Point Configuration Non-invasive PCE Method for Random Plasma Electromagnetic Scattering Characteristics

Abstract

The point collocation non-intrusive PCE method for the electromagnetic scattering characteristics of random plasmas includes the following steps: determining the random variables and their distribution types, taking the Gaussian distribution as an example for the random variables: writing the random variables as #imgabs0# The probability distribution type of the random variables is determined, and the obtained response quantity RCS is expanded with orthonormal Hermite polynomials; performing N random samplings on ξ and substituting them into the random variable expression #imgabs1# in turn to obtain the model parameters {ω p1 , ω p2 , … ω pN}; for the parameter value ω p after each sampling, use the electromagnetic simulation algorithm to calculate the corresponding numerical solution RCS; obtain the corresponding polynomial coefficient C k based on the linear equations obtained by the point collocation method; calculate the RCS statistics: mean and standard deviation according to the polynomial coefficients; this method does not require modifying the control equations and does not require rewriting the program, reducing the complexity; compared with the Monte Carlo method, only a very small number of sample points are needed to obtain the statistical characteristics of the response quantity, and the efficiency is high.

Description

Technical Field

[0001] The present invention belongs to the technical field of computational electromagnetics, and particularly relates to a point collocation non-intrusive PCE method for the electromagnetic scattering characteristics of random plasmas. Background Art

[0002] The random characteristics of the propagation and scattering of electromagnetic waves in random media have been a concern for many years. The Monte Carlo (MC) method, as a golden rule, is widely used in the solution of multi-sample simulation of random problems. However, this method has a large amount of calculation and takes too much time, and is not suitable for complex problems, especially for three-dimensional electromagnetic numerical simulations, such as the finite-difference time-domain (FDTD) numerical simulation. The polynomial chaos expansion method is a promising effective method for quantitatively calculating the statistical characteristics of electromagnetic scattering in random media. This method can be divided into an intrusive polynomial chaos expansion (PCE) method and a non-intrusive PCE method. The intrusive polynomial chaos method expands the physical quantities in the control equation by orthogonal polynomial chaos, and then performs Galerkin mapping to obtain the corresponding control equation. This method cannot utilize existing deterministic programs, while the non-intrusive PCE method regards the response function as a black box. According to the distribution of random variables, a certain sampling method is used in the probability space to obtain a number of sample points. Each sample point is input into the program for numerical solution to obtain the electromagnetic response quantity corresponding to each sample point of the input parameters. It only focuses on the mapping relationship between the input and output, constructs a chaotic polynomial surrogate model of the output response, and thus quantitatively evaluates the uncertain process. Compared with the intrusive polynomial chaos method, the non-intrusive PCE method does not require modification of the control equation and does not need to rewrite the program, greatly reducing the complexity. Compared with the Monte Carlo method, the non-intrusive polynomial chaos method only needs a very small number of sample points to obtain the statistical characteristics of the response quantity, with high efficiency.

[0003] It should be noted that this part aims to provide background or context for the embodiments of the present invention stated in the claims. The description herein is not admitted to be prior art merely because it is included in this part. Summary of the Invention

[0004] To overcome the deficiencies of the above prior art, the purpose of the present invention is to provide a point collocation non-intrusive PCE method for the electromagnetic scattering characteristics of random plasmas, which has a fast calculation speed and high accuracy.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is:

[0006] A point collocation non-intrusive PCE method for the electromagnetic scattering characteristics of random plasmas, comprising the following steps:

[0007] Step 1, determine the random variable and its distribution type through the random variable expression ω, where ω p is the random plasma angular frequency, is the mean value, is the standard deviation, and ξ follows a standard normal distribution;

[0008] Step 2, expand the response quantity RCS determined by the probability distribution type of the random variable using orthonormal Hermite polynomials, and expand it into P terms: where Y is the random response quantity, and ψ k (ξ) is the k-th order expansion term of the orthogonal polynomial basis function determined by the distribution function of the random variable ξ, C k is the undetermined chaos polynomial coefficient, P is the number of polynomial expansion terms, and k is the k-th term in the polynomial expansion terms;

