Fast Analysis Method for Multidimensional Uncertainty Problems of Rotationally Symmetric Bodies in Inhomogeneous Media

Abstract

The present invention discloses a rapid analysis method for multi-dimensional uncertainty problems of a non-uniform medium rotationally symmetric body, and the steps are as follows: First, the mother plane of the rotationally symmetric body is modeled using non-uniform rational B-spline technology, reducing the previous three-dimensional volume meshing of the non-uniform medium target to two-dimensional surface meshing of the mother plane, and making the shape of the model be controlled by control points; Subsequently, a VIE-BOR matrix equation with shape and permittivity random variables is obtained by the perturbation method; Finally, different variation amounts are obtained through random sampling, and the radar cross section after the micro-variation of the shape and permittivity of the rotationally symmetric body, as well as the statistical mean and variance of the radar cross section are obtained. Compared with the VIE method, the present invention has a faster calculation speed and less memory usage, and can simultaneously handle multi-dimensional uncertainty problems of the geometric shape and permittivity of non-uniform media.

Description

Technical Field

[0001] The present invention belongs to the technical field of numerical calculation of target electromagnetic scattering characteristics, and particularly relates to a fast analysis method for multi-dimensional uncertainty problems of non-uniform dielectric rotationally symmetric bodies. Background Art

[0002] In general electromagnetic characteristic analysis problems, the geometric shape and dielectric constant of the target remain unchanged, and the electromagnetic scattering characteristics of the determined target are analyzed. However, due to the influence of factors such as manufacturing processes, external environments, and materials, in actual situations, there will be uncertainties in the geometric shape and dielectric constant of the target, that is, the actual model may have slight changes in geometric shape or dielectric constant compared to the ideal model. To solve the uncertainty problem of the target, the Monte Carlo method (MC) is the most used numerical calculation method. This method represents the possible uncertainty problems of the target by randomly taking values within a specified range, decomposes several uncertain problems into multiple deterministic problems, calculates and analyzes these deterministic problems, and finally uses the statistical value of multiple calculations as an approximation of the numerical solution of the uncertain problem. Therefore, the Monte Carlo method needs to calculate the result from scratch again for each randomly generated value, resulting in a long time required for its solution result.

[0003] To quickly and accurately solve the uncertainty problem, the perturbation method is proposed to solve the uncertainty problem. The accuracy of the perturbation method is slightly lower than that of the Monte Carlo method, but within an acceptable range. The perturbation method only needs to solve the median matrix and perturbation matrix of the original model once, and does not need to recalculate for each random number. Therefore, based on this, the perturbation method greatly reduces the calculation time compared to the Monte Carlo method. Currently, the existing perturbation methods are mainly used to solve the uncertainty problems of metal targets and pure dielectric targets.

[0004] For analyzing the uncertainties of the geometric shape and dielectric constant of non-uniform dielectric targets, there is a perturbation method of the volume integral equation (VIE). The volume integral equation can solve the electromagnetic scattering characteristics problem of non-uniform dielectrics. The volume integral equation is applicable to any general dielectric structure without demanding the particularity of the model. However, the volume integral equation uses a volume discretization form, has many unknowns, and has a slow calculation speed. Summary of the Invention

[0005] The purpose of the present invention is to provide a fast analysis method for multi-dimensional uncertainty problems of non-uniform dielectric rotationally symmetric bodies based on VIE-BOR.

[0006] The technical solution for achieving the purpose of the present invention is as follows: In the first aspect, the present invention provides a fast analysis method for multi-dimensional uncertainty problems of non-uniform dielectric rotationally symmetric bodies, and the steps are as follows:

[0007] Step 1. Control the shape of the axisymmetric body by NURBS: First, use the NURBS method and rectangular basis functions to model the mother plane of the axisymmetric body. Rotate the mother plane 360 degrees around the axis of rotation to obtain the target model. Based on this, the geometric shape of the axisymmetric body can be controlled by multiple control points, and the shape random variable α is introduced into the VIE-BOR matrix equation, and any point in the mother plane of the axisymmetric body is represented by the shape random variable α.

