 | Geiser J. Coupled systems: theory, models, and applications in engineering. - Boca Raton: CRC Press, 2014. - xxiv, 289 p.: ill. - (Chapman & Hall/CRC numerical analysis and scientific computing series). - Bibliogr.: p.257-283. - Ind.: p.285-289. - ISBN 978-1-4665-7801-2
Шифр: (И/Ж12-G33) 02
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List of Figures .............................................. xiii
List of Tables ................................................ xix
Introduction .................................................. xxi
Preface ..................................................... xxiii
1 Introduction ............................................... 1
1.1 Outline of the Book ........................................ 1
1.1.1 The Mathematical Part ............................... 2
1.1.2 The Algorithmic Part ................................ 3
1.1.3 The Practical Part .................................. 3
1.2 Coupled Systems as Interdisciplinary Research .............. 4
1.2.1 Embedding Coupled Systems to Engineering Research ... 6
1.2.2 Computational Engineering ........................... 6
1.2.3 Multiphysics ........................................ 7
1.2.4 Multiscale Modeling ................................. 8
1.2.5 Computational Sciences .............................. 8
1.2.6 Outline of the Monograph ............................ 9
2 General Principle for Coupled Systems ..................... 13
2.1 Coupling Analysis ......................................... 13
2.1.1 Decomposition Idea of Weakly Coupled Systems ...... 14
2.1.1.1 Decomposable Evolution Equations ................. 14
2.1.1.2 Weakly Decomposable Evolution Equations .......... 18
2.1.1.3 Non-Decomposable Evolution Equations ............. 19
2.2 Multiscale Analysis ....................................... 20
2.2.1 Multiscale Averaging (averaging fast scales) ....... 21
2.2.1.1 Averaging a Transport Problem .................. 21
2.2.1.2 Ordinary Differential Equations (kinetic
problems) ...................................... 23
2.2.1.3 Stochastic Ordinary Differential Equations ..... 24
2.2.2 Multiscale Expansion (embedding of the fast
scales) ............................................ 25
2.2.3 Self-Similar Solutions (embedding self-similar
scales) ............................................ 30
3 Numerical Methods ......................................... 35
3.1 Classical Methods ......................................... 35
3.1.1 Multigrid Methods .................................. 35
3.1.2 Iterative Splitting Methods ........................ 38
3.1.2.1 Multi-Iteration Idea, Developing the
Expansion ...................................... 41
3.1.3 Multiresolution: Wavelet Ideas ..................... 44
3.2 Modern Methods ............................................ 49
3.2.1 Iterative Splitting Method with Embedded
Multigrid Method ................................... 49
3.2.1.1 Error Analysis of the Multiscale Method ........ 51
3.2.2 Multiscale Iterative Splitting methods ............. 53
3.2.2.1 Error Analysis for the Multiscale Iterative
Splitting Method ............................... 55
4 Applications .............................................. 59
4.1 Applications to Multiscale Expansions ..................... 59
4.1.1 Application for an Asymmetric Rigid Body
(Levitron) ......................................... 61
4.1.1.1 Model Problem .................................. 61
4.1.1.2 Multiscale Analysis ............................ 64
4.1.1.3 Numerical Results with the Multiscale
Equations ...................................... 69
4.2 Non-linear Reaction Example: Averaging .................... 71
4.3 PECVD-Process: Upscaled Reaction Process .................. 72
4.3.1 Numerical Experiment ............................... 76
4.4 Stochastic Differential Equations: Particle Simulation
for Coulomb Collisions .................................... 79
4.4.1 Model Problem ...................................... 79
4.4.2 Application to a Scalar Langevin Equation .......... 81
4.4.3 Coulomb Test Particle Problem (vectorial problem
of the Langevin equations) ......................... 83
4.5 Particle-In-Cell: Multiscale Method with Applications ..... 88
4.5.1 Mathematical Model for a Simple Plasma Model ....... 89
4.5.2 Error Estimates for the Full PIC Cycle ............. 91
4.5.3 1D Error Estimates for Adaptive Grids .............. 93
4.5.4 Numerical Example: a Many-Particle Experiment
with ID PIC Code ................................... 95
4.6 Application to Multiscale Problem in Transport-Reaction
Problems .................................................. 98
4.6.1 Multiscale Methods and Assembling of the
Splitting and Multigrid Method ..................... 99
4.6.1.1 Multilevel and Multigrid Method ................ 99
4.6.2 Numerical Experiments for the Embedded Methods .... 104
