CN111832157B - Large-scale quasi-periodic structure electromagnetic scattering characteristic analysis method based on sub-global basis function method - Google Patents
Large-scale quasi-periodic structure electromagnetic scattering characteristic analysis method based on sub-global basis function method Download PDFInfo
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Abstract
The invention discloses a large-scale quasi-periodic structure electromagnetic scattering characteristic analysis method based on a sub-global basis function method, which fully utilizes the physical characteristics of a quasi-periodic structure and extracts a main characteristic current mode as a sub-global basis function through a characteristic mode, thereby greatly reducing the complexity of solving and improving the calculation efficiency.
Description
Technical Field
The invention relates to a large-scale quasi-periodic structure electromagnetic scattering characteristic analysis method based on a sub-global basis function method, and belongs to the field of periodic structure electromagnetic characteristic numerical calculation.
Background
In recent years, periodic structures have been widely used in phased array radars, frequency selective surfaces and metamaterials. With the development, the quasi-periodic structure has more and more important applications in reflective array antennas and super surfaces. Most of electromagnetic numerical calculation methods aiming at the quasi-periodic array structure utilize a full-wave method and an acceleration algorithm mixed algorithm, and do not utilize structural characteristics and physical characteristics in the array structure.
The sub-gamut basis function method is always dedicated to efficiently extracting the gamut basis functions by using physical characteristics in an array structure. The traditional sub-global basis function method is to use the excitation source of the solution to excite the 3 x 3 sub-array, and each unit has only one global basis function, thus greatly reducing the equivalent unknown quantity of the periodic array structure and improving the calculation efficiency. However, this method can only process the cycle structures of the units that are completely the same, and the application scenarios are limited. With the research on the sub-global basis function method, the sub-global basis function method is mostly accelerated, and the conventional method for establishing the sub-global basis function is relatively fixed, so that the analyzed objects are all in a periodic array structure with completely identical units.
In summary, the basis function establishing scheme adopted by the existing sub-universe basis function method is limited and not the optimal scheme.
Disclosure of Invention
The invention aims to: in order to break through the limit that the sub-global basis function method can only process the periodic array structure with completely identical units, a large-scale quasi-periodic structure electromagnetic scattering characteristic analysis method based on the sub-global basis function method is provided.
The technical scheme is as follows: the large-scale quasi-periodic structure electromagnetic scattering characteristic analysis method based on the sub-universe basis function method comprises the following steps:
s100: selecting a reference unit to carry out geometric modeling and mesh subdivision, carrying out spatial rotation and scaling on the reference unit to obtain a plurality of other units and corresponding mesh information thereof, and arranging the other units along the unit expansion direction to form a quasi-periodic array with the reference unit;
s200: taking out the a multiplied by a sub-array positioned at the center position of the quasi-periodic array, and carrying out position correspondence on units contained in the sub-array and units of the whole quasi-periodic array;
s300: establishing a generalized characteristic value equation based on RWG basis functions and an electric field integral equation for the a x a sub-array, and solving the generalized characteristic value equation to obtain a characteristic current mode which can be expanded by the RWG basis functions;
s400: performing truncation selection on all characteristic current modes obtained by the a x a sub-array according to the importance of the modes to obtain K characteristic current modes, and establishing a group of sub-full-domain basis functions of each unit according to the corresponding relation between the K characteristic current modes and the positions;
s500: dispersing the current distribution of the whole quasiperiodic array by utilizing the sub-global basis function to obtain a reduced impedance matrix equation, and solving the impedance matrix equation to obtain the surface current distribution of the whole quasiperiodic array and a far-field radar scattering cross section;
s600: and analyzing the electromagnetic scattering characteristics of the alignment periodic array based on the current distribution of the whole quasi-periodic array and the radar scattering cross section.
Further, in S100, the cell closest to the origin of the quasi-periodic array structure is taken as a reference cell.
Further, in S100, there are a uniform cell scale and a cell rotation scale for adjacent cells in the cell expanding direction.
Further, in S100, a triangular mesh is generated for the reference cell.
Further, the step of performing position correspondence between the units included in the sub-array and the units of the whole quasi-periodic array structure in S200 includes: the corner units in the sub-array correspond to the corner units in the whole quasi-periodic array, the edge units in the sub-array correspond to the edge units in the whole quasi-periodic array, and the central units in the sub-array correspond to the central units in the whole quasi-periodic array.