[0008] Step 2. Introduce the shape random variable α and the dielectric constant random variable ε r into the VIE-BOR integral equation: The coordinates of any point in the mother plane of the axisymmetric body are represented by the shape random variable α, and the dielectric constant of the medium in the axisymmetric body is represented by the dielectric constant random variable ε r . Combine the VIE-BOR integral equation with the random variables α and ε r . Obtain the current after the slight change in the shape or dielectric constant of the axisymmetric body through the VIE-BOR integral equation with random variables and the Taylor formula.

[0009] Step 3. Obtain the current and RCS after the slight change and conduct statistical analysis: Randomly generate the change amounts of the shape and dielectric constant of the axisymmetric body within a fixed range, and then combine the perturbation impedance matrix and the perturbation right vector to obtain the perturbation current corresponding to this change amount. Then, use the median current and the perturbation current to obtain the current after the slight change of the model and solve for the RCS response. Generate multiple groups of random change amounts and obtain the RCS corresponding to the change amounts. Conduct statistical analysis on all RCS responses to obtain their mean and variance, and compare the results with the Monte Carlo method.

[0010] In a second aspect, the present application also provides a computer device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the method described in the first aspect above is implemented.

[0011] In a third aspect, the present application also provides a computer-readable storage medium, on which a computer program is stored. When the program is executed by a processor, the method described in the first aspect above is implemented.

[0012] In a fourth aspect, the present application also provides a computer program product, including a computer program, characterized in that when the computer program is executed by a processor, the method described in the first aspect above is implemented.

[0013] Compared with the prior art, the remarkable advantages of the present invention are as follows: (1) Based on VIE-BOR to calculate the electromagnetic scattering characteristics of a rotational symmetric body, it has a faster calculation speed and less memory usage compared with the VIE method; (2) Based on the principle of the perturbation method, by combining the shape uncertainty and the permittivity uncertainty with the VIE-BOR integral equation, only the median matrix and the perturbation matrix of the initial rotational symmetric body need to be calculated, instead of recalculating for each random number, which greatly reduces the calculation time compared with the Monte Carlo method; (3) The VIE-BOR integral equation can handle the rotational symmetric body of non-uniform media. By combining the shape uncertainty and the permittivity uncertainty with the VIE-BOR integral equation, the multi-dimensional uncertainty problems of the geometric shape and the permittivity of non-uniform media can be handled simultaneously. Description of the Drawings

[0014] Figure 1 It is a schematic diagram of rectangular basis functions.

[0015] Figure 2 It is a schematic diagram of hybrid basis functions.

[0016] Figure 3 It is a schematic diagram of triangular basis functions.

[0017] Figure 4 It is a model diagram of a cylinder.

[0018] Figure 5 It is a schematic diagram of the NURBS surface of a cylinder.

[0019] Figure 6 It is a comparison diagram of the mean and variance of the RCS of a dielectric cylinder model.

[0020] Figure 7 It is a model diagram of a frustum of a cone.

[0021] Figure 8 It is a schematic diagram of the NURBS surface of a frustum of a cone.

[0022] Figure 9 It is a comparison diagram of the mean and variance of the RCS of a dielectric frustum of a cone model.

[0023] Tables 1-2 are the comparison of the memory and calculation time between the present method and the Monte Carlo method. Detailed Embodiments

[0024] For this special type of model of rotational symmetric bodies, in order to solve the problem of slow calculation speed with a large number of unknowns in the volume integral equation perturbation method, the volume integral equation for rotational symmetric bodies (VIE-BOR) is proposed to analyze the uncertainty problems of rotational symmetric bodies. The present invention analyzes the electromagnetic characteristics of non-uniform media rotational symmetric bodies with multi-dimensional uncertainties based on VIE-BOR.

[0025] The present invention proposes a fast analysis method for multi-dimensional uncertainty problems of a rotationally symmetric body with non-uniform media, and the steps are as follows: First, the mother plane of the rotationally symmetric body is modeled using non-uniform rational B-spline technology, reducing the previous three-dimensional volume dissection of the non-uniform media target to a two-dimensional surface dissection of the mother plane, and making the shape of the model controlled by control points. Subsequently, a VIE-BOR matrix equation with shape and permittivity random variables is obtained through the perturbation method. Finally, different variation amounts are obtained through random sampling, and the radar cross-section after the micro-variation of the shape and permittivity of the rotationally symmetric body, as well as the statistical mean and variance of the radar cross-section, are obtained.

