Broadband design method for single-layer frequency selective surfaces based on topology optimization

Abstract

The present invention discloses a single-layer frequency selective surface broadband design method based on topology optimization method, which belongs to the field of electromagnetic spatial filtering and stealth technology. The method of the present invention derives the accompanying sensitivity formula of the periodic unit wave transmission coefficient based on the frequency domain finite element method, which can optimize the frequency selectivity of the periodic unit; by interpolating and fitting the wave transmission coefficient, the frequency-varying S is realized. 21 The rapid calculation of the curve improves the optimization efficiency and performance of the topology optimization method for broadband frequency selective surfaces. The designed unit operating frequency band completely covers the entire Ku band, which effectively achieves the design goals. It takes into account broadband wave transmission and wide-angle stability while maintaining a simple structure. In addition, this method, as an optimization and supplement to the general topology optimization method, is suitable for the design of various types of thin-layer periodic units.

Description

Technical Field

[0001] The present invention belongs to the field of electromagnetic spatial filtering and stealth technology, and specifically relates to a single-layer frequency selective surface broadband design method based on a topology optimization method. Background Art

[0002] A frequency selective surface (FSS) is a periodic structure arranged on a two-dimensional plane, primarily composed of metal layers supported by a single or multi-layer dielectric substrate. As a spatial filter, it exhibits selectivity for electromagnetic waves and is widely used in reflective antennas, radomes, and on the surfaces of ships and aircraft. As the electromagnetic environment becomes increasingly complex, broadband bandpass FSSs with excellent angular stability have attracted increasing attention. However, in practical applications, the thickness of FSSs is often significantly limited. Thin, simple FSSs are often unaffected by processing techniques and the operating environment, allowing them to more stably exert their filtering properties.

[0003] Chinese patent application number CN202311780852.2 discloses an ultra-wideband Ka-band frequency selective surface with steeply dropping sidebands. This surface achieves broadband passband by introducing multiple transmission poles within the passband and rapid roll-off into the stopband by introducing transmission zeros on either side of the passband. While achieving good passband edge roll-off, the surface utilizes a complex structure with three metal layers and two dielectric layers, resulting in poor angular stability.

[0004] Chinese patent application number CN202310809939.1 discloses a high-order, broadband, miniaturized frequency selective surface (MFSS) based on a knitted structure. This technology improves filtering characteristics by replacing the traditional metal wireframe with a knitted structure based on a conventional MFSS. This prior art exhibits good broadband bandpass characteristics and mitigates the effects of TM waves at wide angles. However, its numerous metal layers and complex interlayer structure limit its application scenarios.

[0005] Chinese patent application number CN202311391179.3 discloses a low-insertion-loss, wide-bandwidth adjustable active frequency selective surface (AFSS). This technology utilizes a varactor diode and a feed metal structure to achieve wideband passband tunability. This prior art boasts a wide adjustable bandwidth, low insertion loss, and covers both the C-band and S-band. However, varactor diodes are more expensive and less reliable than FSSs, hindering widespread civilian use and limiting their application scenarios.

[0006] Chinese patent application number CN202211598303.9 discloses an ultra-wideband reconfigurable frequency selective surface with wide incidence angles and polarization insensitivity. By adjusting the on / off state of the PIN transistors, broadband shielding and transmission can be switched. This prior art achieves full polarization shielding and transmission of electromagnetic waves, and its three-layer structure offers wide bandwidth and good oblique-incidence performance. However, the structure lacks a bias line for voltage adjustment, requiring an external bias line to switch the PIN transistors on and off, which is not conducive to practical application.

