Low-frequency electromagnetic wave propagation high wave number simulation method and system
By converting the high-wave number Helmholtz equation into a non-local integral form, the problem of degradation of simulation accuracy under high-wave number is solved, and higher numerical stability and accuracy are achieved, which is suitable for low-frequency electromagnetic wave propagation simulation.
Patent Information
- Application Number
- CN202411192569.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-08-28
- Publication Date
- 2024-12-17
AI Technical Summary
When dealing with the high-wave Helmholtz equation, the traditional discrete differential format will lead to numerical oscillation and the ‘high-wave number pollution effect’, resulting in a decrease in simulation accuracy.
By determining the parameter characteristics based on the high-wave number Helmholtz equation, the electromagnetic field fluctuation equation is transformed into a non-local integral form, and a linear system of equations is constructed to reduce digital errors and improve simulation accuracy.
Effectively reduce the digital errors generated by high wave numbers, improve the accuracy of the simulation, avoid numerical oscillation, and enhance the approximation ability to the electromagnetic wave propagation process.
Smart Images
Figure CN119149878A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of electromagnetic wave simulation, and in particular to a method and system for simulating low-frequency electromagnetic wave propagation with high wave numbers. Background Art
[0002] The Helmholtz equation is an elliptic partial differential equation used to describe electromagnetic waves. It has important practical application value by simulating the propagation characteristics of waves through high-order accurate and efficient numerical methods.
[0003] When dealing with complex wave phenomena, especially when high wave numbers are involved, numerical calculations will face great challenges as the wave number increases, and the quality of the solution will often decrease significantly. When the traditional discrete difference format is used to solve the Helmholtz equation with high wave numbers, violent numerical oscillations may occur, leading to the "high wave number contamination effect".
[0004] Therefore, there is an urgent need for a high-wavenumber simulation method and system for low-frequency electromagnetic wave propagation, which can effectively solve the numerical errors caused by high wavenumbers in the existing technology and improve the simulation accuracy. Summary of the invention
[0005] One of the purposes of the present invention is to provide a method and system for simulating low-frequency electromagnetic wave propagation with high wave numbers, which can effectively solve the numerical errors caused by high wave numbers in the prior art and improve the simulation accuracy.
[0006] In order to achieve the above object, a method for simulating low-frequency electromagnetic wave propagation with high wavenumber is provided, comprising the following steps:
[0007] S1. Based on the high-wavenumber Helmholtz equation, determine the parameter characteristics of the corresponding high-wavenumber Helmholtz equation; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wave number k and source term f(x, y);
[0008] S2. According to the determined parameter characteristics of the high-wavenumber Helmholtz equation, the corresponding electromagnetic field wave equation is determined, and the boundary conditions corresponding to the electromagnetic field wave equation are set;
[0009] S3. discretizing the determined electromagnetic field wave equation based on a preset grid spacing to form a discretized electromagnetic field wave equation;
[0010] S4. According to the discretized electromagnetic field wave equation and the corresponding boundary conditions, the partial differential equations corresponding to the electromagnetic field wave equation and the boundary conditions are converted into non-local integral forms, and the converted electromagnetic field wave equation and the boundary conditions are discretized, and the corresponding linear equations are constructed;
[0011] S5. Solve the constructed linear equations and output the amplitude values of each position in the corresponding solution domain.
[0012] Technical principles and effects of this scheme: In this scheme, firstly, based on the high-wavenumber Helmholtz equation, the parameter characteristics of the corresponding high-wavenumber Helmholtz equation are determined; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wavenumber k and source term f(x, y); by determining the parameter characteristics, the propagation characteristics of electromagnetic waves can be understood.
[0013] Then, according to the determined parameter characteristics, the electromagnetic field wave equation is determined and the boundary conditions corresponding to the electromagnetic field wave equation are set. In order to facilitate the solution, the electromagnetic field wave equation is discretized through the preset network spacing, and then the discretized electromagnetic field wave equation and the partial differential equation of the boundary condition are converted into a non-local integral. This conversion method makes it possible to better capture the influence of non-local effects in the process of electromagnetic wave propagation. Then, the electromagnetic field wave equation and the boundary conditions in the non-local integral form are discretized to realize the construction of the corresponding linear equation group. By solving the linear equation group, the amplitude values at each position in the solution domain can be obtained.
