Solving Scattering from Conducting Body Coated by Thin-Layer Material by Hybrid Shell Vector Element with Boundary Integral Method

Abstract

The finite element boundary integral (FEM-BI) method is widely used in the scattering and radiating problems. But for the conducting body coated by thin-layer material, plenty of fine meshes are required to discretize the geometry in the traditional FEM. It requires very expensive storage and CPU time. In this paper, the hybrid shell vector element with the boundary integral method is used to expedite the solution of thin coating problems. The shell vector elements are used to discretize thin-layer material instead of traditional tetrahedral elements. Consequently, the volume integral can be simplified into surface integral. This method reduces the number of unknowns greatly and is also extended into the complicated case of multi-thin-layer coating materials. Several numerical results are presented to prove the accuracy and efficiency of this present method.

1. Introduction

The electromagnetic analysis of the conducting body coated by thin-layer material has received much attention because these composite dielectric and conductor structures are used in many applications, such as the evaluation of echo from stealth aircraft and microwave integrated circuits design. Plenty of research work has been done on the efficient solution of scattering from conducting body coated by thin-layer material.

For the analysis of conductor structures coated by thin-layer material, there are many methods available. Integral equation methods based on thin dielectric sheet approximation (TDS) [1–7] avoid the volumetric discretization of material region, the computational region is only limited as the surface of conductor. A multilevel-TDS extension is also proposed for solution of conducting body coated by multi-thin-layer materials [8–10]. For lossy thin-layer coating, the impedance boundary condition (IBC) is also developed to simplify the electromagnetic analysis by building the relation between the equivalent surface magnetic current and equivalent surface electric current on the surface of conductor [11]. The application of the IBC into the conducting bodies of revolution coated with thin magnetic materials is also developed successfully [12]. A rigorous moment method solution of composite bodies with thin features can achieve accurate results for thin coating problems if the near singularity for near-self term integral can be tackled carefully, as shown in [13, 14].

Besides the integral equation methods, the finite element method (FEM) is also widely used for analysis of composite conducting body and dielectric because of its powerful ability of modeling complex inhomogeneous materials [15]. As well known, the absorption boundary condition (ABC) is required in FEM in order to truncate the computational region. In order to combine together the advantage of FEM and integral equation, the hybrid FEM with the boundary integral method (FEM-BI) is proposed [15–23]. No ABC is needed in the FEM-BI, and the number of unknowns is reduced greatly.

Though FEM-BI has a good computational property for composite conducting body and dielectric, it is deficient for the analysis of conductor structures coated by thin-layer material. This is because plenty of fine meshes will be required for modeling thin layers if using traditional elements like tetrahedral elements. To further reduce the total number of unknowns, prism elements are used in FEM when solving thin-coating structures [23]. In order to further enhance the computational efficiency, shell vector elements are developed in FEM, instead of prism elements [24, 25]. The volume integral can be simplified into surface integral by using shell vector elements.

In this paper, we combine the shell vector element method with the boundary integral method in order to achieve faster solution of scattering from conducting body coated by thin-layer material. The present method does not only reduce the number of unknowns but is also extended into more complicated cases of multi-thin-layer coating. This paper is organized as follows. The basic principle of the traditional FEM-BI method is reviewed in section II. Next, we show the characteristics of the shell elements, and the hybrid shell vector element method with the boundary integral method is implemented. Finally, several examples are shown to demonstrate the efficiency and accuracy of this present method. The conclusions are also given.

2. Traditional FEM-BI Method

Considering the electromagnetic scattering problem shown in Figure 1, the structure is illuminated by an incident plane wave 𝐸^inc or 𝐻^inc, the 𝑛 is the unit normal vector on surface S and S is the surface enclosing volume 𝑉.

Figure 1 

The 3D electromagnetic scattering problem by the FEM-BI.

The 𝐸 field inside volume 𝑉 satisfies the following equation:1∇×𝜇_𝑟∇×𝐸−𝑘²₀𝜀_𝑟𝐸=0, (1) where 𝜇_𝑟 and 𝜀_𝑟 denote the relative permeability and permittivity of the media, respectively, and 𝑘₀ is the wave number in free space.

