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
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, 2] to deal with scattering from multiple objects buried under a rough surface. Subsequently, Lawrence and Sarabandi  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 , 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  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)  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  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)  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.
The finite element method (FEM) is extensively applied to analyze electromagnetic scattering from the complicated inhomogeneous models, because it is very efficient in modeling scattering geometries which contain both PEC and complicated inhomogeneous bodies. In published papers, the truncated boundaries, such as the absorbing boundary condition (ABC) [9, 10] and the perfectly matched layer (PML) [11–14], were usually used as constrained boundaries in applications of FEM. However, these approximate absorbing boundaries often need to be set far enough away from the model surface to keep their precision, and they are often invalid to some particular problems. To improve the precision and versatility of FEM, the integral boundaries were developed as the artificial truncated boundaries. A hybrid finite element-surface integral equation method (FE-SIE)  was presented by Alavikia and Ramahi to study the scattering from cylindrical objects above a perfectly electric conducting plane surface. Demarcke and Rogier  applied the poincare-steklov operator notation to provide a substantial theoretical insight into the analysis of spurious solutions in hybrid finite element-boundary integral equation formulations (FE-BIE), and applied the theory on a scattering example to predict the breakdown frequencies of different hybrid formulations. Combined with the domain decomposition method , Hu and Wang adopted preconditioned formulation of finite element-boundary integral method (FE-BI) to analyze electromagnetic scattering from cavities. Peng and Sheng applied a higher order FE-BI with multilevel fast multipole algorithm (MLFMA)  to study the scattering of a large body with deep cavities. Although many valuable papers have been contributed to electromagnetic scattering simulations in some important publications and journals, there are still few papers to study the scattering from multiple dielectric objects buried under a rough ground using FEM because of the large-scale scattering model and the complicated formulations. In the previous papers, most work based on FEM was limited to the scattering problems of the regular objects or the PEC rough surfaces.
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. The incident wave impinges on the composite model with an incidence angle , and is scattered with a scattering angle . The symbol denotes the unit normal vector on the artificial boundaries. is the truncated boundary of the th object, while denotes the truncated part of the rough surface. The interior region is defined as the domains enclosed by the boundaries containing the objects, and each subdomain of the interior region is expressed by . Symbols and show the region above the rough surface and the region below the rough surface, respectively. For the region of each dielectric object, the boundary is also applied as the truncated boundary of FEM region.
The incident wave for TM polarization is assumed to be invariant along the axis, and the electric field only has a component along axis. Above the rough ground, the total electric field satisfies the Helmholtz equation (a time factor has been assumed and suppressed), which can be written as
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  can be expressed as
In domain , 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
Figure 2 shows the integral paths of BIE in the hybrid method above and below the rough surface. There is a distance between the integral boundaries (, and ) 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 , integrating over , and invoking the second Green's scalar theorem, a boundary integral equation can be obtainedEq. (5).
In the region , the Helmholtz equation is still satisfiedEq. (6) is zero. The Green's function is introduced in region , and it also satisfies the Sommerfeld radiation condition in the infinite distance and the following differential equation
As deductions of Eq. (5), the integral equation in can be expressed as
On artificial boundaries of the FEM domain, the explicit boundary condition is unknown at present. However, the boundary condition can be assumed as follows
As shown in Figs. 1 and 2, the whole computational space in is separated into many isolate interior subdomains . Based on the published work , 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
For every isolated computational subdomain , the form of can be expressed as
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  to discretize unknowns in Eqs. (5), (9) and (11), the field and the normal derivation on boundaries are expanded piecewise using the linear interpolating functions
Therefore, discretize the integral equations by the linear interpolating basis functions, and Eq. (5) can be represented in matrix notation as
The field integral equation in can be expressed as19]
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 . Assuming that rough surface is sampled at points with spacing over a truncated length of . Results with the desired statistic properties can be generated at points by the following equation
Numerical results are presented for the scattering from multiple objects buried beneath a rough ground. The truncated length of the rough ground is . A carefully tapered incident beam with 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. 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 . The square cylinders with a length are buried at a depth beneath the Gaussian rough surface. Centers of two square cylinder are , and , . The relative permittivity of both dielectric objects is . The incident angle of the tapered incident wave is set as . The root mean square height of the Gaussian rough ground is , and the correlation length is . 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. 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 are located at a depth under a rough ground. The relative dielectric constant of the ground is assumed to be . Three objects are located at , , and . All of three objects are assumed to have the same material parameter . The tapered incident wave impinges upon the model with an incident angle , and rough parameters of the ground are assumed to be and . Figure 4 shows numerical comparisons of the total electric field and BSC between different methods. 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. 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 . A circular cylinder with a radius and is buried at ,, while a square cylinder with a length and is assumed to be located at , . 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. The incident angle of a tapered incident wave is . The computational region in x-y plane with a size of 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 and , and a rough surface with and , 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 . The circular cylinder with a radius of and is buried at and . The square cylinder with a length of and is located at and . The root mean square height of the rough surface is , and the correlative length of the rough surface is . The tapered incident wave impinges on the rough surface with an incident angle , , and 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 , 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. 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 , and the correlative length of the rough surface is . The tapered incident wave impinges upon the rough surface with an incident angle . Figure 7 gives the distribution of the total electric field and the comparison of BSC for a rough surfaces of in Fig. 7(a), in Fig. 7(b), and in Fig. 7(c). The material of the ground has a great influence on the scattering pattern. The imaginary part of the permittivity relates to the energy loss of the ground, and the real part of the permittivity is concerned with the reflectivity and transmissivity of the ground. When the imaginary part of 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 increases.
Figure 8 illustrates scattering results of the absolute value of the total electric field and BSC when dielectric objects have different permittivity. Dielectric objects are buried in a more dry soil under a rough interface with and . The relative dielectric constant of the ground is assumed to be . 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 . 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 and in Fig. 8(a), and in Fig. 8(b), and in Fig. 8(c), respectively. The imaginary part of the permittivity of the objects mainly relates to the energy loss inside the objects, while the real part of the permittivity is mainly concerned with the reflectivity and transmissivity on the surface of the objects.
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.
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).
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