# A Domain Decomposition Method for Hybrid Shell Vector Element with Boundary Integral Method

Authors: Lin Lei; Jun Hu; Hao-Quan Hu
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

For the conducting body coated with thin-layer material, plenty of fine meshes are required in general. In this paper, shell vector element (SVE) is used for modeling of thin coating dielectric. Further, a domain decomposition (DD) method for hybrid shell vector element method boundary integral (SVE-BI) is proposed for analysis of electromagnetic problem of multiple three-dimensional thin-coating objects. By this method, the whole computational domains are divided into sub-SVE domains and boundary element domains. With shell element, not only the unknowns are far less than the one by traditional vector element method, but only surface integral is required. The DDM framework used for hybrid SVE-BI also enhances the computational efficiency of solving scattering from multiple coating objects greatly. Finally, several numerical examples are presented to prove the accuracy and efficiency of this DDM-SVE-BI method.


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## Body

## 1. Introduction

Recently electromagnetic scattering from multiple conducting bodies coated by thin-layer dielectric has attracted more and more interest. Typical applications can be found in scattering from composite conductor and dielectric, microwave integrated circuits design, analysis of antenna array, and so on.For conductor structures coated by thin-layer material, many numerical methods have been developed. Integral equation methods based on thin dielectric sheet approximation (TDS) [1–5] 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 multithin-layer materials [6, 7]. 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 magnetic current and electric current on the surface of conductor [8]. A rigorous moment method solution of composite bodies with thin coating dielectric is also shown in [9].As well known, finite element method (FEM) is also widely used for analysis of composite conducting body and dielectric because of its powerful ability of modeling inhomogeneous materials. In order to combine together the advantage of FEM and integral equation, the hybrid FEM with boundary integral method (FEM-BI) is proposed [10–12]. Though FEM-BI has a good computational property for composite structures, 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 required in FEM, shell vector elements (SVE) [13] are developed, respectively. The SVEs are also extended into FEM-BI framework by us [14]. A remarkable advantage of using SVEs is that the volume integral can be simplified into surface integral.To realize efficient analysis of complex structures, domain decomposition method (DDM) is developed based on FEM-BI framework [15, 16]. The FEM is chosen for interior regions of subdomains and the BI is chosen for the rest of the system. In this paper, a novel DDM is developed based on hybrid SVE and BI method (SVE-BI) in order to realize fast solution of multiple conducting bodies coated by thin-layer dielectric. By this method, the original problem is divided into many internal domains and external domains. For internal domains, SVE is used to reduce the number of unknowns. For external domains, BI method is adopted to take into account all interactions between equivalent electric current and magnetic current on the boundary of each object.The rest of the paper is organized as follows: the hybrid SVE-BI method is briefly introduced in Section2.  In Section 3, we demonstrate the DDM-SVE-BI method. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of this method. The conclusions are also given.


