Characteristic mode analysis method based on periodic green's function and application thereof

Abstract

This invention discloses an eigenmode analysis method based on periodic Green's functions and its application. The method is characterized by: obtaining the element structure model of the target element and performing triangular mesh generation; constructing RWG basis functions; obtaining the periodic Green's function and using the EWALD transform to decompose it into spatial and spectral parts; constructing an impedance matrix; decomposing the impedance matrix to construct generalized eigenvalue equations and solving for the eigenvalues, thereby obtaining the mode importance of the target element and completing the eigenmode analysis. The advantages are: by using the periodic Green's function, the coupling between the target element and other elements in the periodic structure is directly calculated, accurately reflecting the true modes under infinite periods; using the EWALD transform to decompose the periodic Green's function into spatial and spectral parts, and then performing convergence in the spatial and spectral parts respectively, followed by summing, greatly accelerates the convergence speed, thereby significantly reducing computational complexity and improving computational efficiency.

Description

Technical Field

[0001] This invention relates to a characteristic modulus analysis method, and more particularly to a characteristic modulus analysis method based on a periodic Green's function and its application. Background Technology

[0002] Characteristic Mode Analysis (CMA) holds promise for bringing crucial technological innovations to emerging fields such as antenna design, electromagnetic compatibility, and metasurface design. While CMA techniques for individual components are mature and offer related functionality in commercial software like FEKO and CST, the coupling of other elements to the target element in the case of periodic arrays cannot be calculated. Therefore, accurately and quickly calculating periodic characteristic modes to guide antenna design has become a crucial issue in CMA.

[0003] The earliest known electromagnetic model (CMA) can be traced back to Garbacz in the last century, who proposed a clear definition of characteristic modes: characteristic modes constitute a set of surface currents and radiation fields that characterize the properties of a target, and these surface currents and radiation fields possess a key characteristic: they are independent of any external source. This definition emphasizes the essential nature of characteristic modes as inherent electromagnetic properties of an object, unaffected by external excitation sources, laying a solid foundation for subsequent in-depth research on characteristic mode theory. As an important analytical method in the field of electromagnetics, it can clearly reveal the resonance characteristics and radiation mechanisms inherent in electromagnetic structures. Using characteristic mode analysis, the resonant states of electromagnetic structures at different frequencies can be directly observed. Simultaneously, key information such as the radiation patterns corresponding to each mode can be obtained, and calculations can be performed over a wide frequency band. This information is of great significance for a deeper understanding of the electromagnetic properties of electromagnetic structures. Currently, CMA is widely used in antenna element analysis and feed location design.

[0004] The fundamental problem faced by traditional CMA in dealing with periodic structures lies in the inherent conflict between its theoretical framework and the physical nature of periodic problems. Traditional methods are based on finitely truncated array models, which inevitably introduces severe edge effects. This leads to serious non-physical coupling of array boundary elements, severely contaminating the mode spectrum and distorting the calculated eigenvalues ​​and mode currents, failing to accurately reflect the true modes under infinite periods. Simultaneously, traditional methods cannot incorporate Floquet periodic boundary conditions; they solve for the eigenmodes of a finite array in free space, rather than the eigenvalues ​​of elements satisfying periodic phase relationships, resulting in a misalignment of physical meaning. Furthermore, the high computational complexity of eigenvalue solving is drastically amplified by the large number of unknowns required by the model truncation, leading to extremely low efficiency. Ultimately, these factors collectively result in ambiguous mode interpretations, making it difficult to distinguish between array collective resonances and element periodic modes, thus losing the core value of eigenmode analysis in guiding the design of periodic structures. Therefore, although traditional CMA is a powerful numerical method, its fundamental assumption of "finite objects" causes it to struggle with both computational efficiency and physical accuracy when dealing with "infinite periodic" problems. Summary of the Invention

[0005] The technical problem to be solved by this invention is to provide a method for eigenmode analysis based on periodic Green's function and its application, which can accurately and efficiently calculate the eigenmode of periodic structures and significantly reduce computational complexity.

[0006] The technical solution adopted by this invention to solve the above-mentioned technical problems is: a method for eigenmode analysis based on periodic Green's functions, comprising the following steps:

[0007] Step 1: Perform 3D modeling of the target element to obtain the element structure model, and set material properties, incident wave properties and periodic boundary conditions for the element structure model;

[0008] Step 2: Divide the unit structure model into triangular meshes, and construct RWG basis functions for each pair of triangles with a common edge.

