Solution method for transmission characteristics of periodic metamaterials based on the finite element port method

Abstract

The present invention provides a method for solving the transmission characteristics of a periodic metamaterial based on the finite element Floquet port method. The implementation steps include: 1) constructing a geometric model of the metamaterial unit in the periodic metamaterial; 2) setting the materials and boundary conditions of the geometric model; 3) calculating the excited electromagnetic fields at the Floquet ports of the geometric model; 4) performing mesh division on the geometric model; 5) solving the electric fields of each mesh unit in the tetrahedral mesh; 6) obtaining the solution results of the transmission characteristics of the periodic metamaterial. Through mathematical processing in the master-slave boundary conditions, this method transforms the analysis of an infinitely large two-dimensional periodic three-dimensional metamaterial into the analysis of a single computational three-dimensional metamaterial unit geometric model. In addition, a plane wave of a specific mode is introduced at the Floquet ports, and the Floquet port boundary conditions are set to analyze the transmission characteristics of the periodic complex medium metamaterial under a specific mode, which has a wide range of applications.

Description

Technical Field

[0001] The present invention belongs to the technical field of electromagnetic simulation, and relates to a method for solving the transmission characteristics of a periodic metamaterial, in particular to a method for solving the transmission characteristics of a periodic metamaterial based on the finite element Floquet port method. Background Art

[0002] In recent years, due to its unique transmission characteristics, metamaterials have received increasing attention in the fields of solid physics, materials science, optics, and applied electromagnetics. Generally, a metamaterial is synthesized by two-dimensional periodic arrangement of unit cells, and the periodic synthesis method is beneficial to the analysis and fabrication of the structure. A metamaterial belongs to a special periodic structure, often a hybrid medium, where the size of the unit cell and the interval between cells are much smaller than the working wavelength of the incident wave, and the spatial variation of the electromagnetic wave in the entire metamaterial structure is much greater than the local variation of the electromagnetic wave caused by the unit discontinuity. The electromagnetic periodic structure shows the characteristics of "more and more unit cells and more and more complex unit structures", resulting in an increasing amount of computing resources required to solve the electromagnetic parameters of the electromagnetic periodic structure, which brings great difficulties to the numerical analysis of the transmission characteristics of the electromagnetic periodic structure.

[0003] Due to the periodicity of the metamaterial, traditional periodic theories such as the Floquet-Bloch theory can be used for analysis. The basis of this method is to equivalent the physical structure of the metamaterial unit cell to a lumped equivalent circuit model, transform the analysis of the electromagnetic field into the analysis of the circuit, and then analyze the inherent dispersion and impedance characteristics of the metamaterial. This method is often used to analyze one-dimensional and two-dimensional metamaterials with relatively simple unit structures. However, in the face of complex metamaterial structures, such as the case of a hybrid medium, it is difficult to extract the lumped equivalent circuit. The full-wave analysis method provides a new approach for analyzing and designing complex metamaterial structures. The main idea of the full-wave analysis method is to use relatively mature microwave numerical algorithms such as the method of moments, the finite element method, and the finite-difference time-domain method. The method of moments obtains an integral equation based on the interaction of surface currents, and then discretizes the obtained integral equation to solve the electromagnetic problem. When solving open-domain problems, it has high accuracy, but the coefficient matrix of the solved equation is dense, resulting in generally large computing resources required to solve high-dimensional matrices. The finite-difference time-domain method uses a difference method to discretize the differential equation, and the equation is relatively simple and intuitive. However, it generally uses a hexahedron grid for discretization, so there is a relatively large error when simulating complex geometric models, and there is an obvious numerical dispersion effect, which is not conducive to solving.

[0004] Anhui University discloses a method for analyzing the characteristic modes of a large-scale finite periodic array structure in its patent document "A Method for Analyzing the Characteristic Modes of a Large-scale Finite Periodic Array Structure" (application number CN201810720625.3, publication number CN10895977A). The method includes the following steps: geometric modeling and meshing of the large-scale finite periodic array structure; establishing an eigenvalue equation based on the electric field integral equation for the reference unit; calculating the characteristic modes of the entire large-scale finite periodic array structure, providing a method for analyzing the characteristic modes of a large-scale finite periodic array structure. However, this invention is an analysis method based on the method of moments, and the method of moments is relatively difficult in analyzing periodic structures of complex media and is difficult to handle periodic mixed-media dielectrics. This invention only solves the method for analyzing the characteristic modes of a two-dimensional finite number of periodic structures and cannot handle the problem of infinitely large periodic metamaterials. This invention can only calculate the distribution under the characteristic mode current and cannot calculate the transmission characteristics of periodic metamaterials under plane waves with arbitrary incident angles and arbitrary polarization modes. Summary of the Invention

[0005] The object of the present invention is to overcome the defects existing in the above-mentioned prior art and propose a method for solving the transmission characteristics of periodic metamaterials based on the finite element Floquet port method, which is used to realize the solution of the transmission characteristics of three-dimensional periodic metamaterials under a specific polarization mode in two dimensions.