[0007] Existing FSS design methods mostly rely on experience to select the initial structure and further optimize dimensional parameters using equivalent circuit methods. Since the metal layers in the initial structure typically use conventional patterns, there are fewer dimensional parameters that can be optimized, and the performance of single-layer metal patterns is limited. Designing broadband bandpass frequency selective surfaces often relies on multilayer structures or tunable diodes to achieve functionality. These existing technologies have certain limitations, primarily due to their overly complex structures and their inability to balance angular stability, often affected by the operating environment. Therefore, existing design methods are not universally applicable. In summary, achieving wide-angle stability within a simple structure has become a challenge in broadband FSS design. Summary of the Invention

[0008] The purpose of the present invention is to overcome the defects of the above-mentioned prior art and provide a single-layer frequency selective surface broadband design method based on the topology optimization method. The topology optimization method is used to realize the broadband design of the single-layer frequency selective surface in the Ku band. The structure is simple while taking into account the angular stability. It is independent of the frequency band and the dielectric substrate and is suitable for the design of various types of frequency selective surfaces.

[0009] The technical problem proposed by the present invention is solved as follows:

[0010] A single-layer frequency selective surface broadband design method based on a topology optimization method includes the following steps:

[0011] Step 1: The frequency selective surface model to be optimized includes a dielectric substrate, an upper surface metal layer located on the upper surface of the dielectric substrate, and a lower surface metal layer located on the lower surface of the dielectric substrate; the upper surface metal layer and the lower surface metal layer of the frequency selective surface model to be optimized are selected as the area to be optimized, and an initial model of the area to be optimized is established;

[0012] Use the frequency domain finite element method to discretize the Maxwell equations into a linear equation system corresponding to the initial model:

[0013] Kc=r (1)

[0014] Where K is the system matrix of the linear equations, c is the coefficient vector of the electric field basis function, r is the excitation vector of the linear equations, and the corresponding adjoint equations are:

[0015] Kv=b (2)

[0016] Where v is the coefficient of the electric field basis function in the adjoint equation, and b is the excitation vector of the adjoint equation;

[0017] Step 2: Set the frequency selection optimization function objective function f0:

[0018]

[0019] Where Re[·] indicates taking the real part, superscript * indicates taking the conjugate, superscript m indicates the m-mode of the Floquet port, and E m represents the modal electric field of the Floquet port in the m-mode, H m represents the modal magnetic field of the Floquet port in the m-mode, dS is the differential vector element, ∫ port (·)dS represents the integration of the Floquet port;

[0020] The excitation vector r of the linear equations corresponding to the frequency selection optimization function f0 and the excitation vector b of the adjoint equation are:

[0021]

[0022]

[0023] Where T represents the finite element global basis function, the subscript in represents the incident source, is the tangential current source in m mode at the Floquet port, is the tangential magnetic current source in the m-mode at the Floquet port, J e，in is the incident current source at the Floquet port, J m，in is the incident magnetic flux source at the Floquet port; is the Hamiltonian operator, is the normal vector of the corresponding surface element of the Floquet port, j is the sign of the imaginary part, ω represents the angular frequency, and μ represents the magnetic permeability;

[0024] Step 3: Construct the inverse of the transmission coefficient of the frequency selective surface model to be optimized

[0025]

[0026] Among them, Q(ω) represents the The interpolation function for interpolating the imaginary part of n Represents the nth order coefficient of the angular frequency ω;

[0027] The fitting is achieved by solving the least square solution of Q(ω) by interpolation, and then the S under different angular frequencies is realized by formula (12). 21 Calculation of

[0028] Step 4: Calculate the gradient of the frequency selection optimization function

[0029]

[0030] Where p is the optimization variable vector, c T Indicates taking the transpose of c;

[0031] Step 5: Use the variable density method and transition boundary conditions to establish the optimization model of the area to be optimized, and use the conductivity of the area to be optimized as the optimization variable vector;

[0032] The optimization model is expressed as:

[0033]

[0034]

[0035] The optimization model of the conductive film with variable conductivity corresponding to the upper surface metal layer or the lower surface metal layer in the area to be optimized is given by equations (15) and (16), where J s1 and J s2 Respectively represents the current intensity on both sides of the conductive film corresponding to the upper surface metal layer or the lower surface metal layer, E t1 and E t2 represents the tangential electric field intensity on both sides of the conductive film corresponding to the upper surface metal layer or the lower surface metal layer, d is the set virtual thickness of the conductive film, and σ is the conductivity of the conductive film;

[0036] Step 6: Substitute the excitation vector r corresponding to formula (4) and the excitation vector b corresponding to formula (5) into the linear equations corresponding to formula (1) and the adjoint equation corresponding to formula (2) in step 1 to obtain the coefficient vector c of the electric field basis function and the solution of the coefficient vector v of the electric field basis function in the adjoint equation. Substitute the solutions of vector c and vector v into step 4 to obtain the gradient of the frequency selection optimization function with respect to the optimization variable vector.

