An electromagnetic simulation method and system based on the variational finite element method

Abstract

The present invention discloses an electromagnetic simulation method and system based on the variational finite element method. The present invention sets a calculation region; defines two auxiliary vector fields on the artificial truncation boundary; projects the two auxiliary vector fields in each far-field direction using the finite element basis function #imgabs0# to calculate the expansion coefficients in each far-field direction; constructs the stiffness matrix #imgabs1# of the finite element equation and the boundary matrix; calculates the electromagnetic far-field components in each far-field direction according to the near-field electric field, the stiffness matrix, the boundary matrix, and the expansion coefficients of the two auxiliary vector fields in the current far-field direction. By introducing the variational finite element method, the present invention eliminates the accuracy loss caused by the geometric boundary complexity or medium inhomogeneity in the traditional integral method. The auxiliary vector field directly embeds the phase factor to accurately characterize the electromagnetic wave propagation characteristics, ensuring the calculation reliability. At the same time, by improving the near-field to far-field transformation process, it can meet the electromagnetic simulation requirements of large-scale simulation domains.

Description

Technical Field

[0001] The present invention belongs to the electromagnetic field, and particularly relates to an electromagnetic simulation method and system based on the variational finite element method. Background Art

[0002] With the development of wireless communication, radar detection, electronic countermeasure, and high-speed circuit systems, electromagnetic field analysis technology has been increasingly widely used in modern engineering fields. Electromagnetic simulation technology is a technical means that uses computer numerical methods to simulate the propagation characteristics of electromagnetic fields in different media and structures. Its core is to solve Maxwell's equations to obtain the distribution, propagation path of the electromagnetic field, and its interaction characteristics with objects. However, with the continuous improvement of the requirements for simulation accuracy and scenario complexity in engineering applications, the existing electromagnetic simulation methods at least face the following challenges:

[0003] 1) Insufficient modeling accuracy for complex geometries and multi-scale structures

[0004] When traditional finite element method (FEM) and finite difference time domain method (FDTD) are used to process complex geometries with sharp edges, curved surfaces, or multi-layer media, the accuracy often decreases due to discretization errors or boundary condition simplifications. In addition, multi-scale problems pose extremely high requirements for the adaptability of spatial discretization. Existing methods often need to sacrifice resolution or increase computing resources to maintain accuracy.

[0005] 2) Limited numerical stability for non-uniform media and high-frequency scenarios

[0006] In non-uniform media or high-frequency electromagnetic fields, traditional integral equation methods (such as the method of moments) lead to a sharp decline in matrix filling and solution efficiency due to the high-oscillation characteristics of the Green's function. Although the finite element method can partially alleviate the problem through adaptive meshing, under high-frequency conditions, due to the shortening of the wavelength, the mesh density needs to increase exponentially, significantly increasing the computational burden. At the same time, the phase-sensitive characteristics of high-frequency fields require strict modeling of the propagation direction and boundary coupling. Approximate treatments in existing methods are prone to introduce pattern distortion.

[0007] 3) Dilemma of trade-off between computing resources and efficiency

[0008] Although high-performance computing (HPC) and parallel algorithms have partially alleviated the computational pressure of large-scale problems, the matrix size and condition number of existing high-precision methods (such as high-order FEM) still grow superlinearly with the problem complexity. Traditional methods consume a large amount of memory and computing time and are difficult to meet the real-time requirements of engineering design.

[0009] In addition, the far field refers to the region far from the radiation source, and its characteristics are that the spatial distributions of the electric field and magnetic field satisfy the plane wave approximation condition. Generally, it is considered that when the distance satisfies where is the size of the radiation source, is the wavelength), it can be considered that the far - field region is entered. In the far - field region, the amplitudes of the electric and magnetic fields decay with the square of the distance, and the electric field, magnetic field, and propagation direction are mutually perpendicular. The main objectives of far - field calculations are to obtain the radiation pattern, gain, and scattering characteristics. However, there are contradictions in the computational efficiency and accuracy of existing electromagnetic simulation and far - field calculation methods. High - precision methods require high computational resources and are difficult to handle large - scale problems; while efficient approximate methods have deficiencies in accuracy.

