Method for improving the speed of solving the electromagnetic scattering field of a rotating sphere without singularity

Abstract

A method for improving the solving speed of electromagnetic scattering field of rotating sphere without singularity belongs to the field of electromagnetic engineering. Aiming at the electromagnetic scattering problem of rotating structure in the actual engineering field, a method for improving the solving speed of electromagnetic scattering field of rotating sphere without singularity is proposed, and the effective analysis of rotating body is realized based on the method of moments. Rotating structure has important application value in electromagnetic radiation and scattering research due to its geometric symmetry, but under the condition of long wavelength, the traditional surface integral method often has singularity problem, which affects the numerical stability and accuracy. Therefore, an effective method is designed to eliminate the singularity integral, which can convert the dense matrix to be solved into a sparse matrix, significantly improve the calculation efficiency and reduce the memory consumption. Through derivation, the surface integral equation without singularity is obtained, and the simulation and numerical simulation are carried out for specific electromagnetic rotating body device model, which provides an efficient numerical solution for electromagnetic scattering problem and has wide engineering application prospect.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of electromagnetic engineering, and particularly relates to a method for improving the solving speed of an electromagnetic scattering field of a rotating sphere without singularity. BACKGROUND

[0002] In the numerical solution of electromagnetic scattering problems, the method of moments (MoM) is widely used due to its advantages in solving open domain problems. Compared with the commonly used finite element method (FEM) and finite difference time domain method (FDTD), the method of moments (MoM) has a significant advantage in dealing with complex geometric structures. By sampling and densely dividing the surface of the object, the number of unknowns to be solved is reduced, and finally the matrix equation is solved. However, the disadvantages are that the memory occupation is too large and the calculation time is significantly increased (Jin J. The Finite Element Method in Electromagnetics [J]. Journal of the Japan Society of Applied Electromagnetics, 2002).

[0003] The electromagnetic radiation and scattering problem of a rotating structure (BoR) has always been an important research direction in the field of computational electromagnetics. Rotating structures are symmetric geometric bodies formed by rotating generatrix around the symmetry axis. In practical engineering, the analysis and research of the electromagnetic scattering and radiation characteristics of rotating structures are always one of the focuses and difficulties in engineering fields (Chen Fangtao, Wang Weili, Liu Qi. Analysis and Design of High-Performance Microwave Transmission Antenna Based on Rotating Body Method of Moments [J]. Modern Electronic Technology, 2009, 32(11): 4). Practical engineering application fields such as radar, missiles, satellite navigation antennas, etc. all belong to the category of electromagnetic radiation and scattering. Therefore, the BoR method can convert complex three-dimensional problems into a series of simple two-dimensional problems for solving, simplifying the electromagnetic calculation process. However, in the solving process, the influence of the Green function will cause singularity, and a large amount of time is spent in solving the surface integral equation of the rotating body to eliminate the influence of singularity, significantly increasing the calculation time. Therefore, it is necessary to solve the electromagnetic scattering field without singularity for this singularity problem to improve the analysis efficiency and reliability of radar antenna surfaces, missile cross sections, and antenna problems involving rotating bodies.

[0004] The surface integral method widely used in current research is mostly based on the Stratton-Chu or Poggio-Miller-Chew-Harrington-Wu-Tsai (PMCHWT) formula, and usually selects the surface current density J and the magnetic current density M as unknown quantities in the boundary problem (Y. Chang and R. Harrington, A surface formulation for characteristic modes of material bodies [J]. IEEE Transactions on Antennas and Propagation, vol. 25, no. 6, pp. 789-795, November 1977). In order to ensure the conservation of current on each surface element, the Rao-Wilton-Glisson (RWG) vector basis function is usually used to represent the current density (S. Rao, D. Wilton and A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape [J]. IEEE Transactions on Antennas and Propagation, vol. 30, no. 3, pp. 409-418, May 1982). At the same time, in order to solve the problem of singularity of Green's function, the singularity subtraction technique is usually used to decompose it into a singular part that can be handled by an analytical method, and a smooth non-singular part that can be numerically integrated. However, this method is not only tedious and laborious, but also may not get accurate results near the scatterer, so the traditional MoM usually faces the problem of singularity when applied to PEC target body. SUMMARY

[0005] The purpose of the present application is to provide a method for improving the solving speed of the electromagnetic scattering field of a rotating sphere without singularity, and the basic technical problem solved by the present application is that the calculation of the electromagnetic device scattering field based on the BoR-MoM method is limited by the singular point of the Green's function. By introducing auxiliary equations and scalar Helmholtz equations, the singularity in the solving process is eliminated, the calculation speed is improved, and the amount of calculation is reduced.