[0009] Step 3, conduct N random samplings on ξ, and substitute them into the random variable expression in Step 1 in sequence to obtain the model parameters {ω p1 , ω p2 , … ω pN}, where ω p is the random plasma angular frequency, and N is the number of random samplings;

[0010] Step 4, for the parameter values {ω p1 , ω p2 , … ω pN} after each sampling, use the electromagnetic simulation algorithm to calculate the corresponding numerical solution RCS;

[0011] Step 5, solve the linear equations obtained from the formula based on the point collocation method to obtain the corresponding polynomial coefficient C k , where k is the k-th term in the polynomial expansion terms;

[0012] Step 6, calculate the RCS statistics: mean value and standard deviation according to the polynomial coefficients.

[0013] For the described Step 1, the specific approach is:

[0014] Write the random variable as ω, where ω p is the random plasma angular frequency, is the mean value, is the standard deviation, and ξ follows a standard normal distribution: pdf(ξ) ~ N(0, 1).

[0015] For the described Step 5, the specific approach is:

[0016] According to the polynomial chaos method, the polynomial chaos expansion of random variables is carried out. P + 1 vectors (ξ i =[ξ1, ξ2, …, ξ n ) i , i = 0, 1, 2, …, P) are selected as samples of the P-order polynomial chaos in the random space. Each sample corresponds to a deterministic calculation, where: i is the i-th vector selected in the random space, n is the number of random variables, P is the number of polynomial expansion terms, and P + 1 deterministic solutions are carried out to obtain the solutions of each response parameter. Substituting the deterministic solutions into the formula , a set of linear equations can be obtained:

[0017]

[0018] By solving this set of linear equations, the values of the polynomial coefficients [C0, C1, …, C P ) T can be obtained, where Y = [g(ξ0), g(ξ1), …, g(ξ P )] T is the random response quantity.

[0019] For step 6 described above, the specific method is as follows:

[0020] Based on the obtained chaos polynomial coefficients C k , directly calculate the random probability characteristics of the output response. Based on the coefficients of each term of the non-intrusive PCE expansion, the statistical characteristics of the random function RCS can be calculated, and

[0021] μ Y = C0

[0022]

[0023] where μ Y is the mean value of the random response quantity, is the variance of the random response quantity.

[0024] Compared with the prior art, the beneficial effects of the present invention:

[0025] The present invention is based on the point collocation non-intrusive PCE method and extends it to the problem of the electromagnetic scattering characteristics of random plasmas. According to the distribution of random variables, a number of sample points are obtained in the probability space through a certain sampling method. Each sample point is input into the program for numerical solution to obtain the electromagnetic response corresponding to each sample point of the input parameters. Based on the non-intrusive PCE method, the statistical characteristics of the response are obtained. Compared with the intrusive polynomial chaos method, the present invention does not require modification of the governing equations and does not need to rewrite the program, greatly reducing the complexity. Compared with the Monte Carlo method, the non-intrusive polynomial chaos method only needs a very small number of sample points to obtain the statistical characteristics of the response, with high efficiency. Description of the Drawings

[0026] Figure 1 is the flowchart of the present invention.

[0027] Figure 2 is the structural schematic diagram of the calculation model in the embodiment of the present invention.

[0028] Figure 3 is the comparison diagram of the mean RCS calculated by the method of the present invention and the MC method.

[0029] Figure 4 is the comparison diagram of the standard deviation of RCS calculated by the method of the present invention and the MC method.

[0030] Figure 5 is the structural schematic diagram of the calculation model in the embodiment of the present invention.

[0031] Figure 6 is the comparison diagram of the mean RCS calculated by the method of the present invention and the MC method.

[0032] Figure 7 is the comparison diagram of the standard deviation of RCS calculated by the method of the present invention and the MC method.

[0033] Figure 8 is the structural schematic diagram of the calculation model in the embodiment of the present invention.

[0034] Figure 9 is the comparison diagram of the mean RCS calculated by the method of the present invention and the PCE and MC methods.