[0010] Therefore, there is an urgent need for an electromagnetic simulation and far - field calculation method and system that can improve computational efficiency while ensuring computational accuracy to meet the requirements of high - precision electromagnetic analysis in modern communication, radar, and electronic countermeasure fields. Summary of the Invention

[0011] The purpose of the present invention is to solve the low - efficiency and accuracy problems of existing electromagnetic simulation methods in far - field calculations, and to propose an electromagnetic simulation method and system based on the variational finite - element method. By improving the near - field to far - field transformation process, a more efficient and accurate calculation method is provided, especially capable of meeting the electromagnetic simulation requirements of large - scale simulation domains.

[0012] In a first aspect, the present invention provides an electromagnetic simulation method based on variational finite - elements, and the method includes:

[0013] Set the computational domain , and divide the computational domain into an internal region and an external region , where the internal region contains scatterers or dielectrics, and the external region is free space; there is an artificial truncation boundary between the internal region and the external region ;

[0014] Define a first auxiliary vector field v and a second auxiliary vector field u on the artificial truncation boundary ;

[0015] For each far - field direction, use the finite - element basis functions to project the first auxiliary vector field v and the second auxiliary vector field u in each far - field direction, and calculate the expansion coefficients and ;

[0016] Construct the stiffness matrix of the finite - element equation and the boundary matrix , and use the stiffness matrix Describe the internal region of the field distribution, the boundary matrix Describe the artificial truncation boundary of the field distribution thereon;

[0017] According to the near-field electric field E, the stiffness matrix , the boundary matrix , and the expansion coefficients of the first auxiliary vector field v and the second auxiliary vector field u in the current far-field direction, calculate the electromagnetic far-field components in each far-field direction .

[0018] In one implementation, the artificial truncation boundary and the Dirichlet boundary together constitute the boundary of the internal region .

[0019] In one implementation, the expressions of the first auxiliary vector field v and the second auxiliary vector field u are:

[0020] (1)

[0021] (2)

[0022] Among them, represents the phase factor, reflecting the propagation characteristics of electromagnetic waves; among them is an imaginary number, is the incident wave number, is the observation unit vector in the current far-field direction, is the artificial truncation boundary on the local coordinates, represents the direction unit vector, is the unit normal vector, pointing to the external region ;

[0023] In one implementation, the process of constructing the stiffness matrix of the finite element equation and the boundary matrix is:

[0024] In free space, the strong form of the electric field distribution is described by the vector wave equation of Maxwell's equations; through the variational finite element method, after further taking the inner product with the first auxiliary vector field v and integrating in the internal region , we get

[0025] (3)

[0026] Among them represents the curl operator; denotes the relative permittivity; denotes a tiny volume element;

[0027] For terms, construct an identity:

[0028] (4)

[0029] After substituting the identity (4) into formula (3), and then using the divergence theorem to convert the terms about divergence into boundary integrals, and rewrite the weak form:

[0030] (5)

[0031] where is the integration volume, is the boundary area, is the surface integral on the artificial truncation boundary Σ, is the surface integral on the Dirichlet boundary ;

[0032] Construct the stiffness matrix respectively through the first term and the third term of the weak form equation (5):

[0033] (6)

[0034] (7)

[0035] In a second aspect, the present invention provides an electromagnetic simulation system, including:

[0036] An acquisition module, configured to acquire a calculation region ;

[0037] A first setting module, configured to define a first auxiliary vector field v and a second auxiliary vector field u on the artificial truncation boundary of the calculation region ;

[0038] A first calculation module, configured to project the first auxiliary vector field v and the second auxiliary vector field u in each far-field direction using the finite element basis function to calculate the expansion coefficients and in each far-field direction;

[0039] A second setting module, configured to construct the stiffness matrix of the finite element equation and the boundary matrix , and use the stiffness matrix to describe the field distribution in the internal region , and the boundary matrix Describe the artificial truncation boundary The field distribution on;

[0040] A second calculation module, configured to calculate the electromagnetic far-field components in each far-field direction according to the near-field electric field E, the stiffness matrix , the boundary matrix , and the expansion coefficients of the first auxiliary vector field v and the second auxiliary vector field u in the current far-field direction .

