Radar cross section calculation method based on three-dimensional parabolic equation

Abstract

The invention discloses a radar cross section calculation method based on a three-dimensional parabolic equation. The radar cross section calculation method comprises the following steps: step 1, performing three-dimensional analysis by using the three-dimensional parabolic equation; step 2, constructing an integral domain with an ideal matching layer PML absorption boundary; step 3, discretizing the three-dimensional parabolic equation; and step 4, radar cross section calculation. According to the method disclosed by the invention, the three-dimensional parabolic equation method is introduced into the calculation of the radar cross section of the aircraft, so that the problems of high calculation cost and long time of a full-wave method, inaccurate calculation result of an asymptotic method and the like are effectively solved, the scattered field of the aircraft can be accurately and quickly calculated, and compared with a traditional calculation method, the calculation efficiency is improved. The method has the advantages that the calculation speed is high, the scatterer in any shape can be accurately modeled, and few calculation resources are consumed.

Description

Technical Field

[0001] The present invention belongs to the technical field of high-speed aircraft detection and warning, and particularly relates to a method for calculating radar cross section based on three-dimensional parabolic equation in this field, which can accurately calculate the radar cross sections of various types of aircraft under different incident angles. Background Technique

[0002] To study the scattering field (scattering cross section) of an aircraft, it is necessary to accurately model the boundary of the aircraft. Due to the complex boundary conditions of the aircraft, accurate modeling is difficult, and the existing calculation methods have the following problems:

[0003] (1) Strict calculation methods require accurate grid division of the calculation object. Otherwise, there will be a large error in the calculation result. At the same time, the method itself has high requirements for computer hardware, making it difficult for strict calculation methods to quickly and accurately calculate the scattering field of large-scale objects.

[0004] (2) Approximation methods model the boundary conditions of the scatterer and need to invert large sparse matrices, resulting in a large amount of calculation and unable to provide accurate calculation results. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a method for calculating radar cross section based on three-dimensional parabolic equation.

[0006] The present invention adopts the following technical solutions:

[0007] A method for calculating radar cross section based on three-dimensional parabolic equation, the improvement lies in that it includes the following steps:

[0008] Step 1, perform three-dimensional analysis using the three-dimensional parabolic equation;

[0009] Step 2, build an integration domain with a perfectly matched layer (PML) absorbing boundary;

[0010] Step 3, discretize the three-dimensional parabolic equation;

[0011] Step 4, calculate the radar cross section.

[0012] Further, the specific content of step 1 includes:

[0013] In the three-dimensional rectangular coordinate system x, y, z, the three-dimensional parabolic equation is expressed as follows:

[0014]

[0015] In the above formula, k is the wave number in vacuum, n is the refractive index in the calculation region, and ψ is the introduced field quantity;

[0016] Select the positive x-axis direction as the paraxial direction, and define the electromagnetic field propagating in the positive x-axis direction as the simplified field u:

[0017] u(x,y,z) = e -ikx ψ(x,y,z) (2)

[0018] In the above formula, i represents the imaginary number;

[0019] Combining equations (1) and (2), the three-dimensional parabolic equation expressed in terms of u is:

[0020]

[0021] Assume Simplify equation (3) to:

[0022]

[0023] Decompose equation (4) into:

[0024]

[0025] From equation (5), the following equation is obtained:

[0026]

[0027] The solution of equation (6-1) corresponds to the wave propagating forward, and the form of the solution is given by the following formula:

[0028] u(x + Δx,y,z) = e ikΔx(Q-1) ·u(x,y,z)

[0029] In the above formula, Δx represents the calculation step taken along the x-axis direction.

[0030] Furthermore, the specific steps of step 2 include:

[0031] Equation (6-1) is obtained by using the first-order Taylor series expansion. Assume Q is:

[0032]

[0033] In the above formula, Assume that the refractive index only depends on the height H change, then:

[0034]

[0035] In the above formula, ~ represents approximately equal to;

[0036] Substitute the first-order Taylor series of each radical in equation (8) into (6-1) to obtain:

[0037]

[0038] According to the definitions of Y and H, Equation (9) is transformed into the following form:

[0039]

[0040] The scattered field of the object and the radar cross section are calculated from Equation (10), and the perfectly matched layer (PML) absorbing boundary condition is introduced as the absorbing boundary condition for the Maxwell equations.

