Volume integral equation broadband self-adaptive encryption electromagnetic simulation method

Abstract

The application discloses a volume integral equation wide-band adaptive encryption electromagnetic simulation method and belongs to the technical field of electromagnetic simulation.The method is used for completing wide-band electromagnetic characteristic calculation of a complex multi-scale non-uniform target, dividing adaptive encryption frequency bands by analyzing the correlation of grid error distribution at different frequency points, adaptively generating high-quality grids under different adaptive encryption frequency bands, and greatly improving the wide-band calculation precision and calculation efficiency.The method is particularly suitable for calculating the electromagnetic characteristics of the complex multi-scale non-uniform target, generating optimal grids in each adaptive encryption frequency band, finally calculating more accurate results with fewer unknown quantities on the whole wide-band, and realizing electromagnetic simulation fast simulation.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation technology, specifically relating to a wideband adaptive encrypted electromagnetic simulation method for volume integral equations. Background Technology

[0002] For multi-scale objects such as chips and antenna arrays with radomes, as well as complex non-uniform medium targets such as flocks of birds and locusts, the field of modern computational electromagnetics is increasingly focusing on the accurate analysis of their electromagnetic properties. These targets are collectively referred to as complex multi-scale non-uniform targets, and the analysis of their electromagnetic properties across different frequency bands has become a key focus and challenge in this field. The existing technology, "An Efficient Wide-Angle / Wideband RCS Prediction Method for PEC Targets With Indeterminate Shape," discloses a wideband computation method that can achieve rapid calculations across the entire frequency band. However, this existing technology uses a fixed mesh across the entire frequency band and does not consider the impact of differences in the electromagnetic properties of the target across different frequency bands on the optimal mesh, thus severely limiting the efficiency and accuracy of the computation. The prior art, "Automatic Localized Nonconformal Mesh Refinement for Surface Integral Equations," discloses a mesh error estimator for integral equations. In this prior art, the error estimator can identify mesh errors and obtain high-quality meshes by combining adaptive refinement methods. However, this error estimator is only suitable for mesh error analysis at a single frequency point and cannot achieve mesh error evaluation over a wide frequency range. Summary of the Invention

[0003] The purpose of this invention is to overcome the shortcomings of the prior art and provide a broadband adaptive encrypted electromagnetic simulation method for volume integral equations, used to calculate the broadband electromagnetic properties of complex multi-scale non-uniform targets. The method of this invention analyzes the correlation of grid error distribution at different frequency points, divides the frequency bands into adaptive encrypted bands, and adaptively generates high-quality grids under different adaptive encrypted frequency bands, significantly improving the calculation accuracy and efficiency of broadband electromagnetic properties.

[0004] The technical problem addressed by this invention is solved as follows:

[0005] A broadband adaptive encrypted electromagnetic simulation method for volume integral equations includes the following steps:

[0006] Step 1: Perform tetrahedral meshing on the target based on the set initial mesh size;

[0007] Step 2: Set the frequency point to be calculated within the target frequency band, set the wideband sampling order, and determine the sampling frequency point;

[0008] Step 3: Select the basis function and calculate the electric displacement vector coefficient of the target at each sampling frequency point in turn; expand the electric displacement vector coefficient using the Chebyshev function to obtain the expression for the electric displacement vector coefficient in the entire target frequency band;

[0009] Step 4: Substitute the frequency points to be calculated into the expression for the electric displacement vector coefficients in the entire target frequency band to obtain the electric displacement vector coefficients at each frequency point to be calculated; substitute the electric displacement vector coefficients at each frequency point to be calculated into the SWG basis function to obtain the electric displacement vector distribution at each frequency point to be calculated, and then calculate the far field in the entire target frequency band.

[0010] Step 5: Construct an error estimator using the electric displacement vector distribution; use the error estimator to perform error analysis on each frequency point to be calculated, and obtain the grid error distribution of the target at each frequency point to be calculated;

[0011] The frequency points to be calculated are arranged in ascending order to construct a frequency point sequence. Starting from the minimum frequency point in the sequence, frequency points are extracted sequentially in ascending order. The correlation of the grid error distribution between the minimum frequency point and the extracted frequency points is calculated. If the correlation is less than 0.8, the current extracted frequency point is recorded. An adaptive encrypted frequency band is constructed using the minimum frequency point and the current extracted frequency point. The current extracted frequency point is used as the minimum frequency point, and the above operation is performed until the entire target frequency band is divided into several adaptive encrypted frequency bands.