[0035] Figure 10 is the comparison diagram of the standard deviation of RCS calculated by the method of the present invention and the PCE and MC methods. Detailed Embodiments

[0036] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. However, the exemplary embodiments can be implemented in various forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that the present invention will be more comprehensive and complete, and the concept of the exemplary embodiments will be fully conveyed to those skilled in the art. The described features or characteristics can be combined in one or more embodiments in any suitable manner.

[0037] The present invention is a PCE (Polynomial Chaos Expansion) method based on point configuration non-invasive PCE method applied to the study of the electromagnetic scattering characteristics of random plasmas. The principle is as follows: First, obtain the polynomial expansion of the electromagnetic scattering model according to the distribution of random variables, perform random sampling on the random variables to obtain the corresponding model parameters, then substitute the model parameters into the electromagnetic simulation algorithm to obtain the corresponding response quantities, and form a linear equation system from the polynomial obtained from the random variables and the corresponding response quantities. By solving this linear equation system, the polynomial coefficient C k value can be obtained. Finally, based on the polynomial coefficient C k solve the statistical characteristics of the response quantity, including the mean and standard deviation.

[0038] The present invention is described based on the polynomial chaos method. Therefore, the principle of the polynomial chaos method is introduced first. The polynomial chaos method estimates the coefficients in the polynomial chaos expansion based on some deterministic solutions, and then obtains the statistical characteristics of the response quantity.

[0039] Taking the response function Y = g(ξ) (ξ = [ξ1, ξ2, …, ξ n ) as an example, the random response quantity Y can be expressed as a P-order truncated polynomial basis function expansion:

[0040]

[0041] where: C k is the undetermined chaos polynomial coefficient; ψ k (ξ) is the k-th order expansion term of the orthogonal polynomial basis function determined by the distribution function of the random variable ξ. The choice of the orthogonal polynomial depends on the probability density function of the random variable. According to the Askey rule, for different probability density functions, there are different optimal expansion polynomials, as shown in Table 1.

[0042] Table 1 Chaos polynomial corresponding to random variable type

[0043]

[0044] The commonly used form of the Hermite polynomial in probability theory is:

[0045]

[0046] As the number of random variables increases, the number of expansion terms of the same highest order polynomial will increase, which means that more coefficients must be calculated. The total number of terms of the orthogonal polynomial is:

[0047]

[0048] where d is the highest order in the polynomial expansion and n is the number of random variables. It can be seen from the above formula that when there is only one random variable, the number of expansion terms P is equal to the highest order d of the polynomial, that is, P = d. Table 2 gives the PCE basis functions with only one random variable and the highest order of the polynomial d = 4. When there are two or more random variables, the number of expansion terms P will increase exponentially. Table 3 shows the PCE basis functions with the total order d = 4 and two random parameters.

[0049] Table 2: Hermite polynomial basis functions with the total order d = 4 and one random variable

[0050]

[0051] The non-intrusive PCE method can be regarded as a combination of the polynomial chaos method and the traditional sampling method, and its idea is to estimate the coefficients in the polynomial chaos expansion based on some deterministic solutions. The non-intrusive PCE expansion process is as Figure 1 shown.

[0052] According to the polynomial chaos method, the polynomial chaos expansion of the random variables is performed as shown in Equation (1). P + 1 vectors (ξ i = [ξ1, ξ2, …, ξ n ) i , i = 0, 1, 2, …, P) are selected as the samples of the P-order polynomial chaos in the random space. Each sample corresponds to a deterministic calculation. After performing P + 1 deterministic solutions, the solutions of each response parameter can be obtained. Substituting the deterministic solutions into Equation (1) gives a set of linear equations

[0053]

[0054] By solving this set of linear equations, the values of the polynomial coefficients C k in Equation (1) can be obtained.

[0055] Once the chaos polynomial coefficients C k are obtained, the random probability characteristics of the output response can be directly calculated.

