A general calculation method for infinite periodic structures based on the method of moments

Abstract

This invention discloses a general calculation method for infinitely large periodic structures based on the method of moments (MoM), comprising: obtaining a periodic structural unit model based on the MoM, and then performing triangular meshing; constructing RWG basis functions for each pair of triangles; discretizing the surface current and magnetic current using the RWG basis functions; labeling the outward region of the periodic structural unit as periodic space and the other regions as free space; constructing the solution formulas for the free space Green's function and the periodic Green's function; introducing virtual patch technology for each target triangle within the periodic structural unit, and constructing a new set of RWG basis functions using the target triangle and its corresponding updated adjacent target triangles; filling the impedance matrix and excitation vector based on all RWG basis functions; solving for the surface current and magnetic current of the periodic structural unit and converting them into electric and magnetic fields to calculate the scattering characteristics. This invention can solve the problem of solving for electromagnetic parameters when there are open metal surfaces at the connection points of periodic structures.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation technology, specifically relating to a general calculation method for infinitely large periodic structures based on the method of moments. Background Technology

[0002] Periodic structure arrays are arrays composed of units of the same material and shape arranged according to a specific pattern. They have wide applications in electromagnetic fields and microwave technology, such as frequency-selective surfaces and phased array antennas. Accurately solving the electromagnetic scattering of periodic objects is of great significance in various applications.

[0003] Currently, there is considerable research on algorithms for composite element bi-periodic structures, such as differential methods based on FEM (Finite Element Method), FDTD (Finite-Difference Time-Domain) methods, and hybrid methods based on FE-BI (Finite Element Boundary Integral) schemes. To reduce the number of unknowns, methods based on surface integral equations are one of the better choices.

[0004] The problem of infinitely large periodic structures refers to the electromagnetic properties exhibited by periodic structural elements throughout an infinitely expanding periodic structure array. In existing technologies, there are two solutions for solving the problem of infinitely large electromagnetic periodic structures using the method of moments (MoM): one is based on the traditional spatial domain MoM. This solution cannot handle the problem of different material properties and meshes at the connections of the periodic structures. For cases where there is open metal at the interfaces of the periodic structure connections, the traditional spatial domain MoM addresses this issue by introducing modified basis functions, but this increases the complexity of the problem. The other solution is the periodic connection region modeling method PCRM. This method uses periodic Green's functions within the periodic structural elements and free-space Green's functions outside the periodic structural elements, reducing the solution time of the periodic Green's functions and compensating for the shortcomings of the traditional spatial domain MoM. However, this method does not consider the current continuity problem on the open metal at the interfaces of the periodic structural elements, and therefore cannot handle the problem of open metal surfaces at the interfaces.

[0005] Therefore, how to improve current periodic connectivity modeling methods to make the calculation of infinitely large periodic structures more universal is an urgent problem to be solved in this field. Summary of the Invention

[0006] To address the aforementioned problems in the existing technology, this invention provides a general calculation method for infinitely large periodic structures based on the method of moments. The technical problem to be solved by this invention is achieved through the following technical solution:

[0007] A general calculation method for infinitely large periodic structures based on the method of moments includes:

[0008] Based on the method of moments, a three-dimensional model of the periodic structural unit is obtained. Material properties and incident wave properties are set for the periodic structural unit model, and the periodic structural unit is set to expand in an edge-fitting manner.

[0009] The obtained periodic structural unit model is divided into triangular meshes; RWG basis functions are constructed for each pair of triangles with common edges obtained by the meshing; the surface current and surface magnetic current of the periodic structural unit are discretized using all the constructed RWG basis functions;

[0010] The region facing outwards from the periodic structural unit is labeled as the periodic space, and the other regions are labeled as free space. A solution expression for the free space Green's function is constructed for the free space, and a solution expression for the periodic Green's function is constructed for the periodic space.

[0011] For each target triangle within the periodic structural unit, a matching triangle within the periodic structural unit is copied to the position of the target neighboring triangle corresponding to the target triangle by introducing virtual patching (PFP) technology, which serves as the updated target neighboring triangle. A new set of RWG basis functions is constructed using the target triangle and the corresponding updated target neighboring triangle. The target neighboring triangle corresponding to any target triangle is located within the adjacent periodic structural unit of the periodic structural unit.

[0012] Based on all the obtained RWG basis functions, fill in the impedance matrix and excitation vector;

[0013] Based on the impedance matrix, the excitation vector, and the discretized representation of the surface current and surface magnetic current of the periodic structural unit, the surface current and surface magnetic current of the periodic structural unit are solved.

[0014] The surface current and surface magnetic current of the periodic structural unit are converted into electric and magnetic fields, and the scattering characteristics are calculated from the obtained electric and magnetic fields.

[0015] In one embodiment of the present invention, the step of performing three-dimensional modeling of periodic structural elements based on the method of moments to obtain a periodic structural element model, setting material properties and incident wave properties for the periodic structural element model, and setting the periodic structural elements to expand in an edge-fitting manner includes:

[0016] The periodic structural unit is modeled in three dimensions using electromagnetic modeling software based on the method of moments, thus obtaining the model of the periodic structural unit.

