translation Method 433
16.2.6 Appendix: Calculation of the Bounds 436
16.3 Isotropic Planar Polycrystals 447
16.3.1 Bounds 448
16.3.2 Extremal Structures: Differential Scheme 450
16.3.3 Extremal Structures: Fixed Point Scheme 454
17 Some Problems of Structural Optimization 459
17.1 Properties of Optimal Layouts 459
17.1.1 Necessary Conditions 460
17.1.2 Remarks on Instabilities 463
17.2 Optimization of the Sum of Elastic Energies 464
17.2.1 Minimization of the Sum of Elastic Energies 465
17.2.2 Optimal Design of Periodic Structures 469
17.3 Arbitrary Goal Functionals 472
17.3.1 Statement 472
17.3.2 Local Problem 473
17.3.3 Asymptotics 475
17.4 Optimization under Uncertain Loading 477
17.4.1 The Formulation 477
17.4.2 Eigenvalue Problem 480
17.4.3 Multiple Eigenvalues 484
17.5 Conclusion 491
References 495
Author/Editor Index 525
Subject Index 533
|
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| isbn | 0387984623 |
| language | English |
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| physical | XXVI, 545 S. graph. Darst. |
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| spelling | Cherkaev, Andrej 1950- Verfasser (DE-588)11560832X aut Variational methods for structural optimization Andrej Cherkaev New York [u.a.] Springer 2000 XXVI, 545 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 140 Strukturoptimierung (DE-588)4183811-7 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Strukturoptimierung (DE-588)4183811-7 s Variationsrechnung (DE-588)4062355-5 s DE-604 Applied mathematical sciences 140 (DE-604)BV000005274 140 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009053133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cherkaev, Andrej 1950- Variational methods for structural optimization Applied mathematical sciences Strukturoptimierung (DE-588)4183811-7 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4183811-7 (DE-588)4062355-5 |
| title | Variational methods for structural optimization |
| title_auth | Variational methods for structural optimization |
| title_exact_search | Variational methods for structural optimization |
| title_full | Variational methods for structural optimization Andrej Cherkaev |
| title_fullStr | Variational methods for structural optimization Andrej Cherkaev |
| title_full_unstemmed | Variational methods for structural optimization Andrej Cherkaev |
| title_short | Variational methods for structural optimization |
| title_sort | variational methods for structural optimization |
| topic | Strukturoptimierung (DE-588)4183811-7 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Strukturoptimierung Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009053133&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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