[0006] To achieve the above object, the technical solution adopted by the present invention includes the following steps:

[0007] (1) Construct the geometric model of the metamaterial unit in the periodic metamaterial:

[0008] According to the design dimensions of any one metamaterial unit in the periodic metamaterial composed of a plurality of metamaterial units arranged periodically along the x-axis and y-axis directions, and the continuity of the interfaces between this metamaterial unit and other metamaterial units, construct the geometric model corresponding to this metamaterial unit and the design requirements;

[0009] (2) Set the materials and boundary conditions of the geometric model:

[0010] Set the material properties corresponding to the actual design requirements of the periodic metamaterial on each geometric body in the geometric model of the anisotropic medium unit. At the same time, set the boundary conditions of the geometric model, including the boundary conditions on the geometric surface, the master-slave boundary conditions, and the port boundary conditions. Set the boundary conditions corresponding to the actual design of the periodic metamaterial on the geometric surface in the geometric model. For a pair of side surfaces in the x direction of the geometric model, set the master boundary condition and the slave boundary condition of the master-slave boundary in the x direction. For a pair of side surfaces in the y direction of the geometric model, set the master boundary condition and the slave boundary condition of the master-slave boundary in the y direction. Set the Floquet port boundary conditions on the top and bottom surfaces. The normal directions of the top and bottom Floquet port boundary surfaces are respectively and Both the master-slave boundary conditions and the Floquet port boundary conditions belong to the Cauchy boundary conditions. The equation of the Floquet port boundary conditions is:

[0011] (3) Calculate the excited electromagnetic fields at the Floquet ports of the geometric model:

[0012] Introduce a polarized plane wave of TE mode, TM mode or TEM mode through the Floquet port boundary conditions of the geometric model, and calculate the excited electric field and magnetic field at the ports of the geometric model corresponding to these three modes according to the mode and incident angle of the polarized plane wave and and E TEM and H TEM :

[0013]

[0014]

[0015]

[0016]

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] m and n are respectively the mode labels of the polarized plane wave, SU is the surface area of the port, is the incident angle of the polarized plane wave, k xmn , k ymn and k zmn are the wave numbers of the polarized plane wave in the x, y, and z directions, a1 and a2 are the edge vectors of the two-dimensional port, k1 and k2 are the frequency domain vectors of the polarized plane wave under the shape of this port, k is the wave number in the dielectric space, k t (n,m) is the transverse field wave number;

[0024] (4) Perform mesh generation on the geometric model:

[0025] Use a mesh generator to divide the geometric model of the metamaterial unit into multiple closely connected tetrahedral elements to obtain a tetrahedral mesh;

[0026] (5) Solve the electric field of each mesh element in the tetrahedral mesh:

[0027] (5a) Use basis functions to expand and express the electric field E (e) inside each mesh element:

[0028]

[0029] where dof is the number of basis functions inside the tetrahedral element, x i (e) is the expansion coefficient to be solved for the i-th basis function N i ;

[0030] (5b) Establish a finite element variational equation based on the material properties and boundary conditions of the tetrahedral mesh, and construct a system matrix K (e) , b (e) using the coefficients in this finite element variational equation, and then use a matrix solver to solve the system matrix to obtain the electric field of each mesh element in the tetrahedral mesh;

[0031] (5c) Obtain the electric field of the complete periodic metamaterial through the master-slave boundary condition relation;

[0032] (6) Obtain the solution result of the transmission characteristics of the periodic metamaterial:

[0033] Use a post-processing calculator for transmission characteristics to convert the electric field into the transmission characteristics of the periodic metamaterial.