[0009] Step 3: Obtain the periodic Green's function of the target element based on the material properties, incident wave properties, and periodic boundary conditions;

[0010] Step 4: Use the EWALD transform to decompose the periodic Green's function to obtain the spatial and spectral parts and the corresponding gradients.

[0011] Step 5: Based on the RWG basis functions constructed in Step 2 and the spatial and spectral parts obtained in Step 4, as well as the corresponding gradients, construct the impedance matrix of the target element under periodic boundary conditions through a hybrid integral equation.

[0012] Step 6: Decompose the impedance matrix to obtain the Emeryst matrix, and construct the generalized eigenvalue equation based on the Emeryst matrix;

[0013] Step 7: Based on the generalized eigenvalue equation, solve for the eigenvalues ​​of the target element under periodic boundary conditions, and obtain the mode importance of the target element according to the eigenvalues ​​to complete the analysis of the eigenmode.

[0014] Compared with existing technologies, the advantages of this invention are as follows: by employing a periodic Green's function, the coupling between the target element and other elements in a periodic structure is directly calculated, accurately reflecting the true modes under infinite periods and eliminating the edge effects introduced by finite truncation in traditional methods; by embedding periodic boundary conditions, the solved mode importance is ensured to satisfy the periodic phase relationship, solving the problem of physical meaning misalignment and making the mode importance and eigenvalues ​​more consistent with the physical essence of the periodic structure; by using the EWALD transform to decompose the periodic Green's function into a spatial part and a spectral part, and then performing convergence in the spatial part and the spectral part respectively, and then adding them together, the convergence speed is greatly accelerated, thereby significantly reducing the computational complexity and improving the computational efficiency; based on the above technologies, by combining the RWG basis functions to construct the impedance matrix and solve the generalized eigenvalue equation, the mode importance of the target element under periodic arrays can be accurately obtained, providing reliable theoretical guidance and design basis for applications such as antenna design, electromagnetic compatibility, and metasurface design.

[0015] Furthermore, the specific operation process of step 1 is as follows:

[0016] Step 1-1: Use the electromagnetic modeling software ANSYS to perform three-dimensional modeling of the target element to obtain the element structure model;

[0017] Steps 1-2: Set the material properties of the unit structure model to perfect electrical conductor;

[0018] Steps 1-3: Set the incident wave properties, including the frequency of the incident wave during the solution process;

[0019] Steps 1-4: Set periodic boundary conditions, including the length of the unit cell in the x and y directions, and the tilt angle of the unit cell relative to the XOY plane. Accurate 3D modeling and material property settings using ANSYS software ensure the accuracy of the unit structure model. Setting plane wave incident and periodic boundary conditions lays the foundation for the subsequent application of periodic Green's functions, enabling accurate simulation of electromagnetic behavior under infinitely large periodic structures and avoiding edge effects in traditional methods.

[0020] Furthermore, in step 2, the constructed RWG basis functions are denoted as... , ,in, Indicates the first A pair of triangles, This represents the coordinate vector of the sampling points inside the triangular mesh. Indicates the first The length of the common side of a pair of triangles and They represent the first Two triangles in a pair of triangles, Represents a triangle area, Indicates from triangle The vector whose vertex points to the internal sampling point. Represents a triangle area, Indicates from triangle The vector of the internal sampling point in the vector points to the vertex, where the vertex is the vertex opposite to the common edge.

[0021] Furthermore, in step 3, the periodic Green's function is denoted as... , in, Indicates the field point. Indicates the source point, This represents the number of unit cells in the x-direction under periodic boundary conditions. This represents the number of unit cells in the y-direction under periodic boundary conditions. Represents the imaginary unit. This represents the distance between the field point and the source point. Indicates the incident wave number. , Indicates the frequency of the incident wave. Represents the speed of light. Represents the dielectric constant of an object. This represents the magnetic permeability of the object. Because the material properties of the unit cell model are perfect electrical conductors, therefore... , , , This represents the x-coordinate of each cell in the array. This represents the distance between unit cells in the x-direction under periodic boundary conditions. This represents the y-coordinate of each cell in the array. This represents the distance between unit cells in the y-direction under periodic boundary conditions. and These represent the incident angles of the incident waves. By defining a periodic Green's function, the coupling of all elements in the periodic array to the target element is directly considered, accurately reflecting the electromagnetic interaction under an infinite period, and solving the problem of misalignment of physical meaning in traditional methods.

