I once considered publishing a book on the finite-difference time-domain (FDTD) method based on notes I wrote for a course I taught.  But, why go through the hassle of publishing through a publisher when you can give away something for free?  (Okay, I can think of several reasons, but I'm going to ignore them.)

So, here is what I have written.  In addition to my students, several people who have stumbled across these notes via a search engine have told me this material has been quite helpful to them.  Hopefully if you read through this you'll find it helpful too.  But, here are a few things to note:

• This material is presented "as is."  One of the reasons I'm not trying to publish this via the traditional route is that I lack the time to put all the finishing touches on this material (such as adding proper citations to previous work).
• Despite the previous statement, I sincerely welcome any and all suggestion to improve this material.  If you catch a typo or think there is some weakness in this material, please let me know (by sending me email).
• If you find this material useful and see an opportunity to cite it in one of your publications, I would be ever so appreciative of the citation.  Please cite this work as: Understanding the Finite-Difference Time-Domain Method, John B. Schneider, www.eecs.wsu.edu/~schneidj/ufdtd, 2010.
• The first two chapters are weak (and have very little to do with FDTD). The third chapter is where the FDTD material really starts and I believe (and hope you agree) that things are pretty decent beginning from there.

• Chapter 1:  Numeric Artifacts.
  A simple overview of some of the errors inherent in digital computation.

  Chapter 1 contents:

  • 1.1 Introduction
  • 1.2 Finite Precision
  • 1.3 Symbolic Manipulation

• Chapter 2:  Brief Review of Electromagnetics.
  This book isn't the place to learn about the fundamentals of electromagnetics, but it was necessary to include some background material.

  Chapter 2 contents:

  • 2.1 Introduction
  • 2.2 Coulomb's Law and Electric Field
  • 2.3 Electric Flux Density
  • 2.4 Static Electric Fields
  • 2.5 Gradient, Divergence, and Curl
  • 2.6 Laplacian
  • 2.7 Gauss's and Stokes' Theorems
  • 2.8 Electric Field Boundary Conditions
  • 2.9 Conductivity and Perfect Electric Conductors
  • 2.10 Magnetic Fields
  • 2.11 Magnetic Field Boundary Conditions
  • 2.12 Summary of Static Fields
  • 2.13 Time Varying Fields
  • 2.14 Summary of Time-Varying Fields
  • 2.15 Wave Equation in a Source-Free Region
  • 2.16 One-Dimensional Solutions to the Wave Equation

• Chapter 3:  Introduction to the Finite-Difference Time-Domain Method:  FDTD in 1D.
  This is where things really start.  You can skip the previous two chapters, but not this one!

  Chapter 3 contents:

  • 3.1 Introduction
  • 3.2 The Yee Algorithm
  • 3.3 Update Equations in 1D
  • 3.4 Computer Implementation of a One-Dimensional FDTD Simulation
  • 3.5 Bare-Bones Simulation
  • 3.6 PMC Boundary in One Dimension
    Animation (courtesy of John Coady)
  • 3.7 Snapshots of the Field
  • 3.8 Additive Source
  • 3.9 Terminating the Grid
  • 3.10 Total-Field/Scattered-Field Boundary
    Animation (courtesy of John Coady)
  • 3.11 Inhomogeneities
    Animation (courtesy of John Coady)
  • 3.12 Lossy Material
    Animation (courtesy of John Coady)

• Chapter 4:  Improving the FDTD Code.
  The goal of this book is to enable you to write fast, efficient FDTD code in the C language.  The material in this chapter discusses a way to "modularize" the code using structures.  (Although it isn't necessarily pretty, the FDTD code in this book is much, much faster than Matlab-based code!)

  Chapter 4 contents:

  • 4.1 Introduction
  • 4.2 Arrays and Dynamic Memory Allocation
  • 4.3 Macros
  • 4.4 Structures
  • 4.5 Improvement Number One
  • 4.6 Modular Design and Initialization Functions
  • 4.7 Improvement Number Two
  • 4.8 Compiling Modular Code
  • 4.9 Improvement Number Three

• Chapter 5:  Scaling FDTD Simulations to Any Frequency.
  So many people talk about a particular frequency when performing an FDTD simulation.  Generally there is no need to do that.  (Granted, sometimes one is interested in a particular frequency, but I find is most convenient to think dimensionlessly, where the points per wavelength in the primary metric describing a simulation.)

  Chapter 5 contents:

  • 5.1 Introduction
  • 5.2 Sources
    • 5.2.1 Gaussian Pulse
    • 5.2.2 Harmonic Sources
    • 5.2.3 The Ricker Wavelet
  • 5.3 Mapping Frequencies to Discrete Fourier Transforms
  • 5.4 Running Discrete Fourier Transform (DFT)
  • 5.5 Real Signals and DFT's
  • 5.6 Amplitude and Phase from Two Time-Domain Samples
  • 5.7 Conductivity
  • 5.8 Transmission Coefficient for a Planar Interface
    • 5.8.1 Transmission through Planar Interface
    • 5.8.2 Measuring the Transmission Coefficient Using FDTD

• Chapter 6:  Differential-Equation Based Absorbing Boundary Conditions.
  The absorbing boundary conditions (ABC's) described here are decidedly old-fashioned.  Still, there is some useful information here and the operator notation that is developed comes in handy at various times.

