Understanding the Finite-Difference Time-Domain Method

1 Understanding the Finite-Difference Time-Domain Method John B. Schneider July 6, 2020. ii Contents 1 Numeric Artifacts 7. Introduction .. 7. finite Precision .. 8. Symbolic Manipulation .. 11. 2 Brief Review of Electromagnetics 13. Introduction .. 13. Coulomb's Law and Electric Field .. 13. Electric Flux Density .. 15. Static Electric Fields .. 17. Gradient, Divergence, and Curl .. 18. Laplacian .. 21. Gauss's and Stokes' Theorems .. 24. Electric Field Boundary Conditions .. 25. Conductivity and Perfect Electric Conductors .. 25. Magnetic Fields .. 26. Magnetic Field Boundary Conditions .. 27. Summary of Static Fields .. 27. time Varying Fields .. 28. Summary of time -Varying Fields .. 29. Wave Equation in a Source-Free Region .. 29. One-Dimensional Solutions to the Wave Equation .. 30. 3 Introduction to the FDTD Method 33. Introduction.

2 33. The Yee Algorithm .. 34. Update Equations in 1D .. 35. Computer Implementation of a One-Dimensional FDTD Simulation .. 39. Bare-Bones Simulation .. 41. PMC Boundary in One Dimension .. 44. Snapshots of the Field .. 45. Additive Source .. 48. Terminating the Grid .. 50. Total-Field/Scattered-Field Boundary .. 53. iii iv CONTENTS. Inhomogeneities .. 60. Lossy Material .. 66. 4 Improving the FDTD Code 75. Introduction .. 75. Arrays and Dynamic Memory Allocation .. 75. Macros .. 77. Structures .. 80. Improvement Number One .. 86. Modular Design and Initialization Functions .. 90. Improvement Number Two .. 95. Compiling Modular Code .. 102. Improvement Number Three .. 103. 5 Scaling FDTD Simulations to Any Frequency 115. Introduction .. 115. Sources .. 115. Gaussian Pulse .. 115. Harmonic Sources .. 116. The Ricker Wavelet.

3 117. Mapping Frequencies to Discrete Fourier Transforms .. 120. Running Discrete Fourier Transform (DFT) .. 121. Real Signals and DFT's .. 123. Amplitude and Phase from Two Time-Domain Samples .. 126. Conductivity .. 128. Transmission Coefficient for a Planar Interface .. 132. Transmission through Planar Interface .. 134. Measuring the Transmission Coefficient Using FDTD .. 135. 6 Differential-Equation Based ABC's 145. Introduction .. 145. The Advection Equation .. 145. Terminating the Grid .. 146. Implementation of a First-Order ABC .. 148. ABC Expressed Using Operator Notation .. 153. Second-Order ABC .. 156. Implementation of a Second-Order ABC .. 158. 7 Dispersion, Impedance, Reflection, and Transmission 161. Introduction .. 161. Dispersion in the Continuous World .. 161. Harmonic Representation of the FDTD Method .. 162.

4 Dispersion in the FDTD Grid .. 165. Numeric Impedance .. 169. Analytic FDTD Reflection and Transmission Coefficients .. 169. CONTENTS v Reflection from a PEC .. 173. Interface Aligned with an Electric-Field Node .. 175. 8 Two-Dimensional FDTD Simulations 181. Introduction .. 181. Multidimensional Arrays .. 181. Two Dimensions: TMz Polarization .. 185. TMz Example .. 189. The TFSF Boundary for TMz Polarization .. 202. TMz TFSF Boundary Example .. 208. TEz Polarization .. 220. PEC's in TEz and TMz Simulations .. 224. TEz Example .. 227. 9 Three-Dimensional FDTD 241. Introduction .. 241. 3D Arrays in C .. 241. Governing Equations and the 3D Grid .. 244. 3D Example .. 252. TFSF Boundary .. 267. TFSF Demonstration .. 272. Unequal Spatial Steps .. 282. 10 Dispersive Material 289. Introduction .. 289. Constitutive Relations and Dispersive Media.