[0006] The method of the present application makes the solution of the rotating spherical electromagnetic device no longer affected by the singular point. The singularity of the matrix equation of the spherical electromagnetic rotating body device is eliminated by introducing auxiliary equations during the calculation of the spherical electromagnetic device. The non-divergence condition is replaced by a scalar Helmholtz equation related to the position vector r, which together with the scalar Helmholtz equations of the three Cartesian components of the electric field forms a set of four coupled scalar wave equations. Through formula derivation, the complex solving equation of the electromagnetic scattering field is converted into a linear equation set, at this time the impedance matrix will change from a dense matrix to a sparse matrix, greatly improving the speed of the numerical simulation research of the electromagnetic rotating body device, and having practical application value for related device simulation technology.

[0007] The present application comprises the following steps:

[0008] 1) Based on the method of moments, the radiation and scattering problem of the PEC (Perfect Electric Conductor) rotating body of the spherical body is solved: according to the frequency domain Maxwell equation and the vector identity, the vector wave equation is derived; the homogeneous scalar wave equation (scalar Helmholtz equation) is obtained by using the vector value equation and the continuity equation change; the Green function is calculated according to the solution of the boundary integral equation form;

[0009] 2) Eliminate the singular integral term in the solving equation: introduce two auxiliary functions that satisfy the Helmholtz equation, and use these functions to remove the solid angle and singularity in the boundary integral equation; through derivation, the electromagnetic scattering solving equation without singularity is obtained;

[0010] 3) Establish the simulation device of the PEC spherical body: on the basis of the above, the scattering problem of the PEC spherical body is specifically solved; the specific form of the auxiliary function is set, and four groups of integral equations containing four scalar unknowns are obtained;

[0011] 4) Solve by using the method of moments: in the cylindrical coordinate system, the basis function and the test function along the generatrix and the azimuth angle direction are set; the four unknowns are expanded by the basis function, substituted into the integral equation, and simplified and arranged into a linear equation set;

[0012] 5) Set the excitation source and establish the linear equation set: the incident wave is a uniform plane wave vertically incident on the spherical electromagnetic device; the unknown coefficients in the solving equation of the electromagnetic device scattering field are arranged into a column vector, and expressed in the form of a linear equation set; the linear equation set is solved to obtain the coefficient matrix, and the scattering field is reconstructed.

[0013] In step 1), the PEC (Perfect Electric Conductor) rotation body radiation and scattering problem of the sphere is solved based on the method of moments, specifically: the PEC rotation body radiation and scattering problem of the sphere is solved based on the method of moments, and the formula is specifically derived through the scattering of the PEC to the incident plane wave; in the frequency domain with time dependence exp(-iωt), the vector wave equation can be obtained according to the frequency domain Maxwell equation and the vector identity:

[0014]

[0015] wherein and Any one of the above formulae is complete, so only one of them needs to be solved; then, by using the vector identity and the continuity equation, one of the Cartesian components of or or is represented, then satisfies the homogeneous scalar wave equation, that is, the scalar Helmholtz equation:

[0016]

[0017] According to the above formula, and by using the vector identity and the vector wave equation, the following formula can be obtained:

[0018]

[0019] From (4), it can be obtained that also satisfies the homogeneous scalar wave equation (3), therefore, the eight scalars about and satisfy the homogeneous scalar wave equation (3), and the eight scalars are respectively: E x , E y , E z , H x , H y , H z , The solution of the boundary integral equation form of the scalar wave equation can be obtained from the second identity:

[0020]

[0021] wherein c0 is the solid angle of , and the specific form of the Green's function is:

[0022]

[0023] In step 2), the singular integral term in the equation is eliminated, specifically using two auxiliary functions satisfying helmholtz equation And That is formula (7), while satisfying (8):

[0024]

[0025] Using the above conditions to meet And The solid angle and singularity in the boundary integral equation (5) can be removed, define function (9), at this time the equation to solve the electromagnetic scattering will not have to consider the influence of singularity, that is:

[0026]

[0027] According to formula (7), (8) continue to derive:

[0028]

[0029] From the above conditions can be derived Satisfy the boundary integral equation (11), and by formula (9)-(11) after simplification to get (12):

