Coupled Systems: Theory, Models, and Applications in Engineering
[Book Description]
Efficient Methods to Solve Complex Coupled Systems Coupled SystemsTheory, Models, and Applications in Engineering explains how to solve complicated coupled models in engineering using analytical and numerical methods. It presents splitting multiscale methods to solve multiscale and multiphysics problems and describes analytical and numerical methods in time and space for evolution equations arising in engineering problems. The book discusses the effectiveness, simplicity, stability, and consistency of the methods in solving problems that occur in real-life engineering tasks. It shows how MATLAB(R) and Simulink(R) are used to implement the methods. The author also covers the coupling of separate, multiple, and logical scales in applications, including microscale, macroscale, multiscale, and multiphysics problems. Covering mathematical, algorithmic, and practical aspects, this book brings together innovative ideas in coupled systems and extends standard engineering tools to coupled models in materials and flow problems with respect to their scale dependencies and their influence on each time and spatial scale.
[Table of Contents]
List of Figures xiii
List of Tables xix
Introduction xxi
Preface xxiii
1 Introduction 1 (12)
1.1 Outline of the Book 1 (3)
1.1.1 The Mathematical Part 2 (1)
1.1.2 The Algorithmic Part 3 (1)
1.1.3 The Practical Part 3 (1)
1.2 Coupled Systems as Interdisciplinary 4 (9)
Research
1.2.1 Embedding Coupled Systems to 6 (1)
Engineering Research
1.2.2 Computational Engineering 6 (1)
1.2.3 Multiphysics 7 (1)
1.2.4 Multiscale Modeling 8 (1)
1.2.5 Computational Sciences 8 (1)
1.2.6 Outline of the Monograph 9 (4)
2 General Principle for Coupled Systems 13 (22)
2.1 Coupling Analysis 13 (7)
2.1.1 Decomposition Idea of Weakly Coupled 14 (6)
Systems
2.1.1.1 Decomposable Evolution Equations 14 (4)
2.1.1.2 Weakly Decomposable Evolution 18 (1)
Equations
2.1.1.3 Non-Decomposable Evolution 19 (1)
Equations
2.2 Multiscale Analysis 20 (15)
2.2.1 Multiscale Averaging (averaging fast 21 (4)
scales)
2.2.1.1 Averaging a Transport Problem 21 (2)
2.2.1.2 Ordinary Differential Equations 23 (1)
(kinetic problems)
2.2.1.3 Stochastic Ordinary Differential 24 (1)
Equations
2.2.2 Multiscale Expansion (embedding of 25 (5)
the fast scales)
2.2.3 Self-Similar Solutions (embedding 30 (5)
self-similar scales)
3 Numerical Methods 35 (24)
3.1 Classical Methods 35 (14)
3.1.1 Multigrid Methods 35 (3)
3.1.2 Iterative Splitting Methods 38 (6)
3.1.2.1 Multi-Iteration Idea, Developing 41 (3)
the Expansion
3.1.3 Multiresolution: Wavelet Ideas 44 (5)
3.2 Modern Methods 49 (10)
3.2.1 Iterative Splitting Method with 49 (4)
Embedded Multigrid Method
3.2.1.1 Error Analysis of the Multiscale 51 (2)
Method
3.2.2 Multiscale Iterative Splitting methods 53 (6)
3.2.2.1 Error Analysis for the Multiscale 55 (4)
Iterative Splitting Method
4 Applications 59 (180)
4.1 Applications to Multiscale Expansions 59 (12)
4.1.1 Application for an Asymmetric Rigid 61 (11)
Body (Levitron)
4.1.1.1 Model Problem 61 (3)
4.1.1.2 Multiscale Analysis 64 (5)
4.1.1.3 Numerical Results with the 69 (2)
Multiscale Equations
4.2 Non-linear Reaction Example: Averaging 71 (1)
4.3 PECVD-Process: Upscaled Reaction Process 72 (7)
4.3.1 Numerical Experiment 76 (3)
4.4 Stochastic Differential Equations: 79 (9)
Particle Simulation for Coulomb Collisions
4.4.1 Model Problem 79 (2)
4.4.2 Application to a Scalar Langevin 81 (2)
Equation
4.4.3 Coulomb Test Particle Problem 83 (5)
(vectorial problem of the Langevin
equations)
4.5 Particle-In-Cell: Multiscale Method with 88 (10)
Applications
