Calculation method of electromagnetic scattering characteristics of metasurface based on finite element method and generalized thin sheet transmission condition

Abstract

The present invention discloses a method for calculating the electromagnetic scattering characteristics of a super surface based on the finite element method and generalized thin sheet transmission conditions, and belongs to the field of electromagnetic simulation technology. The method of the present invention performs infinite periodic boundary condition simulation on a unit to obtain the surface polarizability corresponding to the unit; the super surface to be solved is equivalently modeled as a zero-thickness thin sheet, the target structure to be solved is constructed, and the integral equation boundary surface is established to determine the target solution area; the target structure to be solved is meshed based on the finite element method; an incident plane wave is configured; the generalized thin sheet transmission condition is incorporated into the finite element solution framework, the FEM‑GSTCs system matrix equation is constructed and the electric field of the target solution area is solved, and then the radar scattering cross section of the target structure to be solved in the far field area is calculated. The method of the present invention simplifies the pre-processing process of the electromagnetic modeling of the super surface, improves the efficiency of the solution of the electromagnetic scattering of the super surface, and improves the ability of the finite element method to solve the electromagnetic characteristics of complex targets.

Description

Technical Field

[0001] The present invention belongs to the technical field of electromagnetic simulation, and in particular relates to a method for calculating the electromagnetic scattering characteristics of a super surface based on a finite element method and generalized thin-sheet transmission conditions. Background Art

[0002] Metasurfaces can control inherent properties such as amplitude, phase, polarization, and frequency, retain and improve the performance of many metamaterials, actively control wavefronts according to actual needs, integrate functional materials, and significantly enhance nonlinear responses. Metasurfaces have innovative application results and development prospects in the fields of electromagnetic imaging, communications, and antenna design.

[0003] Electromagnetic simulation technology can effectively solve complex electromagnetic problems such as antenna design and performance analysis, radar target feature analysis and detection, and mobile communication systems. However, limited by the actual conditions of computer simulation, current electromagnetic simulation technology cannot meet the needs of efficient and accurate solution of electromagnetic scattering characteristics of metasurfaces and larger-scale systems including metasurface detailed structures.

[0004] Metasurfaces have attracted much attention and research due to their ultra-thinness and easy processing. Global electrically large-scale metasurfaces are composed of a large number of subwavelength units with fine geometric structures. This type of multi-scale structure needs to be discretized using a finer grid during electromagnetic simulation, which will result in the need to process huge unknown quantities, consume a lot of computing time and computing memory, and even cause the grid shape to be distorted, making the solution unsatisfactory.

[0005] Generalized sheet transition conditions (GSTCs) can describe complex boundary effects such as electromagnetic field discontinuities caused by metasurfaces. The prior art "Computational Analysis of Metasurfaces" discloses the application of GSTCs in frequency domain / time domain finite difference, spectral domain integral equation method, and finite element method. However, these methodological research works only solve the problem of solving the electromagnetic characteristics of metasurfaces in part of one-dimensional and two-dimensional solution domains, that is, constructing a surface polarizability tensor that meets the required electromagnetic function from a theoretical level, and then substituting this parameter into the electromagnetic field numerical method for solution, and verifying whether the expected electromagnetic function is achieved based on the field distribution obtained by the solution. It is unable to solve the electromagnetic scattering calculation problem of metasurfaces with actual physical structures in a three-dimensional solution domain.

[0006] A Chinese patent application entitled “An Efficient Three-Dimensional Spectral Element Metasurface Electromagnetic Simulation Technology” discloses a metasurface electromagnetic simulation technology that combines the spectral element method with generalized thin-film transmission conditions to achieve three-dimensional metasurface electromagnetic simulation. Although the method described in this prior art reduces the density of the subdivided grid during the near-field scattering simulation of an independent metasurface, improves the simulation efficiency, and is applicable to a limited solution area. However, this prior art uses absorbing boundary conditions to truncate the calculation area, introducing additional calculations. The calculation results will also produce errors due to the angle of the incident wave, polarization characteristics, and geometric structure and material differences of the target. It has not been applied to composite targets containing metasurfaces and far-field scattering scenarios to solve the technical problems of calculation accuracy and efficiency. Summary of the invention

[0007] The purpose of the present invention is to overcome the defects of the above-mentioned prior art and provide a method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and generalized thin-sheet transmission conditions, and to efficiently solve the scattering problem in the electromagnetic characteristics of the metasurface. The present invention can efficiently and accurately calculate the electromagnetic scattering field of an independent metasurface or a composite target including a metasurface detail structure, further expanding the applicable scenarios of the finite element method.