[0030]

[0031] In step 3), the establishment of PEC ball simulation device, on the basis of the above specific solution to the scattering problem of the target body, the formula (12) in Substitute for And Can be:

[0032]

[0033] Because the tangential component of PEC total field is 0, so:

[0034]

[0035] The formula (15) into (13)-(14), can be obtained in solving the electromagnetic device scattering field need to solve the four unknowns respectively are In the above formula, the integral surface S is the union of BoR target body surface and S, The positive direction is from the surface of the electromagnetic device to the interior of the target body; when the specific form of the auxiliary function is set, the integral along S can be calculated according to the Sommerfeld radiation condition; when the specific form of the auxiliary function is set as formula (16), the integral region of formula (12) is only the surface of the BOR target body, and an item p(r0) will be added to the left side of the equation; at this time, four groups of integral equations containing four scalar unknowns can be obtained, that is, formulas (13)-(14) can be rewritten as:

[0036]

[0037] In step 4), the moment method is used for solving, and based on the above derivation of the simulation device electromagnetic scattering calculation, the position variable r is (p, phi, z) in the cylindrical coordinate system, t(r) represents the length of the generatrix at r, and the following formula is abbreviated as t, t e (0, T), wherein T is the total length of the generatrix, and the basis functions in the generatrix and the azimuth angle direction are set as f u (t) and wherein wherein u, v = 1, 2, 3... Nt, m, n = -M, -(M-1),..., (M-1), M; at this time, the basis function of the equation is The test function f u (t) δ(t) is set, at this time, the four unknowns required in the above derivation are expanded by using the basis functions to obtain:

[0038]

[0039] The unknowns expanded by using the basis functions are substituted into formulas (17)-(18), and are simplified and arranged to obtain:

[0040]

[0041] In step 5), the excitation source is set, and a linear equation set is established, specifically: the excitation source is set, the incident wave is a uniform plane wave vertically incident on the electromagnetic device of the sphere, and a linear equation set is established; after the above derivation is completed, the unknown coefficients X in the electromagnetic device scattering field equation are arranged to form a column vector, containing N = (2M+1)Nt elements, x mu is the (m+M)Nt+u element of x0; similarly, x The column vector formed by arranging all the unknowns is denoted as x, and formulas (20)-(21) can be written in the form of a linear equation set: Ax = b; at this time, the equation can be expressed as:

[0042]

[0043] By rewriting the complex equations into a system of linear equations, it can be seen that there are many zero matrices in the impedance matrix A. This is why the method of this invention can make dense matrices sparse, greatly reducing the amount of computation and improving the computation speed. Therefore, it is only necessary to first calculate the values ​​of the excitation matrix b and the impedance matrix A obtained from the solution, and then solve the system of linear equations to obtain the coefficient matrix x. The scattered field can be reconstructed from the solved values ​​to obtain the scattered field of the electromagnetic device. The scattered field obtained is compared with the analytical solution of MIE, and the error is 0.042%, which is within a reasonable range. Therefore, this invention has practical application value for the calculation of electromagnetic scattered fields, and also brings about an improvement in computational efficiency.

[0044] Compared with the prior art, the technical effects and outstanding advantages of the present invention are as follows:

[0045] 1. This invention successfully eliminates the Green's function singularity problem encountered when solving for the electromagnetic scattering field of a rotating sphere by introducing two auxiliary functions that satisfy the Helmholtz equation. This improves the accuracy of the calculation and avoids calculation errors or instability caused by singularities.

[0046] 2. This invention combines the scalar Helmholtz equations for the three Cartesian components of the electric field with a scalar Helmholtz equation related to the position vector r, forming a set of four coupled scalar wave equations. This simplifies the solution process for the electromagnetic scattering field while maintaining computational accuracy.

[0047] 3. This invention transforms the complex electromagnetic scattering field equations into a system of linear equations, thereby changing the impedance matrix from a dense matrix to a sparse matrix. The storage and computational efficiency of sparse matrices is far higher than that of dense matrices; therefore, this transformation significantly reduces computational load and memory requirements.

[0048] 4. The method of this invention is not only applicable to simple rotating spherical electromagnetic devices, but can also be extended to more complex structures, such as rotating spheres with coatings or defects. This invention has broad application prospects in various electromagnetic scattering field calculations and analyses. The method of this invention provides a new and effective approach for the rapid and accurate calculation of electromagnetic scattering fields from rotating spheres. Attached Figure Description

[0049] Figure 1 This is a schematic diagram showing the three-dimensional coordinates and related parameters when solving the scattering field using the method of this invention.

