ee757 numerical techniques in electromagnetics nnzz x exe x exh ix h i j h i j e i j c i j e i j c i...
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EE757
Numerical Techniques in Electromagnetics
Lecture 8
2 EE757, 2016, Dr. Mohamed Bakr
2D FDTD
1= ex z
x x ix
x
E HE J
t y
1=
y ezy y iy
y
E HE J
t x
1=
y exzz z iz
z
H HEE J
t x y
=x z
x
H E
t y
=y z
y
H E
t x
1=
yxz
z
EEH
t y x
TEz TMz
3 EE757, 2016, Dr. Mohamed Bakr
TEz Case
two electric field components and one magnetic component
1=
yxz
z
EEH
t y x
1=
y ezy y iy
y
E HE J
t x
1= ex z
x x ix
x
E HE J
t y
4 EE757, 2016, Dr. Mohamed Bakr
TEz Case (Cont’d)
1 112 2
1 2( , ) ( , 1)
( , ) = ( , ) ( , ) ( , ) ( , )
n nn
n n z zx exe x exh ix
H i j H i jE i j C i j E i j C i j J i j
y
1 112 2
1 2( , ) ( 1, )
( , ) = ( , ) ( , ) ( , ) ( , )
n nn
n n z zy eye y eyh iy
H i j H i jE i j C i j E i j C i j J i j
x
1 1
2 2( 1, ) ( , )( , 1) ( , )
( , ) = ( , ) ( , )
n nn nn n y yx xz z hze
E i j E i jE i j E i jH i j H i j C i j
y x
(1/z) 0 in 3D update equations
5 EE757, 2016, Dr. Mohamed Bakr
TMz Case
=y z
y
H E
t x
=x z
x
H E
t y
1=
y exzz z iz
z
H HEE J
t x y
6 EE757, 2016, Dr. Mohamed Bakr
TMz Case (Cont’d)
1
1 1 1 112 2 2 22
( , ) = ( , ) ( , )
( , ) ( 1, ) ( , ) ( , 1)( , ) ( , )
n n
z eze z
n n n nny y x x
ezh iz
E i j C i j E i j
H i j H i j H i j H i jC i j J i j
x y
1 1
2 2( , 1) ( , )
( , ) = ( , ) ( , )n n
n nz z
x x hxe
E i j E i jH i j H i j C i j
y
1 1
2 2( 1, ) ( , )
( , ) = ( , ) ( , ) ,n n
n nz z
y y hye
E i j E i jH i j H i j C i j
x
(1/z) 0 in 3D update equations
7 EE757, 2016, Dr. Mohamed Bakr
1D Maxwell’s Equations
1= ,
y ezz z iz
z
HEE J
t x
=
y z
y
H E
t x
1= ,
y ezy y iy
y
E HE J
t x
= .
yz
z
EH
t x
+ve x
-ve x
8 EE757, 2016, Dr. Mohamed Bakr
1D FDTD
1 112 2
1 2( ) ( 1)
( ) = ( ) ( ) ( ) ( ) ,
n nn
n n z zy eye y eyh iy
H i H iE i C i E i C i J i
x
1 1
2 2( 1) ( )
( ) = ( ) ( )
n nn n y y
z z hze
E i E iH i H i C i
x
set (1/y)0 and (1/z)0 in 3D update equations
9 EE757, 2016, Dr. Mohamed Bakr
1D FDTD (cont’d)
1 1
12 21 2
( ) ( 1)( ) = ( ) ( ) ( ) ( )
n n
ny yn n
z eze z ezh iz
H i H iE i C i E i C i J i
x
1 1
2 2( 1) ( )
( ) = ( ) ( )n n
n nz z
y y hye
E i E iH i H i C i
x
set (1/y)0 and (1/z)0 in 3D update equations
10 EE757, 2016, Dr. Mohamed Bakr
The Courant-Friedrich-Levy (CFL) Limit
𝑡1
𝑐 1𝑥2
+1
𝑦2+
1𝑧2
if x=y=z=h, 𝑡ℎ
𝑐 3
the FDTD time-marching scheme becomes unstable if the time step
exceeds the Courant limit
usually, we choose t=0.9 CFL
CFL for 2D and 1D FDTD?
11 EE757, 2016, Dr. Mohamed Bakr
Boundary Conditions
PEC
PMC
Absorbing Boundary Conditions
Mur’s First-order boundary condition
Mur’s Second-order boundary condition
Liao’s boundary condition
Introduction to PML
12 EE757, 2016, Dr. Mohamed Bakr
PEC: TEz Case
set all tangential E-field
components at the
boundary to zero for all
time steps
FDTD update equations
are applied only to interior
electric and magnetic field
components
PEC
13 EE757, 2016, Dr. Mohamed Bakr
PEC: TMz Case
set the Ez components
at the electrical wall
to zero
14 EE757, 2016, Dr. Mohamed Bakr
PMC
where to put the boundary
magnetic walls?