[0026] Step 1. Controlling the shape of the rotationally symmetric body by NURBS: First, use the non-uniform rational B-spline (NURBS) method and rectangular basis functions to model the mother plane of the rotationally symmetric body (BOR). Rotate the mother plane 360 degrees around the rotation axis to obtain the target model. Based on this, the geometric shape of the rotationally symmetric body can be controlled by multiple control points, and the shape random variable α is introduced into the VIE-BOR matrix equation. Any point in the mother plane of the rotationally symmetric body can be represented by the shape random variable α, as follows:

[0027] First, use NURBS technology to model the mother plane of the rotationally symmetric body. The geometric shape of the mother plane of the rotationally symmetric body can be controlled by multiple control points. Then, the coordinate expression of any point on the NURBS surface is written as:

[0028]

[0029] where U and V respectively represent the number of control points in the u and v directions; p and q are the corresponding orders; P ij =[P ijx ,P ijy ,P ijz respectively represent the coordinates of the control points in the x, y, and z directions; w ij is its corresponding weight; N i,p (u) is the p-order normalized B-spline basis function obtained from the knot vector U = [u0, u1,..., u n+k+1 ; N j,q (v) is the q-order normalized B-spline basis function obtained from the knot vector V = [v0, v1,..., v n+k+1 ; the shape of the mother plane of the rotationally symmetric body is represented by the NURBS surface, and the shape random variable α is the coordinate of the control point.

[0030] Step 2. Incorporating the shape random variable α and the permittivity random variable ε rIntroduced into the VIE-BOR integral equation: The coordinates of any point in the mother plane of the axisymmetric body can be represented by the shape random variable α, and the dielectric constant of the medium in the axisymmetric body can be represented by the random variable ε r is represented, and the VIE-BOR integral equation is combined with the random variables α and ε r ; Through the VIE-BOR integral equation with random variables and the Taylor formula, the current after the slight change of the shape or dielectric constant of the axisymmetric body can be obtained, as follows:

[0031] First, introduce the types of basis functions used in VIE-BOR; In VIE-BOR, the mother plane of the axisymmetric body is meshed. The meshing of VIE-BOR is divided into three basis functions: z-type rectangular basis function, mixed basis function, and triangular basis function. The schematic diagrams of the three basis functions are as Figures 1 to 3 shown.

[0032] A. When the mode m is not equal to 0, the expression of the electric charge density D is as follows:

[0033]

[0034] Among them, M represents the number of modes, g mi (ρ, z), h mi (ρ, z), and f mi (ρ, z) represent the expressions of the z-type rectangular basis function, mixed basis function, and triangular basis function respectively, N z , N h , and N t represent the numbers of the z-type rectangular basis function, mixed basis function, and triangular basis function respectively, and represent the expansion coefficients of the three basis functions respectively. The expressions of the basis functions in each direction are as follows:

[0035] 4) z-type rectangular basis function

[0036]

[0037] Among them, are the unit vectors in the z direction and φ direction respectively, as Figure 1 shown, represents the length of the upper rectangle of the rectangular basis function, that is, is the length of the lower rectangle of the rectangular basis function, that is, m represents the current mode.

[0038] 5) Mixed basis function

[0039]

[0040] Among them, is the unit vector in the ρ direction, as Figure 2 shown, is the vector pointing to the vertex of the triangle, is the left rectangular area, is the right triangular area, represents the length in the ρ direction in the left rectangle, that is, represents the area of the triangle.

[0041] 6) Triangular basis function

[0042]

[0043] Among them, as Figure 3 shown, and are the vectors in the triangle, and represent the upper triangle and the lower triangle respectively, and represent the areas of the upper triangle and the lower triangle respectively. Due to the existence of 1 / ρ in the triangular basis function, the current near the z-axis is inaccurate. In order to make the current near the z-axis more accurate, the above-mentioned rectangular basis function is introduced.

