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Abstract
Electrical Engineering Julius O. Smith III Ivan R. Linscott Perry R. Cook Robert M. Gray
Abstract
Introduction
An Overview of Scattering Methods
Case Study: The Kelly-Lochbaum Digital Speech Synthesis Model
Concatenated Acoustic Tube Model of the Vocal Tract
Wave Propagation in a Tube of Constant Cross-Sectional Area
Junctions Between Two Uniform Acoustic Tubes
Power Conservation at Scattering Junctions
Discrete-time Vocal Tract Model
Relationship to Digital Filters
Digital Waveguide Networks
Waveguide Meshes and the Wave Equation
DWNs and Numerical Integration
A General Approach: Multidimensional Circuit Representations and Wave Digital Filters
WDFs and the Numerical Integration of ODEs
Multidimensional WDFs as PDE Simulators
Questions
Summary and Results
Chapter 2: Wave Digital Filters
Chapter 3: Multidimensional Wave Digital Filters
Chapter 4: Digital Waveguide Networks
Chapter 5: Applications in Vibrational Mechanics
Chapter 6: Conclusions and Future Directions
Appendix A: Finite Difference Schemes for the Wave Equation
Appendix B: Applications in Fluid Dynamics
Wave Digital Filters
Introductory Remarks
Classical Network Theory
N-ports
Power and Passivity
Kirchoff's Laws
Circuit Elements
Wave Digital Elements and Connections
The Bilinear Transform
Wave Variables
Pseudopower and Pseudopassivity
Wave Digital Elements
The Unit Element
Adaptors
Scattering Matrices for Adaptors
Signal and Coefficient Quantization
Vector Wave Variables
Coupled Inductances and Capacitances
Multidimensional Wave Digital Filters
Introductory Remarks
Symmetric Hyperbolic Systems
Note on Boundary Conditions
Phase and Group Velocity
Coordinate Changes and Grid Generation
Structure of Coordinate Changes
Coordinate Changes in (1+1)D
Coordinate Changes in Higher Dimensions
Embeddings
MD-passivity
MD Circuit Elements
The MD Inductor
Other MD Elements
Discretization in the Spectral Domain
Other Spectral Mappings
The (1+1)D Advection Equation
A Multidimensional Kirchoff Circuit
Stability
An Upwind Form
The (1+1)D Transmission Line
Phase and Group Velocity
MDKC for the (1+1)D Transmission Line Equations
Digression: Derivation of an Inductive Lattice Two-port
A MDWD Network for the (1+1)D Transmission Line
Energetic Interpretation
Simplified Networks
The (2+1)D Parallel-plate System
Phase and Group Velocity
MDKC and MDWD Network
Finite Difference Interpretation
MDWD Networks as Multi-step Schemes
Numerical Phase Velocity and Parasitic Modes
Initial Conditions
Boundary Conditions
Note on Perfectly Matched Layers
Balanced Forms
Extension to (2+1)D
Higher-order Accuracy
Digital Waveguide Networks
Introductory Remarks
FDTD and TLM
Digital Waveguides
The Bidirectional Delay Line
Impedance
Wave Equation Interpretation
Note on the Different Definitions of Wave Quantities
Scattering Junctions
Vector Waveguides and Scattering Junctions
Vector/Scalar Waveguide Coupling
Music and Audio Applications of Digital Waveguides
Transitional Note
The (1+1)D Transmission Line
First-order System and the Wave Equation
Centered Difference Schemes and Grid Decimation
A (1+1)D Waveguide Network
Waveguide Network and the Wave Equation
Comment on Numerical Instability
An Interleaved Waveguide Network
Varying Coefficients
Type I: Voltage-centered Network
Type II: Current-centered Network
Type III: Mixed Network
Comment: Passivity and Stability
Incorporating Losses and Sources
Numerical Phase Velocity and Dispersion
Boundary Conditions
The (2+1)D Parallel-plate System
Defining Equations and Centered Differences
The Waveguide Mesh
Losses, Sources, and Spatially-varying Coefficients
Type I: Voltage-centered Mesh