[0014] This solution effectively reduces the digital errors caused by high wave numbers by converting the electromagnetic field wave equation into a non-local integral form, and solves the problem in the prior art that, in the case of high wave numbers, as the wave number increases, numerical calculations will face huge challenges, and the quality of the solution will often decrease significantly. When the traditional discrete difference format is used to solve the He1mholtz equation with high wave numbers, violent numerical oscillations may occur, leading to the problem of "high wave number contamination effect", which can better simulate electromagnetic wave propagation and improve the accuracy of the simulation.
[0015] Furthermore, the electromagnetic field wave equation is:
[0016]
[0017] In the formula, u(x, y) is the amplitude corresponding to a point (x, y) in two-dimensional space, f(x, y) is the source term, that is, the excitation source at the position (x, y), and k 2 is the square of the wave speed.
[0018] Beneficial effects: By using the shortcomings of the electromagnetic field wave equation, the propagation of electromagnetic waves in space can be better understood in the presence of source terms.
[0019] Further, the conversion of the electromagnetic field wave equation in S4 into a non-local integral form comprises the following steps:
[0020] The amplitude u(x, y) at the midpoint (x, y) in the two-dimensional space is expanded by Nth order Taylor, and the formula after expansion is:
[0021]
[0022] Where R(N, x) is the remainder, m1 and m2 are the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 are the unit vectors in the x and y directions, ξ1 and ξ2 are the projections of the length between two material points in the x and y directions;
[0023] According to the properties of orthogonal functions, the function Satisfy the conversion of partial differential into integral; the specific conversion formula is:
[0024]
[0025] Where, 1≤p1+p2≤N, H x is the non-local range of point x; is the weight function, is the unknown coefficient matrix;
[0026] According to the second calculation formula, the corresponding as well as
[0027] The second calculation formula is:
[0028]
[0029] In the formula, is the known coefficient matrix, is the shape matrix;
[0030] According to the above content, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows:
[0031] Beneficial effects: In this scheme, Taylor expansion is first performed, and then the differential form is transformed into the integral form based on the properties of orthogonal functions. After that, the unknown coefficients in the integral form are determined. Through this series of operations, the electromagnetic field wave equation is transformed into a non-local integral form. The transformation of the non-local form can more accurately approximate the actual physical process, facilitate parallel computing, and further improve the flexibility of the calculation.
[0032] Further, the corresponding linear equations are constructed as follows:
[0033]
[0034] Where the matrices M and N represent the coefficients resulting from differentiating the equation, the vector d represents the known value of the discrete point at the boundary, the vector u is the position amplitude, and ε is the Lagrangian operator.
[0035] Beneficial effect: By constructing a system of linear equations, existing numerical methods can be used to solve them, which is more stable than directly solving partial differential equations.
[0036] The present invention also provides a low-frequency electromagnetic wave propagation high-wave number simulation system, comprising:
[0037] A parameter determination module, used for determining the parameter characteristics of the corresponding high-wavenumber Helmholtz equation based on the high-wavenumber Helmholtz equation; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wave number k and source term f(x, y);
[0038] An equation determination module, used to determine the corresponding electromagnetic field wave equation according to the parameter characteristics of the determined high-wavenumber Helmholtz equation, and to set the boundary conditions corresponding to the electromagnetic field wave equation;
[0039] An equation discretization module is used to discretize the determined electromagnetic field wave equation based on a preset grid spacing to form a discretized electromagnetic field wave equation;
[0040] An equation conversion module is used to convert the electromagnetic field wave equation and the partial differential equation corresponding to the boundary conditions into a non-local integral form according to the discretized electromagnetic field wave equation and the corresponding boundary conditions, discretize the converted electromagnetic field wave equation and the boundary conditions, and construct the corresponding linear equation group;
[0041] The amplitude output module is used to solve the constructed linear equations and output the amplitude values at each position of the corresponding solution domain.
[0042] Technical principles and effects of this scheme: In this scheme, firstly, the parameter characteristics of the corresponding high-wavenumber Helmholtz equation are determined based on the high-wavenumber Helmholtz equation through the parameter determination module; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wavenumber k and source term f(x, y); by determining the parameter characteristics, the propagation characteristics of electromagnetic waves can be understood.
