A radar cross-section calculation method based on the spectral element method and the Green's function integral method

Abstract

A method for calculating radar cross section based on the spectral element method and the Green's function integral method, which relates to the field of radar data processing. It includes: constructing a model of the scatterer in the layered background medium and grid meshing; obtaining the scattered field inside the scatterer based on the spectral element method controlled by the scattered field equation; obtaining the incident field at the corresponding position through the Green's function method, and obtaining the total field at any position inside the scatterer according to the superposition principle; obtaining the scattered field at any point in space according to the volume equivalence principle and the superposition principle; calculating the RCS of the target through the definition of radar cross section. It overcomes the limitations of the differential method in the calculation region and the difficulty of the integral method in calculating high-contrast problems, and establishes a set of multi-scale and high-contrast RCS numerical simulation algorithms, which can accurately calculate the RCS at any position in the plane layered medium without being restricted by the calculation region. It solves the problem that the field source in the traditional spectral element method is limited in the calculation region.

Description

Technical Field

[0001] The present invention relates to the field of radar data processing, and in particular to a radar cross-sectional area calculation method based on a spectral element method and a Green's function integral method, which is a method for determining a radar cross-sectional area by combining a spectral element method (differential method) and a Green's function method (integral method). Background Art

[0002] Radar cross section (RCS) is a main characteristic of radar targets. It is a physical quantity that characterizes the intensity of the echo generated by an object when it is irradiated by radar waves, and reflects the scattering ability of the target to the electromagnetic waves irradiated on it. The analysis and research of the scattering characteristics of electromagnetic targets are of great significance in the fields of radar target stealth, radar target recognition, and radar imaging. Therefore, in recent years, the accurate calculation of the RCS of scatterers has always been a research hotspot.

[0003] For the numerical calculation of RCS, in the field of computational electromagnetics, the moment method (MOM), finite element method, finite difference time domain method (FDTD) and other methods can be used for calculation. However, these methods have their own limitations. For example, the impedance matrix generated by the moment method is a dense complex matrix. Due to memory limitations, it can only handle small-sized targets. Although FDTD can solve large targets and has high calculation accuracy, the time step selection is limited by the size of the grid, which increases the calculation cost. At the same time, there are also problems such as inconsistent time and space convergence speed. The finite element method is a differential method. Due to the limitation of computing resources, it is necessary to truncate the calculation area through boundary conditions to include both the source and the receiving point in the calculation area, which greatly restricts the calculation of RCS.

[0004] The spectral element method combines the flexible applicability of the finite element method with the high precision of the spectral method. It is efficient and accurate in the calculation of electromagnetic fields in complex inhomogeneous media, can solve high-contrast problems, and is widely used to solve problems in the field of electromagnetic fields. However, as a differential method, like the finite element method, the solution of RCS is greatly limited. Although the integral equation method based on the Green's function method has no limitation on the calculation area, it cannot solve high-contrast problems, which limits its application in solving RCS. The combination of the two methods effectively retains the advantages of the two methods, and then solves the RCS of high-contrast scatterers at any position in free space / plane layered media. Summary of the invention

[0005] The purpose of the present invention is to overcome the limitation of the calculation area of the differential method and the difficulty of the integral method in calculating the high contrast problem, and to provide a radar cross-sectional area calculation method based on the spectral element method and the Green's function integration method, and a multi-scale, high-contrast RCS numerical simulation, which can accurately calculate the RCS of a high-contrast target at any position without being limited by the calculation area.