[0034] Compared with the prior art, the present invention has the following advantages:

[0035] First, since the finite element Floquet port method is used in the present invention and plane waves with arbitrary angles and arbitrary polarization modes are introduced at the port boundaries, the problem of periodic hybrid medium metamaterials under arbitrary angles and arbitrary polarization modes, which cannot be solved by the prior art, is overcome. This enables the present invention to analyze the problem of periodic hybrid medium metamaterials based on specific mode analysis. According to the corresponding transmission characteristics, available transmission modes and evanescent modes can be selected, and it has a wide range of applications.

[0036] Second, due to the processing in the boundary conditions of the present invention, the analysis of infinite two-dimensional periodic three-dimensional metamaterials is transformed into the analysis of the geometric model of a single three-dimensional metamaterial unit, overcoming the problem of the difficulty in overall geometric modeling and mesh generation of two-dimensional periodic three-dimensional metamaterials in the prior art, and simplifying the solution process of the transmission characteristics of periodic metamaterials. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 is a flowchart for the implementation of the present invention;

[0038] Figure 2 is a schematic diagram of the unit geometric modeling of the periodic metamaterials of the present invention;

[0039] Figure 3 is a schematic diagram of the specific dimensions of the unit structure of the periodic metamaterials of the present invention;

[0040] Figure 4 is a diagram of the calculation results of the periodic metamaterials of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0041] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0042] Referring to Figure 1 , the present invention includes the following steps:

[0043] Step 1. Construct the geometric model of the metamaterial unit in the periodic metamaterials.

[0044] According to the design dimensions of any one metamaterial unit in the periodic metamaterials composed of a plurality of metamaterial units arranged periodically along the x-axis and y-axis directions, and the continuity of the interfaces between this metamaterial unit and other metamaterial units, construct the geometric model of this metamaterial unit corresponding to the design requirements.

[0045] The following further describes the specific steps of the created unit geometric model with reference to Figure 2 the schematic diagram of the unit geometric modeling of the periodic metamaterials.

[0046] From Figure 2 It can be seen that the selected unit structure is in the middle of the model, and the relative dielectric constant ε of the dielectric substrate r = 2.0 - 0.2j, the relative permeability μ r = 1.0, and the thickness of the dielectric plate is 100 mm. The cross-shaped geometric surface at the center of the upper surface of the unit structure is a metal patch. Combining Figure 3 with the specific dimension schematic diagram of the periodic metamaterial unit structure and Table 1, the specific dimensions of the unit structure are further described.

[0047] Table 1

[0048] Parameter <![CDATA[L1]]> <![CDATA[L2]]> <![CDATA[L3]]> <![CDATA[W1]]> <![CDATA[W2]]> Length / mm 100 100 100 500 100

[0049] Above and below the unit structure itself, 1 air body with the same width and length as the unit structure is created respectively. Its relative dielectric constant ε r = 1.0, the relative permeability μ r = 1.0, and the height is one-quarter wavelength.

[0050] Step 2. Set the material and boundary conditions of the geometric model.

[0051] Set the material properties corresponding to the actual design requirements of the periodic metamaterial on each geometric body in the geometric model of the metamaterial unit, and at the same time set the boundary conditions of the geometric model, including the boundary conditions on the geometric surface, the master-slave boundary conditions, and the port boundary conditions. Set the boundary conditions corresponding to the actual design of the periodic metamaterial on the geometric surface in the geometric model. For a pair of side surfaces in the x direction of the geometric model, set the master boundary condition and the slave boundary condition of the master-slave boundary in the x direction. For a pair of side surfaces in the y direction of the geometric model, set the master boundary condition and the slave boundary condition of the master-slave boundary in the y direction. Set the Floquet port boundary conditions on the top surface and the bottom surface. The normal directions of the top and bottom Floquet port boundary surfaces are respectively and Both the master-slave boundary conditions and the Floquet port boundary conditions belong to the Cauchy boundary conditions. The equation of the Floquet port boundary condition is:

[0052]

[0053] Where is the Hamiltonian operator, k0 is the wave number in free space, θ inc is the incident angle of the incident polarized plane wave, is the angle between the plane formed by the incident polarized plane wave and the z-axis and the xoz plane.

[0054] Next, combining Figure 2A schematic diagram of the unit geometric modeling of the periodic anisotropic medium is further described for the specific settings of the master-slave boundary conditions of the unit model. As Figure 2 shown, the geometric model is divided into an upper air medium layer, a unit structure layer, and a lower air medium layer from top to bottom. The left and right surfaces in the upper air medium layer are respectively set as the master boundary 1 and the slave boundary 1, and the front and back surfaces are respectively set as the slave boundary surface 2 and the master boundary surface 2. The left and right surfaces in the unit structure body are respectively set as the master boundary 3 and the slave boundary 3, and the front and back surfaces are respectively set as the slave boundary surface 4 and the master boundary surface 4. The left and right surfaces in the lower air medium layer are respectively set as the master boundary 5 and the slave boundary 5, and the front and back surfaces are respectively set as the slave boundary surface 6 and the master boundary surface 6. The top and top surfaces of the geometric model are set as the Floquet port 1 and the Floquet port 2.

