CN104657527A - Electromagnetic scattering analysis method of ultrahigh-speed thin-coated stealthy flying target - Google Patents

Abstract

The invention discloses an electromagnetic scattering analysis method of an ultrahigh-speed thin-coated stealthy flying target. For the non-uniformity characteristics of plasmas covering the ultrahigh-speed thin-coated stealthy flying target, a volume integral equation method is adopted for analysis; a surface integral-based thin coating analysis method is adopted to perform modeling analysis on the metal body of the stealthy flying target and the stealthy material coated on the metal body. Compared with the traditional method of adopting volume integral equation for all the medium parts, the electromagnetic scattering analysis method is capable of saving the calculation resources; meanwhile, as the green function adopted in the equation is the green function in vacuum, the multi-level fast multipole algorithm is used for further acceleration and calculation, and therefore, less calculation memory and calculation time are needed for solving the scattering problem of the ultrahigh-speed thin-coated stealthy flying target.

Description

The Analysis of Electromagnetic Scattering method of stealthy airbound target is applied in hypervelocity scumbling

Technical field

The invention belongs to the quick computing technique field of electromagnetic characteristic of scattering, be particularly a kind ofly applied to the Analysis of Electromagnetic Scattering method that stealthy airbound target is applied in hypervelocity scumbling.

Background technology

Hypervelocity flight target is owing to having very fast flying speed (more than 3 Mach) and higher flying height (more than 20Km), the Aerodynamic Heating of several thousand degrees Celsius can be produced with windage during its flight, make the existence in ionic condition due to ionization of its surrounding air.When degree of ionization acquires a certain degree, ionized gas has plasma properties.Now be commonly called the coated flow field of plasma in the coated flow field of airbound target near surface, reenter plasma or plasma valve jacket, now be equivalent to airbound target cover by plasma (normal rain. supersonic speed/hypersonic plasma Field Flow Numerical Simulation and electromagnetic property research thereof, National University of Defense technology's PhD dissertation, 2009).Meanwhile, in order to consider stealthy object, usually at aircraft surface coating stealth material, the effect of reduction RCS can be played and reaches stealthy object.

Be ionized the unevenness of the plasma relative dielectric constant of formation due to air, and the absorbing material of aircraft surface coating is very thin, this all gives and uses the electromagnetic scattering problems of numerical method analysis airbound target to bring certain difficulty.Found by research, the plasma valve jacket being positioned at aircraft tip portion has larger equivalent relative dielectric constant, and gas ions valve jacket other parts dielectric permittivity is close to air, this results in the uneven characteristic of plasma relative to dielectric parameter.For uneven plasma valve jacket, use volume integral equations method (Schaubert D, Wilton D andGlisson A.A tetrahedral modeling method for electromagnetic scattering by arbitrarily shapedinhomogeneous dielectric bodies.IEEE Transaction on Antennas and Propagation, 1984,32 (1): 77 – 85.) analyze, for the coating structure on metal body and surface thereof, metallic member is processed by as perfect electronic conductor (PEC) usually, and easily quilt cover integral Equation Methods (SIE) carrys out analysis and solution, wherein RWG basis function (Rao M, Wilton D and Glisson A.Electromagnetic scattering by surfaces of arbitrary shape.IEEE Transaction on Antennas and Propagation, 1982, 30 (3): 409 – 418.) launched the basis function of unknown current because its dirigibility is used as usually, utilize the feature that coating stealth material is thin simultaneously, the faradic form that volume charge in stealthy coating material and body electric current are converted to metal surface is solved.But the electromagnetic scattering problems using the scumbling of above methods analyst hypervelocity to apply stealthy airbound target is faced with the large problem of unknown quantity, brings thus in solution procedure and needs a large amount of computing times and internal memory.

Summary of the invention

The object of the present invention is to provide a kind of hypervelocity scumbling to apply the Analysis of Electromagnetic Scattering method of stealthy airbound target, thus realize obtaining the Electromagnetic Scattering Characteristics parameter that stealthy airbound target is applied in hypervelocity scumbling fast.