[0008] The technical problem proposed by the present invention is solved in this way:

[0009] A method for calculating the electromagnetic scattering characteristics of a metasurface based on a finite element method and a generalized thin-sheet transmission condition comprises the following steps:

[0010] Step 1. The metasurface to be solved includes a number of units arranged in two dimensions. An infinite periodic boundary condition simulation is performed on the units to obtain the transmission coefficient of the units, and the surface polarizability corresponding to the units is extracted according to the transmission coefficient;

[0011] Step 2. Equivalently model the metasurface to be solved as a zero-thickness thin sheet, and the physical characteristics of the zero-thickness thin sheet are characterized by the surface polarizability in step 1; load dielectric substrates on the upper and lower surfaces of the zero-thickness thin sheet, and use the overall structure as the target structure to be solved; establish an integral equation boundary surface on the surface adjacent to the target structure to be solved, truncate the infinite solution area, and use the integral equation boundary surface and the internal area as the target solution area;

[0012] Step 3. Mesh the target structure to be solved based on the finite element method, and export the mesh information of the meshed model;

[0013] Step 4. Configure the frequency, amplitude, phase, angle and polarization direction information of the incident plane wave;

[0014] Step 5. Read the grid information; incorporate the generalized thin film transport condition into the finite element solution framework and construct the FEM-GSTCs system matrix equation The matrix The elements of the sum vector b are the exact integral solutions obtained by the global matrix method; in the matrix equation of the FEM-GSTCs system, the unknown electric field in the target solution area, the unknown surface current and surface tangential electric field of the auxiliary metasurface, the unknown surface current and surface magnetic current on the boundary surface of the integral equation are combined as the unknown quantity x, and the unknown quantity on each grid is expanded by the basis function; the expansion coefficients of the unknown quantities are iteratively solved by the generalized minimum residual method, and the electric field values ​​of the unknown electric field in the target solution area are inferred based on the expansion coefficients of the unknown quantities.

[0015] Step 6. Use the electric field values ​​of the unknown electric field quantities in the target solution area to calculate the radar scattering cross section of the target structure to be solved in the far field area.

[0016] Furthermore, in step 1, the unit of the metasurface to be solved is extended along the x-axis and y-axis directions, and the unit includes a thin dielectric substrate and a metal disc on its upper surface; the unit period is 3 mm, and the radius of the metal disc is 1.3 mm; the relative dielectric constant of the thin dielectric substrate is 4, and the thickness is 0.3 mm; the transmission coefficient of the unit is simulated at a frequency of 10 GHz, where the electric susceptibility tensor of the electric field excitation is It is expressed as:

[0017]

[0018] Simplify the nine components in the above formula into two components, namely and The remaining components are set to zero, and are expressed as follows:

[0019]

[0020]

[0021] Where j is the sign of the imaginary part, k0 is the free space wave number; R TM (0) represents the reflection coefficient of the vertically incident wave along TM polarization, T TM (0) represents the transmission coefficient of the vertically incident wave along TM polarization; R TE (0) represents the reflection coefficient of the vertical incident wave along TE polarization, T TE (0) represents the transmission coefficient for a vertically incident wave along TE polarization.

[0022] Furthermore, in step 2, the unit is expanded 16 times along the x-axis and y-axis respectively to form a 16×16 two-dimensional arrangement of the metasurface to be solved, which is placed between two dielectric substrates with a thickness of 3 mm and a size of 48 mm×48 mm. The overall structure is used as the target structure to be solved; the two dielectric substrates have the same structure and the relative dielectric constant is 2.2.

[0023] Furthermore, in step 3, the zero-thickness sheet and the integral equation boundary surface are meshed with triangles, and the other areas in the target solution area, namely the areas where the two dielectric substrates are located, are meshed with tetrahedral meshes; the areas where the two dielectric substrates are located are meshed independently.

[0024] Furthermore, in step 4, the frequency of the incident plane wave is set to 10 GHz and the incident angle is and θ are the azimuth and elevation angles respectively.

[0025] Furthermore, in step 5, in the matrix In the process of solving the elements of , if the interface meshes on both sides of the metasurface to be solved, i.e. the interface meshes of the two dielectric substrates, are non-conformal, the matrix The corresponding elements in are solved by the bonded fine mesh method with mesh refinement.

