Methods to improve the impedance matrix filling speed when solving electromagnetic scattering fields of rotating bodies

Abstract

This method, belonging to the field of electromagnetic engineering, aims to improve the speed of impedance matrix filling when solving electromagnetic scattering fields of rotating bodies. Based on a higher-order Taylor series method, it obtains more accurate impedance matrix elements with less sampling data, thus significantly saving computational resources and improving design efficiency. Simulations of various rotating electromagnetic devices demonstrate the superior performance in terms of impedance matrix filling accuracy, speed, and memory consumption, as well as its universality for different models. The method derives the modal Green's function expression after higher-order integration and performs simulations using specific electromagnetic rotating device models. The increased order of the method has practical application value in improving the accuracy of device model simulations. The higher-order numerical integration method also brings efficiency improvements, saving memory consumption and significantly increasing the impedance matrix filling speed compared to the traditional T-shaped integration method.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic engineering, and in particular relates to a method for improving the impedance matrix filling speed when solving the electromagnetic scattering field of a rotating body. Background Technology

[0002] Numerical methods for solving electromagnetic fields are divided into two categories: time-domain algorithms and frequency-domain algorithms. Typical time-domain algorithms include the Finite-Difference Time-Domain (FDTD) method and the Finite-Element Time-Domain (FETD) method. Frequency-domain algorithms, based on the different forms of the equations to be solved, are further divided into integral equation methods and differential equation methods. The former is represented by the Finite-Element Method (FEM), which can simulate complex boundaries and is suitable for electromagnetic problems in non-uniform media and enclosed environments. The latter is represented by the Method of Moments (MoM), which performs dense subdivision of the object's surface to reduce the number of unknowns and ultimately solves the matrix equations, achieving high accuracy. However, its drawbacks include increased memory consumption and computation time. The design method proposed in this invention aims to solve this problem.

[0003] In the history of computational electromagnetics, electromagnetic problems involving bodies of revolution (BoRs) have received widespread attention. BoRs are geometric shapes formed by rotating a curve 360° around a coplanar axis of symmetry, with rotational symmetry being their most prominent characteristic. In practical engineering applications, objects with rotating structures such as missiles and radomes can be categorized as electromagnetic radiation and scattering problems involving BoRs. During electromagnetic modeling and solution, appropriate simplification can be made, reducing the three-dimensional problem to a series of two-dimensional problems. This method, known as the BoR method, offers unparalleled advantages in reducing computation time and memory consumption. When using the method of moments (MoM) to solve BoR problems, the modal Green's function (MGF) is employed. The computation time spent solving the integral equations of BoRs largely depends on the calculation of the modal Green's function. Accurate and efficient numerical calculation of the modal Green's function is essential for radar cross-section, scattering, and antenna problems involving rotating bodies.