[0022] Furthermore, in step 4, the airspace portion is denoted as... , ,in, This represents the complement error function. , indicating the starting term of the cleavage. Represents the area of ​​the unit cell. , This indicates the tilt angle of the unit cell relative to the XOY plane;

[0023] Let the gradient of the spatial region be denoted as , ,in, Indicates positive or negative sign;

[0024] The spectral domain portion is denoted as , ,in, , This represents the displacement vector from the field point to the source point. , This represents the component of the displacement vector from the field point to the source point along the z-axis.

[0025] The gradient in the spectral domain is denoted as... , ,in, , , , , , Let x, y, and z represent the unit vectors of the x-axis, y-axis, and z-axis, respectively. The EWALD transformation decomposes the infinite summation of the periodic Green's function into spatial and spectral parts, significantly reducing the summation terms to a finite number of terms, thereby greatly reducing computational complexity, improving computational efficiency, and maintaining computational accuracy.

[0026] Furthermore, in step 5, the impedance matrix is ​​denoted as... , ,in, Represents the wave impedance in free space. , , This indicates taking the inner product. Let represent the RWG trial function in the M-th triangle pair. Since Galerkin's method is used, the specific formula of the trial function is the same as that of the RWG basis functions. , Represents the RWG basis functions in the Nth triangle. Represents the imaginary unit. Indicates the incident wave number. This represents the area of ​​the triangle containing the field point. Represents the basis functions of RWG Take the divergence, The inner product is obtained Indicates the trial function Take the divergence, , This represents the area of ​​the triangle containing the source point. This represents the unit outward normal vector of the target element surface. , By constructing the impedance matrix using the Galerkin method and periodic Green's functions, the electromagnetic problem under periodic boundary conditions is accurately discretized, providing a reliable mathematical framework for subsequent eigenvalue solving.

[0027] Furthermore, the specific operation process of step 6 is as follows:

[0028] Step 6-1: Convert the impedance matrix Decomposed into an Emyer matrix, its expression is: ,in, Denotes the real part of the Emilt matrix. Denotes the imaginary part of the Emilt matrix. Representing the impedance matrix The conjugate transpose of;

[0029] Step 6-2: Construct the generalized eigenvalue equation based on the Emilt matrix, its expression is: ,in, Represents eigenvalues. The mode current is represented by the impedance matrix. By decomposing the impedance matrix into an Emitter matrix, the eigenvalue problem becomes easier to solve. The constructed generalized eigenvalue equation directly relates the mode current and the eigenvalue, providing a core tool for eigenmode analysis.

[0030] Furthermore, the specific operation process of step 7 is as follows:

[0031] Step 7-1: Based on the generalized eigenvalue equation, solve for the eigenvalues ​​of the target element under periodic boundary conditions. , ,in, Represents mode current The conjugate transpose of;

[0032] Step 7-2: Obtain the mode importance of the target unit under periodic boundary conditions based on eigenvalues. MS , ,in, By solving for eigenvalues ​​and calculating mode importance, the contribution of different modes to the periodic structure can be accurately assessed, providing crucial guidance for antenna design and optimization.

[0033] An application of a characteristic mode analysis method based on periodic Green's functions is presented. This method identifies resonant frequencies and modes based on the distribution of mode importance, and then designs effective feed structures using the current distribution of the obtained resonant modes. Through mode importance analysis, resonant frequencies and modes can be identified, thereby guiding the design of effective feed structures, improving antenna performance, and enabling the efficient application of periodic structures. Attached Figure Description

[0034] Figure 1 This is a schematic diagram of the overall process of the present invention;

[0035] Figure 2 This is a schematic diagram of the unit structure model of the open-loop chiral structure used in the simulation of this embodiment;

[0036] Figure 3 This is a schematic diagram of the triangular mesh model of the open-loop chiral structure used in the simulation of this embodiment;

[0037] Figure 4 This is a schematic diagram of a virtual patch of the chiral structure of the open-loop resonator used in the simulation of this embodiment;

[0038] Figure 5 This is a schematic diagram of the chiral structure of the open-loop resonator used in the simulation of this embodiment under periodic boundary conditions;

[0039] Figure 6 Comparison of bistatic radar cross sections of target cells under periodic boundary conditions obtained using the method proposed in this invention and obtained using the commercial software FEKO;

[0040] Figure 7 The figure shows the simulation results of the mode importance of the target element under periodic boundary conditions obtained using the method proposed in this invention.

