OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6844–6858
« Show journal navigation

An efficient hybrid method for scattering from arbitrary dielectric objects buried under a rough surface: TM case

Run-Wen Xu and Li-Xin Guo  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6844-6858 (2014)
http://dx.doi.org/10.1364/OE.22.006844


View Full Text Article

Acrobat PDF (2323 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

1. Introduction

Scattering from objects buried under a rough ground is a subject of increasing interests in scientific and operational applications, such as buried landmines, pipes, submarines, and the other buried objects of interest beneath a rough surface. A variety of techniques have been developed to study scattering properties of perfectly electric conducting (PEC) or dielectric objects buried under a rough ground. A steepest descent fast multipole method was used in [1

1. M. El-Shenawee, C. Rappaport, E. L. Miller, and 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

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]

] to deal with scattering from multiple objects buried under a rough surface. Subsequently, Lawrence and Sarabandi [3

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

] have studied scattering problems of a dielectric cylinder buried beneath a slightly rough surface by an analytical solution of the small perturbation method (SPM). In [4

4. Y. Altuncu, A. Yapar, and 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]

], an approach based on method of moments (MoM) was applied to discuss scattering properties of cylindrical bodies with arbitrary materials and cross sections buried beneath a rough interface, and a novel method was developed to obtain the Green's function of two half-spaces mediums with an arbitrary rough interface. The extended boundary condition method (EBCM) and a scattering matrix technique were developed in [5

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

] to analyze electromagnetic scattering of a buried cylinder in the layered media with rough interfaces. An efficient propagation-inside-layer-expansion method (PILEM) combined with the physical optics approximation (PO) [6

6. C. Bourlier, N. Pinel, and 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]

] was proposed to simulate scattering from coated cylinders, a rough layer, and an object buried below a rough ground. An analytical-numerical technique was presented in [7

7. M. A. Fiaz, F. Frezza, L. Pajewski, C. Ponti, and 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]

] based on the cylindrical wave approach (CWA) for the scattering problem of a cylinder buried under a rough surface, and rough deviations on the interface were dealt with the small perturbation method (SPM). In order to analyze electromagnetic scattering of a randomly rough surface with a buried target, a method combining the extended propagation-inside-layer expansion (EPILE) with the forward-backward method (FBM) [8

8. L. X. Guo, Y. Liang, and 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]

] was presented. The above methods are almost based on boundary integral methods, and they can easily and exactly be applied in electromagnetic simulations for the simple and homogeneous structures, especially for the PEC models. However, the boundary integral methods are hardly extended to the scattering problems of complicated inhomogeneous models, and applications of these methods are limited by their inherent characteristic.

In this paper, a hybrid efficient method combining FEM with BIE is presented to investigate two-dimensional (2D) scattering from dielectric objects buried beneath a dielectric rough surface. Above the rough surface, the region is a half-open homogenous region, and this region can be depicted by one boundary integral equation with the Green's function in free space. The earlier work based on FEM always used an artificial boundary to enclose the whole scattering model, which consumes largely on the computational time and memory. Here, based on domain decomposition technique, the region below the rough surface is divided into multiple interior regions containing each object and the region exterior to all the objects. FEM is applied only inside the interior regions, while another boundary integral equation is applied in the exterior region. On the artificial boundaries between the interior region and the exterior region, the equations of different domains are coupled by the continuous boundary conditions. This paper is the first attempt to implement and put in practice the theoretical developments of the mathematical deductions for the scattering from multiple dielectric objects buried under a rough surface based on the hybrid FEM-BIE. The hybrid algorithm presented here shows effectiveness and efficiency in terms of computing resources, computational time, and versatile applications.

2. Modeling and theoretical formulations

Figure 1 shows the 2D scattering problem of multiple dielectric objects with an arbitrary shape buried under a dielectric rough surface.
Fig. 1 2D scattering problem of multiple dielectric objects buried beneath a rough ground.
The incident wave Φinc impinges on the composite model with an incidence angle θinc, and is scattered with a scattering angle θscat. The symbol n^ denotes the unit normal vector on the artificial boundaries. Γoi is the truncated boundary of the ith object, while Γs denotes the truncated part of the rough surface. The interior region is defined as the domains enclosed by the boundaries Γoi containing the objects, and each subdomain of the interior region is expressed by Ωoi. Symbols Ωa and Ωb show the region above the rough surface and the region below the rough surface, respectively. For the region Ωoi of each dielectric object, the boundary Γoi is also applied as the truncated boundary of FEM region.

