Volume Integral Equation Mesh Adaptive Refinement Electromagnetic Simulation Method Based on Discontinuity Error Estimator

Abstract

The present invention discloses a volume integral equation grid adaptive refinement electromagnetic simulation method based on a discontinuity error estimator, belonging to the technical field of electromagnetic simulation. The method of the present invention is used to complete the electromagnetic characteristic calculation of complex multi-scale targets, provides five volume integral equation discontinuity error estimators, can accurately estimate the errors of the grid, and can be realized in cooperation with an adaptive refinement method of an encryption strategy, so as to increase a small amount of computational unknowns, greatly improve the grid quality, and improve the calculation accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field of electromagnetic simulation, and particularly relates to a volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator. Background Art

[0002] In recent years, complex multi-scale targets, including electrically large targets such as aircraft, aircraft groups, warship models, and fine targets such as antennas, filters, and chips, have gradually become the focus of modern computational electromagnetics. These target structures are becoming increasingly complex, and their materials exhibit non-uniformity, posing higher requirements for the efficiency and accuracy of electromagnetic simulation and calculation. To solve such problems, on the one hand, different calculation methods have their applicable ranges, so it is necessary to select a calculation method suitable for the target; on the other hand, the calculation grid of the target is also an important influencing factor, and the quality of the grid directly affects the efficiency and accuracy of the calculation. How to obtain a high-quality electromagnetic calculation grid has become a research hotspot in the field of computational electromagnetics.

[0003] The prior art "Evaluation of Local Error Estimators for the RWG-Based EFIE" discloses grid error estimators for surface integrals. These error estimators are very efficient and do not consume computing resources, with a time complexity of (O). However, these error estimators can only analyze triangular grids and cannot analyze tetrahedral grids. The prior art "A Posterior h / p Adaptive Refinement Algorithm for 3-D Goal-Oriented Electromagnetic Problems" discloses an adaptive encryption method for tetrahedral grids, but the error estimator used is a goal-oriented error estimator, which is accurate but very resource-consuming. Summary of the Invention

[0004] The object of the present invention is to overcome the defects of the above prior art and provide a volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator, which is used to complete the calculation of the electromagnetic characteristics of complex multi-scale targets. By accurately estimating the errors of the grid and reasonably encrypting it, a small number of unknowns for calculation are increased, the grid quality is greatly improved, and the calculation accuracy is improved.

[0005] The technical problems proposed by the present invention are solved as follows:

[0006] A volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator, comprising the following steps:

[0007] Step 1. Discretize the target to be calculated into tetrahedral meshes in physical modeling software, set the initial dissection length of the tetrahedrons to 0.2 - 0.4 wavelengths, and export the serial numbers and vertex coordinates of each tetrahedral mesh;

[0008] Step 2. Use the volume equivalence principle to construct a volume integral equation within the target calculation region; record the central coordinates and volumes of each tetrahedral mesh, form a tetrahedron pair by combining adjacent tetrahedral meshes, number and record the number N of tetrahedron pairs, and record the area of the common surface; establish SWG basis functions on each tetrahedron pair;

[0009] Step 3. Expand the electric displacement vector within the target calculation region using SWG basis functions and substitute it into the volume integral equation to obtain the discretized volume integral equation; use the Galerkin test method to perform an inner product on both ends of the volume integral equation with N SWG basis functions simultaneously, and expand the number of volume integral equations to N to form a linear equation system containing N equations;

[0010] Define the coefficient term on the left side of the equal sign of the linear equation system as the impedance matrix Z, where Z is an N - dimensional square matrix; regard all the expansion coefficients of the basis functions as the coefficients to be solved and form the coefficient vector I of the electric displacement vector to be solved, where I is an N×1 - dimensional column vector; define the coefficient term on the right side of the equal sign of the linear equation system as the excitation term b, where b is an N×1 - dimensional column vector; the linear equation system is expressed as ZI = b;

[0011] Step 4. Use the Krylov subspace iteration method to solve the linear equation system ZI = b to solve the coefficient vector I of the electric displacement vector; combine the coefficient vector I of the electric displacement vector with the SWG basis functions to solve the distribution of the electric displacement vector in the target region, and further calculate the far - field;