## 2. Hybrid Shell Vector Element and Boundary Integral Method (SVE-BI)

For sake of simplicity, 3D electromagnetic scattering problem of conducting object coated by thin dielectric illuminated by an incident waveEinc is considered, as shown in Figure 1. The n^ is the unit normal vector on surface of the object. In the SVE-BI method, the SVE is used in dielectric domain of the object, the BI is used on the surface of the object. Here, Ju, Mu are the equivalent surface electric and magnetic currents on the surface.Figure 1
The 3D electromagnetic scattering problem of conducting object coated by thin dielectric.TheE field inside dielectric domain of object satisfies the following equation:
(1)∇×(μr-1∇×E)-k02εrE=0,
where μr, εr denotes the relative permeability and permittivity of the dielectric, respectively. k0 is the wave number in free space.The boundary conditions on the surface of object are written as:(2)n^×E|S-=n^×E|S+n^×(1μr∇×E)|S-=-jk0n^×H-|S+,
where H-=η0H, η0 is the wave impedance in free space.The functionalF(E) is expressed as follow:
(3)F(E)=12∭V[μr-1(∇×E)⋅(∇×E)-k02εrE⋅E]dV+jk0η0∬Sn^⋅(E×Hu)dS.Finally, the FEM matrix equation of the object is yielded from (3):
(4)[K]{E}+[B]{Hu}={0},
where E is the electric field expansion coefficient inside the object. Hu is the magnetic field expansion coefficient on the object surface. The matrix K comes from the volume integration, matrix B comes from the surface integration.In the SVE-BI method, integral equation method is applied on the surface of the object. As shown in [14], the matrix is written as follows:
(5)[P]{Eu}+[Q]{Hu}={b},
where P and Q come from the surface integration.Combing (4) and (5), all unknown coefficients can be solved.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 as follows:
(6)Hu=∑j=13HjsNjs,
where Hjs, Njs are the unknown coefficient and the basis function of the jth edge, respectively.By the SVE, the electric field is expanded as follows:(7)Ee=∑j=13(EjeβNj+Ej′eβ′Nj')+∑j=13EnjeLjen^,
where β′=1-β,  Nj=(Lj1e∇Lj2e-Lj2e∇Lj1e)lje,  Nj′=(Lj1e∇Lj2e-Lj2e∇Lj1e)lje,  Eje: the expansion coefficient of the jth edge vector in upper triangle, Ej′e: the expansion coefficient of the jth edge vector in bottom triangle, and Enje: the expansion coefficient of normal vector at the node-j.The shell element is the degenerated prism element. As shown in Figure2, there are total six edge vectors along the corresponding edges in the upper triangle and bottom triangle, and three normal vectors.The structure of the prism vector element and the shell vector element.
(a)
the prism vector element
(b)
the shell vector elementA linear functionβ(ς) was used to describe the variation of the field along the normal direction, and ∇β=-n^/d. The βNj(j=1,2,3), β′Nj′(j=1,2,3) are the edge basis functions in the upper and bottom triangle, respectively. The Lj,j=1,2,3 is the normal basis function at the node-j.For the SVE, as shown in [14], the integration along the normal direction can be calculated analytically. So the volume integral can be simplified into surface integral. More details about the SVE-BI can be found in [14].For conducting objects coated by thin-layer material, the electric field in the bottom surface of shell vector element must be zero, only the integral in the upper surface is needed.


## 3. The DDM-SVE-BI Method

For multiple conducting bodies coated with thin dielectric as shown in Figure3, the whole computational domains are divided into a lot of sub-SVE domains and boundary element domains. For each sub-SVE domain, namely, volume domain of each object, equation (4) can be expressed as follows:
(8)[KnnKnuKunKuu]m{EnmEum}=-[0B]m{Hum},
where Enm is the normal electric field expansion coefficient at each node of the SVEs of the mth object, which is a column matrix of Nnm×1. Eum is the electric field expansion coefficient along each edge of the SVEs of the mth object, which is a column matrix of Num×1. In (8), Knn is a square matrix  of Nnm×Nnm, Knu is  a matrix of Nnm×Num, Kun is  a matrix of Num×Nnm, Kuu is a square matrix  of Num×Num, B is a square matrix  of Num×Num, Hum is  a column matrix of Num×1. Nnm, Num is the number of nodes, the number of edges of all SVEs of the mth object, respectively.Figure 3
The 3D electromagnetic scattering problem of multiple conducting bodies coated with thin layer dielectric.Equation (8) can be written as:
(9){EnmEum}=-[KnnKnuKunKuu]m-1[0B]m{Hum}.Extracting surface electric field expansion coefficientEumfrom (9):
(10){Eum}=[XCm]{Hum},
where XCm is a square matrix of Num×Num. According to the present DDM, the computation of XCm can be easily parallelized, the computational time can be saved greatly. Because of the sparse property of matrix [K], the iterative method usually is used to attain XCm. But it is not always successful, and it costs too much time when the condition number of matrix [K] is not very well. In this paper, the inverse of matrix is used for computation of XCm to keep good accuracy. Here, the direct Gaussian Elimination is used for attaining XCm.Substitute (10) into (5), we obtain,
(11)([P][XC1XC2⋯XCm]+[Q])[Hu]=[b].Here,[P] and [Q] is a square matrix of Nu×Nu,  Hu is a column matrix of Nu×1,Nu=Nu1+Nu2+⋯Numdenotes the summation of surface unknowns along the edges about all the objects. For solving (11), direct Gaussian Elimination technique is used to attain accurate results.