[0041] In a third aspect, the present invention provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed on a computer, the computer is made to execute the method described above.

[0042] In a fourth aspect, the present invention provides a computing device, including a memory and a processor. An executable code is stored in the memory. When the processor executes the executable code, the method described above is implemented.

[0043] The beneficial effects of the present invention at least include:

[0044] By introducing the variational finite element method, the present invention transforms the strong form of the Maxwell equation into a weak form, and combines the projection expansion of the auxiliary vector fields v and u on the artificial truncation boundary, eliminating the accuracy loss caused by the geometric boundary complexity or medium inhomogeneity in the traditional integral method. The auxiliary vector fields directly embed the phase factors, accurately characterizing the electromagnetic wave propagation characteristics and ensuring the calculation reliability in high-frequency bands, multi-scatterer, and non-free space scenarios.

[0045] By jointly defining the Dirichlet boundary and the artificial truncation boundary, the present invention effectively constrains the electromagnetic coupling effect of the calculation domain. The localized projection of the auxiliary fields on the boundary avoids the sensitivity of the global integral to complex geometries, and is particularly suitable for electromagnetic simulation problems of multi-scale structures or those with sharp edges and surfaces.

[0046] The present invention directly correlates the far-field direction with the local coordinates through the auxiliary vector fields, and accurately captures the electromagnetic field phase distribution on the boundary Σ through the projection expansion of the finite element basis functions. This method overcomes the far-field pattern distortion caused by phase approximation in traditional methods, and is particularly suitable for wide-band, multi-incident angle, and polarization-sensitive scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] In order to more clearly illustrate the technical solutions of the present invention, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0048] Figure 1Flowchart of the electromagnetic simulation method provided by an embodiment of the present application.

[0049] Figure 2 For the calculation region in the electromagnetic simulation method provided by an embodiment of the present application Schematic diagram.

[0050] Figure 3 In (a)- Figure 3 In (b) are respectively the front view and the perspective view of the test sample (Vivaldi antenna unit) of an embodiment of the present application.

[0051] Figure 4 For the comparison diagram of the radar cross section obtained from the variational far-field calculation result and the actual measurement value of the present invention in an embodiment of the present application.

[0052] Figure 5 For the comparison diagram of the RMS error obtained from the variational far-field calculation and the traditional integral far-field calculation of the present invention in an embodiment of the present application.

[0053] Figure 6 For the comparison diagram of the time required for the variational far-field calculation and the traditional integral far-field calculation of the present invention in an embodiment of the present application. Detailed implementation manners

[0054] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention.

[0055] Referring to the attached Figure 1 As shown, an embodiment of the present invention provides an electromagnetic simulation method based on variational finite element, including the following steps:

[0056] Step S1, define the calculation region , and set the boundary conditions;

[0057] Referring to the attached Figure 2 , divide the calculation region into an internal region and an external region , the internal region contains a scatterer or a medium (which can be a multi-scale structure, such as a Vivaldi antenna, referring to Figure 3 in (a)- Figure 3 in (b)), the external region is free space.

[0058] There is an artificial truncation boundary between the internal region and the external region , and the artificial truncation boundary and the Dirichlet boundary together constitute the internal region The boundary of

[0059] is the outer boundary of the entire computational domain; is the unit normal vector; is the relative permittivity, is the relative permeability, is the permittivity of free space, is the permeability of free space.

[0060] At this stage, the position and scope of the computational domain are clarified, and reasonable settings are ensured for all computational boundaries. This is a crucial step in the entire electromagnetic field simulation because the choice of the computational domain directly affects the accuracy and efficiency of the final calculation results. This step ensures that the electromagnetic field problem can be effectively solved within a known and limited area, while providing a clear computational framework for subsequent steps.

