Electromagnetic simulation method based on high-order modified Legendre stack basis function

Abstract

The application discloses a kind of electromagnetic simulation methods of moment method based on high-order modified Legendre stack base function, the method includes the following steps: S1, target geometry modeling, using square element to carry out mesh division to target;S2, construct high-order modified Legendre base function;S3, establish integral equation;S4, impedance matrix equation construction;S5, impedance matrix equation solution.The application expresses current with higher polynomial, thereby reducing the number of unknown quantity to the requirement of grid density reduction;While aiming at the problem of low frequency collapse when sweeping frequency, a kind of scalar vector potential separation high-order base function combination coefficient construction loop base function method is innovatively proposed, effectively solve the low frequency problem, reduce the memory of large-scale problem, improve the calculation efficiency, ensure the stability of algorithm.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology, and in particular to an electromagnetic simulation method based on the method of moments (MoM) of higher-order modified Legendre cascaded basis functions. Background Technology

[0002] The Method of Moments (MoM) is a classic and widely used numerical method in computational electromagnetics. Its core idea is to linearly expand an unknown function in a continuous domain (such as the induced current on the surface of a metallic conductor) using a set of known basis functions. The integral equation is then transformed into a system of algebraic equations using the weighted residual method, allowing the electromagnetic response characteristics to be solved. The MoM serves as a crucial bridge between classical electromagnetic theory and modern high-performance computing, and is widely applied in engineering fields such as antenna design, electromagnetic compatibility, and radar cross section (RCS) assessment. Currently, commonly used basis functions in the MoM include RWG basis functions, pulse basis functions, and roof basis functions. Among these, RWG basis functions, defined on pairs of triangles sharing a common edge, are among the most widely used basis functions in engineering practice.

[0003] For very large-scale cases, the use of RWG basis functions requires a precise mesh, which results in an excessively large number of unknowns. This leads to severe degradation in the storage and solution of the method of moments, not only significantly increasing memory usage but also making the filling and solution of the impedance matrix extremely inefficient, making it difficult to meet the requirements of computational speed and scale in engineering applications.

[0004] To address this problem, researchers have proposed a higher-order basis function method to reduce the number of unknowns. Higher-order basis functions use higher-order polynomials to express complex current distributions within a region and are broadly classified into two categories: higher-order interpolation basis functions and higher-order stacked basis functions. Higher-order interpolation basis functions require the same order to be defined across all meshes, lacking flexibility in engineering applications. Higher-order stacked basis functions, on the other hand, cover lower-order basis functions with higher-order ones, offering convenience for engineering applications. However, during broadband frequency sweeps, the conventional RWG basis function method of moments suffers from low-frequency collapse; that is, at low frequencies, the scalar potential values ​​in the impedance matrix become too large, overwhelming the smaller vector potentials and causing significant deviations in the calculation results. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides an electromagnetic simulation method based on higher-order modified Legendre stacked basis functions using the method of moments. This method expresses current using higher polynomials, thereby reducing the requirements for mesh density and the number of unknowns. Furthermore, to address the low-frequency collapse problem during frequency sweeps, this invention innovatively proposes a method for constructing loop basis functions by combining coefficients of higher-order basis functions with scalar vector potential separation. This effectively solves the low-frequency problem, reduces memory usage for large-scale problems, improves computational efficiency, and ensures the stability of the algorithm.

[0006] This invention provides the following solutions:

[0007] This invention provides an electromagnetic simulation method based on higher-order modified Legendre stacked basis functions using the method of moments, the method comprising the following steps:

[0008] S1. Target geometric modeling: The target is meshed using rectangular elements.

[0009] S2. Construct higher-order modified Legendre basis functions;

[0010] S3. Establish the electric field integral equation;

[0011] S4. Construction of the impedance matrix equation;

[0012] S5. Solving the impedance matrix equation.

[0013] Furthermore, step S1 includes the following process: based on the target structure of the electromagnetic problem to be solved, establish a geometric model and use rectangular elements to discretize the target into a mesh.