Further, in S300, establishing a generalized eigenvalue equation based on RWG basis functions and electric field integral equations for the a × a sub-array specifically includes the following steps:
s310: based on the a × a sub-array and the grid information thereof, obtaining an impedance matrix equation according to the RWG basis function, the electric field integral equation and the Galerkin test:
Z SED I SED ＝V SED (1)
the impedance matrix elements are:
wherein,
respectively representing a field point and a source point,
RWG basis functions at the field point and the source point, respectively, j is the imaginary unit, k is the wave constant in free space, η is the free-space wave impedance,
the Green function is a free space, and the expression is as follows:
s320: obtaining a generalized eigenvalue equation from an impedance matrix equation:
wherein, X SED And R SED Is an impedance matrix Z SED The imaginary and real parts of (a),
RWG basis function coefficient vector, λ, for the kth characteristic current mode of the subarray k Is the corresponding characteristic value.
Further, in S300, an implicit restart method is used to solve the impedance matrix equation to obtain a generalized eigenvalue equation.
Further, the S400 specifically includes the following steps:
the mode importance MS is defined as:
in the formula of lambda k Is the corresponding characteristic value;
according to the importance of the mode, all characteristic current modes obtained by the a x a sub-array are cut off and selected, and K characteristic current modes are obtained and are expressed as:
wherein,
representing RWG basis function coefficient vectors corresponding to the qth characteristic current mode of the pth unit in the subarray;
according to the position corresponding relation, a group of sub-universe basis functions of each unit is established, and the sub-universe basis functions are expressed as follows:
wherein,
is the kth sub-global basis function of the mth cell in the entire quasiperiodic array,
is the m-th unit of N 0 RWG basis function,/ m The assignment is carried out according to the corresponding relation between the subarray and the whole quasiperiodic array, and the value range of M is 1,2, \8230Om.
Further, the S500 specifically includes the following steps:
discretizing the current distribution of the entire quasiperiodic array using the sub-global basis functions to obtain a reduced impedance matrix equation represented as:
Z RED α＝V RED (9)
wherein Z is RED For a reduced impedance matrix, V RED Is a reduced voltage vector;
wherein the sub-matrix is represented as:
obtaining sub-global basis function coefficients alpha by solving reduced matrix equations directly or iteratively mk ；
Based on sub-global basis function coefficient alpha mk And a group of sub-global basis functions of each unit in the sub-array to obtain the surface current distribution J (r) of the quasi-periodic array structure:
the far field radar scattering cross section is expressed as:
wherein E is inc Is incident plane wave, E sca The fringe field, which is the surface current in the case of far field approximation, is expressed as:
has the advantages that: the method mainly breaks through the limitation that a sub-global basis function method cannot process a quasi-periodic structure, and can greatly reduce the unknown quantity in a matrix impedance matrix and calculate the required memory compared with the traditional full-wave method; the method specifically comprises the following advantages:
(1) The modeling is simple, only one reference unit is selected for geometric modeling and mesh generation, and models of other units can be obtained through translation, scaling and rotation;
(2) The physical characteristics of the quasi-periodic structure are fully utilized, the physical characteristics of the array structure can be utilized to a certain degree by the sub-global basis function method, and then the main characteristic current mode is extracted through the characteristic mode to be used as the sub-global basis function, so that the solving complexity is greatly reduced, and the calculation efficiency is improved.
Drawings
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is a schematic diagram of a grid of reference cells scaled by rotation to obtain a grid of other cells;
FIG. 3 is a diagram of a quasi-periodic array structure with fixed scale factor and rotation factor;
FIG. 4 is a schematic diagram of the extraction of a 3 × 3 sub-array from a pre-quasiperiodic array structure;
FIG. 5 is a schematic diagram of a 20 × 20 square ring quasiperiodic array structure;
FIG. 6 is a graph illustrating the pattern importance distribution of a 3 × 3 sub-array calculated by the method of the present invention;
FIG. 7 is a plot of the surface current distribution calculated by the moment method;
FIG. 8 is a plot of the surface current distribution calculated using the method of the present invention;
FIG. 9 is a diagram of a comparison of dual-station RCS of normal incidence and oblique incidence of plane waves at 45 degrees for a quasi-periodic array model under 300MHz frequency for a conventional moment method and the method of the present invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings and the embodiment.