[0044] B. When the mode m is equal to 0, the expression of the charge density D is as follows:

[0045]

[0046] Among them, φ 0i (ρ, z) represents the circumferential basis function composed of each rectangular or triangular unit, represent the number of circumferential basis functions in the zero mode respectively, represent the expansion coefficients of the circumferential basis functions in the zero mode respectively. The expressions of the basis functions in each zero mode are as follows:

[0047] 5) z-type rectangular basis function

[0048]

[0049] 6) Mixed basis function

[0050]

[0051] 7) Triangular basis function

[0052]

[0053] 8) Circumferential basis function

[0054]

[0055] According to the above basis functions, the mother plane of the rotationally symmetric body can be divided into a combination of several rectangles and triangles. Subsequently, the VIE-BOR integral equation of the rotationally symmetric body with an uncertain shape and an uncertain dielectric constant is analyzed, which is divided into two cases where the mode m is equal to 0 and not equal to 0.

[0056] C. When m is not equal to 0

[0057] In this case, the shape random variable α and the dielectric constant random variable ε r are introduced into the VIE-BOR matrix equation to obtain the following equation:

[0058]

[0059] where t represents the triangular basis function, h represents the mixed basis function, and z represents the rectangular basis function. The impedance matrix consists of nine parts, which are the current coefficients of the three basis functions, and is the right-hand vector, and the right-hand vector is independent of the dielectric constant.

[0060] D. When m is equal to 0

[0061] The following VIE-BOR matrix equation with the shape random variable can be obtained, where represents the circumferential basis function:

[0062]

[0063] The impedance matrix Z with the shape random variable α and the dielectric constant random variable ε r can be expressed by the following formula:

[0064]

[0065] where W i is the test basis function, p j (r') is the basis function expansion expression of the charge density D(r'). ω is the angular frequency, μ0 is the magnetic permeability of the ideal conductor, ε0 is the vacuum permittivity, ε(r′) is the dielectric constant of the medium, V is the integration region, Ω is the outer surface of the integration region, n is the unit vector perpendicular to the surface, G(r,r′) is the free space Green's function, k j (r′), are quantities related to the dielectric constant, and their expressions are as follows:

[0066]

[0067] The following is the derivation of each impedance matrix in the formula. For Substitute the corresponding triangular basis function and the expansion expression of the basis function of D(r') to obtain:

[0068]

[0069] Through the integral formula converted from the rectangular coordinate system to the cylindrical coordinate system The above formula can be transformed into:

[0070]

[0071] A. When the mode m is not equal to 0, because Therefore, the following formula can be obtained:

[0072]

[0073] Where:

[0074]

[0075]

[0076] B. When the mode m is equal to 0, the following impedance matrix can be obtained:

[0077]

[0078] From the expression of G m It can be seen that G -1 = G1, so:

[0079]

[0080] Similarly, according to the above steps, other impedance matrices with shape random variables can be obtained, just replace the corresponding basis functions.

[0081] Similar to the impedance matrix, substitute the shape random variable α into the right-hand vector to obtain the right-hand vector with the shape random variable, and the following formula is obtained:

[0082] A. In the mode where m is not equal to 0

[0083]

[0084] Where m represents the Fourier mode number, W mi represents the test basis function, t, h, z represent three basis functions, and E θ represents the incident plane wave of θ polarization, and the expression is as follows:

[0085]

[0086] The right - hand vector with shape random variables in the non - zero mode Expanding it gives:

[0087]

[0088] In the above formula, J m is the Bessel function. The Bessel function with shape random variables is as follows:

[0089]

[0090] B. When the mode is equal to 0, the right - hand vector is as follows:

[0091]

[0092] For its expansion expression is as follows:

[0093]

[0094] Similar to the non - zero mode, substituting different basis functions W 0i and the incident plane - wave E θ can obtain the expansion form of the right - hand vector in the 0 - mode.

[0095] So far, the VIE - BOR equation with shape random variables and permittivity random variables has been obtained. Next, use this equation to solve the uncertainty problem. VIE - BOR can be uniformly written in the following form:

[0096] Z m (α, ε r )·D m (α, ε r )=V m (α)(70)

[0097] Using the first - order Taylor formula for the above formula and expanding it at α = α c and where α c and are the medians when determining the model, the following expansion form is obtained:

[0098]

[0099] Because the perturbation current is related to the shape random variables and permittivity random variables, using ΔD mα (α) and ΔD mε (ε r ) to represent the two kinds of perturbation currents, the above formula can be written as:

[0100]

[0101] Next, it is necessary to solve for ΔD mα (α) and ΔD mε (ε r ). By omitting the high-order terms in the above formula, the following can be obtained respectively:

[0102]

[0103]

[0104] Among them, Δα i is the change value of the geometric shape, Δα max is the maximum value of the shape random variable, and the value range of the shape change is [α c -Δα max , α c +Δα max . Δε r is the change value of the dielectric constant, Δε max is the maximum value of the change of the dielectric constant random variable, is the change range of the dielectric constant, where i = 1,...n, n represents the number of shape random variables, and r = 1,...l, l represents the number of dielectric constant random variables.

