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.

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) [

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 [

To realize efficient analysis of complex structures, domain decomposition method (DDM) is developed based on FEM-BI framework [

The rest of the paper is organized as follows: the hybrid SVE-BI method is briefly introduced in Section

For sake of simplicity, 3D electromagnetic scattering problem of conducting object coated by thin dielectric illuminated by an incident wave

The 3D electromagnetic scattering problem of conducting object coated by thin dielectric.

The

The boundary conditions on the surface of object are written as:

The functional

Finally, the FEM matrix equation of the object is yielded from (

In the SVE-BI method, integral equation method is applied on the surface of the object. As shown in [

Combing (

For the surface integral term in (

By the SVE, the electric field is expanded as follows:

The shell element is the degenerated prism element. As shown in Figure

The structure of the prism vector element and the shell vector element.

the prism vector element

the shell vector element

A linear function

For the SVE, as shown in [

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.

For multiple conducting bodies coated with thin dielectric as shown in Figure

The 3D electromagnetic scattering problem of multiple conducting bodies coated with thin layer dielectric.

Equation (

Extracting surface electric field expansion coefficient

Substitute (

Here,

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

The first example is two separated dielectric spheres, shown in Figure

Two dielectric spheres.

The Bistatic RCS of two dielectric spheres.

The second example is to solve scattering of

As shown in Figure

The Bistatic RCS of

The comparison of the DDM-FE-BI and DDM-SVE-BI is shown in Table

The comparison of the DDM-FE-BI and DDM-SVE-BI. Bistatic RCS of

Method | DDM-FE-BI | DDM-SVE-BI |
---|---|---|

Unknowns of single PEC coating sphere | 523 | 274 |

The total unknowns | 3138 | 1644 |

Memory of matrix |
2.2 Mb | 0.6 Mb |

Computational time for matrix |
22.4 s | 4.3 s |

Obviously, 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 same

The Bistatic RCS of

To further investigate the influence of different coating materials on the RCS of this

The Bistatic RCS of

The third example is two coating PEC cubes, shown in Figure

Two coating PEC cubes.

The Bistatic RCS of two coating PEC cubes.

The comparison of the DDM-SVE-BI over the DDM-FE-BI is shown in Table

The comparison of the DDM-FEM-BI and DDM-SVE-BI. Bistatic RCS of two coating PEC cubes. (mesh density with

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 matrix |
29846.8 s | 7702.7 s |

Obviously, compared with the DDM-FE-BI, the DDM-SVE-BI can save the memory and CPU time greatly for multiple thin-coating objects.

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.

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