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Finite-Difference Time-DomainMethod
Dennis Sullivan, Ph.D.Professor of Electrical Engineering
University of IdahoMoscow, ID USA
83844-1023
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Outline (continued)
Interpolationacross boundaries
Frequency dependent materials
Advanced Topics
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E
t
1
0 H
Electromagnetic radiation is governed
by the Maxwells equations
H
t 10 E
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Ex
t
1
0
Hy
z
Hy
t 1
0
Ex
z
In one dimension in free space they become
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1/ 2 1/ 2
, ( ) ( )n n
x x xE z t E k E k t t
, ( 1/ 2) ( 1/ 2)n n
y y yH z t H k H kz x
To put these equations in a computer, take the
finite-differenceapproximations of thepartial derivatives in time and space
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1/ 2 1/ 2
0
( ) ( )
( 1/ 2) ( 1/ 2)
n n
x x
n n
y y
E k E k
tH k H k
x
1
1/ 2 1/ 2
0
( 1/ 2) ( 1/ 2)
( 1) ( )
n n
y y
n n
x x
H k H kt
E k E kx
This is a time-domainmethod. Each new
value of the electric fieldEor the magneticfieldH is determined by the previous values
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Thekrepresents the location in an array in a
computer whilenrepresents time
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E 0
0
E
t x
2 c0
We make a change of variables so E and H have the
same order of magnitude:
Once the cell size is chosen, the time steps must
be chose small enough for stability
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ex[k] = ex[k] + 0.5*( hy[k-1] - hy[k])
hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1])
This results in the following two equations of
code in the C program language
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Calculate En+1/2
Calculate Hn+1
n=n+1
Each time step represents an increment in
the total time T = n t.
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The following is a one-dimensional
simulation of an EM pulse propagation
in free space. {T represents the numberof time steps.}
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E
t
1
0r H
0rE
Media like those found in human tissue are
specified by:
1. Relative dielectric constant
2. Conductivity
r
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These are included in the FDTD formulation:
Exn1/ 2
k
Exn 1/ 2
k t 1
r 00
Hyn
k1 / 2 Hy
n
k 1 / 2 x
r0
Exn 1/ 2 k Ex
n1/ 2 k
2
.
Note that the last term is written as the
average over two cells
Exn1/ 2
k 1
t
2r0
1 t2r0
Exn1/ 2
k 1 / 2
r 1 t2r0
Hynk1 / 2 Hy
nk1 / 2
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This leads to the following C computer code:
ex[k] = ca[k]*ex[k] + cb[k] *( hy[k-1] - hy[k] )
hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1] )
eaf = dt*sigma/(2*epsz*epsilon)
ca[k] = (1. - eaf)/(1. + eaf)
cb[k] =0.5/(epsilon*(1. + eaf)).
Specify the parameters in the cells:
Computer codein the main loop:
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The following simulation shows an EMpulse propagating in free space and then
striking a material with = 5,= 0.05.
(Approximately the values for human fator bone.)
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The cells between 100 and 200 have been
assigned the properties
5 .05r
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Most EM sources, like those used in
hyperthermia, produce sinusoidal radiation
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Once steady state has been reached, the rate of absorption
of energy in the tissue is determined by the specific
absorption rate (SAR)
2
max
1
2SAR E
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However, detemining SAR in this manner is difficult
for two reasons:
1. Sinusoidal sources are difficult2. Infromation can only be obtained for one
frequency at a time.
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The best method to determine the magnitude
of the E field for a sinusoidal source at a certain
frequency:
Use a pulse source in the simulation and then take
the discrete Fourier transform at each cell, at each
cell at each frequency of interest.
112100
T Ni f n T i f t
n
E f E t e dt E n T e
1 1
0 0
( ) cos(2 ) ( ) sin(2 )T T
n n
E n t f t n i E n t f t n
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Remember, this must be done at each cell
where the SAR is to be known. In a 3D
simulation, this is typically many thousands
of cells. It would be impossible to store thetime-domain data and then take the Fourier
transforms.
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1
1
( ) ( 1) ( ) cos(2 )
( ) ( 1) ( ) sin(2 )
real real
imag imag
E n E n E n t f t n
E n E n E n t f t n
However, I can calculate a running
Fourier transform during the simulation,and it only requires two more computer words
per cell, per frequency:
At the end of the simulation, the amplitude is
calculated from:
1/ 2
2 2
real imag Amp E E
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The following simulation shows an EMwave interaction with a section of
dielectric material. The results are
calculated for two different frequencies,
50 MHz and 500 MHz. Note that a
pulse can be used to determine what the
results will be for sinusoidal sources.
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The last slide shows the very different patterns thatcan occur at different frequencies having different
wavelengths.
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The Maxwells equations can also be written
*
0
0
( ) ( ) ( )1
r
t
t
D H
D E
HE
The medium characteristics are in the middle equation
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Calculate Dn+1/2
Calculate Hn+1
n=n+1
This adds another step to each increment
Calculate En+1/2 from Dn+1/2
D
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*
0
0
( ) ( ) ( )
1
r
t
t
DH
D E
HE
There are two main reasons for using this formulation:
1. It is easier to formulate frequency-dependent media
(We will discuss this under advanced topics.)
2. It is easier to formulate the perfectly matched layer
(PML) at the boundaries.