4.6.2.1 Heat Equation ................................. 104
4.6.2.2 Transport-Reaction Equation ................... 105
4.7 Application to Multiscale Problem in Heat Transfer in
Porous Media ............................................. 110
4.7.1 Multiscale Modeling ............................... 110
4.7.1.1 Flow Field .................................... 111
4.7.1.2 Transport Systems (multiphase equations) ...... 111
4.7.2 Discretization and Solver Methods ................. 114
4.7.3 Numerical Simulations of the Heat-Flow Problem .... 116
4.7.3.1 Benchmark Problem: Two-Phase Example .......... 116
4.7.3.2 Parameters of the Model Equations ............. 118
4.7.3.3 Temperature in an Underlying Rock with
Permeable and Less Permeable Layers ........... 119
4.8 Application to a Multiscale Problem in Porous Media
Based on a Model of a Parallel Plate PECVD Apparatus ..... 123
4.8.1 Multiscale Model .................................. 124
4.8.1.1 Model for Small Knudsen Numbers (far-field
model) ........................................ 124
4.8.1.2 Model for Large Knudsen Numbers (near-field
model) ........................................ 126
4.8.1.3 Simplified Model for Large Knudsen Numbers
(near-field model) ............................ 127
4.8.2 Numerical Methods: Multiscale Solvers ............. 128
4.8.2.1 Embedding of Analytical Solution of Reaction
Equations ..................................... 128
4.8.3 Approximation to the Real-Life Experiment ......... 130
4.8.4 Numerical Experiments of the Deposition Process ... 132
4.8.4.1 Flow Field Experiments ........................ 132
4.8.4.2 Delicate Geometries ........................... 133
4.8.4.3 Regression Experiments ........................ 134
4.9 Monte Carlo Simulations Concerning Modeling DC and High
Power Pulsed Magnetron Sputtering ........................ 137
4.9.1 Mathematical Model ................................ 139
4.9.1.1 Ideal and Real Gases .......................... 139
4.9.2 Scattering from a Screened Coulomb Potential
(ion-ion interaction) ............................. 141
4.9.2.1 Implantation Model ............................ 142
4.9.3 Monte Carlo Simulations of the Sputter Process .... 145
4.9.3.1 Sputtering from Target ........................ 145
4.9.3.2 DC Sputtering ................................. 145
4.9.3.3 HIPIMS Sputtering ............................. 147
4.9.3.4 Delicate Deposition Geometries ................ 149
4.10 Splitting Methods as Coupling Schemes: Theory and
Application to Electro-Magnetic Fields ................... 153
4.10.1 Mathematical Model ................................ 153
4.10.2 Numerical Methods ................................. 154
4.10.1 Discretization of the Maxwell Equation: Yee's
Scheme ............................................ 154
4.10.2.2 Discretization of the Momentum Equation ....... 155
4.10.2.3 Multiscale Method: Coupling of the Equations .. 156
4.10.3 Numerical Experiments ............................. 157
4.10.4 Test Experiment 1: Pure Maxwell Equation .......... 157
4.10.4.1 Test Example 2: Pure Momentum Equation
(molecular flow) .............................. 159
4.10.4.2 Test Example 3: Coupled Momentum and
Maxwell Equations ............................. 161
4.11 Improvement of Multiscale Methods via Zassenhaus
Expansion: Theory and Application to Multiphase
Problems ................................................. 166
4.11.1 Modelling and Numerical Motivation ................ 166
4.11.2 Splitting Methods ................................. 168
4.11.2.1 Basic Algorithm: Iterative Splitting Method ... 168
4.11.2.2 Embedded Algorithm: Zassenhaus Formula ........ 170
4.11.2.3 Extended Algorithm: Iterative Splitting with
Zassenhaus Formula ............................ 171
4.11.3 Numerical Examples ................................ 172
4.11.3.1 One-Phase Example ............................. 172
4.11.3.2 Two-Phase Example ............................. 175
4.12 Improvement of Multiscale Methods via Disentanglement
of Exponential Operators ................................. 180
4.12.1 Modelling Problems ................................ 180
4.12.2 Iterative Splitting Methods ....................... 181
4.12.3 Improvement via Zassenhaus Formula ................ 182
4.12.4 Disentanglement of Exponential Operators .......... 182
4.12.5 Numerical Examples ................................ 184
4.12.6 Test Example: Finite Difference Operators ......... 184
4.12.7 Test-Example: Multidimensional Finite Difference
Operators ......................................... 188
4.13 Multiscale Problem with Embedded Analytical Solutions
of the Micro-Scale Part .................................. 193
4.13.1 Introduction to the Multiscale Model of Time-
Dependent Transport Problems ...................... 193
