Abstract

A hybrid method combining the finite element method (FEM) with the boundary integral equation (BIE) is presented in this paper to investigate two-dimensional (2D) electromagnetic scattering properties of multiple dielectric objects buried beneath a dielectric rough ground for TM case. In traditional FEM simulation, the artificial boundaries, such as perfectly matched layer (PML) and the absorbing boundary conditions (ABC), are usually adopted as truncated boundaries to enclose the whole model. However, the enclosed computational domain increases quickly in size for a rough surface with a large scale, especially for the scattering model of objects away from the rough surface. In the hybrid FEM-BIE method, one boundary integral equation is adopt to depict the scattering above the rough surface based on Green's function. Based on the domain decomposition technique, the computational region below the rough ground is divided into multiple isolated interior regions containing each object and the exterior region. Finite element formulations are only applied inside interior regions to derive a set of linear systems, and another boundary integral formula is developed below the rough surface which also acts as the boundary constraint of the FEM region. Compared with traditional FEM, the hybrid technique presented here is highly efficient in terms of computational memory, time, and versatility. Numerical simulations are carried out based on hybrid FEM-BIE to study the scattering from multiple dielectric objects buried beneath a rough ground.

© 2014 Optical Society of America

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References

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  1. M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
    [CrossRef]
  2. M. El-Shenawee, “Scattering from multiple objects buried beneath two-dimensional random rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. 51(4), 802–809 (2003).
    [CrossRef]
  3. D. E. Lawrence, K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50(10), 1368–1376 (2002).
    [CrossRef]
  4. Y. Altuncu, A. Yapar, I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44(6), 1435–1443 (2006).
    [CrossRef]
  5. C.-H. Kuo, M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54(8), 2392–2401 (2006).
    [CrossRef]
  6. C. Bourlier, N. Pinel, G. Kubické, “Propagation-inside-layer-expansion method combined with physical optics for scattering by coated cylinders, a rough layer, and an object below a rough surface,” J. Opt. Soc. Am. A 30(9), 1727–1737 (2013).
    [CrossRef] [PubMed]
  7. M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
    [CrossRef]
  8. L. X. Guo, Y. Liang, Z. S. Wu, “A study of electromagnetic scattering from conducting targets above and below the dielectric rough surface,” Opt. Express 19(7), 5785–5801 (2011).
    [CrossRef] [PubMed]
  9. M. M. Botha, D. B. Davidson, “Rigorous, auxiliary variable-based implementation of a second-order ABC for the vector FEM,” IEEE Trans. Antennas Propag. 54(11), 3499–3504 (2006).
    [CrossRef]
  10. L. E. R. Petersson, J.-M. Jin, “Analysis of periodic structures via a time-domain finite-element formulation with a floquet ABC,” IEEE Trans. Antennas Propag. 54(3), 933–944 (2006).
    [CrossRef]
  11. Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41(3), 799–826 (2003).
    [CrossRef]
  12. P. Liu, Y.-Q. Jin, “Numerical simulation of bistatic scattering from a target at low altitude above rough sea surface under an EM-wave incidence at low grazing angle by using the finite element method,” IEEE Trans. Antennas Propag. 52(5), 1205–1210 (2004).
    [CrossRef]
  13. E. J. Alles, K. W. van Dongen, “Perfectly matched layers for frequency-domain integral equation acoustic scattering problems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58(5), 1077–1086 (2011).
    [CrossRef] [PubMed]
  14. S. H. Lou, L. Tsang, C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Complex 1, 287–307 (1991).
  15. B. Alavikia, O. M. Ramahi, “Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method,” J. Opt. Soc. Am. A 28(12), 2510–2518 (2011).
    [CrossRef] [PubMed]
  16. P. Demarcke, H. Rogier, “The poincare-steklov operator in hybrid finite element-boundary integral equation formulations,” IEEE Antennas Wireless Propag. Lett. 10, 503–506 (2011).
    [CrossRef]
  17. F.-G. Hu, C.-F. Wang, “Preconditioned formulation of FE-BI equations with domain decomposition method for calculation of electromagnetic scattering from cavities,” IEEE Trans. Antennas Propag. 57(8), 2506–2511 (2009).
    [CrossRef]
  18. Z. Peng, X.-Q. Sheng, “A flexible and efficient higher order FE-BI-MLFMA for scattering by a large body with deep cavities,” IEEE Trans. Antennas Propag. 56(7), 2031–2042 (2008).
    [CrossRef]
  19. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
    [CrossRef]
  20. J. M. Jin, The Finite Element Method in Electromagnetics (John Wiley, 2002).
  21. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Wave: Numerical Simulations (John Wiley, 2001).