[0012] Step 6: Set the encryption number threshold for each adaptive encryption frequency band; perform adaptive encryption on each adaptive encryption frequency band in order from low frequency to high frequency to obtain the encrypted tetrahedral mesh, and return to execute steps 3 and 4; after all adaptive encryption frequency bands have completed adaptive encryption, the far field of the entire target frequency band after adaptive encryption of the tetrahedral mesh is finally obtained.

[0013] Furthermore, in step 1, the initial grid size is set to 0.2 times the wavelength corresponding to the center frequency.

[0014] Furthermore, in step 2, the broadband sampling order D is:

[0015]

[0016] in, For adjustment coefficients, The highest frequency in the target frequency band and lowest frequency The increase in wavenumber between The maximum size of the target along the propagation direction;

[0017] The method for calculating the sampling frequency is as follows:

[0018]

[0019] Where 1≤d≤D, This is the d-th sampling frequency point.

[0020] Furthermore, in step 3, the method for calculating the electric displacement vector coefficient for any sampling frequency point is as follows:

[0021] Based on the principle of volume equivalence, a volume integral equation is constructed within the target region. SWG basis functions are defined on a tetrahedral mesh to discretize the electric displacement vector within the target region, and the electric displacement vector coefficients are used to characterize the electric displacement vector. The SWG basis functions are substituted into the volume integral equation to construct a discrete form of the volume integral equation. A linear equation system is constructed using the Galerkin test method. The Krylov subspace iteration method is used to solve the linear equation system to obtain the electric displacement vector coefficients.

[0022] Furthermore, in step 3, the expression for the electric displacement vector coefficients across the entire target frequency band is:

[0023]

[0024] in, This represents any frequency point within the target frequency band. Frequency point The electric displacement vector coefficient at that point, for normalized frequency, Frequency point The value of the Chebyshev polynomial of order d at point d. These are the expansion coefficients of the dth order;

[0025] Normalized frequency for:

[0026]

[0027] d-th order expansion coefficients for:

[0028]

[0029] in, , For the first Each sampling frequency point for normalized frequency, For the first The electric displacement vector coefficients at each sampling frequency point Indicates sampling frequency The value of the Chebyshev polynomial of order d.

[0030] Furthermore, the specific process of constructing the error estimator using the electric displacement vector distribution in step 5 is as follows:

[0031] The error estimate for any tetrahedral mesh is:

[0032]

[0033] in, This represents the error estimate of the i-th tetrahedral mesh; 1 ≤ p ≤ 4. and For the tetrahedral mesh in the th Tangential electric displacement vector and relative permittivity at the interface and For the tetrahedral mesh in the th Tangential electric displacement vector and relative permittivity at adjacent tetrahedral meshes at the interface. For the current tetrahedral mesh, the first Unit normal vector at each face, This represents the maximum value of the ratio of the tangential electric displacement vector to the relative permittivity at all tetrahedral mesh interfaces. This indicates taking the modulus.

[0034] Furthermore, in step 5, the process for calculating the correlation of the grid error distribution is as follows:

[0035]

[0036] in, and These are the minimum frequency point and the extracted frequency point, respectively. minimum frequency point With extraction frequency points The correlation of mesh error distribution between them; M is the total number of tetrahedral meshes, 1≤i≤M; This indicates that the i-th tetrahedral mesh is at the minimum frequency point. The error estimate below, This indicates that all tetrahedral meshes are at their minimum frequency. The average of the error estimates under the given conditions; This indicates that the i-th tetrahedral mesh is at the decimation frequency point. The error estimate below, This indicates that all tetrahedral meshes are at the extracted frequency points. The average of the error estimates.

[0037] Furthermore, in step 6, the specific process of adaptive encryption for the adaptive encryption frequency band is as follows:

[0038] Determine whether the current encryption count has reached the encryption count threshold of the current adaptive encryption frequency band. If it has reached the encryption count threshold, determine whether the current adaptive encryption frequency band is the highest frequency band. If it is the highest frequency band, end the process; otherwise, proceed to the next adaptive encryption frequency band.

