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ОбложкаGeiser 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
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
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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