2013

2012

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

2011

L. X. Guo, Y. Liang, Z. S. Wu, “A study of electromagnetic scattering from conducting targets above and below the dielectric rough surface,” Opt. Express 19(7), 5785–5801 (2011).
[CrossRef] [PubMed]

E. J. Alles, K. W. van Dongen, “Perfectly matched layers for frequency-domain integral equation acoustic scattering problems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58(5), 1077–1086 (2011).
[CrossRef] [PubMed]

B. Alavikia, O. M. Ramahi, “Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method,” J. Opt. Soc. Am. A 28(12), 2510–2518 (2011).
[CrossRef] [PubMed]

P. Demarcke, H. Rogier, “The poincare-steklov operator in hybrid finite element-boundary integral equation formulations,” IEEE Antennas Wireless Propag. Lett. 10, 503–506 (2011).
[CrossRef]

2009

F.-G. Hu, C.-F. Wang, “Preconditioned formulation of FE-BI equations with domain decomposition method for calculation of electromagnetic scattering from cavities,” IEEE Trans. Antennas Propag. 57(8), 2506–2511 (2009).
[CrossRef]

2008

Z. Peng, X.-Q. Sheng, “A flexible and efficient higher order FE-BI-MLFMA for scattering by a large body with deep cavities,” IEEE Trans. Antennas Propag. 56(7), 2031–2042 (2008).
[CrossRef]

2006

M. M. Botha, D. B. Davidson, “Rigorous, auxiliary variable-based implementation of a second-order ABC for the vector FEM,” IEEE Trans. Antennas Propag. 54(11), 3499–3504 (2006).
[CrossRef]

L. E. R. Petersson, J.-M. Jin, “Analysis of periodic structures via a time-domain finite-element formulation with a floquet ABC,” IEEE Trans. Antennas Propag. 54(3), 933–944 (2006).
[CrossRef]

Y. Altuncu, A. Yapar, I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44(6), 1435–1443 (2006).
[CrossRef]

C.-H. Kuo, M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54(8), 2392–2401 (2006).
[CrossRef]

2004

P. Liu, Y.-Q. Jin, “Numerical simulation of bistatic scattering from a target at low altitude above rough sea surface under an EM-wave incidence at low grazing angle by using the finite element method,” IEEE Trans. Antennas Propag. 52(5), 1205–1210 (2004).
[CrossRef]

2003

M. El-Shenawee, “Scattering from multiple objects buried beneath two-dimensional random rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. 51(4), 802–809 (2003).
[CrossRef]

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41(3), 799–826 (2003).
[CrossRef]

2002

D. E. Lawrence, K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50(10), 1368–1376 (2002).
[CrossRef]

2001

M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
[CrossRef]

1991

S. H. Lou, L. Tsang, C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Complex 1, 287–307 (1991).

1988

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

Akduman, I.

Y. Altuncu, A. Yapar, I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44(6), 1435–1443 (2006).
[CrossRef]

Alavikia, B.

Alles, E. J.

E. J. Alles, K. W. van Dongen, “Perfectly matched layers for frequency-domain integral equation acoustic scattering problems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58(5), 1077–1086 (2011).
[CrossRef] [PubMed]

Altuncu, Y.

Y. Altuncu, A. Yapar, I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44(6), 1435–1443 (2006).
[CrossRef]

Botha, M. M.

M. M. Botha, D. B. Davidson, “Rigorous, auxiliary variable-based implementation of a second-order ABC for the vector FEM,” IEEE Trans. Antennas Propag. 54(11), 3499–3504 (2006).
[CrossRef]

Bourlier, C.

Chan, C. H.

S. H. Lou, L. Tsang, C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Complex 1, 287–307 (1991).

Chen, Z.

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41(3), 799–826 (2003).
[CrossRef]

Davidson, D. B.

M. M. Botha, D. B. Davidson, “Rigorous, auxiliary variable-based implementation of a second-order ABC for the vector FEM,” IEEE Trans. Antennas Propag. 54(11), 3499–3504 (2006).
[CrossRef]

Demarcke, P.

P. Demarcke, H. Rogier, “The poincare-steklov operator in hybrid finite element-boundary integral equation formulations,” IEEE Antennas Wireless Propag. Lett. 10, 503–506 (2011).
[CrossRef]

El-Shenawee, M.

M. El-Shenawee, “Scattering from multiple objects buried beneath two-dimensional random rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. 51(4), 802–809 (2003).
[CrossRef]

M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
[CrossRef]

Fiaz, M. A.