The incident wave Φinc for TM polarization is assumed to be invariant along the z axis, and the electric field only has a component along z axis. Above the rough ground, the total electric field satisfies the Helmholtz equation (a time factor ejωt has been assumed and suppressed), which can be written as
2Φ(r)+ka2Φ(r)=f(r)
(1)
where Φ(r) denotes the total electric field, ka is the wavenumber of the space Ωa, f(r) relates to the current Jz and f(r)=jkaηJz(r), η is the characteristic impedance.

Due to the infinite scale of a rough surface, it needs to be truncated into a limited length in our simulation. This can introduce the artificial truncated effect at the ends of the rough surface. To reduce this effect, the tapered incident wave is chose as an incident wave which decreases to a very small value at the ends of the rough surface. The form of the incident wave [19

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]

] can be expressed as
Φinc(r)=exp[jkr(1+w(r))]exp[(xycotθinc)2/g2]
(2)
where w(r)=[2(xycotθinc)2/g21]/(kgsinθinc)2, k is the vector of the wavenumber in the space, g is the tapered factor, and r is the position vector. On the other hand, the application of the tapered wave can ensure only the truncated part of the rough surface is shined by the incident wave, and this can provide a convenience in the following formulations.

In domain Ωa, we introduce the free space Green's function. It satisfies the Sommerfeld radiation condition at an infinite distance from the model and the following differential equation
2Ga(r,r)+ka2Ga(r,r)=δ(rr)
(3)
The Green’s function of the free space Ga(r,r) can be easily found to be a zeroth-order Hankel function of the second kind which can be written as

Ga(r,r)=14jH02(ka|r-r|)
(4)

Figure 2 shows the integral paths of BIE in the hybrid method above and below the rough surface.
Fig. 2 Integral paths of hybrid method.
There is a distance between the integral boundaries (Γoi, Γs+ and Γs) and the model in Fig. 2 just to illuminate integral paths of FEM-BIE, and they are set on the surfaces of the objects and the rough ground in our formulations. Because all sources and objects are immersed in free space and located within a finite distance from the origin of a coordinate system, the total field satisfies the Sommerfeld radiation conditions. Multiplying Eq. (1) with Ga, integrating over Ωa, and invoking the second Green's scalar theorem, a boundary integral equation can be obtained
ΦΓs+(r)=Φinc(r)+Γs+[Φ(r)Ga(r,r)nsGa(r,r)Φ(r)ns]dΓ
(5)
where Φinc(r)=Ωa[Ga(r,r)f(r)]dΩ, Γs+ denotes the truncated domain of a rough surface (+ denotes the side of a rough surface in domain Ωa), ΦΓs+(r) is the total field on Γs+. On the infinite boundary Γ above the rough surface, both Φ(r) and Ga(r,r) satisfy the Sommerfeld radiation conditions, while the incident field on parts of the rough surface which overflow from Γs are almost zero due to the adoption of a tapered incident wave. As a result, only the boundary integral on Γs+ remains in Eq. (5).

In the region Ωb, the Helmholtz equation is still satisfied
2Φ(r)+kb2Φ(r)=0
(6)
where kb is the wave number of Ωb. There is no source inside the soil, so the right part of Eq. (6) is zero. The Green's function Gb(r,r) is introduced in region Ωb, and it also satisfies the Sommerfeld radiation condition in the infinite distance and the following differential equation
2Gb(r,r)+kb2Gb(r,r)=δ(rr)
(7)
where Gb(r,r) is the zeroth-order Hankel function of the second kind below a rough ground which can be written as

Gb(r,r)=14jH02(kb|r-r|)
(8)

As deductions of Eq. (5), the integral equation in Ωb can be expressed as
ΦΓs.or.oi(r)=Γs[Φ(r)Ga(r,r)nsGa(r,r)Φ(r)ns]dΓ+i=1nΓoi[Φ(r)Ga(r,r)noiGa(r,r)Φ(r)noi]dΓ
(9)
where Γs is the truncated domain of a rough surface ( denotes the side of the rough surface in domain Ωb), and Γoi denotes an artificial boundary of the subdomain containing the ith object.

On artificial boundaries of the FEM domain, the explicit boundary condition is unknown at present. However, the boundary condition can be assumed as follows
1μrΦn|Γ=ψ
(10)
where the minus sign has been used simply for convenience.

