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...

1

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


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


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


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


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


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


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


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


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?


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


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


PEC: TMz Case

set the Ez components

at the electrical wall

to zero


PMC

where to put the boundary

magnetic walls?


PMC (Cont’d)

half a cell away from the

boundary

PMC


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


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


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


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


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


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)


Second-Order Mur (Cont’d)

y and z derivatives can be ignored to yield a simpler expression


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


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

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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