[0026] Furthermore, in step 5, the specific process of incorporating the generalized thin-sheet transport condition into the finite element solution framework to construct the FEM-GSTCs system matrix equation is as follows:

[0027] For three-dimensional vector finite element electromagnetic scattering problems, the following vector wave equation is satisfied:

[0028]

[0029] in, represents the curl to be solved; μ r and ε r are the relative magnetic permeability and relative dielectric constant of the dielectric substrate respectively; Ω i is the internal sub-region of the i-th finite element, i = 1, 2, Ω1 is the region where the dielectric substrate is located above the metasurface to be solved, and Ω2 is the region where the dielectric substrate is located below the metasurface to be solved; E i For the region Ω i The electric field inside is unknown;

[0030] For a zero-thickness sheet, i.e., the interface grid of two dielectric substrates, the generalized sheet transmission condition is applied to describe the mutation introduced by the metasurface to be solved. The generalized sheet transmission condition is written as follows:

[0031]

[0032]

[0033] in, is the unit vector in the z direction, E′ i For the region Ω i The electric field on the inner edge of the metasurface to be solved, H i For the region Ω iThe magnetic field of the inner edge of the metasurface to be solved; μ and ε are the magnetic permeability and dielectric constant of the dielectric substrate, and ω is the angular frequency; is the electric susceptibility tensor excited by the magnetic field, is the magnetic susceptibility tensor of electric field excitation, is the magnetic susceptibility tensor of magnetic field excitation, all of which are zero matrices;

[0034] The unknown surface tangential electric field e is introduced, and the unknown surface tangential electric field along the hypersurface to be solved in regions Ω1 and Ω2 is expressed as e1 = n1 × E′1 × n1 and e2 = n2 × E′2 × n2 respectively; the auxiliary surface current unknown J is introduced, and the unknown surface current along the hypersurface to be solved in regions Ω1 and Ω2 is expressed as n1 is the unit normal vector of the hypersurface pointing to the region Ω1, and n2 is the unit normal vector of the hypersurface pointing to the region Ω2;

[0035] Furthermore, the generalized thin-sheet transport condition can be simplified as:

[0036]

[0037] e1=e2

[0038] Where η0 is the free space wave impedance;

[0039] The surface integral equations EFIE and MFIE on the integral equation boundary surface in step 2 are expressed as:

[0040]

[0041]

[0042] in, is the unit normal vector pointing from the boundary surface of the integral equation to the infinite external region; E is the unknown electric field in the infinite external region of the boundary surface of the integral equation; J and M are the unknown surface current and surface magnetic current on the boundary surface of the integral equation, respectively; E inc is the electric field of the incident plane wave, J inc To generate the electric field E inc The current source; L and are integral operators, respectively expressed as:

[0043]

[0044]

[0045] in, is the boundary surface of the target solution area, for Any grid of; X is the independent variable, r and r′ represent the field point vector and the source point vector respectively; represents the divergence of r′, It means to find the gradient; represents the removal of singular terms of the integral operator K; G0(r, r′) is the scalar Green's function of free space, expressed as:

[0046]

[0047] Among them, || is the modulo operation;

[0048] The Galerkin test is performed on the above-mentioned vector wave equation, generalized thin sheet transport condition and surface integral equation on the boundary surface of the integral equation. The unknown quantities in the equation are expanded using basis functions, and the following FEM-GSTCs equation group is obtained, which is expressed as:

[0049]

[0050]

[0051]

[0052]

[0053] in,

[0054]

[0055] Where N m and N n They represent the vector edge element basis functions defined on the mth edge and the nth edge in any grid within the target solution area Ω; m and Λ n They represent the RWG basis functions with the mth edge and the nth edge as common edges in the target solution region Ω; and Respectively represent the region Ω i The vector edge element basis functions defined on the mth edge and the nth edge in any grid; and Respectively represent the region Ω i The RWG basis function with the mth edge and the nth edge as the common edge; <, > means to find the inner product; Represents area Ω i The boundary surface, dΩ i Represents area Ω i Any grid within; x E is the expansion coefficient of the unknown electric field in the target solution area, x j is the expansion coefficient of the unknown surface current, xM is the expansion coefficient of the unknown surface magnetic current, is the expansion coefficient of the unknown quantity of the surface tangential electric field;

[0056] The FEM-GSTCs equations can be rewritten as form, as the matrix equation of the FEM-GSTCs system.