[0041] Figure 8 The figure shows the simulation results of the reflection coefficient of the target element under periodic boundary conditions using the commercial software CST. Detailed Implementation

[0042] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0043] like Figure 1 As shown, a method for eigenmode analysis based on periodic Green's functions includes the following steps:

[0044] Step 1: Perform 3D modeling of the target element to obtain the element structure model, and set material properties, incident wave properties and periodic boundary conditions for the element structure model;

[0045] The specific operation process of step 1 is as follows:

[0046] Step 1-1: Use the electromagnetic modeling software ANSYS to perform three-dimensional modeling of the target element to obtain the element structure model;

[0047] Steps 1-2: Set the material properties of the unit structure model to perfect electrical conductor;

[0048] Steps 1-3: Set the incident wave properties, including the frequency of the incident wave during the solution; wherein, the incident wave is a uniform plane wave, and the electric field intensity and magnetic field intensity at each point in the wavefront of the uniform plane wave are equal, which facilitates the calculation;

[0049] Steps 1-4: Set periodic boundary conditions, including the length of the unit cell in the x-direction and the length in the y-direction, as well as the tilt angle of the unit cell relative to the XOY plane. The origin is the center point of the lower surface of the target object. The direction from the origin to the missing lower ring arm is the negative x-axis, and the direction from the origin to the center of the upper surface is the positive z-axis. The negative x-axis is rotated 90 degrees clockwise around the axis to become the positive y-axis. The unit cells are arranged in the XOY plane.

[0050] Step 2: Divide the unit structure model into triangular meshes, and construct RWG basis functions for each pair of triangles with a common edge. For the triangles divided on the surface of the unit structure model, identify each pair of triangles with a common edge. Number each pair of triangles sequentially from the outer edge to the inner edge. The RWG basis function is a function defined on the common edge of a pair of triangles, having a specific value only within adjacent triangles and zero in other locations. The constructed RWG basis functions are denoted as... , ,in, Indicates the first A pair of triangles, This represents the coordinate vector of the sampling points inside the triangular mesh. Indicates the first The length of the common side of a pair of triangles and They represent the first Two triangles in a pair of triangles, Represents a triangle area, Indicates from triangle The vector whose vertex points to the internal sampling point. Represents a triangle area, Indicates from triangle The vectors that point to the vertices within the sampling points of the vector, where a vertex is the vertex opposite to the common edge;

[0051] In the above definition of RWG basis functions, each vector represents a three-dimensional coordinate vector. The expression of RWG basis functions shows that current and magnetic current flow from a positive triangle to a negative triangle. In a triangle pair, if one triangle is a positive triangle, then the other triangle is a negative triangle.

[0052] Step 3: Based on the material properties, incident wave properties, and periodic boundary conditions, obtain the periodic Green's function of the target element, and denote the periodic Green's function as follows: , in, Indicates the field point. Indicates the source point, This represents the number of unit cells in the x-direction under periodic boundary conditions. This represents the number of unit cells in the y-direction under periodic boundary conditions. Represents the imaginary unit. This represents the distance between the field point and the source point. Indicates the incident wave number. , Indicates the frequency of the incident wave. Represents the speed of light. Represents the dielectric constant of an object. This represents the magnetic permeability of the object. Because the material properties of the unit cell model are perfect electrical conductors, therefore... , , , This represents the x-coordinate of each cell in the array. This represents the distance between unit cells in the x-direction under periodic boundary conditions. This represents the y-coordinate of each cell in the array. This represents the distance between unit cells in the y-direction under periodic boundary conditions. and These two parameters represent the incident angles of the wave. In spherical coordinates, the angle of an incident wave is uniquely determined by these two parameters.