[0037] Step 7: Use the moving asymptote method as the gradient optimization method, and use the gradient obtained in step 6 to update the optimization variable vector iteratively; after each generation of update, use step 3 to solve S 21 The curve is used to obtain the -3dB bandwidth of the current frequency selective surface model; when the set cutoff condition is reached, the update iteration is stopped and the current frequency selective surface model is output as the final frequency selective surface model.

[0038] Furthermore, in step 3, the equivalent impedance of the wave-transmitting frequency selective surface model is expressed as:

[0039]

[0040] Among them, Z F is the equivalent impedance of the frequency selective surface, C is the equivalent capacitance of the frequency selective surface, L1 and L2 are the equivalent inductors of the frequency selective surface respectively;

[0041] Transmission coefficient S 21 The expression is:

[0042]

[0043] Among them, S 21 represents the transmission coefficient between Floquet port 1 and port 2, Z0 is the free space wave impedance, and S 21 The reciprocal of is:

[0044]

[0045] Take the above formula in The fourth-order Taylor expansion at is:

[0046]

[0047] Taking the fourth-order Taylor expansion of the above formula at ω=ω1=0, we get:

[0048]

[0049] Take the above formula in The fourth-order Taylor expansion at is:

[0050]

[0051] Among them, in formula (9), formula (10) and formula (11), Indicates that m The coefficient of the nth-order term expanded at ;

[0052] Combining equations (9), (10) and (11), Construct it and get:

[0053]

[0054] Furthermore, in step 3, the least squares solution of Q(ω) is obtained by interpolation to achieve fitting:

[0055]

[0056] in, Indicates the kth angular frequency interpolation point, 1≤k≤K, K≥6, express The corresponding Q(ω) value;

[0057] The formula (13) is used to realize The fitting of S at different angular frequencies is then realized through formula (12). 21 Calculation.

[0058] Furthermore, in step 7, the cutoff condition is set as follows: S 21 The -3dB bandwidth covers the Ku band.

[0059] The beneficial effects of the present invention are:

[0060] Based on the frequency domain finite element method, the present invention derives the sensitivity formula of the frequency selective surface transmission coefficient (i.e., formula (4) and formula (5)), and combines step 3 to quickly solve S 21 This method can obtain the bandwidth of the frequency selective surface without frequency sweeping, which can greatly reduce the time of topology optimization and achieve broadband wave transmission and wide-angle stability while keeping the structure simple.

[0061] Compared with the existing technology, the present invention optimizes structural parameters instead of dimensional parameters, can optimize more variables, and has better performance of single-layer metal patterns. Only one layer of medium can achieve broadband wave transmission, the structure is simpler and more reliable, and the design also takes into account angular stability.

[0062] Compared with the global optimization algorithm and proxy model method in the prior art, the optimization convergence time required by the method described in the present invention is shorter; compared with the existing topology optimization algorithm, the method described in the present invention converges faster and the final structural performance obtained by optimization is better. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 This is a circuit model diagram of the frequency selective surface equivalent impedance in step 3 of the method described in the embodiment;

[0064] Figure 2 It is a structural schematic diagram of an optimization example in the method described in the embodiment;

[0065] Figure 3 Schematic diagram of the top surface patch structure of the optimized example in the method described in the embodiment;

[0066] Figure 4 Schematic diagram of the bottom surface patch structure of the optimized example in the method described in the embodiment;

[0067] Figure 5 TE polarization performance curves of the optimized example at different oblique incident angles in the method described in the embodiment;

[0068] Figure 6 The figure is a performance curve diagram of TM polarization at different oblique incident angles of the optimized example in the method described in the embodiment. DETAILED DESCRIPTION

[0069] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0070] This embodiment provides a single-layer frequency selective surface broadband design method based on a topology optimization method. Under periodic boundaries, sensitivity analysis of the frequency selective surface is implemented, and the wave transmission passband of the frequency selective surface is fitted in combination with an interpolation algorithm, thereby achieving broadband wave transmission while maintaining wide-angle stability.