[0039] If the encryption count threshold is not reached, calculate the broadband error of the current adaptive encryption frequency band; select the 30% tetrahedral grid with the largest broadband error for encryption.

[0040] Furthermore, in step 6, the method for calculating the broadband error of the adaptive encryption band is as follows:

[0041]

[0042] in, Let be the broadband error of the i-th tetrahedral mesh. This represents the number of frequency points to be calculated within the current adaptive encryption frequency band. Let i be the volume of the i-th tetrahedral mesh. The average volume of all tetrahedral meshes, 1 ≤ n ≤ , This indicates the nth frequency point to be calculated within the current adaptive encryption frequency band for the i-th tetrahedral mesh. The following is the error estimate calculated by the error estimator.

[0043] The beneficial effects of this invention are:

[0044] The electromagnetic simulation method described in this invention is used to calculate the broadband electromagnetic characteristics of complex multi-scale non-uniform targets. It provides a broadband adaptive densification method, which divides the target into frequency bands according to the different electromagnetic characteristics of the target under different frequency bands, and adaptively generates high-quality meshes under different frequency bands, which greatly improves the broadband calculation accuracy and calculation efficiency. Attached Figure Description

[0045] Figure 1 This is a flowchart illustrating the method described in this invention;

[0046] Figure 2 This is a top view of the target of the method described in the embodiment;

[0047] Figure 3 This is a comparison diagram of the radar cross section (RCS) of a single station in the method described in the embodiment;

[0048] Figure 4 The graph shows the convergence curve of the root mean square error (RMSE) in the method described in the embodiment. Detailed Implementation

[0049] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0050] This embodiment provides a wideband adaptive encrypted electromagnetic simulation method for volume fraction equations, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0051] Step 1: Perform tetrahedral meshing on the target. The initial size of the tetrahedral mesh is 0.2 times the wavelength corresponding to the center frequency point.

[0052] In this embodiment, the target is a non-uniform dielectric cone with a height of 120 mm and a base radius of 60 mm. The target is divided into 8 regions, and the relative permittivity of regions 1 to 8 are 5, 8, 9, 4, 7, 2, 3, and 6, respectively. Figure 2 As shown. The incident wave frequency is 800MHz-4GHz, the incident direction is along the z-direction, and the electric field polarization direction is along the x-direction. The monostatic RCS of the entire frequency band is calculated. The average mesh size of the initial conical mesh is 15mm.

[0053] Step 2: Set the frequency point to be calculated within the target frequency band, starting from the highest frequency. and lowest frequency Confirm; Set the wideband sampling order D and determine the sampling frequency. ;

[0054] In this embodiment, the highest frequency lowest frequency The number of frequency points to be calculated is 361.

[0055] In this embodiment, the sampling order D of the medium-wideband is:

[0056]

[0057] in, The highest frequency and lowest frequency The increase in wavenumber between The maximum size of the target along the propagation direction. This is an adjustment coefficient, with a value ranging from 5 to 10. The more complex the calculation target, the larger the value.

[0058] Sampling frequency The calculation method is as follows:

[0059]

[0060] in, Let d be the d-th sampling frequency point, where 1 ≤ d ≤ D.

[0061] In this embodiment, the sampling order D is 65, and 65 sampling frequency points need to be calculated.

[0062] Step 3: Select the basis functions and calculate the electric displacement vector coefficients of the target at each sampling frequency point in turn; expand the electric displacement vector coefficients using Chebyshev functions to obtain the expression for the electric displacement vector coefficients in the entire target frequency band.

[0063] In this embodiment, the electric displacement vector coefficient is calculated as follows for any sampling frequency point:

[0064] Based on the principle of volume equivalence, a volume integral equation is constructed within the target region. SWG basis functions are defined on a tetrahedral mesh to discretize the electric displacement vector within the target region, and electric displacement vector coefficients are used to characterize the electric displacement vector. The SWG basis functions are substituted into the volume integral equation to construct a discrete form of the equation. A linear system of equations is constructed using the Galerkin test method. The Krylov subspace iteration method is used to solve the linear system of equations to obtain the electric displacement vector coefficients. .