[0041] Further, Step 3 specifically includes:

[0042] Define the range of the region (mΔx, y, z) as m, where m is a natural number;

[0043] The discrete form of Equation (10) is:

[0044]

[0045] In the above formula, i represents the i-th row, j represents the j-th column, Δy represents the calculation step size taken along the y-axis direction, and Δz represents the calculation step size taken along the z-axis direction; the fields of ranges m and m - 1 are calculated using Equation (11) to determine the positions of the grid points.

[0046] Further, Step 4 specifically includes:

[0047] In free space, the function with x0 as the field space is:

[0048]

[0049] In the above formula, (x0, y, z) and (x’, y’, z’) are any two points in three-dimensional space, i represents the imaginary number;

[0050] The radar cross section is defined as:

[0051]

[0052] In a three-dimensional rectangular coordinate system, the angle between any point in space and the positive x-axis is θ, and the angle between it and the positive z-axis is The distance from the origin is r. In the above formula, x = rcosθ, y = sinθ, u i represents the incident wave, and u s represents the reflected wave;

[0053] (x, y, z) tends to infinity along the direction given by Equation (12), and from Equation (13), we get:

[0054]

[0055] In the above formula, i represents the imaginary number, and u s(x, y, z) is the scattered field.

[0056] The beneficial effects of the present invention are as follows:

[0057] The method disclosed by the present invention introduces the three-dimensional parabolic equation method into the calculation of the radar cross section of an aircraft, effectively solving the problems of high calculation cost and long time in the full-wave method, and inaccurate calculation results in the asymptotic method. It can accurately and quickly calculate the scattered field of the aircraft. Compared with the traditional calculation method, this method has the advantages of fast calculation speed, accurate modeling of scatterers with arbitrary shapes, and less consumption of computing resources. While accurately analyzing the characteristics of the scattered field of the aircraft, it can ensure the accuracy of the calculation results and save computer resources at the same time.

[0058] The method disclosed by the present invention provides a new idea for studying the Radar Cross Section (RCS) of an aircraft, and has important scientific significance and application value. Description of the Drawings

[0059] Figure 1 is a schematic flow diagram of the method of the present invention;

[0060] Figure 2 is the integration domain with an ideal matched layer PML absorbing boundary;

[0061] Figure 3 is the position map of the three-dimensional parabolic equation grid points;

[0062] Figure 4 is the amplitude distribution map of the initial field of the antenna;

[0063] Figure 5 are the scattered field distribution and the radar cross section values of F22 under different views;

[0064] Figure 6 are the scattered field distribution and the radar cross section values of F117 under different views;

[0065] Figure 7 is the schematic diagram of the model partition of F117. Detailed Embodiments

[0066] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0067] The parabolic equation (PE) method is derived from the wave equation and has been widely used in solving radio wave propagation problems. With the deepening of research, it is found that the PE method has unique technical advantages in calculating electromagnetic scattering problems, and can meet higher efficiency and accuracy requirements under limited computing resources. It can give accurate results when calculating the scattering of objects with a wavelength to dozens of wavelengths. The three-dimensional parabolic equation can accurately model the boundaries of the aircraft, and it has advantages in calculating the scattering field of objects larger than the wavelength.

[0068] The PE method has the following advantages in calculating the scattered field of an aircraft:

[0069] (1) The PE method can minimize the area for solving sparse matrices and significantly reduce the amount of calculation. In addition, the stability of the PE method allows it to be used on extremely irregular surfaces.

[0070] (2) The PE method can process very large objects on a computer. It does not need to divide the object into regular-shaped primitives, but can directly process objects of arbitrary shapes. The PE method has both numerical efficiency and good stability when computing resources are limited.

[0071] Embodiment 1, this embodiment discloses a radar cross section calculation method based on a three-dimensional parabolic equation, such as Figure 1 As shown, the following steps are included:

[0072] Step 1: Perform three-dimensional analysis using the three-dimensional parabolic equation:

[0073] In the three-dimensional rectangular coordinate system x, y, z, the three-dimensional parabolic equation is expressed as follows:

[0074]

[0075] In the above formula, k is the wave number in vacuum, n is the refractive index in the calculation area, and ψ is the introduced field quantity;

[0076] The positive direction of the x-axis is selected as the paraxial direction, and the electromagnetic field propagating along the positive direction of the x-axis is defined as the simplified field u:

[0077] u(x,y,z)=e -ikx ψ(x,y,z) (2)

[0078] In the above formula, i represents an imaginary number;

[0079] Combining equations (1) and (2), the three-dimensional parabolic equation represented by u is:

[0080]

[0081] Assumptions Simplify Equation (3) to:

[0082]

[0083] Decompose Equation (4) into:

[0084]

[0085] From Equation (5), the following equation is obtained:

[0086]

[0087] The solution of Equation (6-1) corresponds to the forward-propagating wave, and the form of the solution is given by:

[0088] u(x + Δx, y, z) = e ikΔx(Q-1) ·u(x, y, z)

[0089] In the above equation, Δx represents the calculation step taken along the x-axis direction.

