Finite Difference Time Domain Modeling in Dispersive Media

Finite Difference Time Domain Modeling in Dispersive Media

Sarah Brown, Sherrette Yeates, & Carey Rappaport

Gordon Center for Subsurface Sensing & Imaging SystemsNortheastern University, Boston, MA

Sarah Brown, Sherrette Yeates, & Carey Rappaport

Gordon Center for Subsurface Sensing & Imaging SystemsNortheastern University, Boston, MA

1PIERS Cambridge, MA, July 2010

OutlineOutline

• The Four-Zero FDTD conductivity model• Stability considerations• FDTD validation• Time reversal with artificial gain• Results

• The Four-Zero FDTD conductivity model• Stability considerations• FDTD validation• Time reversal with artificial gain• Results

2

FDTD Modeling with DispersionFDTD Modeling with Dispersion• Frequency dependent media usually requires (expensive)

convolution in time domain• Model for biological tissue -> Medical Imaging Systems• Model for soil -> Geophysical Sensing

• The Four Zero model of conductivity uses supplemental equation J(t) = F [ E(t) , (t) ] instead of convolution

• Easy finite difference form• Integer exponents easier to convert to FDTD than

traditional Cole-Cole model• Model fits wideband measured, published data• Forward and Reverse FDTD Simulations show proper behavior

• Frequency dependent media usually requires (expensive) convolution in time domain• Model for biological tissue -> Medical Imaging Systems• Model for soil -> Geophysical Sensing

• The Four Zero model of conductivity uses supplemental equation J(t) = F [ E(t) , (t) ] instead of convolution

• Easy finite difference form• Integer exponents easier to convert to FDTD than

traditional Cole-Cole model• Model fits wideband measured, published data• Forward and Reverse FDTD Simulations show proper behavior

3

Four Zero Conductivity ModelFour Zero Conductivity Model

4

j tZ e

Model frequency-dependent conductivity onlyChoose Z-transform variable for easy conversion to discrete time domain

Model frequency-dependent conductivity onlyChoose Z-transform variable for easy conversion to discrete time domain

Choose constant real relative permittivityChoose constant real relative permittivity

Actual real relative permittivity has frequency dependence due to complex conductivity: Actual real relative permittivity has frequency dependence due to complex conductivity:

( ) Im{ ( )} /av

( ) avZ

Four Zero Conductivity Model:Design StepsFour Zero Conductivity Model:Design Steps

Set Δt (= 5ps) Match model to measured data by solving for b0,

b1, b2, and b3, in terms of a1

Optimize a1 and Δz for stability

Calculate k = β – j α from data & model Compare α and β from model to data

Set Δt (= 5ps) Match model to measured data by solving for b0,

b1, b2, and b3, in terms of a1

Optimize a1 and Δz for stability

Calculate k = β – j α from data & model Compare α and β from model to data

5

Stability AnalysisStability Analysis

6

Von Neumann Stability Analysis: Find zeros of Characteristic Equation for lossy media

Estimate step size Δz:

Courant Stability condition Maintain 10 points / λ

where 1 and 2 are the smallest and largest values of real dielectric constant, f is highest frequency in the freq. range

0' 2 1

0

0 12 '( 2 ) ( 1)t

r Z Z Z

Developing the Forward ModelDeveloping the Forward Model

7

Find Model Parameters

Discrete Time Domain Model

Wave propagated through tissue

Compare α and β to measured data

FDTD

FF

T

Fit Model in Terms of Wave NumberFit Model in Terms of Wave Number

8

9.0 9.5 10.0

0.010

0.005

0.000

0.005

Log Freq. Hz

Diff.

Beta

9.0 9.5 10.0

0.08

0.06

0.04

0.02

0.00

0.02

Log Freq. HzDiff.

Alpha

9.5 10.0Log Freq.

80

60

40

20

Alpha -

9.5 10.0Log Freq.

200

400

600

800Beta

0( )

( ) avk j j

Check Stability of Forward Model by Selecting a1 andzCheck Stability of Forward Model by Selecting a1 andz

9

Δz min = 0.0012m

Δzmax = 0.005m

a1 = [-0.5, -0.9]

Δzmin = 0.0012m

Δzmax = 0.005m

a1 = -0.7

1.0 0.5 0.5 1.0Real

1.0

0.5

0.5

1.0

Imaginary

1- D FDTD Incorporating Four-Zero Model1- D FDTD Incorporating Four-Zero Model

10

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45

50

1.2m .6m 7.2m

Excitation

Mur Absorbing Boundary Condition

AirTissue

0 0.5 1 1.5 2 2.5 3 3.5

x 10-9

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Am

plitu

de [

V/m

]

Excitation

Forward SimulationForward Simulation

11

Validation of SimulationValidation of Simulation

FFT Ex Field at air-tissue boundary FFT Ex Field at 0.60m into tissue, center of

pulse at 9ns Divide, take log, divide by (j 0.60), separate α

and β

FFT Ex Field at air-tissue boundary FFT Ex Field at 0.60m into tissue, center of

pulse at 9ns Divide, take log, divide by (j 0.60), separate α

and β

12

Validation of Forward SimulationValidation of Forward Simulation

13

8.6 8.8 9 9.2 9.4 9.6 9.8 10 10.2-140

-120

-100

-80

-60

-40

-20

0

20

40

Log Frequency [Hz]