[0055] Step 3. Calculate the excited electromagnetic fields at the Floquet ports of the geometric model.

[0056] Introduce a polarized plane wave of TE mode, TM mode, or TEM mode through the Floquet port boundary conditions of the geometric model, and calculate the excited electric field and magnetic field at the ports of the geometric model corresponding to these three modes according to the mode and incident angle of the polarized plane wave and and E TEM and H TEM :

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[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067] m and n are respectively the mode labels of the polarized plane wave, S U is the surface area of the port, is the incident angle of the polarized plane wave, k xmn , k ymn and k zmn are the wave numbers of the polarized plane wave in the x, y, and z directions. a1 and a2 are the edge vectors of the two-dimensional port, k1 and k2 are the frequency-domain vectors of the polarized plane wave under the shape of this port, k is the wave number in the dielectric space, k t (n,m) is the transverse field wave number. The type of plane wave introduced here is set to TE 00 mode, and the incident angle is normal incidence. Then the transmission characteristics of the periodic metamaterial under normal-incidence TE 00 mode can be calculated. This is an analysis method based on mode analysis, which can analyze the frequency-domain responses of different modes on the periodic metamaterial, and then a periodic metamaterial with specific transmission characteristics meeting the expectations can be designed. Therefore, it has a wide range of applications.

[0068] Step 4. Mesh the geometric model.

[0069] The geometric model of the metamaterial unit is meshed into multiple closely connected tetrahedral elements through a mesh generator to obtain a tetrahedral mesh.

[0070] Step 5. Solve the electric field of each mesh element in the tetrahedral mesh.

[0071] (5a) Expand and express the electric field E (e) in each mesh element using basis functions:

[0072]

[0073] where dof is the number of basis functions in the tetrahedral element, and x i (e) is the unknown expansion coefficient of the i-th basis function N i .

[0074] (5b) Establish a finite element variational equation according to the material properties and boundary conditions of the tetrahedral mesh:

[0075]

[0076] i = 1 to dof

[0077]

[0078]

[0079] where J is the current density, M is the magnetic current density, is the tensor form of the relative magnetic permeability, is the tensor form of the relative permittivity, f is the excitation function, φ is the Cauchy boundary excitation function, Einc is the exciting electric field of the incident plane wave, Ω is the spatial calculation region, and S is the surface of the outer boundary of the solution region Ω. is the normal vector of the outer boundary surface S, with the direction facing outwards, S C is the boundary surface of the Cauchy boundary condition.

[0080] (5c) Construct the system matrix equations through the coefficients in this finite element variational equation:

[0081] K (e) x (e) = b (e)

[0082] K (e) , b (e) , x (e) are the system matrices, where the matrix elements are:

[0083]

[0084]

[0085]

[0086] Then use the matrix solver to solve the system matrix, obtain the electric field of each grid cell in the tetrahedral mesh, and then combine the electric fields of each grid to form the electric field distribution in the entire unit structure. Therefore, such a method of solving the electric field distribution can analyze the structure with complex media.

[0087] (5d) Through the master-slave boundary condition relationships, including the relationships on the edges in the slave boundary conditions and the relationships on the master and slave surfaces in the master-slave boundary conditions, where the relationships on the edges in the master-slave boundary conditions are:

[0088]

[0089] ψ x = k x D x

[0090] ψ y = k y D y

[0091] where K fi , b fi are the edge system matrix elements between the master boundary surface in the master-slave boundary in the x direction and the master boundary surface in the master-slave boundary in the y direction. K bi , b bi are the edge system matrix elements between the slave boundary surface in the master-slave boundary in the x direction and the master boundary surface in the master-slave boundary in the y direction. Kli , b li is the edge system matrix element between the master boundary surface in the x - direction and the slave boundary surface in the y - direction. K ri , b ri is the edge system matrix element between the slave boundary surface in the x - direction and the slave boundary surface in the y - direction. k x is the component of the wave number k0 in the x - direction, D x is the distance of the master - slave boundary in the x - direction, k y is the component of the wave number k0 in the y - direction, D y is the distance of the master - slave boundary in the y - direction.

