Understanding the Finite-Difference Time-Domain Method

1 Understanding the Finite-Difference Time-DomainMethodJohn B. SchneiderMay 28, 2021iiContents1 Numeric .. Precision .. Manipulation .. 112 Brief Review of .. s Law and Electric Field .. Flux Density .. Electric Fields .. , Divergence, and Curl .. s and Stokes Theorems .. Field Boundary Conditions .. and Perfect Electric Conductors .. Magnetic Fields .. Magnetic Field Boundary Conditions .. Summary of Static Fields .. Time Varying Fields .. Summary of Time-Varying Fields .. Wave Equation in a Source-Free Region .. One-Dimensional Solutions to the Wave Equation.. 303 Introduction to the FDTD .. Yee Algorithm .. Equations in 1D.. Implementation of a One-DimensionalFDTD Simulation .. Simulation .. Boundary in One Dimension.

2 Of the Field .. Source.. the Grid .. Total-Field/Scattered-Field Boundary .. Inhomogeneities.. Lossy Material .. 664 Improving the FDTD .. and Dynamic Memory Allocation .. Number One .. Design and Initialization Functions .. Number Two .. Modular Code.. Number Three .. 1035 Scaling FDTD Simulations to Any .. Pulse .. Sources .. Ricker Wavelet.. Frequencies to Discrete Fourier Transforms .. Discrete Fourier Transform (DFT) .. Signals and DFT s .. and Phase from Two Time-Domain Samples.. Coefficient for a Planar Interface .. through Planar Interface .. the Transmission Coefficient Using FDTD .. 1356 Differential-Equation Based ABC .. Advection Equation .. the Grid .. of a First-Order ABC .. Expressed Using Operator Notation.

3 ABC .. of a Second-Order ABC.. 1587 Dispersion, Impedance, Reflection, and .. in the Continuous World .. Representation of the FDTD Method .. in the FDTD Grid.. Impedance .. FDTD Reflection and Transmission Coefficients .. from a PEC .. Aligned with an Electric-Field Node .. 1758 Two-Dimensional FDTD .. Arrays .. Dimensions: TMzPolarization .. TFSF Boundary for TMzPolarization .. Boundary Example .. s in TEzand TMzSimulations .. 2279 Three-Dimensional .. Arrays in C .. Equations and the 3D Grid .. Example .. Boundary .. Demonstration .. Spatial Steps .. 28210 Dispersive Introduction .. Constitutive Relations and Dispersive Media .. Drude Materials .. Lorentz Material .. Debye Material .. Debye Materials Using the ADE Method .

4 Drude Materials Using the ADE Method .. Magnetically Dispersive Material .. Piecewise Linear Recursive Convolution .. PLRC for Debye Material .. 30511 Perfectly Matched Introduction .. Lossy Layer, 1D .. Lossy Layer, 2D .. Split-Field Perfectly Matched Layer .. Un-Split PML .. FDTD Implementation of Un-Split PML .. 318viCONTENTS12 Acoustic FDTD Introduction .. Governing FDTD Equations .. Two-Dimensional Implementation .. 32813 Parallel Threads .. Thread Examples .. Message Passing Interface .. Open MPI Basics .. Rank and Size .. Communicating Between Processes .. 34414 Near-to-Far-Field Introduction .. The Equivalence Principle .. Vector Potentials .. Electric Field in the Far-Field .. Simpson s Composite Integration .. Collocating the Electric and Magnetic Fields: The Geometric Mean.

5 NTFF Transformations Using the Geometric Mean .. Double-Slit Radiation .. scattering from a Circular Cylinder .. scattering from a Strongly Forward- scattering Sphere .. 371A Construction of Fourth-Order Central Generating a Waterfall Plot and Rendering and Animating Two-Dimensional PostScript Introduction .. The PostScript File .. PostScript Basic Commands .. 1 Numeric IntroductionVirtually all solutions to problems in electromagnetics require the use of a computer. Even whenan analytic or closed form solution is available which is nominally exact, one typically must usea computer to translate that solution into numeric values for a given set of parameters . Becauseof inherent limitations in the way numbers are stored in computers, some errors will invariably bepresent in the resulting solution.

6 These errors will typically be small but they are an artifact aboutwhich one should be aware. Here we will discuss a few of the consequences of finite we will be discussing numeric solutions to electromagnetic problems which are basedon the Finite-Difference Time-Domain (FDTD) Method . The FDTD Method makes approximationsthat force the solutions to be approximate, , the Method is inherently approximate. The resultsobtained from the FDTD Method would be approximate even if we used computers that offeredinfinite numeric precision. The inherent approximations in the FDTD Method will be discussed insubsequent numerical methods there is one note of caution which one should always keep in the implementation of a solution does not fail catastrophically, a computer is alwayswilling to give you a result. You will probably find there are times when, to get your programsimply to run, the debugging process is incredibly arduous.

7 When your program does run, thenatural assumption is that all the bugs have been fixed. Unfortunately that often is not the the program to run is one thing, getting correct results is another. And, in fact, gettingaccurate 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 , the more ways you have to test your implementation and your solution, the better. Forexample, a solution may be obtained at one level of discretization and then another solution usinga finer discretization. If the two solutions are not sufficiently close, one has not yet converged tothe true solution and a finer discretization must be used or, perhaps, there is some systemic errorin the implementation. The bottom line: just because a computer gives you an answer does notmean that answer is notes by John 1.

8 NUMERIC finite PrecisionIf 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? Considerthe C program shown in Program : Test if1/11 + 1/11 + 1/11 + 1/11 + 1/11 + 1/11 + 1/11 +1/11 + 1/11 + 1 *Is summing ten times == */2#include < >34int main() {5float a;67a = / ;89if (a + a + a + a + a + a + a + a + a + a + a == )10printf("Equal.\n");11else12printf("No t equal.\n");1314return 0;15}In this program the float variableais set to one-eleventh. In line 9 the sum of elevena s iscompared to one. If they are equal, the program prints Equal but prints Not equal output of this program is Not equal. Thus, to a computer (at least one running a languagetypically used in the solution of electromagnetics problems), the sum of one-eleventh eleven timesis not equal to one.

9 It is worth noting that had line 9 been writtena=1/11;,awould have been setto zero since integer math would be used to evaluate the division. By usinga = / ;,the computer uses floating-point floating-point data types in C or FORTRAN can only store a finite number of digits. Onmost machines four bytes (32 binary digits or bits) are used for single-precision numbers andeight bytes (64 digits) are used for double precision. Returning to the sum of one-elevenths, asan extreme example, assumed that a computer can only store two decimal digits. One eleventh isequal to .. Thus, to two decimal places one-eleventh would be approximated by this eleven times + + + + + + + + + + = is clearly not equal to one. If the number is stored with more digits, the result becomescloser to one, but it never gets there. Both the decimal and binary floating-point representationof one-eleventh have an infinite number of digits.

10 Thus, when attempting to store finite PRECISION9in a computer the number has to be truncated so that the computer stores an approximation ofone-eleventh. Because of this truncation summing one-eleventh eleven times does not yield equal to , it might appear this number can be stored with a finite numberof digits. Although one-tenth has a finite number of digits when written in base ten (decimalrepresentation), it has an infinite number of digits when written in base two (binary representation).In a floating-point decimal number each digit represents the number of a particular power often. Letting a blank represent a digit, a decimal number can be thought of in the follow way:..10310210110010 110 210 310 4 Each digits tells how many of a particular power of10there is in a number. The decimal pointserves as the dividing line between negative and non-negative exponents.