[0056] Table 3: Hermite polynomial basis functions with the total order d = 4 and two random variables

[0057]

[0058]

[0059] Based on the point collocation non - intrusive PCE to expand each coefficient, the statistical characteristics of the random function Y can be calculated, and we can get:

[0060] The mean value of the random function:

[0061] μ Y = C0(5)

[0062] The variance of the random function:

[0063]

[0064] The present invention is a point collocation non - intrusive PCE method for random plasma electromagnetic scattering characteristics, which is specifically implemented according to the following steps:

[0065] Step 1, determine the random variables and their distribution types through the random variable expression ω p is the random plasma angular frequency, is the mean value, is the standard deviation, and ξ follows a standard normal distribution;

[0066] Step 2, expand the response quantity RCS determined by the probability distribution type of the random variable with orthonormal Hermite polynomials, and expand it into P terms: where Y is the random response quantity, ψ k (ξ) is the k - th order expansion term of the orthogonal polynomial basis function determined by the distribution function of the random variable ξ, C k is the chaos polynomial coefficient to be solved, P is the number of polynomial expansion terms, and k is the k - th term in the polynomial expansion terms;

[0067] Step 3, conduct N random samplings on ξ and substitute them into the random variable expression in Step 1 in sequence to obtain the model parameters {ω p1 , ω p2 , … ω pN}, where ω p is the random plasma angular frequency, and N is the number of random samplings;

[0068] Step 4, for the parameter values {ω p1 , ω p2 , … ω pN} after each sampling, use the electromagnetic simulation algorithm to calculate the corresponding numerical solution RCS;

[0069] Step 5, solve the linear equations obtained from the formula based on the point collocation method to obtain the corresponding polynomial coefficients Ck , where k is the k-th term in the polynomial expansion;

[0070] Step 6, calculate the RCS statistics according to the polynomial coefficients: mean and standard deviation.

[0071] For the said Step 1, the specific method is as follows:

[0072] Write the random variable as ω p is the random plasma angular frequency, is the mean value, is the standard deviation, and ξ follows a standard normal distribution: pdf(ξ) ~ N(0, 1).

[0073] For the said Step 5, the specific method is as follows:

[0074] According to the polynomial chaos method, perform polynomial chaos expansion on the random variable. Select P + 1 vectors (ξ i = [ξ1, ξ2, …, ξ n ) i , i = 0, 1, 2, …, P) in the random space as samples of the P-order polynomial chaos, where each sample corresponds to a deterministic calculation, where: i is the i-th vector selected in the random space, n is the number of random variables, P is the number of polynomial expansion terms, perform P + 1 deterministic solutions to obtain the solutions of each response parameter, and substitute the deterministic solutions into the formula to obtain a set of linear equations:

[0075]

[0076] By solving this set of linear equations, the values of the polynomial coefficients [C0, C1, …, C P ) T can be obtained, where Y = [g(ξ0), g(ξ1), …, g(ξ P )] T is the random response quantity.

[0077] For the said Step 6, the specific method is as follows:

[0078] Based on the obtained chaos polynomial coefficients C k , directly calculate the random probability characteristics of the output response. Based on the coefficients of each term in the non-intrusive PCE expansion, the statistical characteristics of the random function RCS can be calculated, and

[0079] μ Y = C0

[0080]

[0081] where μ Yis the mean value of the random response quantity, is the variance of the random response quantity.

[0082] Calculation completed, drawing graphs, ending.

[0083] Embodiment

[0084] The following three examples are used to verify the correctness of the algorithm.

[0085] Example 1: Model 1 discussed in this paper is as Figure 2 shown. The radius of the metal sphere is a = 100 mm, the outer radius of the covered random plasma is b = 120 mm, the thickness of the covered random plasma is d = 20 mm, the center of the sphere is located at the origin of the rectangular coordinate system, the incident plane wave is incident along the positive z-axis direction, and the electric field polarization direction is along the positive x-axis direction. Assume that the plasma angular frequency ω p is a random variable subject to a normal distribution, the mean value of the plasma angular frequency is μ{ω p} = 1.8032×10 11 rad / s, the standard deviation σ{ω p} = 0.1×μ{ω p}, the collision frequency is v = 20×10 9 rad / s. The simulation results respectively take the number of expansion terms of the d = {1, 2, 4} order Hermite polynomial, and at the same time, the results also adopt the results of the Monte Carlo method with 10,000 simulations as a comparison. Among them, each simulation of the Monte Carlo generates the plasma angular frequency according to the expression of the normal distribution.