[0017] Material properties are set for the dielectric and metallic materials in the periodic structural unit model, wherein the material properties include dielectric constant and magnetic permeability;

[0018] The incident wave is set to a plane wave, and its frequency f is set; the incident electric field is set to E. inc The incident magnetic field is H inc ;

[0019] The expansion direction of the periodic structural unit is set, and the expansion distance is set as the side length of the periodic structural unit in the expansion direction, so that the periodic structural unit expands along the expansion direction in an edge-fitting manner.

[0020] In one embodiment of the present invention, the RWG basis function constructed for each pair of triangles with a common edge obtained from the subdivision is expressed as:

[0021]

[0022] Where n represents the number of the triangle pair; l n Let represent the side length of the common side of the triangle pair; A represent the area of ​​the triangle; r represent the coordinate vector of the sampling point inside the triangle mesh; ρ represent the position vector from the vertex corresponding to the common side of the triangle pair to the sampling point r inside; ± correspond to the two triangles in the triangle pair, respectively denoted as . and

[0023] In one embodiment of the present invention, the discretization representation of the surface current and surface magnetic current of the periodic structural unit using all constructed RWG basis functions includes:

[0024] Using all the constructed RWG basis functions, the surface current and surface magnetic current of the periodic structural unit are discretized based on the preset current discretization formula and the preset magnetic current discretization formula.

[0025] The preset current discretization formula is as follows:

[0026]

[0027] The preset magnetic flux discretization formula is as follows:

[0028]

[0029] Where η0 represents the free-space wave impedance; N represents the total number of triangle pairs obtained after triangular mesh partitioning; α n and β nLet n represent the current weighting coefficient and magnetocurrent weighting coefficient corresponding to the triangle pair numbered n, respectively. The current weighting coefficient and magnetocurrent weighting coefficient corresponding to each triangle pair are unknowns to be determined.

[0030] In one embodiment of the present invention, constructing a solution expression for the free space Green's function for the free space includes:

[0031] Using the medium parameters of the free space, the expression for the free space Green's function can be constructed as follows:

[0032]

[0033] Where G0 represents the Green's function in free space; j represents the imaginary unit; k represents the wavenumber in free space; the coordinate vector r of the sampling points inside the triangular mesh represents the position of the field point, and r' represents the position of the source point.

[0034] In one embodiment of the present invention, constructing a solution expression for the periodic Green's function for the periodic space includes:

[0035] Using the medium parameters of the periodic space, the solution expression for the periodic Green's function is constructed for the periodic space as follows:

[0036]

[0037] Among them, G prd Represents a periodic Green's function; ρ represents the gradient operator; β represents the propagation constant; δ represents the impulse function; ba ρ is the distance between the a-th periodic structural unit and the b-th periodic structural unit. ba Represents ρ ba The modulus.

[0038] In one embodiment of the present invention, the formula used for filling the impedance matrix is:

[0039]

[0040] Where h and u represent the indices of the h-th and u-th triangle pairs corresponding to all obtained RWG basis functions, including the new RWG basis function; J represents the surface current; M represents the surface magnetic current; and the impedance matrix... Z JJ Represents the current-current impedance matrix; Z JM Z represents the current-magnetic impedance matrix; MJ Represents the magnetocurrent-current impedance matrix; Z MM Z represents the magnetocurrent-magnetic impedance matrix; JJ Z JMZ MJ Z MM The element in the h-th row and u-th column is represented as Z. JJ,hu Z JM,hu Z MJ,hu Z MM,hu ;f h (r) and f u (r) represents the RWG basis functions of the h-th and u-th triangle pairs, respectively; η1 and η2 represent the wave impedance of the periodic space and the free space, respectively; L and K represent two operators, respectively; the subscripts 1 and 2 in η1, η2, L1, L2, K1, and K2 represent the periodic space and the free space, respectively;

[0041]

[0042]

[0043]

[0044]

[0045] Where f(r) represents the RWG basis function corresponding to the current sampling point r; G0(r) represents the free space Green's function, G prd (r) represents the Green's function in a periodic space.

[0046] In one embodiment of the present invention, the formula used to fill the excitation vector is:

[0047]

[0048] Wherein, the excitation vector V = [V J V M ], V J V represents the current excitation vector. M V represents the magnetocurrent excitation vector; J This includes the current excitation for each triangle pair, where the current excitation for the h-th triangle pair is in the form of V. J,h Indicates; V M This includes the magnetofluid excitation for each triangle pair, where the magnetofluid excitation for the h-th triangle pair is V. M,h Indicates; E inc H represents the incident electric field; inc This indicates the incident magnetic field.