## 4. Numerical Results

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


### 4.1. Two Dielectric Sphere

The first example is two separated dielectric spheres, shown in Figure4. The frequency is 300 MHz. The radius of the two dielectric spheres are 0.2 m. The distance between the centers of two spheres is 0.8 m. The relative permittivity is εr=2.0. The incident angle of plane wave is θ=45°,φ=0°，the bistatic RCS in horizontal polarization is computed. Here, DDM for FEM-BI is used to calculate scattering from two dielectric spheres in order to prove the accuracy of DDM. From Figure 5, it is shown clearly that the results by DDM-FE-BI agree well with the one by commercial software FEKO based on the surface integral equation with PMCHWT formulation [17].Figure 4
Two dielectric spheres.Figure 5
The Bistatic RCS of two dielectric spheres.


### 4.2.3×2 Array of Coating PEC Spheres

The second example is to solve scattering of3×2 array of coating PEC spheres located in the x-y  plane, as shown in Figure 6. The radius of PEC sphere is 0.2 m. The thickness of dielectric coating is 0.05 m with εr=2.0,μr=1.0. The distance between the centers of two elements is 0.8λ0 in both the x- and y-dimension as shown in Figure 6. The incident electric field is x^ polarized plane wave propagating into the negative z^ direction at 0.3 GHz. The results by Moment of Method (MoM) are also given for comparison.Figure 6
3
×
2 array of coating PEC spheres.As shown in Figure7, the results by DDM-SVE-BI agree with the one by DDM-FE-BI and the result of the MoM very well.Figure 7
The Bistatic RCS of3×2 array of coating PEC spheres.The comparison of the DDM-FE-BI and DDM-SVE-BI is shown in Table1. Table 1 demonstrates that the unknowns, the memory requirement of matrix K for single coating PEC sphere with the SVEs is only 52%, 27% of the traditional tetrahedral elements, respectively. The CPU time for attaining single XC by the SVEs is only 20% of the one by traditional tetrahedral elements.Table 1
The comparison of the DDM-FE-BI and DDM-SVE-BI. Bistatic RCS of3×2 array of coating PEC spheres. (mesh density with λ0/10).
Method
DDM-FE-BI
DDM-SVE-BI
Unknowns of single PEC coating sphere
523
274
The total unknowns
3138
1644
Memory of matrixK for single coating PEC sphere
2.2 Mb
0.6 Mb
Computational time for matrixXC
22.4 s
4.3 sObviously, the advantages of the DDM-SVE-BI over the DDM-FE-BI on reducing the number of unknowns and computational time are very remarkable.To further prove the accuracy of the present method, the RCS of this same3×2 array with magnetic material coating is also computed, the material parameters are εr=1.0,μr=2.0-j1.5. The same incident angle with x^ polarized is analyzed. As shown in Figure 8, the results of DDM-SVE-BI also agree with the results of DDM-FE-BI very well.Figure 8
The Bistatic RCS of3×2 array of coating PEC spheres with magnetic material.To further investigate the influence of different coating materials on the RCS of this3×2 array, four different coating materials with εr=2.0,μr=1.0, εr=2.0-j1.8,μr=1.0, εr=1.0,μr=2.0-j1.5 including PEC are evaluated. As shown in Figure 9, compared with the RCS from 3×2 PEC array, two lossy coating materials reduce the backscattering RCS greatly. The RCS reduction achieves 2.8 dB, 6.9 dB for the case of εr=2.0-j1.8,μr=1.0, εr=1.0,μr=2.0-j1.5, respectively. For this structure, lossless coating material with εr=2.0,μr=1.0 has little influence on the backscattering RCS. For the forward RCS, it is enhanced about 3 dBSW in all four cases. It is also demonstrated from Figure 9 that the magnetic lossy coating material has great influence on the RCS of this array.Figure 9
The Bistatic RCS of3×2 array of coating PEC spheres with different materials.


