FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s)

Absorbing Boundary Conditions:

The FDTD method has been applied to different types of

problems successfully in electromagnetics, including the

open region problems.

In order to model open region problems, absorbing

boundary conditions (ABCs) are often used to truncate the

computational domain since the tangential components of

the electric field along the outer boundary of the

computational domain cannot be updated by using the basic

Yee algorithm. Mainly two types of ABC are used to

truncate the computational domain.

1) Differential Type ABC’s: Differential based ABCs are

generally obtained by factoring the wave equation and

allowing a solution, which permits only outgoing

waves.

2) Material Type ABC’s (PML): Material-based ABCs,

on the other hand, are constructed so that fields are

dampened as they propagate into an absorbing

medium.

First, we will consider ABC’s developed by A. Taflove

which is a special case for the differential type ABC’s and

is very easy to apply.

The condition relates the values of the field components at

the truncation planes or points to the field components at

planes or points to one or more space steps and time steps

within the solution region (lattice).

For One dimensional Wave Propagation:

Assume that waves have only Ex and Hy components and

propagate in the +ve and –ve z directions, then;

1 11 2x x

n nE E 1 1 1

x x

n nE kN E kN

when the lattice extends form z=0 to z= z k.

Store a value of Ex(2) for two time steps and then put it in

Ex(1).

B.C.’s in Matlab computer code are as follows when we

assume that the domain stars at k=1 and ends at k=KE:

(1) _ _ 2;

_ _ 2 _ _ 1;

_ _ 1 (2);

( ) _ _ 2;

_ _ 2 _ _ 1;

_ _ 1 ( 1);

ex ex low m

ex low m ex low m

ex low m ex

ex KE ex high m

ex high m ex high m

ex high m ex KE

The full Matlab code is as follows:

clear all

% 1d fdtd with ABC's

KE=500;

c=3e8;

% Initialize the variables for the ABC's

ex_low_m1=0;

ex_low_m2=0;

ex_high_m1=0;

ex_high_m2=0;

% Initialize the field arrays

for k=1:KE

ex(k)=0;

hy(k)=0;

end

% Define the space and time step sizes by

using the stability criteria

dz=0.05;

dt=dz/(2*c);

% Define other constants

eps0=8.85e-12; mu0=1.25663e-6;

nStop=500;

dtedz=dt/(eps0*dz);

dtmdz=dt/(mu0*dz);

% time stepping

for n=1:nStop

%plot the electric field

exn(n)=ex(150);

if mod(n,1)==0

ntime=num2str(n);

plot(exn)

% update the electric field

for k=2:KE

ex(k)=ex(k)-dtedz*(hy(k)-hy(k-1));

% Save the field at t=600dt

if(n==600)

ex600(k)=ex(k);

end

if(n==700)

ex700(k)=ex(k);

end

if(n==500)

ex500(k)=ex(k);

end

end

% Apply the ABC's at k=1 and k=KE

ex(1) = ex_low_m2;

ex_low_m2 = ex_low_m1;

ex_low_m1 = ex(2);

ex(KE) = ex_high_m2;

ex_high_m2 = ex_high_m1;

ex_high_m1 = ex(KE-1);

% apply the source at k=3

ex(3)=exp(-(n-30)^2/100);

%update the magnetic field

for k=1:KE-1

hy(k)=hy(k)-dtmdz*(ex(k+1)-ex(k));

end

title(ntime);

mov(n)=getframe;

end

end

Now consider that half of the medium is a

dielectric with relative permittivity er:

clear all

% 1d fdtd with ABC's

KE=500;

c=3e8;

% Initialize the variables for the ABC's

ex_low_m1=0;

ex_low_m2=0;

ex_high_m1=0;

ex_high_m2=0;

% Initialize the field arrays

for k=1:KE

ex(k)=0;

hy(k)=0;

end

% Define the stace and time step sizes by

using the stability criteria

dz=0.05;

dt=dz/(2*c);

% Define other constants

eps0=8.85e-12; mu0=1.25663e-6;

nStop=1000;

dtedz=dt/(eps0*dz);

dtmdz=dt/(mu0*dz);

% time stepping

for n=1:nStop

%plot the electric field

exn(n)=ex(150);

if mod(n,1)==0

ntime=num2str(n);

plot(exn)

% mesh(ezs(1:nx1,1:ny1,3));