[0004] The study of BoR-MoM in computational electromagnetics began with the introduction of basis functions for rotating bodies by MG Andreasen in his 1965 paper on the analysis of scattering from bodies of revolution (MG Andreasen. Scattering from bodies of revolution[J]. IEEE Transactions on Antennas and Propagation, 1965, 13(2): 303-310). Later, JR Mautz and RF Harrington provided many details on singularity handling and error analysis in BoR-MoM calculations in their paper on electromagnetic radiation and scattering from bodies of revolution (Mautz JR, Harrington R F. Radiation and scattering from bodies of revolution[J]. Applied Scientific Research, 1969, 20(1): 405-435), thus perfecting the theory of the BoR method. In subsequent research, scholars have almost exhausted the applications of rotating bodies in all numerical methods. Since the rapid development of fast computation methods, Professor R. Mittra used the Fast Fourier Transform (FFT) technique to accelerate the calculation of modal Green's functions (SDGedney, R. Mittra. The use of the FFT for the efficient solution of the problem of electromagnetic scattering by a body of revolution[J]. IEEE transactions on antennas and propagation, 1990, 38(3): 313-322), and used CBFM to solve many electrically large body of revolution problems. However, due to the semi-global basis characteristics of bodies of revolution, designing acceleration algorithms is very difficult. Since the beginning of the new century, AKAbdelmageed accelerated the calculation of modal Green's functions in 2000 by using spherical Bessel function expansion (AKAbdelmageed. Efficient evaluation of modal Green's functions arising in EM scattering by bodies of revolution[J]. Progress in Electromagnetics Research, 2000, 27: 337-356), improving the efficiency of calculating problems of bodies of revolution structures.However, the computational complexity of these methods remains high, and they still consume a lot of computational resources for solving electrically large problems. In recent years, research on the BoR method in China has also developed rapidly. Among them, for the calculation of the far field of the scattering field of a rotating target, Guo Jingli and Liu Qizhong proposed a fast calculation method based on the reciprocity theorem (Zou Yanlin, Guo Jingli, Liu Qizhong. Fast calculation of the far field of scattering of a rotating target [J]. Journal of Xi'an University of Electronic Science and Technology, 2009, 36(02): 285-288.). This method uses the exponential distribution characteristics of the surface equivalent electromagnetic current and the structural characteristics of the scatterer in each mode to derive the analytical expression of the azimuth integral, thereby reducing the double integral to a single arc length integral that is easy to calculate, thus speeding up the calculation and saving the calculation time. Summary of the Invention

[0005] The purpose of this invention is to address the limitation of k|Rp-Rpq|<=pi / 20 when designing rotating electromagnetic devices using the impedance matrix filling method based on the first-order numerical method (MG Andreasen. Scattering from bodies of revolution[J].IEEE Transactions on Antennas and Propagation, 1965, 13(2): 303-310). This invention provides a new method to improve the impedance matrix filling speed when solving for the electromagnetic scattering field of a rotating body using a higher-order numerical integration method. This method eliminates this limitation when designing rotating electromagnetic devices. The method calculates the modal Green's function based on the higher-order numerical integration method, derives the expression of the modal Green's function after higher-order integration, and studies the computational accuracy of this method using a specific electromagnetic rotating device through numerical simulation. The increase in order has practical application value in improving the simulation accuracy of the device model.

[0006] This invention includes the following steps:

[0007] 1) For the radiation and scattering problem of a perfectly conductive rotating body of arbitrary shape, when solving the rotating body problem using the method of moments, the surface integral equations employ electric field (EFIE), magnetic field (MFIE), or combined field (CFIE). Utilizing the rotational symmetry of the rotating body, Fourier series are introduced to transform the original three-dimensional problem caused by the incident plane wave into a series of two-dimensional problems: Fourier components or Fourier modes. For the nth Fourier component, the unit source can be considered as a loop with a current distribution of cos nφ, and its radiation field is called the mode Green's function.

[0008]

[0009] In the formula, φ is the rotation azimuth angle, k corresponds to the angular wavenumber, and R represents the distance between the field point and the source point;

[0010] In the study of radiation and scattering from bodies of revolution by Mautz JR et al. (Mautz JR, Harrington RF. Radiation and scattering from bodies of revolution[J] Applied Scientific Research, 1969, 20(1): 405-435), a detailed expression for calculating the generalized impedance of an object is given:

[0011]

[0012] In the formula, k corresponds to the angular wavenumber, R represents the distance between the field point and the source point, and W... i For the test function, j j It is a basis function, Z ij These are elements of the impedance matrix;

[0013] This works for objects of any shape, but for solids of revolution:

[0014]

[0015] In the formula, φ is the rotation azimuth angle, t is the rotation generatrix, and s is the surface of the rotating body.

