Finite element fast frequency sweeping method and device based on MPVL algorithm

Abstract

The application discloses a kind of based on MPVL algorithm's finite element fast sweep method and device, belong to electromagnetic full-wave simulation technical field, method includes: according to the port type and / or boundary type of electromagnetic field model, construct finite element boundary value equation and finite element variational equation;The finite element boundary value equation is substituted into finite element variational equation to obtain new finite element variational equation;According to the characteristics of wave number in new finite element variational equation, wave number or square root of wave number is extracted as the frequency-related common factor of each term in new finite element variational equation and is arranged, to obtain arrangement equation;Arrange equation is reconstructed as the matrix equation that can use MPVL algorithm to solve, and based on MPVL algorithm, the matrix equation at given expansion frequency point is solved to obtain electromagnetic field value under different excitation.The application realizes fast sweep, and sweep result has higher accuracy.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of electromagnetic full-wave simulation, and particularly relates to a finite element fast frequency sweeping method and device based on an MPVL (Matrix Padé Via Lanczos) algorithm. BACKGROUND

[0002] Electromagnetics and electromagnetic simulation software provide powerful simulation guarantee for engineering design and scientific research, and are widely used in many fields such as communication, electronics, aerospace and the like. With the rapid development of electronic technology, antenna and filter models gradually become complex, and the solving ability and solving speed of electromagnetic simulation tools are also constantly improved. In view of the demand of software users for reducing simulation time and improving work efficiency, fast frequency sweeping technology becomes an indispensable link.

[0003] In the process of electromagnetic full-wave simulation analysis, it is often necessary to observe the relationship between the electromagnetic field distribution, network parameters, directional coefficients and the like of a device and frequency, and to perform frequency sweeping in a given frequency band to obtain the electromagnetic field value of the electromagnetic field equation at each frequency point. The electromagnetic finite element analysis method needs to solve a sparse matrix equation at an expanded frequency point, and the analysis and decomposition of the sparse matrix in this process is the most time-consuming. At present, most domestic electromagnetic simulation software adopts a discrete frequency sweeping method, and the analysis and decomposition of the sparse matrix at all expanded frequency points are performed one by one to obtain the electromagnetic field value at each expanded frequency point.

[0004] However, the discrete frequency sweeping method has high accuracy but low efficiency, and when there are many expanded frequency points to be solved, it takes a very long time. SUMMARY

[0005] In order to solve the above problems in the prior art, the application provides a finite element fast frequency sweeping method and device based on an MPVL algorithm. The technical problems to be solved by the application are solved through the following technical solutions.

[0006] In a first aspect, the application embodiment provides a finite element fast frequency sweeping method based on an MPVL algorithm, which comprises the following steps.

[0007] S10. Constructing a finite element boundary value equation and a finite element variation equation according to the port type and / or the boundary type of an electromagnetic field model;

[0008] S20. Substituting the finite element boundary value equation into the finite element variation equation to obtain a new finite element variation equation;

[0009] S30. According to the characteristics of the wave number in the new finite element variation equation, extracting and arranging the wave number or the square root of the wave number as a frequency-related common factor of each term in the new finite element variation equation to obtain an arranged equation.

[0010] S40, reconstructing the arrangement equation into a matrix equation capable of being solved by using the MPVL algorithm, and solving the matrix equation at a given expansion frequency point based on the MPVL algorithm to obtain electromagnetic field values under different excitations.

[0011] In a second aspect, an embodiment of the present application provides a finite element fast frequency sweeping device based on an MPVL algorithm, the device comprising:

[0012] A construction module is configured to construct a finite element boundary value equation and a finite element variational equation according to a port type and / or a boundary type of an electromagnetic field model.

[0013] A substitution module is configured to substitute the finite element boundary value equation into the finite element variational equation to obtain a new finite element variational equation.

[0014] An extraction and arrangement module is configured to extract and arrange a wave number or a square root of the wave number as a frequency-related common factor of each term in the new finite element variational equation according to characteristics of the wave number in the new finite element variational equation to obtain an arrangement equation.

[0015] A reconstruction and solving module is configured to reconstruct the arrangement equation into a matrix equation capable of being solved by using the MPVL algorithm, and solve the matrix equation at a given expansion frequency point based on the MPVL algorithm to obtain electromagnetic field values under different excitations.

[0016] The present application has the following advantages:

[0017] The finite element fast frequency sweeping method based on the MPVL algorithm provided by the present application innovatively applies the MPVL algorithm to electromagnetic field finite element simulation, specifically: by analyzing a port type and / or a boundary type of an electromagnetic field model, a corresponding finite element boundary value equation and a finite element variational equation are constructed, the finite element boundary value equation is substituted into the finite element variational equation to obtain a new finite element variational equation, according to characteristics of a wave number in the new finite element variational equation, the wave number or a square root of the wave number is extracted and arranged as a frequency-related common factor of each term in the new finite element variational equation to obtain an arrangement equation, the arrangement equation is reconstructed into a matrix equation capable of being solved by using the MPVL algorithm, and the matrix equation at a given expansion frequency point is solved based on the MPVL algorithm to obtain electromagnetic field values under different excitations. It can be seen that the present application takes the port type and the boundary type in the simulation analysis process into account, which not only takes simple boundary conditions into account, but also takes complex boundary conditions into account, and considers the complex dependence of some complex boundary conditions on the frequency in electromagnetic simulation, and simplifies and arranges them with correct theoretical basis, improves the calculation efficiency through model order reduction, and then reconstructs the simplified arrangement matrix into a matrix equation capable of being solved by using the MPVL algorithm, the MPVL algorithm realizes fast frequency sweeping, and ensures the accuracy of the frequency sweeping result.

