阅读说明：本技术 一种地下多层介质涂层导体圆柱的隐身设计方法 (Stealth design method for underground multilayer dielectric coating conductor cylinder ) 是由 陈波 耿友林 尹川 张鹏泉 于 2021-06-07 设计创作，主要内容包括：本发明公开了一种地下多层介质涂层导体圆柱的隐身设计方法,以圆柱波函数为基础提出了一种埋在地下多层介质涂层导体圆柱的电磁散射的解析解,第一次给出了计算结果,并基于遗传算法,实现了地下多层涂层导体圆柱隐身。在选取材料方面选取了更容易实现的各向同性材料,通过遗传算法进行优化,得到最优的材料和涂层厚度,从而使涂层圆柱在地面的散射强度最小。本发明适用于精确、快速的计算地下涂层柱体目标的散射特性,并拓宽了圆柱隐身的应用场合。本发明方法与其他方法相比较,有精确,快速,高效的求解柱类目标的电磁散射特性,并具有可以有效实现电磁隐身的效果。(The invention discloses a stealth design method of an underground multilayer medium coating conductor cylinder, which provides an analytic solution of electromagnetic scattering of the underground multilayer medium coating conductor cylinder on the basis of a cylindrical wave function, gives a calculation result for the first time, and realizes the stealth of the underground multilayer medium coating conductor cylinder on the basis of a genetic algorithm. Isotropic materials which are easier to realize are selected in the aspect of selecting materials, and optimization is carried out through a genetic algorithm to obtain the optimal materials and the optimal coating thickness, so that the scattering intensity of the coating cylinder on the ground is minimum. The method is suitable for accurately and quickly calculating the scattering characteristic of the underground coating cylinder target, and widens the application occasions of cylinder stealth. Compared with other methods, the method has the advantages of accurately, quickly and efficiently solving the electromagnetic scattering characteristic of the columnar target and effectively realizing the electromagnetic stealth effect.)

1. A stealth design method for underground multilayer dielectric coating conductor cylinders is characterized by comprising the following steps:

step 1, calculating specific expressions of transmitted waves and scattered waves;

assuming a TM wave propagates in free space along the + z-axis, normal incidence to an infinite uniform dielectric half-space at z-d, where z is<The relative dielectric constant present at d is ε_r1Wherein d is the buried depth; solving incident wave by using passive Maxwell equation set and boundary conditionThe transmission coefficient T and the scattering coefficient R at the position where z is-d are obtained, and then the transmitted wave is obtainedAnd scattered wavesThe specific expression of (1);

step 2, calculating the integral scattering field outside the dielectric coating conductor cylinder;

will transmit the waveThe incident wave is taken as the incident wave of the dielectric coating conductor cylinder, and the incident wave is converted into a standard wave function under a cylindrical coordinate system; the whole infinite-length dielectric coating conductor cylinder is regarded as a whole, the field of each layer in the multilayer dielectric coating can be regarded as the superposition form of a first Bessel function and a second Bessel function, and then the total scattered field outside the coating cylinder is calculated by each layer by utilizing boundary conditions and a recurrence formula

Step 3, scattering waves of the infinite-length dielectric coating conductor cylinderThe scattered wave is used as the incident wave to be scattered and transmitted by the vertical incidence ground plane, and the scattered wave is used as the incident wave of the dielectric coating conductor cylinder again and circularly reciprocates; in the whole process, the expressions of scattered waves and transmitted waves on the interface are obtained according to the mirror image method of the medium interface, and then the scattering field under the local coordinate system is obtained by utilizing the simultaneous equations of the boundary conditionsAnd a transmission field in a global coordinate system

And 4, according to the calculation method of the underground multilayer medium coating cylinder in the steps 1, 2 and 3, selecting the sum of transmitted waves generated when each scattered wave in the step 3 is incident to the ground plane as a fitness function, and performing optimization processing by using a genetic algorithm to calculate the optimal relative dielectric constant and thickness of each layer of material.