[0105] First, analyze the shape perturbation current ΔD mα (α). From the formula, it can be seen that it is necessary to find the derivatives of and two terms; the expression of the VIE-BOR integral equation at α = α c is as follows: when

[0106]

[0107] The simplification method of this equation is the same as before. The derivative formula of the impedance matrix with a non-zero mode is as follows:

[0108]

[0109] The derivative formula of the impedance matrix when the mode is equal to 0 is as follows:

[0110]

[0111] Just substitute the corresponding basis functions to obtain the derivative forms of other impedance matrices.

[0112] Next, find the derivative of the right vector When the mode m is not equal to 0:

[0113]

[0114] For Its expression form is as follows:

[0115]

[0116] The derivative of the right vector when the mode m is equal to 0 is as follows:

[0117]

[0118] For Its expression is:

[0119]

[0120] The expression forms of other right vectors are the same as Substitute the corresponding basis functions. According to the above steps, after obtaining the derivative of the shape impedance matrix and the derivative of the right vector under the mode m, the perturbed current ΔD mα (α) can be calculated. Finally, through the median current under each mode and the perturbed current ΔD mα (α), the RCS after the random change of the shape can be obtained.

[0121] Next, analyze the electromagnetic scattering characteristics when the permittivity is uncertain. It is necessary to solve the permittivity perturbed current ΔD mε (ε r ). Since the right vector is independent of the permittivity, only the expression of the perturbed impedance matrix of VIE - BOR at needs to be solved, which is as follows:

[0122]

[0123] The derivative formula of the impedance matrix when the mode is not 0 is as follows:

[0124]

[0125] The derivative formula of the impedance matrix when the mode is equal to 0 is as follows:

[0126]

[0127] Just substitute the corresponding basis functions to obtain the derivative forms of other impedance matrices.

[0128] Since the right vector is independent of the permittivity, according to the permittivity perturbed current ΔD mε (ε r ) can be solved.

[0129] According to the shape perturbation current ΔD mα (α) and the permittivity perturbation current ΔD mε (ε r ), the total current when the shape and permittivity change simultaneously can be obtained, and thus the RCS after the change can be solved.

[0130] Step 3: Obtain the slightly changed current and RCS and conduct statistical analysis. Randomly generate the change amounts of the shape and permittivity of the rotationally symmetric body within a fixed range. Then, combined with the perturbation impedance matrix and the perturbation right-hand vector, the perturbation current corresponding to this change amount can be obtained. Next, use the median current and the perturbation current to obtain the slightly changed current of the model and solve for the RCS response. Generate multiple groups of random change amounts and obtain the corresponding RCSs. Conduct statistical analysis on all RCS responses to obtain their mean and variance, and compare the results with the Monte Carlo method.

[0131] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0132] Embodiment 1

[0133] This embodiment analyzes a dielectric cylinder with an uncertain shape and an uncertain permittivity. This embodiment is implemented on a computing platform with an Inter(R) Core(TM) i7-7700K CPU @ 3.6GHz and 8GB of memory. The size parameters of the dielectric cylinder model are as Figure 4 shown, which is composed of three dielectrics with permittivities of ε1 = (4,0), ε2 = (3,-0.05), and ε3 = (1.5,-0.6) respectively. By using the NURBS surface modeling method for modeling, the parent plane of the cylinder model is as Figure 5 shown, which is composed of 3 NURBS surfaces and controlled by 8 control points. The x coordinate change range of points 4 and 8 of the cylinder model is [-0.05m, 0.05m], and the permittivity change range of dielectric #2 is [-0.1, 0.1]. The incident frequency of the plane wave is 300 MHz, the direction is θ = 0°, the Fourier mode number is 1, the observation angle is θ = 0 to 180°, and the mean and variance of the RCS are obtained by sampling 1000 times. Compare the Monte Carlo method with this method, and the mean and variance results of the RCS are as Figure 6 shown. It can be seen that the results of the perturbation method are basically the same as those of the Monte Carlo method, and this method can be well used to solve the uncertainty problem of non-uniform dielectrics. The time and memory comparison of the two methods are shown in Table 1.