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

Frezza, F.

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

Guo, L. X.

Hu, F.-G.

F.-G. Hu, C.-F. Wang, “Preconditioned formulation of FE-BI equations with domain decomposition method for calculation of electromagnetic scattering from cavities,” IEEE Trans. Antennas Propag. 57(8), 2506–2511 (2009).
[CrossRef]

Jin, J.-M.

L. E. R. Petersson, J.-M. Jin, “Analysis of periodic structures via a time-domain finite-element formulation with a floquet ABC,” IEEE Trans. Antennas Propag. 54(3), 933–944 (2006).
[CrossRef]

Jin, Y.-Q.

P. Liu, Y.-Q. Jin, “Numerical simulation of bistatic scattering from a target at low altitude above rough sea surface under an EM-wave incidence at low grazing angle by using the finite element method,” IEEE Trans. Antennas Propag. 52(5), 1205–1210 (2004).
[CrossRef]

Kubické, G.

Kuo, C.-H.

C.-H. Kuo, M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54(8), 2392–2401 (2006).
[CrossRef]

Lawrence, D. E.

D. E. Lawrence, K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50(10), 1368–1376 (2002).
[CrossRef]

Liang, Y.

Liu, P.

P. Liu, Y.-Q. Jin, “Numerical simulation of bistatic scattering from a target at low altitude above rough sea surface under an EM-wave incidence at low grazing angle by using the finite element method,” IEEE Trans. Antennas Propag. 52(5), 1205–1210 (2004).
[CrossRef]

Lou, S. H.

S. H. Lou, L. Tsang, C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Complex 1, 287–307 (1991).

Miller, E. L.

M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
[CrossRef]

Moghaddam, M.

C.-H. Kuo, M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54(8), 2392–2401 (2006).
[CrossRef]

Pajewski, L.

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

Peng, Z.

Z. Peng, X.-Q. Sheng, “A flexible and efficient higher order FE-BI-MLFMA for scattering by a large body with deep cavities,” IEEE Trans. Antennas Propag. 56(7), 2031–2042 (2008).
[CrossRef]

Petersson, L. E. R.

L. E. R. Petersson, J.-M. Jin, “Analysis of periodic structures via a time-domain finite-element formulation with a floquet ABC,” IEEE Trans. Antennas Propag. 54(3), 933–944 (2006).
[CrossRef]

Pinel, N.

Ponti, C.

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

Ramahi, O. M.

Rappaport, C.

M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
[CrossRef]

Rogier, H.

P. Demarcke, H. Rogier, “The poincare-steklov operator in hybrid finite element-boundary integral equation formulations,” IEEE Antennas Wireless Propag. Lett. 10, 503–506 (2011).
[CrossRef]

Sarabandi, K.

D. E. Lawrence, K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50(10), 1368–1376 (2002).
[CrossRef]

Schettini, G.

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

Sheng, X.-Q.

Z. Peng, X.-Q. Sheng, “A flexible and efficient higher order FE-BI-MLFMA for scattering by a large body with deep cavities,” IEEE Trans. Antennas Propag. 56(7), 2031–2042 (2008).
[CrossRef]

Silevitch, M. B.

M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
[CrossRef]

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

Tsang, L.

S. H. Lou, L. Tsang, C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Complex 1, 287–307 (1991).

van Dongen, K. W.

E. J. Alles, K. W. van Dongen, “Perfectly matched layers for frequency-domain integral equation acoustic scattering problems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58(5), 1077–1086 (2011).
[CrossRef] [PubMed]

Wang, C.-F.

F.-G. Hu, C.-F. Wang, “Preconditioned formulation of FE-BI equations with domain decomposition method for calculation of electromagnetic scattering from cavities,” IEEE Trans. Antennas Propag. 57(8), 2506–2511 (2009).
[CrossRef]

Wu, H.

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41(3), 799–826 (2003).
[CrossRef]

Wu, Z. S.

Yapar, A.

Y. Altuncu, A. Yapar, I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44(6), 1435–1443 (2006).
[CrossRef]

IEEE Antennas Wireless Propag. Lett.

P. Demarcke, H. Rogier, “The poincare-steklov operator in hybrid finite element-boundary integral equation formulations,” IEEE Antennas Wireless Propag. Lett. 10, 503–506 (2011).
[CrossRef]

IEEE Trans. Antennas Propag.