[0092] Relations of the master - slave boundary conditions on the master and slave surfaces:

[0093]

[0094]

[0095] where is the system matrix element on the master boundary surface in the master - slave boundary in the x - direction. is the system matrix element on the slave boundary surface in the master - slave boundary in the x - direction. is the system matrix element on the master boundary surface in the master - slave boundary in the y - direction. is the system matrix element on the slave boundary surface in the master - slave boundary in the y - direction. It can be known from the above steps that the electric - field distribution corresponding to the system matrix within a single - period structure is transformed into the electric - field distribution corresponding to the system matrix of other unit structures through the relations under the master - slave boundary conditions. By combining the electric - field distributions of each unit, the electric - field distribution of the complete - period metamaterial is further solved, achieving

[0096] Step 6. Obtain the solution result of the transmission characteristics of the periodic metamaterial.

[0097] Using the transmission - characteristic post - processor, the electric field is transformed into the transmission characteristics of the periodic metamaterial.

[0098] The following further illustrates the effect of the present invention in combination with simulation experiments:

[0099] 1. Simulation experiment conditions:

[0100] The hardware platform for the simulation experiment of the present invention is: blade nodes, with 4 12 - core Intel(R)Xeon(R)Gold 5215 CPUs, the main frequency is 2.50 GHz, and the memory is 1024 GB.

[0101] The software platform for the simulation experiment of the present invention is: Red Hat Enterprise Linux Server release 7.4 operating system and Fortran 90.

[0102] 2. Simulation content and result analysis:

[0103] The simulation experiment of the present invention uses the technology of the present invention to calculate the transmission characteristics of an infinite periodic structure composed of unit models as Figure 2 shown. The simulation experiment of the present invention selects the transmission parameters S 11 and S 21 as the results to analyze the structure of the transmission characteristics. As Figure 4 shown, the frequency range is from 300 MHz to 500 MHz. It is proved that this method can analyze the problem of periodic hybrid medium metamaterials based on mode analysis, and can select available transmission modes and evanescent modes according to the corresponding transmission characteristics, with wide applications.

[0104] The above description of the embodiments is to enable those of ordinary skill in the art to understand and apply the technology of this case. It is obvious that those who are familiar with the technology in this field can easily make various modifications to these examples and apply the general principles described here to other embodiments without creative labor. Therefore, this case is not limited to the above embodiments, and the improvements and modifications made by those skilled in the art to this case according to the disclosure of this case should be within the protection scope of this case.

Claims

1. A method for solving the transmission characteristics of periodic metamaterials based on the finite element Floquet port method, characterized in that It includes the following steps: (1) Construct the geometric model of the metamaterial unit in the periodic metamaterial: According to the design dimensions of any one metamaterial unit in the periodic metamaterial composed of multiple metamaterial units arranged periodically along the x-axis and y-axis directions, and the continuity of the boundary surfaces between this metamaterial unit and other metamaterial units, construct the geometric model corresponding to this metamaterial unit and the design requirements; (2) Set the material and boundary conditions of the geometric model: Set the material properties corresponding to the actual design requirements of the periodic metamaterial on each geometric body in the geometric model of the anisotropic medium unit. At the same time, set the boundary conditions of the geometric model, including the boundary conditions on the geometric surface, the master-slave boundary conditions, and the port boundary conditions. Set the boundary conditions corresponding to the actual design of the periodic metamaterial on the geometric surface in the geometric model. For a pair of side surfaces in the x direction of the geometric model, set the master boundary condition and the slave boundary condition of the master-slave boundary with respect to the x direction. For a pair of side surfaces in the y direction of the geometric model, set the master boundary condition and the slave boundary condition of the master-slave boundary with respect to the y direction. Set the Floquet port boundary conditions on the top surface and the bottom surface. The normal directions of the top and bottom Floquet port boundary surfaces are respectively and Both the master-slave boundary condition and the Floquet port boundary condition belong to the Cauchy boundary condition. The equation of the Floquet port boundary condition is: wherein is the Hamiltonian operator, k0 is the wave number in free space, θ inc is the incident angle of the incident polarized plane wave, is the angle between the plane formed by the incident polarized plane wave and the z-axis and the xoz plane; (3) Calculate the excited electromagnetic field at the Floquet ports of the geometric model: Introduce polarized plane waves of TE mode, TM mode or TEM mode through the Floquet port boundary conditions of the geometric model, and calculate the excitation electric field and magnetic field at the ports of the geometric model corresponding to these three modes according to the mode of the polarized plane wave and the incident angle. and and E TEM and H TEM : m and n are the mode labels of the polarized plane wave, and S U is the surface area of the port, is the incident angle of the polarized plane wave, and k xmn , k ymn and k zmn are the wavenumbers of the polarized plane wave in the x, y, and z directions. a1 and a2 are the edge vectors of the two-dimensional port, k1 and k2 are the frequency-domain vectors of the polarized plane wave under the shape of this port, k is the wavenumber in the dielectric space, and k t (n, m) is the transverse field wavenumber; (4) Perform mesh division on the geometric model: Use a mesh generator to divide the geometric model of the metamaterial unit into multiple closely connected tetrahedral elements to obtain a tetrahedral mesh; (5) Solve the electric field of each mesh unit in the tetrahedral mesh: (5a) Expand and express the electric field E within each grid cell using basis functions (e) as follows: where dof is the number of basis functions within the tetrahedral element, and x i (e) is the expansion coefficient to be solved for the \(i\)-th basis function \(N\) i ; (5b) Establish a finite element variational equation based on the material properties and boundary conditions of the tetrahedral mesh, and construct a system matrix equation set \(K\) through the coefficients in the finite element variational equation. (e) x (e) = b (e) , and then use a matrix solver to solve the system matrix equation set to obtain the electric field of each mesh element in the tetrahedral mesh. (5c) Obtain the electric field of the complete periodic metamaterial through the master-slave boundary condition relation; (6) Obtain the solution results of the transmission characteristics of the periodic metamaterial: Use a transmission characteristic post-processing calculator to convert the electric field into the transmission characteristics of the periodic metamaterial.