[0049] In one embodiment of the present invention, the surface current and surface magnetic current of the periodic structural unit are solved based on the impedance matrix, the excitation vector, and the discretized representation of the surface current and surface magnetic current of the periodic structural unit, including:

[0050] Based on the impedance matrix Z and the excitation vector V, the impedance matrix equation is generated as follows:

[0051]

[0052] Among them, I d I represents the current weighting coefficient vector. d ={α1,α2,α3,…,α N1};M d M represents the magnetic flux weighting coefficient vector. d ={β1,β2,β3,…,β N1}; N1 represents the total number of triangle pairs corresponding to all obtained RWG basis functions;

[0053] The current weighting coefficient vector and the magnetic current weighting coefficient vector are solved from the impedance matrix equation;

[0054] The surface current J(r) of the periodic structural unit is solved using the solved current weighting coefficient vector and the preset current discretization formula.

[0055] The surface magnetic current M(r) of the periodic structural unit is solved by using the solved magnetic current weighting coefficient vector and the preset magnetic current discretization formula.

[0056] In one embodiment of the present invention, the surface current and surface magnetic current of the periodic structural unit are correspondingly converted into electric and magnetic fields, including:

[0057] The surface current J(r) and surface magnetic current M(r) of the periodic structural unit are converted into electric and magnetic fields using a preset field calculation formula; wherein,

[0058] The preset field calculation formula includes:

[0059]

[0060]

[0061] Where the subscript 'type' indicates the type of medium space, and when 'r' is in free space, G... type (r,r')=G0(r,r'); when r is in a periodic space, G type (r,r')=G prd (r,r').

[0062] The beneficial effects of this invention are:

[0063] This invention provides a general calculation method for infinitely large periodic structures based on the method of moments (MoM). First, preprocessing is performed using MoM to create a 3D model of the periodic structural unit, obtaining a model of the periodic structural unit. Material properties and incident wave properties are set, and the periodic structural unit is expanded in an edge-fitting manner, thus enabling the representation of an infinitely large periodic structure in the form of an array. Second, triangular meshing is performed, and the current and magnetic current on the surface of the periodic structural unit are discretized using RWG basis functions defined on each pair of triangles. Third, the outward-facing region of the periodic structural unit is marked as periodic space, and other regions are marked as free space. Free-space Green's functions are used in free space, and periodic Green's functions are used in periodic space, thus ensuring the integrity of the unit during calculation. Interfaces at connection points are also considered. Material properties and mesh generation are no longer restricted. Next, when open metal surfaces exist at the junctions of periodic structures, virtual patch technology is used to form new RWG basis functions on the metal surfaces at the junctions of periodic structures, ensuring the continuity of current at the metal surface junctions. This allows for the use of a method based on modeling periodic connection regions to solve more general periodic structure problems. Then, based on all the obtained RWG basis functions, the impedance matrix and excitation vector are filled. Subsequently, based on the impedance matrix, the excitation vector, and the discretized representations of the surface current and surface magnetic current of the periodic structural unit, the surface current and surface magnetic current of the periodic structural unit are solved. Finally, the surface current and surface magnetic current of the periodic structural unit are converted into electric and magnetic fields, and the scattering characteristics are calculated from the obtained electric and magnetic fields.

[0064] This invention provides a general treatment for connected and disconnected periodic structures, reducing the number of interpolation tables compared to existing techniques. Furthermore, it ensures that meshing of periodic structural elements does not require maintaining consistent meshing on both sides of the periodic boundary, thus lowering the requirements for mesh generation. When adjacent portions of contacting periodic elements belong to different media, existing techniques cannot easily remove the interface at the connection point, increasing the difficulty of modeling complex periodic structures. This invention offers a simple and flexible approach to analyzing such problems. Moreover, addressing the limitation of existing techniques that can only solve periodic structures with the same material properties and meshing at the connection points, and cannot handle cases where open metal exists at the connection points, this invention introduces virtual patch technology at the interface for processing and recalculation. This solves the problem of solving electromagnetic parameters when there is an open metal surface at the interface of the periodic structure connection, ensuring the continuity of the interface current at the open metal connection and making the periodic structure modeling method more universal. Attached Figure Description

[0065] Figure 1 This is a flowchart illustrating a general calculation method for infinitely large periodic structures based on the method of moments provided in an embodiment of the present invention.

[0066] Figure 2 These are the multi-layer periodic structure unit model diagram and mesh model diagram used in the simulation of this invention embodiment;

[0067] Figure 3 This is a schematic diagram of the virtual patch technology PFP according to an embodiment of the present invention;

[0068] Figure 4 This is a simulation result of the amplitude and phase of the reflection coefficient of the multilayer periodic structure unit obtained in the embodiment of the present invention. Detailed Implementation

[0069] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0070] This invention provides a general calculation method for infinitely large periodic structures based on the method of moments, such as... Figure 1 As shown, the method may include the following steps:

[0071] S1. Based on the method of moments, a three-dimensional model of the periodic structural unit is obtained. Material properties and incident wave properties are set for the periodic structural unit model, and the periodic structural unit is set to expand in an edge-fitting manner.

[0072] Specifically, S1 may include the following steps:

[0073] S11. Using electromagnetic modeling software based on the method of moments, the periodic structural unit is modeled in three dimensions to obtain the model of the periodic structural unit.

[0074] The electromagnetic modeling software can be GID software, etc., and no specific restrictions are made here.