2. The stealth design method for the underground multilayer dielectric coating conductor cylinder according to claim 1, wherein the specific method of the step 1 is as follows;

required scattering coefficient R and transmission coefficient T, and the scattering field at z-dAnd a transmission fieldThe specific calculation of (a) is as follows:

the area above the ground is an infinite free space with a range z<D, its magnetic permeability is μ₀＝4π×10^-7H/m, dielectric constant of epsilon₀＝1/36π×10^-9F/m, wherein the incident wave is TM wave with amplitude of 1 and vertically incident to the ground plane along the positive direction of the z axis, the polarization direction is the positive direction of the x axis, and the parameter of mu is filled below the ground₁＝μ₀μ_1r，ε₁＝ε₀ε_1rIn which mu_1r1 and ε_1rRelative permeability and relative permittivity of the subsurface medium, respectively; the expressions of the electric and magnetic fields of the incident wave, the scattered wave, the transmitted wave at the ground surface where z-d is expressed as follows:

in the above formula, R and T are a scattering coefficient and a transmission coefficient at an interface, respectively; transmitted waveThe space is the medium 1, and the range is z>-d, the space being a lossless medium; eta₀Is an intrinsic impedance of free space of valueThe intrinsic impedance of the lossless medium isIs the wave number of the plane wave in free space,is the wave number of medium 1, j is an imaginary unit;

a multi-layer dielectric coating conductor cylinder with radius a and total coating thickness b-a is buried in the medium 1, and each layer thickness of the coating is d₂,d₃...d_mThe relative dielectric constant of each coating is sequentially epsilon from inside to outside_2r,ε_3r...ε_mrThe relative magnetic permeability is 1; the system of equations can be listed from the boundary conditions at the medium interface of z-d:

substituting z-d into equation (7) can solve the expressions of transmission coefficient T and scattering coefficient R at the interface of the medium, which are respectively:

and (3) calculating values of the scattering coefficient R and the transmission coefficient T, and substituting the obtained values of the scattering coefficient R and the transmission coefficient T into the formula (7) to obtain electric field values of the scattered wave and the transmitted wave on the surface of the medium.

3. The stealth design method for an underground multilayer dielectric coating conductor cylinder according to claim 2, wherein the calculation process of the scattering field of the whole external dielectric coating conductor cylinder in the step 2 is as follows:

the planar transmitted wave is used as the incident waveIncident along the positive direction of an x axis, and solving a homogeneous scalar Helmholtz equation set of the transmitted wave based on a traditional separation variable method, so as to obtain an isotropic cylindrical scalar wave function under a cylindrical coordinate system, wherein the expressions of an electric field and a magnetic field are as follows:

wherein A is_n＝TE₀j^-nThe space of the outermost layer of the cylinder is set as an m +1 layer; in the mth layer, the electric and magnetic fields are:

in the above formulaIs the wave number in the m-th coating,is the intrinsic impedance in the dielectric coating, Y_nIs the second classA Bessel function; b is_mnAnd C_mnUnknown coefficients of the electric field for each layer; the electric and magnetic field patterns of the scattered waves outside the dielectric coating are shown in the following formula:

in the above formula D_nUnknown coefficients of scattered waves;is a second class of hank functions; without loss of generality, it can be determined that the non-normalized amplitude of the electric field in the first layer is:

B_1,n′＝1,C_1,n′＝0 (15)

the coefficient B of the electric field of each layer can be obtained by utilizing boundary conditions at the interface of each coating layer and recursion_m,nAnd C_m,nAnd the electric field coefficient B of the other layer_m+1,nAnd C_m+1,nThe relationship of (a) is as follows:

wherein:

U_mn＝μ_mk_m+1J_n(k_mr_m)Y′_n(k_m+1r_m)-μ_m+1k_mJ′_n(k_mr_m)Y_n(k_m+1r_m)

V_mn＝μ_m+1k_mJ_n(k_m+1r_m)J′_n(k_mr_m)-μ_mk_m+1J′_n(k_m+1r_m)J_n(k_mr_m)

W_mn＝μ_mk_m+1Y_n(k_mr_m)Y′_n(k_m+1r_m)-μ_m+1k_mY′_n(k_mr_m)Y_n(k_m+1r_m)

X_mn＝μ_m+1k_mY′_n(k_mr_m)J_m(k_m+1r_m)-μ_mk_m+1Y_n(k_mr_m)J′_n(k_m+1r_m) (17)

j 'in the formula'_nAnd Y'_nDerivatives of Bessel functions of a first type and a second type, respectively; based on the boundary conditions of the coating interface, one can obtain:

can find outOf (2) undetermined D_nThe following were used:

D_n＝jC_m+1,n (20)

B_m+1,n-jC_m+1,n＝j^-n (21)

due to B_m+1,nAnd C_m+1,nAlso iterated from equation (16), the coefficients of the final scattered field can be calculated by the joint equations (20), (21):