[0065] The expression for the electric displacement vector coefficients across the entire target frequency band is:

[0066]

[0067] in, This represents any frequency point within the target frequency band. Frequency point The electric displacement vector coefficient at that point, Represents the normalized frequency. Frequency point The value of the Chebyshev polynomial of order d at point d. These are the expansion coefficients of the dth order;

[0068] Among them, normalized frequency The calculation method is as follows:

[0069]

[0070] d-th order expansion coefficients Calculation method:

[0071]

[0072] Where 1≤l≤D, For the l-th sampling frequency point, The electric displacement vector coefficient at the l-th sampling frequency point is... Indicates sampling frequency The value of the Chebyshev polynomial of order d.

[0073] Step 4: Substitute the frequency point to be calculated into the expression for the electric displacement vector coefficient within the entire target frequency band. The electric displacement vector coefficients at each frequency point to be calculated are obtained. These coefficients are then substituted into the SWG basis functions to obtain the electric displacement vector distribution at each frequency point, and the far-field within the entire target frequency band is calculated.

[0074] Step 5: Construct an error estimator using the electric displacement vector distribution; use the error estimator to perform error analysis on each frequency point to be calculated, and obtain the grid error distribution of the target at each frequency point to be calculated;

[0075] After arranging all the frequency points to be calculated in ascending order, a sequence of frequency points to be calculated is constructed. Starting from the minimum frequency point in the sequence, frequency points are extracted in ascending order. The correlation of grid error distribution between the minimum frequency point and the extracted frequency points is calculated. If the correlation is less than 0.8, the current extracted frequency point is recorded. An adaptive encrypted frequency band is constructed using the minimum frequency point and the current extracted frequency point. The current extracted frequency point is used as the minimum frequency point. The above operation is performed until the entire target frequency band is divided into several adaptive encrypted frequency bands.

[0076] In this embodiment, the specific process of constructing an error estimator using the electric displacement vector distribution is as follows:

[0077] The electric displacement vector exhibits a jump discontinuity between two adjacent tetrahedral meshes, and an error estimator can be constructed based on this discontinuity.

[0078] The error estimate for any tetrahedral mesh is:

[0079]

[0080] in, This represents the error estimate of the i-th tetrahedral mesh; 1 ≤ p ≤ 4. and For the tetrahedral mesh in the th Tangential electric displacement vector and relative permittivity at the interface and For the tetrahedral mesh in the th Tangential electric displacement vector and relative permittivity at adjacent tetrahedral meshes at the interface. For the current tetrahedral mesh, the first Unit normal vector at each face, This represents the maximum value of the ratio of the tangential electric displacement vector to the relative permittivity at the interfaces of all tetrahedral meshes.

[0081] The correlation calculation process for grid error distributions at different frequencies is expressed as follows:

[0082]

[0083] in, minimum frequency point With extraction frequency points The correlation of mesh error distribution between them; M is the total number of tetrahedral meshes, 1≤i≤M; Indicates that the i-th tetrahedral mesh is in Error estimate at frequency, Indicates all tetrahedral meshes in The average value of the error estimate at the frequency; Indicates that the i-th tetrahedral mesh is in Error estimate at frequency, Indicates all tetrahedral meshes in The average value of the error estimate at the specified frequency.

[0084] The method described in this embodiment divides the frequency bands into three bands: 800MHz-2.25GHz, 2.25GHz-3.1GHz, and 3.1GHz-4GHz.

[0085] Step 6: Set the encryption number threshold for each adaptive encryption frequency band; perform adaptive encryption on each adaptive encryption frequency band in order from low frequency to high frequency to obtain the encrypted tetrahedral mesh, and return to execute steps 3 and 4; after all adaptive encryption frequency bands have completed adaptive encryption, the far field of the entire target frequency band after adaptive encryption of the tetrahedral mesh is finally obtained.

[0086] In this embodiment, the specific process of adaptive encryption for the adaptive encryption frequency band is as follows:

[0087] Determine whether the current encryption count has reached the encryption count threshold of the current adaptive encryption frequency band. If it has reached the encryption count threshold, determine whether the current adaptive encryption frequency band is the highest frequency band. If it is the highest frequency band, end the process; otherwise, proceed to the next adaptive encryption frequency band.

[0088] If the encryption count threshold is not reached, calculate the broadband error of the current adaptive encryption frequency band; select the 30% tetrahedral grid with the largest broadband error for encryption.