As shown in fig. 1, in order to break through the limitation that the sub-global basis function method cannot handle the quasiperiodic structure, and to greatly reduce the unknown quantity in the matrix impedance matrix compared with the conventional full-wave method, and reduce the memory required for calculation, the embodiment provides a method for analyzing the electromagnetic scattering characteristics of the large-scale quasiperiodic structure based on the sub-global basis function method, which specifically includes the following steps:
s100: performing geometric modeling and triangular mesh subdivision on a reference unit in a large-scale quasi-periodic structure, and exporting meshes, as shown in fig. 2, performing spatial rotation, scaling and the like on the meshes to obtain mesh information of other units, wherein the triangular meshes among the units have a one-to-one corresponding relationship;
in this step, the geometric modeling and mesh generation of the large-scale quasi-periodic structure comprises the following steps:
s110: the unit expansion directions of the quasi-periodic array are set to be the + x-axis direction and the + y-axis direction, and the number of the units along the + x-axis direction is N x The number of the units along the + y-axis direction is N y And thus the total number of cells of the array is M = N x ×N y The distances between the centers of the adjacent units along the + x axis and the + y axis are respectively D x And D y 。
S120: in the quasi-periodic array, the cell closest to the origin is used as the reference starting cell, and the cell scaling α of the neighboring cells in the quasi-periodic array along the + x-axis direction and the + y-axis direction is consistent x 、α y Proportion of rotation of unit
FIG. 3 is a schematic diagram of a finite large scale quasi-periodic structure.
S200: as shown in fig. 4, performing eigenmode analysis on a 3 × 3 sub-array at the center of the whole quasi-periodic array, taking out the 3 × 3 sub-array located at the center of the whole quasi-periodic array, and performing position correspondence between nine units of the sub-array and units of the whole quasi-periodic array structure, where the specific correspondence relationship is as follows:
corner cells (top left, top right, bottom left and bottom right) in the 3 × 3 sub-array correspond to corner cells (top left, top right, bottom left and bottom right) in the full quasi-periodic array; the edge units (upper edge unit, left edge unit, right edge unit and lower edge unit) in the 3 × 3 sub-array correspond to the edge units (upper edge unit, left edge unit, right edge unit and lower edge unit) in the entire quasi-periodic array; the center cell in the 3 x 3 sub-array corresponds to the center cell in the entire quasi-periodic array.
S300: establishing a characteristic value equation based on an electric field integral equation and RWG basis functions for the 3 × 3 sub-array in the S200, and solving, wherein the obtained characteristic current can be expanded by the RWG basis functions;
the specific operation steps are as follows:
s310: selecting a 3 multiplied by 3 sub array and a grid thereof at the center of the quasi-periodic array, and obtaining an impedance matrix equation based on RWG basis functions, an electric field integral equation and Galerkin test:
Z SED I SED ＝V SED (1)
the impedance matrix elements are:
wherein,
respectively representing a field point and a source point,
respectively field point and sourceRWG basis functions at points, j is the imaginary unit, k is the wave constant in free space, η is the free-space wave impedance,
the Green function is a free space, and the expression is as follows:
where pi is the circumferential ratio.
S320: the generalized eigenvalue equation of the impedance matrix established by the 3 × 3 sub-array can be obtained:
wherein X SED And R SED Is an impedance matrix Z SED The imaginary and real parts of (a),
RWG basis function coefficient vector, λ, for the kth characteristic current mode of a 3 × 3 sub-array k The pattern importance (MS) is defined here as the corresponding eigenvalue:
the generalized eigenvalue equation can be solved by using an implicit restart method (IRAM).
S400: cutting off and selecting all characteristic current modes obtained by a 3 multiplied by 3 subarray according to the importance of the modes, and establishing a reduced matrix equation for the whole quasi-periodic array, wherein the specific steps comprise the following steps:
s410: taking out K characteristic current modes according to the importance of the modes
Wherein
Represents RWG basis function coefficient vectors corresponding to the qth characteristic current mode of the pth unit in a 3 multiplied by 3 subarray, the value range of p is 1,2 \8230, 9, and the value range of K is 1,2 \8230, K.
S420: according to the position correspondence, the sub-global basis functions on all the blocks can be expressed as:
wherein,
is the kth sub-global basis function of the mth cell in the entire quasiperiodic array structure,
is the m-th unit of N 0 RWG basis function, l m The value of M is correspondingly selected from 1,2 \82309and 9 according to the corresponding relation between the 3 multiplied by 3 subarray and the whole quasi-periodic structure, and the value range of M is 1,2 \8230M.