[0057] The beneficial effects of the present invention are:

[0058] The method described in the present invention greatly improves the efficiency of solving the far-field scattering of electromagnetic metasurfaces, fully utilizes the respective advantages of the finite element method and the generalized thin-film transmission condition, gives full play to their respective advantages, simplifies the pre-processing process of metasurface electromagnetic modeling, improves the efficiency of solving the electromagnetic scattering of metasurfaces, and improves the ability of the finite element method to solve the electromagnetic characteristics of complex targets. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a schematic diagram of the process of the method of the present invention;

[0060] Figure 2 Schematic diagram of the unit structure of the hypersurface to be solved in the method described in the embodiment;

[0061] Figure 3 A schematic diagram of the simulation operation of the super surface to be solved in the method described in the embodiment;

[0062] Figure 4 A schematic diagram of a target structure to be solved in the method described in the embodiment;

[0063] Figure 5 A schematic diagram of an equivalent model of surface polarizability of a metasurface divided by a coarse grid in the method described in the embodiment;

[0064] Figure 6 This is a comparison chart of the dual-station RCS calculation results of the present invention and electromagnetic simulation using an actual physical model and the finite element method. DETAILED DESCRIPTION

[0065] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0066] This embodiment provides a method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition. The flowchart is shown in FIG. Figure 1 As shown, the following steps are included:

[0067] Step 1. The metasurface to be solved includes a number of units arranged in two dimensions. An infinite periodic boundary condition simulation is performed on the units to obtain the transmission coefficient of the units, and the surface polarizability corresponding to the units is extracted according to the transmission coefficient;

[0068] In this embodiment, the structural diagram of the unit in the hypersurface to be solved is as follows: Figure 2 As shown, the simulation operation is as follows Figure 3 shown.

[0069] The unit is extended along the x-axis and y-axis directions, and the unit includes a thin dielectric substrate and a metal disc on its upper surface; the unit period is 3 mm, and the radius of the metal disc is 1.3 mm; the relative dielectric constant of the thin dielectric plate loaded at the bottom is 4, and the thickness is 0.3 mm; the transmission coefficient of the unit described in this embodiment is obtained by simulation at a frequency of 10 GHz, wherein the electric susceptibility tensor of the electric field excitation is It is expressed as:

[0070]

[0071] Among them, for a metasurface with sufficiently symmetric and thin units, the nine components in the above formula are simplified to two components, namely and The remaining components are set to zero, and are expressed as follows:

[0072]

[0073]

[0074] Where j is the sign of the imaginary part, k0 is the free space wave number; R TM (0) represents the reflection coefficient of the vertically incident wave along TM polarization, T TM (0) represents the transmission coefficient of the vertically incident wave along TM polarization; R TE (0) represents the reflection coefficient of the vertical incident wave along TE polarization, T TE (0) represents the transmission coefficient for vertical incident wave along TE polarization;

[0075] In this embodiment, the surface polarizability component corresponding to the metasurface unit is extracted according to the above analytical formula:

[0076]

[0077] Step 2. Equivalently model the metasurface to be solved as a zero-thickness thin sheet, and the physical characteristics of the zero-thickness thin sheet are characterized by the surface polarizability in step 1; load dielectric substrates on the upper and lower surfaces of the zero-thickness thin sheet, and use the overall structure as the target structure to be solved; establish an integral equation boundary surface on the surface adjacent to the target structure to be solved, truncate the infinite solution area, and use the integral equation boundary surface and the internal area as the target solution area;

[0078] In this embodiment, the target structure to be solved is as follows: Figure 4 As shown, Figure 2The units shown are extended 16 times along the x-axis and y-axis to form a 16×16 two-dimensional arrangement of the metasurface to be solved, which is placed between two dielectric substrates with a thickness of 3mm and a size of 48mm×48mm. The overall structure is used as the target structure to be solved; the two dielectric substrates have the same structure and the relative dielectric constant is 2.2. The metasurface to be solved is equivalently modeled as a zero-thickness thin sheet connecting different target solution areas, and an integral equation boundary surface is established on the surface adjacent to the target structure to be solved to truncate the infinite solution area.

[0079] Step 3. Mesh the target structure to be solved based on the finite element method, and export the mesh information of the meshed model.

[0080] In step 3, the zero-thickness sheet and the integral equation boundary surface are triangularly meshed, and the other areas within the target solution area, namely the areas where the two dielectric substrates are located, are tetrahedral meshed; the areas where the two dielectric substrates are located are allowed to be meshed independently, because the mutation introduced by the hypersurface to be solved is reflected through the variational form of the generalized thin-sheet transport condition mapping, and there is no need to impose conformal restrictions on the meshes on both sides of the hypersurface to be solved.