[0061] Step S2. Define the first auxiliary vector field v and the second auxiliary vector field u on the artificial truncation boundary The expressions are as follows:

[0062] (1)

[0063] (2)

[0064] where represents the phase factor, reflecting the propagation characteristics of electromagnetic waves; where is an imaginary number, is the incident wave number, is the observation unit vector in the current far-field direction, is the artificial truncation boundary is the local coordinate on represents the direction unit vector, is the unit normal vector, pointing to the external region ;

[0065] The first auxiliary vector field v and the second auxiliary vector field u are only defined on the boundary to simplify the treatment of boundary conditions, thereby achieving the purpose of ensuring numerical stability and improving computational efficiency.

[0066] To obtain the projections of the first auxiliary vector field v and the second auxiliary vector field u in each direction, the finite element basis function is used to project v and u according to formulas (3)-(4) respectively, and the corresponding expansion coefficients and are calculated.

[0067] (3)

[0068] (4)

[0069] in express The i-th element in express The i-th element in ;

[0070] Expand formula (3) and construct the following projection conditions:

[0071] For any test basis function ,have

[0072] (5)

[0073] in Indicates artificial truncation boundary The number of Gaussian points on Represents a surface element;

[0074] According to formula (5), the artificial truncation boundary is obtained The inner product relationship between the basis functions is described above ,Right now:

[0075] (6)

[0076] According to formula (5), the artificial truncation boundary is obtained The inner product relationship between the basis function and the first auxiliary vector field v is described above ,Right now:

[0077] (7)

[0078] Combining formulas (6)-(7), formula (5) can be written as a matrix equation:

[0079] (8)

[0080] in, , T represents transpose;

[0081] Similarly, formula (4) is expanded to construct the following projection conditions:

[0082] (9)

[0083] According to formula (8), the artificial truncation boundary is obtained The basis functions and the second auxiliary vector field are described above The inner product relationship between ,Right now:

[0084] (10)

[0085] The artificial truncation boundary obtained from formula (9) describing the inner product relationship between basis functions is the same as formula (6), and then based on the inner product relationship and it can be known that:

[0086] (11)

[0087] where .

[0088] Divide the artificial truncation boundary into several units, establish integration points for each unit, and obtain the expressions of the basis functions on these units. For each pair of basis functions and , that is:

[0089] (12)

[0090] where M represents the mass matrix, calculated using the Gaussian integration method; represents the number of each Gaussian point during numerical integration on the unit , is the integration point on the unit , is the corresponding weight, represents the value of the basis function at , represents the value of the basis function at .

[0091] For each test basis function , calculate the inner product relationship according to formulas (13)-(14):

[0092] (13)

[0093] (14)

[0094] where represents the value of the first auxiliary vector field v at , represents the value of the second auxiliary vector field u at ;

[0095] The inner product relationship Substitute into formulas (8) and (11), and then solve to obtain the expansion coefficients of the projections of the first auxiliary vector field v and the second auxiliary vector field u on the basis functions. and .

[0096] In the traditional finite element method , those involved in the formula belong to low-regularity spaces, making it impossible to accurately represent them through conventional basis functions, thus affecting the accuracy of far-field calculations, especially on complex geometric boundaries. In the example of the present invention, the projection calculation through the expansion of basis functions can effectively simplify the handling of boundary conditions and reduce the computational complexity; by introducing the auxiliary field vectors, it is only necessary to handle the auxiliary fields on the boundary, avoiding the complexity of global integration.

[0097] Meanwhile, in the traditional finite element method, the need for high-order integration and multi-directional integral calculations lead to a significant increase in the computational amount for solving complex boundaries and high-resolution grids. For large-scale computational domains, the integration calculation time increases quadratically and linearly with the increase of the domain size and far-field direction, resulting in low computational efficiency. In the example of the present invention, by introducing the auxiliary field, the handling of boundary conditions can be transformed into matrix-vector products, avoiding complex boundary integral calculations. By processing the boundary conditions through matrix operations, the computational efficiency is greatly improved.