As shown in Figs. 1 and 2, the whole computational space in Ωb is separated into many isolate interior subdomains Ωoi. Based on the published work [20

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

], the electromagnetic scattering problems in every closed subdomain of the objects can be formulated into an equivalent variational problem, which are given by the following equation

δFoi(Φ)=0i=1,2,,n
(11)

For every isolated computational subdomain Ωoi, the form of Foi(Φ) can be expressed as
Foi(Φ)=12Ωoi[1μr(Φx)2+1μr(Φy)2k02εrΦ2]dΩ+ΓoiΦψdΓ
(12)
where symbols Ωoi and Γoi denote the interior domain and its boundary of the ith object respectively. Scattered fields in subdomains Ωo1, Ωo2 ⋅⋅⋅, Ωo(n1), and Ωon can be calculated by the finite element theory.

In the interior region of the FEM domain, the domain can be discretized into small triangles with three nodes, and the boundary can be discretized into short line segments with two nodes. Choosing linear interpolating functions as in [20

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

] to discretize unknowns in Eqs. (5), (9) and (11), the field Φ and the normal derivation ψ on boundaries are expanded piecewise using the linear interpolating functions
Φe(r)=i=13NieΦie
(13)
Φs(r)=i=12NisΦis
(14)
ψs(r)=i=12Nisψis
(15)
where the superscript e shows the surface element in the interior region, and the superscripts s expresses the boundary element on Γoi and Γs, i denotes the ith nodes on the segments.

Therefore, discretize the integral equations by the linear interpolating basis functions, and Eq. (5) can be represented in matrix notation as
[Φi]=[S1+][ΦΓs+]+[S2+][ψΓs+]
(16)
where [S1] and [S2] are given by

Φi=Nis,Φinc
(17)
Sij1+=Nis,NjsΓs+[NjsGa(r,r)n]dΓ
(18)
Sij2+=Nis,Γs+NjsGa(r,r)dΓ
(19)

The field integral equation in Ωb can be expressed as
[0]=[S1][ΦΓs]+[S2][ψΓs]+Σm=1n[O1][ΦΓom]+Σm=1n[O2][ψΓom]
(20)
where elements of [S1], [S2], [O1], and [O2] are defined as

Sij1=Nis,NjsΓs+[NjsGb(r,r)n]dΓ
(21)
Sij2=Nis,ΓsNjsGb(r,r)dΓ
(22)
Oij1=Nis,Njs+Γo[NjsGb(r,r)n]dΓ
(23)
Oij2=Nis,ΓoNjsGb(r,r)dΓ
(24)

In subdomains Ωo1, Ωo2, ⋅⋅⋅, Ωo(n1), and Ωon, the scattering field can be calculated by the finite element method. Based on Eqs. (13)(15), the variational function of Eq. (11) can be generally arranged as
[MoiI][ΦoiI]+[MoiB][ψoiB]=[0]i=1,2,...,n
(25)
where the superscript I denotes the surface elements in each subdomain Ωoi, the superscript B is defined as the boundary elements on the artificial boundary Γoi of the subdomains, and the subscript oi indicates the computational domains of the ith target. The forms of matrixes [MoiB] and [MoiB] can be obtained from

MijI=Ωe[1μrNiexNjex+NieyNjeyk02εrNieNje]dxdy
(26)
MijB=ΓNisNjsdΓ
(27)

Equations (16), (20) and (25) are coupled by the continuity conditions on integral boundaries Γs, Γo1, Γo2, ⋅⋅⋅, Γo(n1), and Γon, which can be expressed as follows
Φ|Γ+=Φ|Γ
(28)
1μr+Φn|Γ+=1μrΦn|Γ
(29)
where Γ+ and Γ denote the observe point approaching the artificial boundary from both side of every interface. The element matrix can be evaluated using midpoint integration, and when integration points are at the same subsegment can be evaluated analytically. The linear system of equations in this paper is solved by the complex sparse system of linear equation by Gaussian elimination. The bistatic scattering coefficient (BSC) for the tapered incidence wave can be calculated which is defined as [19

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]

]
BSC=limr2πr|Φscat|2Pinc
(30)
where

Pinc=π2gsinθinc(11+2cot2θinc2(k0gsinθinc)2)
(31)

3. Numerical results and discussions

3.1 Validation of the hybrid method

In traditional FEM based on the truncated boundaries of ABC or PML, to keep their precision, the truncated boundaries should be set far enough from the scattering bodies to enclose a larger additional region. This leads to a prohibitive increase in the computational cost, especially for a large scale scattering model. Compared with published papers based on FEM employing ABC or PML, there is no need in hybrid FEM-BIE to fully enclose the scattering geometries to truncate the computational region. In our hybrid method, only the complex dielectric target need to be dealt with FEM, while BIE is applied to analyze the scattering from the rough surface. The interactions between dielectric objects and the rough surface are taken into account by boundary integral equations. What is more, the truncated boundaries enclosed the objects based on BIE can be set on the surface of the objects, which has no effect on the computational precision. Here, the validity of the hybrid method in this paper is verified by FEM-PML, and then the scattering properties of multiple dielectric objects buried beneath a ground are studied.