[0081] Step 4. Configure the frequency, amplitude, phase, angle and polarization direction information of the incident plane wave, and set the scattered field information to be solved;

[0082] In this embodiment, a 10 GHz uniform plane wave is set along the angle The incident target has an electric field polarized along the x-axis. The bistatic radar cross section RCS of the xoz plane (θ=0°)~(θ=180°) is: and θ are the azimuth and elevation angles respectively.

[0083] Step 5. Read the grid information; the unknown quantity on each grid can be expanded through the basis function containing the grid coordinate information, the generalized thin film transport condition is incorporated into the finite element solution framework, and the FEM-GSTCs system matrix equation is constructed The matrix The elements of the sum vector b are the exact integral solutions obtained by the global matrix method, the matrix The specific position of the element of the sum vector b in the system matrix is ​​determined by the grid numbering information; in the FEM-GSTCs system matrix equation, the unknown electric field in the target solution area, the surface current unknowns and the surface tangential electric field unknowns of the auxiliary metasurface to be solved, and the surface current unknowns and the surface magnetic current unknowns on the boundary surface of the integral equation are combined as the unknown quantity x; the expansion coefficients of the unknown quantities are iteratively solved by the generalized minimum residual method, and the electric field values ​​of the unknown electric field in the target solution area are inferred based on the expansion coefficients of the unknown quantities.

[0084] In step 5, during the process of solving the matrix elements, if the interface meshes on both sides of the metasurface to be solved, i.e., the interface meshes of the two dielectric substrates, are non-conformal, the matrix The corresponding elements in the grid are accurately solved through the union mesh technology of mesh encryption.

[0085] In step 5, the specific process of incorporating the generalized thin-sheet transport condition into the finite element solution framework to construct the FEM-GSTCs system matrix equation is as follows:

[0086] For three-dimensional vector finite element electromagnetic scattering problems, the following vector wave equation is satisfied:

[0087]

[0088] in, represents the curl to be solved; μ r and ε r are the relative magnetic permeability and relative dielectric constant of the dielectric substrate respectively; Ω i is the internal sub-region of the i-th finite element, i = 1, 2, Ω1 is the region where the dielectric substrate is located above the metasurface to be solved, and Ω2 is the region where the dielectric substrate is located below the metasurface to be solved; E i For the region Ω i The electric field inside is unknown;

[0089] Applying the generalized thin-sheet transmission condition to the zero-thickness thin sheet, i.e., the interface grid of two dielectric substrates, can describe the mutation introduced by the metasurface to be solved. The generalized thin-sheet transmission condition introduces the surface polarizability and combines it with the Maxwell equations to establish a set of strict boundary conditions to describe the interaction between the metasurface to be solved and the electromagnetic wave. The metasurface to be solved is placed perpendicular to the z-axis, its thickness is set to be approximately zero, and the effect of the normal polarization component is ignored, which can be written as follows:

[0090]

[0091]

[0092] in, is the unit vector in the z direction, E′ i For the region Ω i The electric field on the inner edge of the metasurface to be solved, H i For the region Ω i The magnetic field of the inner edge of the metasurface to be solved; μ and ε are the magnetic permeability and dielectric constant of the dielectric substrate, and ω is the angular frequency; is the electric susceptibility tensor excited by the magnetic field, is the magnetic susceptibility tensor of electric field excitation, is the magnetic susceptibility tensor of magnetic field excitation, all of which are zero matrices;

[0093] The unknown surface tangential electric field e is introduced, and the unknown surface tangential electric field along the hypersurface to be solved in regions Ω1 and Ω2 is expressed as e1 = n1 × E′1 × n1 and e2 = n2 × E′2 × n2 respectively; the auxiliary surface current unknown J is introduced, and the unknown surface current along the hypersurface to be solved in regions Ω1 and Ω2 is expressed as n1 is the unit normal vector of the hypersurface pointing to the area Ω1, and n2 is the unit normal vector of the hypersurface pointing to the area Ω2.

[0094] Furthermore, the generalized thin-sheet transport condition can be simplified as:

[0095]

[0096] e1=e2

[0097] Where η0 is the free space wave impedance.