[0075] S12, set material properties for the dielectric material and metallic material in the periodic structural unit model, wherein the material properties include dielectric constant and magnetic permeability;

[0076] For ease of understanding, to Figure 2 The following explanation uses a three-dimensional model of the periodic structural unit shown as an example. Figure 2 The left image shows the structure of a periodic structural unit model, which includes three dielectric plates from top to bottom and an open rectangular metal sheet. Different colored dielectric plates represent different relative permittivity ε. r The dielectric material, the relative permittivity of the dielectric plate is all within Figure 2The relative permeability μ of all dielectric plates is indicated. r All are 1. Each substrate has an open metal sheet of a different shape attached, as shown in the yellow area of ​​the three substrates. Figure 2 The bottom of the diagram shows a coordinate system, where red, green, and blue represent x, y, and z, respectively. It can be understood that the dielectric constant ε is the relative dielectric constant ε. r The product of the absolute permittivity ε0 in vacuum and the permeability μ. The permeability μ is the relative permeability μ. r The product of the vacuum permeability μ0 and the permeability of vacuum.

[0077] S13, Set the incident wave for the solution to a plane wave, and set the frequency f of the incident wave; set the incident electric field to E. inc The incident magnetic field is H inc ;

[0078] This step involves setting the incident wave properties, specifically including whether the incident wave is a plane wave, the frequency of the incident wave, and marking the incident electric field E. inc Incident magnetic field H inc .

[0079] S14, set the expansion direction of the periodic structural unit, and set the expansion distance as the side length of the periodic structural unit in the expansion direction, so that the periodic structural unit expands along the expansion direction in an edge-fitting manner.

[0080] against Figure 2 The periodic structural unit shown is set to extend infinitely in the +x and +y directions. The extension distance is the length of the periodic structural unit in the x and y directions, that is, the corresponding side length, so that the extended periodic structural units can fit together with each other, so as to represent that several extended periodic structural units can form an infinitely large periodic structure in the form of an array.

[0081] 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 field and microwave technology, such as frequency selective surfaces. Therefore, the general calculation method proposed in this invention for infinitely large periodic structures has high application value.

[0082] S2, perform triangular meshing on the obtained periodic structural unit model; construct RWG basis functions for each pair of triangles with common edges obtained from the meshing; use all the constructed RWG basis functions to discretize and represent the surface current and surface magnetic current of the periodic structural unit.

[0083] Specifically, a mesh generator can be used to perform triangular meshing on the periodic structural element model, generating triangular meshes for the periodic structural elements. (See [link to documentation]). Figure 2As can be understood from the right figure, each triangular mesh has material properties, which are the material properties of the dielectric material and the open metal sheet in S12. The material properties include two parameters: dielectric constant ε and magnetic permeability μ.

[0084] For the triangular mesh generated by the periodic structural unit model, pairs of triangles sharing a common edge can be identified, and these pairs are assigned numbers according to a specific order. Based on the triangular mesh generated by the meshing, the surface current and surface magnetic current of the periodic structural unit are discretized using RWG basis functions defined on a pair of triangles. The RWG basis functions are functions defined on the common edge of a pair of triangles, having specific values ​​only within adjacent triangles and zero elsewhere.

[0085] Specifically, for each pair of triangles with a common edge obtained from the partitioning, the constructed RWG basis function is expressed as:

[0086]

[0087] Where n represents the number of the triangle pair; l n Let A represent the side length of the common side of the triangle pair; let A represent the area of ​​the triangle; let r represent the coordinate vector of the sampling point inside the triangle mesh. Specifically, a sampling point is taken inside the triangle, and r represents the three-dimensional coordinate vector (x, y, z) of this sampling point. Generally, the sampling point represented by r is connected to the three vertices of the triangle, thus trisecting the area of ​​the triangle. Let ρ represent the position vector from the vertex corresponding to the common side of the triangle pair to the internal sampling point r; ± correspond to the two triangles in the triangle pair, respectively denoted as T. n + and T n - .

[0088] In the above definition of RWG basis functions, each vector represents a three-dimensional coordinate vector. The expression of RWG basis functions states that current and magnetic current flow from a positive triangle to a negative triangle, where if one triangle in a triangle pair is a positive triangle, then the other triangle is a negative triangle.

[0089] In this embodiment of the invention, the discretization of the surface current and surface magnetic current of the periodic structural unit using all constructed RWG basis functions may include:

[0090] Using all the constructed RWG basis functions, the surface current and surface magnetic current of the periodic structural unit are discretized based on the preset current discretization formula and the preset magnetic current discretization formula.

[0091] The preset current discretization formula is as follows:

[0092]

[0093] The preset magnetic flux discretization formula is as follows:

[0094]

[0095] Where η0 represents the free-space wave impedance, with a value of 377 ohms; N represents the total number of triangle pairs obtained after triangular mesh partitioning; α n and β n Let n represent the current weighting coefficient and magnetocurrent weighting coefficient corresponding to the triangle pair numbered n, respectively. The current weighting coefficient and magnetocurrent weighting coefficient corresponding to each triangle pair are unknowns to be determined.