[0086]

[0087] The graphs comparing the mean value and standard deviation of the RCS calculated by the method of the present invention with the traditional Monte Carlo method are respectively as Figure 3 and Figure 4 shown. Figure 3 The average value of the monostatic radar cross-section calculated by the MC and non-intrusive PCE methods at frequencies from 0 to 2.5 GHz is compared, Figure 4 and the comparison of the standard deviation of the radar cross-section calculated by different methods is shown. It can be seen that within the entire frequency range, the difference in the mean value of the radar cross-section calculated by the non-intrusive PCE method and the MC method is not significant. When the number of expansion terms is P = d = 1, very consistent results can be obtained. As the order increases, the mean value curve of the RCS gets closer and closer to the MC value. The value of the standard deviation has a relatively large deviation, especially when P = d = 1. When the number of expansion terms is P = d = 4, it can be seen that the results of the non-intrusive PCE method and the MC results can be well matched.

[0088] Example 2: Model 2 discussed in this paper is as Figure 5As shown. The radius of the metal sphere is a = 50 mm, the outer radii of the random plasma coatings are b = 60 mm and c = 70 mm respectively, the thicknesses of the random plasma coatings are d1 = d2 = 10 mm respectively, the center of the sphere is located at the origin of the rectangular coordinate system, the incident plane wave is incident along the positive z-axis direction, and the electric field polarization direction is along the positive x-axis direction. Assume that the plasma angular frequency is a random variable following a normal distribution, and the plasma angular frequency means are μ{ω p1} = 3.0159×10 10 rad / s, μ{ω p2} = 3.1416×10 10 rad / s, the standard deviations are σ{ω p1} = 0.1×μ{ω p1}, σ{ω p2} = 0.1×μ{ω p2}, and the collision frequency is v = 1.5708×10 10 rad / s. The simulation results respectively take the expansion term numbers of the d = {1, 2, 4} order Hermite polynomials, and at the same time, the results also adopt the results of the Monte Carlo method with 10,000 simulations as a comparison. Among them, each simulation of the Monte Carlo generates the plasma angular frequency according to the expression of the normal distribution.

[0089]

[0090]

[0091] The figures comparing the mean and standard deviation of the RCS calculated by the method of the present invention with the traditional Monte Carlo method are respectively as Figure 6 and Figure 7 shown. Figure 6 Compares the mean of the monostatic radar cross-section calculated by the MC and non-intrusive PCE methods at frequencies from 0 to 5 GHz, Figure 7 shows the comparison of the standard deviation of the radar cross-section calculated by different methods. It can be seen that within the entire frequency range, the difference in the mean of the radar cross-section calculated by the non-intrusive PCE method and the MC method is not significant. When the expansion term number is d = 1, very consistent results can be obtained. As the order increases, the mean curve of the RCS gets closer and closer to the MC value. The standard deviation values also match well. Especially when the expansion term number is d = 4, it can be seen that the results of the non-intrusive PCE and the MC match well.

[0092] Example 3: The model discussed in this article is as Figure 8As shown. The simulation region is a 200×200 uniform grid with the grid size set as Δx = Δy = 2.5 mm. There is a metal cylinder with a radius of r = 100 mm, and the metal cylinder is located at the center of the FDTD grid. A layer of plasma with a thickness of d = 50 mm is coated outside the metal cylinder. A modulated Gaussian pulse source with a carrier frequency of 3 GHz is added to the total field / scattered field boundary. Assume that the electron density n of the plasma e is a random variable following a normal distribution, and the mean and standard deviation of n e are μ{n e} = 1.0×10 18 m -3 and σ{n e} = 0.05×μ{n e}. Similarly, the collision frequency v of the plasma is v = 1.0×10 9 rad / s. Among them, for each Monte Carlo simulation, the electron density n of the plasma medium is generated according to the expression of the normal distribution.