[0089] The calculation method for the wideband error of the adaptive encryption frequency band is as follows:

[0090]

[0091] in, Let be the broadband error of the i-th tetrahedral mesh. This represents the number of frequency points to be calculated within the current adaptive encryption frequency band. Let i be the volume of the i-th tetrahedral mesh. The average volume of all tetrahedral meshes, 1 ≤ n ≤ , This indicates the nth frequency point to be calculated within the current adaptive encryption frequency band for the i-th tetrahedral mesh. The following is the error estimate calculated by the error estimator.

[0092] In this embodiment, the encryption threshold for each adaptive encryption band is set to 1 time. Each adaptive encryption band undergoes one adaptive encryption, for a total of 3 adaptive encryption operations. Figure 3 As shown, the single-station RCS calculated by the adaptive encryption mesh is more accurate than that calculated by the direct mesh in the prior art. However, the number of unknowns in the adaptive mesh is 20048, while the number of unknowns in the directly meshed mesh is 24980. The broadband adaptive encryption method described in this embodiment generates a mesh with fewer unknowns and calculates more accurate results, demonstrating that the method described in this embodiment is particularly suitable for calculating the broadband electromagnetic properties of complex multi-scale non-uniform targets.

[0093] The broadband adaptive encryption method described in this embodiment is compared with the existing global encryption technology. The single-station RCS value calculated after each grid encryption is recorded sequentially. The single-station RCS calculated after the last global encryption is taken as the reference solution. Then, the root mean square error (RMSE) is calculated between the single-station RCS value calculated after each grid encryption and the reference solution. Figure 4 As shown, the mesh obtained by the broadband adaptive encryption method described in this embodiment, compared with the mesh obtained by the global encryption method, calculates more accurate results with fewer unknowns. This proves that the method described in this embodiment is reliable in calculating broadband errors, and also proves that adaptively generating high-quality meshes in different frequency bands can significantly improve the accuracy and efficiency of broadband calculations.

[0094] The above description is merely a specific embodiment of the present invention. Any feature disclosed in this specification may be replaced by other equivalent or similar alternative features unless otherwise specified. All disclosed features, or steps in all methods or processes, except for mutually exclusive features and / or steps, may be used.

Claims

1. A broadband adaptive encrypted electromagnetic simulation method for volume integral equations, characterized in that, Includes the following steps: Step 1: Perform tetrahedral meshing on the target based on the set initial mesh size; Step 2: Set the frequency point to be calculated within the target frequency band, set the wideband sampling order, and determine the sampling frequency point; Step 3: Select the basis function and calculate the electric displacement vector coefficient of the target at each sampling frequency point in turn; The electric displacement vector coefficients are expanded using Chebyshev functions to obtain the expression for the electric displacement vector coefficients throughout the entire target frequency band. Step 4: Substitute the frequency points to be calculated into the expression for the electric displacement vector coefficients in the entire target frequency band to obtain the electric displacement vector coefficients at each frequency point to be calculated. Substitute the electric displacement vector coefficients at each frequency point to be calculated into the SWG basis function to obtain the electric displacement vector distribution at each frequency point to be calculated, and then calculate the far field in the entire target frequency band. Step 5: Construct an error estimator using the electric displacement vector distribution; use the error estimator to perform error analysis on each frequency point to be calculated, and obtain the grid error distribution of the target at each frequency point to be calculated; The frequency points to be calculated are arranged in ascending order to construct a frequency point sequence. Starting from the minimum frequency point in the sequence, frequency points are extracted sequentially in ascending order. The correlation of the grid error distribution between the minimum frequency point and the extracted frequency points is calculated. If the correlation is less than 0.8, the current extracted frequency point is recorded. An adaptive encrypted frequency band is constructed using the minimum frequency point and the current extracted frequency point. The current extracted frequency point is used as the minimum frequency point, and the above operation is performed until the entire target frequency band is divided into several adaptive encrypted frequency bands. Step 6: Set the encryption number threshold for each adaptive encryption frequency band; perform adaptive encryption on each adaptive encryption frequency band in order from low frequency to high frequency to obtain the encrypted tetrahedral mesh, and return to execute steps 3 and 4; after all adaptive encryption frequency bands have completed adaptive encryption, the far field of the entire target frequency band after adaptive encryption of the tetrahedral mesh is finally obtained.

2. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 1, characterized in that, In step 1, the initial grid size is set to 0.2 times the wavelength corresponding to the center frequency.

3. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 1, characterized in that, In step 2, the broadband sampling order D is: in, For adjustment coefficients, The highest frequency in the target frequency band and lowest frequency The increase in wavenumber between The maximum size of the target along the propagation direction; The method for calculating the sampling frequency is as follows: Where 1≤d≤D, This is the d-th sampling frequency point.

4. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 1, characterized in that, In step 3, the electric displacement vector coefficient is calculated for any sampling frequency point as follows: Based on the principle of volume equivalence, a volume integral equation is constructed within the target region; SWG basis functions are defined on a tetrahedral mesh to discretize the electric displacement vector within the target region, and the electric displacement vector coefficients are used to characterize the electric displacement vector. The SWG basis functions are substituted into the volume integral equations to construct a discrete form of the volume integral equations; a linear system of equations is constructed using the Galerkin test method; and the electric displacement vector coefficients are obtained by solving the linear system of equations using the Krylov subspace iteration method.

5. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 1, characterized in that, In step 3, the expression for the electric displacement vector coefficients across the entire target frequency band is: in, This represents any frequency point within the target frequency band. Frequency point The electric displacement vector coefficient at that point, for normalized frequency, Frequency point The value of the Chebyshev polynomial of order d at point d. These are the expansion coefficients of the dth order; Normalized frequency for: d-th order expansion coefficients for: in, , For the first Each sampling frequency point for normalized frequency, For the first The electric displacement vector coefficients at each sampling frequency point Indicates sampling frequency The value of the Chebyshev polynomial of order d.

6. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 1, characterized in that, The specific process of constructing the error estimator using the electric displacement vector distribution in step 5 is as follows: The error estimate for any tetrahedral mesh is: in, This represents the error estimate of the i-th tetrahedral mesh; 1 ≤ p ≤ 4. and For the tetrahedral mesh in the th Tangential electric displacement vector and relative permittivity at the interface and For the tetrahedral mesh in the th Tangential electric displacement vector and relative permittivity at adjacent tetrahedral meshes at the interface. For the current tetrahedral mesh, the first Unit normal vector at each face, This represents the maximum value of the ratio of the tangential electric displacement vector to the relative permittivity at all tetrahedral mesh interfaces. This indicates taking the modulus.

7. The wideband adaptive encrypted electromagnetic simulation method for volume integral equations according to claim 1, characterized in that, In step 5, the process of calculating the correlation of grid error distribution is as follows: in, and These are the minimum frequency point and the extracted frequency point, respectively. minimum frequency point With extraction frequency points The correlation of mesh error distribution between them; M is the total number of tetrahedral meshes, 1≤i≤M; This indicates that the i-th tetrahedral mesh is at the minimum frequency point. The error estimate below, This indicates that all tetrahedral meshes are at their minimum frequency. The average of the error estimates under the given conditions; This indicates that the i-th tetrahedral mesh is at the decimation frequency point. The error estimate below, This indicates that all tetrahedral meshes are at the extracted frequency points. The average of the error estimates.

8. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 7, characterized in that, In step 6, the specific process of adaptive encryption for the adaptive encryption frequency band is as follows: Determine whether the current encryption count has reached the encryption count threshold of the current adaptive encryption frequency band. If it has reached the encryption count threshold, determine whether the current adaptive encryption frequency band is the highest frequency band. If it is the highest frequency band, end the process; otherwise, proceed to the next adaptive encryption frequency band. If the encryption count threshold is not reached, calculate the broadband error of the current adaptive encryption frequency band; select the 30% tetrahedral grid with the largest broadband error for encryption.

9. The wideband adaptive encrypted electromagnetic simulation method for volume fraction equations according to claim 8, characterized in that, In step 6, the method for calculating the wideband error of the adaptive encryption band is as follows: in, Let be the broadband error of the i-th tetrahedral mesh. This represents the number of frequency points to be calculated within the current adaptive encryption frequency band. Let i be the volume of the i-th tetrahedral mesh. The average volume of all tetrahedral meshes, 1 ≤ n ≤ , This indicates the nth frequency point to be calculated within the current adaptive encryption frequency band for the i-th tetrahedral mesh. The following is the error estimate calculated by the error estimator.

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