[0098] Step S3: In free space, describe the strong form of the electric field distribution through the vector wave equation of Maxwell's equations, which is for

[0099] (15)

[0100] Through the variational finite element method, after taking the inner product of equation (15) with the first auxiliary vector field v and integrating within the internal region , we obtain

[0101] (16)

[0102] where represents the curl operator; represents the relative permittivity, which describes the response of the material to the electric field; represents the infinitesimal volume element;

[0103] For the term, construct the identity:

[0104] (17)

[0105] After substituting the identity (17) into formula (16), we get:

[0106] (18)

[0107] By using the divergence theorem again, the terms related to divergence in formula (18) are transformed into boundary integrals, obtaining:

[0108] (19)

[0109] where is the outer boundary of the internal region for applying the far - field radiation condition.

[0110] Rewrite formula (19) into the weak form as

[0111] (20)

[0112] The boundary of the internal region includes the artificial truncation boundary and the Dirichlet boundary , so the boundary integral term can be decomposed into

[0113] (21)

[0114] where is the surface integral on the artificial truncation boundary Σ, is the surface integral on the Dirichlet boundary , is the surface integral on the outer boundary of the entire internal region;

[0115] According to formula (21), rewrite the weak - form equation (20) as:

[0116] (22)

[0117] where is the integration volume, is the boundary area;

[0118] Construct the stiffness matrices from the first and third terms of the weak - form equation (22) respectively:

[0119] (23)

[0120] (24)

[0121] Finally, use the stiffness matrix to describe the field distribution in the internal region , and the boundary matrix to describe the field distribution on the artificial truncation boundary . After constructing the matrices, the sparse matrix Store for subsequent calculation use.

[0122] Step S4: Expand the near-field electric field E according to formula (25) as .

[0123] (25)

[0124] Based on the stiffness matrix , the boundary matrix , and the expansion coefficients and , perform a matrix-vector product to calculate the electromagnetic far-field component :

[0125] (26)

[0126] In the traditional finite element method, when directly calculating the far field, it is necessary to handle complex phase factors and boundary conditions, with a large amount of calculation and difficult to implement. In the example of the present invention, the components of the far-field electromagnetic field are directly obtained through matrix operations, where the matrix is a sparse matrix, and the complexity of the matrix-vector product is , so the calculation efficiency is relatively high. Since an auxiliary vector field is used to handle the boundary conditions, the integral calculation of the boundary conditions is effectively simplified, and the calculation process is more efficient.

[0127] Step S5: For each far-field direction, repeat Step S2 to update the expansion coefficients , , and Step S4 to update the electromagnetic far-field component . Finally, the electromagnetic far-field components in all far-field directions are obtained, and the electromagnetic simulation is completed. Specifically, first calculate the projection coefficients and in each far-field direction, and then perform a matrix-vector product to obtain the far-field electric field component in the corresponding direction. Since the complexity of the projection calculation is , where is the number of grid cells on the boundary, the calculation complexity is relatively low. The traditional finite element method directly performs integration and projection calculations on the boundary, with a high complexity. Especially in high-frequency electromagnetic problems, the complexity of boundary condition handling will increase rapidly. In the example of the present invention, due to the introduction of the auxiliary field vector, it can be directly expanded based on the finite element basis function, and the complexity of the projection calculation is significantly reduced. At the same time, through the matrix-vector product method, the overhead of directly calculating high-dimensional boundary integrals is avoided.

[0128] Figure 4It shows that the radar cross section obtained by the variational far-field calculation method of the present invention is very close to the actual measured value, and the error value is very small, proving that the electromagnetic simulation method given by the present invention can achieve the simulation goal.

[0129] Figure 5 It shows that under the condition of the same grid discretization accuracy, the RMS error obtained by the variational far-field calculation of the present invention is much smaller than that of the far-field calculation based on the traditional integral method, proving that the electromagnetic simulation method given by the present invention is superior to the existing methods in terms of calculation error and has a significant improvement in calculation accuracy.