[0071] The method described in this embodiment specifically includes the following steps:

[0072] Step 1: Select the upper surface metal layer and the lower surface metal layer of the frequency selective surface model to be optimized as the area to be optimized, and establish an initial model of the area to be optimized;

[0073] Use the frequency domain finite element method to discretize the Maxwell equations into a linear equation system corresponding to the initial model:

[0074] Kc=r (1)

[0075] Where K is the system matrix of the linear equations, c is the coefficient vector of the electric field basis function, r is the excitation vector of the linear equations, and the corresponding adjoint equations are:

[0076] Kv=b (2) Where v is the coefficient of the electric field basis function in the adjoint equation, and b is the excitation vector of the adjoint equation.

[0077] Step 2: Set the objective function:

[0078]

[0079] Where f0 is the frequency selection optimization function, Re[·] means taking the real part, superscript * means taking the conjugate, superscript m means the m mode of the Floquet port, E m represents the modal electric field of the Floquet port in the m-mode, H m represents the modal magnetic field of the Floquet port in the m-mode, dS is the differential vector element, ∫ port (·)dS represents the integration of the Floquet port. The above formula is used to express the power received by the Floquet port.

[0080] The excitation vector r of the linear equations corresponding to the frequency selection optimization function f0 and the excitation vector b of the adjoint equation are:

[0081]

[0082] Where T represents the finite element global basis function, subscript in represents the incident source, and J e，in is the incident current source at the Floquet port, J m，in is the incident magnetic flux source at the Floquet port, is the tangential current source in m mode at the Floquet port, is the tangential magnetic current source in the m-mode at the Floquet port, is the Hamiltonian operator, is the normal vector of the surface element corresponding to the Floquet port, j is the sign of the imaginary part, ω represents the angular frequency, and μ represents the magnetic permeability.

[0083] Step 3: Quickly calculate the wave transmission coefficient of the frequency selective surface in the working frequency band;

[0084] The equivalent circuit model of the general wave-transmitting FSS model is as follows: Figure 1 As shown, the equivalent impedance can be expressed as:

[0085]

[0086] Among them, Z F is the equivalent impedance of the frequency selective surface, C is the equivalent capacitance of the equivalent circuit model, L1 and L2 are the equivalent inductors in the equivalent circuit model;

[0087] Corresponding transmission coefficient S 21 The expression is:

[0088]

[0089] Among them, S 21 represents the transmission coefficient between Floquet port 1 and port 2, Z0 is the free space wave impedance, and S 21 The reciprocal of is:

[0090]

[0091] Take the above formula in The fourth-order Taylor expansion at can be obtained:

[0092]

[0093] Taking the fourth-order Taylor expansion of the above formula at ω=ω1=0, we can get:

[0094]

[0095] Take the above formula in The fourth-order Taylor expansion at can be obtained:

[0096]

[0097] Among them, in formula (9), formula (10) and formula (11), Indicates that m The coefficient of the nth-order term expanded at .

[0098] Combining equations (9), (10) and (11), Construct it and get:

[0099]

[0100] Among them, Q(ω) represents the The interpolation function for interpolating the imaginary part of n Represents the coefficient of the nth-order term.

[0101] The fitting can be achieved by interpolating a sufficient number of points and solving the least squares solution:

[0102]

[0103] in, Indicates the kth angular frequency interpolation point, 1≤k≤K, K≥6, express The corresponding Q(ω) value.

[0104] Formula (13) can be used to achieve The fitting of S at different angular frequencies is further realized. 21 Fast calculation of .

[0105] Step 4: Calculate the gradient of the objective function:

[0106]

[0107] Where p is the optimization variable vector, c T It means taking the transpose of c.