% update the electric field

for k=2:KE

if(k<250)

er=1;

elseif(k==250)

er=(1.+2)/2.;

else

er=2;

end

ex(k)=ex(k)-(dtedz/er)*(hy(k)-hy(k-1));

if(n==600)

ex600(k)=ex(k);

end

if(n==700)

ex700(k)=ex(k);

end

if(n==500)

ex500(k)=ex(k);

end

end

% Apply the ABC's

ex(1) = ex_low_m2;

ex_low_m2 = ex_low_m1;

ex_low_m1 = ex(2);

ex(KE) = ex_high_m2;

ex_high_m2 = ex_high_m1;

ex_high_m1 = ex(KE-1);

% apply the source

ex(3)=exp(-(n-30)^2/100);

%update the magnetic field

for k=1:KE-1

hy(k)=hy(k)-dtmdz*(ex(k+1)-ex(k));

end

title(ntime);

mov(n)=getframe;

end

end

Exersices:

Now, write a complete Matlab program to simulate one-

dimensional wave propagation. Assume that the source is at the center of the problem

space. Observe the wave propagation. Then modify your program by applying the source at k=3. Observe the wave propagation. Observe the wave propagation w.r.t. the time and space.

Change the number of the time steps and observe the propagation.

Change the stability factor from 0.5 to 1, 1.1 and 0.25. What happens?

Differential Type ABC’s

A simple absorbing boundary condition (ABC) was used to terminate the grid. It is based on the fact that the fields were propagating in one dimension and the speed of propagation was such that the fields moved one spatial step for every time step

(i.e., the Courant number was unity c t

z

=1).

The node on the boundary was updated using the value of the adjacent interior node from the previous time steps. However, when a dielectric was introduced, and the local speed of propagation was not equal to c, this ABC failed to work properly. One would also find that in high dimensions this simple ABC would not work even in free space. The wave equation which governs the propagation of the electric field in one dimension is:

2 2

2 20x xE E

z t

Similarly, the following form represents the equation in terms of an operator operating on Ex.

This operator can be factored as:

0z t

Their solutions:

i.e a wave traveling in the negative z direction.

i.e a wave traveling in the positive z direction.

Grid Termination: Let us now consider how the wave

equation can be used to provide an update equation for

a node at the end of the computational domain.

Consider field at z=0.

2 2

2 20xE

z t

0z t

0 ( ) (1)x xE E t zz t

0 ( ) (2)x xE E t zz t

1(1)n

xE

Remember that the interior nodes can be updated

before the boundary node.

Assume that all the adjacent nodes in space-time are

known.

i.e. are known.

At the left end of the grid, the fields should only be

traveling to the left.

Thus the fields satisfy the equation given by (1).

The finite-difference approximation of this equation

provides the necessary update equation, but the way to

discretize the equation is not entirely obvious.

Equation (1) is expanded about the space-time point

To obtain an approximation of:

Similarly:

Therefore the temporal derivative can be approximated by

the following finite difference:

1(2), (1), (2)n n n

x x xE E E

3, ( 1)

2

zn t

1 1 13 1( ) (1) (2)2 2

n n n

x x xE E E

3 1( ) (1) (2)2 2

n n n

x x xE E E

1 1

3 1,( )

2 2

(1) (2) (1) (2)

2 2

n n n n

x x x x

x

zn t

E E E EE

t t

The spatial derivative can be approximated as:

Combination of two difference equations:

Letting and solving for :

This equation provides a first-order absorbing boundary

condition which updates the field on the boundary using the

values of past and interior fields.

Note that when 1pv t

z

is unity, this equation reduces to

the ABC used before (Taflove).

1 1

3 1,( )

2 2

(2) (2) (1) (1)

2 2

n n n n

x x x x

x

zn t

E E E EE

z z

1 1 1 1(1) (2) (1) (2) (2) (2) (1) (1)

2 2 2 2 0

n n n n n n n n

x x x x x x x xE E E E E E E E

t z

1pv

1(1)n

xE

1 1(1) (2) ( (2) (1))pn n n n

x x x x

p

v t zE E E E

v t z

1(1) (2)n n

x xE E

A nearly identical equation can be obtained for the other

end of the grid.

Equation (2) would be expanded in the neighbourhood of

the last node of the grid.

Although (1) and (2) differ in the sign of one term, when

(2) is applied it is “looking” in the negative z direction.

That effectively cancels the sign change. Hence the update

equation for the last node in the grid, which is identified

here as is:

1( )n

zE KE

1 1( ) ( 1) ( ( 1) ( ))pn n n n

x x x x

p

v t zE KE E KE E KE E KE

v t z