(c) Based on the sub-global basis function, the current distribution J (r) of the entire quasiperiodic array structure can be expanded as:
wherein alpha is mk Are the sub-gamut basis function coefficients that need to be solved.
Based on the moment method and the Galerkin test method, a reduced matrix equation can be obtained:
Z RED α＝V RED (9)
wherein Z RED To reduce the impedance matrix, V RED For a reduced voltage vector, the reduced impedance matrix can be written as:
wherein the sub-matrix:
obtaining a sub-universe basis function coefficient by directly or iteratively solving the formula (9), and obtaining the surface current distribution J (r) of the quasiperiodic array structure according to the formula (8), thereby obtaining the far-field RCS of the quasiperiodic array structure:
in the formula, E inc For incident plane waves, their amplitude is usually normalized | E inc |＝1，E sca The scattered field, which is the surface current in the case of a far field approximation, is expressed as:
in order to verify the accuracy and the efficiency of the invention, taking quasiperiodic array structure analysis of a square ring unit as an example, the method is completed on a personal computer with the main frequency of 2.8GHz and the internal memory of 32 GB.
As shown in fig. 5, in a 20 × 20 quasiperiodic array structure of square ring units, the solution frequency is 300MHz, the outer side length of the square ring is 0.5 λ, and the inner side length is 0.3 λ. Cell center spacing D x ＝D y =0.8 λ, cell scaled by α in + x-axis direction x =0.95 reduction in order, scaling α in + y-axis direction y =1.0, cell in + y-axis direction by twiddle factor
Sequentially rotated, unit twiddle factor in + x direction
The importance distribution graph of the model obtained by analyzing the 3 × 3 sub-array is shown in fig. 6, and the first 15 models are taken to establish sub-global basis functions.
The excitation source is a plane wave with the amplitude of 1V from the + z axis to the-z axis and the polarization direction is the-x axis.
Fig. 7 and 8 show current distribution diagrams of the embodiment calculated by the moment method and the method of the invention, and it can be seen that the coincidence degree of the two is high.
FIG. 9 shows the angle of incidence θ of the excited plane wave inc =0 ° and θ inc When the angle is 45 degrees, a two-station RCS comparison graph calculated by a moment method and the method provided by the invention shows that the moment method and the method provided by the invention keep high goodness of fit, and the accuracy of the method provided by the invention is proved.
Table 1 shows the time consumption and memory consumption required to calculate an embodiment of the method of the invention and the method of the moment
| Time consumption | Memory consumption | |
| Method of moment | 597s | 5.49GB |
| The method of the invention | 124s | 244.14MB |
As can be seen from table 1, the time consumption and the memory consumption of the method of the present invention are less compared with the moment method, and the efficiency of the method of the present invention is further demonstrated.
Claims (9)
1. The method for analyzing the electromagnetic scattering characteristics of the large-scale quasi-periodic structure based on the sub-global basis function method is characterized by comprising the following steps of: the method comprises the following steps:
s100: selecting a reference unit to perform geometric modeling and mesh subdivision, performing spatial rotation and scaling on the reference unit to obtain a plurality of other units and corresponding mesh information thereof, and arranging the other units along the unit expansion direction to form a quasi-periodic array with the reference unit;
s200: taking out the a multiplied by a sub-array positioned at the center position of the quasi-periodic array, and carrying out position correspondence on units contained in the sub-array and units of the whole quasi-periodic array;
s300: establishing a generalized characteristic value equation based on RWG basis functions and an electric field integral equation for the a x a sub-array, and solving the generalized characteristic value equation to obtain a characteristic current mode which can be expanded by the RWG basis functions;
s400: cutting off and selecting all characteristic current modes obtained by the a multiplied by a sub array according to the importance of the modes to obtain K characteristic current modes, and establishing a group of sub universe basis functions of each unit according to the corresponding relation between the K characteristic current modes and positions;
s500: dispersing the current distribution of the whole quasiperiodic array by utilizing the sub-global basis function to obtain a reduced impedance matrix equation, and solving the impedance matrix equation to obtain the surface current distribution of the whole quasiperiodic array and a far-field radar scattering cross section;
s600: and analyzing the electromagnetic scattering characteristics of the alignment periodic array based on the current distribution of the whole quasi-periodic array and the radar scattering cross section.