[0053] Step 4: Use the EWALD transform to decompose the periodic Green's function, obtaining the spatial and spectral domain parts and their corresponding gradients. The spatial part is denoted as... , ,in, This represents the complement error function. , indicating the starting term of the cleavage. Represents the area of ​​the unit cell. , This indicates the tilt angle of the unit cell relative to the XOY plane;

[0054] Let the gradient of the spatial region be denoted as , ,in, Indicates positive or negative sign;

[0055] The spectral domain portion is denoted as , ,in, , This represents the displacement vector from the field point to the source point. , This represents the component of the displacement vector from the field point to the source point along the z-axis.

[0056] The gradient in the spectral domain is denoted as... , ,in, , , , , , These represent the unit vectors of the x-axis, y-axis, and z-axis, respectively.

[0057] Step 5: Based on the RWG basis functions constructed in Step 2 and the spatial and spectral domain parts obtained in Step 4, along with the corresponding gradients, construct the impedance matrix of the target element under periodic boundary conditions using a hybrid integral equation. The impedance matrix is ​​denoted as... , ,in, Represents the wave impedance in free space. , , This indicates taking the inner product. Let represent the RWG trial function in the M-th triangle pair. Since Galerkin's method is used, the specific formula of the trial function is the same as that of the RWG basis functions. , Represents the RWG basis functions in the Nth triangle. Represents the imaginary unit. Indicates the incident wave number. This represents the area of ​​the triangle containing the field point. Represents the basis functions of RWG Take the divergence, The inner product is obtained Indicates the trial function Take the divergence, , This represents the area of ​​the triangle containing the source point. This represents the unit outward normal vector of the target element surface. , ;

[0058] Step 6: Decompose the impedance matrix to obtain the Emeryst matrix, and construct the generalized eigenvalue equation based on the Emeryst matrix;

[0059] The specific operation process for step 6 is as follows:

[0060] Step 6-1: Convert the impedance matrix Decomposed into an Emyer matrix, its expression is: ,in, Denotes the real part of the Emilt matrix. Denotes the imaginary part of the Emilt matrix. Representing the impedance matrix The conjugate transpose of;

[0061] Step 6-2: Construct the generalized eigenvalue equation based on the Emilt matrix, its expression is: ,in, Represents eigenvalues. Indicates the mode current.

[0062] Step 7: Based on the generalized eigenvalue equation, solve for the eigenvalues ​​of the target element under periodic boundary conditions, and obtain the mode importance of the target element according to the eigenvalues ​​to complete the analysis of the eigenmodes;

[0063] The specific operation process for step 7 is as follows:

[0064] Step 7-1: Based on the generalized eigenvalue equation, solve for the eigenvalues ​​of the target element under periodic boundary conditions. , ,in, Represents mode current The conjugate transpose of;

[0065] Step 7-2: Obtain the mode importance of the target unit under periodic boundary conditions based on eigenvalues. MS , ,in, By solving for eigenvalues ​​and calculating mode importance, the contribution of different modes to the periodic structure can be accurately assessed, providing crucial guidance for antenna design and optimization.

[0066] An application of a characteristic mode analysis method based on periodic Green's function is presented. The resonant frequency and resonant mode are found according to the distribution of mode importance. Then, an effective feeding structure is designed by the current distribution of the obtained resonant mode. The resonant frequency when MS is 1 is the resonant point, and the mode it represents is the resonant mode. The effective manifestation is that the impedance is matched with the system and aligned with the current direction by tuning.

[0067] The following is a practical example using the method proposed in this invention to illustrate the relevant results. The eigenmode analysis method based on the periodic Green's function proposed in this invention is implemented on the periodic Green's function algorithm and is named PGF.

[0068] This practical example uses, for example Figure 2 The periodic unit structure shown is a two-layer chiral resonator structure, with both layers being perfect conductors. In steps 1-4, it is set to extend infinitely in both the +x and +y directions. The extension distance should be greater than the length of the unit structure in the x and y directions, i.e., the corresponding side length, to avoid current discontinuities at the boundaries. This characterizes the ability of several extended periodic unit structures to form an infinitely large periodic structure in an array. Currently, this type of periodic structure array, composed of units of the same material and shape arranged according to a specific rule, has wide applications in electromagnetic fields and microwave technologies such as frequency-selective surfaces. Therefore, the characteristic mode analysis of periodic structures in this embodiment of the invention has high application value. Figure 3 As shown, each triangular mesh has material properties, which are the material properties set in steps 1-2. In this embodiment, the dielectric constant used is 1, that is, a perfect electrical conductor.