2. The method for solving the transmission characteristics of a periodic metamaterial based on the finite element Floquet port method according to claim 1, wherein The tetrahedral mesh described in step (4), where the electric field E in any tetrahedral element (e) has the following expansion: where dof is the number of basis functions within the tetrahedral element, and x i (e) is the \(i\)-th basis function for which the expansion coefficients to be solved 3. The method for solving the transmission characteristics of a periodic metamaterial based on the finite element Floquet port method according to claim 2, characterized in that The finite element variational equation in step (5b) is: where J is the current density, M is the magnetic current density, is the tensor form of the relative magnetic permeability, is the tensor form of the relative permittivity, f is the excitation function, φ is the Cauchy boundary excitation function, E inc is the introduced plane-wave excitation electric field, Ω is the computational domain space, S is the surface of the outer boundary of the solution domain Ω, is the normal vector of the outer boundary surface S, with the direction pointing outwards, S C is the boundary surface of the Cauchy boundary condition.

4. The method for solving the transmission characteristics of a periodic metamaterial based on the finite element Floquet port method according to claim 3, wherein The system matrix equation set in step (5b) is: K (e) x (e) = b (e) K (e) 、b (e) 、x (e) are system matrices, where the matrix elements are:

5. The method for solving the transmission characteristics of a periodic metamaterial based on the finite element Floquet port method according to claim 1, characterized in that The relation on the edge in the master-slave boundary condition in step (5c): ψ x = k x D x ψ y = k y D y where K fi and b fi are the edge system matrix elements between the master surface of the master-slave boundary in the x-direction and the master surface of the master-slave boundary in the y-direction. K bi and b bi are the edge system matrix elements between the slave surface of the master-slave boundary in the x-direction and the master surface of the master-slave boundary in the y-direction. K li and b li are the edge system matrix elements between the master surface of the master-slave boundary in the x-direction and the slave surface of the master-slave boundary in the y-direction. K ri and b ri are the edge system matrix elements between the slave surface of the master-slave boundary in the x-direction and the slave surface of the master-slave boundary in the y-direction. k x is the component of the wave number k0 in the x-direction. D x is the distance of the master-slave boundary in the x-direction. k y is the component of the wave number k0 in the y-direction. D y is the distance of the master-slave boundary in the y-direction. The relations of the master-slave boundary condition on the master and slave surfaces: wherein is the system matrix element on the master boundary surface in the master-slave boundary in the x direction, is the system matrix element on the slave boundary surface in the master-slave boundary in the x direction, is the system matrix element on the master boundary surface in the master-slave boundary in the y direction, is the system matrix element on the slave boundary surface in the master-slave boundary in the y direction.

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