[0043] Then the equation is determined through the equation determination module. Specifically, according to the determined parameter characteristics, the electromagnetic field wave equation is determined and the boundary conditions corresponding to the electromagnetic field wave equation are set. In order to facilitate the solution, the electromagnetic field wave equation is discretized through the preset network spacing. Then the equation transformation module is used to perform integral transformation on the discretized equation. Specifically, the discretized electromagnetic field wave equation and the partial differential equation of the boundary condition are transformed into a non-local integral. This transformation method can better capture the influence of non-local effects in the process of electromagnetic wave propagation. Then the electromagnetic field wave equation and the boundary conditions in the form of non-local integral are discretized to realize the construction of the corresponding linear equation group. By solving the linear equation group, the amplitude value of each position in the solution domain can be obtained.
[0044] This solution effectively reduces the digital errors caused by high wave numbers by converting the electromagnetic field wave equation into a non-local integral form, and solves the problem in the prior art that, in the case of high wave numbers, as the wave number increases, numerical calculations will face huge challenges, and the quality of the solution will often decrease significantly. When the traditional discrete difference format is used to solve the Helmholtz equation with high wave numbers, violent numerical oscillations may occur, leading to the problem of "high wave number contamination effect", which can better simulate electromagnetic wave propagation and improve the accuracy of the simulation.
[0045] Furthermore, the electromagnetic field wave equation is:
[0046]
[0047] In the formula, u(x,y) is the amplitude corresponding to a point (x,y) in two-dimensional space, f(x,y) is the source term, that is, the excitation source at the position (x,y), and k 2 is the square of the wave speed.
[0048] Furthermore, the conversion of the electromagnetic field wave equation into a non-local integral form in the equation conversion module comprises the following steps:
[0049] The amplitude u(x,y) at the midpoint (x,y) in the two-dimensional space is subjected to an N-order Taylor expansion, and the formula after expansion is:
[0050]
[0051] Where R(N,x) is the remainder, m1 and m2 are the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 are the unit vectors in the x and y directions, ξ1 and ξ2 are the projections of the length between two material points in the x and y directions;
[0052] According to the properties of orthogonal functions, the function Satisfy the conversion of partial differential into integral; the specific conversion formula is:
[0053]
[0054] Where, 1≤p1+p2≤N, H x is the non-local range of point x; is the weight function, is the unknown coefficient matrix;
[0055] According to the second calculation formula, the corresponding as well as
[0056] The second calculation formula is:
[0057]
[0058] In the formula, is the known coefficient matrix, is the shape matrix;
[0059] According to the above content, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows:
[0060] Further, the corresponding linear equations are constructed as follows:
[0061]
[0062] Where the matrices M and N represent the coefficients resulting from differentiating the equation, the vector d represents the known value of the discrete point at the boundary, the vector u is the position amplitude, and ε is the Lagrangian operator. BRIEF DESCRIPTION OF THE DRAWINGS
[0063] Figure 1 This is a flow chart of a method for simulating low-frequency electromagnetic wave propagation with high wave numbers in Embodiment 1 of the present invention;
[0064] Figure 2 This is a logic block diagram of a high wave number simulation system for low frequency electromagnetic wave propagation in Embodiment 1 of the present invention;
[0065] Figure 3 The calculated wave amplitude cloud diagram when the wave number k = 100;
[0066] Figure 4 This is the absolute error cloud diagram between the calculated nonlocal solution and the exact solution when the wave number k = 100;
[0067] Figure 5 The amplitude cloud diagram of the wave calculated when the wave number k = 1000;
[0068] Figure 6This is the absolute error cloud diagram between the calculated nonlocal solution and the exact solution when the wave number k = 1000;
[0069] Figure 7 The calculated wave amplitude cloud diagram when the wave number k = 10000;
[0070] Figure 8 This is the absolute error cloud diagram between the calculated nonlocal solution and the exact solution when the wave number k = 10000. DETAILED DESCRIPTION
[0071] The following is further described in detail through specific implementation methods:
[0072] Embodiment 1
[0073] A method for simulating the propagation of low-frequency electromagnetic waves with high wave numbers, basically as Figure 1 As shown, the following steps are included:
[0074] S1. Based on the high-wavenumber Helmholtz equation, determine the parameter characteristics of the corresponding high-wavenumber Helmholtz equation; the parameter characteristics include directional displacements x and y, amplitude u, length L and width H of the solution domain, wavenumber k and source term f(x, y); in this embodiment, x, y∈(0, 1).