The profile of rough ground can be generated on the basis of Monte Carlo method [21

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

]. Assuming that rough surface is sampled at N points with spacing Δx over a truncated length of (N1)Δx. Results with the desired statistic properties can be generated at points xn=(n1)Δx(n=1,,N) by the following equation
yn=f(xn)=1Li=N/2N/21F(ki)ejkixn
(32)
where for i0,
F(ki)=2πLW(ki){N(0,1)i=0,N/2N(0,1)+jN(0,1)2other
(33)
For i<0, F(ki)=F(ki), where the asterisk implies complex conjugate. N(0,1) is a random number with a Gaussian distribution of zero mean and unit variance. The profile of the dielectric ground is characterized with Gaussian statistics in our simulations. W(ki) is the Gaussian power spectrum function.
W(ki)=δ2l2πexp(ki2l24)
(34)
where δ represents RMS height, l represents correlation length, and ki=2πi/L.

Numerical results are presented for the scattering from multiple objects buried beneath a rough ground. The truncated length of the rough ground is Lrs=25.6λ. A carefully tapered incident beam with g=Lrs/4 is used for excitation to eliminate truncated effects of a rough ground. In Figs. 3(a) and 3(b), the computer code of FEM-BIE is compared with traditional FEM-PML.
Fig. 3 Scattering from two dielectric square cylinders buried under a Gaussian rough ground: (a) the absolute value of the field; (b) BSC.
Two square cylinders are firstly considered to be buried under a rough ground. The relative dielectric constant of the Gaussian ground is assumed to be εr=2.5j0.18. The square cylinders with a length ls=1.6λ are buried at a depth d=2.5λ beneath the Gaussian rough surface. Centers of two square cylinder are x=2λ,y=2.5λ and x=2λ, y=2.5λ. The relative permittivity of both dielectric objects is εr=5.5j0.15. The incident angle of the tapered incident wave is set as θinc=90. The root mean square height of the Gaussian rough ground is δ=0.15λ, and the correlation length is l=0.6λ. The distribution of the total electric field on a square cylinder is plotted in Fig. 3(a), and BSC of two dielectric objects buried under the ground is shown in Fig. 3(b). It can be seen from Figs. 3(a) and 3(b) that two method agree with each other very well.

Table 1 shows the number of unknowns and the solution time for two different method.

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

table-icon
View This Table
| View All Tables
The results are calculated by a computer with a 2.50GHz processor (Intel (R) Core (TM)2 Quad CPU), 3.47GB memory. The number of unknowns for FEM-BIE is about 0.92% of that using traditional FEM-PML, while the time consumed in hybrid method is 20% of that using FEM-PML. In traditional FEM based on PML, PML should enclose the whole model, and need to be set far enough from the model in FEM simulations. Unlike traditional FEM based on PML, the artificial boundaries do not need to enclose the entire scattering model, and the integral boundary can be arranged in a very close distance from the model with an arbitrary shape. What is more, only BIE is used to simulate the scattering from the homogeneous rough surface, while FEM is just used to analyze the scattering of the dielectric objects. This can greatly reduce the computational domain, so the number of unknowns in the hybrid algorithm is less than FEM-PML. As a result, the time consumed in FEM-BIE is less than that of traditional FEM.

Considering that three circular cylinders are buried under a Gaussian rough surface, three cylinders with radius r=0.6λ are located at a depth d=1.5λ under a rough ground. The relative dielectric constant of the ground is assumed to be εr=2.5j0.18. Three objects are located at (2.5λ,-1.5λ), (0,1.5λ), and (2.5λ,1.5λ). All of three objects are assumed to have the same material parameter εr=3.5j0.05. The tapered incident wave impinges upon the model with an incident angle θinc=90, and rough parameters of the ground are assumed to be δ=0.05λ and l=0.8λ. Figure 4 shows numerical comparisons of the total electric field and BSC between different methods.
Fig. 4 Scattering from three dielectric circular cylinders buried under a Gaussian rough ground: (a) the absolute value of the field; (b) BSC.
The well matched results in two simulations guarantee a feasibility of the hybrid FEM-BIE again. Because FEM-BIE is based on differential equations, increasing the mesh density or using higher order basis functions can improve the precision of the hybrid method if the computer cost is not considered.