[0018] The application will be described in further detail below with reference to the drawings and embodiments. BRIEF DESCRIPTION OF DRAWINGS

[0019] Figure 1 is a flowchart of a finite element fast frequency sweeping method based on the MPVL algorithm provided by an embodiment of the application;

[0020] Figure 2 is a flowchart of another finite element fast frequency sweeping method based on the MPVL algorithm provided by an embodiment of the application;

[0021] Figure 3 is a flowchart of still another finite element fast frequency sweeping method based on the MPVL algorithm provided by an embodiment of the application;

[0022] Figure 4 is a model diagram of a model of an electromagnetic field provided by an embodiment of the application, in which the port type of the model is a horn antenna;

[0023] Figure 5 is a structural diagram of a finite element fast frequency sweeping device based on the MPVL algorithm provided by an embodiment of the application. DETAILED DESCRIPTION

[0024] The application will be described in further detail below with reference to the drawings and embodiments.

[0025] MPVL technique is first used for calculating the frequency response function of circuit system, and Tianjin University proposes an electromagnetic sensitivity analysis method for fast frequency analysis of electromagnetic field finite element in its applied patent document "Electromagnetic sensitivity analysis method for fast frequency analysis of electromagnetic field finite element" (application number CN202110992300.2). The method combines MPVL algorithm with electromagnetic sensitivity analysis, obtains accurate electromagnetic field sensitivity of the entire frequency band by solving electromagnetic field sensitivity equation at only a single frequency, and deduces the accompanying / autocompanion formula of model reduction technology based on finite element, thereby accelerating the speed of electromagnetic sensitivity analysis. The method aims to accelerate the process of electromagnetic sensitivity analysis, and faster obtain the influence degree of geometric or material parameters on device electromagnetic parameters to guide design and processing. It can be seen that the above electromagnetic sensitivity analysis method, although MPVL algorithm is applied to electromagnetic sensitivity analysis to accelerate matrix equation solving, its technical focus is how to quickly obtain the influence of a design variable of the model on the electromagnetic field, and then guide the model optimization, and little attention is paid to various boundary conditions in actual problems, and it cannot be used to accelerate the full-wave simulation analysis process of electromagnetic finite element, and cannot adapt to various boundary conditions often encountered in electromagnetic analysis. Meanwhile, when the port type is a general wave port, the absorbing boundary condition has complex frequency dependence, and more considerations are needed to realize fast frequency sweeping under the premise of ensuring solution accuracy, and the method does not consider this.

[0026] In the electromagnetic analysis process, the finite element variational equation can be established by Galerkin method as follows:

[0027] (1);

[0028] wherein, represents a three-dimensional region of an electromagnetic field model to be solved, represents a Galerkin test function, represents a Hamiltonian operator, represents relative permeability, represents an electromagnetic field vector, represents wave number in vacuum, , represents frequency, , respectively represent permeability and dielectric constant in vacuum, represents relative dielectric constant, S represents a three-dimensional region boundary surface of an electromagnetic field model to be solved, represents an outer normal unit vector of the boundary, represents a volume element, represents an area element. In order to apply the MPVL algorithm to the fast solution of the electromagnetic field, it is necessary to make the dependence of each term in formula (1) on the frequency explicit, separate the common factor related to the frequency, and reconstruct the matrix equation. It can be seen from formula (1) that: the two volume integral terms of the three-dimensional region, that is, the integral terms related to Ω, are very clear in terms of the frequency dependence, while the area integral term, that is, the integral term related to S Ω, is not clear in terms of the frequency dependence due to the existence of various boundary conditions and ports, and sometimes is very complex.

[0029] Based on the above analysis, the embodiment of the present application provides a finite element fast sweep frequency solving method and device based on the MPVL technology, which is used for realizing the finite element fast simulation analysis of a model with various common boundary conditions and port types.

[0030] In a first aspect, referring to Figure 1 , the embodiment of the present application provides a finite element fast sweep frequency method based on the MPVL algorithm, which specifically includes the following steps:

[0031] S10, constructing a finite element boundary value equation and a finite element variation equation according to the port type and / or the boundary type of the electromagnetic field model;

[0032] S20, substituting the finite element boundary value equation into the finite element variation equation to obtain a new finite element variation equation;

[0033] S30, extracting and arranging the wave number or the square root of the wave number as a frequency-related common factor of each term in the new finite element variation equation according to the characteristics of the wave number in the new finite element variation equation, to obtain an arranged equation;

[0034] S40, reconstructing the arranged equation into a matrix equation that can be solved by using the MPVL algorithm, and solving the matrix equation at a given expansion frequency point based on the MPVL algorithm to obtain the electromagnetic field values under different excitations.

[0035] For different port types and boundary types of the electromagnetic field model, the equation design involved in S10-S40 is different, and the detailed design process of the three common scenarios is introduced as follows.