[0016] This formula is correct. The integral part is extracted as equation (1); in the calculation of equation (2), a trigonometric function T is introduced, and the derivative of T is approximated as four pulses, with the coordinate parameter value within each pulse approximating the coordinate value of the midpoint of the pulse; thus, the Green's function G of the pulse is defined. n :

[0017]

[0018] Dividing the azimuth angle φ into equal intervals M, equation (4) can be further simplified to:

[0019]

[0020] Further expressed as:

[0021]

[0022] in:

[0023]

[0024] Where, φ m = (m-1 / 2)π / M;

[0025] 2) Establish a model in cylindrical coordinates and derive the Green's function of the model; the parameters ρ, φ, and z are the coordinate parameters in cylindrical coordinates, while t is the generatrix that generates the surface S of the solid of revolution and is a length variable; the parameter v represents the angle between the tangent direction of the generatrix t and the positive z-axis, and is positive when t is far from the z-axis and negative otherwise; p and q represent any two points on the surface S of the solid of revolution, which can be represented as permutations and combinations in a matrix in numerical operations; ρ p z p v p and ρ q z q v q Represent the ρ, z, and v parameter values ​​at points p and q, respectively; R in equation (7) p This represents the distance between point p and any point on t [t1, t2].

[0026]

[0027] If point q is located at the midpoint of this interval, then the distance R between points p and q can be defined. pq for:

[0028]

[0029] For ease of integration calculation, equation (8) cannot be used directly, but rather R is used. pq Let t be an expression; define d as the distance from point p to t. q If the length of the perpendicular segment of the tangent is given, then the length from the perpendicular point to point q is t0.

[0030] t0=|(z p -z q cosv q +(ρ p cosφ m -ρ q sin v q | (10)

[0031] d 2 =R pq 2 -t0 2 (11)

[0032] At this time R p It can be represented as:

[0033]

[0034] Where t is an integral variable that changes approximately linearly in the neighborhood of t0, while d does not change with t when points p and q are determined, thus further simplifying the calculation of equation (7).

[0035] 3) Establish the general formula for filling the impedance matrix of the simulation device; for the integral formula (7), even if substituting... The geometric relationship also makes it difficult to derive the definite integral expression with respect to t; according to Taylor's mean value theorem, if the function f(x) has an nth derivative at x0, then there exists a neighborhood of x0 such that for any x in the neighborhood, we have:

[0036]

[0037] Then, when the integration interval in equation (7) is small enough, we have R p →R pq For the exponent term We can then perform an nth-order Taylor expansion and set the remainder to 0 to perform an approximate calculation, i.e.:

[0038]

[0039] The original integration problem is then transformed into an integration problem involving the expansion of multi-term functions, which can be derived and calculated using the definite integral expression of the nth-order function expansion. To study the impact of increasing the order on the error, it is necessary to derive the integral result of the multi-term function. Polynomials of orders one through four (first to fourth) are selected for calculation, retaining the first to fourth-order terms of the Taylor expansion. The general formula of the expansion is defined as:

[0040]

[0041] Results for n = 1, 2, 3, 4 respectively:

[0042]

[0043] This invention focuses on the mode Green's function expression of the pulses generated during the calculation of rotating bodies using the method of moments, and derives it based on a high-order numerical integration method. The increased order in this method has practical application value in improving computational accuracy, while also improving computational efficiency, saving memory consumption and reducing computation time compared to the traditional T-shaped integral method. This effect is applicable to rotating bodies of arbitrary shapes. Simulations of various rotating electromagnetic devices demonstrate the excellent effects on the accuracy, speed, and memory consumption of the impedance matrix filling, as well as the universality of the model. By deriving the mode Green's function expression after high-order integration, numerical simulations are performed using specific electromagnetic rotating device models. The increased order in the method has practical application value in improving the accuracy of device model simulation. The high-order numerical integration method also improves efficiency, saving memory consumption and reducing filling time compared to the traditional T-shaped integral method. These model simulations use models such as circular radomes, curved missiles, and fighter jet jet engines as references. The model shapes are designed and relevant parameters are set. Simulation inputs are performed according to the invention design, and numerical comparisons and calculations are performed based on the output results to ultimately verify the design's effectiveness. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of the principle of calculating the solid of revolution and a coordinate diagram showing the geometric meaning of the corresponding parameters. Among them, (a) is a three-dimensional coordinate diagram of the model when calculating the modal Green's function of the inventive method and the corresponding related parameters, and (b) is a two-dimensional schematic diagram of any face, reflecting the geometric meaning of parameters such as Rpq.