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The Yee Cell
Three-dimensional Simulation
The E and Hfields are interwoven
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1/ 2 1/ 2
0 0
( , , 1/ 2) ( , , 1/ 2)
( ( 1/ 2, , 1/ 2) ( 1/ 2, , 1/ 2)
( , 1/ 2, 1/ 2) ( , 1/ 2, 1/ 2) )
n n
z z
n n
y y
n n
x x
D i j k D i j k
t
H i j k H i j kx
H i j k H i j k
In the FDTD formulation, the Maxwell equations
become six interwoven field calculations
as well as three equation to get the E field from
the flux densities in the x, y, and z directions
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Up until now, we have not discussed the boundary
conditions at the edges of the problem space.
These are necessary to keep unwanted reflections
from coming back.
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Probably the best solution is the perfectly
matched layer (PML) which absorbs
out-going waves.
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The reflection of an outgoing wave is determined
by the reflection coefficient
A BA B
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There are two conditions to form a PML:
0 m
*Fx
*Fx
1. The impedance going from the background
medium to the PML must be constant:
2. The direction perpendicular to the boundary,
the x direction for example, must be the inverse of
the other directions:
*Fx
1
*Fy
*Fx 1
*Fy
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F
t
EH
* D E
Ft
H E
To implement the PML, we will assume that there are
fictitious values of and that we can attach to the
Maxwells equations.
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We assume and are complex. It isthe imaginary part that leads to
absorption
*Fm Fm
Dm
j0
*Fm Fm
Hm
j0
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The following selection of parameterssatisfies these requirements:
Fm Fm1
Dm
0
Hm
0
D
0
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0 m *Fx
*Fx
1 (x) /j0
1 (x) /j01
An outgoing wave sees a constant impedance
as its going into the PML. The conductivitycauses it to be absorbed ounce its in the
PML.
The values of are gradually increased as
they go into the PML
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The following two-dimensional
simulation shows the radiation
from a point source. The radiationis absorbed by the PML.
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Remember that the fictitious values of
and that are used to implement the PML
have nothing to do with the real values
of * that specify the body being
simulated. Therefore, they can overlap.
Applicators can be simulated in FDTD by specifying
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Applicators can be simulated in FDTD by specifyingthe material and the source of energy. Here is a
simple dipole antenna.
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The portion that is to be metal can be specified byjust holding the E field to zero.
The excitation comes
from just specifying
the E field in the gap.
The FDTD method will
determine the H field,
which is an indicationof the current in the
dipole
It also calculates the E field that would radiate
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It also calculates the E field that would radiate
out from the antenna.
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The following simulation is from a three-dimensional simulation of a dipole radiating in
free space.
The first set of slides shows the H fields nextto the metal of the dipole arms, which are an
indication of the current.
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The next set of slides shows the E field in
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The next set of slides shows the E field inthe plane of the gap as it radiates away
from the antenna
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Each cell in an FDTD simulation can be
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specified as a different tissue type.
FDTD does not require an elaborate mesh
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to specify the boundaries. Each cell is
composed of a material.
If a simple in or out approach is used, a
t i i ff t lt
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stair casing effect results.
This can be improved by decreasing the cell size,
b t th t ill i t
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but that will require more computer resources.
Another possibility is to average across the cells.
This gives a better representation without
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This gives a better representation without
increasing computer resources.
The applicators and the body to be
di t d b i l d d i th i l ti
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radiated can be included in the simulation.
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Advanced Topics:
1. Interpolation to improveaccuracy
2. Frequency dependentFDTD formulation
The basic FDTD method assumes that the E
field is perpendicular from the plane
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HxHx
Hy
HyEz
field is perpendicular from the plane
containing the H fields.
In the vicinity of dielectric boundaries, the
actual E field could be substantially different.
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Hx
Hx
Hy
Ez
actual E field could be substantially different.
Ez(actual)
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A High-Resolution Interpolation at
Arbitrary Interfaces for the FDTD Method
J. Nadobny, D. Sullivan, P. Wust, M. Seebass,P. Dueflhard, and R. Felix
IEEE Transactions on Microwave Theory and Techniques,
Vol. 46, Novmeber 1998.
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At the end of the simulation the
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At the end of the simulation, the
FDTD value of each field,is corrected by the second term on
the right.
norm1z
z z z
z
E E E
zE
A program simulates the radiation of a layered sphere with a
plane wave. The results can be compared with the analytic
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p p y
calculate of the E fields using a Bessel function expansion
method.
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Accuracy of Ezfield simulation on the 45 degree axis.
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Accuracy of Eyfield simulation on the 45 degree axis.
Frequency dependent methods
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Frequency dependent methods
*0
0
( ) ( ) ( )
1
r
t
t
DH
D E
HE
The medium characteristics are in the middle equation
Biological tissues can have properties
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Biological tissues can have properties
that vary at different frequencies. Thefollowing table shows the values for
human muscle.
40 97 0.69
100 72 0.89
200 56 1.28
300 54 1.37
433 53 1.43
Frequency (MHz) r
The dielectric constant and conductivity can be
i l di l i
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written as a complex dielectric constant
r*
() r
j0
r*
() r
j0
1
1 jt0
However, most tissues are frequency dependent
and have one or more additional terms
This could not be incorporated in the previous method
The frequency dependent term must be taken
h i d i h i b l i
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to the time-domain where it becomes a convolution
This can be formulated in FDTD by the following
S(t) 1
t0e
(t' t)/ t0E(t' ) dt'0
t
Sn
1
t
t0En et/ t0Sn1
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0
0
/1 1
1
0 0
1
0
/ 1
1
0
t tn n n
n
r
n n n
t tn n n
D I e S
E t t
t
tI I E
t
S e S E t
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The End