[0050] Figure 2 This is a comparison chart showing the error between the calculation method of this invention and the MIE solution for solving the coefficient matrix.

[0051] Figure 3 The diagram shows an actual spherical radome model for the application of the method of this invention. Detailed Implementation

[0052] In order to make the objects, technical solutions and advantages of the present application clearer, the following embodiments will further illustrate the present application with reference to the drawings. It should be understood that the specific embodiments described herein are only used to explain the present application and should not be used to limit the present application. On the contrary, the present application covers any alternatives, modifications, equivalent methods and solutions defined by the claims within the spirit and scope of the present application.

[0053] The embodiments of the present application include the following steps:

[0054] Step 1: Solving the radiation and scattering problem of the PEC (perfect electric conductor) sphere by using the method of moments

[0055] The method of moments is mainly used to solve the radiation and scattering problem of the PEC (perfect electric conductor) sphere. First, the vector wave equation is derived according to the frequency domain Maxwell equation and the vector identity. Then, by changing the vector value equation and the continuity equation, the Cartesian components of the electric field or magnetic field that satisfy the homogeneous scalar wave equation (i.e. the scalar Helmholtz equation) can be obtained. Then, by using the vector identity and the vector wave equation, it is further derived that other related components also satisfy the homogeneous scalar wave equation. The homogeneous scalar wave equation satisfied by the eight scalar components of the electric field and magnetic field is obtained. Specifically:

[0056] Based on the method of moments, the radiation and scattering problem of the PEC sphere is solved by specifically deriving the formula through the scattering of the incident plane wave by the PEC. In the frequency domain with time dependence exp(-iωt), the vector wave equation can be obtained according to the frequency domain Maxwell equation and the vector identity:

[0057]

[0058] In the formula and Any one of the above formulae is complete, so only one of them needs to be solved. Then, by changing the vector value equation and the continuity equation, one of the Cartesian components of is represented by or , then satisfies the homogeneous scalar wave equation, i.e. the scalar Helmholtz equation:

[0059]

[0060] According to the above formula, and by using the vector identity and the vector wave equation, we can get:

[0061]

[0062] From (4), we can get It also satisfies the homogeneous scalar wave equation (3), therefore, regarding and The eight scalars satisfy the homogeneous scalar wave equation (3), and these eight scalars are: E x E y E z , H x H y H z , The solution to the scalar wave equation in the form of the boundary integral equation can be obtained from the second identity:

[0063]

[0064] In the formula, c0 is The solid angle, and the specific form of the Green's function is:

[0065]

[0066] Step 2: Eliminate singular integral terms in the equation

[0067] During calculations, the behavior of the Green's function becomes extremely complex when the observation point approaches the source point, potentially leading to numerical instability or decreased accuracy. Therefore, the influence of the Green's function can cause singularities, significantly increasing the computation time required to solve the surface integral equations of solids of revolution. To address this issue, this invention introduces two auxiliary functions that satisfy the Helmholtz equations. These auxiliary functions help eliminate solid angles and singularities in the boundary integral equations, simplifying the calculation process and improving accuracy. By defining a new function and substituting the auxiliary functions into the boundary integral equations, a new equation without singular integral terms can be obtained. The influence of singularities then no longer needs to be considered when solving the electromagnetic scattering equations. Specifically:

[0068] To address this singularity problem, this invention uses two auxiliary functions that satisfy the Helmholtz equation. and That is, formula (7) also satisfies (8):

[0069]

[0070] Using those that meet the above conditions and The solid angle and singularity in the boundary integral equation (5) can be removed, and the function (9) can be defined. At this time, solving the equation for electromagnetic scattering will not require considering the effect of singularity, that is:

[0071]

[0072] Continuing to derive according to formula (7), (8) is:

[0073]

[0074] From the above conditions, it can be deduced that Satisfy the boundary integral equation (11), and after simplifying formula (9)-(11) get (12):

[0075]

[0076] Step 3: Establish the simulation device of PEC sphere and solve the scattering problem

[0077] On the basis of the above, the simulation device of PEC sphere is established, and the scattering problem of the target body is specifically solved. First, the new equation obtained in step 2 is substituted into the specific solving process. Then, by using the characteristic that the tangential component of the PEC total field is 0, the formula is simplified. Through the simplification, four unknowns needed to solve when solving the scattering field of electromagnetic devices can be obtained. These unknowns are the basis for subsequent solving of linear equations. Specifically:

[0078] On the basis of the above, the scattering problem of the target body is specifically solved, and In formula (12) is substituted for And Can be obtained:

[0079]

[0080] Because the tangential component of the PEC total field is 0, so:

[0081]

[0082] Substitute formula (15) into (13)-(14), four unknowns needed to solve when solving the scattering field of electromagnetic devices are respectively In the above formula, the integral surface S is the union of the surface of the BoR target body and S, The electromagnetic device surface points to the inside of the target body as positive. Figure 1 The three-dimensional coordinate and related parameters of the method for solving the scattering field of the present application are shown in the schematic diagram. When the specific form of the auxiliary function is determined, the integral along S can be calculated according to Sommerfeld radiation condition. In the embodiment, the specific form of the auxiliary function is set as formula (16). At this time, the integral region of formula (12) is only the surface of the BoR target body, and there is an additional term p(r0) on the left side of the equation. At this time, four groups of integral equations containing four scalar unknowns can be obtained, that is, formula (13)-(14) can be rewritten as:

[0083]

[0084] Step 4: Continue solving by using the method of moments

[0085] Based on the above derivation of the simulation of electromagnetic scattering of devices, the method of moments is continued to be used for solving. The position variable r is represented in the cylindrical coordinate system, and the basis functions and test functions along the generatrix and azimuthal angle directions are set. The four unknowns required are expanded by the basis functions and substituted into the related formulas for simplification and arrangement. Through simplification and arrangement, a linear equation set is obtained. This linear equation set contains key information such as impedance matrix A and excitation matrix b. Specifically:

[0086] Based on the above derivation of the simulation of electromagnetic scattering of devices, the method of moments is continued to be used, and the position variable r is in the cylindrical coordinate system (p, phi, z). t(r) represents the length of the generatrix at r, and the following formulas are abbreviated as t, t ∈ (0, T), where T is the total length of the generatrix. The basis functions along the generatrix and azimuthal angle directions are set as f u (t) and where u, v = 1, 2, 3... Nt, m, n = -M, -(M-1),..., (M-1), M. At this time, the basis function of the equation is The test function f u (t) δ(t) is set, and at this time the four unknowns required in the above derivation are expanded by the basis functions to obtain:

[0087]

[0088] At this time, the unknowns expanded by the basis functions are substituted into formulas (17-18), and they are simplified and arranged to obtain:

[0089]

[0090] Step 5: Set the excitation source and establish a linear equation set

[0091] The incident wave is set as a uniform plane wave vertically incident on the electromagnetic device as the excitation source. The unknown coefficients in the equation of the electromagnetic device scattering field are arranged into a column vector, and they are associated with the impedance matrix A and the excitation matrix b. By solving this linear equation set, the values of the unknown coefficients can be obtained. Finally, the solved values are reconstructed into the scattering field, and the scattering field of the electromagnetic device to be solved can be obtained. Specifically:

[0092] The excitation source is set, the incident wave is a uniform plane wave vertically incident on the spherical electromagnetic device, and a linear equation set is established. After the above derivation, the unknown coefficients x in the electromagnetic device scattering field equation are arranged to form a column vector, containing N=(2M+1)Nt elements, x mu is the (m+M)Nt+u element of x0. Similarly, x All unknowns are arranged into a column vector x, and (20)-(21) can be written in the form of a linear equation set: Ax=b. At this time, the equation can be expressed as:

[0093]

[0094] By rewriting the complex equation into the form of a linear equation set, it can be seen that there are many zero matrices in the impedance matrix A, which is also the method of using the design invention that can make the dense matrix sparse, greatly reducing the calculation amount and improving the calculation speed.

[0095] In order to verify the accuracy of the method of the invention in solving the electromagnetic scattering field of the spherical radar antenna cover, a spherical model in cylindrical coordinates is established as a simulation object, as Figure 3 shown is a spherical radar antenna cover simulation model, the sphere is taken as the axis of the z-axis, located in the positive half of the z-axis and tangent to the x0y plane, and its generatrix is a semicircle. The generatrix is subdivided into N segments, and the related coordinate parameters are:

[0096]

[0097] where θ is the pitch angle (polar angle), and φ is the azimuth angle. The spherical radius is set to 2m, the incident wave frequency Fre is 1*10 8 HZ, the generatrix length T is 2π, ppw is 20, the generatrix sampling point is obtained as 84 by Nth=T / λ*ppw / 2)*2, the azimuth angle sampling point Nph is set to 42, the length of each segment after the generatrix is subdivided is dn=2π / Nth, and the length of each segment after the azimuth angle is subdivided is dp=2π / Nph. The rest of the parameters are set with reference to Table 1.