Am

plitu

de [

dB]

At air-tissue boundary0.600 m into tissue

Simulation v. DataSimulation v. Data

14

8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8-50

0

50

100

150

200

250Beta

Log Frequency [Hz]

Pha

se S

hift

[1/

m]

From simulationFrom literature

8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8-18

-16

-14

-12

-10

-8

-6

-4

-2

Log Frequency [Hz]

Dec

ay r

ate

[1/m

]

Alpha

From simulationFrom literature

Time Reversal with Artificial Loss to Maintain StabilityTime Reversal with Artificial Loss to Maintain Stability•Forward dispersive model is stable•Zeros are all within unit circle•Time reversed model, Z -> Z-1, maps zeros to

exterior of unit circle•Growth is expected, but unstable explosive

growth may result•Add artificial loss, σ0, for time reversal

•Re-establish stability•Post-multiply by ent/0, 0 = 0/0, after n time

steps, to compensate for added loss

•Forward dispersive model is stable•Zeros are all within unit circle•Time reversed model, Z -> Z-1, maps zeros to

exterior of unit circle•Growth is expected, but unstable explosive

growth may result•Add artificial loss, σ0, for time reversal

•Re-establish stability•Post-multiply by ent/0, 0 = 0/0, after n time

steps, to compensate for added loss

15

Artificial LossArtificial Loss

16

Incorporate artificial loss, σ0, into both Ampere’s and Faraday’s Laws:

Revised Stability condition:

0

00

0

'

0( )E

H E Et

0 0

00

HE H

t

2 1

0

221

0 0 0 0

( )0 12 '( 2 ) ( 1)

( )' ( ) '

Z tr Z Z Z

Z tt tZ Z

Time Reversed Model: Selecting Artificial LossTime Reversed Model: Selecting Artificial Loss

17

Δz min = 0.0012m Δzmax = 0.005ma1 = 0.7σ0 = .08

Δz min = 0.0012mΔzmax = 0.005ma1 = [-0.5, -0.8]σ0 = .08

1.0 0.5 0.5 1.0

Real

1.0

0.5

0.5

1.0

Imaginary

FDTD Time Reversal ValidationFDTD Time Reversal Validation

Excite with recording from z = 2.4m Apply artificial loss at each step Remove artificial loss at air-tissue boundary Compare with original signal

Excite with recording from z = 2.4m Apply artificial loss at each step Remove artificial loss at air-tissue boundary Compare with original signal

18

Verification of Time ReversalVerification of Time Reversal

19

0 0.5 1 1.5 2 2.5

x 10-8

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time [s]

Am

plitu

de [

V/m

]

Air-tissue Boundary from Forward

0 0.5 1 1.5 2 2.5

x 10-8

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time [s]

Am

plitu

de [

V/m

]

Air-tissue Boundary from Reverse

Small phase error due to assuming t/0 << 1

20

ConclusionsConclusions

Four Zero model is stable for many biological tissues

Forward FDTD agrees with frequency domain analysis of tissue properties

Time Reversed FDTD with artificial loss for gain removal is stable

Four Zero model is stable for many biological tissues

Forward FDTD agrees with frequency domain analysis of tissue properties

Time Reversed FDTD with artificial loss for gain removal is stable

Acknowledgements: Gordon-CenSSIS (ERC Award Number EEC-9986821)

the New England Louis Stokes Alliance for Minority Participation

ReferencesReferences

21

1. C. Rappaport and S. Winton, “Modeling dispersive soil for FDTD computation by fitting conductivity parameters,” in 12th Annu. Rev. Progress Applied Computational Electromagnetic Symp. Dig., Mar. 1997, pp. 112–118.

2. W. Weedon, and Rappaport, C., “A General Method for FDTD Modeling of Wave Propagation in Arbitrary Frequency-Dependent Media,” IEEE T. Ant. Prop., vol. 45, Mar. 1997, pp. 401-410.

3. C. Rappaport, E. Bishop and P. Kosmas “Modeling FDTD wave propagation in dispersive biological tissue using a single pole Z-transform function”, International Conference of the IEEE Eng. in Medicine & Biology Society, Cancun, Mexico, September 2003.

4. M. Jalalinia, Rappaport, C., “Propagation and Stabilization in Dispersive Media,” Progress in Electromagnetics Research Symposium, Cambridge, MA, July 2008, p. 62.

1. C. Rappaport and S. Winton, “Modeling dispersive soil for FDTD computation by fitting conductivity parameters,” in 12th Annu. Rev. Progress Applied Computational Electromagnetic Symp. Dig., Mar. 1997, pp. 112–118.

2. W. Weedon, and Rappaport, C., “A General Method for FDTD Modeling of Wave Propagation in Arbitrary Frequency-Dependent Media,” IEEE T. Ant. Prop., vol. 45, Mar. 1997, pp. 401-410.

3. C. Rappaport, E. Bishop and P. Kosmas “Modeling FDTD wave propagation in dispersive biological tissue using a single pole Z-transform function”, International Conference of the IEEE Eng. in Medicine & Biology Society, Cancun, Mexico, September 2003.

4. M. Jalalinia, Rappaport, C., “Propagation and Stabilization in Dispersive Media,” Progress in Electromagnetics Research Symposium, Cambridge, MA, July 2008, p. 62.