[0069] like Figure 4 The diagram provides a more intuitive view of how the subdivision is performed and how the virtual tiles are combined.

[0070] like Figure 5 The diagram illustrates how the unit is expanded into a periodic structure.

[0071] like Figure 6 As shown, the vertical axis represents the bistatic radar cross section in dBsm, and the horizontal axis represents the angle. The mesh is divided according to the wavelength corresponding to 1 / 10 of the highest frequency. The bistatic radar cross section values ​​and distribution angles obtained by the method (PGF) described in this invention are consistent with the results obtained by commercial software (FEKO), which proves that the impedance matrix of the target element under periodic conditions can be accurately calculated using the periodic Green's function. This is a prerequisite for calculating the characteristic mode.

[0072] Furthermore, in terms of computational efficiency: at 150 terahertz, the computation time of the impedance matrix is ​​reduced from 213 minutes to 116 minutes after using the method proposed in this invention, which is an improvement of 45.5%; at 350 terahertz, the computation time of the impedance matrix is ​​reduced from 278 minutes to 157 minutes after using the method proposed in this invention, which is an improvement of 43.5%.

[0073] like Figure 7 As shown, the vertical axis represents the importance of the mode, and the horizontal axis represents the frequency. After using the method proposed in this invention, resonant points appear in mode two, mode three, and mode four, respectively, with corresponding frequencies of 245 terahertz, 285 terahertz, and 350 terahertz. Figure 8As shown, the vertical axis represents the S-parameters, and the horizontal axis represents the frequency. After using the commercial software CST, the resonant points were observed through the s21-x polarization and s21-y polarization curves. It was found that the frequencies of the resonant points were 245 terahertz, 285 terahertz, and 350 terahertz, respectively. It can be seen that the resonant points of the two can match, which verifies the feasibility of the method proposed in this invention.

[0074] Definitions of terms in this patent:

[0075] The EWALD transform is a mathematical method for efficiently calculating the Coulomb potential under periodic boundary conditions. It significantly improves the convergence speed by decomposing long-range interactions into real space and reciprocal space.

Claims

1. A method for eigenmode analysis based on periodic Green's functions, characterized in that... Includes the following steps: Step 1: Perform 3D modeling of the target element to obtain the element structure model, and set material properties, incident wave properties and periodic boundary conditions for the element structure model; Step 2: Divide the unit structure model into triangular meshes, and construct RWG basis functions for each pair of triangles with a common edge. Step 3: Obtain the periodic Green's function of the target element based on the material properties, incident wave properties, and periodic boundary conditions; Step 4: Use the EWALD transform to decompose the periodic Green's function to obtain the spatial and spectral parts and the corresponding gradients. In step 4, the airspace portion is denoted as , ,in, This represents the number of unit cells in the x-direction under periodic boundary conditions. This represents the number of unit cells in the y-direction under periodic boundary conditions. This represents the distance between the field point and the source point. Represents the imaginary unit. Indicates the incident wave number. , Indicates the frequency of the incident wave. Represents the speed of light. Represents the dielectric constant of an object. This represents the magnetic permeability of the object. Because the material properties of the unit cell model are perfect electrical conductors, therefore... , and These represent the incident angles of the incident waves, This represents the distance between unit cells in the x-direction under periodic boundary conditions. This represents the distance between unit cells in the y-direction under periodic boundary conditions. , This represents the complement error function. , indicates the starting term of the cleavage. Represents the area of ​​the unit cell. , This indicates the tilt angle of the unit cell relative to the XOY plane; Let the gradient of the spatial region be denoted as , ,in, Indicates positive or negative sign; The spectral domain portion is denoted as , ,in, , This represents the displacement vector from the field point to the source point. , This represents the component of the displacement vector from the field point to the source point along the z-axis. The gradient in the spectral domain is denoted as... , ,in, , , , , , These represent the unit vectors of the x-axis, y-axis, and z-axis, respectively. Step 5: Based on the RWG basis functions constructed in Step 2 and the spatial and spectral parts obtained in Step 4, as well as the corresponding gradients, construct the impedance matrix of the target element under periodic boundary conditions through a hybrid integral equation. Step 6: Decompose the impedance matrix to obtain the Emeryst matrix, and construct the generalized eigenvalue equation based on the Emeryst matrix; Step 7: Based on the generalized eigenvalue equation, solve for the eigenvalues ​​of the target element under periodic boundary conditions, and obtain the mode importance of the target element according to the eigenvalues ​​to complete the analysis of the eigenmode.

