Wave and Scattering Methods for Numerical Simulation

Preface xi

Foreword xv

1 Introduction 1

1.1 An Overview of Scattering Methods 3

1.1.1 Remarks on Passivity 3

1.1.2 Case Study: The Kelly–Lochbaum Digital Speech Synthesis Model 4

1.1.3 Digital Waveguide Networks 12

1.1.4 A General Approach: Multidimensional Circuit Representations and Wave Digital Filters 18

1.2 Questions 24

2 Wave Digital Filters 25

2.1 Classical Network Theory 27

2.1.1 N-ports 27

2.1.2 Power and Passivity 28

2.1.3 Kirchhoff’s Laws 30

2.1.4 Circuit Elements 31

2.2 Wave Digital Elements and Connections 32

2.2.1 The Bilinear Transform 33

2.2.2 Wave Variables 35

2.2.3 Pseudopower and Pseudopassivity 36

2.2.4 Wave Digital Elements 37

2.2.5 Adaptors 41

2.2.6 Signal and Coefficient Quantization 43

2.2.7 VectorWave Variables 45

2.3 Wave Digital Filters and Finite Differences 48

3 Multidimensional Wave Digital Networks 53

3.1 Symmetric Hyperbolic Systems 55

3.2 Coordinate Changes and Grid Generation 60

3.2.1 Structure of Coordinate Changes 61

3.2.2 Coordinate Changes in (1 +1)D 61

3.2.3 Coordinate Changes in Higher Dimensions 62

3.3 MD-passivity 65

3.4 MD Circuit Elements 68

3.4.1 The MD Inductor 68

3.4.2 Other MD Elements 70

3.4.3 Discretization in the Spectral Domain 71

3.4.4 Other Spectral Mappings 73

3.5 The (1 + 1)D Advection Equation 74

3.5.1 A Multidimensional Kirchhoff Circuit 75

3.5.2 Stability 76

3.5.3 An Upwind Form 77

3.6 The (1 +1)D Transmission Line 79

3.6.1 MDKC for the (1 + 1)D Transmission Line Equations 80

3.6.2 Digression: The Inductive Lattice Two-port 82

3.6.3 Energetic Interpretation 83

3.6.4 An MDWD Network for the (1 + 1)D Transmission Line 83

3.6.5 Simplified Networks 85

3.7 The (2 +1)D Parallel-plate System 86

3.7.1 MDKC and MDWD Network 87

3.8 Finite Difference Interpretation 89

3.8.1 MDWD Networks as Multistep Schemes 90

3.8.2 Numerical Phase Velocity and Parasitic Modes 93

3.9 Initial Conditions 97

3.10 Boundary Conditions 99

3.10.1 MDKC Modeling of Boundaries 101

3.11 Balanced Forms 105

3.12 Higher-order Accuracy 108

4 Digital Waveguide Networks 115

4.1 FDTD and TLM 117

4.2 Digital Waveguides  118

4.2.1 The Bidirectional Delay Line 118

4.2.2 Impedance 119

4.2.3 Wave Equation Interpretation 120

4.2.4 Note on the Different Definitions of Wave Quantities 121

4.2.5 Scattering Junctions 122

4.2.6 Vector Waveguides and Scattering Junctions 124

4.2.7 Transitional Note 126

4.3 The (1 +1)D Transmission Line 127

4.3.1 First-order System and the Wave Equation 127

4.3.2 Centered Difference Schemes and Grid Decimation 127

4.3.3 A (1+1)D Waveguide Network 129

4.3.4 Waveguide Network and the Wave Equation 131

4.3.5 An Interleaved Waveguide Network 133

4.3.6 Varying Coefficients 135

4.3.7 Incorporating Losses and Sources 141

4.3.8 Numerical Phase Velocity and Dispersion 143

4.3.9 Boundary Conditions 144

4.4 The (2 +1)D Parallel-plate System . 146

4.4.1 Defining Equations and Centered Differences 146

4.4.2 The Waveguide Mesh 149

4.4.3 Reduced Computational Complexity and Memory Requirements in the Standard Form of the Waveguide Mesh 156

4.4.4 Boundary Conditions 158

4.5 Initial Conditions 162

4.6 Music and Audio Applications of Digital Waveguides 164

5 Extensions of Digital Waveguide Networks 169

5.1 Alternative Grids in (2 +1)D  169

5.1.1 Hexagonal and Triangular Grids 170

5.1.2 The Waveguide Mesh in Radial Coordinates 173

5.2 The (3 + 1)D Wave Equation and Waveguide Meshes 180