[0036] The first scenario provided by the embodiment of the present application is that if the port type of the electromagnetic field model is a lumped port or a coaxial wave port, and the boundary type is an absorbing boundary or an ideal conductor boundary or an impedance boundary or a lumped boundary, since the ideal conductor boundary among the four boundary types belongs to the Dirichlet boundary condition, and the remaining three belong to the Cauchy boundary condition, the finite element boundary value equation constructed in S10 corresponds to the formula:

[0037] (2);​​

[0038] wherein, denotes the boundary surface where the boundary type belongs to the Dirichlet boundary condition, denotes the outer normal unit vector of the boundary, denotes the electromagnetic field vector, denotes the boundary surface where the boundary type belongs to the Cauchy boundary condition, denotes the relative permeability, denotes the Hamiltonian operator, denotes the imaginary unit, denotes the electromagnetic wave number, denotes a constant, which takes different values for different boundary types, and is specifically determined according to the actual scene. As can be seen from formula (2), the Dirichlet boundary condition is irrelevant to the wave number k , and the Cauchy boundary condition has a simple linear relationship with the wave number k .

[0039] For the first scenario, formula (1) is rewritten to obtain the finite element variational equation in S10, which is represented by formula:

[0040] (3);

[0041] wherein, Ω denotes a three-dimensional region of an electromagnetic field model to be solved, denotes the Galerkin test function, denotes the wave number in vacuum, denotes the frequency, , denote the permeability and the dielectric constant in vacuum respectively, denotes the relative dielectric constant, denotes the volume element, denotes the area element.

[0042] Formula (2) is substituted into formula (3) to obtain a new finite element variational equation in S20, which is represented by formula:

[0043] (4);

[0044] wherein, denotes a constant matrix composed of .

[0045] The wave number in formula (4) is analyzed, and it can be seen that for the first scenario: the wave number k can be extracted as a frequency-related common factor of each term in the new finite element variational equation, and the arrangement equation is obtained, which is represented by formula: ​

[0046] (5);

[0047] in, , , Each represents one A sparse matrix, with corresponding subscripts representing wavenumbers. k Corresponding power, This indicates the number of frequency points in the electromagnetic field model. This represents the electromagnetic field values ​​to be solved under different excitations. , Indicates the first The electromagnetic field value to be solved under a given excitation. Represents the activation matrix, , Indicates the first An incentive.

[0048] The matrix equation is reconstructed to a form solvable using the MPVL algorithm. Finally, in S40, the matrix equation at a given expanded frequency point, such as the center frequency point, is solved using the following formula:

[0049] (6);

[0050] in, , Each represents one The block matrix, which is independent of frequency. , , express The zero matrix,

[0051] express The identity matrix,

[0052] , Represents the reconstructed first... The electromagnetic field value to be solved under a given excitation. This represents the reconstructed activation matrix. , express The zero matrix, Indicates a given expanded frequency point Electromagnetic wave number at that location , , These represent magnetic permeability and dielectric constant, respectively.

[0053] Further, the second scenario provided by the embodiment of the present application is that if the port type of the electromagnetic field model is a lumped port or a coaxial wave port and the boundary type is a finite conductivity boundary, the finite element boundary value equation constructed in S10 is expressed by the following formula:

[0054] (7) ;

[0055] wherein, Z represents an impedance value, μ0 represents the magnetic permeability in vacuum, k represents the wave number of electromagnetic waves, μ0 represents the magnetic permeability in vacuum, c represents the speed of light, σ represents the electrical conductivity, H represents the magnetic field intensity, , j represents the imaginary unit, ω represents the angular frequency, H represents the Hamiltonian operator, E represents the electromagnetic field vector, n represents the outer normal unit vector of the boundary.

[0056] Since the boundary type is a finite conductivity boundary, it also belongs to the Cauchy boundary condition, and the finite element variational equation in S10 is obtained by rewriting formula (1), which is expressed by the following formula:

[0057] (8) ;

[0058] wherein Ω represents a three-dimensional region of the electromagnetic field model to be solved, φ represents the Galerkin test function, μr represents the relative magnetic permeability, k represents the wave number in vacuum, , ω represents the frequency, , μ0 and ε0 represent the magnetic permeability and the dielectric constant in vacuum respectively, εr represents the relative dielectric constant, S represents the boundary surface of which the boundary type belongs to the Cauchy boundary condition, dV represents the volume element, dS represents the area element.

[0059] Substituting formula (7) into formula (8), the new finite element variational equation obtained by substituting in S20 is expressed by the following formula:

[0060] (9) ;

[0061] wherein, Z0 represents the electromagnetic wave impedance in vacuum, , represents a constant matrix composed of .

[0062] analysis wave number In formula (9), it can be seen that for the second scenario: the square root of the wave number can be extracted as a frequency-related common factor of each term in the new finite element variational equation, and the arrangement equation is obtained, which is expressed as:

[0063] (10);

[0064] wherein, , , , respectively represent a sparse matrix, represent the number of expansion frequency points in the electromagnetic field model, represent corresponding sparse matrix, represent the electromagnetic field values to be solved under different excitations, , represent the electromagnetic field values to be solved under the excitation, represent the excitation matrix, , represent the excitation.