[0098] Table 1

[0099]

[0100] After the model is established and the parameters are set, the excitation matrix b and the impedance matrix A are calculated using the method of the invention. The calculated values of the excitation matrix b and the impedance matrix A are used to reconstruct the scattering field, and the scattering field of the electromagnetic device is obtained. In order to verify the accuracy of the results, the calculated scattering field is compared with the MIE analytical solution, and the error is 0.042%, which is within a reasonable range, as Figure 2The method has high precision and practicability in solving electromagnetic scattering field, and therefore has practical application value in calculation of electromagnetic scattering field and brings improvement in calculation efficiency. The method can be used as a reference for a spherical antenna cover and a missile circular cross section model, a model shape is designed, relevant parameters in a program are changed, simulation input is performed according to the design of the method, the electromagnetic scattering field of the model is quickly solved, numerical comparison and calculation are performed according to the output result, and finally the design effect is verified.

[0101] The method of the application eliminates singularity of a matrix equation when solving a rotating spherical electromagnetic device by introducing an auxiliary equation and a scalar Helmholtz equation, improves calculation speed and precision, and changes the impedance matrix from a dense matrix to a sparse matrix, thereby significantly reducing calculation amount and memory requirement. The method is not only suitable for electromagnetic scattering field calculation of a spherical antenna cover, but also can be widely applied to electromagnetic scattering analysis of a missile circular cross section model. By designing models with different shapes and changing relevant parameters in a program, electromagnetic scattering fields of various complex models can be quickly solved. The method provides a powerful tool and support for research in the fields of electromagnetic compatibility analysis and radar stealth technology.

[0102] The above examples are only preferred embodiments of the application and should not be considered as limiting the scope of the application. Any equivalent changes and improvements made within the scope of the application should still belong to the patent coverage of the application.

Claims

1. A method for improving the solution speed of electromagnetic scattering fields without singularities of a rotating sphere, characterized in that... Includes the following steps: 1) Solving the radiation and scattering problem of a sphere-based PEC rotating body based on the method of moments: derive the vector wave equation based on the frequency domain Maxwell's equations and vector identities; obtain the homogeneous scalar wave equation, i.e., the scalar Helmholtz equation, by utilizing the vector value equation and the transformation of the continuity equation; calculate the solution using the Green's function based on the solution in the form of the boundary integral equation. 2) Eliminating singular integral terms in the solution equations: Introduce two auxiliary functions that satisfy the Helmholtz equations, and use these functions to remove solid angles and singularities in the boundary integral equations; through derivation, the electromagnetic scattering solution equations without singularities are obtained; 3) Establish a simulation device for the PEC sphere: Based on the above, specifically solve the scattering problem of the PEC sphere; define the specific form of the auxiliary function to obtain four sets of integral equations containing four scalar unknowns; 4) Solve using the method of moments: In cylindrical coordinates, define the basis functions and test functions along the generatrix and azimuth directions; expand the four unknowns using the basis functions, substitute them into the integral equation, and simplify and rearrange them into a system of linear equations; 5) Set up the excitation source and establish a linear equation system: The incident wave is a uniform plane wave that is perpendicularly incident on the spherical electromagnetic device; arrange the unknown coefficients in the equation for solving the scattering field of the electromagnetic device into a column vector and express it in the form of a linear equation system; solve the linear equation system to obtain the coefficient matrix and reconstruct the scattering field.

