Estimation method of electromagnetic scattering characteristics of inhomogeneous medium targets based on PMCHWT integral equation

Abstract

This paper discloses a method for estimating the electromagnetic scattering characteristics of inhomogeneous dielectric targets based on the PMCHWT integral equation. The method includes: modeling using FEKO software; introducing random variables related to the dielectric constant into the PMCHWT integral equation; expanding the impedance matrix, right-hand side vector, and current at the median of the random variables according to a Taylor series; deriving the current moment vector expression through moment coefficient matching; applying the pseudospectral method to the AWE to calculate the current moment vector of each order; and calculating the current and radar cross section of targets in different dielectrics through rational Padé approximation, combining Taylor series and Padé polynomials. Finally, the mean and variance of all RCS responses are obtained through multiple sampling. Compared with the Taylor method, this method has a larger convergence radius and can significantly improve efficiency compared with the Monte Carlo method.

Description

Technical Field

[0001] The invention belongs to the technical field of numerical calculation of target electromagnetic scattering characteristics, in particular to a method for estimating electromagnetic scattering characteristics of a non-uniform medium target by using a PMCHWT integral equation. Background Art

[0002] In electromagnetic analysis, the dielectric constant of a target ideally remains constant. Therefore, the electromagnetic properties of a target with a defined dielectric constant are typically analyzed. However, in real life, the dielectric constant of a target can vary due to factors such as manufacturing processes, the external environment, and human factors, leading to uncertainty in the dielectric constant. To account for this uncertainty, the PMCHWT integral equation is introduced, which can address the electromagnetic scattering characteristics of multi-medium targets.

[0003] The Monte Carlo (MC) method can be used to solve these uncertain electromagnetic scattering problems. Specifically, the Monte Carlo method generates a random number each time to represent the uncertainty, transforming the uncertainty into a deterministic problem. The mean and variance of the RCS response are then calculated based on multiple samplings. For multiple sampling problems, this method requires a new solution each time. While this method offers high accuracy, it also takes a long time to compute.

[0004] In order to solve the shortcomings of the above-mentioned Monte Carlo method, the perturbation method can sacrifice a little accuracy within an acceptable range to improve the calculation speed of uncertainty problems. The traditional perturbation method based on Taylor series expansion has a small convergence radius, a very complex derivation formula, and the memory required for this method is also very large. Therefore, a perturbation method based on AWE technology is proposed. This method introduces the rational Padé approximation, and on this basis, the impedance matrix and the right-hand side vector are differentiated by the pseudo-spectral method. Compared with the traditional Taylor method, this method has a larger convergence radius and a simple and clear derivation method, which greatly reduces the workload of derivation; compared with the Monte Carlo method, this method greatly reduces the time required for multiple sampling. Based on the PMCHWT integral equation, the asymptotic waveform estimation technology (AWE) and the pseudo-spectral method, the present invention can solve the electromagnetic scattering characteristics of medium targets with uncertainty and inhomogeneity. Summary of the Invention

[0005] The object of the present invention is to provide a method for estimating the electromagnetic scattering characteristics of a non-uniform medium target based on the PMCHWT integral equation.

[0006] The technical solution to achieve the purpose of the present invention is as follows: First, the present invention provides a method for estimating the electromagnetic scattering characteristics of a target in a non-uniform medium based on the PMCHWT integral equation, the steps of which are as follows:

[0007] Step 1: Modeling is done using FEKO software. Various parameters are set in FEKO and the model is divided. The set model information is exported as a NAS file. The RWG basis function used in the PMCHWT integral equation is combined with it. The dielectric constant of the medium in any target is expressed as ε. r express;

[0008] Step 2: Dielectric constant random variable ε r Introduced into the PMCHWT equation: The impedance matrix, right-hand side vector, and current are expanded using Taylor series at the median of the random variable. The expression for the current moment vector is derived by moment coefficient matching. The pseudospectral method is then introduced into the AWE. The D matrix is ​​determined by the range of the dielectric constant. The derivative of the product of the impedance matrix and the moment vector is obtained from the D matrix. Through the rational Padé approximation, the equation of the Taylor series and the Padé polynomial is established to calculate the current and the corresponding RCS.