D_n＝-j^-n(C_m+1,n/(C_m+1,n+jB_m+1,n)) (22)。

4. the stealth design method for an underground multilayer dielectric coating conductor cylinder according to claim 3, wherein the scattering field in the local coordinate system and the transmission field in the global coordinate system in step 3 are specifically calculated as follows:

global coordinate system: the three coordinate components in the cylindrical coordinates are r,z；

local coordinate system: the three coordinate components under the cylindrical coordinates are r_img，z_img

Any point in cylindrical coordinate systemIs the intersection of three coordinate surfaces: r ═ r₀The cylindrical surface of (1) and the included angle formed by the included z-axis and the xz plane isSemi-plane of (2), z ═ z₀A plane of (a); the expression for converting the rectangular coordinate system into the cylindrical coordinate system is as follows:

electric field of the scattered field of the formulas (13) and (14)And a magnetic fieldThe scattering wave in the local coordinate system and the transmitted wave in the global coordinate system are obtained by using the mirror image principle of the medium plane, wherein the scattering field is taken as an incident wave and is vertically incident to the ground, and the expressions of the scattering wave in the local coordinate system and the transmitted wave in the global coordinate system are as follows:

in the above formula E_nAnd F_nAre the unknown coefficients of the scattered and transmitted waves at the interface; from the boundary condition, the electric field is continuous, the tangential component of the magnetic field in the y-axis is continuous, and the magnetic field can be obtained at any pointThe following expressions are listed:

solving to obtain:

wherein

The transformation of the local coordinate system to the global coordinate system using the additive theorem is:

wherein

Then will beIncident as incident wave to the dielectric coating conductor cylinder, repeating the step 2 and the step 3, and establishing a database to obtain the transmitted wave for storing each P pointFinally, the field strength expression of the point P can be obtained as follows:

in the above formulaThe method comprises the steps of representing the transmitted wave generated at the interface of the medium every time, solving a specific value of an electric field through an MATLAB programming language, establishing a database through iterative loop, finally obtaining the electric field value at a point P, namely the electric field value obtained by a program, and comparing the obtained numerical value result with an FDTD result to verify the accuracy of derivation.

5. The stealth design method for the underground multilayer dielectric coating conductor cylinder according to claim 4, wherein the specific method in the step 4 is as follows:

the target prototype under consideration is the sum of the generated transmitted waves of each scattered wave incident on the ground plane of the subsurface multi-layer dielectric coating cylinder, i.e.The genetic algorithm optimization comprises the following specific steps:

(1) optimizing variables

The optimized variable is the relative dielectric constant epsilon of each dielectric coating₂,ε₃,...,ε_mAnd thickness { d }₂,d₃...d_mAnd setting an optimization variable to meet the following conditions

(2) Initial population

In the case of satisfying the above-mentioned constrainer conditions, a random initial population is generated with an initial size of 50

(3) Genetic generation

In order to ensure higher accuracy of the genetic algorithm, the target model is selected from 100 genetic generations;

(4) basic operation

Determining a coding scheme, setting a cross probability, a mutation probability, and determining a fitness function, the fitness function F being set toThe higher the fitness is, the smaller the sum value is at the 1-10GHz sum;

Technical Field

The invention belongs to the technical field of electromagnetic scattering, and particularly relates to a stealth design method of an underground multilayer dielectric coating conductor cylinder.

Background

When an electromagnetic wave encounters an obstacle such as a columnar medium, surface currents and surface charges are induced thereon, an electromagnetic field generated by the induced currents and the charges is called a scattering field, and this physical phenomenon is called electromagnetic scattering, and the obstacle is called a scatterer. To achieve electromagnetic cloaking, the basic idea is to "bend the wave around the protected area and then bend it back to propagate in the original direction". In 2006, j.pendry et al proposed the theory of transforming optics that could be used to freely control the propagation of electromagnetic waves, with the most important application being the realization of electromagnetic stealth through anisotropic media. Wood et al also proposed in the same year that thin, alternating metal and dielectric layered structures could be used to simulate ideal anisotropic materials.

In 2011, after-resonance et al have proposed that multilayer isotropic materials with a relative dielectric constant less than 1 can be used to achieve stealth in free space. In the prior art, almost no design method for calculating the scattering property of the underground multilayer dielectric coating conductor cylinder and realizing stealth exists.

Disclosure of Invention

The invention aims to provide a stealth design method of an underground multilayer dielectric coating conductor cylinder, which is accurate, rapid and efficient in solving the electromagnetic scattering property of a cylinder target and has the effect of effectively realizing electromagnetic stealth compared with other methods.

