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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