[0093] e

[0094]

[0095] The figures comparing the mean and standard deviation of the RCS calculated by the method of the present invention with the intrusive polynomial chaos method and the traditional Monte Carlo method are respectively as shown in Figure 9 and Figure 10 . Figure 9 The average values of the radar cross-section calculated by the MC, intrusive PCE, and non-intrusive PCE methods at a frequency of 3 GHz are compared, Figure 10 and the comparison of the standard deviation of the radar cross-section calculated by different methods is shown. It can be seen that within the entire angular range, when the number of expansion terms is d = 4, the difference in the mean values of the radar cross-section calculated by the non-intrusive PCE method and the intrusive PCE method is not significant, and it can be well matched with the results obtained by the MC simulation. When the number of expansion terms is d = 1, the standard deviation obtained by the non-intrusive PCE method differs greatly from the MC result, while when d = 4, it is basically in agreement with the MC result.

[0096] After considering the specification and practicing the invention disclosed herein, those skilled in the art will readily think of other embodiments of the present invention. This application is intended to cover any variations, uses, or adaptations of the present invention, and these uses or adaptations follow the general principles of the present invention and include the common general knowledge or conventional technical means in the technical field not disclosed in the present invention. The specification and examples are only regarded as exemplary, and the true scope and spirit of the present invention are pointed out by the appended claims.

Claims

1. A point configuration non-invasive PCE method for the electromagnetic scattering characteristics of random plasmas, characterized in that Including the following steps: Step 1, through the random variable expression Determine the random variable and its distribution type, ω p is the random plasma angular frequency, is the mean value, is the standard deviation, and ξ is the standard normal distribution; Step 2: Expand the response quantity RCS obtained by determining the probability distribution type of the random variable using orthonormal Hermite polynomials, and expand it into P terms: where Y is the random response quantity, and ψ k (ξ) is the k-th order expansion term of the orthogonal polynomial basis function determined by the distribution function of the random variable ξ, and C k is the coefficient of the chaos polynomial to be determined, P is the number of polynomial expansion terms, and k is the k-th term in the polynomial expansion terms; Step 3: Conduct N random samplings on ξ and substitute them into the random variable expression in Step 1 in sequence to obtain the model parameters {ω p1 , ω p2 , … ω pN}, where ω p is the random plasma angular frequency and N is the number of random samplings; Step 4. For the parameter values {ω p1 , ω p2 , … ω pN} after each sampling, use the electromagnetic simulation algorithm to calculate the corresponding numerical solution RCS; Step 5, solve the linear equations obtained from the formula to obtain the corresponding polynomial coefficients C k , where k is the k-th term in the polynomial expansion; Step 6, calculating the RCS statistics: mean and standard deviation according to the polynomial coefficients.

2. The point configuration non-invasive PCE method for the random plasma electromagnetic scattering characteristics according to claim 1, wherein For the specific implementation of the said Step 1: Write the random variable as ω p as the random plasma angular frequency, is the mean value, is the standard deviation, and ξ is the standard normal distribution: pdf(ξ) ~ N(0, 1).

3. The point configuration non-invasive PCE method for random plasma electromagnetic scattering characteristics according to claim 1, characterized in that, For the specific implementation of the said Step 5: According to the polynomial chaos method, the polynomial chaos expansion of random variables is carried out, and P + 1 vectors ξ are selected in the random space i as samples of the Pth-order polynomial chaos, ξ i = [ξ1, ξ2, …, ξ n i , i = 0, 1, 2, …, P, where each sample corresponds to a deterministic calculation, where: i is the ith vector selected in the random space, n is the number of random variables, P is the number of polynomial expansion terms, and P + 1 deterministic solutions are obtained through P + 1 deterministic solutions. Substituting the deterministic solutions into the formula results in a set of linear equations:​ By solving this linear equation system, the polynomial coefficients [C0, C1, …, C P T can be obtained, where Y = [g(ξ0), g(ξ1), …, g(ξ P )] T is the random response quantity.​ 4. The non-invasive PCE method for point configuration of random plasma electromagnetic scattering characteristics according to claim 1, characterized in that For the specific implementation of the said Step 6: According to the obtained polynomial coefficients C k , the random probability characteristics of the output response are directly calculated. Based on the non-intrusive PCE to expand each coefficient, the statistical characteristics of the random function RCS can be calculated, and we can get: μ Y = C0 Among them, μ Y is the mean value of the random response quantity, and is the variance of the random response quantity.

Images