4.5.1 Mathematical Model for a Simple 89 (2)
Plasma Model
4.5.2 Error Estimates for the Full PIC Cycle 91 (2)
4.5.3 1D Error Estimates for Adaptive Grids 93 (2)
4.5.4 Numerical Example: a Many-Particle 95 (3)
Experiment with 1D PIC Code
4.6 Application to Multiscale Problem in 98 (12)
Transport-Reaction Problems
4.6.1 Multiscale Methods and Assembling of 99 (5)
the Splitting and Multigrid Method
4.6.1.1 Multilevel and Multigrid Method 99 (5)
4.6.2 Numerical Experiments for the 104 (6)
Embedded Methods
4.6.2.1 Heat Equation 104 (1)
4.6.2.2 Transport-Reaction Equation 105 (5)
4.7 Application to Multiscale Problem in Heat 110 (13)
Transfer in Porous Media
4.7.1 Multiscale Modeling 110 (4)
4.7.1.1 Flow Field 111 (1)
4.7.1.2 Transport Systems (multiphase 111 (3)
equations)
4.7.2 Discretization and Solver Methods 114 (2)
4.7.3 Numerical Simulations of the 116 (7)
Heat-Flow Problem
4.7.3.1 Benchmark Problem: Two-Phase 116 (2)
Example
4.7.3.2 Parameters of the Model Equations 118 (1)
4.7.3.3 Temperature in an Underlying Rock 119 (4)
with Permeable and Less Permeable Layers
4.8 Application to a Multiscale Problem in 123 (14)
Porous Media Based on a Model of a Parallel
Plate PECVD Apparatus
4.8.1 Multiscale Model 124 (4)
4.8.1.1 Model for Small Knudsen Numbers 124 (2)
(far-field model)
4.8.1.2 Model for Large Knudsen Numbers 126 (1)
(near-field model)
4.8.1.3 Simplified Model for Large 127 (1)
Knudsen Numbers (near-field model)
4.8.2 Numerical Methods: Multiscale Solvers 128 (2)
4.8.2.1 Embedding of Analytical Solution 128 (2)
of Reaction Equations
4.8.3 Approximation to the Real-Life 130 (2)
Experiment
4.8.4 Numerical Experiments of the 132 (5)
Deposition Process
4.8.4.1 Flow Field Experiments 132 (1)
4.8.4.2 Delicate Geometries 133 (1)
4.8.4.3 Regression Experiments 134 (3)
4.9 Monte Carlo Simulations Concerning 137 (16)
Modeling DC and High Power Pulsed Magnetron
Sputtering
4.9.1 Mathematical Model 139 (2)
4.9.1.1 Ideal and Real Gases 139 (2)
4.9.2 Scattering from a Screened Coulomb 141 (4)
Potential (ion-ion interaction)
4.9.2.1 Implantation Model 142 (3)
4.9.3 Monte Carlo Simulations of the 145 (8)
Sputter Process
4.9.3.1 Sputtering from Target 145 (1)
4.9.3.2 DC Sputtering 145 (2)
4.9.3.3 HIPIMS Sputtering 147 (2)
4.9.3.4 Delicate Deposition Geometries 149 (4)
4.10 Splitting Methods as Coupling Schemes: 153 (13)
Theory and Application to Electro-Magnetic
Fields
4.10.1 Mathematical Model 153 (1)
4.10.2 Numerical Methods 154 (3)
4.10.2.1 Discretization of the Maxwell 154 (1)
Equation: Yee's Scheme
4.10.2.2 Discretization of the Momentum 155 (1)
Equation
4.10.2.3 Multiscale Method: Coupling of 156 (1)
the Equations
4.10.3 Numerical Experiments 157 (1)
4.10.4 Test Experiment 1: Pure Maxwell 157 (9)
Equation
4.10.4.1 Test Example 2: Pure Momentum 159 (2)
Equation (molecular flow)
4.10.4.2 Test Example 3: Coupled Momentum 161 (5)
and Maxwell Equations
4.11 Improvement of Multiscale Methods via 166 (14)
Zassenhaus Expansion: Theory and Application
to Multiphase Problems
4.11.1 Modelling and Numerical Motivation 166 (2)
4.11.2 Splitting Methods 168 (4)
4.11.2.1 Basic Algorithm: Iterative 168 (2)
Splitting Method
4.11.2.2 Embedded Algorithm: Zassenhaus 170 (1)
Formula
4.11.2.3 Extended Algorithm: Iterative 171 (1)
Splitting with Zassenhaus Formula
4.11.3 Numerical Examples 172 (8)
4.11.3.1 One-Phase Example 172 (3)
4.11.3.2 Two-Phase Example 175 (5)
4.12 Improvement of Multiscale Methods via 180 (13)
Disentanglement of Exponential Operators
4.12.1 Modelling Problems 180 (1)
4.12.2 Iterative Splitting Methods 181 (1)
4.12.3 Improvement via Zassenhaus Formula 182 (1)
4.12.4 Disentanglement of Exponential 182 (2)
Operators