The boundary conditions on surface 𝑆 are written as𝑛×𝐸|_𝑆−=𝑛×𝐸|_𝑆+,1𝑛×𝜇_𝑟∇×𝐸_𝑆−=−𝑗𝑘₀𝑛×𝐻_0𝑆+, (2) where 𝐻₀=𝜂₀𝐻 and 𝜂₀ is the wave impedance in free space.

The FEM matrix can be constructed based on the variational method; the functional 𝐹(𝐸) is expressed as follows:𝐹(𝐸)=12_𝑉1𝜇_𝑟(∇×𝐸)⋅(∇×𝐸)−𝑘²₀𝜀_𝑟𝐸⋅𝐸𝑑𝑉+𝑗𝑘₀𝜂_0𝑆𝑛⋅𝐸×𝐻_𝑠𝑑𝑆. (3)

The tetrahedral element is most widely applied in traditional FEM and the electric field 𝐸 can be expanded by the tetrahedral element as𝐸=⁶_𝑗=1𝐸^𝑒_𝑗𝑁^𝑒_𝑗, (4) where 𝐸^𝑒_𝑗 and 𝑁^𝑒_𝑗 are the unknown coefficient and the basis function of the jth edge, respectively.

For the surface integral term in (3), the magnetic field on the surface can be expanded by three edges of planar triangle on surface 𝑆 as𝐻_𝑠=³_𝑗=1𝐻^𝑠_𝑗𝑁^𝑠_𝑗, (5) where 𝐻^𝑠_𝑗 and 𝑁^𝑠_𝑗 are the unknown coefficient and the basis function of the jth edge, respectively.

Finally, the following FEM matrix equation is yielded from (3):[𝐾]{𝐸}+[𝐵]{𝐻}={0}. (6)

In the FEM-BI method, surface integral equation method is applied on the surface 𝑆. The electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) are used as follows:𝐸^inc=𝐿𝐽−𝐾(𝑀),𝐻^inc=𝐾𝐽+𝐿(𝑀). (7)

The equivalent surface electric and magnetic currents are defined as𝐽=𝑛×𝐻,𝑀=𝐸×𝑛,𝐽=𝑛×𝐻. (8) here 𝐻^inc=𝜂₀𝐻^inc. The integral operators 𝐿 and 𝐾 are expressed, respectively, as𝐿(𝑋)=𝑗𝑘_0𝑠′𝑋𝑟^′+1𝑘²₀∇∇^′⋅𝑋𝑟^′𝐺𝑟,𝑟^′𝑑𝑠^′,𝐾(𝑋)=TY(𝑟)+_𝑠′𝑋𝑟^′×∇𝐺𝑟,𝑟^′𝑑𝑠^′. (9)(10)

In (10), 𝑌(𝑟)=𝑋(𝑟)×𝑛, 𝑇=1−Ω/4𝜋. For the smooth surface, Ω=2𝜋. 𝐺=𝑒^{−𝑗𝑘0|𝑟−𝑟′|}/4𝜋|𝑟−𝑟′| is Green’s function in free space. The integral terms with bars in (10) denote the principal value integrals. The singularity is excluded.

By choosing 𝑆^𝑠_𝑖=𝑛×𝑁_𝑖 as the weighting function, the TE formulation can be derived as𝑃^TE𝐸_𝑆+𝑄^TE𝐻_𝑆=𝑏^TE, (11) where𝑃^TE_𝑖𝑗=−_𝑆𝑆^𝑠_𝑖⋅𝐾𝑁_𝑗𝑄×𝑛𝑑𝑠,_TE𝑖𝑗=_𝑆𝑆^𝑠_𝑖⋅𝐿𝜂₀𝑛×𝑁_𝑗𝑏𝑑𝑠,_TE𝑖=_𝑆𝑆^𝑠_𝑖⋅𝐸^𝑖𝑑𝑠. (12)

Similarly, the NE formulation can be derived by choosing 𝑛×𝑆^𝑠_𝑖 as the weighting function as𝑃^NE𝐸_𝑆+𝑄^NE𝐻_𝑆=𝑏^NE. (13)

The TH formulation can be derived by choosing 𝑆^𝑠_𝑖 as the weighting function as:𝑃^TH𝐸_𝑆+𝑄^TH𝐻_𝑆=𝑏^TH. (14)