F.-G. Hu, C.-F. Wang, “Preconditioned formulation of FE-BI equations with domain decomposition method for calculation of electromagnetic scattering from cavities,” IEEE Trans. Antennas Propag. 57(8), 2506–2511 (2009).
[CrossRef]

Z. Peng, X.-Q. Sheng, “A flexible and efficient higher order FE-BI-MLFMA for scattering by a large body with deep cavities,” IEEE Trans. Antennas Propag. 56(7), 2031–2042 (2008).
[CrossRef]

P. Liu, Y.-Q. Jin, “Numerical simulation of bistatic scattering from a target at low altitude above rough sea surface under an EM-wave incidence at low grazing angle by using the finite element method,” IEEE Trans. Antennas Propag. 52(5), 1205–1210 (2004).
[CrossRef]

M. El-Shenawee, “Scattering from multiple objects buried beneath two-dimensional random rough surface using the steepest descent fast multipole method,” IEEE Trans. Antennas Propag. 51(4), 802–809 (2003).
[CrossRef]

D. E. Lawrence, K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50(10), 1368–1376 (2002).
[CrossRef]

C.-H. Kuo, M. Moghaddam, “Electromagnetic scattering from a buried cylinder in layered media with rough interfaces,” IEEE Trans. Antennas Propag. 54(8), 2392–2401 (2006).
[CrossRef]

M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, G. Schettini, “Scattering by a circular cylinder buried under a slightly rough surface: The cylindrical-wave approach,” IEEE Trans. Antennas Propag. 60(6), 2834–2842 (2012).
[CrossRef]

M. M. Botha, D. B. Davidson, “Rigorous, auxiliary variable-based implementation of a second-order ABC for the vector FEM,” IEEE Trans. Antennas Propag. 54(11), 3499–3504 (2006).
[CrossRef]

L. E. R. Petersson, J.-M. Jin, “Analysis of periodic structures via a time-domain finite-element formulation with a floquet ABC,” IEEE Trans. Antennas Propag. 54(3), 933–944 (2006).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

Y. Altuncu, A. Yapar, I. Akduman, “On the scattering of electromagnetic waves by bodies buried in a half-space with locally rough interface,” IEEE Trans. Geosci. Remote Sens. 44(6), 1435–1443 (2006).
[CrossRef]

M. El-Shenawee, C. Rappaport, E. L. Miller, M. B. Silevitch, “Three-dimensional subsurface analysis of electromagnetic scattering from penetrable/PEC objects buried under rough surfaces: Use of the steepest descent fast multipole method,” IEEE Trans. Geosci. Remote Sens. 39(6), 1174–1182 (2001).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

E. J. Alles, K. W. van Dongen, “Perfectly matched layers for frequency-domain integral equation acoustic scattering problems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58(5), 1077–1086 (2011).
[CrossRef] [PubMed]

J. Acoust. Soc. Am.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83(1), 78–92 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

SIAM J. Numer. Anal.

Z. Chen, H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41(3), 799–826 (2003).
[CrossRef]

Waves Random Complex

S. H. Lou, L. Tsang, C. H. Chan, “Application of finite element method to Monte Carlo simulations of scattering of waves by random rough surfaces: penetrable case,” Waves Random Complex 1, 287–307 (1991).

Other

J. M. Jin, The Finite Element Method in Electromagnetics (John Wiley, 2002).

L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Wave: Numerical Simulations (John Wiley, 2001).

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Figures (8)

Fig. 1
Fig. 1

2D scattering problem of multiple dielectric objects buried beneath a rough ground.

Fig. 2
Fig. 2

Integral paths of hybrid method.

Fig. 3
Fig. 3

Scattering from two dielectric square cylinders buried under a Gaussian rough ground: (a) the absolute value of the field; (b) BSC.

Fig. 4
Fig. 4

Scattering from three dielectric circular cylinders buried under a Gaussian rough ground: (a) the absolute value of the field; (b) BSC.

Fig. 5
Fig. 5

Scattering from two objects buried under a rough ground with different δ and l: (a) the absolute value of the field for a plane surface; (b) the absolute value of the field for δ = 0.1 λ and l = 0.8 λ ; (d) the absolute value of the field for δ = 0.18 λ and l = 0.6 λ ; (d) BSC.

Fig. 6
Fig. 6

Scattering from two dielectric objects buried under a rough ground for different incident angle θ i n c : (a) the absolute value of the field for θ i n c = 90 ; (b) the absolute value of the field for θ i n c = 60 ; (d) the absolute value of the field for θ i n c = 30 ; (d) BSC.