The comparisons of solution time and the number of unknowns in two methods are given in Table 2.

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

table-icon
View This Table
| View All Tables
The traditional FEM-PML yields more number of unknowns in the modeling and takes more time than that of FEM-BIE. The number of unknowns for FEM-BIE is reduced to 4.77% of that in FEM-PML, while the time consumed in hybrid method is 7.5% of those using FEM based on PML. Comparing Table 2 with 1, the number of unknowns and the solution time for FEM-PML have a little decrease in some extent. This is because the depth of the objects for Table 2 is less than that of Table 1. When the distance between the buried objects and the horizontal surface increases, the number of unknowns and the solution time in FEM-PML will have a sharp rise. The number of unknowns and the time consumed in Table 2 have a noticeable decrease for FEM-BIE compared with Table 1, and they are mainly determined by the total areas of objects. Therefore, the efficiency of FEM-BIE is almost independent of the distance between objects and the ground. The hybrid technique is highly efficient in terms of computational memory, time, and versatility, especially for the scattering problem of a large-scale rough surface or the objects away from the ground.

3.2 Numerical results

To simulate a more general case, a scattering model of different objects buried beneath a Gaussian rough ground are assumed in the following simulations. The relative dielectric constant of a rough ground is assumed to be εr=2.5j0.08. A circular cylinder with a radius r=λ and εr=3.5j0.05 is buried at x=2λ,y=2.5λ, while a square cylinder with a length ls=2λ and εr=5.5j0.15 is assumed to be located at x=3λ, y=3.5λ. To see the influence of roughness on the distribution of total field and BSC, images based on the absolute magnitude of the total field are presented in 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=0.1λ and l=0.8λ; (d) the absolute value of the field for δ=0.18λ and l=0.6λ; (d) BSC.
The incident angle of a tapered incident wave is θinc=90. The computational region in x-y plane with a size of 25.6λ×25.6λ is simulated to show the distribution of the total electric field. The images shown in Figs. 5(a)5(c) are for two objects buried under a plane surface, a rough surface with δ=0.1λ and l=0.8λ, and a rough surface with δ=0.18λ and l=0.6λ, respectively. Figure 5(d) gives a comparison of BSC corresponding for the scattering models with different roughness. With increase of the ground roughness, the specular scattering energy decreases, and the scattering energy in non-specular direction shows a rise.

Variation of the absolute total electric field versus different incident angles is demonstrated in Fig. 6 for two different dielectric objects buried under a Gaussian ground with εr=2.5j0.08.
Fig. 6 Scattering from two dielectric objects buried under a rough ground for different incident angle θinc: (a) the absolute value of the field for θinc=90; (b) the absolute value of the field for θinc=60; (d) the absolute value of the field for θinc=30; (d) BSC.
The circular cylinder with a radius of r=λ and εr=3.5j0.05 is buried at x=2λ and y=2.5λ. The square cylinder with a length of ls=2λ and εr=5.5j0.15 is located at x=3λ and y=3.5λ. The root mean square height of the rough surface is δ=0.12λ, and the correlative length of the rough surface is l=0.75λ. The tapered incident wave impinges on the rough surface with an incident angle θinc=90, θinc=60, and θinc=30 in Figs. 6(a)6(c), respectively. It can be seen from Fig. 6, the scattering results of BSC reach to a peak value at the corresponding specular angle for different incident angles. When the tapered incident wave impinges the middle of the rough surface by an incident angle of θinc=30, the transmissive wave inside the soil almost impinges upon the square, and the energy impinges on the circular cylinder is very little.