The NH formulation can be derived by choosing 𝑛×𝑆^𝑠_𝑖 as the weighting function as:𝑃^NH𝐸_𝑆+𝑄^NH𝐻_𝑆=𝑏^NH. (15) To avoid the interior resonance that occurred in some cases, the combined field integral equation (CFIE), which combines the EFIE and MFIE, can be used,[𝑃]𝐸_𝑆+[𝑄]𝐻_𝑆={𝑏}, (16) where[𝑃]=𝛼𝑃^TE+𝛽𝑃^NE+𝛾𝑃^NH,[𝑄]=𝛼𝑄^TE+𝛽𝑄^NE+𝛾𝑄^NH,{𝑏}=𝛼𝑏^TE+𝛽𝑏^NE+𝛾𝑏^NH. (17)

The parameters 𝛼, 𝛽, and 𝛾 satisfy: 𝛼+𝛽+𝛾=1.

By combining (6) and (16), we get the matrix equation of the FEM-BI method as follows:𝐾_𝐼𝐼𝐾_𝐼𝑆0𝐾_𝑆𝐼𝐾_𝑆𝑆𝐵𝐸0𝑃𝑄^𝐼𝐸_𝑆𝐻_𝑆=00𝑏, (18) where 𝐸_𝐼, 𝐸_𝑆, and 𝐻_𝑆 are the coefficient of the electric field inside volume 𝑉, on surface 𝑆, and the magnetic field on surface 𝑆, respectively.

3. Hybrid Shell Element with Boundary Integral (Hybrid Shell Element-BI)

The shell element is the degenerated prism element. As shown in Figure 2, there are six edge vectors along the corresponding edges in the upper triangle and bottom triangle and three normal vectors. A linear function 𝛽(𝜍) was used to describe the variation of the field along the normal direction, and ∇𝛽=−𝑛/𝑑. The 𝛽𝑁_𝑗(𝑗=1,2,3) and 𝛽^′𝑁^′_𝑗(𝑗=1,2,3) are the edge basis functions in the upper and bottom triangle, respectively. The 𝐿_𝑗, 𝑗=1, 2, 3 is the normal basis function at node 𝑗.

(a) The prism element

(b) The shell element

Figure 2 

The structure of the prism element and the shell element.

By the shell element, the electric field is expanded as follows:𝐸^𝑒=³_𝑗=1𝐸^𝑒_𝑗𝛽𝑁_𝑗+𝐸^′𝑒_𝑗𝛽^′𝑁^′_𝑗+³_𝑗=1𝐸^𝑒_𝑛𝑗𝐿^𝑒_𝑗𝑛, (19) where 𝛽′=1−𝛽, 𝑁_𝑗=(𝐿^𝑒_𝑗1∇𝐿^𝑒_𝑗2−𝐿^𝑒_𝑗2∇𝐿^𝑒_𝑗1)𝑙^𝑒_𝑗, 𝑁^′_𝑗=(𝐿^𝑒_𝑗1∇𝐿^𝑒_𝑗2−𝐿^𝑒_𝑗2∇𝐿^𝑒_𝑗1)𝑙^𝑒_𝑗,  𝐸^𝑒_𝑗 is the expansion coefficient of the 𝑗th edge vector in the upper triangle, 𝐸_𝑗^′𝑒 is the expansion coefficient of the jth edge vector in the bottom triangle, 𝐸^𝑒_𝑛𝑗 is the expansion coefficient of the normal vector at node 𝑗.

Based on (3) and (19), we attain_𝑣∇×𝛽𝑁^𝑒_𝑖+𝛽′𝑁^𝑒_𝑖+𝐿^𝑒_𝑖×∑𝑛⋅∇_3𝑗=1𝐸^𝑒_𝑗𝛽𝑁^𝑒_𝑗+𝐸_𝑗^′𝑒𝛽^′𝑁^𝑒_𝑗+∑³_𝑗=1𝐸^𝑒_𝑛𝑗𝐿^𝑒_𝑗𝑛𝜇_𝑟−𝑘²₀𝜀_𝑟𝛽𝑁^𝑒_𝑖+𝛽′𝑁^𝑒_𝑖+𝐿^𝑒_𝑖⋅_3𝑗=1𝑛𝐸^𝑒_𝑗𝛽𝑁^𝑒_𝑗+𝐸^′𝑒_𝑗𝛽^′𝑁^𝑒_𝑗+³_𝑗=1𝐸^𝑒_𝑛𝑗𝐿^𝑒_𝑗𝑛𝑑𝑉=𝑗𝑘₀𝜂₀³_𝑗=1𝐻^𝑠_𝑗𝑠𝑁^𝑠_𝑖⋅𝑛×𝑁^𝑠_𝑗𝑑𝑠. (20)