In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:

a stealth design method for underground multilayer dielectric coating conductor cylinders comprises the following steps:

step 1, calculating specific expressions of transmitted waves and scattered waves;

assuming a TM wave propagates in free space along the + z-axis, normal incidence to an infinite uniform dielectric half-space at z-d, where z is<The relative dielectric constant present at d is ε_r1Wherein d is the buried depth. Solving incident wave by using passive Maxwell equation set and boundary conditionThe transmission coefficient T and the scattering coefficient R at the position where z is-d are obtained, and then the transmitted wave is obtainedAnd scattered wavesThe specific expression of (1).

Step 2, calculating the integral scattering field outside the dielectric coating conductor cylinder;

will transmit the waveThe method is used as the incident wave of a dielectric coating conductor cylinder, and the incident wave is converted into a standard wave function under a cylindrical coordinate system. The whole infinite-length dielectric coating conductor cylinder is regarded as a whole, the field of each layer in the multilayer dielectric coating can be regarded as the superposition form of a first Bessel function and a second Bessel function, and then the total scattered field outside the coating cylinder is calculated by each layer by utilizing boundary conditions and a recurrence formula

Step 3, scattering waves of the infinite-length dielectric coating conductor cylinderThe scattered wave is used as the incident wave to be scattered and transmitted by the vertical incidence ground plane, and the scattered wave is used as the incident wave of the dielectric coating conductor cylinder again and circularly reciprocates; in the whole process, the expressions of scattered waves and transmitted waves on the interface are obtained according to the mirror image method of the medium interface, and then the scattering field under the local coordinate system is obtained by utilizing the simultaneous equations of the boundary conditionsAnd a transmission field in a global coordinate system

And 4, according to the calculation method of the underground multilayer medium coating cylinder in the steps 1, 2 and 3, selecting the sum of transmitted waves generated when each scattered wave in the step 3 is incident to the ground plane as a fitness function, and performing optimization processing by using a genetic algorithm to calculate the optimal relative dielectric constant and thickness of each layer of material.

Further, step 1, calculating specific expressions of the transmitted wave and the scattered wave;

required scattering coefficient R and transmission coefficient T, and the scattering field at z-dAnd a transmission fieldThe specific calculation of (a) is as follows:

the area above the ground is an infinite free space with a range z<D, its magnetic permeability is μ₀＝4π×10^{- 7}H/m, dielectric constant of epsilon₀＝1/36π×10^-9F/m, wherein the incident wave is TM wave with amplitude of 1 and vertically incident to the ground plane along the positive direction of the z axis, the polarization direction is the positive direction of the x axis, and the parameter of mu is filled below the ground₁＝μ₀μ_1r，ε₁＝ε₀ε_1rIn which mu_1r1 and ε_1rThe relative permeability and relative permittivity of the subsurface medium, respectively. At the ground surface where z-d, the expressions of the electric and magnetic fields of the incident wave, the scattered wave, the transmitted wave are as follows:

where R and T are the scattering coefficient and transmission coefficient, respectively, at the interface. Transmitted waveThe space is the medium 1, and the range is z>-d, the space being a lossless medium. Eta₀Is an intrinsic impedance of free space of valueThe intrinsic impedance of the lossless medium isIs the wave number of the plane wave in free space,is the wave number of medium 1, j is an imaginary unit.

A multi-layer dielectric coating conductor cylinder with radius a and total coating thickness b-a is buried in the medium 1, and each layer thickness of the coating is d₂,d₃...d_mThe relative dielectric constant of each coating is sequentially epsilon from inside to outside_2r,ε_3r...ε_mrThe relative permeability was 1. The system of equations can be listed from the boundary conditions at the medium interface of z-d:

substituting z-d into equation (7) can solve the expressions of transmission coefficient T and scattering coefficient R at the interface of the medium, which are respectively:

and (3) calculating values of the scattering coefficient R and the transmission coefficient T, and substituting the obtained values of the scattering coefficient R and the transmission coefficient T into the formula (7) to obtain electric field values of the scattered wave and the transmitted wave on the surface of the medium.