The technical solution realizing the object of the invention is: the Analysis of Electromagnetic Scattering method of stealthy airbound target is applied in hypervelocity scumbling, and step is as follows:

The first step, set up the stealthy airbound target of high speed and plasma valve jacket model, the mainly determination of the electromagnetic parameter model of plasma valve jacket, it is relevant with the flight environment of vehicle of high-speed flight target, as flying height, flying speed and airbound target ambient atmosphere pressure and temperature etc.

Second step, grid process.Triangle subdivision is adopted for metal and coating part thereof, Tetrahedron subdivision is adopted for plasma valve jacket part.

3rd step, sets up mixture Line Integral equation.According to the scattering properties of mixed structure, the incident field that resultant field in target equals and all scattered field sums, incident electric fields is known excitation, and uniform plane wave is usually used to as incident electric fields, and scattering electric field can represent with electric flux density to be asked and induced current density.

4th step, launches Green function in free space based on addition theorem, and the expression formula of combination Line Integral equation, provides the polymerizing factor of far-field portion, and transfer factor embodies form with the configuration factor.

5th step, matrix equation solves and the calculating of electromagnetic scattering parameter.

The present invention compared with prior art, its remarkable advantage: 1. unknown quantity is few.Apply interior polarization charge due to scumbling and electric current all represents with the electric current on metal body, allly do not need extra unknown quantity to describe electromagnetic parameter in coating.2. solving speed is fast.Due to the non-homogeneous plasma that have employed honorable integration equation analysis hypervelocity flight target and be wrapped in outside hypervelocity flight target, Green function used is the Green function of free space, facilitates the introducing of the quick multistage sub-technology of multilayer, accelerates Matrix Solving.

Accompanying drawing explanation

Stealthy airbound target electromagnetic model schematic diagram is applied in the scumbling of Fig. 1 hypervelocity.

Stealthy airbound target structural representation is applied in the scumbling of Fig. 2 hypervelocity.

The RCS figure of stealthy airbound target is applied in the scumbling of Fig. 3 hypervelocity.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail.

The first step, set up hypervelocity flight target and plasma valve jacket model, the mainly determination of the electromagnetic parameter model of plasma valve jacket, it is relevant with the flight environment of vehicle of hypervelocity flight target, as flying height, flying speed and airbound target ambient atmosphere pressure and temperature etc.By the flying height of airbound target, the angle of attack and flight Mach number parameter, business software ANSYS is used to carry out pneumatic analog calculating to object module, obtain the electron number densitiy of target, temperature, pressure information data, obtain plasma characteristics frequency and collision frequency thus, then obtain the equivalent relative dielectric constant of plasma valve jacket by following formula

${\mathrm{\ϵ}}_r=1-\frac{{{\mathrm{\ω}}^2}_{\mathrm{pe}}}{{\mathrm{\ω}}^2+v^2}-j\frac{v}{\mathrm{\ω}}\frac{{{\mathrm{\ω}}^2}_{\mathrm{pe}}}{{\mathrm{\ω}}^2+v^2}---\left(1\right)$

Wherein ω _pefor plasma characteristics frequency, ω is wave frequency, and v is plasma collision frequency.

Second step, adopts triangle subdivision for metal and coating part thereof, adopts Tetrahedron subdivision for plasma valve jacket part.

3rd step, according to the scattering properties of parcel plasma valve jacket high hypervelocity airbound target structure, adopt method of moment basic theory, obtain honorable integral equation, its matrix form equation is:

$\left[\begin{array}{cc}{Z}_{\mathrm{mn}}^{\mathrm{DD}}& Z_{\mathrm{mn}}^{\mathrm{MD}}\\ Z_{\mathrm{mn}}^{\mathrm{DM}}& Z_{\mathrm{mn}}^{\mathrm{MM}}\end{array}\right]\left[\begin{array}{c}{D}_n\\ I_n\end{array}\right]=\left[\begin{array}{c}{v}_m^V\\ v_m^S\end{array}\right]---\left(2\right)$