[0090] Step 2, construct an integration domain with an ideal perfectly matched layer (PML) absorption boundary:

[0091] Equation (6-1) is obtained by a first-order Taylor series expansion. Assume Q is:

[0092]

[0093] In the above equation, Assume that the refractive index only depends on the height H, then:

[0094]

[0095] In the above equation, ~ means approximately equal to;

[0096] Substitute the first-order Taylor series of each radical in Equation (8) into (6-1) to obtain:

[0097]

[0098] According to the definitions of Y and H, transform Equation (9) into the following form:

[0099]

[0100] Calculate the scattering field of the object and the radar cross section (RCS) from Equation (10). The integration domain is considered to be a box containing the object. The integration domain must be truncated in the transverse plane. To achieve this, introduce the ideal perfectly matched layer (PML) absorption boundary condition as the absorption boundary condition for Maxwell's equations. An important advantage of PML is that it is effective for all incident angles at several grid points of the integration domain. The integration domain with a PML absorption boundary is asFigure 2 as shown

[0101] Step 3, discretize the three-dimensional parabolic equation:

[0102] Usually, the Crank-Nicolson format is adopted. In the present invention, another format with better stability than Crank-Nicolson is adopted:

[0103] Define the range of the region (mΔx, y, z) as m, where m is a natural number. Although in the Crank-Nicolson scheme, the second-order derivatives of y and z are calculated by taking the average value between the range m and the range m - 1, this scheme only calculates the second-order derivatives in the range m.

[0104] The discrete form of the free-space equation in Equation (10) is:

[0105]

[0106] In the above formula, i represents the i-th row, j represents the j-th column, Δy represents the calculation step size taken along the y-axis direction, and Δz represents the calculation step size taken along the z-axis direction; as Figure 3 shown, use Equation (11) to calculate the fields in the ranges m and m - 1 and determine the positions of the grid points.

[0107] When analyzing with the parabolic equation, it is necessary to invert a triangular matrix to obtain u at the range m. In the three-dimensional case, the sparse matrix is a very large matrix and cannot be directly inverted to solve the equation.

[0108] Step 4, radar cross-section calculation:

[0109] After calculating the fields in the entire calculation domain, the fields in any region x can be calculated. In free space, the function of the field space with x0 is:

[0110]

[0111] In the above formula, (x0, y, z), (x’, y’, z’) are any two points in the three-dimensional space, i represents the imaginary number;

[0112] The radar cross-section is defined as:

[0113]

[0114] In the three-dimensional rectangular coordinate system, the angle between any point in space and the positive x-axis is θ, and the angle between it and the positive z-axis is The distance from the origin is r. In the above formula, x = rcosθ, y = sinθ, u i represents the incident wave, us Indicates the reflected wave;

[0115] (x, y, z) tends to infinity in the direction given by Equation (12), and from Equation (13), we get:

[0116]

[0117] In the above formula, i represents the imaginary number, and u s (x, y, z) is the scattered field.

[0118] The initial field of the antenna adopts a Gaussian pattern function. As Figure 4 shown, the main radiation energy of the antenna is concentrated in the main lobe, and the radiation characteristics of the antenna can be seen. Using the Gaussian pattern function, the antenna can control beamforming and beam pointing, and can achieve focused monitoring of a specific area. By adjusting the phase and amplitude of the antenna, enhanced beam scanning in a specific direction can be achieved, significantly improving the detection ability of the target. The antenna using the Gaussian pattern function can further reduce the side lobe and enhance the radiation in the direction of the antenna symmetry axis, so a higher gain than that of a linear antenna can be obtained.

[0119] To calculate the scattered field of the aircraft, a stepped model of the aircraft is adopted. The integral domain sizes for calculating the scattered field of the aircraft in the x, y, and z directions are 40λ, 30λ, and 30λ respectively. Assume that the grid spacings in the x, y, and z directions are and

[0120] Figure 5 are the scattered field distribution and the radar cross-section values of F22 under different views; Figure 6 are the scattered field distribution and the radar cross-section values of F117 under different views.

[0121] The incident wave adopts a Gaussian pattern function, and the incident frequency is 300 MHz. The calculation results give the field strength of the scattered field of the aircraft under different incident angles. It can be seen that the numerical strength of the aircraft RCS changes significantly with the incident direction because the movement of the aircraft will cause the change of the line-of-sight angle, thus affecting the RCS.