[0009] Step 3: During multiple samplings, the change in the dielectric constant is randomly generated within a certain range. The corresponding current and RCS are obtained through the random change generated by each sampling. After obtaining each RCS, all RCS responses are statistically analyzed to obtain the electromagnetic scattering characteristics of the target with uncertain inhomogeneous media, and the statistical mean and variance of the RCS response are calculated.

[0010] In a second aspect, the present invention provides a computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the method described in the first aspect when executing the program.

[0011] In a third aspect, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the program, when executed by a processor, implements the steps of the method described in the first aspect.

[0012] In a fourth aspect, the present invention provides a computer program product, comprising a computer program, characterized in that when the computer program is executed by a processor, the steps of the method described in the first aspect are implemented.

[0013] Compared with the existing technology, the present invention has the following significant advantages: (1) Compared with the Monte Carlo method, this invention only needs to fill the impedance matrix once, which greatly improves the calculation efficiency; (2) This invention uses the pseudo-spectral method, which saves a lot of memory consumption compared with the traditional perturbation method and has a larger convergence radius. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a rectangular model diagram.

[0015] Figure 2 This is a comparison chart of the RCS statistical results (mean and variance) of the rectangular parallelepiped model with uncertain dielectric constant. DETAILED DESCRIPTION

[0016] This paper proposes a method for estimating the electromagnetic scattering characteristics of inhomogeneous dielectric targets based on the PMCHWT integral equation. The steps are as follows: First, a model is created using FEKO software, and the dielectric constant of the medium can be controlled by a random variable of the dielectric constant. Unlike the application of AWE technology at frequency and angle, this paper applies AWE technology to the medium. Then, the random variable of the dielectric constant is introduced into the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation. The impedance matrix, right-hand side vector, and current are expanded using Taylor series at the median of the random variable. The expression for the current moment vector is derived using the moment coefficient matching method. Next, the pseudospectral method is applied to AWE to calculate the product of the impedance matrix derivative and the moment vector, thereby calculating the current moment vector of each order. Then, through the rational Padé approximation, the Taylor series and Padé polynomials are combined to calculate the current and radar cross section (RCS) of different dielectric targets. Finally, through multiple sampling, the mean and variance of all RCS responses are obtained. Compared to the Taylor method, this method has a larger convergence radius and significantly improves efficiency compared to the Monte Carlo method. This method can also take into account the impact of changes in the target dielectric constant.

[0017] The present invention is further described in detail below with reference to the accompanying drawings.

[0018] Step 1: Modeling is done using FEKO software. Various parameters are set in FEKO and the model is divided. The set model information is exported as a NAS file so that the algorithm can read the model information. The RWG basis function used in the PMCHWT integral equation is combined with it. The dielectric constant of the medium in any target can be expressed as ε r express;

[0019] Step 2: Dielectric constant random variable ε rIntroduced into the PMCHWT equation: The impedance matrix, right-hand side vector, and current are expanded according to the Taylor series at the median of the random variable. The expression of the current moment vector is derived by the moment coefficient matching method. Then, the pseudo-spectral method is introduced into the AWE. The D matrix is ​​determined by the range of the dielectric constant. The derivative of the product of the impedance matrix and the moment vector can be obtained through the D matrix. Since the right-hand side vector is a quantity independent of the dielectric constant, the derivative of the right-hand side vector is 0. Then, through the rational Padé approximation, the equation of the Taylor series and the Padé polynomial is established to calculate the current and the corresponding RCS.

[0020] Step 3: During multiple sampling, the change in the dielectric constant is randomly generated within a certain range. The random change generated by each sampling can be used to obtain the corresponding current and RCS. After obtaining each RCS, all RCS responses are statistically analyzed to obtain the electromagnetic scattering characteristics of the target with uncertain inhomogeneous dielectric. The statistical mean and variance of the RCS response are calculated and compared with the statistical mean and variance of the Monte Carlo method to verify the accuracy of this method.