Wherein:

$\begin{array}{c}{Z}^{\mathrm{DD}}=\underset{V}{\∫}\frac{1}{{\mathrm{\ϵ}}_n}{f}_m^V\·f_n^V\mathrm{dV}-{\mathrm{\ω}}^2{\mathrm{\μ}}_0\underset{V_m}{\∫}\underset{V_n}{\∫}{K}_n{f}_m^V\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}\\ +\frac{1}{{\mathrm{\ϵ}}_0}\underset{V_m}{\∫}\underset{V_n}{\∫}{K}_n\▿\·f_m^V{\▿}^{\′}\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}-\frac{1}{{\mathrm{\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{V_n}{\∫}{K}_n\hat{n}\·f_m^V{\▿}^{\′}\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}\\ +\frac{1}{{\mathrm{\ϵ}}_0}\underset{V_m}{\∫}\underset{V_n}{\∫}\▿\·f_m^V(\▿K_n)\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}-\frac{1}{{\mathrm{\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{V_n}{\∫}\hat{n}\·f_m^V(\▿K_n)\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}\end{array}---\left(3\right)$

$\begin{array}{c}{Z}^{\mathrm{MD}}={\mathrm{j\ω\μ}}_0\underset{V_m}{\∫}\underset{S_n}{\∫}{f}_m^V\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dV}-\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{V_m}{\∫}\underset{S_n}{\∫}\▿\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dV}\\ +\frac{1}{{\mathrm{j\ω\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{S_n}{\∫}\hat{n}\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dV}-{\mathrm{j\ω\μ}}_0\underset{V_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{K}_n{f}_m^V\·\widehat{n^{\′}}{\▿}^{\′}\·f_n^S{\mathrm{GdV}}^{\′}\mathrm{dV}\\ -\frac{1}{{\mathrm{j\ω\ϵ}}_0}\underset{V_m}{\∫}\underset{S_n^{\▿}}{\∫}{K}_n\▿\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\▿}_{\′}}\mathrm{dV}+\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{S_n^{\▿}}{\∫}{K}_n\hat{n}\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\▿}_{\′}}\mathrm{dV}\\ +\frac{1}{j{\mathrm{\ω\ϵ}}_0}\underset{V_m}{\∫}\underset{S_n^{\mathrm{\Δ}}}{\∫}{K}_n\▿\·f_m^V{\▿}^{\′}\·f_n^{S}G{\mathrm{dS}}^{{\mathrm{\Δ}}_{\′}}\mathrm{dV}-\frac{1}{{\mathrm{j\ω\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{S_n^{\mathrm{\Δ}}}{\∫}{K}_n\hat{n}\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\mathrm{\Δ}}_{\′}}\mathrm{dV}\\ +{\mathrm{j\ω\μ}}_0({\mathrm{\μ}}_r-1)\underset{V_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{f}_m^V.\widehat{n^{\′}}\×f_n^S\×\▿{\mathrm{GdV}}^{\′}\mathrm{dV}\end{array}---\left(4\right)$

$\begin{array}{c}{Z}^{\mathrm{DM}}=-{\mathrm{\ω}}^2{\mathrm{\μ}}_0\underset{S_m}{\∫}\underset{V_n}{\∫}{K}_n{f}_m^S\·f_n^V{\mathrm{GdV}}^{\′}\mathrm{dS}+\frac{1}{{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{V_n}{\∫}{K}_n\▿\·f_m^S{\▿}^{\′}\·f_n^V{\mathrm{GdV}}^{\′}\mathrm{dS}\\ +\frac{1}{{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{V_n}{\∫}\▿\·f_m^S(\▿K_n)\·f_n^V{\mathrm{GdV}}^{\′}\mathrm{dS}\end{array}---\left(5\right)$