[0108] Step 5: Use the variable density method and transition boundary conditions to establish the optimization model of the area to be optimized, and use the conductivity of the area to be optimized as the optimization variable vector;

[0109] The expression of the optimization model is:

[0110]

[0111] The area to be optimized on the upper or lower surface of the frequency selective surface unit can be regarded as an optimization model of a conductive film with variable conductivity through formulas (15) and (16), where J s1 、J s2Respectively represents the current intensity on both sides of the conductive film corresponding to the upper surface or lower surface, E t1 、E t2 They represent the tangential electric field strength on both sides of the conductive film corresponding to the upper surface or lower surface, d is the virtual thickness of the conductive film (the actual thickness is 0), and σ is the conductivity of the conductive film.

[0112] Step 6: Substitute the excitation vector r corresponding to formula (4) and the excitation vector b corresponding to formula (5) into the linear equations corresponding to formula (1) and the adjoint equation corresponding to formula (2) in step 1 to obtain the coefficient vector c of the electric field basis function and the solution of the coefficient vector v of the electric field basis function in the adjoint equation. Substitute the solution into step 4 to obtain the gradient of the objective function with respect to the optimization variable vector.

[0113] Step 7: Use the moving asymptote method as the gradient optimization method, and use the gradient obtained in step 6 to update the optimization variable vector iteratively; after each generation of update, use the S in step 3 21 Fast calculation method to solve S 21 The curve is used to further obtain the -3dB bandwidth. When the oblique incidence is 60°, S 21 -3dB bandwidth is used as the criterion for judging convergence; until the newly obtained S 21 Until the -3dB bandwidth covers the Ku band, the new optimization variable vector meets the convergence condition and a frequency selective surface model that meets the requirements is obtained.

[0114] In this embodiment, the metal distribution on the upper and lower surfaces of the frequency selective surface is optimized. When using the variable density method for topology optimization of metal, its material properties are usually optimized. The conductivity of the area to be optimized is used as the optimization variable. By using an exponential function for function mapping, the conductivity can smoothly transition between air (conductivity is 0) and metal (conductivity is close to infinity). Therefore, the conductivity of the area to be optimized can be expressed as:

[0115]

[0116] Among them, p∈[0,1] is the optimization variable, and are the upper and lower limits of conductivity, respectively.

[0117] The optimized frequency selective surface example model is a square periodic structure with a dielectric parameter of 3.2, a loss tangent of 0.005, a thickness of 0.25 mm, and a unit length and width of 5.5 mm. The optimized area is the metal on the upper and lower surfaces of the frequency selective surface unit. By optimizing the metal arrangement, the transmission band is fully covered in the Ku band under the 0N60° oblique incidence range, achieving in-band transmission and out-of-band suppression. The objective function is set to

[0118] ftotal =-(-f 10GHz -f 11.5GHz +f 12.5GHz +f 14.5GHz +f 15GHz

[0119] +f 16.5GHz +f 17.5GHz -f 18.5GHz -f 20GHz )

[0120] Among them, f total represents the overall objective function, f αGHz represents the wave transmission coefficient at a frequency of α GHz. This objective function is to improve the wave transmission efficiency at 12.5 GHz, 14.5 GHz, 15 GHz, 16.5 GHz, and 17.5 GHz, and reduce the wave transmission efficiency at 10 GHz, 11.5 GHz, 18.5 GHz, and 20 GHz, thereby achieving the goals of in-band transmission and out-of-band suppression in the Ku band.

[0121] In this example, the frequency domain finite element method is selected as the numerical simulation method, and the linear equation group (1) corresponding to the optimized frequency selective surface model can be obtained by discretizing the Maxwell equation group; combined with the variable density method and the transition boundary condition, the optimization model of the metal layer on the upper surface and the lower surface can be established.

[0122] Taking 10 GHz, 11.5 GHz, 12.5 GHz, 14.5 GHz, 15 GHz, 16.5 GHz, 17.5 GHz, 18.5 GHz and 20 GHz as sampling points, the excitation vector r of the linear equation group and the excitation vector b of the adjoint equation at each frequency are obtained according to formula (3). Substituting them into the linear equation group and its adjoint equation in step 1, a pair of solutions c and v are obtained.