[0021] Furthermore, the model is modeled using FEKO software as described in step 1. Various parameters are set in FEKO and the model is divided. The set model information is exported as a NAS file so that the algorithm can read the model information. The RWG basis function used in the PMCHWT integral equation is combined with it, and the dielectric constant of the medium in any target can be expressed as ε r Indicates that the terms related to the triangular medium in the PMCHWT equation are combined with the random variable of the dielectric constant Δε, as follows:

[0022] First, build a model in FEKO software, set the corresponding frequency, plane wave source, and far field parameters, and then split the set model; export the split model as a NAS file, and integrate the exported NAS file and MAP file as the input file; by giving the dielectric constant ε r A random variable Δε is introduced to change the target dielectric constant; where ε r =[ε r 1 ,ε r 2 …ε r t ], Δε=[Δε1, Δε2…Δε t ]; Finally, the PMCHWT integral equation is established based on the RWG basis function.

[0023] Furthermore, in step 2, the dielectric constant random variable ε rIntroduced into the PMCHWT equation: The impedance matrix, right-hand side vector, and current are expanded according to the Taylor series at the median of the random variable. The expression of the current moment vector is derived by the moment coefficient matching method. Then the pseudo-spectral method is introduced into AWE. The D matrix is ​​determined by the range of dielectric constant variation. The derivative of the impedance matrix and the matrix-vector product is obtained through the D matrix. Since the right-hand side vector is a quantity independent of the dielectric constant, the derivative of the right-hand side vector is 0. Then, through the rational Padé approximation, the equation of the Taylor series and the Padé polynomial is established to calculate the current and the corresponding RCS; the details are as follows:

[0024] First, we establish a random variable containing the dielectric constant ε r The PMCHWT equation:

[0025]

[0026] Among them, Z(ε r ), I(ε r ), V is expressed as an impedance matrix with random variables of dielectric parameters, current coefficient and right side vector (the right side vector does not contain quantities related to dielectric constant);

[0027] Formula (1) can be written as follows:

[0028] Z(ε r )·I(ε r )=V (2)

[0029] Then, the impedance matrix and current are expanded according to the Taylor series at the median of the dielectric constant random variable:

[0030]

[0031]

[0032] in They represent: the maximum value of i, the unknown quantity with the random variable of the medium parameter, the initial value of the random variable of the medium parameter, the i-th order derivative of the impedance matrix with the random variable, the n-th order moment vector, a i The order of b j The order of .

[0033] Substituting (3), (4) and the right vector into (2), we have:

[0034]

[0035] The moment coefficients of the left and right orders in the matching formula can be obtained by recursion:

[0036]

[0037] Approximate I(ε) using a reduced-order rational Padé function:

[0038]

[0039]

[0040] Combining the two equations, we can get:

[0041]

[0042] By matching the moment coefficients of each order in (9), we can get a i , b j The expression:

[0043]

[0044]

[0045] Next, we consider a non-uniform pseudo-spectral method that satisfies the Gauss-Labatto-Chebyshve (GLC) interpolation formula, considering the interval ε∈[ε0-Δε,ε0+Δε] and N+1 sampling points in this interval that satisfy the following relationship

[0046]

[0047] At the same time, through the Gauss-Labatto-Chebyshve (GLC) interpolation formula, the matrix Z(ε) can be approximated as

[0048]

[0049]

[0050] in is the Chebyshev polynomial T N The derivative of (x), the coefficients of the GLC interpolation formula are

[0051]

[0052]

[0053] Here the function T N (x) is a Chebyshev polynomial

[0054]

[0055] Here [N / 2] is an integer operation; so the i-th order derivative of the matrix Z(ε) can be written as

[0056]

[0057] The first-order derivative at the GLC point (37) chosen here can therefore be written as

[0058]

[0059] The elements of the matrix D are Therefore, the matrix D can be recorded as the pseudo-spectral derivative matrix; written in matrix form as