[0006] The present invention includes the following steps:

[0007] Step 1: Construct a 3D model of the scatterer under test in the background medium and perform mesh division on the calculation space;

[0008] Step 2: Calculate the scattered field inside the scatterer based on the spectral element method controlled by the scattering field equation, obtain the incident field at the corresponding position through the Green's function method, and obtain the total field at any position inside the scatterer according to the superposition principle;

[0009] Step 3: According to the volume equivalence theory, the equivalent source in any small volume inside the scatterer can be obtained, and then according to the superposition principle, the scattered field at any place in space can be obtained;

[0010] Step 4: The incident field at any place emitted by the point source can be obtained through the Green's function method. According to the definition formula of the radar cross section, the radar cross-sectional area σ at each point can be directly calculated:

[0011]

[0012] In Step 1, the process of modeling the calculation region containing the target is as follows:

[0013] (1) Model the calculation region according to the size of the calculation target, the background medium it is in, and its spatial position. Due to the advantages of the method, the calculation region only needs to contain a small amount of the background region, which is divided into different regions according to different materials, and hexahedral mesh division is performed.

[0014] (2) Configure the model parameters, including: the position, amplitude, and frequency of the emission source; the electromagnetic characteristic parameters (relative permittivity, conductivity, permeability) of various materials in the calculation region; the boundary conditions of the spectral element method; the receiving point coordinates, etc.

[0015] In Step 2, the specific steps include:

[0016] (1) The control equation of the scattered magnetic field H s is the vector Helmholtz equation and Gauss's theorem:

[0017]

[0018]

[0019] k0 and ε0 are the wave number and permittivity in vacuum, ω is the angular frequency, I is the unit vector, and and are the complex relative permittivity and permeability in the calculation region respectively, and They are the complex relative permittivity and permeability of the background medium respectively. After obtaining the scattered magnetic field, similarly, the scattered electric field E can be obtained through the duality theorem. s .

[0020] (2) In the spectral element method, the curl vector basis function Φ n is used to expand the scattered magnetic field H s , to achieve exponential convergence (i.e., the error decreases exponentially with the increase of the order of the basis function), and the nodal basis function is used to expand the scalar function p. Through the Galerkin method, the following discrete linear system can be obtained:

[0021]

[0022] Among them, S is called the stiffness matrix, M is called the mass matrix, K is called the constraint term, and K T is the transpose of K; the vectors x and y represent the unknowns of H s and p respectively; b and c s are the load vectors. The element terms in the above system matrix are as follows:

[0023]

[0024]

[0025]

[0026]

[0027]

[0028] where the subscripts m and n represent the indices of the test function and the basis function respectively. α is a scaling factor used to balance Gauss's theorem and the vector Helmholtz equation.

[0029] (3) Solve the incident field through the Green's function method (integral method). In free space, the Green's function can be obtained through an analytical solution, while the Green's function of the stratified medium can be obtained from the literature (Fields and Waves in Inhomogeneous Media [M]. Publishing House of Electronics Industry, written by Weng Cho Chew (USA), 1992). Through the superposition principle, the total field E tot inside the scatterer is:

[0030] E tpt = E i + E s

[0031] In step 3, based on the volume equivalence theory, the equivalent sources inside any small volume of the scatterer are obtained, and then through the superposition principle, the scattered field at any point in space can be obtained through integral calculation. The scattered field at any position r is:

[0032]

[0033] The dyadic Green's function generated by a unit current point source located at r' at r. In a uniaxial anisotropic background, it can be obtained through an analytical solution, while in a layered background medium, it can be obtained from the literature (Fields and Waves in Inhomogeneous Media [M]. Publishing House of Electronics Industry, written by Weng Cho Chew (American), 1992). is the dyadic Green's function generated by a unit magnetic current point source at r'. Its analytical solution can be obtained through the duality theory from The equivalent sources eq (r) and eq have the following relationships with the dielectric / magnetic parameters of the target scatterer and the total internal field:

[0034]

[0035]

[0036] In step 4, the incident field at any point emitted by the point source can be calculated by the Green's function method, which is the same as the method for calculating the incident field in steps 2 and 3.

[0037] The present invention includes constructing a model of the scatterer in a layered background medium and grid meshing; obtaining the scattered field inside the scatterer based on the spectral element method controlled by the scattered field equation; obtaining the scattered field at any point in space according to the volume equivalence principle and the superposition principle; calculating the RCS of the target through the definition of the radar cross section.