For the shell element model, the volume integral can be simplified as:_𝑉∇×𝛽𝑁^𝑒_𝑖⋅∇×𝛽𝑁^𝑒_𝑗𝑑𝑣=^1/2_−1/2𝛽𝛽𝑑𝜍_𝑠∇×𝑁^𝑒_𝑖⋅∇×𝑁^𝑒_𝑗−𝑑𝑠_1/2−1/2𝛽𝑑𝑑𝜍_𝑠∇×𝑁^𝑒_𝑖⋅𝑆^𝑒_𝑗−𝑑𝑠_1/2−1/2𝛽𝑑𝑑𝜍_𝑠𝑆^𝑒_𝑖∇×𝑁^𝑒_𝑗𝑑𝑠+^1/2_−1/21𝑑²𝑑𝜍_𝑠𝑆^𝑒_𝑖⋅𝑆^𝑒_𝑗𝑑𝑠, (21)_𝑉∇×𝐿^𝑒_𝑖𝑛⋅∇×𝛽^′𝑁^𝑒_𝑗𝑑𝑣=^1/2_−1/2𝛽^′𝑑𝜍_𝑠∇𝐿^𝑒_𝑖×𝑛⋅∇×𝑁^𝑒_𝑗+𝑑𝑠_1/2−1/21𝑑𝑑𝜍_𝑠∇𝐿^𝑒_𝑖×𝑛⋅𝑆^𝑒_𝑗𝑑𝑠, (22) where 𝑆^𝑒_𝑖=𝑛×𝑁^𝑒_𝑖, 𝑆^𝑒_𝑗=𝑛×𝑁^𝑒_𝑗 and 𝑑 is the thickness of the thin-layer material.

Obviously, the integration along the normal direction can be calculated analytically. So the volume integral can be simplified into the surface integral.

For the conducting object coated by thin-layer material, the electric field in the bottom surface of shell element must be zero, only the integral in the upper surface is needed.

The hybrid shell element-BI can also be extended into the conducting body coated by multi-thin-layered material. For example, for the conducting body coated by three-thin-layer material, the matrix equation is expressed as:𝐾_𝑛𝑛1𝐾_𝑛𝑢10𝐾_𝑢𝑛1𝐾_{𝑢𝑢1+𝑏𝑏2}𝐾_𝑏𝑛2𝐾_𝑏𝑢2𝐾_𝑛𝑏2𝐾_𝑛𝑛2𝐾_𝑛𝑢2𝐾_𝑢𝑏2𝐾_𝑢𝑛2𝐾_{𝑢𝑢2+𝑏𝑏3}𝐾_𝑏𝑛3𝐾_𝑏𝑢3𝐾_𝑛𝑏3𝐾_𝑛𝑛3𝐾_𝑛𝑢3𝐾_𝑢𝑏3𝐾_𝑢𝑛3𝐾_𝑢𝑢3𝐵𝐸0𝑃𝑄^𝑛1𝐸_𝑢1𝐸_𝑛2𝐸_𝑢2𝐸_𝑛3𝐸_𝑢3𝐻_𝑠=000000𝑏, (23) where 𝐸_𝑛1 and 𝐸_𝑢1 are the expansion coefficients of the electric field in the normal and upper surface of the first layer, respectively; 𝐸_𝑛2 and 𝐸_𝑢2 are the expansion coefficients of the electric field in the normal and upper surface of the second layer, respectively, 𝐸_𝑛3 and 𝐸_𝑢3 are the expansion coefficients of the electric field in the normal and upper surface of the third layer, respectively.

4. Numerical Results

To demonstrate the accuracy and efficiency of the present method, some typical numerical results are shown here.