Type II: Current-centered Mesh
Type III: Mixed Mesh
Reduced Computational Complexity and Memory Requirements in the Standard Form of the Waveguide Mesh
Boundary Conditions
Grid Arrangement Requiring Voltage and Tangential Current Density Component on Boundary
Grid Arrangement with Normal Current Density Component Required on Boundary
Initial Conditions
Alternative Grids in (2+1)D
Hexagonal and Triangular Grids
The Waveguide Mesh in Radial Coordinates
Type I: Voltage-Centered Mesh
Type II: Current-Centered Mesh
Central Gridpoint in a Radial Mesh
Simulation: Circular Region with Varying Inductance
The (3+1)D Wave Equation and Waveguide Meshes
The Waveguide Mesh in General Curvilinear Coordinates
Interfaces Between Grids
Doubled Grid Density Across an Interface
Corners
Simulation
Progressive Grid Density Doubling
Grid Density Quadrupling
Connecting Rectilinear and Radial Grids
Simulation: Solving the Acoustic Wave Equation in a U-Shaped Tube
Grid Density Doubling in (3+1)D
Note
Incorporating the DWN into the MDWD Framework
Multidimensional Unit Elements
Hybrid Form of the Multidimensional Unit Element
Alternative MDKC for the (1+1)D Transmission Line
Alternative MDKC for (2+1)D Parallel-plate System
Higher-order Accuracy Revisited
Maxwell's Equations
Phase and Group Velocity
Scattering Networks for Maxwell's Equations
Applications in Vibrational Mechanics
Transverse Motion of the Ideal Beam
Phase and Group Velocity
Finite Differences
Waveguide Network for the Euler-Bernoulli System
Type I: Voltage-centered Network
Type II: Current-centered Network
Type III: Mixed Network
Boundary Conditions in the Waveguide Network
Fixed End, Allowed to Pivot
Fixed Clamped End
Free End
Timoshenko's Beam Equations
Dispersion
MDKC and MDWDF for Timoshenko's System
Waveguide Network for Timoshenko's System
Other Waveguide Networks for Timoshenko's System
Boundary Conditions in the DWN
Simulation: Timoshenko's System for Beams of Uniform and Varying Cross-sectional Areas
Improved MDKC for Timoshenko's System Via Balancing
Longitudinal and Torsional Waves in Rods
Plates
Maximum Group Velocity
MDKCs and Scattering Networks for Mindlin's System
Boundary Termination of the Mindlin Plate
Free Edge
Simply Supported Edge (1)
Simply Supported Edge (2)
Clamped Edge
Simulation: Mindlin's System, for Plates of Uniform and Varying Thickness
Cylindrical Shells
The Membrane Shell
The Naghdi-Cooper System II Formulation
Elastic Solids
Phase and Group Velocities
Scattering Networks for the Navier System
Boundary Conditions
Conclusions and Future Directions
Answers
Future Directions
Passivity vs. Stability
Higher-order Accuracy
MDKC Modeling of Boundaries
Multi-grid Methods Using MDKCs
Spectral Mappings and Network Transformations
Finite Arithmetic Testing
Time-varying Systems
Afterword
Finite Difference Schemes for the Wave Equation
Von Neumann Analysis of Difference Schemes
One-step Schemes
Multi-step Schemes
Vector Schemes
Numerical Phase Velocity
Finite Difference Schemes for the (2+1)D Wave Equation
The Rectilinear Scheme
The Interpolated Rectilinear Scheme
Optimally direction-independent numerical dispersion
The Triangular Scheme
The Hexagonal Scheme
A Fourth-order Scheme
Finite Difference Schemes for the (3+1)D Wave Equation
The Cubic Rectilinear Scheme
The Octahedral Scheme
The (3+1)D Interpolated Rectilinear Scheme
The Tetrahedral Scheme
Applications in Fluid Dynamics
Nonlinear Circuit Elements
Burger's Equation
The Gas Dynamics Equations
MDKC and MDWDF for the Gas Dynamics Equations
An Alternate MDKC and Scattering Network
Entropy Variables
An open problem
Bibliography
About this document ...
Stefan Bilbao 2002-01-22