[0098] The surface integral equations EFIE and MFIE on the integral equation boundary surface in step 2 can accurately satisfy the Sommerfeld far-field radiation boundary condition, which can be expressed as:

[0099]

[0100]

[0101] in, is the unit normal vector pointing from the boundary surface of the integral equation to the infinite external region; E is the unknown electric field in the infinite external region of the boundary surface of the integral equation; J and M are the unknown surface current and surface magnetic current on the boundary surface of the integral equation, respectively; E inc is the electric field of the incident plane wave, J inc To generate the electric field E inc The current source; L and are integral operators, respectively expressed as:

[0102]

[0103]

[0104] in, is the boundary surface of the target solution area, for Any grid of; X is the independent variable, r and r′ represent the field point vector and the source point vector respectively; represents the divergence of r′, It means to find the gradient; represents the removal of singular terms of the integral operator K; G0(r, r′) is the scalar Green's function of free space, which is expressed as:

[0105]

[0106] Among them, || is the modulo operation.

[0107] The Galerkin test is performed on the above-mentioned vector wave equation, generalized thin sheet transport condition and surface integral equation on the boundary surface of the integral equation. The unknown quantities in the equation are expanded using basis functions, and the following FEM-GSTCs equation group is obtained, which is expressed as:

[0108]

[0109]

[0110]

[0111]

[0112] in,

[0113]

[0114] Where N m and N n They represent the vector edge element basis functions defined on the mth edge and the nth edge in any grid within the target solution area Ω; m and Λ n They represent the RWG basis functions with the mth edge and the nth edge as common edges in the target solution region Ω; and Respectively represent the region Ω i The vector edge element basis functions defined on the mth edge and the nth edge in any grid; and Respectively represent the region Ω i The RWG basis function with the mth edge and the nth edge as the common edge; <, > means to find the inner product; Represents area Ω i The boundary surface, dΩ i Represents area Ω i Any grid within; x E is the expansion coefficient of the unknown electric field in the target solution area, x j is the expansion coefficient of the unknown surface current, x M is the expansion coefficient of the unknown surface magnetic current, is the expansion coefficient of the unknown quantity of the surface tangential electric field.

[0115] The FEM-GSTCs equations can be rewritten as form as the matrix equation of the FEM-GSTCs system; the generalized minimum residual method GMRES is used to iteratively solve the matrix equation of the FEM-GSTCs system, and the iterative convergence threshold is set to 0.001 to obtain the expansion coefficients of the unknown quantities; the electric field values ​​of the unknown quantities in the target solution area are inferred based on the expansion coefficients of the unknown quantities.

[0116] Step 6. Use the electric field values ​​of the unknown electric field quantities in the target solution area to calculate the radar scattering cross section of the target structure to be solved in the far field area.

[0117] In this embodiment, the solved electric field value is substituted into the far-field scattering field formula to output the RCS distribution. The number of unknowns in this embodiment is 75857, the iteration is 40 steps to converge, the calculation time is 236s, the peak memory usage is 2.6GB, and the dual-station RCS calculation results using the actual physical model and finite element method for electromagnetic simulation are compared as follows Figure 6 The results are highly consistent. The computation time and memory consumed by the two methods are compared, as shown in Table 1.

[0118] Table 1 Comparison of peak calculation memory and calculation time corresponding to calculation methods

[0119] Calculation method Peak computing memory (GB) Calculation time(s) FEM 11.4 1012 FEM-GSTCs 2.6 236

[0120] In this embodiment, after the metasurface is equivalently modeled as a zero-thickness thin sheet, the surface polarizability tensor equivalent model discretized by a coarse grid can be used to replace the actual physical metasurface discretized by a fine grid, such as Figure 5 As shown in the figure, the meshing of the tangential part of the hypersurface can be significantly reduced. Since the hypersurface exists in the composite solution area as a zero-thickness sheet, the longitudinal meshing is reduced overall. This homogenization method can not only simplify the pre-processing process, but also avoid the generation of dense meshes, saving computing time and memory consumption.

[0121] The finite element method can accurately and efficiently solve electromagnetic computing problems of complex targets, and the generalized thin-sheet transport condition can better describe the interaction between the metasurface and the electromagnetic field compared to other boundary conditions. The new computing method that combines the finite element method and the generalized thin-sheet transport condition can efficiently and accurately analyze the electromagnetic scattering characteristics of thin-layer metasurface structures or larger-scale systems that include metasurface detailed structures, providing a feasible new approach for electromagnetic simulation calculations of complex structural targets.