The first example is a conducting sphere coated by magnetic material, shown in Figure 3. The frequency is 300 MHz. The radius of the conducting sphere is 0.3367 m, and the thickness of the coating layer is 0.001 m. The relative permeability is 𝜇_𝑟=4 and the relative permittivity is 𝜀_𝑟=1. In the hybrid shell element with BI, the mesh density of 15 elements per wavelength is enough for achieving reasonable accuracy, but in traditional FEM, at least 32 elements per wavelength are required in order to mesh this very thin layer successfully. Table 1 demonstrates the comparisons between the two methods. As shown in Figure 4, the results by the hybrid shell element with BI agree with the ones of MIE method very well. And, the number of the unknowns by the traditional FEM-BI and the present method is 15167 and 2249, respectively. Obviously, the advantage of the present method over the traditional FEM-BI in reducing the number of unknowns is very remarkable.

Table 1 

The comparison of the FEM-BI and the hybrid shell element-BI. Bistatic RCS of the conducting sphere coated by magnetic material is calculated.

Figure 3 

Conducting sphere coated by magnetic material.

Figure 4 

The bistatic RCS of the conducting sphere coated by magnetic material by the hybrid shell element-BI.

To further investigate the accuracy of the hybrid shell element-BI, Figure 6 shows the bistatic RCS of the conducting sphere coated by thin dielectric material, the geometry is shown in Figure 5, 𝑘₀𝑎=6.28, the thickness of the thin layer is 𝑎/100, and the relative permittivity is 𝜀_𝑟=4, where 𝑘₀ is the wave number in free space and 𝑎 is the radius of the conducting sphere. The mesh density used here is only 6 elements per wavelength. As shown in Figure 6, the results by the hybrid shell element -BI agree well with the ones of the MIE method. In this case, the number of unknowns is 10530.

Figure 5 

Coated metal sphere.

Figure 6 

The bistatic RCS of the conducting sphere coated by thin dielectric material by the hybrid shell element with BI.

Figure 8 shows the bistatic RCS of the conducting cube coated by thin material in horizontal polarization. The geometry is shown in Figure 7. The length of the cube is 0.1m, and the thickness of the coated layer is 0.001m. The wavelength is 0.1m.The incident angle of the plane wave is 𝜃^inc=45^∘,𝜑^inc=0^∘.

Figure 7 

Conducting cube coated by thin material.

Figure 8 

The bistatic RCS of the coated metallic cube.

From Figure 8, it is shown clearly that the results by hybrid shell element-BI agree well with the ones by the commercial software FEKO. The number of unknowns in the hybrid shell element-BI is 5630, but the one in the traditional FEM-BI is 7760.

The solving CPU time by the present method is only one-third of the one by the traditional FEM-BI. The details are summarized in Table 2. Obviously, compared with the FEM-BI, the hybrid shell element with BI can save memory and CPU time greatly.

Table 2 

The comparison of the FEM-BI and the hybrid shell element-BI. Bistatic RCS of the conducting cube coated by thin material (mesh density with 10 elements/λ).

Figure 9 shows a conducting sphere with radius of 0.5𝜆₀ coated by three-thin-layer material, the thickness of each layer is 0.01𝜆₀, 𝜇_𝑟1=1.0, 𝜀_𝑟1=2.0, 𝜇_𝑟2=2.0−𝑗2.0, 𝜀_𝑟2=2.0−𝑗2.0, 𝜇_𝑟3=2.0, and 𝜀_𝑟3=2.0. The number of unknowns is 9591 and the total memory required is 736 Mb, but the number of unknowns in the traditional FEM-BI is 18775 and the total memory required is 2.9 Gb. The bistatic RCS computed by the hybrid shell element-BI is shown in Figure 10, the one by the MIE method is also shown for comparison. The lines marked with dot and square denote the curves of results in HH polarization and VV polarization, respectively. Obviously, they agree well with each other.

Figure 9 

Conducting sphere coated by three-thin-layer materials.

Figure 10 

The bistatic RCS of the conducting sphere coated by three-thin-layer material.

5. Conclusions

In this paper, the hybrid shell element with BI is developed to achieve efficient solution of scattering from the conducting body coated by thin-layer electric or magnetic material. Compared with the traditional FEM-BI, the present method reduces the number of unknowns greatly. Numerical results show that the hybrid shell element with BI is accurate and effective in solving the scattering problems of the conducting body coated with thin layer, also valid for the case of multi-thin-layer material coating.

Acknowledgment

The work is supported by the Nature Science Foundation of China (no. 60971032).