[0065] Similarly, formula (10) is reconstructed into a form that can be solved using the MPVL algorithm, and finally the matrix equation at the given expansion frequency point in S40 is solved, which is expressed as:

[0066] (11);

[0067] wherein, , respectively represent a block matrix, , , represent zero matrix, represent identity matrix, , represent the electromagnetic field values to be solved under the reconstructed excitation, represent the reconstructed excitation matrix, , represent zero matrix, represent the electromagnetic wave wave number at the given expansion frequency point, , , Let represent the permeability and dielectric constant, respectively. This treatment increases the time and computational costs, but the speedup is still significant when the model has a large number of solution frequencies, and the results have very small errors compared to the exact solution.

[0068] Furthermore, the third scenario provided in this embodiment of the invention is when the electromagnetic field model has general waveports, where general waveports specifically refer to rectangular waveguide ports and circular waveguide ports. If the port type of the electromagnetic field model is a rectangular waveguide port or a circular waveguide port, please refer to [link to relevant documentation]. Figure 2 The finite element fast frequency sweep method based on the MPVL algorithm provided in this embodiment of the invention further includes, after S20:

[0069] S50. Extract the integral terms of the boundary surfaces affected by the propagation constant in the new finite element variational equation and express them as matrix functions. Then, perform interpolation on the matrix functions to obtain new matrix functions.

[0070] S60. Reapply the new matrix function as an integral term of the boundary surface affected by the propagation constant to update the new finite element variational equation, and return to step S30. Specifically:

[0071] For the third scenario, both rectangular and circular waveguide ports still fall under Cauchy boundary conditions, but the finite element boundary value equations constructed in S10 are expressed as follows:

[0072] (12);

[0073] in, The outward normal unit vector of the boundary. Represents relative permeability. Represents the Hamiltonian operator. Represents the electromagnetic field vector. Represents the imaginary unit. Describe a constant. Represents the wave number of electromagnetic waves. Indicates in The propagation constant at that point, , This indicates the waveguide cutoff wavenumber.

[0074] The finite element variational equations constructed for the third scenario are the same as those for the second scenario, and are expressed in the same way as formula (8), as follows:

[0075] ;

[0076] Where Ω represents the three-dimensional region of the electromagnetic field model to be solved. This represents the Galerkin test function. represents wave number in vacuum, , represents frequency, , respectively represent magnetic permeability and dielectric constant in vacuum, represents relative dielectric constant, represents a boundary surface of which boundary type belongs to Cauchy boundary condition, represents volume microelement, represents area microelement.

[0077] Substitute formula (12) into formula (8), so that the new finite element variation equation obtained by substitution in S20 is expressed as:

[0078] (13);

[0079] wherein, represents a constant matrix composed of . It can be seen from formula (13) that the boundary condition of a general wave port has complex frequency dependence, and wave number k cannot be extracted. In view of this, the embodiment of the present application proposes to extract the integral term of the boundary surface affected by the propagation constant from formula (13) and express it as a matrix function, denoted as , that is:

[0080] (14);

[0081] wherein, represents the matrix function at .

[0082] Take the lowest frequency point, the center frequency point and the highest frequency point of the electromagnetic field value to be solved to perform second-order Newton interpolation processing on the matrix function , and the new matrix function obtained by interpolation processing is expressed as:

[0083] (15);

[0084] wherein, represents the new matrix function at , , , respectively represent interpolation polynomial coefficient matrix, , , respectively represent the wave number of the electromagnetic wave at the lowest frequency point, the center frequency point and the highest frequency point of the electromagnetic field value to be solved, , , .

[0085] Substitute formula (15) into formula (13) to update the new finite element variational equation as the integral term of the boundary affected by the propagation constant, at this time the wave number k Extract the frequency-related common factor of each term in the new finite element variational equation, and organize the equation to obtain the organized equation, which is expressed as formula (16):

[0086] (16);

[0087] wherein, , , respectively represent a sparse matrix of, represents the number of expansion frequency points in the electromagnetic field model, represents the electromagnetic field value to be solved under different excitations, , represents the electromagnetic field value to be solved under the m-th excitation, represents an excitation matrix, , , represents the m-th excitation.

[0088] Similarly, reconstruct formula (16) into a form that can be solved using the MPVL algorithm, and solve the matrix equation at the given expansion frequency point in S40, which is expressed as formula (17):

[0089] (17);

[0090] wherein, , respectively represent a block matrix of, , , , represents a zero matrix of, represents a unit matrix of, , represents the electromagnetic field value to be solved under the m-th excitation after reconstruction, represents the excitation matrix after reconstruction, , represents a zero matrix of, represents the electromagnetic wave wave number at the given expansion frequency point , , represents a zero matrix of, , , , , respectively represent permeability and permittivity.