$\begin{array}{c}{Z}^{\mathrm{MM}}={\mathrm{j\ω\μ}}_0\underset{S_m}{\∫}\underset{S_n}{\∫}{f}_m^S\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dS}+\frac{1}{j{\mathrm{\ω\ϵ}}_0}\underset{S_m}{\∫}\underset{S_n}{\∫}\▿\·f_m^S{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dS}\\ -{\mathrm{j\ω\μ}}_0\underset{S_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{K}_n{f}_m^S\·\widehat{n^{\′}}{\▿}^{\′}\·f_n^S{\mathrm{GdV}}^{\′}\mathrm{dS}-\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{S_n^{\▿}}{\∫}{K}_n\▿\·f_m^S{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\▿}_{\′}}\mathrm{dS}\\ +\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{S_n^{\mathrm{\Δ}}}{\∫}{K}_n\▿\·f_m^S{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\mathrm{\Δ}}_{\′}}\mathrm{dS}+{\mathrm{j\ω\μ}}_0({\mathrm{\μ}}_r-1)\underset{S_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{f}_m^S\·\widehat{n^{\′}}\×f_n^S\×\▿{\mathrm{GdV}}^{\′}\mathrm{dS}\end{array}---\left(6\right)$

Wherein, with represent body and face test basis function respectively, V represents areas of dielectric, and S represents metal surface, and VTDS represents thin dielectric region, S ^▽represent thin-medium lower surface, S ^Δrepresent thin-medium upper surface, ω is electromagnetic wave angular frequency, constant coefficient $K=\frac{\mathrm{\ϵ}-{\mathrm{\ϵ}}_0}{\mathrm{\ϵ}}=1-\frac{1}{{\mathrm{\ϵ}}_r},G(r,r^{\′})=\frac{e^{-\mathrm{jk}(r-r^{\′})}}{4\mathrm{\π}|r-r^{\′}|}$ It is the Green function of free space.

In above formula, the right vector is produced by plane wave, can be write as

$v_m^V=\underset{V}{\∫}{f}_m^V\left(r\right)\·E^i\mathrm{dV}---\left(7\right)$

$v_m^S=\underset{V}{\∫}{f}_m^S\left(r\right)\·E^i\mathrm{dV}---\left(8\right)$

E ⁱit is incident electric fields.

Z ^dDrepresent the effect of medium to medium, Z ^dMrepresent the effect of medium to metal and coating, Z ^mDall represent metal and the effect of coating to medium, Z ^mMrepresent the effect of metal to metal;

4th step, solution matrix equation, obtains current coefficient, then calculates electromagnetic scattering parameter according to reciprocal theorem by current coefficient, as RCS, nearly far field Electric Field Distribution.

In order to the feasibility of verification method, shown below is the example of the electromagnetic scattering of hypervelocity flight target.This example to be a radius be Metal Ball of 0.2 meter, be coated with the film dielectric layer that a layer thickness is 0.02 meter, parameter is ε _r=3-j0.006, μ _r=2, wrap up again in the outside of film dielectric layer the medium that a layer thickness is 0.08 meter, its relative dielectric constant ε _r=2.Replace the special valve jacket of plasma with uniform dielectric in the present embodiment, convenient and existing business software is compared.As shown in Figure 3, for RCS parameter, the RCS that the inventive method calculates conforms to the result of calculation of business software FEKO, demonstrates the validity of the inventive method.

Claims (2)

1. an Analysis of Electromagnetic Scattering method for stealthy airbound target is applied in hypervelocity scumbling, it is characterized in that step is as follows:

The first step, set up hypervelocity scumbling and apply stealthy airbound target plasma valve jacket model, according to the flying height of airbound target, the angle of attack and flight Mach number parameter, pneumatic analog calculating is carried out to hypervelocity flight target, obtain the electron number densitiy of target, temperature and pressure information data, obtain plasma characteristics frequency and collision frequency thus, then obtain the equivalent relative dielectric constant of each locus of plasma valve jacket by following formula

${\mathrm{\ϵ}}_r=1-\frac{{{\mathrm{\ω}}^2}_{\mathrm{pe}}}{{\mathrm{\ω}}^2+v^2}-j\frac{v}{\mathrm{\ω}}\frac{{{\mathrm{\ω}}^2}_{\mathrm{pe}}}{{\mathrm{\ω}}^2+v^2}---\left(1\right)$