[0012] Step 5. According to the solved coefficient vector I of the electric displacement vector, select and construct an error estimator for the calculation target, perform error analysis on each tetrahedral mesh of the calculation target, and calculate the global prediction error G err ;

[0013] Step 6. Set a threshold value. When the global prediction error G err is greater than the threshold value, enter Step 7; otherwise, end;

[0014] Step 7. Set an encryption strategy; perform volume weighting on the error estimation values of each tetrahedral mesh and correct the mesh error values at the boundary; set a coefficient a, where 0 < a < 1, then sort the meshes in descending order of the error estimation values to form a set C; mark the meshes in the set C in sequence, and the number of marked meshes is a×N; use the edge refinement method to perform conformal encryption on the marked meshes, and return to execute Step 2.

[0015] Furthermore, the error estimator in step 5 is an electric displacement vector discontinuity error estimator:

[0016] There is a jump discontinuity in the electric displacement vector obtained according to vector I and the SWG basis function between two adjacent tetrahedral meshes. For any tetrahedral mesh, the error estimate value LE Dtan is:

[0017]

[0018] where 1 ≤ p ≤ 4, and are the tangential electric displacement vector and the relative permittivity of the p-th face of the current tetrahedral mesh respectively, and are the tangential electric displacement vector and the relative permittivity of the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh respectively, is the unit tangential vector of the p-th face of the current tetrahedral mesh, and Dtmax represents the maximum value of the ratio of the tangential point displacement vector to the relative permittivity at the midpoint of the four faces of all tetrahedral meshes.

[0019] Furthermore, the error estimator in step 5 is a charge discontinuity error estimator:

[0020] For the error estimate value LE at any tetrahedral mesh ρ is:

[0021]

[0022] where ρ represents the volume charge density of the current tetrahedral mesh, ρ q，p represents the volume charge density of the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh, and ρ max represents the maximum value of the volume charge density of all tetrahedral meshes.

[0023] Furthermore, the error estimator in step 5 is a combined discontinuity error estimator:

[0024] For the error estimate value LE at any tetrahedral mesh Comb is:

[0025] LE Comb = max{LE Dtan , LE ρ}

[0026] where max represents taking the maximum value.

[0027] Furthermore, the error estimator in step 5 is an electric displacement vector weighted charge discontinuity error estimator:

[0028] The error estimation value LE at any tetrahedral mesh FWC is as follows:

[0029]

[0030] where V and V q，p represent the volumes of the current tetrahedral mesh and the tetrahedral meshes adjacent to the p-th face of the current tetrahedral mesh respectively, and FWCmax represents the maximum value of the sum of the inner product of the electric displacement vector and the unit tangential vector of the four faces and the volume charge amount in all tetrahedral meshes.

[0031] Furthermore, the error estimator in step 5 is the charge recovery error estimator:

[0032] For a vertex K with a total of T common vertices of tetrahedrons, the charge density ρ at vertex K K is the average of the centroid charge densities of T tetrahedral meshes, expressed as:

[0033]

[0034] where 1 ≤ t ≤ T, ρ t represents the volume charge density of the t-th tetrahedral mesh adjacent to vertex K;

[0035] The Lagrange interpolation method is used to solve the charge density ρ at any position of the mesh Recovery :

[0036]

[0037] where θ t (r) is the t-th order Lagrange interpolation function formed by the known positions of the known charge densities;

[0038] The error estimation value LE at any tetrahedral mesh recovery is as follows:

[0039]

[0040] where ρVmax represents the maximum value of the volume charge amounts in all tetrahedral meshes.

[0041] Furthermore, the specific process of constructing the global prediction error in step 5 is as follows:

[0042] The local discontinuity error estimator can be used to calculate the global error, and its definition is:

[0043]

[0044] where 1 ≤ s ≤ Ncells, Ncells is the total number of tetrahedral meshes of the current calculation target; LEs The error calculated by the error estimator for the s-th tetrahedral mesh, A s The volume of the s-th tetrahedral mesh, A tot The total volume of the calculation target.