In Fig. 7, the relative permittivity of a rough ground is changed to discuss their influence on the absolute value of near field and BSC.
Fig. 7 Scattering from two objects with different permittivity εr buried under a rough ground: (a) the absolute value of the field for a rough surface with εr=2.5j0.01; (b) the absolute value of the field for a rough surface with εr=2.5j0.25; (d) the absolute value of the field for a rough surface with εr=6.5j0.01; (d) BSC.
The parameters of dielectric objects buried under the ground are the same as in Fig. 6. The root mean square height of the rough surface is δ=0.2λ, and the correlative length of the rough surface is l=0.5λ. The tapered incident wave impinges upon the rough surface with an incident angle θinc=60. Figure 7 gives the distribution of the total electric field and the comparison of BSC for a rough surfaces of εr=2.5j0.01 in Fig. 7(a), εr=2.5j0.25 in Fig. 7(b), and εr=6.5j0.01 in Fig. 7(c). The material of the ground has a great influence on the scattering pattern. The imaginary part of the permittivity εr relates to the energy loss of the ground, and the real part of the permittivity εr is concerned with the reflectivity and transmissivity of the ground. When the imaginary part of εr increases, the transmissive wave quickly decays with the depth increasing. The scattering energy above the rough surface becomes strong as shown in Figs. 7(c) and 7(d), while the transmissive energy decreases when the real part of εr increases.

Figure 8 illustrates scattering results of the absolute value of the total electric field and BSC when dielectric objects have different permittivity.
Fig. 8 Scattering from two objects buried under a rough ground with different permittivity εr: (a) the absolute value of the field for a buried square cylinder with εr=3.5j0.01 and a circular cylinder withεr=5.5j0.05; (b) the absolute value of the field for a buried square cylinder with εr=3.5j0.15 and a circular cylinder with εr=5.5j0.45; (d) the absolute value of the field for a square cylinder with εr=6.5j0.01 and a circular cylinder with εr=9.5j0.05; (d) BSC.
Dielectric objects are buried in a more dry soil under a rough interface with δ=0.2λ and l=0.5λ. The relative dielectric constant of the ground is assumed to be εr=2.5j0.01. The rate of decrease in the transmitted wave is very small in this case, so its transmissive energy is very strong inside the soil. The incident angle is assumed to be θinc=60. The model sizes and locations of two objects are the same as Fig. 6, and the materials of the square cylinder and circular cylinder are assumed to be εr=3.5j0.01 and εr=5.5j0.05 in Fig. 8(a), εr=3.5j0.15 and εr=5.5j0.45 in Fig. 8(b), εr=6.5j0.01 and εr=9.5j0.05 in Fig. 8(c), respectively. The imaginary part of the permittivity εr of the objects mainly relates to the energy loss inside the objects, while the real part of the permittivity εr is mainly concerned with the reflectivity and transmissivity on the surface of the objects.

4. Conclusion

In this work, an efficient hybrid method combining FEM with BIE is developed, and the scattering from multiple objects buried beneath a rough surface is investigated. In the simulations, the whole computational domain is divided into multiple domains containing each object and the rough surface. FEM is applied only inside the regions of objects, while the domain above the rough surface and the domain inside the soil exterior to the objects are analyzed by BIE. Compared with published works of traditional FEM based on PML and ABC, the hybrid method can reduce the computational domain of the scattering problem and achieve a more precise result because BIE in the hybrid method incorporates the Sommerfeld radiation condition through the use of an appropriate Green’s function. It can be well used to the scattering problem of multiple objects below a rough surface with a large scale. Validated by classical FEM-PML, the hybrid technique shows highly efficient in terms of computational memory, time, and versatility. The scattering properties of two different objects buried under the ground is discussed in detail based on hybrid method. If combining FEM-BIE with the parallel technology or the optimization of the sparse matrix storage method, the hybrid method can be expected to get a more efficient result. In the future, most work will be focused on the application of hybrid FEM-BIE for a three-dimensional (3D) scattering problem and the accelerated treatments on the hybrid method.

Acknowledgments

This work was supported by the National Natural Science Foundation for Distinguished Young Scholars of China (Grant No. 61225002), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20100203110016), and the Fundamental Research Funds for the Central Universities (Grant No. K5051007001).

References and links

1.

M. El-Shenawee, C. Rappaport, E. L. Miller, and 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 and 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, and 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 and 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, and 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, and 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, and 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 and 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 and 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 and 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 and 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 and 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, and 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 and 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 and 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 and 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 and 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).

OCIS Codes
(290.5880) Scattering : Scattering, rough surfaces
(280.1350) Remote sensing and sensors : Backscattering

ToC Category:
Scattering

History
Original Manuscript: January 20, 2014
Revised Manuscript: February 22, 2014
Manuscript Accepted: March 6, 2014
Published: March 17, 2014

Citation
Run-Wen Xu and Li-Xin Guo, "An efficient hybrid method for scattering from arbitrary dielectric objects buried under a rough surface: TM case," Opt. Express 22, 6844-6858 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6844


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited