COMPUTATION OF SCATTERING USING THE FINITE
DIFFERENCE TIME DOMAIN (FDTD) METHOD
Introduction
The finite difference time domain (FDTD) method is a means of determining the scattered field
from objects by solving Maxwell’s equations in the time domain. The spatial and temporal
differential operators are approximated by finite differences. Update equations can be derived
(see reference [1]) that give the field at a point in space and time as a function of the field at the
same and neighboring points at previous times. Therefore the solution is said to “march in time.”
As with most numerical solutions, the computational region is discretized into appropriate
subdomains. For example, in 3-dimensional space they might be cubes. The incident wave is
introduced into the computational grid and the scattered fields computed throughout the grid as a
function of time. The fields at the boundaries of the computational grid are used to compute
equivalent currents, which, in turn, are used in the radiation integrals to compute the far field.
The resulting far fields are a function of time. They can be Fourier transformed to obtain the
RCS as a function of frequency,  ( ) .
Reference [1] describes the limitations, restrictions and tradeoffs involved in the computation.
Important computational parameters and tradeoff issues include:





grid size, 
time step, t : determines the highest frequency of the computed RCS,  ( f max )
time window, Tmax  Nt ( N time steps): frequency resolution (i.e., spacing between
frequency samples after the Fourier transform) depends on the length of the time
window. In most cases the fast Fourier transform (FFT) algorithm is used, which
requires N  2 M , M an integer. The frequency resolution is f  1 / Tmax
 and t cannot be chosen independently and the Nyquist sampling criterion must
be satisfied
the effects of terminating the computational grid must be addressed
The incident waveform is a Gaussian pulse. The pulse parameters are:



effective time duration, Teff
effective bandwidth, Beff
dynamic range, Rdyn


time duration, To
signal-to-noise ratio,
SNR 
POWER IN FREQUEN CIES BELOW f max
POWER IN FREQUEN CIES ABOVE f max
A Gaussian pulse is shown below:
1
NORMALIZED TRUNCATED
GAUSSIAN PULSE, g(t)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2
0
2
4
6
8
10
TIME, t (ns)
The computer code fdtdpanel.m solves for the field scattered by a panel of inifinite extent, but
finite thickness. The codes rcs2dbir.m rcs2inc.m, rcs2far.m and rcs2scat.m are used sequentially
in order to compute the field scattered by an infinitely long square cylinder.
One-dimensional Solution: fdtdpanel.m
The program fdtdpanel.m uses the FDTD method to compute the scattering from an infinite
panel of thickness d . The waveform is a Gaussian pulse and the computational parameters are
assigned in the code as shown below. The panel extends from diel_start to diel_stop.
The conductivity, relative dielectric constant, and relative permeability can be chosen by the
user. For each time step the fields are plotted. Lines representing the faces of the panel are also
plotted as a reference. The oversampling ratio is K (if K=1 the Nyquist rate is used).
Setup for the program fdtdpanel.m:
%%% INPUT PARAMETERS %%%
K
= 4;
%
N_samples= 2048*K;
%
v_rat
= 1;
%
T_max
= 1024*10^(-9);
%
dT
= T_max/N_samples;
%
dL
= 0.15/K;
%
N_nodes = K*101;
%
L
= (N_nodes-1)*dL;
%
T0
= K*16*dT;
%
T_shift = T0/2;
%
B_eff
= 10^9;
%
f_max
= 1/(2*dT);
%
N_obs
= round(N_nodes/(K*2));%
c
= 3e8;
eta0
= 377;
ugrid
= dL/dT;
%%% ASSIGN INPUT VARIABLES %%%
Oversampling Ratio
Time Samples
Velocity Ratio (=c/ugrid)
Time Window
Time Step
Spatial Step
Number of E-nodes
Spatial Domain Length
Truncated Gaussian Pulse Duration
Gaussian Pulse Time Shift
Effective Bandwidth
Nyquist Frequency
Observation Point, Total Field
eps_rel =
mu_rel
=
sigma
=
diel_start
diel_stop
4;
1;
0;
= 6;
= 7;
%
%
%
%
%
Relative permittivity is fixed
Relative permeability is fixed
Conductivity inside of panel
Left edge of dielectric
Right edge of dielectric
%%% SET UP PLOT RANGE AND DRAW THE WALL &&&
Lmax=15;
Amax=0;
Amin=-40;
Xw1 = [diel_start, diel_start];
Xw2 = [diel_stop, diel_stop];
A snapshot of the computation is shown below for the data listed above. Note that the reflections
from the front and rear faces are traveling in the reverse direction, while the transmitted wave is
traveling in the forward direction.
0
Relative Power (dB)
-5
-10
-15
-20
-25
-30
-35
-40
0
5
10
15
Distance (m)
Two-dimensional Scattering
The scattering from an infinitely long cylinder is computed using the four “rcs2xxx.m” codes in
the following order:
rcs2dbir.m
rcs2inc.m (typical: 1-10 MHz for resolution; 0.5-1 oversampling ratio)
rcs2scat.m (a movie can be generated by using getframe with the array Mov)
rcs2far.m
A brief description of the setup (variable assignment) section of each code follows. More
detailed explanations are given in reference [1], Chapter 4. Several helpful hints:
1. Some of these programs were written under Matlab 4.x. Some warnings may be
encountered, but the programs will run with Matlab 5.x. If the warnings are a nuisance they
can be turned off by typing “warnings off”
2. The programs may run a long time if a large number of FFT points are used.
The discretization of the cylinder is illustrated below:
^
ki
Ei
L
Hi
L