  Chapter 6 contents:

  • 6.1 Introduction
  • 6.2 The Advection Equation
  • 6.3 Terminating the Grid
  • 6.4 Implementation of a First-Order ABC
  • 6.5 ABC Expressed Using Operator Notation
  • 6.6 Second-Order ABC
  • 6.7 Implementation of a Second-Order ABC

• Chapter 7:  Dispersion, Impedance, Reflection, and Transmission.
  This chapter is something of a grab-bag analysis of the FDTD method and the ways in which it differs from the continuous world.

  Chapter 7 contents:

  • 7.1 Introduction
  • 7.2 Dispersion in the Continuous World
  • 7.3 Harmonic Representation of the FDTD Method
  • 7.4 Dispersion in the FDTD Grid
  • 7.5 Numeric Impedance
  • 7.6 Analytic FDTD Reflection and Transmission Coefficients
  • 7.7 Reflection from a PEC
  • 7.8 Interface Aligned with an Electric-Field Node

• Chapter 8:  Two-Dimensional FDTD Simulations.
  Finally we get beyond 1D!

  Chapter 8 contents:

  • 8.1 Introduction
  • 8.2 Multidimensional Arrays
  • 8.3 Two Dimensions:  TM^z Polarization
  • 8.4 TM^z Example
    Animation (courtesy of John Coady)
  • 8.5 The TFSF Boundary for TM^z Polarization
  • 8.6 TM^z TFSF Boundary Example
  • 8.7 TE^z Polarization
  • 8.8 PEC's in TE^z and TM^z Simulations
  • 8.9 TE^z Example

• Chapter 9:  Three-Dimensional FDTD.
  If you understood FDTD in 1D, then making the transition to 2D and 3D is truly simple.

  Chapter 9 contents:

  • 9.1 Introduction
  • 9.2 3D Arrays in C
  • 9.3 Governing Equations and the 3D Grid
  • 9.4 3D Example
  • 9.5 TFSF Boundary
  • 9.6 TFSF Demonstration
  • 9.7 Unequal Spatial Steps

• Chapter 10:  Dispersive Material.
  Yes, the FDTD grid is itself dispersive, but here we are trying to model media that are dispersive in the continuous world.

  Chapter 10 contents:

  • 10.1 Introduction
  • 10.2 Constitutive Relations and Dispersive Media
    • 10.2.1 Drude Materials
    • 10.2.2 Lorentz Material
    • 10.2.3 Debye Material
  • 10.3 Debye Materials Using the ADE Method
  • 10.4 Drude Materials Using the ADE Method
  • 10.5 Magnetically Dispersive Material
  • 10.6 Piecewise Linear Recursive Convolution
  • 10.7 PLRC for Debye Material

• Chapter 11:  Perfectly Matched Layer.
  Now we're ready to tackle a perfectly matched layer (PML) which is arguably the current state-of-the-art when it comes to ABC's.

  Chapter 11 contents:

  • 11.1 Introduction
  • 11.2 Lossy Layer, 1D
  • 11.3 Lossy Layer, 2D
  • 11.4 Split-Field Perfectly Matched Layer
  • 11.5 Un-Split PML
  • 11.6 FDTD Implementation of Un-Split PML

• Chapter 12:  Acoustic FDTD Simulations.
  FDTD can be used for more than just Maxwell's equations.  This chapter gives a brief overview of the application of the FDTD method to small-signal linear acoustics.

  Chapter 12 contents:

  • 12.1 Introduction
  • 12.2 Governing FDTD Equations
  • 12.3 Two-Dimensional Implementation

• Chapter 13:  Parallel Processing.
  The FDTD method is a computational hog.  To help handle that, one can parallelize the algorithm (the FDTD method is said to be "trivially parallelizable").  This chapter provides a brief discussion of threading and the Message Passing Interface (MPI) as means of parallelizing code.

  Chapter 13 contents:

  • 13.1 Threads
  • 13.2 Thread Examples
  • 13.3 Message Passing Interface
  • 13.4 Open MPI Basics
  • 13.5 Rank and Size
  • 13.6 Communicating Between Processes

• Chapter 14:  Near-to-Far-Field Transformation.
  Discussion of how one can use the fields in the FDTD grid to determine the fields at any "distant" point.

  Chapter 14 contents:

  • 14.1 Introduction
  • 14.2 The Equivalence Principle
  • 14.3 Vector Potentials
  • 14.4 Electric Field in the Far-Field
  • 14.5 Simpson's Composite Integration
  • 14.6 Collocating the Electric and Magnetic Fields:  The Geometric Mean
  • 14.7 NTFF Transformations Using the Gemoetric Mean
    • 14.7.1 Double-Slit Radiation
    • 14.7.2 Scattering from a Circular Cylinder
    • 14.7.3 Scattering from a Strongly Forward-Scattering Sphere

• Appendices
  Miscellaneous material including a PostScript primer that has very little to do with FDTD, but I have found it useful at times to have my code directly draw output in PostScript so that I can visualize what is happening in the grid.

  Appendices:

  • A Construction of Fourth-Order Central Differences
  • B Generating a Waterfall Plot and Animation
  • C Rendering and Animating Two-Dimensional Data
  • D Notation
  • E PostScript Primer
    • E.1 Introduction
    • E.2 The PostScript File
    • E.3 PostScript Basic Commands

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Last updated:  May 15 2012, 12:35.