Principles of multiscale modeling:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2011
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Cover image Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XVII, 466 S. Ill., graph. Darst. |
| ISBN: | 9781107096547 |
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| adam_text | Titel: Principles of multiscale modeling
Autor: E, Weinan
Jahr: 2011
Contents
Preface page xii
1 Introduction 1
1.1 Examples of multiscale problems 1
1.1.1 Multiscale data and their representation 2
1.1.2 Differential equations with multiscale data 2
1.1.3 Differential equations with small parameters 4
1.2 Multi-physics problems 5
1.2.1 Examples of scale-dependent phenomena 5
1.2.2 Deficiencies of the traditional approaches
to modeling 7
1.2.3 The multi-physics modeling hierarchy 10
1.3 Analytical methods 12
1.4 Numerical methods 13
1.4.1 Linear scaling algorithms 13
1.4.2 Sublinear scaling algorithms 13
1.4.3 Type A and type B multiscale problems 14
1.4.4 Concurrent versus sequential coupling 15
1.5 What are the main challenges? 17
1.6 Notes 19
References for Chapter 1 21
2 Analytical methods 25
2.1 Matched asymptotics 26
2.1.1 A simple advection-diffusion equation 26
2.1.2 Boundary layers in incompressible
flows 28
V
vi
Contents
2.1.3 Structure and dynamics of shocks 30
2.1.4 Transition layers in the Allen-Cahn
equation 32
2.2 The WKB method 35
2.3 Averaging methods 37
2.3.1 Oscillatory problems 38
2.3.2 Stochastic ordinary differential equations 41
2.3.3 Stochastic simulation algorithms 4G
2.4 Multiscale expansions 54
2.4.1 Removing secular terms 54
2.4.2 Homogenization of elliptic equations 56
2.4.3 Homogenization of the Hamilton-Jacobi
equations 60
2.4.4 Flows in porous media 63
2.5 Scaling and self-similar solutions 64
2.5.1 Dimensional analysis 65
2.5.2 Self-similar solutions of PDEs 66
2.6 Renormalization group analysis 70
2.6.1 The Ising model and critical exponents 70
2.6.2 An illustration of the renormalization
transformation 74
2.6.3 Renormalization group analysis of the
two-dimensional Ising model 76
2.6.4 A PDE example 80
2.7 The Mori-Zwanzig formalism 82
2.8 Notes 86
References for Chapter 2 87
3 Classical multiscale algorithms 90
3.1 Multigrid method 90
3.2 Fast summation methods 99
3.2.1 Low-rank kernels 100
3.2.2 Hierarchical algorithms 103
3.2.3 The fast multipole method 107
3.3 Adaptive mesh refinement 110
3.3.1 A posteriori error estimates and local
error indicators 111
3.3.2 The moving mesh method 113
Contents vii
3.4 Domain decomposition methods 115
3.4.1 Nonoverlapping domain decomposition
methods 115
3.4.2 Overlapping domain decomposition methods 118
3.5 Multiscale representation 119
3.5.1 Hierarchical bases 120
3.5.2 Multi-resolution analysis and wavelet bases 122
3.5.3 Examples 124
3.6 Notes 129
References for Chapter 3 129
4 The hierarchy of physical models 132
4.1 Continuum mechanics 133
4.1.1 Stress and strain in solids 136
4.1.2 Variational principles in elasticity theory 138
4.1.3 Conservation laws 141
4.1.4 Dynamic theory of solids and thermoelasticity 144
4.1.5 Dynamics of fluids 147
4.2 Molecular dynamics 150
4.2.1 Empirical potentials 151
4.2.2 Equilibrium states and ensembles 156
4.2.3 The elastic continuum limit; the
Cauchy-Born rule 158
4.2.4 Nonequilibrium theory 163
4.2.5 Linear response theory and the Green-Kubo
formula 166
4.3 Kinetic theory 167
4.3.1 The BBGKY hierarchy 168
4.3.2 The Boltzmann equation 170
4.3.3 The equilibrium states 173
4.3.4 Macroscopic conservation laws 176
4.3.5 The hydrodynamic regime 178
4.3.6 Other kinetic models 181
4.4 Electronic structure models 181
4.4.1 The quantum many-body problem 182
4.4.2 Hartree and Hartree Fock approximations 185
4.4.3 Density functional theory 187
4.4.4 Tight-binding models 193
viii
Contents
4.5 Notes 198
References for Chapter 4 198
5 Examples of multi-physics models 201
5.1 Brownian dynamics models of polymer fluids 202
5.2 Extensions of the Cauchy-Born rule 210
5.2.1 High-order, exponential and local
Cauchy-Born rules 211
5.2.2 An example of a one-dimensional chain 211
5.2.3 Sheets and nanotubes 213
5.3 The moving contact line problem 216
5.3.1 Classical continuum theory 217
5.3.2 Improved continuum models 219
5.3.3 Measuring the boundary conditions using
molecular dynamics 224
5.4 Notes 227
References for Chapter 5 228
6 Capturing the macroscale behavior 232
6.1 Some classical examples 235
6.1.1 Car-Parrinello molecular dynamics 235
6.1.2 The quasicontinuum method 237
6.1.3 Kinetic schemes 239
6.1.4 Cloud-resolving convection parametrization 242
6.2 The multigrid and equation-free approaches 243
6.2.1 Extended multigrid method 243
6.2.2 The equation-free approach 245
6.3 The heterogeneous multiscale method 248
6.3.1 The main components of the method 248