[0045] Figure 2 The diagram shows the model and error results of the invention applied to a semi-circular radome model. (a) is a three-dimensional coordinate diagram applied to the semi-circular busbar model and corresponding parameters; (b) is a semi-logarithmic curve plotted based on the relative error after operation; and (c) is the actual model for reference application of the invention method.

[0046] Figure 3 The diagram shows the coordinates and error results of the invention applied to an asymmetric single-peak model. (a) is a schematic diagram of the coordinates of the rotated section applied to the asymmetric single-peak busbar model and the corresponding parameters; (b) is a semi-logarithmic curve plotted based on the relative error after operation; and (c) is the actual model for which the invention method is applied.

[0047] Figure 4 This is a schematic diagram of the rotational section coordinates of the present invention applied to an asymmetric double-peaked busbar model. (a) shows the rotational section coordinates and corresponding parameters applied to the asymmetric double-peaked busbar model, while (b) shows the actual model to which the method of the present invention is applied. Detailed Implementation

[0048] The following embodiments will further illustrate the present invention with reference to the accompanying drawings.

[0049] The embodiments of the present invention include the following steps:

[0050] 1) For the radiation and scattering problem of a perfectly conductive rotating body of arbitrary shape, when solving the rotating body problem using the method of moments, the surface integral equation can be the electric field (EFIE), magnetic field (MFIE), or combined field (CFIE). Utilizing the rotational symmetry of the rotating body, a Fourier series is introduced to transform the original three-dimensional problem caused by the incident plane wave into a series of two-dimensional problems (Fourier components or Fourier modes). For the nth Fourier component (mode n), the unit source can be considered as a current distribution... The loop whose radiation field is usually called the mode Green's function:

[0051]

[0052] In the study of radiation and scattering of rotating bodies by Mautz JR et al. [2] The text provides a detailed expression for calculating the generalized impedance of an object:

[0053]

[0054] This works for objects of any shape; however, for solids of revolution:

[0055]

[0056] The integral of φ in equation (3) is precisely equation (1); the calculation of equation (2) introduces the trigonometric function T, and the derivative of T is approximated as four pulses, with the coordinate parameter value within each pulse approximating the coordinate value of the midpoint of the pulse; thus, the Green's function of the pulse is defined:

[0057]

[0058] azimuth angle Divided into M equal intervals, equation (4) can be further simplified to:

[0059]

[0060] Further expressed as:

[0061]

[0062] in:

[0063]

[0064] Where, φ m = (m-1 / 2)π / M;

[0065] 2) Establish a model in cylindrical coordinates and refine the derivation of the mode Green's function. First, the parameters ρ, z represents the coordinate parameters in cylindrical coordinates, while t is the generatrix that generates the surface S of the solid of revolution, and is a length variable; the parameter v represents the angle between the tangent direction of the generatrix t and the positive z-axis, and is positive when t is far from the z-axis, and negative otherwise; p and q represent any two points on S, which can represent permutations and combinations in matrices in numerical operations; ρ p z p v p and ρ q z q v q Let p, z, and v represent the parameter values ​​at points p and q, respectively; in equation (7)

[0066]

[0067] Let p represent the distance between point p and any point on t [t1, t2]. If q is located at the midpoint of this interval, then the distance between the two points can be defined as follows:

[0068]

[0069] However, for the convenience of integral calculation, equation (8) cannot be used directly, but rather R is used. pq Let t be an expression; define d as the distance from point p to t. q If the length of the perpendicular segment of the tangent is given, then the length from the perpendicular point to point q is t0.

[0070] t0=|(z p -z q cosv q +(ρ p cosφ m -ρ q sin v q | (10)

[0071] d 2 =R pq 2 -t0 2 (11)

[0072] At this time R p It can be represented as:

[0073]

[0074] Where t is an integral variable that changes approximately linearly in the neighborhood of t0, while d does not change with t when points p and q are determined, the calculation of equation (7) is further simplified.