[0045] The beneficial effects of the present invention are:

[0046] The electromagnetic simulation method of the present invention is used to calculate the electromagnetic characteristics of complex multi-scale targets, provides five volume integral equation error estimators, can accurately estimate the errors of the meshes, and is implemented in cooperation with the encryption strategy and the adaptive encryption method to increase a small number of unknowns for calculation, greatly improve the mesh quality, and improve the calculation accuracy. Brief Description of the Drawings

[0047] Figure 1 It is a schematic diagram of the mesh division of the calculation target of the electromagnetic simulation method described in the embodiment;

[0048] Figure 2 It is a curve graph of the RCS calculated adaptively for the last time under five different error estimators in the electromagnetic simulation method described in the embodiment;

[0049] Figure 3 It is a distribution diagram of the errors of the initial mesh of the electric displacement vector discontinuous error estimator in the electromagnetic simulation method described in the embodiment;

[0050] Figure 4 It is a distribution diagram of the errors of the initial mesh of the charge discontinuous error estimator in the electromagnetic simulation method described in the embodiment;

[0051] Figure 5 It is a distribution diagram of the errors of the initial mesh of the combined discontinuous error estimator in the electromagnetic simulation method described in the embodiment;

[0052] Figure 6 It is a distribution diagram of the errors of the initial mesh of the electric displacement vector weighted charge discontinuous error estimator in the electromagnetic simulation method described in the embodiment;

[0053] Figure 7 It is a distribution diagram of the errors of the initial mesh of the charge recovery error estimator in the electromagnetic simulation method described in the embodiment;

[0054] Figure 8 It is a curve graph of the change of the root mean square error of the RCS with the number of encryption times under five different error estimators in the electromagnetic simulation method described in the embodiment. Detailed Embodiment

[0055] The present invention will be further described below in conjunction with the drawings and embodiments.

[0056] This embodiment provides a grid adaptive encryption electromagnetic simulation method based on volume integral equations, such asFigure 1 As shown, it includes the following steps:

[0057] Step 1. In the physical modeling software, discretize the target to be calculated with tetrahedral meshes. The initial dissection length of the tetrahedrons is set to 0.4 - 0.2 wavelengths. The initial mesh should ensure the geometric characteristics of the target. Finally, export the serial numbers of each tetrahedron and the vertex coordinates.

[0058] In this embodiment, the calculation target of the method is a dielectric slab. The length and width of the dielectric slab are 1 m, and the height is 0.1 m. As Figure 1 shown, the relative permittivity is 2.6, the incident wave frequency is 30 MHz, the incident direction is along the x - direction, the electric field polarization direction is along the y - direction, and the initial dissection length of the mesh is 0.2 m.

[0059] Step 2. Use the volume equivalence principle to construct a volume integral equation in the calculation target area. Record the central coordinates and volumes of each tetrahedral mesh. Combine adjacent tetrahedral meshes into a tetrahedron pair, number and record the number N, and record the area of the common surface. Establish SWG basis functions on each tetrahedron pair.

[0060] Step 3. Expand the electric displacement vector in the target area with SWG basis functions, expressed as 1 ≤ n ≤ N, I n represents the expansion coefficient of the basis function of the nth tetrahedron pair; substitute the above formula into the volume integral equation to obtain the discretized volume integral equation. Use the Galerkin test method, that is, take the inner product of both ends of the volume integral equation with N SWG basis functions at the same time, expand the number of equations to N, and form a linear equation system containing N equations.

[0061] Define the coefficient term on the left - hand side of the linear equation system as the impedance matrix Z, and Z is an N - dimensional square matrix; regard all I n as the coefficients to be solved, form the coefficient vector I of the electric displacement vector to be solved, and I is an N×1 - dimensional column vector; define the coefficient term on the right - hand side of the linear equation system as the excitation term b, and b is an N×1 - dimensional column vector; the linear equation system is expressed as ZI = b.

[0062] Step 4. Use the Krylov subspace iteration method to solve the linear equation system ZI = b, and solve the coefficient vector I of the electric displacement vector. The coefficient vector I of the electric displacement vector combined with the SWG basis functions can be used to find the distribution of the electric displacement vector in the target area, and then the far - field can be calculated.