[0096] S3, the region facing outward of the periodic structural unit is marked as periodic space, and the other regions are marked as free space. A solution expression for the free space Green's function is constructed for the free space, and a solution expression for the periodic Green's function is constructed for the periodic space.

[0097] The expression for solving the Green's function in the free space includes:

[0098] Using the medium parameters of the free space, the expression for the free space Green's function can be constructed as follows:

[0099]

[0100] Where G0 represents the Green's function in free space; j represents the imaginary unit; and k represents the wave number in free space. Where w = 2πf, f represents the frequency, ε represents the dielectric constant, and μ represents the permeability; the coordinate vector r of the sampling points inside the triangular grid represents the position of the field point, and r' represents the position of the source point.

[0101] The solution expression for the periodic Green's function is constructed for the periodic space, including:

[0102] Using the medium parameters of the periodic space, the solution expression for the periodic Green's function is constructed for the periodic space as follows:

[0103]

[0104] Among them, G prd Represents a periodic Green's function; ρ represents the gradient operator; β represents the propagation constant; δ represents the impulse function; ba ρ is the distance between the a-th periodic structural unit and the b-th periodic structural unit. ba Represents ρ ba The modulus.

[0105] The present invention relates to a method called PCRM based on periodic connected regions, which uses free-space Green's functions in free space and periodic Green's functions inside the elements. This ensures the integrity of the periodic structural elements during computation and removes restrictions on material properties and meshing at the interfaces of the connections.

[0106] S4, for each target triangle within the periodic structural unit, by introducing virtual patching (PFP) technology, a matching triangle within the periodic structural unit is copied to the position of the target neighboring triangle corresponding to the target triangle, which is then used as the updated target neighboring triangle; a new set of RWG basis functions is constructed using the target triangle and the corresponding updated target neighboring triangle; wherein, the target neighboring triangle corresponding to any target triangle is located within the adjacent periodic structural unit of the periodic structural unit.

[0107] This invention addresses the issue of open metal in the connection regions of periodic structural units by introducing Virtual Patch (PFP) technology. For details, please refer to [link to relevant documentation]. Figure 3 Understanding that there are two types of triangles on the open metal surface of the periodic structural unit connection region: the first type is completely inside the metal surface, such as... Figure 3 triangle T p and T q The second type is contact with the metal surface of other periodic structural units, such as... Figure 3 triangle T m It has a target adjacent triangle (see Figure 3 The red triangle T in the middle n ') is located in triangle T m Within the adjacent periodic structural units of the periodic structural unit, triangle T m This is the target triangle, which shares a common edge with its corresponding adjacent target triangle. In this embodiment of the invention, for the second case, triangle T... m A matching triangle T within the periodic structural unit n Copy to triangle T n At point ', as the adjacent triangle of the updated target, then T m The updated target triangle forms a new triangle pair, and the new triangle pair can be used to construct a new set of RWG basis functions according to formula (1) in step S2.

[0108] Therefore, it can be understood that for each target triangle within the periodic structural unit, the above processing can yield a new pair of triangles and a new set of RWG basis functions.

[0109] The newly obtained triangle pairs in S4 can be merged with the triangle pairs obtained after the original triangle mesh subdivision, and each merged triangle pair is assigned a corresponding triangle pair number. Similarly, the newly obtained RWG basis functions in S4 and the RWG basis functions obtained in S2 are merged to form all the obtained RWG basis functions.

[0110] In this embodiment of the invention, when there is an open metal surface at the connection of a periodic structure, a virtual patch technique is used to form a new RWG basis function on the metal surface at the junction of the periodic structure. This can ensure the continuity of the current at the junction of the metal surfaces and can solve more general periodic structure problems by using a method based on modeling the periodic connection region.

[0111] S5, based on all the obtained RWG basis functions, fill in the impedance matrix and excitation vector;

[0112] In this embodiment of the invention, the formula used for filling the impedance matrix is:

[0113]

[0114] Where h and u represent the indices of the h-th and u-th triangle pairs corresponding to all obtained RWG basis functions, including the new RWG basis function; J represents the surface current; M represents the surface magnetic current; and the impedance matrix... Z JJ Represents the current-current impedance matrix; Z JM Z represents the current-magnetic impedance matrix; MJ Represents the magnetocurrent-current impedance matrix; Z MM Z represents the magnetocurrent-magnetic impedance matrix; JJ Z JM Z MJ Z MM The element in the h-th row and u-th column is represented as Z. JJ,hu Z JM,hu Z MJ,hu Z MM,hu ;f h (r) and f u (r) represents the RWG basis functions of the h-th and u-th triangle pairs, respectively; η1 and η2 represent the wave impedance of the periodic space and the free space, respectively; L and K represent two operators, respectively; the subscripts 1 and 2 in η1, η2, L1, L2, K1, and K2 represent the periodic space and the free space, respectively;

[0115]

[0116]

[0117]

[0118]

[0119] Where f(r) represents the RWG basis function corresponding to the current sampling point r; G0(r) represents the free space Green's function, G prd (r) represents the Green's function in a periodic space.

[0120] Through the above processing, the impedance matrix can be obtained. As will be understood by those skilled in the art, the impedance matrix Z represents the discretized matrix form of the integral equation.