[0130] Figure 6 It shows that under the condition of the same number of far-field calculation angles, the calculation time required by the variational far-field calculation of the present invention is much less than that required by the traditional integral far-field calculation, proving that the electromagnetic simulation method given by the present invention is superior to the existing calculation methods in terms of calculation time and has a significant improvement in efficiency.

[0131] In summary, through the simplified representation of the vector field, the present invention makes the transformation from near field to far field more direct and efficient. Traditional methods rely on complex integral operations to deduce the far field from the near field, while the present invention reduces the complexity of numerical integration through appropriate mathematical transformations of the electromagnetic field. By using the existing finite element solutions and combining with simplified mathematical expressions, the far-field calculation can be completed in a shorter time. The matrix calculation in the finite element method is also optimized. By analyzing the boundary conditions of the electric field and magnetic field in the finite element method, the present invention proposes an efficient calculation method based on matrix operations. Utilizing the characteristics of sparse matrices, the time overhead of matrix operations is reduced. In far-field calculations, the required matrix operations can be pre-calculated once and the results stored, and only a small number of matrix-vector multiplications are required, thus improving the calculation speed.

[0132] In the method of the present invention, the electromagnetic quantities in the near field are decomposed into multiple simple matrix elements. These matrix elements can be directly calculated through finite element solutions, and these solutions can be reused throughout the calculation process. This decomposition method greatly reduces the dependence on complex integrals and enables the far-field calculation to be completed more quickly.

[0133] The method of the present invention supports parallel computing. By decomposing the calculation tasks into multiple subtasks and utilizing the multi-core processing capabilities of modern computers, the calculation efficiency can be significantly improved. When dealing with large-scale electromagnetic simulation tasks, the feature of supporting parallel computing enables the present invention to complete a large number of calculations in a shorter time and adapt to larger-scale calculation requirements.

[0134] The present invention adopts an improved high-order Gaussian integration method to calculate the electromagnetic field quantities on the integration surface. This method can reduce the number of calculations and the number of integration points while ensuring the calculation accuracy. In this way, the present invention can significantly improve the calculation efficiency while ensuring the accuracy.

[0135] The present invention strictly controls the calculation accuracy. Although a variety of simplification methods are adopted to improve the calculation efficiency, all simplification operations strictly ensure the accuracy of the calculation results. By analyzing and controlling the errors in the calculation process, the accuracy of the far-field calculation is ensured. Especially in the high-frequency case, the reliability of the results can still be guaranteed.

[0136] The electromagnetic far-field calculation method proposed by the present invention has significant technical advantages, especially in improving the calculation efficiency and ensuring the calculation accuracy. By introducing an artificial integration surface and simplifying the transformation process from the near field to the far field, the present invention significantly improves the calculation efficiency while retaining the high calculation accuracy. By setting the artificial integration surface, the present invention limits the calculation range and reduces the number of integration points, thus avoiding the complex traversal of the entire electromagnetic space and greatly improving the calculation speed and efficiency.

[0137] On this basis, the present invention adopts an optimized finite element matrix calculation method. By utilizing the characteristics of the sparse matrix and decomposing the electromagnetic field calculation into reusable matrix elements, the redundant calculation is reduced. Especially when dealing with the far-field calculation, by means of the finite element solution that has been calculated, the present invention reduces the dependence on complex integrals and improves the calculation speed. In large-scale simulations, especially in the high-frequency electromagnetic field calculation, the far-field results can be obtained quickly and accurately, thus improving the efficiency of the overall simulation process.

[0138] To ensure the calculation accuracy, the present invention strictly controls the influence of all simplification steps on the result accuracy. Although the complexity of the numerical integration is reduced by optimizing the calculation method, all simplification steps are carried out within the range of controllable accuracy. This enables the present invention to still maintain a high calculation accuracy in the high-frequency band and complex electromagnetic environments, avoiding the problem of reduced accuracy caused by excessive calculation complexity in traditional methods.