5 290. Drude Materials .. 291. Lorentz Material .. 292. Debye Material .. 293. Debye Materials Using the ADE Method .. 294. Drude Materials Using the ADE Method .. 296. Magnetically Dispersive Material .. 298. Piecewise Linear Recursive Convolution .. 301. PLRC for Debye Material .. 305. 11 Perfectly Matched Layer 307. Introduction .. 307. Lossy Layer, 1D .. 308. Lossy Layer, 2D .. 310. Split-Field Perfectly Matched Layer .. 312. Un-Split PML .. 315. FDTD Implementation of Un-Split PML .. 318. vi CONTENTS. 12 Acoustic FDTD Simulations 323. Introduction .. 323. Governing FDTD Equations .. 325. Two-Dimensional Implementation .. 328. 13 Parallel Processing 331. Threads .. 331. Thread Examples .. 333. Message Passing Interface .. 340. Open MPI Basics .. 341. Rank and Size .. 343. Communicating Between Processes .. 344. 14 Near-to-Far-Field Transformation 351.

6 Introduction .. 351. The Equivalence Principle .. 351. Vector Potentials .. 352. Electric Field in the Far-Field .. 359. Simpson's Composite Integration .. 363. Collocating the Electric and Magnetic Fields: The Geometric Mean .. 363. NTFF Transformations Using the Geometric Mean .. 366. Double-Slit Radiation .. 366. Scattering from a Circular Cylinder .. 370. Scattering from a Strongly Forward-Scattering Sphere .. 371. 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 Introduction .. The PostScript File .. PostScript Basic Commands .. Index 403. Chapter 1. Numeric Artifacts Introduction Virtually all solutions to problems in electromagnetics require the use of a computer. Even when an analytic or closed form solution is available which is nominally exact, one typically must use a computer to translate that solution into numeric values for a given set of parameters.

7 Because of inherent limitations in the way numbers are stored in computers, some errors will invariably be present in the resulting solution. These errors will typically be small but they are an artifact about which one should be aware. Here we will discuss a few of the consequences of finite precision. Later we will be discussing numeric solutions to electromagnetic problems which are based on the Finite-Difference Time-Domain (FDTD) Method . The FDTD Method makes approximations that force the solutions to be approximate, , the Method is inherently approximate. The results obtained from the FDTD Method would be approximate even if we used computers that offered infinite numeric precision. The inherent approximations in the FDTD Method will be discussed in subsequent chapters. With numerical methods there is one note of caution which one should always keep in mind.

8 Provided the implementation of a solution does not fail catastrophically, a computer is always willing to give you a result. You will probably find there are times when, to get your program simply to run, the debugging process is incredibly arduous. When your program does run, the natural assumption is that all the bugs have been fixed. Unfortunately that often is not the case. Getting the program to run is one thing, getting correct results is another. And, in fact, getting accurate results is yet another thing your solution may be correct for the given implementation, but the implementation may not be one which is capable of producing sufficiently accurate results. Therefore, the more ways you have to test your implementation and your solution, the better. For example, a solution may be obtained at one level of discretization and then another solution using a finer discretization.

9 If the two solutions are not sufficiently close, one has not yet converged to the true solution and a finer discretization must be used or, perhaps, there is some systemic error in the implementation. The bottom line: just because a computer gives you an answer does not mean that answer is correct. Lecture notes by John Schneider. 7. 8 CHAPTER 1. NUMERIC ARTIFACTS. finite Precision If we sum one-eleventh eleven times we know that the result is one, , 1/11 + 1/11 + 1/11 +. 1/11 + 1/11 + 1/11 + 1/11 + 1/11 + 1/11 + 1/11 = 1. But is that true on a computer? Consider the C program shown in Program Program : Test if 1/11 + 1/11 + 1/11 + 1/11 + 1/11 + 1/11 + 1/11 +. 1/11 + 1/11 + 1/11 equals 1. 1 /* Is summing ten times == */. 2 #include < >. 3. 4 int main() {. 5 float a;. 6. 7 a = / ;. 8. 9 if (a + a + a + a + a + a + a + a + a + a + a == ).}

10 10 printf("Equal.\n");. 11 else 12 printf("Not equal.\n");. 13. 14 return 0;. 15 }. In this program the float variable a is set to one-eleventh. In line 9 the sum of eleven a's is compared to one. If they are equal, the program prints Equal but prints Not equal otherwise. The output of this program is Not equal. Thus, to a computer (at least one running a language typically used in the solution of electromagnetics problems), the sum of one-eleventh eleven times is not equal to one. It is worth noting that had line 9 been written a=1/11;, a would have been set to zero since integer math would be used to evaluate the division. By using a = / ;, the computer uses floating-point math. The floating-point data types in C or FORTRAN can only store a finite number of digits. On most machines four bytes (32 binary digits or bits) are used for single-precision numbers and eight bytes (64 digits) are used for double precision.