[0122] The comparison of the dual-station RCS calculation results of the present invention and the comparison of the computing resources and time consumption using the actual physical model and the finite element method for electromagnetic simulation shows that the method for calculating the electromagnetic scattering characteristics of the metasurface based on the finite element method and the generalized thin-sheet transmission condition of the present invention improves the calculation efficiency of the thin-layer metasurface structure while ensuring accuracy. The non-uniform metasurface is equivalent to a surface model with uniform electromagnetic characteristics, which saves a lot of memory and time resources, is no longer limited by the geometric details of the actual physical model, reduces the consumption of computing resources, expands the use scenarios of the finite element method, and provides a feasible new way for electromagnetic calculation of complex structural targets.

Claims

1. A method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and generalized thin-sheet transmission conditions, characterized in that: The following steps are involved: Step 1. The metasurface to be solved includes a number of units arranged in two dimensions. An infinite periodic boundary condition simulation is performed on the units to obtain the transmission coefficient of the units, and the surface polarizability corresponding to the units is extracted according to the transmission coefficient; Step 2. The metasurface to be solved is equivalently modeled as a zero-thickness thin sheet, and the physical characteristics of the zero-thickness thin sheet are characterized by the surface polarizability in step 1; a dielectric substrate is loaded on the upper and lower surfaces of the zero-thickness thin sheet, and the overall structure is used as the target structure to be solved; On the surface of the target structure to be solved, an integral equation boundary surface is established, the infinite solution area is truncated, and the integral equation boundary surface and the internal area are used as the target solution area; Step 3. Mesh the target structure to be solved based on the finite element method, and export the mesh information of the meshed model; Step 4. Configure the frequency, amplitude, phase, angle and polarization direction information of the incident plane wave; Step 5. Read the grid information; incorporate the generalized thin film transport condition into the finite element solution framework and construct the FEM-GSTCs system matrix equation The matrix The elements of the sum vector b are the exact integral solutions obtained by the global matrix method; In the matrix equation of the FEM-GSTCs system, the unknown electric field in the target solution area, the unknown surface current and surface tangential electric field of the auxiliary metasurface, and the unknown surface current and surface magnetic current on the boundary surface of the integral equation are combined as the unknown quantity x, and the unknown quantity on each grid is expanded through the basis function; the expansion coefficient of the unknown quantity is iteratively solved by the generalized minimum residual method, and the electric field value of the unknown electric field in the target solution area is inferred based on the expansion coefficient of the unknown quantity; Step 6. Use the electric field values ​​of the unknown electric field quantities in the target solution area to calculate the radar scattering cross section of the target structure to be solved in the far field area.

2. The method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition according to claim 1 is characterized in that: In step 1, the unit of the metasurface to be solved is extended along the x-axis and y-axis directions. The unit includes a thin dielectric substrate and a metal disc on its upper surface. The unit period is 3 mm, and the radius of the metal disc is 1.3 mm. The relative dielectric constant of the thin dielectric substrate is 4 and the thickness is 0.3 mm. The transmission coefficient of the unit is simulated at a frequency of 10 GHz, where the electric susceptibility tensor of the electric field excitation is It is expressed as: Simplify the nine components in the above formula into two components, namely and The remaining components are set to zero, and are expressed as follows: Where j is the sign of the imaginary part, k0 is the free space wave number; R TM (0) represents the reflection coefficient of the vertically incident wave along TM polarization, T TM (0) represents the transmission coefficient of the vertically incident wave along TM polarization; R TE (0) represents the reflection coefficient of the vertical incident wave along TE polarization, T TE (0) represents the transmission coefficient for a vertically incident wave along TE polarization.

3. The method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition according to claim 1, characterized in that: In step 2, the unit is expanded 16 times along the x-axis and y-axis respectively to form a 16×16 two-dimensional arrangement of the metasurface to be solved, which is placed between two dielectric substrates with a thickness of 3 mm and a size of 48 mm×48 mm. The overall structure is used as the target structure to be solved; the two dielectric substrates have the same structure and the relative dielectric constant is 2.

2.

4. The method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition according to claim 1, characterized in that: In step 3, the zero-thickness sheet and the integral equation boundary surface are meshed with triangles, and the other areas in the target solution area, namely the areas where the two dielectric substrates are located, are meshed with tetrahedral meshes; the areas where the two dielectric substrates are located are meshed independently.

5. The method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition according to claim 1, characterized in that: In step 4, the frequency of the incident plane wave is set to 10 GHz and the incident angle is and θ are the azimuth and elevation angles respectively.