[0139] The method of the present invention also performs excellently in reducing the consumption of calculation resources. By optimizing the numerical integration method and the matrix calculation process, the resources required for the calculation are effectively controlled. In a large-scale calculation environment, the present invention can complete efficient and accurate far-field calculations at a lower calculation cost, greatly reducing the resource consumption and cost burden caused by a large amount of calculation in traditional methods.

[0140] The above are the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and modifications can be made, and these improvements and modifications are also regarded as the protection scope of the present invention.

Claims

1. An electromagnetic simulation method based on variational finite element, characterized in that The method includes: Set the calculation region , divide the calculation region into an internal region and an external region . The internal region contains scatterers or a medium, and the external region is free space; there is an artificial truncation boundary between the internal region and the external region ; Define a first auxiliary vector field v and a second auxiliary vector field u on the artificial truncation boundary ; For each far-field direction, finite element basis functions are used Project the first auxiliary vector field v and the second auxiliary vector field u in each far-field direction, and calculate the expansion coefficients for each far-field direction and ; Construct the stiffness matrix of the finite element equation and the boundary matrix , and use the stiffness matrix to describe the field distribution in the internal region . The boundary matrix describes the field distribution on the artificial truncation boundary ; According to the near-field electric field E, stiffness matrix , boundary matrix , and the expansion coefficients of the first auxiliary vector field v and the second auxiliary vector field u in the current far-field direction, calculate the electromagnetic far-field components in each far-field direction .

2. The method according to claim 1, wherein Artificially truncated boundary and Dirichlet boundary together form the boundary of the internal region .

3. The method according to claim 1, wherein The expressions of the first auxiliary vector field v and the second auxiliary vector field u are: （1） （2） Among them, represents the phase factor, reflecting the propagation characteristics of electromagnetic waves; among them is an imaginary number, is the incident wave number, is the observation unit vector in the current far-field direction, is the artificial truncation boundary is the local coordinate on it, represents the direction unit vector, is the unit normal vector, pointing to the external region .

4. The method according to claim 1, wherein The stiffness matrix for constructing the finite element equations and the boundary matrix The process is as follows: In free space, the electric field is described by the vector wave equation of Maxwell's equations in the strong form of the distribution; through the variational finite element method, after further taking the inner product with the first auxiliary vector field v and integrating in the internal region the integral is obtained （3） wherein represents the curl operator; represents the relative permittivity; represents a tiny volume element; is the incident wave number; For items, construct an identity: （4） After substituting the identity (4) into formula (3), and then using the divergence theorem to convert the terms about divergence into boundary integrals, and then rewriting the weak form: （5） wherein is the integration volume, is the boundary area, is the surface integral on the artificially truncated boundary Σ, is the surface integral on the Dirichlet boundary ; is the unit normal vector, pointing to the external region ; Construct the stiffness matrix through the first and third terms of the weak form equation (5) respectively : （6） （7） 5. An electromagnetic simulation system for implementing the method according to any one of claims 1-4, characterized in that including: Acquisition module, used to acquire the calculation area ; A first setting module for defining a first auxiliary vector field v and a second auxiliary vector field u on an artificial truncation boundary of a calculation region ; ​ The first calculation module is used to, for each far-field direction, use the finite element basis function to project the first auxiliary vector field v and the second auxiliary vector field u in each far-field direction, and calculate the expansion coefficients in each far-field direction and ; The second setting module is used to construct the stiffness matrix of the finite element equation and the boundary matrix , and use the stiffness matrix to describe the field distribution in the internal region , and the boundary matrix describes the field distribution on the artificial truncation boundary ; A second calculation module, configured to calculate electromagnetic far-field components in each far-field direction according to the near-field electric field E, the stiffness matrix , the boundary matrix , and the expansion coefficients of the first auxiliary vector field v and the second auxiliary vector field u in the current far-field direction .

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed in a computer, the computer is made to execute the method according to any one of claims 1-4.

7. A computing device, comprising a memory and a processor, wherein executable code is stored in the memory, characterized in that, When the processor executes the executable code, the method according to any one of claims 1-4 is implemented.

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