[0063] Further, in Step 4, the specific process of using the Krylov subspace iteration method to solve the linear equation system is as follows:

[0064] The linear equation system to be solved is ZI = b. Set the initial vector I (0) , calculate the initial residual r0 = b - ZI(0) , v (1) = r0 / ||r0||, v (1) represents the basis vectors of the Krylov subspace. And find the best approximation solution I in the Krylov subspace (1) , that is: I (1) ∈ I (0) + K1 = I (0) + span{v (1)}, span{a, b, c} represents any linear combination of vectors a, b, and c.

[0065] Call the Arnoldi or Lanczos process on matrix Z to calculate vector v m , and find the best approximation solution I in the Krylov subspace (m) That is: I (m) ∈ I (0) + K m = I (0) + span{v (1) , …, v (m)}, and calculate the residual r0 = b - ZI (m) . If the residual is less than the set threshold, then I (m) is the coefficient of the electric displacement vector to be solved. Otherwise, repeat this step until the residual is less than the set threshold.

[0066] In the method described in this embodiment, the radar cross section (RCS) is as Figure 2 shown, showing the RCS calculated by the last adaptive calculation and the RCS calculated by the last global encryption under five different error estimators.

[0067] Step 5. According to the solved coefficient vector I of the electric displacement vector, construct a corresponding error estimator for the calculation target, perform error analysis on each tetrahedral mesh of the calculation target, and then calculate the global prediction error G err .

[0068] Furthermore, the specific process of constructing the error estimator in step 5 is as follows:

[0069] Electric displacement vector discontinuity error estimator:

[0070] There will be a jump discontinuity in the electric displacement vector obtained from vector I and the SWG basis function between two adjacent tetrahedral meshes. Based on this discontinuity, an electric displacement vector discontinuity estimator can be constructed. For the error estimate value LE Dtan at any tetrahedral mesh is: the average value of the tangential discontinuity of the electric displacement vector ratio relative to the relative permittivity at the midpoints of the four faces, and then normalized by dividing by the global maximum value of the electric displacement vector ratio relative to the relative permittivity value.

[0071]

[0072] where 1 ≤ p ≤ 4, and are the tangential electric displacement vector and the relative permittivity of the p-th face of the current tetrahedral mesh respectively, and are the tangential electric displacement vector and the relative permittivity of the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh respectively, is the unit tangential vector of the p-th face of the current tetrahedral mesh, and Dtmax represents the maximum value of the ratio of the tangential point displacement vector to the relative permittivity at the midpoint of the four faces of all tetrahedral meshes.

[0073] Charge discontinuity error estimator:

[0074] Since the SWG basis function is a first-order basis function, the free charge density within the mesh is constant and does not vary with position. The smaller the regional error, the smoother the distribution of the free charge density between meshes and the smaller the difference. Generally, the larger the difference in the free charge density between meshes, the larger the regional error, and a denser mesh is required to calculate this region. For the error estimate value LE at any tetrahedral mesh p is: the average value of the difference between the charge density at the current mesh and the charge densities at the four adjacent meshes, expressed as:

[0075]

[0076] where ρ represents the volume charge density of the current tetrahedral mesh, p q，p represents the volume charge density of the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh, and ρ max represents the maximum value of the volume charge density of all tetrahedral meshes.

[0077] Combined discontinuity error estimator:

[0078] Using the electric displacement vector discontinuity error estimator or the charge discontinuity error estimator alone may ignore some regions with excessive errors. The combined discontinuity estimator is obtained from the maximum value of the electric displacement vector discontinuity estimator and the charge discontinuity estimator at this mesh:

[0079] LE Comb = max{LE Dtan , LE ρ}

[0080] Electric displacement vector weighted charge discontinuity error estimator:

[0081] Under certain conditions, discontinuous estimators may fail, and the quantities they process will be forced to cancel due to geometric symmetry or source symmetry. However, these two estimators will fail under different conditions. This indicates that estimators based on these two quantities may be more robust. For the error estimate value LE at any grid FWC is:

[0082]

[0083] where V and V q，p represent the volumes of the current tetrahedral mesh and the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh respectively, and FWCmax represents the maximum value of the sum of the inner product of the electric displacement vector and the unit tangential vector and the volume charge amount on the four faces in all tetrahedral meshes. The overall dimension is kept as Coulomb C (the integral of the electric displacement vector over the area, and the volume charge density weighted by the volume).

