Abstract

Inverse method has wide application on medical diagnosis where non-destructive evaluation is the key factor .Back scattered waves or echoes generated from the forward moving waves has information about its geometry, size and location. In this paper we have investigated how well different geometries of the object is determined from the back scattered waves by a high accuracy Non-Standard Finite Difference Time Inverse (NSFD-TI) Maxwell’s algorithm and how the refractive index of the object plays a deterministic role on its size.

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References

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  1. L. W. Schmerr, Jr., in Fundamentals of Ultrasonic Nondestructive Evaluation: A Modeling Approach, (Plenum Press, New York, 1998).
  2. R. Mickens, E, in Nonstandard finite difference models of differential equations, (World Scientific Publishing Co., Inc., River Edge, NJ, 1994).
  3. K. S. Kunz, and R. J. Luebbers, in The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, New York, 1993).
  4. J. B. Cole, S. Banerjee, and M. Haftel, in Advances in the Applications of Nonstandard Finite Difference Schemes, ed. R. E. Mickens (World Scientific Singapore, 2007), Chap.4.
  5. R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993).
    [CrossRef]
  6. P. W. Barber, and S. C. Hill, in Light Scattering by Particles: Computational Methods, (World Scientific, Singapore, 1990).
  7. J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. 50(9), 1185–1191 (2002).
    [CrossRef]
  8. H. Kudo, et al., “Numerical Dispersion and Stability Condition of the Nonstandard FDTD Method” Electronics and Communications in Japan, 85, 22–30(2002), http://www3.interscience.wiley.com/cgi-bin/fulltext/93514073/PDFSTART .

2002

J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. 50(9), 1185–1191 (2002).
[CrossRef]

1993

R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993).
[CrossRef]

Cole, J. B.

J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. 50(9), 1185–1191 (2002).
[CrossRef]

Mezzanotte, P.

R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993).
[CrossRef]

Roselli, L.

R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993).
[CrossRef]

Sorrentino, R.

R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993).
[CrossRef]

IEEE Microw. Guid. Wave Lett.

R. Sorrentino, L. Roselli, and P. Mezzanotte, “Time reversal in finite difference time domain method,” IEEE Microw. Guid. Wave Lett. 3(11), 402–404 (1993).
[CrossRef]

IEEE Trans. Antenn. Propag.

J. B. Cole, “High-Accuracy Yee algorithm Based on Nonstandard Finite Difference: New Developments and Verifications,” IEEE Trans. Antenn. Propag. 50(9), 1185–1191 (2002).
[CrossRef]

Other

H. Kudo, et al., “Numerical Dispersion and Stability Condition of the Nonstandard FDTD Method” Electronics and Communications in Japan, 85, 22–30(2002), http://www3.interscience.wiley.com/cgi-bin/fulltext/93514073/PDFSTART .

P. W. Barber, and S. C. Hill, in Light Scattering by Particles: Computational Methods, (World Scientific, Singapore, 1990).

L. W. Schmerr, Jr., in Fundamentals of Ultrasonic Nondestructive Evaluation: A Modeling Approach, (Plenum Press, New York, 1998).

R. Mickens, E, in Nonstandard finite difference models of differential equations, (World Scientific Publishing Co., Inc., River Edge, NJ, 1994).

K. S. Kunz, and R. J. Luebbers, in The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, New York, 1993).

J. B. Cole, S. Banerjee, and M. Haftel, in Advances in the Applications of Nonstandard Finite Difference Schemes, ed. R. E. Mickens (World Scientific Singapore, 2007), Chap.4.

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Figures (4)

Fig. 1
Fig. 1

Mie scattering from an infinite dielectric cylinder. Scattered intensity is visualized in shades of red. E is polarized parallel cylinder axis. Infinite plane wave impinges from the left. Cylinder radius = 0.65 λ 0 , λ 0 = vacuum wavelength, refractive index n = 1.8 .

Fig. 2
Fig. 2

Angular distribution of scattered intensity about a circular contour of radius λ 0 centered on the axis of the cylinder. (a) Analytic solution (black) compared with the NSFD calculation (NSFD-8, red) and the SFD one (SFD-8, blue) at λ 0 / h = 8 . (b) Analytic solution (black) compared with the SFD (SFD-24, blue) one at λ 0 / h = 24 .