[0122] To calculate the scattered field of the aircraft, a stepped model of the aircraft is adopted. Here, taking the F117 aircraft as an example, when using the three-dimensional parabolic equation for calculation, as Figure 7 shown, the F117 is decomposed into 53 finite regions, where 26 regions are the same on the left and right sides of the aircraft, and another region is the lower surface. The aircraft is calculated at different azimuths from 0 to 360° and different elevation angles from 0 to 70° with a step size of 10°. After processing, the aperture area of each finite area is calculated to obtain the surface of the F117. The incident angle of each region is calculated from the observation point at the z coordinate of each target position, and then the radar cross-section is obtained as a function of the azimuth angle and the elevation angle.

[0123] In summary, the method disclosed in this embodiment uses the parabolic equation combined with the perfectly matched layer absorption condition to solve the scattering field distribution of the aircraft. The research results show that, for different aircraft models and different incident azimuth angles, the parabolic equation method can quickly and accurately calculate the scattering field distribution of the aircraft, and the calculation results are in good agreement with the theoretical analysis. This method provides a new idea for the calculation of the aircraft scattering field.

Claims

1. A radar cross section calculation method based on a three-dimensional parabolic equation, characterized in that: The steps include: Step 1, using a three-dimensional parabolic equation to perform a three-dimensional analysis; Step 2, construct an integration domain with a PML absorbing boundary; Step 3, discretize the three-dimensional parabolic equation; Step 4: Calculate the radar cross section.

2. The radar cross section calculation method based on the three-dimensional parabolic equation according to claim 1, characterized in that: The step 1 specifically includes: In the three-dimensional rectangular coordinate system x, y, z, the three-dimensional parabolic equation is expressed as follows: In the above formula, k is the wave number in vacuum, n is the refractive index in the calculation area, and ψ is the introduced field quantity; The positive direction of the x-axis is selected as the paraxial direction, and the electromagnetic field propagating along the positive direction of the x-axis is defined as the simplified field u: u(x,y,z)=e -ikx ψ(x,y,z) (2) In the above formula, i represents an imaginary number; Combining equations (1) and (2), the three-dimensional parabolic equation represented by u is: Assumptions Simplify formula (3) to: Decompose equation (4) into: From formula (5), we get the following equation: The solution of equation (6-1) corresponds to the wave propagating forward, and the solution is given by the following equation: u(x+Δx,y,z)=e ikΔx(Q-1) ·u(x,y,z) In the above formula, Δx represents the calculation step taken along the x-axis direction.

3. The radar cross section calculation method based on the three-dimensional parabolic equation according to claim 2, characterized in that: The step 2 specifically includes: Formula (6-1) is obtained by first-order Taylor series expansion, assuming that Q is: In the above formula, Assuming that the refractive index depends only on the height H, then: In the above formula, ~ means approximately equal to; Substituting the first-order Taylor series of each root of equation (8) into (6-1) yields: According to the definitions of Y and H, formula (9) is transformed into the following form: The scattering field and radar cross section of the object are calculated by equation (10), and the ideal matching layer (PML) absorbing boundary condition is introduced as the absorbing boundary condition of Maxwell equations.

4. The radar cross section calculation method based on the three-dimensional parabolic equation according to claim 3 is characterized in that: The step 3 specifically includes: Define the region (mΔx, y, z) to be m, where m is a natural number; The discrete form of formula (10) is: In the above formula, i represents the i-th row, j represents the j-th column, Δy represents the calculation step taken along the y-axis direction, and Δz represents the calculation step taken along the z-axis direction. Formula (11) is used to calculate the field in the range m and m-1 to determine the position of the grid point.

5. The radar cross section calculation method based on the three-dimensional parabolic equation according to claim 4, characterized in that: The step 4 specifically includes: The function in free space with x0 as the field space is: In the above formula, (x0, y, z), (x', y', z') are any two points in three-dimensional space. i represents an imaginary number; The radar cross section is defined as: In a three-dimensional rectangular coordinate system, the angle between any point in space and the positive x-axis is θ, and the angle between any point in space and the positive z-axis is The distance from the origin is r. In the above formula, x = rcosθ, y = sinθ, u i represents the incident wave, u s represents the reflected wave; (x, y, z) tends to infinity along the direction given by equation (12), and from equation (13) we get: In the above formula, i represents an imaginary number, u s (x,y,z) is the scattered field.

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