[0134] Table 1

[0135]

[0136] Example 2

[0137] In this example, an analysis was performed on a dielectric frustum model with an uncertain shape and an uncertain dielectric constant. The parameters of the dielectric frustum model are as follows Figure 7 shown, consisting of frustums of two dielectrics, with the dielectric constants of the dielectrics being ε1 = (4,0) and ε2 = (3,0). The distribution of control points on its mother plane is as follows Figure 8 shown. The range of the z - coordinate of points 1 and 2 of the frustum model is [-0.05 m, 0.05 m]. At the same time, the range of the dielectric constant of dielectric #1 is [-0.1, 0.1], the incident frequency of the plane wave is 300 MHz, and the direction is θ = 10°, the number of Fourier modes is 4, the observation angle is θ = 0 to 180°, and the mean and variance of the RCS are obtained by sampling 1000 times. Comparing the Monte Carlo method with this method, the results of the mean and variance of the RCS are as follows Figure 9 shown. Table 2 shows the comparison of memory and time between the two methods.

[0138] Table 2

[0139]

[0140] It can be seen from Table 1 and Table 2 that the method of the present invention greatly reduces the calculation time on the basis of increasing the memory compared with the Monte Carlo method, and the increased memory is also within an acceptable range, reflecting the high efficiency of this method.

Claims

1. A rapid analysis method for multi-dimensional uncertainty problems of a rotationally symmetric body with inhomogeneous medium, characterized in that, The steps are as follows: Step 1. Controlling the shape of the rotationally symmetric body by NURBS: First, use the NURBS method and rectangular basis functions to model the mother plane of the rotationally symmetric body. Rotate the mother plane 360 degrees around the rotation axis to obtain the target model. Based on this, the geometric shape of the rotationally symmetric body can be controlled by multiple control points, and the shape random variable α is introduced into the VIE - BOR matrix equation. Any point in the mother plane of the rotationally symmetric body is represented by the shape random variable α; Step 2: Introduce the shape random variable α and the permittivity random variable ε r into the VIE-BOR integral equation: The coordinates of any point in the mother plane of the axisymmetric body are represented by the shape random variable α, and the permittivity of the medium in the axisymmetric body is represented by the permittivity random variable ε r ; Combine the VIE-BOR integral equation with the random variables α and ε r ; Obtain the current after the slight change in the shape or permittivity of the axisymmetric body through the VIE-BOR integral equation with random variables and the Taylor formula Obtain the VIE - BOR equation with shape random variables and permittivity random variables, and then use this equation to solve the uncertainty problem. VIE - BOR can be uniformly written in the following form: Z m (α, ε r )·D m (α, ε r ) = V m (α) (28) Using the first-order Taylor formula for the above equation, expand it at α = α c and where α c and are the medians when determining the model, and the following expanded form is obtained: Since the perturbation current is related to the shape random variable and the permittivity random variable, ΔD mα (α) and ΔD mε (ε r ) are used to represent the two perturbation currents, so the above equation can be written as: Step 3. Obtain the slightly changed current and RCS and conduct statistical analysis; Randomly generate the change amounts of the shape and permittivity of the rotationally symmetric body within a fixed range, and then combine the perturbation impedance matrix and the perturbation right - hand vector to obtain the perturbation current corresponding to this change amount. Then, use the median current and the perturbation current to obtain the slightly changed current of the model and solve for the RCS response; Generate multiple groups of random change amounts and obtain the RCS corresponding to the change amounts. Conduct statistical analysis on all RCS responses to obtain their mean and variance, and compare the results with the Monte Carlo method.