[0038] Compared with the prior art, the advantages and technical effects of the present invention are as follows:

[0039] For the calculation of the RCS of the target scatterer, the source needs to be far enough from the target. And due to the large variation in the electromagnetic properties of composite materials, the corresponding contrast will also vary greatly. Therefore, for the calculation of the RCS, it is necessary to solve the problems of large calculation scale and large contrast. Obviously, the traditional differential and integral algorithms are difficult to meet this requirement. That is, the traditional differential method cannot meet the calculation requirements of large scales, while the integral method cannot meet the calculation needs of high contrast. Therefore, in the present invention, the equation of "scattered field and incident field" of the spectral element method is extended and derived to solve the problem that the field source in the traditional spectral element method is limited in the calculation region. Then, based on the volume equivalence theory, an integral method is introduced, and the scattered field at any far point in space is obtained through the convolution of the Green's function and the equivalent source, so that the RCS at any position in space can be accurately calculated. The present invention can be applied to the technical field of radar data processing. Brief Description of the Drawings

[0040] Figure 1Schematic diagram of the computational domain model containing the scatterer to be measured according to an embodiment of the present invention.

[0041] Figure 2 Schematic diagram of the model after the computational domain is meshed with hexahedral meshes according to an embodiment of the present invention.

[0042] Figure 3 Schematic diagram of the three-dimensional coordinate scatter points of the receiving points where the RCS needs to be calculated according to an embodiment of the present invention.

[0043] Figure 4 Schematic diagram of the two-dimensional coordinate cross-section of the receiving points where the RCS needs to be calculated according to an embodiment of the present invention.

[0044] Figure 5 At 8 GHz Figure 2 、 3 Schematic diagram of the three-dimensional surface of the RCS calculation results at the receiving points in

[0045] Figure 6 At 10 GHz Figure 2 、 3 Schematic diagram of the three-dimensional surface of the RCS calculation results at the receiving points in

[0046] Figure 7 For Figure 2 、 3 Line graph of the RCS calculation results at the receiving points in Detailed implementation manner

[0047] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be described more comprehensively below with reference to the relevant drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0048] The present invention proposes a method for calculating the RCS, which combines the spectral element method (differential method) and the Green's function method (integral method). The scattered field inside the target is calculated by the spectral element method controlled by the scattered field equation, the incident field at the corresponding position is obtained by the Green's function method, and the total field at any position inside the scatterer is obtained according to the superposition principle. According to the volume equivalence theory, the equivalent source inside any small volume inside the scatterer can be obtained, and then through the superposition theorem, that is, the convolution of the equivalent source and the Green's function can obtain the scattered field at any position in space. Finally, the radar cross-section of the target scatterer is calculated through the definition formula.

[0049] The method for calculating the radar cross-section (RCS) based on the combination of the spectral element method (differential method) and the Green's function method (integral method) according to the embodiment of the present invention specifically includes the following steps:

[0050] Step 1: Establish a 3D model of a composite medium plate in a layered background medium, mesh it with a hexahedral grid, and configure the model parameters.

[0051] Step 2: Use the spectral element method to calculate the scattered field inside the scatterer in the model, and use the Green's function method to obtain the incident field at the internal receiving point. The total field at the internal receiving point of the scatterer can be obtained by adding them together.

[0052] Step 3: The field inside the scatterer is equivalent to an equivalent source; it is convolved with the Green's function, and the scattering field at the receiving point is obtained through the superposition theorem.

[0053] Step 4: Use Green's function again to find the incident field at the receiving point in free space emitted by the point source. Finally, calculate the RCS of each point using the definition formula.