### 4.3. Two Coating PEC Cubes

The third example is two coating PEC cubes, shown in Figure10. The frequency is 300 MHz. The length of the cube is 0.5 m, the thickness of the coated layer is 0.02 m. The distance between the centers of two cubes is 1.6 m. The relative permittivity εr=2.0. The incident angle of plane wave is θ=45°,φ=0°, the bistatic RCS in horizontal polarization is computed. From Figure 11, it is shown  clearly that the results by the DDM-SVE-BI agree well with the one by the DDM-FEM-BI.Figure 10
Two coating PEC cubes.Figure 11
The Bistatic RCS of two coating PEC cubes.The comparison of the DDM-SVE-BI over the DDM-FE-BI is shown in Table2. The unknowns, the memory of matrix K for single coating PEC cube meshed with the SVEs is only 60%, 36% of the one with traditional tetrahedral elements, respectively. The CPU time for attaining single XC by the SVEs is only 25.8% of the one by tetrahedral elements.Table 2
The comparison of the DDM-FEM-BI and DDM-SVE-BI. Bistatic RCS of two coating PEC cubes. (mesh density withλ0/20).
Method
DDM-FE-BI
DDM-SVE-BI
Unknowns of single PEC coating cube
5344
3218
The total unknowns
10688
6436
Memory of matrix K about single PEC coating cube
228.5 Mb
82.8 Mb
Computational time for matrixXC
29846.8 s
7702.7 sObviously, compared with the DDM-FE-BI, the DDM-SVE-BI can save the memory and CPU time greatly for multiple thin-coating objects.


## 4.1. Two Dielectric Sphere

The first example is two separated dielectric spheres, shown in Figure4. The frequency is 300 MHz. The radius of the two dielectric spheres are 0.2 m. The distance between the centers of two spheres is 0.8 m. The relative permittivity is εr=2.0. The incident angle of plane wave is θ=45°,φ=0°，the bistatic RCS in horizontal polarization is computed. Here, DDM for FEM-BI is used to calculate scattering from two dielectric spheres in order to prove the accuracy of DDM. From Figure 5, it is shown clearly that the results by DDM-FE-BI agree well with the one by commercial software FEKO based on the surface integral equation with PMCHWT formulation [17].Figure 4
Two dielectric spheres.Figure 5
The Bistatic RCS of two dielectric spheres.