[0075] 3) For integral (7), even if we substitute... The geometric relationship also makes it difficult to derive the definite integral expression with respect to t; according to Taylor's mean value theorem, if the function f(x) has an nth derivative at x0, then there exists a neighborhood of x0 such that for any x in the neighborhood, we have:

[0076]

[0077] Then, when the integration interval in equation (7) is small enough, we have R p →R pq For the exponent term We can then perform an nth-order Taylor expansion and set the remainder to 0 to perform an approximate calculation, that is:

[0078]

[0079] The original integration problem is then transformed into an integration problem involving multi-term expansions, which can be derived and calculated using the definite integral expression of an nth-order function expansion. To study the impact of increasing the order on the error, it is necessary to derive the integral result of the multi-term function. Polynomials of orders one through four (first to fourth) are selected for calculation, retaining the first to fourth-order terms of the Taylor expansion polynomial. That is, the following calculation is performed:

[0080]

[0081] Results for n = 1, 2, 3, 4 respectively:

[0082]

[0083]

[0084] This invention provides detailed solution instructions and application processes under different models; the principle diagram of the proposed method is shown below. Figure 1 As shown in Figures (a) and (b), assuming the coordinates of two points p and q on the body of revolution are known, along with other necessary parameters, the values ​​of parameters Rpq, t0, and d can be obtained. A neighborhood of size dn, [t1, t2], is defined along the generatrix of the body of revolution containing point q. When this neighborhood is sufficiently small, it satisfies the condition that the coordinates of point q change approximately linearly with t. Substituting the values ​​of all parameters into the expression for f(φm), and then substituting the above results into equation (6), the model Green's function value can be obtained. Given a predetermined model n, only a suitable number of azimuth intervals M is needed to calculate a model Green's function value for p and q.

[0085] Five different Gn values ​​can be obtained by calculating using the T-shaped integral method and first- to fourth-order integral methods, respectively. However, it is clear that data from only two points is insufficient for reliable error analysis, and the interference caused by the specific point selection cannot be eliminated. Only when considering the calculation of the modal Green's function applied to the entire surface of the solid of revolution does the error analysis result have practical significance. The generatrix of a solid of revolution of arbitrary shape is segmented, and for ease of calculation, it is divided into N segments at equal intervals. Points p and q are the midpoints of the intervals between any two different segments. The non-repeating iterative calculation can obtain N×(N-1) modal Green's function matrices covering the surface of the solid of revolution. On this basis, the calculation results under different modes of the modal Green's function need to be considered. Taking n consecutive modes, the modal Green's function matrix in the loop is expanded to a size of n×N×(N-1). The result matrix Gn0 obtained by the T-shaped integral method is regarded as the accurate value. The result matrices Gn1, Gn2, Gn3, and Gn4 obtained by the five different order integral methods are vectorized in one dimension and then subjected to a L2 norm relative error analysis with the accurate value. This allows us to compare the impact of higher-order numerical integration methods on the accuracy of the calculated modal Green's function.

[0086] like Figure 2 (a) shows a simulation model of a spherical radar radome. To verify whether the invented method can improve the speed and accuracy of solving the electromagnetic scattering field impedance matrix of this radome, a spherical model in cylindrical coordinates is established as the simulation object. The sphere is centered on the z-axis, located in the positive half of the z-axis and tangent to the x0y plane. Its generatrix is ​​a semicircle, which is subdivided into N segments. A point q is located at the midpoint of the i-th segment. Then its relevant coordinate parameters are:

[0087]

[0088] The same principle applies to point p.

[0089] With a radius r = 100 meters and a wavelength λ = 40 meters, the length of each segment after dividing the generatrix is ​​calculated using the semicircle circumference formula: dn = rπ / N, and the angular wave number k = 2π / λ = π / 20. Other parameter settings are shown in Table 1.

[0090] Table 1

[0091]

[0092] If we trust the Green's function value calculation results of the T-type integral method with 1000 sampling points, the final five sets of error data are shown in Table 2.