[0060] Z (1) m n-1 =D·Zm n-1 (20) The specific values ​​of the elements of matrix D are

[0061]

[0062] The parameters here

[0063]

[0064]

[0065] Similarly, higher-order partial derivatives at the GLC point can be written as

[0066] Z (i) m n-1 =D·Z (i-1) m n-1 =D i ·Zm n-1 (twenty four)

[0067] By using this derivation method, the numerical results of high-order derivatives are much better than those of tight difference technology. By using the above method, the impedance matrix and the right-hand side of the matrix equation can be derived without analytical partial differential derivation of the integral kernel of the integral equation, thereby realizing the asymptotic waveform estimation. i 、b j 、m n The total current can be obtained, and then the electromagnetic scattering characteristics of this model can be obtained based on the current.

[0068] Furthermore, in step 3, the change in the dielectric constant is randomly generated within a certain range during multiple samplings. The change is randomly generated within [-Δx, Δx], where Δx is the set change. The corresponding current and RCS can be obtained through the random change generated by each sampling. After obtaining each RCS, all RCS responses are statistically analyzed to obtain the electromagnetic scattering characteristics of the target with uncertain inhomogeneous medium. The details are as follows:

[0069] During the calculation of each current over multiple sampling cycles, a random variable is introduced to change the target's dielectric constant. This allows the radar cross-section (RCS) after the model change to be calculated using the surface current and the re-established model dielectric constant. The mean and variance of the RCS obtained from multiple sampling cycles reveal the electromagnetic scattering characteristics of targets in uncertain inhomogeneous media.

[0070] The Monte Carlo (MC) method requires filling multiple impedance matrices for multiple samplings, which greatly increases the time required. However, this method only needs to calculate the required coefficient vector once, which greatly saves calculation time and improves efficiency.

[0071] The technical solution of the present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0072] Example

[0073] This embodiment performs electromagnetic scattering calculations on a dielectric cube with an uncertain shape. This embodiment is implemented on a 12th Gen Intel(R) Core(TM) i7-12700H 2.30GHz, 16GB memory computing platform. The dielectric cuboid model is as follows: Figure 1 As shown, a cuboid with dimensions of 0.2m×0.2m×0.4m is divided into two parts. The dielectric constant of the first part is ε r1 =(2,0), the dielectric constant of the second part is ε r2 =(3,0). The real part of the dielectric constant of the two dielectrics of the cuboid varies in the range of [-0.5,0.5]. The plane wave incident frequency is 300MHZ and the direction is θ=0°. Where θ is the pitch angle, is the azimuth. The observation angle is θ=0~180°, Using the same incident wave setting, the Monte Carlo method with 1000 sampling times is compared with the method of the present invention. The statistical change results of RCS are as follows: Figure 2 As shown in Figure 1, the two curves agree well. The comparison of the computation time and memory usage of the two methods is shown in Table 1.

[0074] Table 1

[0075]

[0076] As can be seen from Table 1, the memory required by the method of the present invention is larger than that required by the Monte Carlo method, but the calculation time is much shorter than the Monte Carlo method with 1000 samples. This shows that the method of the present invention has the advantage of faster calculation speed than the Monte Carlo method.