[0075] S2. According to the determined parameter characteristics of the high-wavenumber Helmholtz equation, the corresponding electromagnetic field wave equation is determined, and the boundary conditions corresponding to the electromagnetic field wave equation are set;
[0076] The electromagnetic field wave equation is:
[0077]
[0078] In the formula, u(x, y) is the amplitude corresponding to a point (x, y) in two-dimensional space, f(x, y) is the source term, that is, the excitation source at the position (x, y), and k 2 is the square of the wave speed.
[0079] S3. discretizing the determined electromagnetic field wave equation based on a preset grid spacing to form a discretized electromagnetic field wave equation;
[0080] S4. According to the discretized electromagnetic field wave equation and the corresponding boundary conditions, the partial differential equations corresponding to the electromagnetic field wave equation and the boundary conditions are converted into non-local integral forms, and the converted electromagnetic field wave equation and the boundary conditions are discretized, and the corresponding linear equations are constructed;
[0081] The conversion of the electromagnetic field wave equation in S4 into a non-local integral form comprises the following steps:
[0082] The amplitude u(x,y) at the midpoint (x,y) in the two-dimensional space is subjected to an N-order Taylor expansion, and the formula after expansion is:
[0083]
[0084] Where R(N,x) is the remainder, m1 and m2 are the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 are the unit vectors in the x and y directions, ξ1 and ξ2 are the projections of the length between two material points in the x and y directions;
[0085] According to the properties of orthogonal functions, the function Satisfy the conversion of partial differential into integral; the specific conversion formula is:
[0086]
[0087] Where, 1≤p1+p2≤N, H x is the non-local range of point x; is the weight function, is the unknown coefficient matrix;
[0088] According to the second calculation formula, the corresponding as well as
[0089] The second calculation formula is:
[0090]
[0091] In the formula, is the known coefficient matrix, is the shape matrix;
[0092] According to the above content, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows:
[0093] In this embodiment, the specific conversion is:
[0094] The corresponding electromagnetic field wave equation of the high-wavenumber Helmholtz equation is:
[0095]
[0096] The formula for converting partial differentials into integral form is:
[0097]
[0098] Therefore, according to the above conversion formula, each term in the electromagnetic field wave equation is transformed into an integral:
[0099] For the first item Find the second derivative with respect to x, where p1 = 2p2 = 0:
[0100]
[0101] For the second item Find the second derivative of y, where p1 = 0 and p2 = 2:
[0102]
[0103] k 2 u(x,y) does not contain partial differential terms, so k 2 u(x,t)=k 2 y
[0104] f(x,y) is a function independent of the field function u(x,y), therefore, f(x,y) = f(x,y);
[0105] Therefore, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows: Among them, u ′ =u(x+ξ), u=u(x).
[0106] S5. Solve the constructed linear equations and output the amplitude values of each position in the corresponding solution domain.
[0107] The linear equations corresponding to the construction are as follows:
[0108]
[0109] Wherein, the matrices M and N represent the coefficients generated by the differential equation, the vector d represents the known value of the discrete point at the boundary, the vector u is the position amplitude, and ε is the Lagrangian operator. In this embodiment, in order to better illustrate the ability of this embodiment to solve the high-wavenumber Helmholtz equation, the wave speed is k=100, 1000 and 10000 respectively, where the source term is f(x, y)=(-2π 2 k 2 +k 4 )sin(πx)sin(πy). The amplitudes on all four boundaries are all 0, and the analytical solution for this amplitude is: u(x,y)=k 2 sin(πx)sin(πy). Figure 3 , Figure 5 as well as Figure 7 As shown, they are the amplitude cloud diagrams when wave numbers k = 100, 1000 and 10000 respectively. Figure 4 , Figure 6 as well as Figure 8The absolute error value cloud diagram is shown when the wave number k = 100, 1000 and 10000. From the simulation results, it can be obtained that the present embodiment can efficiently and accurately solve the high-wave number Helmholtz equation, and provides a new solution idea for solving the high-wave number Helmholtz equation in the low-frequency electromagnetic wave propagation simulation, which has important reference significance for the research in the fields of wireless through-the-ground communication, etc.