4.12.5 Numerical Examples 184 (1)
4.12.6 Test Example: Finite Difference 184 (4)
Operators
4.12.7 Test-Example: Multidimensional 188 (5)
Finite Difference Operators
4.13 Multiscale Problem with Embedded 193 (20)
Analytical Solutions of the Micro-Scale Part
4.13.1 Introduction to the Multiscale Model 193 (1)
of Time-Dependent Transport Problems
4.13.2 Mathematical Model 194 (2)
4.13.3 Functional Splitting I: Analytical 196 (4)
Solutions of the Microscopic Equations
4.13.4 Functional Splitting II: Analytical 200 (1)
Solutions of the Macroscopic Equations
4.13.5 Transport Part: Time-Dependent 200 (1)
Convection-Diffusion Equations
4.13.6 Multiphase Part: Mobile and Immobile 201 (6)
Sub-Problems
4.13.6.1 Coupling Convection and Reaction 201 (1)
Parts
4.13.6.2 The Iterative Splitting Scheme 202 (1)
4.13.6.3 Analytical Solutions of the 202 (2)
Decoupled Sub-Problems
4.13.6.4 Iterative Coupling of the 204 (1)
Decoupled Sub-Problems
4.13.6.5 Coupling Convection-Diffusion 205 (1)
Equations and Reaction Equations
4.13.6.6 Successive Approximation Scheme 205 (1)
4.13.6.7 Transformed Analytical Solutions 206 (1)
of the Decoupled Sub-Problems
4.13.6.8 Successive Coupling of the 206 (1)
Decoupled Sub-Problems
4.13.7 Numerical Experiments 207 (6)
4.13.7.1 First Benchmark Experiment: 207 (1)
Multispecies Convection-Reaction Equation
4.13.7.2 Second Benchmark Experiment: 208 (5)
Convection-Reaction Equation with General
Initial Conditions
4.14 Multiscale Approaches to Solve 213 (11)
Time-Dependent Burgers' Equations
4.14.1 Motivation to the Multiscale Approach 213 (1)
4.14.2 Meshless Radial Basis Functions 213 (1)
4.14.3 Application of the RBFs to Partial 214 (2)
Differential Equations
4.14.4 Prewavelets and Multiquadratic 216 (1)
Convergence
4.14.5 Decomposition Method: Notations 217 (1)
4.14.6 16 Cubes 218 (1)
4.14.7 Boundary Conditions (Surfaces) 219 (1)
4.14.8 Overlapping Cubes 219 (1)
4.14.9 Decomposition Method: Alternating 220 (1)
Schwarz Waveform Relaxation
4.14.10 Model Four-Dimensional Problem 221 (3)
4.15 Step-Size Control in Simulation of 224 (17)
Diffusive CVD Processes Based on Adaptive
Schemes
4.15.1 Introduction to the Multiscale Model 225 (1)
of an Optimal Control Problem
4.15.2 Approximation and Discretization 226 (1)
4.15.3 Optimal Control Methods 227 (4)
4.15.3.1 Forward Controller (simple 227 (1)
P-controller)
4.15.3.2 PID Controller 228 (3)
4.15.3.3 Adaptive Time Control 231 (1)
4.15.4 Experiment for the CVD Process 231 (10)
4.15.4.1 Simulation of an Optimal Control 231 (2)
of a Diffusion Equation with Heuristic
Choice of the Control Parameters
4.15.4.2 Simulation of an Optimal Control 233 (6)
of a Diffusion Equation with Adaptive
Control
5 Summary and Perspectives 239 (2)
6 Software Tools 241 (10)
6.1 Software Package rｳt 241 (3)
6.1.1 Model Equation in rｳt: Transport 241 (1)
Model of Mobile Immobile and Adsorbed Zones
6.1.2 Conception of rｳt 242 (1)
6.1.3 Application of rｳt 243 (1)
6.2 Benchmark Software: MULTI-OPERA 244 (7)
6.2.1 Fluid Problems (authors: J. Geiser 244 (1)
and Th. Zacher)
6.2.2 Stochastic Differential Equations 245 (1)
(authors: J. Geiser and Th. Zacher)
6.2.3 Improvement of Multiscale Methods via 246 (1)
Zassenhaus Expansion (authors: J. Geiser
and Th. Zacher)
6.2.4 Maxwell Solver: Coupling Schemes 247 (1)
Applied to Electro-Magnetic Fields
(authors: J. Geiser and Th. Zacher)
6.2.5 Multiphase Solver: Splitting Schemes 248 (3)
Applied to Multi-phase Problems (authors:
J. Geiser and Th. Zacher)
Appendix 251 (6)
List of Abbreviations 251 (1)
Symbols 252 (2)
General Notations 254 (1)
Notations in the Models 255 (2)
Bibliography 257 (28)
Index 285
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