Multiscale Methods: Bridging the Scales in Science and Engineering

Contents......Page 10
Preface......Page 6
List of Contributors......Page 22
I: INFORMATION-PASSING MULTISCALE METHODS IN SPACE......Page 26
1.1 Introduction......Page 28
1.2 Preliminaries and motivation......Page 30
1.3 Mixed multiscale finite element method for flow equations......Page 32
1.4 Numerical results......Page 35
1.5 Analysis of global mixed MsFEM on unstructured grid......Page 41
1.6 Conclusions......Page 51
References......Page 52
2.1 Introduction......Page 56
2.2 The geometry of self-similar structures......Page 60
2.3 Basic concepts in fractional calculus......Page 61
2.4 Mechanics boundary value problems on materials with fractal microstructure......Page 66
2.5 Closure......Page 78
References......Page 79
3.1 Introduction......Page 82
3.2 The N-scale mathematical homogenization......Page 83
3.3 Residual-free governing equations at multiple scales......Page 86
3.4 Multiple-scale reduced order model......Page 88
3.5 Reduced order unit cell problems......Page 95
3.6 Summary of the computational procedure......Page 101
3.7 Example: three scale analysis......Page 103
3.8 Appendix......Page 111
References......Page 112
II: CONCURRENT MULTISCALE METHODS IN SPACE......Page 116
4.1 Introduction......Page 118
4.2 Classification of methods......Page 121
4.3 Mechanical equilibrium methods......Page 123
4.4 Molecular dynamics systems......Page 138
4.5 Numerical results......Page 144
References......Page 155
5.1 Coupling atomistic and continuum length scales......Page 159
5.2 Coarse graining formalism......Page 162
5.3 The CGMD Hamiltonian......Page 164
5.4 Non-equilibrium effects......Page 169
5.5 Practical details......Page 171
5.6 Performance tests......Page 173
5.7 Conclusion......Page 185
References......Page 186
6.1 Introduction......Page 190
6.2 Atomistic and continuum models......Page 193
6.3 Force-based blending......Page 197
6.4 Energy-based blending......Page 207
6.5 Generalized continua......Page 210
References......Page 211
III: SPACE-TIME SCALE BRIDGING METHODS......Page 216
7.1 Introduction......Page 218
7.2 Systematic upscaling (SU): an outline......Page 224
7.3 Derivation of coarse equations......Page 227
7.4 Window developments......Page 230
7.5 Some special situations......Page 231
7.6 Extensions in brief......Page 232
References......Page 235
8.1 Introduction......Page 241
8.2 Equation-free multiscale framework......Page 242
8.3 Model homogenization problems......Page 246
8.4 The gap-tooth scheme......Page 248
8.5 Patch dynamics with buffers......Page 253
8.6 General tooth boundary conditions......Page 263
8.8 Acknowledgments......Page 267
References......Page 268
9.1 Introduction......Page 272
9.2 The reference problem......Page 274
9.3 Reformulation of the problem with structure decomposition......Page 277
9.4 Multiscale description in the time-space domain[0, T] × Ω......Page 281
9.5 The multiscale computational strategy......Page 283
9.6 The radial time-space approximation......Page 291
9.7 Conclusions......Page 302
References......Page 303
IV: ADAPTIVITY, ERROR ESTIMATION AND UNCERTAINTY QUANTIFICATION......Page 308
10.1 Problem setting......Page 310
10.2 The general theory of modeling error estimation......Page 312
10.3 A large-scale molecular statics model......Page 314
10.4 The family of six algorithms governing multiscale modeling of polymer densification......Page 315
10.5 Representative results......Page 321
10.6 Extensions......Page 324
References......Page 328
11.1 Introduction......Page 330
11.2 The key is stability. But what is stability . . . and stability of what?......Page 338
11.3 The tools for quantifying stability properties: functionals, duality, and adjoint operators......Page 348
11.4 A posteriori error analysis using adjoints......Page 359
11.5 A posteriori error estimates and adaptive mesh refinement......Page 376
11.6 Multiscale operator decomposition......Page 383
11.7 The effect of iteration......Page 410
11.8 Conclusion......Page 411
References......Page 412
V: MULTISCALE SOFTWARE......Page 416
12.1 Introduction......Page 418
12.2 Abstraction of adaptive multiscale simulation......Page 419
12.3 Multiscale simulation components and structures......Page 424
12.4 Component tools for crystaline materials......Page 430
12.5 A concurrent atomistic/continuum adaptive multiscale simulation tool......Page 435
References......Page 442
VI: SELECTED MULTISCALE APPLICATIONS......Page 448
13.1 Introduction......Page 450
13.2 Finite temperature QC method......Page 452
13.3 Local phonon density of states approach......Page 479
13.4 Conclusion......Page 500
References......Page 501
14.1 Materials modeling and simulation (computational materials)......Page 506
14.2 Atomistic measures of strength and deformation......Page 511
14.3 Atomistic measure of defect mobility......Page 525
14.5 Acknowledgment......Page 535
References......Page 536
15.1 Introduction......Page 539
15.2 Chromatin structure......Page 540
15.3 The first-generation macroscopic chromatin models and results......Page 543
15.4 The second-generation mesoscopic chromatin model and results......Page 545
15.5 Future perspective......Page 554
References......Page 556
16.1 Introduction......Page 561
16.2 Resolving multiscale patterns in heterogeneous systems......Page 563
16.3 Imposing multiscale structure using a nature-inspired approach......Page 570
16.4 Concluding remarks......Page 578
References......Page 579
A......Page 586
B......Page 588
C......Page 590
D......Page 593
E......Page 597
F......Page 599
G......Page 602
H......Page 603
I......Page 604
K......Page 605
L......Page 606
M......Page 607
N......Page 610
O......Page 611
P......Page 612
Q......Page 614
R......Page 615
S......Page 616
T......Page 620
V......Page 622
Z......Page 623