[0014] Furthermore, step S2 includes the following process: for each rectangular grid, the surface current is directed along the U direction r u and V direction r v Expand in two directions; where for r u Direction, establish a set of basis functions on a rectangular grid, let r u and r v The orders of the expansions in the two directions are M respectively. u and M v The discrete expression for the current on a rectangular grid surface is:

[0015] ,

[0016] ,

[0017] ;

[0018] Among them, J s For the current in a rectangular grid, u and v represent the U-direction and V-direction, respectively. To represent the unknown quantity of the current coefficient. Jacobi factor, and C n These are the first and second scale factors, respectively, P n For Legendre polynomials, For the modified Legendre polynomial, C n and The expressions are as follows:

[0019] ;

[0020] n=0,1,2...;

[0021] ;

[0022] ;

[0023] Where m and n represent the orders in the U and V directions, respectively; The basis functions consist of two parts: edge basis functions and surface basis functions. The edge basis functions are expressed in the expanded form of the modified Legendre polynomial, and their expression is: The expression for the surface basis function is: .

[0024] Furthermore, step S3 includes the following processes:

[0025] S3.1. Establish the corresponding electric field integral equation according to the type of electromagnetic problem to be solved. The expression is:

[0026] ,

[0027] in, For the incident electric field, For the unknown current density on the target surface, Let R be the Green's function in free space, j be the imaginary unit, k be the wave number, R = |r - r'| be the distance between the field point and the source point, ω be the angular frequency, μ be the permeability, S be the set of elements after discretization of the target surface, and r and r' be the field point and the source point, respectively.

[0028] S3.2, unknown current density The higher-order modified Legendre cascade basis functions constructed in step S2 are linearly expanded, i.e.:

[0029] ,

[0030] Where M is the total number of basis functions, i.e., the number of unknowns, and I i For unknown current coefficients, Let be the i-th higher-order modified Legendre basis function.

[0031] Furthermore, step S4 includes the following processes:

[0032] S4.1. Using the Galerkin method to perform the inner product test, substitute the discrete current expression from step S2 into the integral equation from step S3.1, and use the same test function as the basis function to perform the inner product test, resulting in the following matrix equation:

[0033] ZI=V,

[0034] Where Z is the impedance matrix, I is the unknown current coefficient vector, and V is the right-hand term excitation;

[0035] S4.2 Impedance Matrix Filling: Derive the formula for calculating the elements of the impedance matrix, substitute them into the current basis function expression to solve for the impedance matrix; the specific expression for the elements of the impedance matrix is ​​as follows:

[0036] ,

[0037] Among them, Z ij S is the element in the i-th row and j-th column of the impedance matrix. i and S j These are the rectangular patches containing the i-th and j-th basis functions, respectively.

[0038] S4.3 Singularity handling: Adaptive numerical integration is used to handle the singular integral of the impedance matrix, and Gaussian sampling points are staggered to avoid integral singularity.

[0039] S4.4. Combining Gaussian numerical integration, parallel acceleration is used in the calculation of impedance matrix elements to improve the efficiency of integration calculation.

[0040] S4.5 For low-frequency problems, based on the divergence of basis functions of different orders, the combination coefficients of the Loop basis functions whose sum of divergences is 0 are obtained, and the scalar potential and vector potential of the impedance matrix are separated based on these coefficients.

[0041] Furthermore, step S5 includes the following process: using the GMRES (Generalized Minimal Residual) method to solve the matrix equation ZI=V to obtain the unknown current coefficient vector I; and calculating the electromagnetic response characteristics of the target based on the obtained current coefficients.

[0042] Furthermore, the electromagnetic response characteristics include radar cross section, radiation pattern, and surface current distribution.

[0043] The beneficial effects of this invention based on its technical solution are as follows:

[0044] (1) Compared with the traditional method of moments, the present invention effectively reduces the number of grids, greatly reduces the scale of unknowns, improves the computational efficiency of matrix equations and reduces memory consumption during the solution process.

[0045] (2) Compared with higher-order interpolation basis functions, the present invention can flexibly define the order of the subdivision unit and properly handle the problem of the vector potential being submerged by the numerical value at low frequencies, thus having good low-frequency stability.

[0046] (3) The algorithm flow of the present invention is easy to be embedded into the code framework of the method of moments, and it is easy to combine with fast algorithms such as MLFMA (Multi Level Fast Multipole Method) and ACA (Adaptive Cross Approximation). Attached Figure Description

[0047] To more clearly illustrate the technical solutions in the embodiments of this specification or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0048] Figure 1 This is a flowchart illustrating the process of this embodiment.

[0049] Figure 2 This is a schematic diagram of higher-order modified Legendre edge basis functions.

[0050] Figure 3 This is a schematic diagram of the basis functions for the higher-order modified Legendre surface.

[0051] Figure 4 A schematic diagram of the combination coefficients for constructing loops for higher-order basis functions. Detailed Implementation

[0052] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention are within the protection scope of the embodiments of the present invention.