6.3.2 Simulating gas dynamics using molecular
dynamics 252
6.3.3 The classical examples from the HMM
viewpoint 255
6.3.4 Modifying traditional algorithms to handle
multiscale problems 257
6.4 Some general remarks 258
6.4.1 Similarities and differences 258
6.4.2 Difficulties with the three approaches 260
6.5 Seamless coupling 262
Contents ix
6.6 Application to fluids 269
6.7 Stability, accuracy and efficiency 278
6.7.1 The heterogeneous multiscale method 279
6.7.2 The boosting algorithm 282
6.7.3 The equation-free approach 284
6.8 Notes 287
References for Chapter 6 291
7 Resolving local events or singularities 296
7.1 Domain decomposition method for type A problems 297
7.1.1 Energy-based formulation 298
7.1.2 Dynamic atomistic and continuum methods
for solids 301
7.1.3 Coupled atomistic and continuum methods
for fluids 302
7.2 Adaptive model refinement or model reduction 305
7.2.1 The nonlocal quasicontinuum method 306
7.2.2 Coupled gas-dynamic-kinetic models 310
7.3 The heterogeneous multiscale method 312
7.4 Stability issues 315
7.5 Consistency questions illustrated using nonlocal QC 320
7.5.1 The appearance of the ghost force 322
7.5.2 Removing the ghost force 323
7.5.3 Truncation error analysis 324
7.6 Notes 327
References for Chapter 7 329
8 Elliptic equations with multiscale coefficients 334
8.1 Introduction 334
8.2 Multiscale finite element methods 337
8.2.1 The generalized finite element method 337
8.2.2 Residual-free bubbles 338
8.2.3 Variational multiscale method 340
8.2.4 Multiscale basis functions 342
8.2.5 Relations between the various methods 344
8.3 Upscaling via successive elimination of fine-scale
components 345
8.4 Sublinear scaling algorithms 349
8.4.1 Finite element HMM 350
X
Contents
8.4.2 The local microscale problem 352
8.4.3 Error estimates 355
8.4.4 Information about the gradients 356
8.5 Notes 357
References for Chapter 8 363
9 Problems that have multiple time scales 368
9.1 ODEs with disparate time scales 368
9.1.1 General setup for limit theorems 368
9.1.2 Implicit methods 371
9.1.3 Stabilized Runge-Kutta methods 372
9.1.4 Heterogeneous multiscale method 378
9.2 Application of HMM to stochastic simulation
algorithms 379
9.3 Coarse-grained molecular dynamics 386
9.4 Notes 393
References for Chapter 9 393
10 Rare events 397
10.1 Introduction 397
10.2 Theoretical background 402
10.2.1 Metastable states and reduction to Markov
chains 402
10.2.2 Transition state theory 403
10.2.3 Large-deviation theory 405
10.2.4 First-exit times 409
10.2.5 Transition path theory 415
10.3 Numerical algorithms 425
10.3.1 Finding transition states 425
10.3.2 Finding the minimal energy path 427
10.3.3 Finding the transition path ensemble or the
transition tubes 433
10.4 Accelerated dynamics and sampling methods 438
10.4.1 TST-based acceleration techniques 438
10.4.2 Met adynamics 440
10.4.3 Temperature-accelerated molecular
dynamics 441
10.5 Notes 442
References for Chapter 10 443
Contents xi
11 Other perspectives 448
11.1 Open problems 448
11.1.1 Variational model reduction 450
11.1.2 Modeling memory effects 455
References for Chapter 11 459
Subject index 461
|
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| spelling | E, Weinan 1963- Verfasser (DE-588)139594116 aut Principles of multiscale modeling Weinan E 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2011 XVII, 466 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturangaben Multiscale modeling MATHEMATICS / General bisacsh Mehrskalenmodell (DE-588)7600619-0 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mehrskalenmodell (DE-588)7600619-0 s Differentialgleichung (DE-588)4012249-9 s DE-604 http://assets.cambridge.org/97811070/96547/cover/9781107096547.jpg Cover image HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024542649&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | E, Weinan 1963- Principles of multiscale modeling Multiscale modeling MATHEMATICS / General bisacsh Mehrskalenmodell (DE-588)7600619-0 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)7600619-0 (DE-588)4012249-9 |
| title | Principles of multiscale modeling |
| title_auth | Principles of multiscale modeling |
| title_exact_search | Principles of multiscale modeling |
| title_full | Principles of multiscale modeling Weinan E |
| title_fullStr | Principles of multiscale modeling Weinan E |
| title_full_unstemmed | Principles of multiscale modeling Weinan E |
| title_short | Principles of multiscale modeling |
| title_sort | principles of multiscale modeling |
| topic | Multiscale modeling MATHEMATICS / General bisacsh Mehrskalenmodell (DE-588)7600619-0 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Multiscale modeling MATHEMATICS / General Mehrskalenmodell Differentialgleichung |
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