5.3 The Waveguide Mesh in General Curvilinear Coordinates 182

5.4 Interfaces between Grids 186

5.4.1 Doubled Grid Density Across an Interface 187

5.4.2 Progressive Grid Density Doubling 193

5.4.3 Grid Density Quadrupling 196

5.4.4 Connecting Rectilinear and Radial Grids 198

5.4.5 Grid Density Doubling in (3 +1)D 202

5.4.6 Note 203

6 Scattering Methods: A Unified Perspective 205

6.1 The (1 +1)D Transmission Line Revisited 206

6.1.1 Multidimensional Unit Elements 207

6.1.2 Hybrid Form of the Multidimensional Unit Element 208

6.1.3 Alternative MDKC for the (1+1) D Transmission Line 210

6.2 Alternative MDKC for the (2 + 1) D Parallel-plate System 212

6.3 Higher-order Accuracy Revisited 214

6.4 Maxwell’s Equations 217

7 Applications to Vibrating Systems 223

7.1 Beam Dynamics 224

7.1.1 MDKC and MDWD network for Timoshenko’s System 226

7.1.2 Waveguide Network for Timoshenko’s System 228

7.1.3 Boundary Conditions in the DWN 230

7.1.4 Simulation: Timoshenko’s System for Beams of Uniform and Varying Cross-sectional Areas 232

7.1.5 Improved MDKC for Timoshenko’s System via Balancing 233

7.2 Plates 235

7.2.1 MDKCs and Scattering Networks for Mindlin’s System 238

7.2.2 Boundary Termination of the Mindlin Plate 242

7.2.3 Simulation: Mindlin’s System for Plates of Uniform and Varying Thickness 246

7.3 Cylindrical Shells 247

7.3.1 The Membrane Shell 248

7.3.2 The Naghdi–Cooper System II Formulation 250

7.4 Elastic Solids 252

7.4.1 Scattering Networks for the Navier System 255

7.4.2 Boundary Conditions 258

8 Time-varying and Nonlinear Systems 261

8.1 Time-varying and Nonlinear Circuit Elements 262

8.1.1 Lumped Elements 262

8.1.2 Distributed Elements 263

8.2 Linear Time-varying Distributed Systems 264

8.2.1 A Time-varying Transmission Line Model 266

8.3 Lumped Nonlinear Systems in Musical Acoustics 267

8.3.1 Piano Hammers 267

8.3.2 The Single Reed 270

8.4 From Wave Digital Principles to Relativity Theory 272

8.4.1 Origin of the Challenge 272

8.4.2 The Principle of Newtonian Limit 274

8.4.3 Newton’s Second Law 274

8.4.4 Newton’s Third Law and Some Consequences 276

8.4.5 Moving Electromagnetic Fields 277

8.4.6 The Bertozzi Experiment 277

8.5 Burger’s Equation 278

8.6 The Gas Dynamics Equations 280

8.6.1 MDKC and MDWD Network for the Gas Dynamics Equations 282

8.6.2 An Alternate MDKC and Scattering Network 283

8.6.3 Entropy Variables 285

9 Concluding Remarks 289

9.1 Answers 289

9.2 Questions 293

A Finite Difference Schemes for the Wave Equation 297

A.1 Von Neumann Analysis of Difference Schemes 298

A.1.1 One-step Schemes 299

A.1.2 Multistep Schemes 300

A.1.3 Vector Schemes 302

A.1.4 Numerical Phase Velocity 302

A.2 Finite Difference Schemes for the (2 + 1)D Wave Equation 303

A.2.1 The Rectilinear Scheme 304

A.2.2 The Interpolated Rectilinear Scheme 305

A.2.3 The Triangular Scheme 309

A.2.4 The Hexagonal Scheme 311

A.2.5 Note on Higher-order Accuracy 314

A.3 Finite Difference Schemes for the (3 + 1)D Wave Equation 315

A.3.1 The Cubic Rectilinear Scheme 315

A.3.2 The Octahedral Scheme 317

A.3.3 The (3 + 1) D Interpolated Rectilinear Scheme 318

A.3.4 The Tetrahedral Scheme 321

B Eigenvalue and Steady State Problems 325

B.1 Introduction 325

B.2 Abstract Time Domain Models 326

B.3 Typical Eigenvalue Distribution of a Discretized PDE 326

B.4 Excitation and Filtering 327

B.5 Partial Similarity Transform 327

B.6 Steady State Problems 329

B.7 Generalization to Multiple Eigenvalues 330

B.8 Numerical Example 331

Bibliography 333

Index 355