[0084] Charge recovery error estimator:

[0085] The charge recovery method adopted is superconvergent patch recovery (SPR), and the specific process is to recover the charge at each tetrahedral vertex. For vertex K, there are T tetrahedral common vertices in total, and the charge density ρ K at vertex K is the average of the centroid charge densities of T tetrahedral meshes, that is:

[0086]

[0087] where 1 ≤ t ≤ T, and ρ t represents the volume charge density of the t-th tetrahedral mesh adjacent to vertex K;

[0088] After recovering the charge of each vertex using SPR, the charge densities at each grid vertex and the centroid are known, and the Lagrange interpolation method is used to solve the charge density at any place in the grid:

[0089]

[0090] where θ t (r) is the t-th order Lagrange interpolation function formed by the known positions of the known charge densities.

[0091] For the error estimate value LE at the current tetrahedral mesh recovery is:

[0092]

[0093] where ρVmax represents the maximum value of the volume charge amounts in all tetrahedral meshes.

[0094] These five error estimators are all very efficient and do not consume computing resources, with a time complexity of (L). Different error estimators can be selected for different computational problems to improve the efficiency of the adaptive encryption program.

[0095] Furthermore, the specific process of constructing the global prediction error in step 5 is as follows:

[0096] The local discontinuity error estimator can be used to calculate the global error, which is defined as:

[0097]

[0098] where 1 ≤ s ≤ Ncells, and Ncells is the total number of tetrahedral meshes of the current computational target; LE s is the error calculated by the 5 different error estimators for the s-th tetrahedral mesh, and A s is the volume of the s-th tetrahedral mesh, and A tot is the total volume of the computational target.

[0099] In the method of this embodiment, five error estimators are sequentially used to perform error analysis on each tetrahedral mesh. Figure 3 shows the error distribution of the initial mesh of the electric displacement vector discontinuity error estimator. Figure 4 shows the error distribution of the initial mesh of the charge discontinuity error estimator. Figure 5 shows the error distribution of the initial mesh of the combined discontinuity error estimator. Figure 6 shows the error distribution of the initial mesh of the electric displacement vector weighted charge discontinuity error estimator. Figure 7 shows the error distribution of the initial mesh of the charge recovery error estimator.

[0100] Step 6. Set a threshold. When the global prediction error G err is greater than the threshold, go to step 7; otherwise, end.

[0101] Step 7. Set the encryption strategy. Perform volume weighting on the error estimation values of each tetrahedral mesh, and correct the mesh error values at the boundaries. Set the coefficient a (range 0 - 1), then sort the meshes in descending order of the error estimation values to form a set C. Mark the meshes in the set C in order, and the number of marked meshes is a × N. Use the refinement edge method to perform conformal encryption on the marked meshes, and return to execute step 2.

[0102] Furthermore, the specific setting process of the encryption strategy in step 7 is as follows:

[0103] Volume weighting:

[0104] Due to the singularity of the target edge, over - encryption occurs at the edge after using the discontinuity error estimator. Therefore, in order to make the errors calculated for each grid better applicable to adaptive encryption, the grids are volume - weighted, and the volume - weighted error Error of the s - th tetrahedral grid s is expressed as:

[0105]

[0106] where A ave represents the average volume of all tetrahedral grids.

[0107] Correction of the error estimate value at the boundary:

[0108] When the tetrahedral grid is on the boundary of the calculation target, the number of grids adjacent to this grid is not 4, but a number smaller than 4. The error estimator is corrected. Suppose the tetrahedral grid on the boundary is adjacent to k grids. At this time, this grid can obtain k discontinuity values with adjacent grids, that is, the error estimate value of this grid. Multiply this error estimate value by k / 4 to make the error of this grid and all grids remain in the same dimension.

[0109] In this embodiment, the coefficient a is set to 0.2.

[0110] In this embodiment, the grid - adaptive encryption electromagnetic scattering simulation method based on the volume integral equation is compared with global encryption, and the RCS values after each grid encryption are recorded in turn. The RCS calculated by the last global encryption is taken as the reference solution. Then, the root - mean - square error (RMSE) is taken between the RCS value after each grid encryption and the reference RCS, as Figure 8 shown. It can be seen that for these five error estimators, regardless of which error estimator is used, the adaptively encrypted grids always calculate more accurate results with fewer unknowns compared to global encryption. This proves that these five error estimators are reliable and also proves that the adaptive method is effective.