Wherein ω _pefor plasma characteristics frequency, ω is wave frequency, and v is plasma collision frequency;

Second step, according to the scattering properties of parcel plasma valve jacket hypervelocity flight object construction, adopt method of moment basic theory, obtain integral equation, its matrix form equation is:

$\left[\begin{array}{cc}{Z}_{\mathrm{mn}}^{\mathrm{DD}}& Z_{\mathrm{mn}}^{\mathrm{MD}}\\ Z_{\mathrm{mn}}^{\mathrm{DM}}& Z_{\mathrm{mn}}^{\mathrm{MM}}\end{array}\right]\left[\begin{array}{c}{D}_n\\ I_n\end{array}\right]=\left[\begin{array}{c}{v}_m^V\\ v_m^S\end{array}\right]---\left(2\right)$

Z ^dDrepresent the effect of medium to medium, Z ^dMrepresent the effect of medium to metal and coating, Z ^mDall represent metal and the effect of coating to medium, Z ^mMrepresent the effect of metal to metal, D _nand I _nunknown current coefficient to be asked, with it is the right vector excitation;

3rd step, solution matrix equation (2), obtains current coefficient D _nand I _n, then calculate electromagnetic scattering parameter according to reciprocal theorem by current coefficient.

2. the Analysis of Electromagnetic Scattering method of hypervelocity flight target according to claim 1, is characterized in that, in step 2, matrix equation to embody form as follows:

$\begin{array}{c}{Z}^{\mathrm{DD}}=\underset{V}{\∫}\frac{1}{{\mathrm{\ϵ}}_n}{f}_m^V\·f_n^V\mathrm{dV}-{\mathrm{\ω}}^2{\mathrm{\μ}}_0\underset{V_m}{\∫}\underset{V_n}{\∫}{K}_n{f}_m^V\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}\\ +\frac{1}{{\mathrm{\ϵ}}_0}\underset{V_m}{\∫}\underset{V_n}{\∫}{K}_n\▿\·f_m^V{\▿}^{\′}\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}-\frac{1}{{\mathrm{\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{V_n}{\∫}{K}_n\hat{n}\·f_m^V{\▿}^{\′}\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}\\ +\frac{1}{{\mathrm{\ϵ}}_0}\underset{V_m}{\∫}\underset{V_n}{\∫}\▿\·f_m^V(\▿K_n)\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}-\frac{1}{{\mathrm{\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{V_n}{\∫}\hat{n}\·f_m^V(\▿K_n)\·f_n^V\mathrm{Gd}{V}^{\′}\mathrm{dV}\end{array}---\left(3\right)$

$\begin{array}{c}{Z}^{\mathrm{MD}}={\mathrm{j\ω\μ}}_0\underset{V_m}{\∫}\underset{S_n}{\∫}{f}_m^V\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dV}-\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{V_m}{\∫}\underset{S_n}{\∫}\▿\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dV}\\ +\frac{1}{{\mathrm{j\ω\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{S_n}{\∫}\hat{n}\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dV}-{\mathrm{j\ω\μ}}_0\underset{V_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{K}_n{f}_m^V\·\widehat{n^{\′}}{\▿}^{\′}\·f_n^S{\mathrm{GdV}}^{\′}\mathrm{dV}\\ -\frac{1}{{\mathrm{j\ω\ϵ}}_0}\underset{V_m}{\∫}\underset{S_n^{\▿}}{\∫}{K}_n\▿\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\▿}_{\′}}\mathrm{dV}+\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{S_n^{\▿}}{\∫}{K}_n\hat{n}\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\▿}_{\′}}\mathrm{dV}\\ +\frac{1}{j{\mathrm{\ω\ϵ}}_0}\underset{V_m}{\∫}\underset{S_n^{\mathrm{\Δ}}}{\∫}{K}_n\▿\·f_m^V{\▿}^{\′}\·f_n^{S}G{\mathrm{dS}}^{{\mathrm{\Δ}}_{\′}}\mathrm{dV}-\frac{1}{{\mathrm{j\ω\ϵ}}_0}\underset{{\mathrm{\Ω}}_m}{\∫}\underset{S_n^{\mathrm{\Δ}}}{\∫}{K}_n\hat{n}\·f_m^V{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\mathrm{\Δ}}_{\′}}\mathrm{dV}\\ +{\mathrm{j\ω\μ}}_0({\mathrm{\μ}}_r-1)\underset{V_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{f}_m^V.\widehat{n^{\′}}\×f_n^S\×\▿{\mathrm{GdV}}^{\′}\mathrm{dV}\end{array}---\left(4\right)$