[0091] ​​Further, the general wave port adopts the way of constructing an interpolation polynomial, which is effective when the frequency band is narrow, but when the frequency band is wide, the interpolation error introduced will affect the accuracy of the result, so it is necessary to adaptively divide the frequency band into small intervals for processing in turn by analyzing the result error, but this will increase the calculation cost and time cost, and the increase degree depends on the specific model and cannot be determined. In view of this situation, by analyzing the function characteristics, the inventor finds that the error generated by formula (17) is derived from the interpolation approximation of the matrix function P(k), and this ultimately is the approximation of the propagation constant , by analyzing the function form, when , it can be regarded as , at this time, it is directly processed according to the steps of the first scenario, when and , the points in the interval can be expanded using the second-order Taylor series in two segments, and a good approximation effect can be obtained, so that the frequency solving interval is determined, without the need for error analysis and adaptive process. Specifically:

[0092] For the third scenario provided, if the port type of the electromagnetic field model is a rectangular waveguide port or a circular waveguide port, please refer to Figure 3 , the finite element fast frequency sweeping method based on the MPVL algorithm provided by the embodiment of the application comprises the following steps before S10:

[0093] S70, according to the waveguide cross-section wave number , the wave number is divided into three intervals; wherein the first interval is , the second interval is , and the third interval is ;

[0094] When the first interval, execute steps S10~S40, that is, the steps of the first scenario, which will not be repeated here;

[0095] When the second interval and the third interval, after S10, further comprise:

[0096] S80, using the second-order Taylor series expansion method, the propagation constant is expressed by the second-order Taylor formula to obtain a new propagation constant;

[0097] S90, updating the finite element boundary value equation according to the new propagation constant, and returning to step S20;

[0098] S100, merging the electromagnetic field values under different excitations corresponding to the three intervals.

[0099] Here, the finite element boundary value equation constructed in S10 is the same as formula (12), which is expressed as follows:

[0100] ​ ;

[0101] wherein, denotes the outward normal unit vector of the boundary, denotes the relative permeability, denotes the Hamiltonian operator, denotes the electromagnetic field vector, denotes the imaginary unit, denotes a constant, denotes the electromagnetic wave number, denotes the propagation constant at , , denotes the waveguide cut-off wave number;

[0102] The finite element variational equation constructed in S10 is shown in formula (8), as follows:

[0103] ;

[0104] wherein, Ω denotes a three-dimensional region of the electromagnetic field model to be solved, denotes the Galerkin test function, denotes the wave number in vacuum, , denotes the frequency, , denotes the permeability and the dielectric constant in vacuum, respectively, denotes the relative dielectric constant, denotes the boundary surface of which the boundary type belongs to the Cauchy boundary condition, denotes the volume infinitesimal element, denotes the area infinitesimal element.

[0105] Next, for the second interval , the third interval , the propagation constant is expressed by the second-order Taylor formula to obtain a new propagation constant by using the second-order Taylor series expansion method in S80, and the formula is shown as follows:

[0106] (18);

[0107] wherein, denotes the new propagation constant at , denotes the electromagnetic wave number at the center frequency point of the electromagnetic field value to be solved, denotes the propagation constant at , denotes the first-order derivative of the propagation constant at , denotes the first-order derivative of the propagation constant at the second derivative of the field at the point.

[0108] Substitute equation (18) into equation (12) to obtain the updated finite element boundary value equation in S90, which is expressed as:

[0109] (19);

[0110] Substitute equation (19) into equation (8) to obtain the new finite element variational equation in S20, which is expressed as:

[0111] (20);

[0112] wherein, is a constant matrix composed of At this time, the wave number k in S30 can be extracted as the frequency-dependent common factor of each term in the new finite element variational equation, and the rearranged equation is obtained, which is expressed as:

[0113] (21);

[0114] wherein, , , respectively represent a sparse matrix, represents the number of expansion frequency points in the electromagnetic field model, represents the electromagnetic field values to be solved under different excitations, , represents the electromagnetic field value to be solved under the th excitation, represents an excitation matrix, , represents the th excitation;

[0115] Similarly, equation (21) is reconstructed into a form that can be solved using the MPVL algorithm, and the matrix equation at the given expansion frequency point is solved in S40, which is expressed as:

[0116] (22);

[0117] wherein, , respectively represent a block matrix, , , represents a zero matrix of , represents a unit matrix of , , denotes the reconstructed electromagnetic field value under the th excitation, denotes the reconstructed excitation matrix, , denotes the zero matrix of , denotes the electromagnetic wave number at a given expansion frequency point , , , denote the magnetic permeability and the dielectric constant, respectively.

[0118] Further, for the above three complex scenarios, formulas (6), (11), (17), and (22) are constructed to be in a form that can be solved using the MPVL algorithm, and then the MPVL algorithm is used for fast solving. For convenience of description, at a given expansion frequency point, the matrix A is denoted as follows:

[0119] (23);

[0120] Two sets of orthogonal vectors and are generated using the MPVL algorithm, which span the Krylov space and reduce the matrix to a three-diagonal matrix to simplify the calculation, denotes the element at the th row and the th column in , q is the order of the MPVL algorithm, i.e., the number of inner vectors in each orthogonal vector. The orthogonal vectors are generated as follows:

[0121] (24);

[0122] (25);

[0123] (26);

[0124] It should be noted that can be any unit vector that does not have an inner product of 0 with , denotes the inverse operation, denotes the transpose operation, denotes the operation of first taking the transpose and then the inverse. Finally, the MPVL algorithm outputs , which are used to approximately solve the electromagnetic field values of the electromagnetic field model under different excitations at the expansion frequency point, and the formula is expressed as:

[0125] (27);

[0126] wherein, , represents the th excitation, , I represents q the th order unit matrix, The electromagnetic field values under different excitations at any given expansion frequency point in the electromagnetic field model can be completed through formulas (23) to (27).