[0111] For the volume integral equation method, it is applicable to solve the electromagnetic target characteristic problems of inhomogeneous media, and is also very suitable for analyzing and solving thin dielectric targets containing a large number of sharp corners and edges. Also, because the volume integral equation method belongs to the second - kind Fredholm integral equation, its matrix system has good conditional behavior and good convergence. Therefore, for complex multi - scale targets, the volume integral equation method is a suitable method. The adaptive grid encryption technology is a numerical calculation method widely used in the fields of computational mechanics, computational fluid, etc. Its core idea is to combine the physical characteristics of the problem to be solved, the numerical calculation method and the grid distribution, and continuously adjust the grid distribution according to the characteristics of the numerical solution to obtain a more accurate solution. The application of the method described in the present invention can effectively improve the calculation accuracy and has important significance for solving complex engineering problems.

[0112] The method of the present invention constructs five different volume integral equation error estimators, namely, the electric displacement vector discontinuity error estimator, the charge discontinuity error estimator, the combined discontinuity error estimator, the weighted charge discontinuity error estimator of the electric displacement vector, and the charge recovery error estimator. The electric displacement vector obtained according to the SWG basis function has a jump discontinuity between two adjacent elements. Through this discontinuity, the electric displacement vector discontinuity estimator can be constructed. Since the SWG basis function is a first-order basis function, the free charge density within the grid is constant and does not change with position. The greater the difference in the free charge density between grids, the greater the regional error, and a denser grid is required to calculate this region. Thus, the charge discontinuity error estimator is constructed. Using only the electric displacement vector discontinuity error estimator or the charge discontinuity error estimator may overlook some regions with excessive errors. Thus, the combined discontinuity error estimator is constructed. Under certain conditions, the discontinuity estimators may fail, and the quantities they process may be forced to cancel due to geometric symmetry or source symmetry. However, these two estimators will fail under different conditions. This indicates that estimators based on these two quantities may be more robust. Thus, the weighted charge discontinuity error estimator of the electric displacement vector is constructed. The SPR method is used to recover the true charge density distribution, and the recovered charge density is compared with the numerically calculated charge density, so that the charge recovery error estimator is constructed. These five error estimators are all very efficient and do not consume computational resources, with a time complexity of (O). After analyzing the error of each grid through the error estimator, an adaptive encryption can be achieved in cooperation with an encryption strategy. Since these five error estimators are all very accurate, the adaptive encryption program can increase a small number of computational unknowns, greatly improve the grid quality, and improve the calculation accuracy.

Claims

1. A volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator, characterized in that, Including the following steps: Step 1. Discretize the target to be calculated into tetrahedral meshes in physical modeling software, set the initial dissection length of the tetrahedrons to 0.2 - 0.4 wavelengths, and export the serial numbers and vertex coordinates of each tetrahedral mesh; Step 2. Use the volume equivalence principle to construct a volume integral equation in the target calculation area; record the central coordinates and volumes of each tetrahedral mesh, form a tetrahedron pair with adjacent tetrahedral meshes, number and record the number N of tetrahedron pairs, and record the area of the common surface; establish SWG basis functions on each tetrahedron pair; Step 3. Expand the electric displacement vector in the target calculation area with SWG basis functions and substitute it into the volume integral equation to obtain the discretized volume integral equation; use the Galerkin test method to take the inner product of both ends of the volume integral equation with N SWG basis functions simultaneously, expand the number of volume integral equations to N, and form a linear equation system containing N equations; Define the coefficient term on the left side of the equal sign of the linear equation system as the impedance matrix Z, and Z is an N - dimensional square matrix; Take all the expansion coefficients of the basis functions as the coefficients to be solved, form the coefficient vector I of the electric displacement vector to be solved, and I is an N×1 - dimensional column vector; Define the coefficient term on the right side of the equal sign of the linear equation system as the excitation term b, and b is an N×1 - dimensional column vector; The linear equation system is expressed as ZI = b; Step 4. Use the Krylov subspace iteration method to solve the linear equation system ZI = b, and solve out the coefficient vector I of the electric displacement vector; combine the coefficient vector I of the electric displacement vector with the SWG basis functions to solve out the distribution of the electric displacement vector in the target area, and then calculate the far - field; Step 5. According to the solved electric displacement vector coefficient vector I, an error estimator is constructed for the calculation target, and error analysis is performed on each tetrahedral mesh of the calculation target to calculate the global prediction error G err ; Step 6. Set a threshold. When the global prediction error G err is greater than the threshold, go to Step 7; otherwise, end. Step 7. Set the encryption strategy; perform volume weighting on the error estimation values of each tetrahedral mesh, and correct the mesh error values at the boundary; set the coefficient a, 0 < a < 1, then sort the meshes in descending order of the error estimation values to form a set C; mark the meshes in order in the set C, and the number of marked meshes is a×N; use the edge refinement method to perform conformal encryption on the marked meshes, and return to execute Step 2.