2. The rapid analysis method for multi-dimensional uncertainty problems of a rotationally symmetric body in a non-uniform medium according to claim 1, characterized in that The control of the shape of the rotationally symmetric body by NURBS described in Step 1 is specifically as follows: Use NURBS technology to model the mother plane of the rotationally symmetric body. The mother plane of the rotationally symmetric body can control its geometric shape through multiple control points. Then, the coordinate expression of any point on the NURBS surface is written as: where U and V respectively represent the number of control points in the u and v directions; p and q are the corresponding orders; P ij = [P ijx , P ijy , P ijz respectively represent the coordinates of the control points in the x, y, and z directions; w ij is its corresponding weight; N i,p (u) is the p-th order normalized B-spline basis function obtained from the knot vector U = [u0, u1,..., u n+k+1 ; N j,q (v) is the q-th order normalized B-spline basis function obtained from the knot vector V = [v0, v1,..., v n+k+1 ; the shape of the mother plane of the rotationally symmetric body is represented by a NURBS surface, and the shape random variable α is the coordinate of the control point.

3. The rapid analysis method for multi-dimensional uncertainty problems of a rotationally symmetric body in a non-uniform medium according to claim 2, characterized in that The shape random variable α and the permittivity random variable ε described in Step 2 r are introduced into the VIE-BOR integral equation as follows: In VIE - BOR, the mother plane of the rotationally symmetric body is meshed. The meshing of VIE - BOR is divided into three basis functions: z - type rectangular basis function, mixed basis function, and triangular basis function; A. When the mode m is not equal to 0, the expression of the electric charge density D is as follows: where M represents the number of modes, g mi (ρ, z), h mi (ρ, z) and f mi (ρ, z) represent the expressions of the z-type rectangular basis function, the mixed basis function, and the triangular basis function respectively, N z , N h and N t represent the numbers of the z-type rectangular basis function, the mixed basis function, and the triangular basis function respectively, and represent the expansion coefficients of the three basis functions respectively; the expressions of the basis functions in each direction are as follows: 1) z - type rectangular basis function Among them, are the unit vectors in the z - direction and φ - direction respectively, represent the maximum and minimum values of the rectangular basis function in the ρ - direction, represents the length of the upper rectangle of the rectangular basis function, is the maximum value of the rectangular basis function in the z - direction, is the length of the lower rectangle of the rectangular basis function, is the minimum value of the rectangular basis function in the z - direction, and m represents the current mode; 2) Mixed basis function Among them, is the unit vector in the ρ direction, is the vector pointing to the triangle vertex, is the rectangular basis function region, is the triangular basis function region, is expressed as the length in the ρ direction in the left rectangle, is the maximum value in the ρ direction of the rectangular basis function, A i - is expressed as the area of the triangle; 3) Triangular basis function Among them, and are vectors in a triangle, and represent the upper triangle and the lower triangle respectively, and represent the areas of the upper triangle and the lower triangle respectively, and ρ is the value of the basis function in the ρ direction; due to the existence of 1 / ρ in the triangular basis function, the current near the z-axis is inaccurate. In order to make the current near the z-axis more accurate, the above rectangular basis function is introduced; B. When the mode m is equal to 0, the expression of the electric charge density D is as follows: where φ 0i (ρ, z) represents the circumferential basis function composed of each rectangular or triangular element, respectively represent the number of circumferential basis functions in the zero mode, respectively represent the expansion coefficients of the circumferential basis functions in the zero mode; the expressions of the basis functions in each zero mode are as follows: 1) z - type rectangular basis function 2) Mixed basis function 3) Triangular basis function 4) Circumferential basis function According to the above basis functions, mesh the mother plane of the rotationally symmetric body into a combination of several rectangles and triangles; Subsequently, analyze the VIE - BOR integral equation of the rotationally symmetric body with uncertain shape and uncertain permittivity, which is divided into two cases where the mode m is equal to 0 and not equal to 0; A. When m is not equal to 0 In this case, the shape random variable α and the dielectric constant random variable ε r are introduced into the VIE-BOR matrix equation to obtain the following equation: where \(t\) represents the triangular basis function, \(h\) represents the hybrid basis function, and \(z\) represents the rectangular basis function; the impedance matrix consists of nine parts, is the current coefficient of the three basis functions, is the right-hand side vector, and the right-hand side vector is independent of the dielectric constant; B. When m is equal to 0 The following VIE-BOR matrix equation with shape random variables can be obtained, where represents the circumferential basis function: With a shape random variable α and a permittivity