15 EE757, 2016, Dr. Mohamed Bakr
PMC (Cont’d)
half a cell away from the
boundary
PMC
16 EE757, 2016, Dr. Mohamed Bakr
Mur’s Boundary Conditions
initial work : B. Engquist and A. Majda, “Absorbing boundary
conditions for the numerical simulation of waves,” Mathematics of
Computation, vol. 31, 1977, pp. 629-651.
starting from the 3D wave equation
with respect to every dimension, the operator L is decomposed into two
operators.
2 2 2 2
22 2 22
10
f f f f
yx c tz
2 2 2 22 2 2 2 2
22 2 22
1y z txL c
yx c tz
Lf=0
17 EE757, 2016, Dr. Mohamed Bakr
Mur’s Boundary (Cont’d)
with respect to the x-dimension, the wave operator is decomposed to
Lf=L+ L- f , where
the operators L+ and L- are pseudo-differential operators and cannot be
applied directly to a function
L-(f)=0 represents a wave traveling along –x
L+(f)=0 represents a wave traveling along +x
Taylor expansion is used to approximate these operators
1 2 1 21 , 1t tx x c S c SL L
2 2
2
1 1
y zS
c t c t
18 EE757, 2016, Dr. Mohamed Bakr
First-order Mur Boundary Condition
For a first-order approximation we use
the partial derivatives with respect to y and z are assumed very small
this is the case for a normally incident plane wave
at x=0, we impose the condition
at x=xmax, we impose the condition
21 1S
1 1,t tx x c cL L
10
f f
x c t
10
f f
x c t
19 EE757, 2016, Dr. Mohamed Bakr
Illustration of 1st order Mur’s ABC for 1D
at the left boundary, we impose the one-way condition
the +ve x wave operator is used to derive the boundary condition at
x=xmax
( 1) ( ) ( 1) ( )( )(0) (1) (1) (0)
( )
n n n ny y y y
c t xE E E E
c t x
20 EE757, 2016, Dr. Mohamed Bakr
Second-order Mur’s boundary conditions
for the second order Mur, we use the approximation
2 21
1 12
S S
2 2
2
1 1
y zS
c ct t
2 2
1
1 1
1 2 2
2 1 2 2 2
11
2
1
2
- 0.5 0
tx
t yy zzx
t
t tt yy zzx
y zcL
c t c t
cf+ fcL
f= f f+ c f+ fcL
21 EE757, 2016, Dr. Mohamed Bakr
Second-order Mur (Cont’d)
1 1
1, , 0, , 1, , 0, ,2
1/2, ,
1
2
n n n nn j k j k j k j k
j k
f f f fftx
t x x
1 1 1 1
0, , 0, , 0, , 1, , 1, , 1, ,2
2 21/2, ,
2 21
2
n n n- n n n-n j k j k j k j k j k j k
j k
f f f f f fft
t t
x
y
z
(0,j,k)
(1,j,k)
(0,j+1,k)
(0,j,k+1)
22 EE757, 2016, Dr. Mohamed Bakr
Second-Order Mur (Cont’d)
y and z derivatives can be ignored to yield a simpler expression
23 EE757, 2016, Dr. Mohamed Bakr
References
[1] A. Elsherbeni and V. Demir, The Finite Difference Time
Domain Method for Electromagnetics with MATLAB
Simulations, ACES Series on Computational Electromagnetics
and Engineering, SciTech Publishing Inc. an Imprint of the IET,
Second Edition, Edison, NJ, 2015.
[2] R.C. Booton, Computational Methods for
Electromagneticsand Microwaves, Wiley, 1992, pp. 59-73
[3] M.N.O. Sadiku, Numerical Techniques in Electromagnetics,
CRC Press, 2001, pp. 159-192
[4] W. Yu et al., Parallel Finite Difference Time Domain Method,
Artech, 2006
24 EE757, 2016, Dr. Mohamed Bakr
References
[5] A. Taflove, Computational Elertodynamics: the Finite-
Difference Time-Domain Method, Artech, 1995
[6]A. Taflove and S.C. Hagness, same as above,2nded., Artech,
2000
[7] K. Kunz and R. Luebbers, Finite-Difference Time-Domain
Method for Electromagnetics, CRC Press, 1993
[8] K.S. Yee, “Numerical solution of initial boundary value
problems involving Maxwell’s equations in isotropic media,”
IEEE Trans. Antennas Propagat., vol. AP-14, No. 3, pp. 302-
307, May 1966
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