[0054] Furthermore, the specific method for establishing the model in step 1 is as follows:

[0055] First, a layered model of half space is established, with the upper layer as the air layer and the lower layer as the ground. Then a medium is established as a scatterer. In this case, it consists of 4 pieces of dielectric materials placed on the ground, and the entire model is divided into hexahedral meshes. Due to the advantages of the spectral element method controlled by the scattered field equation, only a small amount of background area needs to be retained. The point source setting can be placed at any position outside the calculation area, with a position of (50m, 50m, 10m), an amplitude of 1, a polarization direction of (1,1,1), and an operating frequency of 8GHz and 10GHz respectively.

[0056] Furthermore, the calculation parameters of the spectral element method in step 2 are as follows:

[0057] In order to absorb the field at the simulation boundary, the spectral element method uses the PML (perfectly matched layer) boundary condition. And in order to ensure the accuracy of subsequent calculations, the equivalent source inside the scatterer needs to be dense enough, so 100×100×20, a total of 200,000 receiving points, are set inside the scatterer. These receiving points will be equivalent to 200,000 equivalent sources in the subsequent volume equivalent process.

[0058] Specific application embodiments are given below in conjunction with the accompanying drawings.

[0059] The model structure is as follows Figure 1 As shown. The upper layer is the air layer, with a thickness of 0.023m, a length and width of 0.11m, and the scatterer is composed of four different dielectric materials, each with a size of 0.05m×0.05m×0.02m. The scatterer is placed at the center of the ground in the calculation area. The lower part is the ground, with a thickness of 0.03m. The ground and the air layer constitute the background medium. The relative dielectric constant of the ground is set to 4, and the conductivity is 1e -4 S / m, the relative permittivity of air is set to 1, and the conductivity is 1e-6 S / m, and the permeability is 1 for all.

[0060] The electromagnetic characteristic parameters of the four regions of the scatterer calculated are shown in Table 1:

[0061] Table 1 Electromagnetic characteristic parameter table for each region

[0062] upper right region upper left region lower left region lower right region relative permittivity 1.5 6.5 9.0 2.5 conductivity <![CDATA[10 -2.75 S / m]]> <![CDATA[10 -2.15 S / m]]> <![CDATA[10 -3.05 S / m]]> <![CDATA[10 -3.8 S / m]]> relative permeability 1 1 1 1

[0063] The entire calculation region is meshed with hexahedral meshes, and the schematic diagram of the meshed model is as Figure 2 shown.

[0064] To ensure the accuracy of solving the scattered field by the scattered field equation, the density of the equivalent sources inside the scatterer needs to be at least 10 points per wavelength. Therefore, the positions of the receiving points (equivalent sources) inside the scatterer are set at x = -0.0495:0.001:0.0495, y = 0.0495:0.001:0.0495, z = 0.0005:0.001:0.0195, forming an array of 100×100×20.

[0065] The total field inside the scatterer is obtained by superimposing the scattered field obtained by the spectral element method and the incident field obtained by the Green's function method. Further, the corresponding equivalent sources can be obtained. Then, according to the scattered field integral equation, the convolution of the equivalent sources and the Green's function can be used to calculate the scattered field at any arbitrary receiving point far away. This field is the scattered field generated by the field source irradiating the scatterer. The position coordinates of the receiving points are as Figure 3 ,4 shown. Figure 3 It is a three-dimensional position coordinate diagram of the receiving points. Figure 4 It is a cross-sectional diagram of the receiving points when z = 3.

[0066] After obtaining the scattered field at the receiving points, the incident field is obtained by the Green's function method, and the RCS values of these points are obtained according to the definition formula.

[0067] To make the calculation results more intuitive, the RCS calculation results are represented by a three-dimensional surface diagram, as Figure 5 , Figure 6 shown.

[0068] Figure 7 It is a curve diagram of the RCS calculation results of the target scatterer at the receiving points.

[0069] The present invention proposes a method combining the spectral element method (differential method) and the Green's function method (integral method), establishing a set of RCS calculation methods with multi-scale and high contrast, and realizing accurate calculation of the RCS at any position without being restricted by the calculation region.