[0093] Table 2

[0094]

[0095] Figure 2 Figure (b) shows a line graph plotted based on the data results. It can be seen that when the number of bus segments is 25 and the corresponding PPW is 3.18, the relative error of the first-order method is approximately 11%, indicating a certain level of error. However, when the number of bus segments in the fourth-order method is 250 and the PPW is approximately 19.10, the relative error is only 2.9 × 10⁻⁶. -8 This indicates that the latter's calculation results achieve higher accuracy. For integration methods of the same order, the relative error decreases by an almost exponential magnitude as the number of bus segments increases, further verifying the conjecture regarding the application conditions of Taylor expansion. The condition for Rp→Rpq in the model is achieved by increasing the number of bus segments N. The size of the radius r indicates the proportion of the entire model and does not change the relative relationship between points p and q. A sufficiently large N makes the bus segments t within the integration interval closer to line segments, the approximation of the linear changes of coordinate parameters ρ and z with t more reliable, and Vq closer to a constant within the interval. This series of relative relationships makes the integration result of the Taylor expansion polynomial Rp→Rpq closer to the original function.

[0096] This invention can significantly reduce the number of data sampling points while achieving the same effect, thereby improving the calculation speed of impedance matrix filling and increasing the efficiency of electromagnetic simulation and electromagnetic device design.

[0097] like Figure 3 Figure (a) shows a simulation model of a pot-shaped radar radome, intended to verify whether the invented method has good performance in complex and irregular models. Using the missile's nose cone shape as a reference, an asymmetric single-peak generatrix is ​​constructed using a generatrix function. This generatrix can rotate around the Z-axis, resulting in a pot-shaped body of revolution. The generatrix is ​​divided into N equidistant segments, and a MATLAB program is designed to calculate the Z-coordinate parameters corresponding to the midpoint positions of each generatrix. Then, the ρ and v coordinates are calculated based on the function and its derivative.

[0098] ρ=πz sin(πz) (23)

[0099] v=arctan[πsin(πz)+zπ 2 cos(πz)] (24)

[0100] The model's scale is [0, 1] on the z-axis. Assuming a wavelength of λ = 0.7 m, the angular wavenumber k = 2πλ = 20π / 7. The remaining parameters are set the same as before. Finally, the error data for the four sets of modulo Green's functions calculated based on this model are shown in Table 3.

[0101] Table 3

[0102]

[0103] When the power-to-weight ratio (PPW) is 4.54, the error of the first-order method is 5.4%. When the PPW increases by a factor of 6, the error decreases to 0.15%. The fourth-order method, with a PPW of 4.54, reduces the error to 10%. -5 With a 6-fold increase in PPW, the error decreased to 10. -9 The scale. Draw a broken line. Figure 3 (b) Visualize all the results. It can be seen that as the number of segments and PPW increase, the relative error of the mode Green's function decreases significantly, and the increase in order even leads to an exponential change. This further verifies that the present invention still has good results for irregular bodies of revolution.

[0104] like Figure 4 As shown, to verify the effectiveness of the invention method in more complex radar radome device models with larger scales, referencing the shape of a fighter jet's tail jet engine, a quadratic polynomial function can be used to construct a model as shown. Figure 4 The asymmetric bimodal generatrix shown is rotated around the z-axis to obtain a gourd-shaped body of revolution. Assuming wavelength a = 0.1 m, based on the length of the generatrix, the model has approximately 200 wavenumbers. Adjusting the value of N to 500, the corresponding PPW value is calculated to be approximately 2.45. Simultaneously, the number of segments in the t-integral is extended to 3000. Finally, the relative error results for the first to fourth orders are calculated using MATLAB.

[0105] Table 4

[0106]

[0107] As shown in Table 4, when the PPW value is less than 3, the first-order calculation results have a large relative error of approximately 20%. However, when the order increases to fourth order, the error decreases significantly. This demonstrates that higher-order numerical methods are highly effective in large-scale applications, obtaining more accurate impedance matrix elements with less sampled data. This greatly saves computational resources, improves design efficiency, and solves the problem of slow impedance matrix filling calculations that affect the efficiency of related electromagnetic simulations and electromagnetic device design.