[0053] This embodiment provides an electromagnetic simulation method based on the method of moments (MoM) using higher-order modified Legendre cascaded basis functions. The method includes the following steps:

[0054] S1. Based on the target structure of the electromagnetic problem to be solved, establish a geometric model and use rectangular elements to discretize the target into a mesh.

[0055] S2. Construct higher-order modified Legendre basis functions, specifically including the following process: For each rectangular subdivision element, the surface current is oriented according to the U direction r u and V direction r v Expand in two directions; where for r u Direction, establish a set of basis functions on a rectangular grid, let r u and r vThe orders of the expansions in the two directions are M respectively. u and M v The discrete expression for the current on a rectangular grid surface is:

[0056] ,

[0057] ,

[0058] ;

[0059] Among them, J s For the current in a rectangular grid, u and v represent the U-direction and V-direction, respectively. To represent the unknown quantity of the current coefficient. Jacobi factor, and C n These are the first and second scale factors, respectively, P n For Legendre polynomials, For the modified Legendre polynomial, C n and The expressions are as follows:

[0060] ;

[0061] n=0,1,2...;

[0062] ;

[0063] ;

[0064] Where m and n represent the orders in the U and V directions, respectively; For basis functions, including Figure 2 The edge basis functions shown are Figure 3 The surface basis function is shown in two parts. In the figure, the horizontal axis represents the U direction, and the vertical axis represents the V direction. m and n are the expansion orders in the U and V directions, respectively. The edge basis function adopts the expansion form of the modified Legendre polynomial, and the expression is: The expression for the surface basis function is: In the formula, bold text represents vectors, and unbold text represents scalars.

[0065] S3. Establish the integral equation, which includes the following steps:

[0066] S3.1. Establish the corresponding electric field integral equation according to the type of electromagnetic problem to be solved. The expression is:

[0067] ,

[0068] in, For the incident electric field, For the unknown current density on the target surface, Let R be the Green's function in free space, j be the imaginary unit, k be the wave number, R = |r - r'| be the distance between the field point and the source point, ω be the angular frequency, μ be the permeability, S be the set of elements after discretization of the target surface, and r and r' be the field point and the source point, respectively.

[0069] S3.2, unknown current density The higher-order modified Legendre cascade basis functions constructed in step S2 are linearly expanded, i.e.:

[0070] ,

[0071] Where M is the total number of basis functions, i.e., the number of unknowns, and I i For unknown current coefficients, Let be the i-th higher-order modified Legendre basis function.

[0072] S4. Construction of the impedance matrix equation, which includes the following process:

[0073] S4.1. Using the Galerkin method to perform the inner product test, substitute the discrete current expression from step S2 into the integral equation from step S3.1, and use the same test function as the basis function to perform the inner product test, resulting in the following matrix equation:

[0074] ZI=V,

[0075] Where Z is the impedance matrix, I is the unknown current coefficient vector, and V is the right-hand term excitation;

[0076] S4.2 Impedance Matrix Filling: Derive the formula for calculating the elements of the impedance matrix, substitute them into the current basis function expression to solve for the impedance matrix; the specific expression for the elements of the impedance matrix is ​​as follows:

[0077] ,

[0078] Among them, Z ij S is the element in the i-th row and j-th column of the impedance matrix. i and S j These are the rectangular patches containing the i-th and j-th basis functions, respectively.

[0079] S4.3 Singularity handling: Adaptive numerical integration is used to handle the singular integral of the impedance matrix, and Gaussian sampling points are staggered to avoid integral singularity.

[0080] S4.4. Combining Gaussian numerical integration, parallel acceleration is used in the calculation of impedance matrix elements to improve the efficiency of integration calculation.

[0081] S4.5 For low-frequency problems, based on the divergence of basis functions of different orders, the combination coefficients of the Loop basis functions whose sum of divergences is 0 are obtained. These coefficients are used to separate the scalar potential and vector potential of the impedance matrix, such as... Figure 4 As shown, for M u =2,M v The 24 higher-order basis functions with a value of 2 can be used to construct 9 loop basis functions. The combination coefficients of the higher-order basis functions corresponding to each loop basis function are shown in the figure. The green box represents the higher-order basis function and its combination coefficients corresponding to loop2, and the yellow box represents the higher-order basis function and its combination coefficients corresponding to loop6.