[0110] like Figure 2 As shown, this embodiment also discloses a low-frequency electromagnetic wave propagation high wave number simulation system, including:
[0111] A parameter determination module, used for determining the parameter characteristics of the corresponding high-wavenumber Helmholtz equation based on the high-wavenumber Helmholtz equation; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wave number k and source term f(x, y);
[0112] An equation determination module, used to determine the corresponding electromagnetic field wave equation according to the parameter characteristics of the determined high-wavenumber Helmholtz equation, and to set the boundary conditions corresponding to the electromagnetic field wave equation;
[0113] The electromagnetic field wave equation is:
[0114]
[0115] In the formula, u(x,y) is the amplitude corresponding to a point (x,y) in two-dimensional space, f(x,y) is the source term, that is, the excitation source at the position (x,y), and k 2 is the square of the wave speed.
[0116] An equation discretization module is used to discretize the determined electromagnetic field wave equation based on a preset grid spacing to form a discretized electromagnetic field wave equation;
[0117] An equation conversion module is used to convert the electromagnetic field wave equation and the partial differential equation corresponding to the boundary conditions into a non-local integral form according to the discretized electromagnetic field wave equation and the corresponding boundary conditions, discretize the converted electromagnetic field wave equation and the boundary conditions, and construct the corresponding linear equation group;
[0118] The conversion of the electromagnetic field wave equation into a non-local integral form in the equation conversion module comprises the following steps:
[0119] The amplitude u(x,y) at the midpoint (x,y) in the two-dimensional space is subjected to an N-order Taylor expansion, and the formula after expansion is:
[0120]
[0121] Where R(N,x) is the remainder, m1 and m2 are the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 are the unit vectors in the x and y directions, ξ1 and ξ2 are the projections of the length between two material points in the x and y directions;
[0122] According to the properties of orthogonal functions, the function Satisfy the conversion of partial differential into integral; the specific conversion formula is:
[0123]
[0124] Where, 1≤p1+p2≤N, H x is the non-local range of point x; is the weight function, is the unknown coefficient matrix;
[0125] According to the second calculation formula, the corresponding as well as
[0126] The second calculation formula is:
[0127]
[0128] In the formula, is the known coefficient matrix, is the shape matrix;
[0129] According to the above content, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows:
[0130] The amplitude output module is used to solve the constructed linear equations and output the amplitude values at each position of the corresponding solution domain.
[0131] The linear equations corresponding to the construction are as follows:
[0132]
[0133] Where the matrices M and N represent the coefficients resulting from differentiating the equation, the vector d represents the known value of the discrete point at the boundary, the vector u is the position amplitude, and ε is the Lagrangian operator.
[0134] The above is only an embodiment of the present invention. The common sense such as the known specific structure and characteristics in the scheme is described too much here. The ordinary technicians in the relevant field know all the common technical knowledge in the technical field of the invention before the application date or priority date, can know all the existing technologies in the field, and have the ability to apply the conventional experimental means before that date. The ordinary technicians in the relevant field can improve and implement this scheme in combination with their own abilities under the enlightenment given by this application. Some typical known structures or known methods should not become obstacles for ordinary technicians in the relevant field to implement this application. It should be pointed out that for those skilled in the art, without departing from the structure of the present invention, several deformations and improvements can be made, which should also be regarded as the protection scope of the present invention, which will not affect the effect of the implementation of the present invention and the practicality of the patent. The protection scope required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the specification can be used to interpret the content of the claims.
Claims
1. A method for simulating low-frequency electromagnetic wave propagation with high wavenumber, characterized in that: The following steps are involved: S1. Based on the high-wavenumber Helmholtz equation, determine the parameter characteristics of the corresponding high-wavenumber Helmholtz equation; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wave number k and source term f(x, y); S2. According to the determined parameter characteristics of the high-wavenumber Helmholtz equation, the corresponding electromagnetic field wave equation is determined, and the boundary conditions corresponding to the electromagnetic field wave equation are set; S3. discretizing the determined electromagnetic field wave equation based on a preset grid spacing to form a discretized electromagnetic field wave equation; S4. According to the discretized electromagnetic field wave equation and the corresponding boundary conditions, the partial differential equations corresponding to the electromagnetic field wave equation and the boundary conditions are converted into non-local integral forms, and the converted electromagnetic field wave equation and the boundary conditions are discretized, and the corresponding linear equations are constructed; S5. Solve the constructed linear equations and output the amplitude values of each position in the corresponding solution domain.
2. A method for simulating low-frequency electromagnetic wave propagation with high wavenumber according to claim 1, characterized in that: The electromagnetic field wave equation is: In the formula, u(x,y) is the amplitude corresponding to a point (x,y) in two-dimensional space, f(x,y) is the source term, that is, the excitation source at the position (x,y), and k 2 is the square of the wave speed.