[0070] The above embodiments are only preferred embodiments of the present invention and should not be considered as limiting the scope of implementation of the present invention. All equivalent changes and improvements made within the scope of the application of the present invention should still fall within the scope covered by the patent of the present invention.

Claims

1. A method for calculating radar cross section based on the spectral element method and the Green's function integral method, characterized in that It includes the following steps: Step 1: Construct a 3D model of the scatterer under test in the background medium and perform mesh division on the calculation space; Step 2: Calculate the scattered field inside the scatterer based on the spectral element method controlled by the scattered field equation, obtain the incident field at the corresponding position through the Green's function method, and obtain the total field at any position inside the scatterer according to the superposition principle. The specific steps include: (1) Scattering magnetic field H s The governing equations are the vector Helmholtz equation and Gauss's theorem: $k_0$ and $\varepsilon_0$ are the wavenumber and permittivity in vacuum, $\omega$ is the angular frequency, $\mathbf{I}$ is the unit vector, and and are the complex relative permittivity and permeability of the scatterers in the computational domain, respectively, and are the complex relative permittivity and permeability of the background medium, respectively; after obtaining the scattered magnetic field, similarly, the scattered electric field $\mathbf{E}$ is obtained by the duality theorem s ; (2) In the spectral element method, the curl vector basis function Φ n is used to expand the scattered magnetic field H s , so as to achieve exponential convergence, that is, the error decreases exponentially with the increase of the order of the basis function; the nodal basis function is used to expand the scalar function p; through the Galerkin method, the following discrete linear system is obtained: Among them, S is called the stiffness matrix, M is called the mass matrix, K is called the Gaussian constraint term, and K T is the transpose of K; the vectors x and y represent the unknowns of H s and p respectively; b and c a are the load vectors; the element terms in the above system matrix are as follows: where the subscripts m and n represent the indices of the test function and the basis function respectively; α is a scaling factor used to balance Gauss's theorem and the vector Helmholtz equation; (3) Solve the incident field by the Green's function method. In free space, the Green's function is obtained through an analytical solution. By the superposition principle, the total field E inside the scatterer tot is as follows: E tot = E i + E s Step 3: According to the volume equivalence theory, obtain the equivalent source in any small volume inside the scatterer, and then obtain the scattered field at any point in space according to the superposition principle; the scattered field at any position r is: wherein, is the dyadic Green's function generated by a unit current point source located at r' at r. In a uniaxial anisotropic background, is obtained through an analytical solution, while in a layered background medium, it is obtained through existing methods; is the dyadic Green's function generated by a unit magnetic current point source at r'. Through the duality theory, its analytical solution is obtained; the equivalent sources J eq (r) and M eq have the following relationships with the dielectric / magnetic parameters of the target scatterer and the total internal field: Step 4: Obtain the incident field at any point emitted by the point source through the Green's function method, and directly calculate the radar cross-sectional area σ at each point according to the definition formula of the radar cross section:

2. The radar cross-section calculation method based on the spectral element method and the Green's function integration method as described in claim 1, wherein In Step 1, the 3D model of the scatterer under test constructed in the background medium is to model the calculation region containing the target, and the process is as follows: (1) Perform calculation region modeling according to the size, background medium and spatial position of the calculation target. Due to the method advantages, the calculation region only needs to contain a small amount of background region, which is divided into different regions according to different materials and hexahedral mesh division is performed; (2) Configure the model parameters, including: the position, amplitude and frequency of the emission source; the electromagnetic characteristic parameters of various materials in the calculation region, including relative permittivity, conductivity, permeability; the boundary conditions of the spectral element method; the receiving point coordinates.

3. The radar cross-section calculation method based on the spectral element method and the Green's function integration method as described in claim 1, wherein In Step 4, the incident field at any point emitted by the point source is calculated through the Green's function method, and the method is the same as the incident field calculation method in Steps 2 and 3.

Images