Claims

1. A method for estimating electromagnetic scattering characteristics of inhomogeneous medium targets based on PMCHWT integral equation, characterized in that: Here are the steps: Step 1: Modeling is done using FEKO software. Various parameters are set in FEKO and the model is divided. The set model information is exported as a NAS file. The RWG basis function used in the PMCHWT integral equation is combined with it. The dielectric constant of the medium in any target is expressed as ε. r express; Step 2: Dielectric constant random variable ε r Introduced into the PMCHWT equation: The impedance matrix, right-hand side vector, and current are expanded according to the Taylor series at the median of the random variable. The expression of the current moment vector is derived by the moment coefficient matching method. Then, the pseudo-spectral method is introduced into AWE. The D matrix is ​​determined by the range of dielectric constant variation. The derivative of the product of the impedance matrix and the moment vector is obtained through the D matrix. Through the rational Padé approximation, the equation of the Taylor series and the Padé polynomial is established to calculate the current and the corresponding RCS. The details are as follows: First, we establish a random variable containing the dielectric constant ε r The PMCHWT equation: Among them, Z(ε r ), I(ε r ), V is represented as an impedance matrix with random variables of dielectric parameters, current coefficient and right-hand side vector; Equation (1) is written in the following form: Z(e r )·I(e r )=V (2) Then, the impedance matrix and current are expanded according to the Taylor series at the median of the dielectric constant random variable: Where N represents the maximum value of i, ε r represents the unknown quantity with medium parameter random variable, represents the initial value of the medium parameter random variable, Z (i) (ε r ) represents the i-th order derivative of the impedance matrix with random variables, m n represents the n-order moment vector, L represents a i The order of M represents b j The order of Substituting (3), (4) and the right vector into (2), we have: The moment coefficients of the left and right orders in the matching formula can be obtained by recursion: Approximate I(ε) using reduced-order rational Padé functions: Combining the two equations, we can get: Match the moment coefficients of each order in (9), and we get a i , b j The expression: N+1 sampling points in the interval ε∈[ε0-Δε,ε0+Δε] that satisfy the following relationship Through the GLC interpolation formula, the matrix Z(ε) can be approximated as in is the Chebyshev polynomial T N The derivative of (x), the coefficients of the GLC interpolation formula are T N (x) is a Chebyshev polynomial: [N / 2] is an integer operation; the i-th order derivative of the matrix Z(ε) can be written as The first-order derivative of the selected GLC point is written as The elements of the matrix D are Let matrix D be the pseudo-spectral derivative matrix; written in matrix form as The specific values ​​of the elements of matrix D are in Higher-order partial derivatives on the GLC point are written as According to a i 、b j 、m n The total current is obtained, and then the electromagnetic scattering characteristics of the model are obtained based on the current; Step 3: During multiple samplings, the change in the dielectric constant is randomly generated within a certain range. The corresponding current and RCS are obtained through the random change generated by each sampling. After obtaining each RCS, all RCS responses are statistically analyzed to obtain the electromagnetic scattering characteristics of the target with uncertain inhomogeneous media, and the statistical mean and variance of the RCS response are calculated.

2. The method for estimating electromagnetic scattering characteristics of a target in a non-uniform medium based on the PMCHWT integral equation according to claim 1 is characterized in that: As described in step 1, the model is modeled using FEKO software. Various parameters are set in FEKO and the model is divided. The set model information is exported as a NAS file. The RWG basis function used in the PMCHWT integral equation is combined with it, and the dielectric constant of the medium in any target is expressed as ε. r Indicates that the terms related to the triangular medium in the PMCHWT equation are combined with the random variable of the dielectric constant Δε, as follows: First, build a model in FEKO software, set the corresponding frequency, plane wave source, and far field parameters, and then split the set model; export the split model as a NAS file, and integrate the exported NAS file and MAP file as the input file; by giving the dielectric constant ε r A random variable Δε is introduced to change the target dielectric constant; where ε r =[ε r 1 ,ε r 2 …ε r t ], Δε=[Δε1, Δε2…Δε t ]; Finally, the PMCHWT integral equation is established based on the RWG basis function.

3. The method for estimating electromagnetic scattering characteristics of a target in a non-uniform medium based on the PMCHWT integral equation according to claim 1 is characterized in that: Step 3: During multiple sampling, the change in the dielectric constant is randomly generated within [-Δx, Δx], where Δx is the set change. When calculating each current during multiple sampling, a random variable is introduced to change the dielectric constant of the target. The surface current and the re-established model dielectric constant information are used to calculate the radar cross-section after the model is changed once. The mean and variance of RCS obtained by multiple samplings can be used to obtain the electromagnetic scattering characteristics of uncertain inhomogeneous medium targets.

4. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method according to any one of claims 1 to 3 are implemented.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the method according to any one of claims 1 to 3 are implemented.

6. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 3 are implemented.

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