3. A method for simulating low-frequency electromagnetic wave propagation with high wavenumber according to claim 2, characterized in that: The conversion of the electromagnetic field wave equation in S4 into a non-local integral form comprises the following steps: The amplitude u(x,y) at the midpoint (x,y) in the two-dimensional space is expanded by Nth order Taylor, and the formula after expansion is: Where R(N,x) is the remainder, m1 and m2 are the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 are the unit vectors in the x and y directions, ξ1 and ξ2 are the projections of the length between two material points in the x and y directions; According to the properties of orthogonal functions, the function Satisfy the conversion of partial differential into integral; the specific conversion formula is: Where, 1≤p1+p2≤N, H x is the non-local range of point x; w q1q2 (|ξ|) is the weight function, is the unknown coefficient matrix; According to the second calculation formula, the corresponding as well as The second calculation formula is: In the formula, is the known coefficient matrix, is the shape matrix; According to the above content, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows:
4. A method for simulating low-frequency electromagnetic wave propagation with high wavenumber according to claim 3, characterized in that: The linear equations corresponding to the construction are as follows: Where the matrices M and N represent the coefficients resulting from differentiating the equation, the vector d represents the known value of the discrete point at the boundary, the vector u is the position amplitude, and ε is the Lagrangian operator.
5. A low-frequency electromagnetic wave propagation high wave number simulation system, characterized by: include: A parameter determination module, used for determining the parameter characteristics of the corresponding high-wavenumber Helmholtz equation based on the high-wavenumber Helmholtz equation; the parameter characteristics include directional displacement x and y, amplitude u, length L and width H of the solution domain, wave number k and source term f(x, y); An equation determination module, used to determine the corresponding electromagnetic field wave equation according to the parameter characteristics of the determined high-wavenumber Helmholtz equation, and to set the boundary conditions corresponding to the electromagnetic field wave equation; An equation discretization module is used to discretize the determined electromagnetic field wave equation based on a preset grid spacing to form a discretized electromagnetic field wave equation; An equation conversion module is used to convert the electromagnetic field wave equation and the partial differential equation corresponding to the boundary conditions into a non-local integral form according to the discretized electromagnetic field wave equation and the corresponding boundary conditions, discretize the converted electromagnetic field wave equation and the boundary conditions, and construct the corresponding linear equation group; The amplitude output module is used to solve the constructed linear equations and output the amplitude values at each position of the corresponding solution domain.
6. A low-frequency electromagnetic wave propagation high-wavenumber simulation system according to claim 5, characterized in that: The electromagnetic field wave equation is: In the formula, u(x,y) is the amplitude corresponding to a point (x,y) in two-dimensional space, f(x,y) is the source term, that is, the excitation source at the position (x,y), and k 2 is the square of the wave speed.
7. A low-frequency electromagnetic wave propagation high-wavenumber simulation system according to claim 6, characterized in that: The conversion of the electromagnetic field wave equation into a non-local integral form in the equation conversion module comprises the following steps: The amplitude u(x,y) at the midpoint (x,y) in the two-dimensional space is expanded by Nth order Taylor, and the formula after expansion is: Where R(N,x) is the remainder, m1 and m2 are the differential orders with respect to x and y, ξ=ξ1e1+ξ2e2, e1 and e2 are the unit vectors in the x and y directions, ξ1 and ξ2 are the projections of the length between two material points in the x and y directions; According to the properties of orthogonal functions, the function Satisfy the conversion of partial differential into integral; the specific conversion formula is: Where, 1≤p1+p2≤N, H x is the non-local range of point x; is the weight function, is the unknown coefficient matrix; According to the second calculation formula, the corresponding as well as The second calculation formula is: In the formula, is the known coefficient matrix, is the shape matrix; According to the above content, the electromagnetic field wave equation is converted into an integral form through its corresponding differential form. The corresponding integral form is as follows:
8. A low-frequency electromagnetic wave propagation high-wavenumber simulation system according to claim 7, characterized in that: The linear equations corresponding to the construction are as follows: Where the matrices M and N represent the coefficients resulting from differentiating the equation, the vector d represents the known value of the discrete point at the boundary, the vector u is the position amplitude, and ε is the Lagrangian operator.