$\begin{array}{c}{Z}^{\mathrm{DM}}=-{\mathrm{\ω}}^2{\mathrm{\μ}}_0\underset{S_m}{\∫}\underset{V_n}{\∫}{K}_n{f}_m^S\·f_n^V{\mathrm{GdV}}^{\′}\mathrm{dS}+\frac{1}{{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{V_n}{\∫}{K}_n\▿\·f_m^S{\▿}^{\′}\·f_n^V{\mathrm{GdV}}^{\′}\mathrm{dS}\\ +\frac{1}{{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{V_n}{\∫}\▿\·f_m^S(\▿K_n)\·f_n^V{\mathrm{GdV}}^{\′}\mathrm{dS}\end{array}---\left(5\right)$

$\begin{array}{c}{Z}^{\mathrm{MM}}={\mathrm{j\ω\μ}}_0\underset{S_m}{\∫}\underset{S_n}{\∫}{f}_m^S\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dS}+\frac{1}{j{\mathrm{\ω\ϵ}}_0}\underset{S_m}{\∫}\underset{S_n}{\∫}\▿\·f_m^S{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{\′}\mathrm{dS}\\ -{\mathrm{j\ω\μ}}_0\underset{S_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{K}_n{f}_m^S\·\widehat{n^{\′}}{\▿}^{\′}\·f_n^S{\mathrm{GdV}}^{\′}\mathrm{dS}-\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{S_n^{\▿}}{\∫}{K}_n\▿\·f_m^S{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\▿}_{\′}}\mathrm{dS}\\ +\frac{1}{\mathrm{j\ω}{\mathrm{\ϵ}}_0}\underset{S_m}{\∫}\underset{S_n^{\mathrm{\Δ}}}{\∫}{K}_n\▿\·f_m^S{\▿}^{\′}\·f_n^S{\mathrm{GdS}}^{{\mathrm{\Δ}}_{\′}}\mathrm{dS}+{\mathrm{j\ω\μ}}_0({\mathrm{\μ}}_r-1)\underset{S_m}{\∫}\underset{V_{\mathrm{nTDS}}}{\∫}{f}_m^S\·\widehat{n^{\′}}\×f_n^S\×\▿{\mathrm{GdV}}^{\′}\mathrm{dS}\end{array}---\left(6\right)$

Wherein, with represent body and face test basis function respectively, V represents areas of dielectric, and S represents metal surface, V _tDSrepresent thin dielectric region, S ^▽represent thin-medium lower surface, S ^Δrepresent thin-medium upper surface, ω is electromagnetic wave angular frequency, constant coefficient $K=\frac{\mathrm{\ϵ}-{\mathrm{\ϵ}}_0}{\mathrm{\ϵ}}=1-\frac{1}{{\mathrm{\ϵ}}_r},G(r,r^{\′})=\frac{e^{-\mathrm{jk}(r-r^{\′})}}{4\mathrm{\π}|r-r^{\′}|}$ It is the Green function of free space;

In above formula, the right vector is produced by plane wave, is write as

$v_m^V=\underset{V}{\∫}{f}_m^V\left(r\right)\·E^i\mathrm{dV}---\left(7\right)$

$v_m^S=\underset{V}{\∫}{f}_m^S\left(r\right)\·E^i\mathrm{dV}---\left(8\right)$

E ⁱit is incident electric fields.