2. The characteristic modulus analysis method based on a periodic Green's function according to claim 1, characterized in that... The specific operation process of step 1 is as follows: Step 1-1: Use the electromagnetic modeling software ANSYS to perform three-dimensional modeling of the target element to obtain the element structure model; Steps 1-2: Set the material properties of the unit structure model to perfect electrical conductor; Steps 1-3: Set the incident wave properties, including the frequency of the incident wave during the solution process; Steps 1-4: Set periodic boundary conditions, including the length of the unit cell in the x-direction and the length in the y-direction, as well as the tilt angle of the unit cell relative to the XOY plane.

3. The characteristic modulus analysis method based on a periodic Green's function according to claim 1, characterized in that... In step 2, the constructed RWG basis functions are denoted as... , ,in, Indicates the first A pair of triangles, This represents the coordinate vector of the sampling points inside the triangular mesh. Indicates the first The length of the common side of a pair of triangles and They represent the first Two triangles in a pair of triangles, Represents a triangle area, Indicates from triangle The vector whose vertex points to the internal sampling point. Represents a triangle area, Indicates from triangle The vector of the internal sampling point in the vector points to the vertex, where the vertex is the vertex opposite to the common edge.

4. The characteristic modulus analysis method based on a periodic Green's function according to claim 3, characterized in that... In step 3, the periodic Green's function is denoted as... , in, Indicates the field point. Indicates the source point, This represents the number of unit cells in the x-direction under periodic boundary conditions. This represents the number of unit cells in the y-direction under periodic boundary conditions. Represents the imaginary unit. This represents the distance between the field point and the source point. Indicates the incident wave number. , Indicates the frequency of the incident wave. Represents the speed of light. Represents the dielectric constant of an object. This represents the magnetic permeability of the object. Because the material properties of the unit cell model are perfect electrical conductors, therefore... , , , This represents the x-coordinate of each cell in the array. This represents the distance between unit cells in the x-direction under periodic boundary conditions. This represents the y-coordinate of each cell in the array. This represents the distance between unit cells in the y-direction under periodic boundary conditions. and These represent the incident angles of the incident waves.

5. The characteristic modulus analysis method based on a periodic Green's function according to claim 4, characterized in that... In step 5, the impedance matrix is ​​denoted as... , ,in, Represents the wave impedance in free space. , , This indicates taking the inner product. Let represent the RWG trial function in the M-th triangle pair. Since Galerkin's method is used, the specific formula of the trial function is the same as that of the RWG basis functions. , Describe the RWG basis functions in the Nth triangle. Represents the imaginary unit. Indicates the incident wave number. This represents the area of ​​the triangle containing the field point. Represents the basis functions of RWG Take the divergence, The inner product is obtained Indicates the trial function Take the divergence, , This represents the area of ​​the triangle containing the source point. This represents the unit outward normal vector of the target element surface. , .

6. The characteristic modulus analysis method based on a periodic Green's function according to claim 5, characterized in that... The specific operation process for step 6 is as follows: Step 6-1: Convert the impedance matrix Decomposed into an Emilt matrix, its expression is: ,in, Denotes the real part of the Emilt matrix. Denotes the imaginary part of the Emilt matrix. Representing the impedance matrix The conjugate transpose of; Step 6-2: Construct the generalized eigenvalue equation based on the Emilt matrix, its expression is: ,in, Represents eigenvalues. Indicates the mode current.

7. The characteristic modulus analysis method based on a periodic Green's function according to claim 6, characterized in that... The specific operation process for step 7 is as follows: Step 7-1: Based on the generalized eigenvalue equation, solve for the eigenvalues ​​of the target element under periodic boundary conditions. , ,in, Represents mode current The conjugate transpose of; Step 7-2: Obtain the mode importance of the target unit under periodic boundary conditions based on eigenvalues. MS , ,in, .

8. The application of the characteristic modulus analysis method based on periodic Green's function as described in claim 1, characterized in that... The resonant frequency and resonant mode are found based on the distribution of mode importance, and then an effective feeding structure is designed based on the current distribution of the obtained resonant mode.

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