Setup for rcs2dbir.m
N
Nin
Nout
= 111;
= 51;
= Nin + 2;
h_dur = 50;
vr = 1/sqrt(2);
rowsh = 0.5*(N - Nin);
colsh = 0.5*(N - Nin);
irt
= rowsh + 1;
irb
= irt + Nin - 1;
icl
= colsh + 1;
icr
= icl + Nin - 1;
nodes_out = 4*(Nin+1);
nodes_in = 4*(Nin-1);
%
%
%
%
%
%
%
%
%
%
duration of impulse responses
grid "speed"
Row "shift" for the "inside" matrix
Column "shift" for the "inside" matrix
Top row for the "inside" matrix
Bottom row for the "inside" matrix
Leftmost column for the "inside" matrix
Rightmost column for the "inside" matrix
Number of nodes on the boundary
Number of nodes just inside the boundary
rcs2dbir.m writes the files dbir2par and dbir2dat.
Setup for rcs2inc.m:
load dbir2par
% constants
eps0 = 10^(-9)/(36*pi);
mu0 = 4*pi*10^(-7);
v0
= 1/sqrt(mu0*eps0);
% User Input
fmax
= 10^9;
dt_Nyquist = 1/(2*fmax);
eps_rel
= 1;
dt
= dt_Nyquist/sqrt(eps_rel);
Df
= input('Enter the desired frequency resolution in MHz:');
K_over
dt
Df
dl_air
=
=
=
=
input('Enter the oversampling ratio:');
dt/K_over;
Df*10^6;
sqrt(2)*v0*dt;
% Gaussian Pulse Parameters
B_eff
= fmax;
To
= 8/B_eff;
T_shift = To/2;
Gp_dur
= ceil(To/dt);
% Time Samples
Tmax = 1/Df;
N_samples = 2^(ceil(log10(Tmax/dt)/log10(2)));
rcs2inc.m saves several variables which are used in rcs2scat.m
Setup for rcs2scat.m:
load dbir2dat
load dbir2par
load ez_inc
% constants
eps0 = 10^(-9)/(36*pi);
mu0 = 4*pi*10^(-7);
v0
= 1/sqrt(mu0*eps0);
% User Input
fmax
= 10^9;
dt_Nyquist = 1/(2*fmax);
eps_rel
= 1;
dt
= dt_Nyquist/sqrt(eps_rel);
%K_over
=input('Enter the oversampling ratio:');
dt
= dt/K_over;
dl_air
= sqrt(2)*v0*dt;
dl_diel
= dl_air/sqrt(eps_rel);
%%%%% INPUT PARAMETERS %%%%%
N
= Nin + 2;
nodes_out = 4*(N-1);
nodes_in = 4*(Nin-1);
% Time Samples
%N_samples = 512;
Tmax
= N_samples*dt;
% Gaussian Pulse Parameters
B_eff
= fmax;
To
= 8/B_eff;
T_shift = To/2;
% Number of nodes on the boundary
% Number of nodes just inside the boundary
Example: Snapshot of the field for a square cylinder (computational parameters listed above).
Top: mesh plot; bottom: contour plot. The plots are obtained from the program rcs2inc.m
50
y
40
30
20
10
0
0
20
40
x
120
150
90 1
0.8 60
0.6
0.4
0.2
180
30
0
210
330
240
300
270
The normalized scattered far-field pattern is shown above for a frequency of 100 MHz. The
result is obtained by running the program rcs2far.m.
Reference:
[1] D. Jenn, Radar and Laser Cross Section Engineering, AIAA Education Series, 1995