2. The volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator according to claim 1, wherein The error estimator in Step 5 is the electric displacement vector discontinuity error estimator: There is a jump discontinuity in the electric displacement vector obtained from the vector I and the SWG basis functions between two adjacent tetrahedral meshes. For any tetrahedral mesh, the error estimate value LE Dtan is as follows: where 1 ≤ p ≤ 4, and are the tangential electric displacement vector and relative permittivity of the p-th face of the current tetrahedral mesh, respectively, and are the tangential electric displacement vector and relative permittivity of the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh, respectively, is the unit tangential vector of the p-th face of the current tetrahedral mesh, and Dtmax represents the maximum value of the ratio of the tangential point displacement vector to the relative permittivity at the midpoint of the four faces of all tetrahedral meshes.

3. The volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator according to claim 2, wherein The error estimator in Step 5 is the charge discontinuity error estimator: For the error estimate value LE at any tetrahedral mesh ρ It is as follows: where ρ represents the volume charge density of the current tetrahedral mesh, and ρ q,p represents the volume charge density of the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh, and ρ max represents the maximum value of the volume charge density of all tetrahedral meshes.

4. The volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator according to claim 3, characterized in that The error estimator in Step 5 is the combined discontinuity error estimator: For the error estimate value LE at any tetrahedral mesh Comb is as follows: LE Comb = max{LE Dtan , LE ρ} Where, max represents taking the maximum value.

5. The volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator according to claim 4, characterized in that The error estimator in Step 5 is the electric displacement vector weighted charge discontinuity error estimator: The error estimate value LE at any tetrahedral mesh FWC is as follows: where V and V q，p represent the volumes of the current tetrahedral mesh and the tetrahedral mesh adjacent to the p-th face of the current tetrahedral mesh, respectively, and FWCmax represents the maximum value of the sum of the inner product of the electric displacement vector and the unit tangential vector of the four faces and the volume charge amount in all tetrahedral meshes.

6. The volume integral equation grid adaptive encryption electromagnetic simulation method based on a discontinuity error estimator according to claim 5, characterized in that The error estimator in Step 5 is the charge recovery error estimator: For vertex K, there are T tetrahedrons sharing the common vertex, and the charge density ρ at vertex K K is the average of the charge densities at the centroids of the T tetrahedron meshes, expressed as: where 1 ≤ t ≤ T, ρ t represents the volume charge density of the t-th tetrahedral mesh adjacent to vertex K; Use the Lagrange interpolation method to solve for the charge density ρ at any point in the grid Recovery : where θ t (r) is the t-th order Lagrangian interpolation function formed by known charge densities at known positions; The error estimate value LE at any tetrahedral mesh recovery is as follows: Where, ρVmax represents the maximum value of the volume charge amount of all tetrahedral meshes.

7. The volume integral equation mesh adaptive encryption electromagnetic simulation method based on a discontinuity error estimator according to claim 1, characterized in that The specific process of constructing the global prediction error in Step 5 is: The local discontinuity error estimator can be used to calculate the global error, and its definition is: where \(1\leq s\leq N_{cells}\), and \(N_{cells}\) is the total number of tetrahedral meshes of the current calculation target; \(LE\) s is the error calculated by the error estimator for the \(s\)-th tetrahedral mesh, and \(A\) s is the volume of the \(s\)-th tetrahedral mesh, and \(A\) tot is the total volume of the calculation target.

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