[0127] Figure 4 In order to verify the effectiveness of the method proposed in the application, an electromagnetic field model of a horn antenna with a port type as shown in is provided in the embodiment of the application, and the discrete sweep frequency method and the sweep frequency method proposed in the application are compared. The simulation results show that the sweep frequency method proposed in the application has almost the same return loss S11 parameter as the discrete sweep frequency method. At the same time, the time consumption comparison results of the discrete sweep frequency method and the sweep frequency method proposed in the application are shown in Table 1.

[0128] Table 1 Comparison results of time consumption of the discrete sweep frequency method and the sweep frequency method proposed in the application

[0129]

[0130] From Table 1, it can be seen that the sweep frequency method proposed in the application realizes faster sweep frequency, and the more the expansion frequency points are, the more obvious the advantage of fast sweep frequency is.

[0131] In summary, the finite element fast frequency sweeping method based on the MPVL algorithm proposed in the embodiment of the application innovatively applies the MPVL algorithm to electromagnetic field finite element simulation, specifically: by analyzing the port type and / or the boundary type of the electromagnetic field model, the corresponding finite element boundary value equation and finite element variational equation are constructed, the finite element boundary value equation is substituted into the finite element variational equation to obtain a new finite element variational equation, according to the characteristics of the wave number in the new finite element variational equation, the wave number or the square root of the wave number is extracted as a frequency-related common factor of each term in the new finite element variational equation, and the arrangement equation is obtained by arranging, the arrangement equation is reconstructed into a matrix equation that can be solved using the MPVL algorithm, and the electromagnetic field values under different excitations at a given expansion frequency point are obtained based on the MPVL algorithm. It can be seen that the embodiment of the application takes the port type and the boundary type in the simulation analysis process into account, which not only considers the simple boundary condition, but also considers the complex boundary condition, and considers the complex dependence of the frequency on the complex boundary condition in electromagnetic simulation, and simplifies and arranges it with correct theoretical basis, improves the calculation efficiency through model order reduction, and then reconstructs the simplified arrangement matrix into a matrix equation that can be solved using the MPVL algorithm, the MPVL algorithm realizes fast frequency sweeping, and ensures the accuracy of the frequency sweeping result.

[0132] In a second aspect, referring to Figure 5 The embodiment of the application provides a finite element fast frequency sweeping device based on an MPVL algorithm, and the device comprises:

[0133] A construction module is configured to construct a finite element boundary value equation and a finite element variational equation according to a port type and / or a boundary type of an electromagnetic field model.

[0134] A substitution module is configured to substitute the finite element boundary value equation into the finite element variational equation to obtain a new finite element variational equation.

[0135] An extraction and arrangement module is configured to extract and arrange, according to characteristics of a wave number in the new finite element variational equation, the wave number or the square root of the wave number as a frequency-related common factor of each term in the new finite element variational equation to obtain an arrangement equation.

[0136] A reconstruction and solving module is configured to reconstruct the arrangement equation into a matrix equation that can be solved using the MPVL algorithm, and solve the matrix equation at a given expansion frequency point based on the MPVL algorithm to obtain electromagnetic field values under different excitations.

[0137] For the device embodiment of the second aspect, the description is relatively simple because it is basically similar to the method embodiment of the first aspect, and the related parts refer to the part of the method embodiment of the first aspect.

[0138] In the description of the application, it should be understood that the terms "first", "second" are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the technical features indicated. Therefore, the features defined with "first", "second" can explicitly or implicitly include one or more of the features. In the description of the application, the meaning of "a plurality of" is two or more, unless otherwise specifically limited.

[0139] Although the application is described herein in conjunction with specific embodiments thereof, numerous other variations and modifications are possible in light of the description and drawings. In the description, the word "comprising" does not exclude other components or steps, and "a" or "one" does not exclude a plurality. Certain measures are described in mutually different embodiments, but this does not mean that these measures cannot be combined to produce good results.

[0140] The above is a further detailed description of the application in conjunction with specific preferred embodiments, and the specific implementation of the application cannot be limited to these descriptions. For those of ordinary skill in the art to which the application belongs, without departing from the concept of the application, a number of simple deductions or substitutions can be made, which should be considered as falling within the scope of protection of the application.