[0121] In this embodiment of the invention, the formula used to fill the excitation vector is:

[0122]

[0123] Wherein, the excitation vector V = [V J V M ], V J V represents the current excitation vector. M V represents the magnetocurrent excitation vector; J This includes the current excitation for each triangle pair, where the current excitation for the h-th triangle pair is in the form of V. J,h Indicates; V M This includes the magnetofluid excitation for each triangle pair, where the magnetofluid excitation for the h-th triangle pair is V. M,h Indicates; E inc H represents the incident electric field; inc This indicates the incident magnetic field.

[0124] Through the above processing, the excitation vector V = [V J V M This facilitates the subsequent solution of the surface current and surface magnetic current of the periodic structural unit.

[0125] S6. Based on the impedance matrix, the excitation vector, and the discretized representation of the surface current and surface magnetic current of the periodic structural unit, solve for the surface current and surface magnetic current of the periodic structural unit.

[0126] S6 may include the following steps:

[0127] S61, based on the impedance matrix Z and the excitation vector V, the impedance matrix equation is generated as follows:

[0128]

[0129] Among them, I d I represents the current weighting coefficient vector. d={α1,α2,α3,…,α N1};M d M represents the magnetic flux weighting coefficient vector. d ={β1,β2,β3,…,β N1}; N1 represents the total number of triangle pairs corresponding to all obtained RWG basis functions;

[0130] S62, Solve the current weighting coefficient vector and the magnetocurrent weighting coefficient vector from the impedance matrix equation;

[0131] Specifically, I is obtained from formula (12). d and M d From each element, we obtain I. d ={α1,α2,α3,…,α N1} and M d ={β1,β2,β3,…,β N1}

[0132] S63, using the solved current weighting coefficient vector and the preset current discretization formula, the surface current J(r) of the periodic structural unit is solved;

[0133] Specifically, the solved current weighting coefficient vector I d ={α1,α2,α3,…,α N1 Substituting into formula (2), the surface current J(r) of the periodic structural unit can be solved.

[0134] S64. Using the solved magnetic current weighting coefficient vector and the preset magnetic current discretization formula, the surface magnetic current M(r) of the periodic structural unit is solved.

[0135] Specifically, the solved magnetic flux weighting coefficient vector M d ={β1,β2,β3,…,β N1 Substituting into formula (3), the surface magnetic current M(r) of the periodic structural unit can be solved.

[0136] S7, the surface current and surface magnetic current of the periodic structural unit are converted into electric field and magnetic field respectively, and the scattering characteristics are calculated from the obtained electric field and magnetic field.

[0137] Optionally, the surface current and surface magnetic current of the periodic structural unit are converted into electric and magnetic fields, respectively, including:

[0138] The surface current J(r) and surface magnetic current M(r) of the periodic structural unit are converted into electric and magnetic fields using a preset field calculation formula; wherein,

[0139] The preset field calculation formula includes:

[0140]

[0141]

[0142] Wherein, formulas (13) and (14) represent the calculation formulas for the electric field and magnetic field in the preset field calculation formula, respectively; the subscript type indicates the type of medium space, and when r is in free space, G type (r,r')=G0(r,r'); when r is in a periodic space, G type (r,r')=G prd (r,r').

[0143] The present invention proposes the above formulas (13) and (14) as new expressions for obtaining electric and magnetic fields, which can represent the electric and magnetic fields in free space and periodic space.

[0144] In this embodiment of the invention, an electromagnetic property post-processing calculator can be used to perform the above-mentioned conversion calculation of electric and magnetic fields. Then, the relevant parameters of the scattering problem can be obtained from the obtained electric and magnetic fields, and the corresponding scattering characteristics can be derived.

[0145] The following is a practical example using the method of this invention to illustrate the relevant effects. The general calculation method for infinite periodic structures based on the method of moments proposed in this invention is implemented on the PCRM algorithm and named PFP-PCRM.

[0146] This practical example uses, for example Figure 2 The periodic structural unit shown has different material properties for each layer. Figure 4 The results compare the reflection coefficient amplitude and phase calculated using the above method with those obtained using the commercial software Ansys HFSS. Ansys HFSS uses a finite element method based on differential equations, which belongs to a different category from integral equation methods. Figure 4 The solid line represents the calculation results of the commercial software HFSS; the hollow line represents the calculation results of this invention under the wavelength mesh partitioning corresponding to the highest frequency of 1 / 12; the dashed line represents the calculation results of this invention under the wavelength mesh partitioning corresponding to the highest frequency of 1 / 16; where HFSS(0.001) indicates that the convergence residual of HFSS is set to 0.001, phase represents phase, and Magnitude and mag represent amplitude.

[0147] from Figure 4 It can be seen that whether the mesh is divided according to 1 / 12 of the highest frequency corresponding to the broadcast length or according to 1 / 16 of the highest frequency corresponding to the broadcast length, the calculation results of amplitude and phase are consistent with those of the commercial software HFSS.