2. The method for improving the solution speed of electromagnetic scattering fields without singularities of a rotating sphere as described in claim 1, characterized in that... In step 1), the solution to the radiation and scattering problem of the PEC rotating body based on the method of moments is specifically: the formula is derived by scattering the incident plane wave by the PEC; in the frequency domain with time dependence exp(-iωt), the vector wave equation is obtained according to the frequency domain Maxwell's equations and vector identities: In the formula and Derived from Maxwell's equations, since any one of the above equations is complete, only one needs to be solved; further, by transforming the vector value equation and the continuity equation, using... express or One of the Cartesian components, then It satisfies the homogeneous scalar wave equation, i.e., the scalar Helmholtz equation: Based on the above formula, and then using the vector identity... And from the vector wave equation, we get: From (4) we get It also satisfies the homogeneous scalar wave equation (3), therefore, regarding and The eight scalars satisfy the homogeneous scalar wave equation (3), and these eight scalars are: E x E y E z , H x H y H z , The solution to the scalar wave equation in the form of the boundary integral equation is obtained from the second identity: In the formula, c0 is The solid angle, and the specific form of the Green's function is:

3. The method for improving the solution speed of electromagnetic scattering fields without singularities of a rotating sphere as described in claim 1, characterized in that... In step 2), the elimination of singular integral terms in the solution equation is specifically achieved using two auxiliary functions that satisfy the Helmholtz equation. and That is, formula (7) also satisfies (8): Using those that meet the above conditions and Removing the solid angle and singularity from the boundary integral equation (5), and defining the function (9), the solution to the electromagnetic scattering equation no longer needs to consider the effect of singularity, i.e.: Based on formulas (7) and (8), we can further derive: Based on the above conditions, we can deduce that The boundary integral equation (11) is satisfied, and (12) is obtained by simplifying equations (9)-(11):

4. The method for improving the solution speed of electromagnetic scattering fields without singularities of a rotating sphere as described in claim 1, characterized in that... In step 3), the simulation device for establishing the PEC sphere is used to specifically solve the scattering problem of the target body based on the above, and the scattering problem of the target body is solved by using equation (12). Substitute as and have to: Since the tangential square of the total field of PEC is 0, therefore: Substituting formula (15) into (13)-(14), we obtain the four unknowns that need to be solved when solving the scattering field of electromagnetic devices. In the above formula, the integral surface S is the union of the BoR target surface and S. The direction from the surface of the electromagnetic device to the inside of the target body is considered positive. After determining the specific form of the auxiliary function, the integral along S is calculated according to the Sommerfeld radiation condition. The specific form of the auxiliary function is set as shown in formula (16). At this time, the integration region of formula (12) is only the surface of the target body, and an additional term p(r0) will be added to the left side of the equation. Four sets of integral equations containing four scalar unknowns are obtained, that is, formulas (13)-(14) are rewritten as:

5. The method for improving the solution speed of electromagnetic scattering fields without singularities of a rotating sphere as described in claim 1, characterized in that... In step 4), the method of moments is applied to solve the problem. Based on the above derivation of the electromagnetic scattering calculation of the simulated device, the method of moments is applied again. The position variable r in cylindrical coordinates is (p, phi, z), and t(r) represents the length of the generatrix at r, which is abbreviated as t, t∈(0, T) in the following formula, where T is the total length of the generatrix. The basis functions along the generatrix and azimuth directions are set as fi and fj, respectively. u (t) and Where u, v = 1, 2, 3, ..., Nt, m, n = -M, -(M-1), ..., (M-1), M; the basis functions of the equation are... Define the test function f u (t)δ(t), at this point, the four unknowns required in the above derivation are... Expanding using basis functions, we get: Substituting the unknowns obtained by expanding the basis functions into formulas (17)-(18), and simplifying them, we get:

6. The method for improving the solution speed of electromagnetic scattering fields without singularities of a rotating sphere as described in claim 1, characterized in that... In step 5), the excitation source is set and a system of linear equations is established. Specifically, the excitation source is set, and the incident wave is a uniform plane wave perpendicularly incident on the spherical electromagnetic device. Simultaneously, a system of linear equations is established. After completing the above derivation, the unknown coefficients X in solving the electromagnetic device's scattering field equations are arranged into a column vector containing N = (2M+1)Nt elements, x... mu It is the (m+M)Nt+u-th element of x0; similarly denoted... Let x be the column vector formed by arranging all the unknowns a = x, y, z. Then equations (20)-(21) can be written in the form of a system of linear equations: Ax = b. The equations can then be expressed as: By rewriting the complex equations into a system of linear equations, it can be seen that there are many zero matrices in the impedance matrix A, which makes the dense matrix sparse, greatly reducing the amount of calculation and improving the calculation speed. It is only necessary to calculate the values ​​of the excitation matrix b and the impedance matrix A obtained by solving the equations first, and then solve the system of linear equations to obtain the coefficient matrix x. The scattered field is reconstructed from the solved values, and the scattered field of the electromagnetic device is obtained.

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