Further, the calculation process of the scattering field of the whole external part of the dielectric coating conductor cylinder in the step 2 is as follows:

the planar transmitted wave is used as the incident waveIncident along the positive direction of an x axis, and solving a homogeneous scalar Helmholtz equation set of the transmitted wave based on a traditional separation variable method, so as to obtain an isotropic cylindrical scalar wave function under a cylindrical coordinate system, wherein the expressions of an electric field and a magnetic field are as follows:

wherein A is_n＝TE₀j^-nAnd the space at the outermost layer of the cylinder is set as m +1 layers. In the mth layer, the electric and magnetic fields are:

in the above formulaIs the wave number in the m-th coating,is the intrinsic impedance in the dielectric coating, Y_nIs a second type of bessel function. B is_mnAnd C_mnIs the unknown coefficient of the electric field of each layer. The electric and magnetic field patterns of the scattered waves outside the dielectric coating are shown in the following formula:

in the above formula D_nIs the unknown coefficient of the scattered wave.Is a hank function of the second kind. Without loss of generality, it can be determined that the non-normalized amplitude of the electric field in the first layer is:

B_1,n′＝1,C_1,n′＝0 (15)

the coefficient B of the electric field of each layer can be obtained by utilizing boundary conditions at the interface of each coating layer and recursion_m,nAnd C_m,nAnd the electric field coefficient B of the other layer_m+1,nAnd C_m+1,nThe relationship of (a) is as follows:

wherein:

U_mn＝μ_mk_m+1J_n(k_mr_m)Y′_n(k_m+1r_m)-μ_m+1k_mJ′_n(k_mr_m)Y_n(k_m+1r_m)

V_mn＝μ_m+1k_mJ_n(k_m+1r_m)J′_n(k_mr_m)-μ_mk_m+1J′_n(k_m+1r_m)J_n(k_mr_m)

W_mn＝μ_mk_m+1Y_n(k_mr_m)Y′_n(k_m+1r_m)-μ_m+1k_mY′_n(k_mr_m)Y_n(k_m+1r_m)

X_mn＝μ_m+1k_mY′_n(k_mr_m)J_m(k_m+1r_m)-μ_mk_m+1Y_n(k_mr_m)J′_n(k_m+1r_m) (17)

j 'in the formula'_nAnd Y'_nRespectively, the derivatives of the Bessel functions of the first type and the second type are obtained according to the boundary conditions of the coating interface:

can find outOf (2) undetermined D_nThe following were used:

D_n＝jC_m+1,n (20)

B_m+1,n-jC_m+1,n＝j^-n (21)

due to B_m+1,nAnd C_m+1,nAlso iterated from equation (16), the coefficients of the final scattered field can be calculated by the joint equations (20), (21):

D_n＝-j^-n(C_m+1,n/(C_m+1,n+jB_m+1,n)) (22)

further, the scattered field in the local coordinate system and the transmission field in the global coordinate system in step 3 are specifically calculated as follows:

global coordinate system: the three coordinate components in the cylindrical coordinates are r,z；

local coordinate system: the three coordinate components under the cylindrical coordinates are r_img，z_img

Any point in cylindrical coordinate systemIs the intersection of three coordinate surfaces: r ═ r₀The cylindrical surface of (1) and the included angle formed by the included z-axis and the xz plane isSemi-plane of (2), z ═ z₀Of the plane of (a). The expression for converting the rectangular coordinate system into the cylindrical coordinate system is as follows:

electric field of the scattered field of the formulas (13) and (14)And a magnetic fieldIs expressed in a global coordinate system, and the scattered field is taken as an incident wave to be vertically incident to the groundAnd solving the expressions of the scattered wave in a local coordinate system and the transmitted wave in a global coordinate system by using the mirror image method principle of the medium plane as follows:

in the above formula E_nAnd F_nAre the unknown coefficients of the scattered and transmitted waves at the interface. From the boundary condition, the electric field is continuous, the tangential component of the magnetic field in the y-axis is continuous, and the magnetic field can be obtained at any pointThe following expressions are listed:

solving to obtain:

wherein

The transformation of the local coordinate system to the global coordinate system using the additive theorem is:

wherein

Then will beIncident as incident wave to the dielectric coating conductor cylinder, repeating the step 2 and the step 3, and establishing a database to obtain the transmitted wave for storing each P pointFinally, the field strength expression of the point P can be obtained as follows:

in the above formulaThe method comprises the steps of representing the transmitted wave generated at the interface of the medium every time, solving a specific value of an electric field through an MATLAB programming language, establishing a database through iterative loop, finally obtaining the electric field value at a point P, namely the electric field value obtained by a program, and comparing the obtained numerical value result with an FDTD result to verify the accuracy of derivation.