6. The method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition according to claim 2, characterized in that: In step 5, in the matrix In the process of solving the elements of , if the interface meshes on both sides of the metasurface to be solved, i.e. the interface meshes of the two dielectric substrates, are non-conformal, the matrix The corresponding elements in are solved by the bonded fine mesh method with mesh refinement.

7. The method for calculating the electromagnetic scattering characteristics of a metasurface based on the finite element method and the generalized thin-sheet transmission condition according to claim 6, characterized in that: In step 5, the specific process of incorporating the generalized thin-sheet transport condition into the finite element solution framework to construct the FEM-GSTCs system matrix equation is as follows: For three-dimensional vector finite element electromagnetic scattering problems, the following vector wave equation is satisfied: in, represents the curl to be solved; μ r and ε r are the relative magnetic permeability and relative dielectric constant of the dielectric substrate respectively; Ω i is the internal sub-region of the i-th finite element, i = 1, 2, Ω1 is the region where the dielectric substrate is located above the metasurface to be solved, and Ω2 is the region where the dielectric substrate is located below the metasurface to be solved; E i For the region Ω i The electric field inside is unknown; For a zero-thickness sheet, i.e., the interface grid of two dielectric substrates, the generalized sheet transmission condition is applied to describe the mutation introduced by the metasurface to be solved. The generalized sheet transmission condition is written as follows: in, is the unit vector in the z direction, E′ i For the region Ω i The electric field on the inner edge of the metasurface to be solved, H i For the region Ω i The magnetic field of the inner edge of the metasurface to be solved; μ and ε are the magnetic permeability and dielectric constant of the dielectric substrate, and ω is the angular frequency; is the electric susceptibility tensor excited by the magnetic field, is the magnetic susceptibility tensor of electric field excitation, is the magnetic susceptibility tensor of magnetic field excitation, all of which are zero matrices; The unknown surface tangential electric field e is introduced, and the unknown surface tangential electric field along the hypersurface to be solved in regions Ω1 and Ω2 is expressed as e1 = n1 × E′1 × n1 and e2 = n2 × E′2 × n2 respectively; the auxiliary surface current unknown J is introduced, and the unknown surface current along the hypersurface to be solved in regions Ω1 and Ω2 is expressed as n1 is the unit normal vector of the hypersurface pointing to the region Ω1, and n2 is the unit normal vector of the hypersurface pointing to the region Ω2; Furthermore, the generalized thin-sheet transport condition can be simplified as: e1=e2 Where η0 is the free space wave impedance; The surface integral equations EFIE and MFIE on the integral equation boundary surface in step 2 are expressed as: in, is the unit normal vector pointing from the boundary surface of the integral equation to the infinite external region; E is the unknown electric field in the infinite external region of the boundary surface of the integral equation; J and M are the unknown surface current and surface magnetic current on the boundary surface of the integral equation, respectively; E inc is the electric field of the incident plane wave, J inc To generate the electric field E inc The current source L and are integral operators, respectively expressed as: in, is the boundary surface of the target solution area, for Any grid of; X is the independent variable, r and r′ represent the field point vector and the source point vector respectively; represents the divergence of r′, It means to find the gradient; represents the removal of singular terms of the integral operator K; G0(r, r′) is the scalar Green's function of free space, which is expressed as: Among them, || is the modulo operation; The Galerkin test is performed on the above-mentioned vector wave equation, generalized thin sheet transport condition and surface integral equation on the boundary surface of the integral equation. The unknown quantities in the equation are expanded using basis functions, and the following FEM-GSTCs equation group is obtained, which is expressed as: in, Where N m and N n They represent the vector edge element basis functions defined on the mth edge and the nth edge in any grid within the target solution area Ω; m and Λ n They represent the RWG basis functions with the mth edge and the nth edge as common edges in the target solution region Ω; and Respectively represent the region Ω i The vector edge element basis functions defined on the mth edge and the nth edge in any grid; and Represents the region Ω i The RWG basis function with the mth edge and the nth edge as the common edge; <, > means to find the inner product; Represents area Ω i The boundary surface, dΩ i Represents area Ω i Any grid within; x E is the expansion coefficient of the unknown electric field in the target solution area, x j is the expansion coefficient of the unknown surface current, x M is the expansion coefficient of the unknown surface magnetic current, is the expansion coefficient of the unknown quantity of the surface tangential electric field; The FEM-GSTCs equations can be rewritten as form, as the matrix equation of the FEM-GSTCs system.

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