random variable ε r The impedance matrix Z can be expressed by the following formula: Among which W i is the test basis function, p j (r') is the basis function expansion expression of the electric charge density D(r'); ω is the angular frequency, μ0 is the magnetic permeability of the perfect conductor, ε0 is the vacuum permittivity, ε(r′) is the permittivity of the medium, V is the integration region, Ω is the outer surface of the integration region, n is the unit vector perpendicular to the surface, G(r, r′) is the free space Green's function, k j (r′), are quantities related to the permittivity, and their expressions are as follows: The following is the derivation of each impedance matrix in the formula. For Substitute the corresponding expansion expressions of the triangular basis function and the basis function of D(r') to obtain: Integral formula obtained by transforming from rectangular coordinates to cylindrical coordinates Transform the above equation into: A. When the mode m is not equal to 0, because Therefore, the following equation can be obtained: Where: B. When the mode m is equal to 0, the following impedance matrix is obtained: From G m As can be seen from the expression of -1 G = G1, so: Similarly, according to the above steps, obtain other impedance matrices with shape random variables, and only need to replace the corresponding basis functions; Similar to the impedance matrix, substitute the shape random variable α into the right - hand vector to obtain the right - hand vector with shape random variables, and obtain the following formula: A. In the mode where m is not equal to 0 where m represents the number of Fourier modes, and W mi represents the test basis function, and t, h, and z represent three basis functions, and E θ represents the incident plane wave of θ polarization, and the expression is as follows: The right vector with shape random variables in non-zero mode Expanding gives: In the above formula, J m is a Bessel function. The Bessel function with a shape random variable is as follows: B. When the mode is equal to 0, the right - hand vector is as follows: For Its expansion expression is shown as follows: Similar to the non-zero mode, substitute different basis functions W 0i and the incident plane wave E θ to obtain the expansion form of the right vector in the 0 mode; Next, it is necessary to solve for ΔD separately mα (α) and ΔD mε (ε r ). By omitting the higher-order terms in the above formula, we obtain respectively: where, Δα i is the change value of the geometric shape, Δα max is the maximum value of the shape random variable, and the value range of the shape change is [α c -Δα max , α c +Δα max , Δε r is the change value of the dielectric constant, Δε max is the maximum value of the change of the dielectric constant random variable, is the change range of the dielectric constant, where i = 1, …, n, n represents the number of shape random variables, and r = 1, …, l, l represents the number of dielectric constant random variables; First, analyze the shape perturbation current ΔD mα (α). As can be seen from the formula, it is necessary to find out and the two derivatives; the expression of the VIE-BOR integral equation at α = α c is as follows when : The simplification method of this equation is the same as before. The derivative formula of the impedance matrix when the mode is not 0 is as follows: The derivative formula of the impedance matrix when the mode is equal to 0 is as follows: Just substitute the corresponding basis functions to obtain the derivative forms of other impedance matrices; Next, find the derivative of the right vector When the mode m is not equal to 0: For Its expression form is as follows: The derivative of the right vector when the mode m is equal to 0 is as follows: For Its expression is: The expression forms of other right vectors are the same as , and the corresponding basis functions can be substituted; according to the above steps, after obtaining the derivative of the shape impedance matrix and the derivative of the right vector in mode m, the perturbed current ΔD mα (α) in mode m is calculated. Finally, the RCS after the random change of the shape can be obtained through the median current in each mode and the perturbed current ΔD mα (α). Next, analyze the electromagnetic scattering characteristics when the permittivity is uncertain. It is necessary to solve the permittivity perturbation current ΔD mε (ε r ); Since the right vector is independent of the permittivity, only VIE-BOR at The expression of the perturbation impedance matrix is as follows: The derivative formula of the impedance matrix when the mode is not 0 is as follows: The derivative formula of the impedance matrix when the mode is equal to 0 is as follows: Substitute the corresponding basis functions to obtain the derivative forms of other impedance matrices; Since the right vector is independent of the dielectric constant, according to the perturbation current of the dielectric constant ΔD can be solved mε (ε r ); According to the shape perturbation current ΔD mα (α) and the permittivity perturbation current ΔD mε (ε r ), the total current when the shape and permittivity change simultaneously is obtained, and then the RCS after the change can be solved.

4. A computer device, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps of the method described in any one of claims 1-3.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the method described in any one of claims 1-3.

6. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the method described in any one of claims 1-3.

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