2. The method for analyzing electromagnetic scattering properties of large-scale quasi-periodic structure based on sub-global basis function method as claimed in claim 1, wherein: in S100, the closest cell to the origin of the quasi-periodic array structure is taken as a reference cell.
3. The method according to claim 1, wherein the method comprises the following steps: in S100, there are a uniform cell scale and cell rotation scale for adjacent cells in the cell expansion direction.
4. The method for analyzing electromagnetic scattering properties of large-scale quasi-periodic structure based on sub-global basis function method as claimed in claim 1, wherein: in S100, a triangular mesh subdivision is performed on the reference cell.
5. The method for analyzing electromagnetic scattering properties of large-scale quasi-periodic structure based on sub-global basis function method as claimed in claim 1, wherein: in S200, the position mapping of the units included in the sub-array and the units of the whole quasi-periodic array structure includes: the corner units in the sub-array correspond to the corner units in the whole quasi-periodic array, the edge units in the sub-array correspond to the edge units in the whole quasi-periodic array, and the central units in the sub-array correspond to the central units in the whole quasi-periodic array.
6. The method for analyzing electromagnetic scattering properties of large-scale quasi-periodic structure based on sub-global basis function method as claimed in claim 1, wherein: in S300, establishing a generalized eigenvalue equation based on RWG basis functions and electric field integral equations for the a × a sub-array specifically includes the following steps:
s310: based on the a × a sub-array and the grid information thereof, obtaining an impedance matrix equation according to the RWG basis function, the electric field integral equation and the Galerkin test:
Z SED I SED ＝V SED (1)
the impedance matrix elements are:
wherein,
respectively representing a field point and a source point,
RWG basis functions at the field point and the source point, respectively, j is the imaginary unit, k is the wave constant in free space, η is the free-space wave impedance,
the Green function is a free space, and the expression is as follows:
s320: obtaining a generalized eigenvalue equation from an impedance matrix equation:
wherein, X SED And R SED Is an impedance matrix Z SED The imaginary and real parts of (a),
RWG basis function coefficient vector, λ, for the kth characteristic current mode of the subarray k Is the corresponding characteristic value.
7. The method of claim 6, wherein the sub-population basis function method based analysis method for electromagnetic scattering properties of large-scale quasiperiodic structures comprises: in S300, an implicit restart method is used to solve the impedance matrix equation to obtain a generalized eigenvalue equation.
8. The method for analyzing electromagnetic scattering properties of large-scale quasi-periodic structure based on sub-global basis function method as claimed in claim 1, wherein: the S400 specifically includes the following steps:
the mode importance MS is defined as:
in the formula, λ k Is the corresponding characteristic value;
according to the importance of the mode, all characteristic current modes obtained by the a x a sub-array are cut off and selected, and K characteristic current modes are obtained and are expressed as follows:
wherein,
representing RWG basis function coefficient vectors corresponding to the qth characteristic current mode of the pth unit in the subarray;
according to the position corresponding relation, a group of sub-universe basis functions of each unit is established, and the sub-universe basis functions are expressed as follows:
wherein,
is the kth sub-global basis function of the mth cell in the entire quasiperiodic array,
is the m-th cell N 0 RWG basis function, l m The assignment is carried out according to the corresponding relation between the subarray and the whole quasiperiodic array, and the value range of M is 1,2, \8230Om.
9. The method of claim 8, wherein the sub-population basis function method based analysis method for electromagnetic scattering properties of large-scale quasiperiodic structures comprises: the S500 specifically includes the following steps:
using the sub-global basis function to discretize the current distribution of the entire quasiperiodic array to obtain a reduced impedance matrix equation represented as:
Z RED α＝V RED (9)
wherein Z is RED For a reduced impedance matrix, V RED Is a reduced voltage vector;
wherein the sub-matrix is represented as:
obtaining sub-global basis function coefficients alpha by solving reduced matrix equations directly or iteratively mk ；
Based on sub-global basis function coefficient alpha mk And a group of sub-global basis functions of each unit in the sub-array to obtain the surface current distribution J (r) of the quasi-periodic array structure:
the far field radar scattering cross section is expressed as:
wherein, E inc Is incident plane wave, E sca The scattered field, which is the surface current in the case of a far field approximation, is expressed as:
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