[0123] Substitute the c obtained at each frequency into formula (3) to obtain the wave transmission coefficient of the frequency selective surface at each frequency, and use formula (13) to quickly interpolate formula (12) to obtain S that changes with frequency: 21 The curve is used to judge the convergence conditions. If the convergence conditions are met, the structure corresponding to the output parameters is used as the final result. If the convergence conditions are not met, the solutions c and v at each frequency are substituted into the gradient formula of the objective function to obtain the gradient value corresponding to the optimization variable. Then, the moving asymptote method is used as the gradient optimization method to update the old optimization variables and obtain new optimization variables until the new S is obtained. 21 The -3dB bandwidth covers the Ku band.

[0124] Figure 2The model structure obtained after the optimization method described in this embodiment converges, the optimized frequency selective surface unit includes an upper surface metal layer, a dielectric substrate and a lower surface metal layer, Figure 3 and Figure 4 They are top views of the upper surface metal layer and the lower surface metal layer respectively.

[0125] Figure 5 is the S of the TE mode at different incident angles in the optimization example described in this embodiment. 21 curve, Figure 6 is the S of the TM mode at different incident angles in the optimization example described in this embodiment. 21 The curve shows that in the range of TE polarization oblique incidence of 0° to 60°, the -3dB bandwidth is 11.78GHz to 18.48GHz, and in the range of TM polarization oblique incidence of 0° to 60°, the -3dB bandwidth is 10.78GHz to 18.79GHz. The two polarizations completely cover the entire Ku band.

[0126] In summary, the present invention discloses a single-layer frequency selective surface broadband design method based on topology optimization method. The present invention derives the accompanying sensitivity analysis method of the periodic unit wave transmission coefficient and adopts the S 21 Interpolation fitting method to quickly calculate S that changes with frequency 21 The curve greatly reduces the number of sampling points within the broadband, shortens the time required for topology optimization, and further improves the optimization efficiency and performance of broadband frequency selective surfaces. It can be used to design various thin-layer frequency selective surface units, expanding the scope of application of topology optimization methods in frequency selective surface unit design. A case study was designed to achieve full coverage of the Ku band under oblique incidence of 0 to 60 degrees, achieving in-band transmission and out-of-band suppression. The optimized frequency selective surface unit has good performance, with a -3dB bandwidth of 11.78 GHz to 18.48 GHz for TE polarization and 10.78 GHz to 18.79 GHz for TM polarization. Both polarizations of the unit fully cover the entire Ku band, effectively achieving the design goal, while maintaining a simple structure and taking into account broadband transmission and wide-angle stability.