[0148] This invention provides a general calculation method for infinitely large periodic structures based on the method of moments (MoM). First, preprocessing is performed using MoM to create a 3D model of the periodic structural unit, obtaining a model of the periodic structural unit. Material properties and incident wave properties are set, and the periodic structural unit is expanded in an edge-fitting manner, thus enabling the representation of an infinitely large periodic structure in the form of an array. Second, triangular meshing is performed, and the current and magnetic current on the surface of the periodic structural unit are discretized using RWG basis functions defined on each pair of triangles. Third, the outward-facing region of the periodic structural unit is marked as periodic space, and other regions are marked as free space. Free-space Green's functions are used in free space, and periodic Green's functions are used in periodic space, thus ensuring the integrity of the unit during calculation. Interfaces at connection points are also considered. Material properties and mesh generation are no longer restricted. Next, when open metal surfaces exist at the junctions of periodic structures, virtual patch technology is used to form new RWG basis functions on the metal surfaces at the junctions of periodic structures, ensuring the continuity of current at the metal surface junctions. This allows for the use of a method based on modeling periodic connection regions to solve more general periodic structure problems. Then, based on all the obtained RWG basis functions, the impedance matrix and excitation vector are filled. Subsequently, based on the impedance matrix, the excitation vector, and the discretized representations of the surface current and surface magnetic current of the periodic structural unit, the surface current and surface magnetic current of the periodic structural unit are solved. Finally, the surface current and surface magnetic current of the periodic structural unit are converted into electric and magnetic fields, and the scattering characteristics are calculated from the obtained electric and magnetic fields.

[0149] This invention provides a general treatment for connected and disconnected periodic structures, reducing the number of interpolation tables compared to existing techniques. Furthermore, it ensures that meshing of periodic structural elements does not require maintaining consistent meshing on both sides of the periodic boundary, thus lowering the requirements for mesh generation. When adjacent portions of contacting periodic elements belong to different media, existing techniques cannot easily remove the interface at the connection point, increasing the difficulty of modeling complex periodic structures. This invention offers a simple and flexible approach to analyzing such problems. Moreover, addressing the limitation of existing techniques that can only solve periodic structures with the same material properties and meshing at the connection points, and cannot handle cases where open metal exists at the connection points, this invention introduces virtual patch technology at the interface for processing and recalculation. This solves the problem of solving electromagnetic parameters when there is an open metal surface at the interface of the periodic structure connection, ensuring the continuity of the interface current at the open metal connection and making the periodic structure modeling method more universal.

[0150] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A general calculation method for infinitely large periodic structures based on the method of moments, characterized in that, include: Based on the method of moments, a three-dimensional model of the periodic structural unit is obtained. Material properties and incident wave properties are set for the periodic structural unit model, and the periodic structural unit is set to expand in an edge-fitting manner. The obtained periodic structural unit model is subjected to triangular meshing; RWG basis functions are constructed for each pair of triangles with common edges obtained from the meshing. The surface current and surface magnetic current of the periodic structural unit are discretized using all the constructed RWG basis functions; The region facing outwards from the periodic structural unit is labeled as the periodic space, and the other regions are labeled as free space. A solution expression for the free space Green's function is constructed for the free space, and a solution expression for the periodic Green's function is constructed for the periodic space. For each target triangle within the periodic structural unit, a matching triangle within the periodic structural unit is copied to the position of the target neighboring triangle corresponding to the target triangle by introducing virtual patching (PFP) technology, which serves as the updated target neighboring triangle. A new set of RWG basis functions is constructed using the target triangle and the corresponding updated target neighboring triangle. The target neighboring triangle corresponding to any target triangle is located within the adjacent periodic structural unit of the periodic structural unit. Based on all the obtained RWG basis functions, fill in the impedance matrix and excitation vector; Based on the impedance matrix, the excitation vector, and the discretized representation of the surface current and surface magnetic current of the periodic structural unit, the surface current and surface magnetic current of the periodic structural unit are solved. The surface current and surface magnetic current of the periodic structural unit are converted into electric and magnetic fields, and the scattering characteristics are calculated from the obtained electric and magnetic fields.

2. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 1, characterized in that, The method of moments (MoM) is used to perform three-dimensional modeling of periodic structural elements to obtain a periodic structural element model. Material properties and incident wave properties are set for the periodic structural element model, and the periodic structural elements are extended using an edge-fitting method. This includes: The periodic structural unit is modeled in three dimensions using electromagnetic modeling software based on the method of moments, thus obtaining the model of the periodic structural unit. Material properties are set for the dielectric and metallic materials in the periodic structural unit model, wherein the material properties include dielectric constant and magnetic permeability; The incident wave is set to a plane wave, and its frequency f is set; the incident electric field is set to E. inc The incident magnetic field is H inc ; The expansion direction of the periodic structural unit is set, and the expansion distance is set as the side length of the periodic structural unit in the expansion direction, so that the periodic structural unit expands along the expansion direction in an edge-fitting manner.

3. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 2, characterized in that, For each pair of triangles with a common edge obtained from the partitioning, the constructed RWG basis function is expressed as: Where n represents the number of the triangle pair; l n Let represent the side length of the common side of the triangle pair; A represent the area of ​​the triangle; r represent the coordinate vector of the sampling point inside the triangle mesh; ρ represent the position vector from the vertex corresponding to the common side of the triangle pair to the sampling point r inside; ± correspond to the two triangles in the triangle pair, respectively denoted as . and 4. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 3, characterized in that, The discretization of the surface current and surface magnetic current of the periodic structural unit using all constructed RWG basis functions includes: Using all the constructed RWG basis functions, the surface current and surface magnetic current of the periodic structural unit are discretized based on the preset current discretization formula and the preset magnetic current discretization formula. The preset current discretization formula is as follows: The preset magnetic flux discretization formula is as follows: Where η0 represents the free-space wave impedance; N represents the total number of triangle pairs obtained after triangular mesh partitioning; α n and β n Let n represent the current weighting coefficient and magnetocurrent weighting coefficient corresponding to the triangle pair numbered n, respectively. The current weighting coefficient and magnetocurrent weighting coefficient corresponding to each triangle pair are unknowns to be determined.

5. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 4, characterized in that, The solution expression for the free space Green's function is constructed for the free space, including: Using the medium parameters of the free space, the expression for the free space Green's function can be constructed as follows: Where G0 represents the Green's function in free space; j represents the imaginary unit; k represents the wavenumber in free space; the coordinate vector r of the sampling points inside the triangular mesh represents the position of the field point, and r' represents the position of the source point.

6. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 5, characterized in that, The solution expression for the periodic Green's function is constructed for the periodic space, including: Using the medium parameters of the periodic space, the solution expression for the periodic Green's function is constructed for the periodic space as follows: Among them, G prd Represents a periodic Green's function; ρ represents the gradient operator; β represents the propagation constant; δ represents the impulse function; ba ρ is the distance between the a-th periodic structural unit and the b-th periodic structural unit. ba Represents ρ ba The modulus.

7. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 6, characterized in that, The formula used to fill the impedance matrix is: Where h and u represent the indices of the h-th and u-th triangle pairs corresponding to all obtained RWG basis functions, including the new RWG basis function; J represents the surface current; M represents the surface magnetic current; and the impedance matrix... Z JJ Represents the current-current impedance matrix; Z JM Z represents the current-magnetic impedance matrix; MJ Represents the magnetocurrent-current impedance matrix; Z MM Z represents the magnetocurrent-magnetic impedance matrix; JJ Z JM Z MJ Z MM The element in the h-th row and u-th column is represented as Z. JJ,hu Z JM,hu Z MJ,hu Z MM,hu ;f h (r) and f u (r) represents the RWG basis functions of the h-th and u-th triangle pairs, respectively; η1 and η2 represent the wave impedance of the periodic space and the free space, respectively; L and K represent two operators, respectively; the subscripts 1 and 2 in η1, η2, L1, L2, K1, and K2 represent the periodic space and the free space, respectively; Where f(r) represents the RWG basis function corresponding to the current sampling point r; G0(r) represents the free space Green's function, G prd (r) represents the Green's function in a periodic space.

8. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 7, characterized in that, The formula used to fill the excitation vector is: Wherein, the excitation vector V = [V J V M ], V J V represents the current excitation vector. M V represents the magnetocurrent excitation vector; J This includes the current excitation for each triangle pair, where the current excitation for the h-th triangle pair is in the form of V. J,h Indicates; V M This includes the magnetofluid excitation for each triangle pair, where the magnetofluid excitation for the h-th triangle pair is V. M,h Indicates; E inc H represents the incident electric field; inc This indicates the incident magnetic field.

9. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 8, characterized in that, Based on the impedance matrix, the excitation vector, and the discretized representation of the surface current and surface magnetic current of the periodic structural unit, the surface current and surface magnetic current of the periodic structural unit are solved, including: Based on the impedance matrix Z and the excitation vector V, the impedance matrix equation is generated as follows: Among them, I d I represents the current weighting coefficient vector. d ={α1,α2,α3,…,α N1 };M d M represents the magnetic flux weighting coefficient vector. d ={β1,β2,β3,…,β N1 }; N1 represents the total number of triangle pairs corresponding to all obtained RWG basis functions; The current weighting coefficient vector and the magnetic current weighting coefficient vector are solved from the impedance matrix equation; The surface current J(r) of the periodic structural unit is solved using the solved current weighting coefficient vector and the preset current discretization formula. The surface magnetic current M(r) of the periodic structural unit is solved by using the solved magnetic current weighting coefficient vector and the preset magnetic current discretization formula.

10. The general calculation method for infinitely large periodic structures based on the method of moments according to claim 9, characterized in that, The surface current and surface magnetic current of the periodic structural unit are converted into electric and magnetic fields, respectively, including: The surface current J(r) and surface magnetic current M(r) of the periodic structural unit are converted into electric and magnetic fields using a preset field calculation formula; wherein, The preset field calculation formula includes: Where the subscript 'type' indicates the type of medium space, and when 'r' is in free space, G... type (r,r')=G0(r,r'); when r is in a periodic space, G type (r,r')=G prd (r,r').

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