Claims

1. A finite element fast frequency sweeping method based on MPVL algorithm, characterized in that, The method comprises: S10, constructing a finite element boundary value equation and a finite element variation equation according to a port type and / or a boundary type of an electromagnetic field model; S20, substituting the finite element boundary value equation into the finite element variation equation to obtain a new finite element variation equation; S30, extracting and arranging a wave number or a square root of the wave number as a frequency-related common factor of each term in the new finite element variation equation according to characteristics of the wave number in the new finite element variation equation, to obtain an arranged equation; S40, reconstructing the arranged equation into a matrix equation that can be solved using an MPVL algorithm, and solving the matrix equation at a given expansion frequency point based on the MPVL algorithm to obtain electromagnetic field values under different excitations; wherein, if the port type of the electromagnetic field model is a lumped port or a coaxial wave port, and the boundary type is an absorbing boundary or an ideal conductor boundary or an impedance boundary or a lumped boundary: the finite element boundary value equation constructed in S10 is expressed by a formula as: ； wherein, denotes a boundary surface where the boundary type belongs to a Dirichlet boundary condition, denotes an outward unit normal vector of the boundary, denotes an electromagnetic field vector, denotes a boundary surface where the boundary type belongs to a Cauchy boundary condition, denotes a relative permeability, denotes a Hamiltonian operator, denotes an imaginary unit, denotes an electromagnetic wave number, denotes a constant; the finite element variation equation constructed in S10 is expressed by a formula as: ； wherein Ω represents a three-dimensional region of the electromagnetic field model to be solved, denotes a Galerkin test function, denotes a wave number in vacuum, , denotes a frequency, , denote the magnetic permeability and the permittivity in vacuum, respectively, denotes the relative permittivity, denotes a volume infinitesimal, denotes an area infinitesimal; the new finite element variation equation obtained by substitution in S20 is expressed by a formula as: ； wherein represents a constant matrix consisting of ones; the arranged equation obtained by extraction and arrangement in S30 is expressed by a formula as: ； wherein, , , respectively represent a sparse matrix, represents the number of expansion frequency points in the electromagnetic field model, represents the electromagnetic field value to be solved under different excitations, , represents the electromagnetic field value to be solved under the th excitation, represents an excitation matrix, , represents the th excitation; the matrix equation solved at the given expansion frequency point in S40 is expressed by a formula as: ； wherein, , respectively represent a block matrix of , , represent a zero matrix of represent a identity matrix of , represent the reconstructed electromagnetic field value under the excitation, represent the reconstructed excitation matrix, , represent a zero matrix of represent the electromagnetic wave number at a given unfolded frequency point , , , respectively represent the magnetic permeability and the dielectric constant.

2. The MPVL algorithm based finite element fast frequency sweeping method of claim 1, wherein, if the port type of the electromagnetic field model is a lumped port or a coaxial wave port, and the boundary type is a finite electrical conductivity boundary: the finite element boundary value equation constructed in S10 is expressed by a formula as: ； wherein, denotes an impedance value, , denotes an electromagnetic wave number, denotes a magnetic permeability in vacuum, denotes a light speed, denotes an electric conductivity, denotes a magnetic field strength, , denotes an imaginary unit, denotes an angular frequency, denotes a Hamiltonian operator, denotes an electromagnetic field vector, denotes an outer normal unit vector of the boundary; the finite element variation equation constructed in S10 is expressed by a formula as: ； wherein Ω denotes a three-dimensional region of an electromagnetic field model to be solved, denotes a Galerkin test function, denotes a relative permeability, denotes a wave number in vacuum, , denotes a frequency, , denotes a permeability and a dielectric constant in vacuum, respectively, denotes a relative dielectric constant, denotes a boundary surface of which the boundary type belongs to a Cauchy boundary condition, denotes a volume infinitesimal, denotes an area infinitesimal; the new finite element variation equation obtained by substitution in S20 is expressed by a formula as: ； wherein denotes the impedance of an electromagnetic wave in vacuum, , denotes a constant matrix consisting of ​ the arranged equation obtained by extraction and arrangement in S30 is expressed by a formula as: ； wherein, , , , respectively represent a sparse matrix of , represents the number of expansion frequency points in the electromagnetic field model, represents the electromagnetic field value to be solved under different excitations, , represents the electromagnetic field value to be solved under the th excitation, represents an excitation matrix, , represents the th excitation; the matrix equation solved at the given expansion frequency point in S40 is expressed by a formula as: ； wherein, , respectively represent a block matrix of , , zero matrix of identity matrix of , , , represent the reconstructed electromagnetic field value under the th excitation, represent the reconstructed excitation matrix, , zero matrix of , represent the electromagnetic wave number at a given unfolded frequency point , , , respectively represent the magnetic permeability and the dielectric constant.

3. The MPVL algorithm based finite element fast frequency sweeping method of claim 1, wherein, if the port type of the electromagnetic field model is a rectangular waveguide port or a circular waveguide port, after S20, the method further comprises: S50, extracting an integral term of a boundary surface affected by a propagation constant in the new finite element variation equation and expressing the integral term as a matrix function, and performing interpolation processing on the matrix function to obtain a new matrix function; S60, reusing the new matrix function as the integral term of the boundary surface affected by the propagation constant to update the new finite element variation equation, and returning to step S30.