Claims

1. A single-layer frequency selective surface broadband design method based on topology optimization method, characterized in that: The following steps are involved: Step 1: The frequency selective surface model to be optimized includes a dielectric substrate, an upper surface metal layer located on the upper surface of the dielectric substrate, and a lower surface metal layer located on the lower surface of the dielectric substrate; the upper surface metal layer and the lower surface metal layer of the frequency selective surface model to be optimized are selected as the area to be optimized, and an initial model of the area to be optimized is established; Use the frequency domain finite element method to discretize the Maxwell equations into a linear equation system corresponding to the initial model: Kc=r (1) Where K is the system matrix of the linear equations, c is the coefficient vector of the electric field basis function, r is the excitation vector of the linear equations, and the corresponding adjoint equations are: Kv=b(2) Where v is the coefficient of the electric field basis function in the adjoint equation, and b is the excitation vector of the adjoint equation; Step 2: Set the frequency selection optimization function objective function f0: Where Re[.] indicates taking the real part, the superscript * indicates taking the conjugate, the superscript m indicates the m-mode of the Floquet port, and E m represents the modal electric field of the Floquet port in the m-mode, H m represents the modal magnetic field of the Floquet port in the m-mode, dS is the differential vector element, ∫ port (·)dS represents the integration of the Floquet port; The excitation vector r of the linear equations corresponding to the frequency selection optimization function f0 and the excitation vector b of the adjoint equation are: Where T represents the finite element global basis function, the subscript in represents the incident source, is the tangential current source in m mode at the Floquet port, is the tangential magnetic current source in the m-mode at the Floquet port, J e，in is the incident current source at the Floquet port, J m，in is the incident magnetic flux source at the Floquet port; is the Hamiltonian operator, is the normal vector of the corresponding surface element of the Floquet port, j is the sign of the imaginary part, ω represents the angular frequency, and μ represents the magnetic permeability; Step 3: Construct the inverse of the transmission coefficient of the frequency selective surface model to be optimized Among them, Q(ω) represents the The interpolation function for interpolating the imaginary part of n Represents the nth order coefficient of the angular frequency ω; The fitting is achieved by solving the least square solution of Q(ω) by interpolation, and then the S under different angular frequencies is realized by formula (12). 21 Calculation of Step 4: Calculate the gradient of the frequency selection optimization function Where p is the optimization variable vector, c T Indicates taking the transpose of c; Step 5: Use the variable density method and transition boundary conditions to establish the optimization model of the area to be optimized, and use the conductivity of the area to be optimized as the optimization variable vector; The optimization model is expressed as: The optimization model of the conductive film with variable conductivity corresponding to the upper surface metal layer or the lower surface metal layer in the area to be optimized is given by equations (15) and (16), where J s1 and J s2 Respectively represents the current intensity on both sides of the conductive film corresponding to the upper surface metal layer or the lower surface metal layer, E t1 and E t2 represents the tangential electric field intensity on both sides of the conductive film corresponding to the upper surface metal layer or the lower surface metal layer, d is the set virtual thickness of the conductive film, and σ is the conductivity of the conductive film; Step 6: Substitute the excitation vector r corresponding to formula (4) and the excitation vector b corresponding to formula (5) into the linear equations corresponding to formula (1) and the adjoint equation corresponding to formula (2) in step 1 to obtain the coefficient vector c of the electric field basis function and the solution of the coefficient vector v of the electric field basis function in the adjoint equation. Substitute the solutions of vector c and vector v into step 4 to obtain the gradient of the frequency selection optimization function with respect to the optimization variable vector. Step 7: Use the moving asymptote method as the gradient optimization method, and use the gradient obtained in step 6 to update the optimization variable vector iteratively; after each generation of update, use step 3 to solve S 21 The curve is used to obtain the -3dB bandwidth of the current frequency selective surface model; when the set cutoff condition is reached, the update iteration is stopped and the current frequency selective surface model is output as the final frequency selective surface model.

2. The single-layer frequency selective surface broadband design method based on topology optimization method according to claim 1 is characterized in that: In step 3, the equivalent impedance of the wave-transmitting frequency selective surface model is expressed as: Among them, Z F is the equivalent impedance of the frequency selective surface, C is the equivalent capacitance of the frequency selective surface, L1 and L2 are the equivalent inductors of the frequency selective surface respectively; Transmission coefficient S 21 The expression is: Among them, S 21 represents the transmission coefficient between Floquet port 1 and port 2, Z0 is the free space wave impedance, and S 21 The reciprocal of is: Take the above formula in The fourth-order Taylor expansion at is: Taking the fourth-order Taylor expansion of the above formula at ω=ω1=0, we get: Take the above formula in The fourth-order Taylor expansion at is: Among them, in formula (9), formula (10) and formula (11), Indicates that m The coefficient of the nth-order term expanded at ; Combining equations (9), (10) and (11), Construct it and get:

3. The single-layer frequency selective surface broadband design method based on topology optimization method according to claim 1 is characterized in that: In step 3, the fitting is achieved by interpolating the least squares solution of Q(ω): in, Indicates the kth angular frequency interpolation point, 1≤k≤K, K≥6, express The corresponding Q(ω) value; The formula (13) is used to realize The fitting of S at different angular frequencies is then realized through formula (12). 21 Calculation.

4. The single-layer frequency selective surface broadband design method based on topology optimization method according to claim 1 is characterized in that: In step 7, the cutoff condition is set as follows: S 21 The -3dB bandwidth covers the Ku band.

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