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3-D FDTD code with PEC boundaries
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Date of this version: February 2000
This MATLAB M-file implements the finite-difference time-domain
solution of Maxwell's curl equations over a three-dimensional
Cartesian space lattice comprised of uniform cubic grid cells.
To illustrate the algorithm, an air-filled rectangular cavity
resonator is modeled. The length, width, and height of the
cavity are 10.0 cm (x-direction), 4.8 cm (y-direction), and
2.0 cm (z-direction), respectively.
The computational domain is truncated using PEC boundary
conditions:
ex(i,j,k)=0 on the j=1, j=jb, k=1, and k=kb planes
ey(i,j,k)=0 on the i=1, i=ib, k=1, and k=kb planes
ez(i,j,k)=0 on the i=1, i=ib, j=1, and j=jb planes
These PEC boundaries form the outer lossless walls of the cavity.
The cavity is excited by an additive current source oriented
along the z-direction. The source waveform is a differentiated
Gaussian pulse given by
J(t)=-J0*(t-t0)*exp(-(t-t0)^2/tau^2),
where tau=50 ps. The FWHM spectral bandwidth of this zero-dc-
content pulse is approximately 7 GHz. The grid resolution
(dx = 2 mm) was chosen to provide at least 10 samples per
wavelength up through 15 GHz.
To execute this M-file, type "fdtd3D" at the MATLAB prompt.
This M-file displays the FDTD-computed Ez fields at every other
time step, and records those frames in a movie matrix, M, which
is played at the end of the simulation using the "movie" command.
***********************************************************************
clear
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Fundamental constants
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cc=2.99792458e8; %speed of light in free space
muz=4.0*pi*1.0e-7; %permeability of free space
epsz=1.0/(cc*cc*muz); %permittivity of free space
***********************************************************************
Grid parameters
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ie=50; %number of grid cells in x-direction
je=24; %number of grid cells in y-direction
ke=10; %number of grid cells in z-direction
ib=ie+1;
jb=je+1;
kb=ke+1;
is=26; %location of z-directed current source
js=13; %location of z-directed current source
kobs=5;
dx=0.002; %space increment of cubic lattice
dt=dx/(2.0*cc); %time step
nmax=500; %total number of time steps
***********************************************************************
Differentiated Gaussian pulse excitation
***********************************************************************
rtau=50.0e-12;
tau=rtau/dt;
ndelay=3*tau;
srcconst=-dt*3.0e+11;
***********************************************************************
Material parameters
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eps=1.0;
sig=0.0;
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Updating coefficients
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ca=(1.0-(dt*sig)/(2.0*epsz*eps))/(1.0+(dt*sig)/(2.0*epsz*eps));
cb=(dt/epsz/eps/dx)/(1.0+(dt*sig)/(2.0*epsz*eps));
da=1.0;
db=dt/muz/dx;
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Field arrays
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ex=zeros(ie,jb,kb);
ey=zeros(ib,je,kb);
ez=zeros(ib,jb,ke);
hx=zeros(ib,je,ke);
hy=zeros(ie,jb,ke);
hz=zeros(ie,je,kb);
**********************************************************************
Movie initialization
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tview(:,:)=ez(:,:,kobs);
sview(:,:)=ez(:,js,:);
subplot('position',[0.15 0.45 0.7 0.45]),pcolor(tview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j,k=5), time step = 0']);
xlabel('i coordinate');
ylabel('j coordinate');
subplot('position',[0.15 0.10 0.7 0.25]),pcolor(sview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j=13,k), time step = 0']);
xlabel('i coordinate');
ylabel('k coordinate');
rect=get(gcf,'Position');
rect(1:2)=[0 0];
M=moviein(nmax/2,gcf,rect);
***********************************************************************
BEGIN TIME-STEPPING LOOP
***********************************************************************
for n=1:nmax
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Update electric fields
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ex(1:ie,2:je,2:ke)=ca*ex(1:ie,2:je,2:ke)+...
cb*(hz(1:ie,2:je,2:ke)-hz(1:ie,1:je-1,2:ke)+...
hy(1:ie,2:je,1:ke-1)-hy(1:ie,2:je,2:ke));
ey(2:ie,1:je,2:ke)=ca*ey(2:ie,1:je,2:ke)+...
cb*(hx(2:ie,1:je,2:ke)-hx(2:ie,1:je,1:ke-1)+...
hz(1:ie-1,1:je,2:ke)-hz(2:ie,1:je,2:ke));
ez(2:ie,2:je,1:ke)=ca*ez(2:ie,2:je,1:ke)+...
cb*(hx(2:ie,1:je-1,1:ke)-hx(2:ie,2:je,1:ke)+...
hy(2:ie,2:je,1:ke)-hy(1:ie-1,2:je,1:ke));
ez(is,js,1:ke)=ez(is,js,1:ke)+...
srcconst*(n-ndelay)*exp(-((n-ndelay)^2/tau^2));
**********************************************************************
Update magnetic fields
**********************************************************************
hx(2:ie,1:je,1:ke)=hx(2:ie,1:je,1:ke)+...
db*(ey(2:ie,1:je,2:kb)-ey(2:ie,1:je,1:ke)+...
ez(2:ie,1:je,1:ke)-ez(2:ie,2:jb,1:ke));
hy(1:ie,2:je,1:ke)=hy(1:ie,2:je,1:ke)+...
db*(ex(1:ie,2:je,1:ke)-ex(1:ie,2:je,2:kb)+...
ez(2:ib,2:je,1:ke)-ez(1:ie,2:je,1:ke));
hz(1:ie,1:je,2:ke)=hz(1:ie,1:je,2:ke)+...
db*(ex(1:ie,2:jb,2:ke)-ex(1:ie,1:je,2:ke)+...
ey(1:ie,1:je,2:ke)-ey(2:ib,1:je,2:ke));
***********************************************************************
Visualize fields
***********************************************************************
if mod(n,2)==0;
timestep=int2str(n);
tview(:,:)=ez(:,:,kobs);
sview(:,:)=ez(:,js,:);
subplot('position',[0.15 0.45 0.7 0.45]),pcolor(tview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j,k=5), time step = ',timestep]);
xlabel('i coordinate');
ylabel('j coordinate');
subplot('position',[0.15 0.10 0.7 0.25]),pcolor(sview');
shading flat;
caxis([-1.0 1.0]);
colorbar;
axis image;
title(['Ez(i,j=13,k), time step = ',timestep]);
xlabel('i coordinate');
ylabel('k coordinate');
nn=n/2;
M(:,nn)=getframe(gcf,rect);
end;
***********************************************************************
END TIME-STEPPING LOOP
***********************************************************************
end
movie(gcf,M,0,10,rect);