Claims

1. A method for improving the impedance matrix filling speed when solving for the electromagnetic scattering field of a rotating body, characterized in that... Includes the following steps: 1) For the radiation and scattering problem of a perfectly conductive rotating body of arbitrary shape, when using the method of moments to solve the rotating body problem, the surface integral equations employ electric field, magnetic field, or a combination of fields. Utilizing the rotational symmetry of the rotating body, Fourier series are introduced to transform the original three-dimensional problem caused by the incident plane wave into a series of two-dimensional problems: Fourier components or Fourier modes. For the nth Fourier component, the unit source is considered as a loop with a current distribution of cos nφ, and its radiation field is called the mode Green's function. In the formula, φ is the rotation azimuth angle, k corresponds to the angular wavenumber, and R represents the distance between the field point and the source point; The detailed expression for calculating the generalized impedance of an object is as follows: In the formula, k corresponds to the angular wavenumber, R represents the distance between the field point and the source point, and W... i For the test function, J j It is a basis function, Z ij These are elements of the impedance matrix; This works for objects of any shape, but for solids of revolution: In the formula, φ is the rotation azimuth angle, h is the rotation generatrix, and s is the surface of the body of revolution; This formula is correct. The integral part is extracted as equation (1); in the calculation of equation (2), a trigonometric function T is introduced, and the derivative of T is approximated as four pulses, with the coordinate parameter value within each pulse approximating the coordinate value of the midpoint of the pulse; thus, the Green's function G of the pulse is defined. n : Dividing the azimuth angle φ into equal intervals M, equation (4) is further simplified to: Further expressed as: in: among them,f m =(m-1 / 2)π / M; 2) Establish a model in cylindrical coordinates and derive the Green's function of the model; the parameters ρ, φ, and z are the coordinate parameters in cylindrical coordinates, while t is the generatrix that generates the surface S of the solid of revolution and is a length variable; the parameter v represents the angle between the tangent direction of the generatrix t and the positive z-axis, and is positive when t is far from the z-axis and negative otherwise; p and q represent any two points on the surface S of the solid of revolution, which in numerical operations represent permutations and combinations in a matrix; ρ p z p v p and ρ q z q v q Represent the ρ, z, and v parameter values ​​at points p and q, respectively; R in equation (7) p This represents the distance between point p and any point on t between [t1, t2]. If point q is located at the midpoint of the interval, then the distance R between points p and q is defined as follows: pq for: For ease of integration calculation, equation (8) cannot be used directly, but rather R is used. pq Let t be an expression; define d as the distance from point p to t. q The length of the perpendicular segment of the tangent is given by the length from the perpendicular point to point q, which is t0. t0=|(z p -z q )cosv q +(ρ p cosφ m -ρ q )sinv q | (10) d 2 =R pq 2 -t0 2 (11) At this time R p Represented as: Where t is an integral variable that changes approximately linearly in the neighborhood of t0, while d does not change with t when points p and q are determined, thus further simplifying the calculation of equation (7). 3) Establish the general formula for filling the impedance matrix of the simulation device; for the integral formula (7), even if substituting... The geometric relationship also makes it difficult to derive the definite integral expression with respect to t; according to Taylor's mean value theorem, if the function f(x) has an nth derivative at x0, then there exists a neighborhood of x0 such that for any x in the neighborhood, we have: Then, when the integration interval in equation (7) is small enough, we have R p →R pq For the exponent term Perform an nth-order Taylor expansion and set the remainder to 0 to perform an approximate calculation, i.e.: The original integration problem is then transformed into an integration problem involving the expansion of multi-term functions. The definite integral expression of the nth-order function expansion is derived and calculated. To study the impact of increasing the order on the error, it is necessary to derive the integral result of the multi-term function. Polynomials of orders one through four are selected for calculation, retaining the first to fourth degree terms of the Taylor expansion. The general formula of the expansion is defined as: Results for n = 1, 2, 3, 4 respectively:

Images