[0082] S5. Solving the impedance matrix equation, including the following process: Solving the matrix equation ZI=V using the GMRES method to obtain the unknown current coefficient vector. Based on the current coefficient obtained from the solution, the electromagnetic response characteristics of the target, such as radar cross section, radiation pattern and surface current distribution, are calculated.

[0083] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0084] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0085] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. An electromagnetic simulation method based on the method of moments (MoM) using higher-order modified Legendre cascaded basis functions, characterized in that, The method includes the following steps: S1. Target geometric modeling: The target is meshed using rectangular elements. S2. Construct higher-order modified Legendre basis functions; S3. Establish the electric field integral equation; S4. Construction of the impedance matrix equation; S5. Solving the impedance matrix equation.

2. The electromagnetic simulation method based on the method of moments according to claim 1, characterized in that: Step S1 includes the following process: Based on the target structure of the electromagnetic problem to be solved, establish a geometric model and use rectangular elements to discretize the target into a mesh.

3. The electromagnetic simulation method based on the method of moments according to claim 1, characterized in that: Step S2 includes the following process: For each rectangular grid, the surface current is directed along the U direction r u and V direction r v Expand in two directions; where for r u Direction, establish a set of basis functions on a rectangular grid, let r u and r v The orders of the expansions in the two directions are M respectively. u and M v The discrete expression for the current on a rectangular grid surface is: ， ， ； Among them, J s For the current in a rectangular grid, u and v represent the U-direction and V-direction, respectively. To represent the unknown quantity of the current coefficient. Jacobi factor, and C n These are the first and second scale factors, respectively, P n For Legendre polynomials, For the modified Legendre polynomial, C n and The expressions are as follows: ； ，n=0,1,2...； ； ； Where m and n represent the orders in the U and V directions, respectively; The basis functions consist of two parts: edge basis functions and surface basis functions. The edge basis functions are expressed in the expanded form of the modified Legendre polynomial, and their expression is: The expression for the surface basis function is: .

4. The electromagnetic simulation method based on the method of moments according to claim 2, characterized in that: Step S3 includes the following processes: S3.

1. Establish the corresponding electric field integral equation according to the type of electromagnetic problem to be solved. The expression is: ， in, For the incident electric field, For the unknown current density on the target surface, Let R be the Green's function in free space, j be the imaginary unit, k be the wave number, R = |r - r'| be the distance between the field point and the source point, ω be the angular frequency, μ be the permeability, S be the set of elements after discretization of the target surface, and r and r' be the field point and the source point, respectively. S3.2, unknown current density The higher-order modified Legendre cascade basis functions constructed in step S2 are linearly expanded, i.e.: ， Where M is the total number of basis functions, i.e., the number of unknowns, and I i For unknown current coefficients, Let be the i-th higher-order modified Legendre basis function.

5. The electromagnetic simulation method based on the method of moments according to claim 4, characterized in that: Step S4 includes the following processes: S4.

1. Using the Galerkin method to perform the inner product test, substitute the discrete current expression from step S2 into the integral equation from step S3.1, and use the same test function as the basis function to perform the inner product test, resulting in the following matrix equation: ZI=V, Where Z is the impedance matrix, I is the unknown current coefficient vector, and V is the right-hand term excitation; S4.2 Impedance Matrix Filling: Derive the formula for calculating the elements of the impedance matrix, substitute them into the current basis function expression to solve for the impedance matrix; the specific expression for the elements of the impedance matrix is ​​as follows: ， Among them, Z ij S is the element in the i-th row and j-th column of the impedance matrix. i and S j These are the rectangular patches containing the i-th and j-th basis functions, respectively. S4.3 Singularity handling: Adaptive numerical integration is used to handle the singular integral of the impedance matrix, and Gaussian sampling points are staggered to avoid integral singularity. S4.

4. Combining Gaussian numerical integration, parallel acceleration is used in the calculation of impedance matrix elements to improve the efficiency of integration calculation. S4.5 For low-frequency problems, based on the divergence of basis functions of different orders, the combination coefficients of the Loop basis functions whose sum of divergences is 0 are obtained, and the scalar potential and vector potential of the impedance matrix are separated based on these coefficients.

6. The electromagnetic simulation method based on the method of moments according to claim 5, characterized in that: Step S5 includes the following process: using the GMRES method to solve the matrix equation ZI=V to obtain the unknown current coefficient vector I; and calculating the electromagnetic response characteristics of the target based on the obtained current coefficients.

7. The electromagnetic simulation method based on the method of moments according to claim 6, characterized in that: The electromagnetic response characteristics include radar cross section, radiation pattern, and surface current distribution.

Images