## 4.2.3×2 Array of Coating PEC Spheres

The second example is to solve scattering of3×2 array of coating PEC spheres located in the x-y  plane, as shown in Figure 6. The radius of PEC sphere is 0.2 m. The thickness of dielectric coating is 0.05 m with εr=2.0,μr=1.0. The distance between the centers of two elements is 0.8λ0 in both the x- and y-dimension as shown in Figure 6. The incident electric field is x^ polarized plane wave propagating into the negative z^ direction at 0.3 GHz. The results by Moment of Method (MoM) are also given for comparison.Figure 6
3
×
2 array of coating PEC spheres.As shown in Figure7, the results by DDM-SVE-BI agree with the one by DDM-FE-BI and the result of the MoM very well.Figure 7
The Bistatic RCS of3×2 array of coating PEC spheres.The comparison of the DDM-FE-BI and DDM-SVE-BI is shown in Table1. Table 1 demonstrates that the unknowns, the memory requirement of matrix K for single coating PEC sphere with the SVEs is only 52%, 27% of the traditional tetrahedral elements, respectively. The CPU time for attaining single XC by the SVEs is only 20% of the one by traditional tetrahedral elements.Table 1
The comparison of the DDM-FE-BI and DDM-SVE-BI. Bistatic RCS of3×2 array of coating PEC spheres. (mesh density with λ0/10).
Method
DDM-FE-BI
DDM-SVE-BI
Unknowns of single PEC coating sphere
523
274
The total unknowns
3138
1644
Memory of matrixK for single coating PEC sphere
2.2 Mb
0.6 Mb
Computational time for matrixXC
22.4 s
4.3 sObviously, the advantages of the DDM-SVE-BI over the DDM-FE-BI on reducing the number of unknowns and computational time are very remarkable.To further prove the accuracy of the present method, the RCS of this same3×2 array with magnetic material coating is also computed, the material parameters are εr=1.0,μr=2.0-j1.5. The same incident angle with x^ polarized is analyzed. As shown in Figure 8, the results of DDM-SVE-BI also agree with the results of DDM-FE-BI very well.Figure 8
The Bistatic RCS of3×2 array of coating PEC spheres with magnetic material.To further investigate the influence of different coating materials on the RCS of this3×2 array, four different coating materials with εr=2.0,μr=1.0, εr=2.0-j1.8,μr=1.0, εr=1.0,μr=2.0-j1.5 including PEC are evaluated. As shown in Figure 9, compared with the RCS from 3×2 PEC array, two lossy coating materials reduce the backscattering RCS greatly. The RCS reduction achieves 2.8 dB, 6.9 dB for the case of εr=2.0-j1.8,μr=1.0, εr=1.0,μr=2.0-j1.5, respectively. For this structure, lossless coating material with εr=2.0,μr=1.0 has little influence on the backscattering RCS. For the forward RCS, it is enhanced about 3 dBSW in all four cases. It is also demonstrated from Figure 9 that the magnetic lossy coating material has great influence on the RCS of this array.Figure 9
The Bistatic RCS of3×2 array of coating PEC spheres with different materials.


## 4.3. Two Coating PEC Cubes

The third example is two coating PEC cubes, shown in Figure10. The frequency is 300 MHz. The length of the cube is 0.5 m, the thickness of the coated layer is 0.02 m. The distance between the centers of two cubes is 1.6 m. The relative permittivity εr=2.0. The incident angle of plane wave is θ=45°,φ=0°, the bistatic RCS in horizontal polarization is computed. From Figure 11, it is shown  clearly that the results by the DDM-SVE-BI agree well with the one by the DDM-FEM-BI.Figure 10
Two coating PEC cubes.Figure 11
The Bistatic RCS of two coating PEC cubes.The comparison of the DDM-SVE-BI over the DDM-FE-BI is shown in Table2. The unknowns, the memory of matrix K for single coating PEC cube meshed with the SVEs is only 60%, 36% of the one with traditional tetrahedral elements, respectively. The CPU time for attaining single XC by the SVEs is only 25.8% of the one by tetrahedral elements.Table 2
The comparison of the DDM-FEM-BI and DDM-SVE-BI. Bistatic RCS of two coating PEC cubes. (mesh density withλ0/20).
Method
DDM-FE-BI
DDM-SVE-BI
Unknowns of single PEC coating cube
5344
3218
The total unknowns
10688
6436
Memory of matrix K about single PEC coating cube
228.5 Mb
82.8 Mb
Computational time for matrixXC
29846.8 s
7702.7 sObviously, compared with the DDM-FE-BI, the DDM-SVE-BI can save the memory and CPU time greatly for multiple thin-coating objects.


## 5. Conclusions

In this paper, the SVE-BI based on the DDM framework (DDM-SVE-BI) is proposed for scattering analysis of multiple conducting bodies coated with thin layer dielectric. The whole computational domains are divided into sub-SVE domains and boundary element domains. Compared with traditional vector element method, the DDM-SVE-BI reduces the unknowns greatly, enhances the computational efficiency of solving scattering from multiple coating objects.Because linear basis function is used to represent the normal component of electric field in coating materials, this method based on shell vector basis is very efficient for solving thin coating problems, even multiple thin coating problems. It is valid for thin coating materials with thickness of up to 0.1 dielectric wavelength. On the other hand, this method is limited to solve the objects at resonant region. To solve large scale problems, fast methods are necessary to implement into it. This will be our next work.



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*Source: 790164-2012-09-03.xml*