Further, the target prototype considered in step 4 is the sum of the generated transmitted waves of each scattered wave of the underground multi-layer medium coating cylinder incident to the ground plane, namely(n ═ 1, 2, 3.. n), the specific steps of genetic algorithm optimization are as follows:

(1) optimizing variables

The optimized variable is the relative dielectric constant epsilon of each dielectric coating₂,ε₃,...,ε_mAnd thickness { d }₂,d₃...d_mAnd setting an optimization variable to meet the following conditions

(2) Initial population

In the case of satisfying the above-mentioned constrainer conditions, a random initial population is generated with an initial size of 50

(3) Genetic generation

To ensure higher accuracy of the genetic algorithm, the target model was selected for 100 genetic generations.

(4) Basic operation

Determining a coding scheme, setting a cross probability, a mutation probability, and determining a fitness function, the fitness function F being set toThe higher the fitness, the smaller the sum, at a sum of 1-10 GHz.

The invention has the following beneficial effects:

an analytic solution of electromagnetic scattering of the underground multilayer dielectric coating conductor cylinder is provided on the basis of a cylindrical wave function, a calculation result is given for the first time, and the stealth of the underground multilayer coating conductor cylinder is realized on the basis of a genetic algorithm. Isotropic materials which are easier to realize are selected in the aspect of selecting materials, and optimization is carried out through a genetic algorithm to obtain the optimal materials and the optimal coating thickness, so that the scattering intensity of the coating cylinder on the ground is minimum. The method is suitable for accurately and quickly calculating the scattering characteristic of the underground coating cylinder target, and widens the application occasions of cylinder stealth.

Drawings

FIG. 1 is a structural section view of electromagnetic scattering of a subsurface multilayer dielectric coated conductor cylinder according to an embodiment of the invention.

Fig. 2 is a diagram of coordinate transformation of addition theorem based on cylindrical wave function according to an embodiment of the present invention.

FIG. 3 is a basic flow chart of a genetic algorithm according to an embodiment of the present invention.

Fig. 4 is a graph comparing the analytic calculation result and FDTD calculation result of a four-layer dielectric coated conductor cylinder according to an embodiment of the present invention.

FIG. 5 is a comparison of a four layer dielectric coated conductor cylinder and a bare PEC cylinder optimized by a genetic algorithm in accordance with an embodiment of the present invention.

Detailed Description

The present invention is further illustrated by the following examples, which are not intended to limit the invention to these embodiments. It will be appreciated by those skilled in the art that the present invention encompasses all alternatives, modifications and equivalents as may be included within the scope of the claims.

A design method for a cylindrical stealth of an underground multilayer dielectric coating conductor comprises the following specific steps:

step 1. suppose a TM wave propagates in free space along the + z-axis, where it is perpendicularly incident on an infinite uniform dielectric half-space, at z ═ d, where it is incident<The relative dielectric constant present at d is ε_r1Wherein d is the buried depth. Solving incident wave by using passive Maxwell equation set and boundary conditionThe transmission coefficient T and the scattering coefficient R at the position where z is-d are obtained, and then the transmitted wave is obtainedAnd scattered wavesThe specific expression of (1).

Step 2, transmitting the waveThe method is used as the incident wave of a dielectric coating conductor cylinder, and the incident wave is converted into a standard wave function under a cylindrical coordinate system. The whole infinite-length dielectric coating conductor cylinder is regarded as a whole, the field of each layer in the multilayer dielectric coating can be regarded as the superposition form of a first Bessel function and a second Bessel function, and then the total scattered field outside the coating cylinder is calculated by each layer by utilizing boundary conditions and a recurrence formula

Step 3, scattering waves of the infinite-length dielectric coating conductor cylinderThe scattered wave is used as the incident wave to be scattered and transmitted by the vertical incidence ground plane, and the scattered wave is used as the incident wave of the dielectric coating conductor cylinder again and circularly reciprocates; in the whole process, the expressions of scattered waves and transmitted waves on the interface are obtained according to the mirror image method of the medium interface, and then the scattering field under the local coordinate system is obtained by utilizing the simultaneous equations of the boundary conditionsAnd a transmission field in a global coordinate system

And 4, according to the calculation method of the underground multilayer medium coating cylinder in the steps 1, 2 and 3, selecting the sum of transmitted waves generated when each scattered wave in the step 3 is incident to the ground plane as a fitness function, and performing optimization processing by using a genetic algorithm to calculate the optimal relative dielectric constant and thickness of each layer of material.