4. The MPVL algorithm based finite element fast frequency sweeping method of claim 3, wherein, the finite element boundary value equation constructed in S10 is expressed by a formula as: ； wherein, represents an outward normal unit vector to the boundary, represents relative permeability, represents a Hamiltonian operator, represents an electromagnetic field vector, represents the imaginary unit, represents a constant, represents an electromagnetic wave number, represents a propagation constant at , , represents a waveguide cut-off wave number; the finite element variation equation constructed in S10 is expressed by a formula as: ； wherein Ω represents a three-dimensional region of an electromagnetic field model to be solved, represents a Galerkin test function, represents a wave number in vacuum, , represents a frequency, , respectively represent a magnetic permeability and a dielectric constant in vacuum, represents a relative dielectric constant, represents a boundary surface of which a boundary type belongs to a Cauchy boundary condition, represents a volume infinitesimal, represents an area infinitesimal; the new finite element variation equation obtained by substitution in S20 is expressed by a formula as: ； wherein represents a constant matrix consisting of ones; the matrix function extracted and expressed in S50 is expressed by a formula as: ； wherein represents a matrix function at place. the new matrix function obtained by interpolation processing in S50 is expressed by a formula as: ； wherein, denotes a new matrix function at , , , denote interpolation polynomial coefficient matrices, , , denote the lowest, center, and highest frequency points of the electromagnetic wave number of the electromagnetic field values to be solved, , , ; the arranged equation obtained by extraction and arrangement in S30 is expressed by a formula as: ； wherein, , , respectively represent a sparse matrix of a , denotes the number of expansion frequency points in the electromagnetic field model, denotes the electromagnetic field value to be solved under different excitations, , denotes the electromagnetic field value to be solved under the th excitation, denotes an excitation matrix, , denotes the th excitation; the matrix equation solved at the given expansion frequency point in S40 is expressed by a formula as: ； wherein, , respectively represent a block matrix of , , zero matrix of , identity matrix of , , reconstructed electromagnetic field value under the th excitation, reconstructed excitation matrix, , zero matrix of , wave number of electromagnetic wave at a given unfolded frequency point , , , respectively represent magnetic permeability and dielectric constant.

5. The MPVL algorithm based finite element fast frequency sweeping method of claim 1, wherein, if the port type of the electromagnetic field model is a rectangular waveguide port or a circular waveguide port, before S10, the method further comprises: S70, according to the waveguide cutoff wave number , the wave number is divided into three intervals; wherein the first interval is , the second interval is , and the third interval is ; when in the first interval, performing steps S10-S40; when in the second interval and the third interval, after S10, the method further comprises: S80, using a second-order Taylor series expansion method to express the propagation constant as a new propagation constant by a second-order Taylor formula; S90, updating the finite element boundary value equation according to the new propagation constant, and returning to step S20; S100, merging electromagnetic field values under different excitations corresponding to three intervals.

6. The MPVL algorithm based finite element fast frequency sweeping method of claim 5, wherein, The finite element boundary value equation constructed in S10 is expressed by the formula: ； wherein, represents an outward normal unit vector to the boundary, represents relative magnetic permeability, represents a Hamiltonian operator, represents an electromagnetic field vector, represents an imaginary unit, represents a constant, represents an electromagnetic wave number, represents a propagation constant at , , represents a waveguide cutoff wave number; The finite element variational equation constructed in S10 is expressed by the formula: ； wherein Ω represents a three-dimensional region of an electromagnetic field model to be solved, represents a Galerkin test function, represents a wave number in vacuum, , represents a frequency, , respectively represent a magnetic permeability and a dielectric constant in vacuum, represents a relative dielectric constant, represents a boundary surface of which a boundary type belongs to a Cauchy boundary condition, represents a volume infinitesimal, represents an area infinitesimal; The new propagation constant expressed by the second-order Taylor formula in S80 is expressed by the formula: ； wherein, denotes the propagation constant at , denotes the electromagnetic wave number at the center frequency point of the electromagnetic field value to be solved, denotes the propagation constant at , denotes the propagation constant denotes the first derivative at , denotes the propagation constant denotes the second derivative at ; The finite element boundary value equation updated in S90 is expressed by the formula: ； The new finite element variational equation obtained by substituting in S20 is expressed by the formula: ； wherein represents a constant matrix consisting of ones; The arrangement equation obtained by extracting and arranging in S30 is expressed by the formula: ； wherein, , , respectively represent a sparse matrix of a , denotes the number of expansion frequency points in the electromagnetic field model, denotes the electromagnetic field value to be solved under different excitations, , denotes the electromagnetic field value to be solved under the excitation, denotes the excitation matrix, , denotes the excitation; The matrix equation at a given expansion frequency point solved in S40 is expressed by the formula: ； wherein, , respectively represent a block matrix of , , zero matrix of , identity matrix of , , reconstructed electromagnetic field value under the th excitation, reconstructed excitation matrix, , zero matrix of , wave number of electromagnetic wave at a given unfolded frequency point , , , respectively represent magnetic permeability and dielectric constant.

7. A finite element fast sweeping method based on MPVL algorithm device for implementing the finite element fast sweeping method based on MPVL algorithm as claimed in any one of claims 1 to 6, characterized in that, The device comprises: A construction module configured to construct a finite element boundary value equation and a finite element variational equation according to a port type and / or a boundary type of an electromagnetic field model; A substitution module configured to substitute the finite element boundary value equation into the finite element variational equation to obtain a new finite element variational equation; An extraction and arrangement module configured to extract and arrange a wave number or a square root of the wave number as a frequency-related common factor of each term in the new finite element variational equation according to characteristics of the wave number in the new finite element variational equation, to obtain an arrangement equation; A reconstruction and solving module configured to reconstruct the arrangement equation into a matrix equation that can be solved using an MPVL algorithm, and to solve the matrix equation at a given expansion frequency point based on the MPVL algorithm to obtain electromagnetic field values under different excitations.

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