4.13.2 Mathematical Model ................................ 194
4.13.3 Functional Splitting I: Analytical Solutions of
the Microscopic Equations ......................... 196
4.13.4 Functional Splitting II: Analytical Solutions of
the Macroscopic Equations ......................... 200
4.13.5 Transport Part: Time-Dependent Convection-
Diffusion Equations ............................... 200
4.13.6 Multiphase Part: Mobile and Immobile Sub-
Problems .......................................... 201
4.13.6.1 Coupling Convection and Reaction Parts ........ 201
4.13.6.2 The Iterative Splitting Scheme ................ 202
4.13.6.3 Analytical Solutions of the Decoupled
Sub-Problems .................................. 202
4.13.6.4 Iterative Coupling of the Decoupled Sub-
Problems ...................................... 204
4.13.6.5 Coupling Convection-Diffusion Equations and
Reaction Equations ............................ 205
4.13.6.6 Successive Approximation Scheme ............... 205
4.13.6.7 Transformed Analytical Solutions of the
Decoupled Sub-Problems ........................ 206
4.13.6.8 Successive Coupling of the Decoupled Sub-
Problems ...................................... 206
4.13.7 Numerical Experiments ............................. 207
4.13.7.1 First Benchmark Experiment: Multispecies
Convection-Reaction Equation .................. 207
4.13.7.2 Second Benchmark Experiment: Convection-
Reaction Equation with General Initial
Conditions .................................... 208
4.14 Multiscale Approaches to Solve Time-Dependent Burgers'
Equations ................................................ 213
4.14.1 Motivation to the Multiscale Approach ............ 213
4.14.2 Meshless Radial Basis Functions .................. 213
4.14.3 Application of the RBFs to Partial Differential
Equations ........................................ 214
4.14.4 Prewavelets and Multiquadratic Convergence ....... 216
4.14.5 Decomposition Method: Notations .................. 217
4.14.6 16 Cubes ......................................... 218
4.14.7 Boundary Conditions (Surfaces) ................... 219
4.14.8 Overlapping Cubes ................................ 219
4.14.9 Decomposition Method: Alternating Schwarz
Waveform Relaxation .............................. 220
4.14.10 Model Four-Dimensional Problem ................... 221
4.15 Step-Size Control in Simulation of Diffusive CVD
Processes Based on Adaptive Schemes ...................... 224
4.15.1 Introduction to the Multiscale Model of an
Optimal Control Problem ........................... 225
4.15.2 Approximation and Discretization .................. 226
4.15.3 Optimal Control Methods ........................... 227
4.15.3.1 Forward Controller (simple P-controller) ...... 227
4.15.3.2 PID Controller ................................ 228
4.15.3.3 Adaptive Time Control ......................... 231
4.15.4 Experiment for the CVD Process .................... 231
4.15.4.1 Simulation of an Optimal Control of
a Diffusion Equation with Heuristic Choice
of the Control Parameters ..................... 231
4.15.4.2 Simulation of an Optimal Control of
a Diffusion Equation with Adaptive Control .... 233
5 Summary and Perspectives ................................. 239
6 Software Tools ........................................... 241
6.1 Software Package r3t ..................................... 241
6.1.1 Model Equation in r3t: Transport Model of Mobile
Immobile and Adsorbed Zones ....................... 241
6.1.2 Conception of r3t ................................. 242
6.1.3 Application of r3t ................................ 243
6.2 Benchmark Software: MULTI-OPERA .......................... 244
6.2.1 Fluid Problems (authors: J. Geiser and
Th. Zacher) ....................................... 244
6.2.2 Stochastic Differential Equations (authors:
J. Geiser and Th. Zacher) ......................... 245
6.2.3 Improvement of Multiscale Methods via Zassenhaus
Expansion (authors: J. Geiser and Th. Zacher) ..... 246
6.2.4 Maxwell Solver: Coupling Schemes Applied to
Electro-Magnetic Fields (authors: J. Geiser and
Th. Zacher) ....................................... 247
6.2.5 Multiphase Solver: Splitting Schemes Applied to
Multi-phase Problems (authors: J. Geiser and
Th. Zacher) ....................................... 248
Appendix ...................................................... 251
List of Abbreviations ......................................... 251
Symbols ....................................................... 252
General Notations ............................................. 254
Notations in the Models ....................................... 255
Bibliography .................................................. 257
Index ......................................................... 285
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