Fig. 3
Fig. 3

Time Inverse Computed Intensities using NS-FDTD for: (a)Circular scatterer (diameter = 2 λ 0 )(b) Rectangular scatterer (side = 2 λ 0 ) (c)Triangular scatterer(base = 2 λ 0 ,height = 1 λ 0 ),where, λ 0 / h = 10

Fig. 4
Fig. 4

Distinguishable Size vs. Refractive Index

Equations (57)

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t B ( x , t ) = × E ( x , t )
t D ( x , t ) = × H ( x , t ) J ( x , t )
· B ( x , t ) = 0
· D ( x , t ) = ρ ( x )
B ( x , t ) = μ H ( x , t )
D ( x , t ) = ε ( x ) E ( x , t )
μ t H ( x , t ) = × E ( x , t )
ε ( x ) t E ( x , t ) = × H ( x , t )
f ( x ) f ( x + h 2 ) f ( x h 2 ) h
d x f ( x ) = f ( x + h 2 ) f ( x h 2 )
f ( x ) d x h f ( x )
d = ( d x , d y , d z )
f ( x ) d h f ( x )
f ( x ) d x 2 h 2 f ( x ) .
d x 2 f ( x ) = f ( x + h ) + f ( x h ) 2 f ( x )
d 2 = d x 2 + d y 2 + d z 2
2 f ( x ) d 2 h 2 f ( x )
d t H ( x , t ) = 1 μ Δ t h d × E ( x , t )
H ( x , t + Δ t 2 ) = H ( x , t Δ t 2 ) 1 μ Δ t h d × E ( x , t )
d t E ( x , t + Δ t 2 ) = 1 ε ( x ) Δ t h d × H ( x , t + Δ t 2 )
E ( x , t + Δ t ) = E ( x , t ) 1 ε ( x ) Δ t h d × H ( x , t + Δ t 2 )
d t H ( x , t ) = d 1 × E ( x , t ) ,
d t E ( x , t + Δ t 2 ) = 1 ε μ Δ t 2 h 2 d 1 × H ( x , t + Δ t 2 )
d 1 · d 0 = d 0 · d 1 = d 0 2 .
d x ( 0 ) = d x + ( 1 γ 4 ) d x d y 2
d y ( 0 ) = d y + ( 1 γ 4 ) d y d x 2
and d 0 = d 1 ( 1 + 1 γ 4 d x d y )
d t H ( x , t ) = d 1 × E ( x , t )
d t E ( x , t + Δ t 2 ) = u 0 2 ε d 0 ( ε ) × H ( x , t + Δ t 2 )
H ( x , t + Δ t 2 ) = H ( x , t Δ t 2 ) d 1 × E ( x , t )
E ( x , t + Δ t ) = E ( x , t ) + u 0 2 ε d 0 ( ε ) × H ( x , t + Δ t 2 )
H ( x , t Δ t 2 ) = H ( x , t + Δ t 2 ) + d 1 × E ( x , t + Δ t )
E ( x , t ) = E ( x , t + Δ t ) u 0 2 ε d 0 ( ε ) × H ( x , t Δ t 2 )
ψ ( t + Δ t ) = p ψ ( t Δ t ) + 2 q ψ ( t )
ψ n + 1 = p ψ n 1 + 2 q ψ n
λ 2 2 q λ p = 0
λ ± = q ± q 2 + p
ψ n = c + λ + n + c λ n
( t 2 v 2 2 ) ψ ( x , t ) = 0 ,
( d t 2 u 0 2 d 2 ) ψ ( x , t ) = 0
ψ ( x , t + Δ t ) = ψ ( x , t Δ t ) + ( 2 + u 0 2 d 2 ) ψ ( x , t )
d 2 = d 1 2 = 2 sin 2 ( k h / 2 ) : 1Dimension
d 2 = d 1 2 = 2 [ sin 2 ( k x h / 2 ) + sin 2 ( k y h / 2 ) ] : 2Dimensions
d 0 2 = d 1 2 = d x 2 and Max ( d 0 2 ) = 2 : 1Dimension
d 2 2 = 1 cos ( k x h / 2 ) cos ( k x h / 2 ) : 2Dimensions
γ 0 = 1 sin 2 ( k x h / 2 ) + sin 2 ( k y h / 2 ) sin 2 ( k h / 2 ) 2 sin 2 ( k x h / 2 ) sin 2 ( k y h / 2 )
d 0 2 = γ 0 d 1 2 + ( 1 γ 0 ) d 2 2
d 0 2 ( k ) = γ 0 d 1 2 ( k ) + ( 1 γ 0 ) d 2 2 ( k )
φ x t + 1 = φ x t 1 + 2 ( 1 d 2 u 0 2 ) φ x t
λ ± = 1 u 0 2 d 2 ± ( 1 u 0 2 d 2 ) 2 1
u 0 2 2 d 2
u 0 2 2 Max ( d 2 )
sin ( ω Δ t / 2 ) sin ( k h / 2 ) 2 Max( d 0 2 )
sin ( c k ¯ / 2 ) 2 m d n s f d sin ( k ¯ / 2 )
c < 2 d π arcsin [ 2 m d n s f d sin ( π 2 D ) ]
c 1 nsfd = 1 : In One Dimension
c 1 nsfd 2 2 π arcsin [ 3 2 sin ( π 2 2 ) ] 0.84 : In Two Dimensions

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