Fig. 7
Fig. 7

Scattering from two objects with different permittivity ε r buried under a rough ground: (a) the absolute value of the field for a rough surface with ε r = 2.5 j 0.01 ; (b) the absolute value of the field for a rough surface with ε r = 2.5 j 0.25 ; (d) the absolute value of the field for a rough surface with ε r = 6.5 j 0.01 ; (d) BSC.

Fig. 8
Fig. 8

Scattering from two objects buried under a rough ground with different permittivity ε r : (a) the absolute value of the field for a buried square cylinder with ε r = 3.5 j 0.01 and a circular cylinder with ε r = 5.5 j 0.05 ; (b) the absolute value of the field for a buried square cylinder with ε r = 3.5 j 0.15 and a circular cylinder with ε r = 5.5 j 0.45 ; (d) the absolute value of the field for a square cylinder with ε r = 6.5 j 0.01 and a circular cylinder with ε r = 9.5 j 0.05 ; (d) BSC.

Tables (2)

Tables Icon

Table 1 Solution time and Number of Unknowns in FEM-PML and FEM-BIE

Tables Icon

Table 2 Solution time and Number of Unknowns in FEM-PML and FEM-BIE

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

2 Φ( r )+ k a 2 Φ( r )=f( r )
Φ inc (r)=exp[jkr(1+w(r))]exp[ (xycot θ inc ) 2 / g 2 ]
2 G a ( r, r )+ k a 2 G a ( r, r )=δ( r r )
G a (r, r )= 1 4j H 0 2 ( k a | r- r |)
Φ Γ s + ( r ) = Φ i n c ( r ) + Γ s + [ Φ ( r ) G a ( r , r ) n s G a ( r , r ) Φ ( r ) n s ] d Γ
2 Φ( r )+ k b 2 Φ( r )=0
2 G b ( r, r )+ k b 2 G b ( r, r )=δ( r r )
G b (r, r )= 1 4j H 0 2 ( k b | r- r |)
Φ Γ s.or.oi ( r )= Γ s [ Φ( r ) G a (r, r ) n s G a (r, r ) Φ( r ) n s ]d Γ + i=1 n Γ oi [ Φ( r ) G a (r, r ) n oi G a (r, r ) Φ( r ) n oi ]d Γ
1 μ r Φ n | Γ =ψ
δ F oi (Φ)=0i=1,2,,n
F oi (Φ)= 1 2 Ω oi [ 1 μ r ( Φ x ) 2 + 1 μ r ( Φ y ) 2 k 0 2 ε r Φ 2 ]dΩ + Γ oi ΦψdΓ
Φ e ( r )= i=1 3 N i e Φ i e
Φ s ( r )= i=1 2 N i s Φ i s
ψ s ( r )= i=1 2 N i s ψ i s
[ Φ i ]=[ S 1+ ][ Φ Γ s+ ]+[ S 2+ ][ ψ Γ s+ ]
Φ i = N i s , Φ inc
S ij 1+ = N i s , N j s Γ s+ [ N j s G a (r, r ) n ]d Γ
S ij 2+ = N i s , Γ s+ N j s G a (r, r )dΓ
[0]=[ S 1 ][ Φ Γ s ]+[ S 2 ][ ψ Γ s ]+ Σ m=1 n [ O 1 ][ Φ Γ om ]+ Σ m=1 n [ O 2 ][ ψ Γ om ]
S ij 1 = N i s , N j s Γ s+ [ N j s G b (r, r ) n ]d Γ
S ij 2 = N i s , Γ s N j s G b (r, r )d Γ
O ij 1 = N i s , N j s + Γ o [ N j s G b (r, r ) n ]d Γ
O ij 2 = N i s , Γ o N j s G b (r, r )d Γ
[ M oi I ][ Φ oi I ]+[ M oi B ][ ψ oi B ]=[0]i=1,2,...,n
M ij I = Ω e [ 1 μ r N i e x N j e x + N i e y N j e y k 0 2 ε r N i e N j e ]dxdy
M ij B = Γ N i s N j s dΓ
Φ| Γ + = Φ| Γ
1 μ r+ Φ n | Γ + = 1 μ r Φ n | Γ
BSC= lim r 2πr | Φ scat | 2 P inc
P inc = π 2 gsin θ inc ( 1 1+2 cot 2 θ inc 2 ( k 0 gsin θ inc ) 2 )
y n =f( x n )= 1 L i= N /2 N/2 1 F( k i ) e j k i x n
F( k i )= 2πLW( k i ) { N(0,1) i=0,N/2 N(0,1)+jN(0,1) 2 other
W( k i )= δ 2 l 2 π exp( k i 2 l 2 4 )

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