The required scattering system is determined according to the requirements of step 1 of this embodimentNumber and transmission coefficient, and the scattered field at z ═ dAnd a transmission fieldIs specifically calculated as follows

Sectional view of the stealth design of the underground multi-layer cylinders studied in this example referring to FIG. 1, the area above the ground is an infinite free space, in the range z<D, its magnetic permeability is μ₀＝4π×10^-7H/m, dielectric constant of epsilon₀＝1/36π×10^-9F/m, wherein the incident wave is TM wave with amplitude of 1 and vertically incident to the ground plane along the positive direction of the z axis, the polarization direction is the positive direction of the x axis, and the parameter of mu is filled below the ground₁＝μ₀μ_1r，ε₁＝ε₀ε_1rIn which mu_1r1 and ε_1rIs the relative permeability and relative permittivity of the subsurface medium. At the ground surface where z-d, the expressions of the electric and magnetic fields of the incident wave, the scattered wave, the transmitted wave are as follows:

in the above formula, R and T are scattering coefficient and transmission coefficient at the interface, respectively, the space where the transmitted wave is located is medium 1, and the range is z>-d, the space being a lossless medium. Eta₀Is the eigenimpedance of free space with a value ofThe intrinsic impedance of the lossless medium is Is the wave number of the plane wave in free space,is the wave number of medium 1, j is an imaginary unit.

A multi-layer dielectric coating conductor cylinder with radius a and total coating thickness b-a is buried in the medium 1, and each layer thickness of the coating is d₂,d₃...d_mThe relative dielectric constant of each coating is sequentially epsilon from inside to outside_2r,ε_3r...ε_mrThe relative permeability was 1. From the boundary conditions at the medium interface of z-d, the system of equations is set forth:

substituting z-d into formula (3) can solve the expressions of transmission coefficient T and scattering coefficient R at the interface of the medium, which are respectively:

and calculating values of the scattering coefficient R and the transmission coefficient T, and substituting the obtained values of R and T into the formula (43) to obtain electric field values of the scattered and transmitted waves on the surface of the medium.

In this embodiment, the process of calculating the scattering field of the whole external part of the dielectric coated conductor cylinder in step 2 is as follows:

the planar transmitted wave is used as the incident wave E_t1Incident along the positive direction of an x axis, and solving a homogeneous scalar Helmholtz equation set of the transmitted wave based on a traditional separation variable method, so as to obtain an isotropic cylindrical scalar wave function under a cylindrical coordinate system, wherein the expressions of an electric field and a magnetic field are as follows:

wherein A is_n＝TE₀j^-nThe continuous layer of the cylinder is represented as m 2, … m, and the outer space may be provided as m +1 layers. In the mth layer, the field is:

in the above formulaIs the wave number in the m-th coating,is the intrinsic impedance in the dielectric coating, Y_nIs a second type of bessel function. B is_mnAnd C_mnIs the unknown coefficient of the electric field of each layer. Dielectric coating external powderThe electric and magnetic field patterns of the radio wave can be set as follows:

in the above formula D_nIs the unknown coefficient of the scattered wave.Is a hank function of the second kind. Without loss of generality, it can be determined that the non-normalized amplitude of the electric field in the first layer is

B_1,n＝1,C_1,n＝0 (51)

The coefficient B of the electric field of each layer can be obtained by utilizing boundary conditions at the interface of each coating layer and recursion_m,nAnd C_m,nAnd the electric field coefficient B of the other layer_m+1,nAnd C_m+1,nThe relationship of (a) is as follows:

wherein:

U_mn＝μ_mk_m+1J_n(k_mr_m)Y′_n(k_m+1r_m)-μ_m+1k_mJ′_n(k_mr_m)Y_n(k_m+1r_m)

V_mn＝μ_m+1k_mJ_n(k_m+1r_m)J′_n(k_mr_m)-μ_mk_m+1J′_n(k_m+1r_m)J_n(k_mr_m)

W_mn＝μ_mk_m+1Y_n(k_mr_m)Y′_n(k_m+1r_m)-μ_m+1k_mY′_n(k_mr_m)Y_n(k_m+1r_m)

X_mn＝μ_m+1k_mY′_n(k_mr_m)J_m(k_m+1r_m)-μ_mk_m+1Y_n(k_mr_m)J′_n(k_m+1r_m) (53)

j 'in the formula'_nAnd Y'_nRespectively, the derivatives of the Bessel functions of the first type and the second type are obtained according to the boundary conditions of the coating interface:

can find outOf (2) undetermined D_nThe following were used:

D_n＝jC_m+1,n (56)

B_m+1,n-jC_m+1,n＝j^-n (57)

due to B'_mnAnd C'_mnAlso iterated from equation (56), the joint equations (54) (55) can calculate the final scatter field coefficients as:

D_n＝-j^-n(C_m+1,n/(C_m+1,n+jB_m+1,n)) (58)

referring to fig. 2, in this embodiment, the expression of the transmission field in the cylindrical coordinate system in step 3 is in the global coordinate system, and the expression of the scattering field is in the local coordinate system, which is specifically calculated as follows:

global coordinate system: the three coordinate components in the cylindrical coordinates arer，z；

Local coordinate system: the three coordinate components under the cylindrical coordinates are r_img，z_img

Any point in cylindrical coordinate systemIs the intersection of three coordinate surfaces: r ═ r₀The cylindrical surface of (1) and the included angle formed by the included z-axis and the xz plane isSemi-plane of (2), z ═ z₀Of the plane of (a). The expression for converting the rectangular coordinate system into the cylindrical coordinate system is as follows:

electric field of the scattered field of the formulas (47) and (48)And a magnetic fieldThe scattering wave in the local coordinate system and the transmitted wave in the global coordinate system can be obtained by using the mirror image principle of the medium plane as the following expressions:

from the boundary condition, the electric field is continuous, the tangential component of the magnetic field in the y-axis is continuous, and the magnetic field can be obtained at any pointThe following expressions may be listed:

solved to obtain

Wherein

The transformation of the local coordinate system to the global coordinate system using the additive theorem is:

wherein

Then will beIncident as incident wave to the dielectric coating conductor cylinder, repeating the step 2 and the step 3, and establishing a database to obtain the transmitted wave for storing each P pointFinally, the field strength expression of the point P can be obtained as follows:

referring to fig. 3, the results of the analytical calculations are compared with the results of the FDTD calculations, where the solid lines indicate the results of the analytical calculations, the results of the FDTD calculations with asterisks. The frequency is 5-10GHz, the buried depth d is 5mm, the radius of the conductor cylinder is 0.8mm, the thicknesses of the 4 dielectric coatings are equal and are all 0.2mm, the space above the ground is free space, and the relative dielectric constant epsilon of the medium 1_1rThe relative permeability is 2, the relative permeability is 1, the relative dielectric constants of the dielectric coating from inside to outside are 3, 4, 5 and 6 respectively, and the relative permeability is 1. The correctness of the theoretical derivation can be demonstrated from fig. 3.

The target prototype considered in step 4 of the embodiment is the sum of the generated transmitted waves of each scattered wave of the underground multi-layer medium coating cylinder incident to the ground plane, namely(n ═ 1, 2, 3.. n). Referring to fig. 4, a basic flow chart of the genetic algorithm, the genetic algorithm optimization comprises the following specific steps:

(1) optimizing variables

The optimized variable is the relative dielectric constant epsilon of each dielectric coating₂,ε₃,...,ε_mAnd thickness { d }₂,d₃...d_mAnd setting an optimization variable to meet the following conditions

(2) An initial population;

under the condition of meeting the above-mentioned restraint body condition, producing random initial population, and its initial size is 50;

(3) a genetic generation;

in order to ensure higher accuracy of the genetic algorithm, the target model is selected from 100 genetic generations;

(4) basic operation;

determining a coding scheme, setting a cross probability, a mutation probability, and determining a fitness function, the fitness function F being set toThe higher the fitness is, the smaller the sum value is at the 1-10GHz sum;

by analytical calculation, correspondingThe curves with frequency change refer to fig. 5, wherein the dashed line shows the electric field curve of the underground uncoated conductor column, and the solid line shows the optimized curve of the underground 4-layer dielectric coated conductor column. The frequency is 1-10GHz, the buried depth d is 0.5m, the radius of the conductor cylinder is 0.03m, the thickness of the 4 layers of dielectric coatings is 0.0082m,0.0094m,0.007m and 0.0095m respectively, free space is above the ground, and the relative dielectric constant epsilon of the medium 1 is_1rThe relative permeability is 2, the relative dielectric constant of the dielectric coating from inside to outside is 13.806, 0.601, 13.605 and 11.654 respectively, and the relative permeability is 1. As can be seen from the figure, the optimization based on the genetic algorithm can effectively reduce the scattering intensity, particularly reduce the scattering intensity by about 25dB at 5GH, which shows that the structure can realize good stealth effect.

While the preferred embodiments and principles of this invention have been described in detail, it will be apparent to those skilled in the art that variations may be made in the embodiments based on the teachings of the invention and such variations are considered to be within the scope of the invention.