CN108170647B - Self-adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis - Google Patents

Self-adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis Download PDF

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CN108170647B
CN108170647B CN201711331269.8A CN201711331269A CN108170647B CN 108170647 B CN108170647 B CN 108170647B CN 201711331269 A CN201711331269 A CN 201711331269A CN 108170647 B CN108170647 B CN 108170647B
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李猛猛
余欢
陈如山
丁大志
樊振宏
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Abstract

The invention discloses a self-adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis. The method comprises the following steps: firstly, extracting a grid file of an electromagnetic analysis target, and setting incident electromagnetic wave parameters; then, grouping target grids to be analyzed by adopting an octree according to the average grid number in each group, defining discrete grids and the groups where the discrete grids are located, and then counting the number of basis functions in each non-empty group; establishing a membership relationship between groups of adjacent layers, establishing an electric field integral equation in a discrete grid area, and dividing the discrete grid area into a near field area and a far field area according to the distance between the groups; then, calculating an impedance matrix formed between non-adjacent groups by adopting a self-adaptive nested cross approximation method; and finally, obtaining the solution of the electric field integral equation through iterative solution so as to obtain the low-frequency electromagnetic characteristic. The method has the advantages of stable low-frequency numerical value, simple programming realization and high calculation efficiency, and realizes the high-efficiency analysis of the low-frequency electromagnetic characteristic.

Description

Self-adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis
Technical Field
The invention relates to the technical field of electromagnetic simulation, in particular to a self-adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis.
Background
Low-frequency electromagnetic scattering analysis gradually becomes one of the difficulties of the current electromagnetic analysis, and typical problems include analysis of a micro-scale antenna array on a large-size platform, extraction of electromagnetic parameters of a package interconnection line in an integrated circuit, electromagnetic compatibility inside a case and the like. The low-frequency electromagnetic scattering is characterized in that the size of an analyzed target is smaller than the wavelength, the target has a complex structure, and the numerical modeling needs to perform fine grid dispersion on the target, so that the discrete grids contained in a unit square wavelength area reach tens of thousands or even hundreds of thousands. In a traditional fast calculation method such as a multilayer fast multipole method, the grouping size of octree is generally larger than 0.2 wavelength, so that the occupied scale of a near field is huge, and the calculation efficiency is seriously influenced; a typical low-rank compression decomposition method such as an adaptive cross approximation method has the advantages of stable low-frequency numerical values and simple programming realization, but the low-rank matrix decomposition process consumes huge calculation time and memory, and is difficult to meet the requirements of engineering electromagnetic simulation for large calculation unknowns.
Disclosure of Invention
The invention aims to provide a self-adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis, which has stable low-frequency numerical values and simple programming realization, and provides theory and basis for low-frequency electromagnetic analysis in engineering practice.
The technical solution for realizing the purpose of the invention is as follows: an adaptive nested cross approximation method for low frequency electromagnetic property analysis, comprising the steps of:
step 1, extracting a grid file of an electromagnetic analysis target, and setting incident electromagnetic wave parameters;
step 2, grouping target grids to be analyzed by adopting an octree method according to the average grid number in each group, establishing an octree index, defining discrete grids and the groups where the discrete grids are located, then counting the number of basis functions in each non-empty group, and stopping subdivision of the octree when the average value of the number of the basis functions is smaller than a set threshold value; establishing a membership relationship between groups of adjacent layers, establishing an electric field integral equation in a discrete grid area, and dividing the discrete grid area into a near field area and a far field area according to the distance between the groups; for two groups with far interaction, the center of the group is used as the sphere center to establish an equivalent spherical surface, and the directions of uniform distribution on the equivalent spherical surface are established3 mutually perpendicular Rao-Wilton-Glisson basis functions;
step 3, forming a matrix by the basis functions of the groups in the octree and the basis functions on the equivalent spherical surface, compressing the matrix by adopting a self-adaptive cross approximation method, determining the basis functions with main effects in the groups, defining the basis functions as main basis functions, and calculating the interaction between two remote interaction groups;
and 4, solving a current coefficient by using an iterative solver, and calculating the low-frequency electromagnetic characteristic through the current coefficient.
Further, the target grids to be analyzed in step 2 are grouped by using an octree method according to the average grid number in each group, which is specifically as follows:
(2.1) defining a cube containing the three-dimensional object under analysis as layer 0, then equally dividing the cube into 8 microcubes, defining as layer 1, and continuing to equally divide each microcube into 8 microcubes until layer L, so that the number of average discrete edges in each group does not exceed 50;
(2.2) the number of groups contained in the first layer is 8lAll groups in each layer are numbered 1 to 8 in sequence according to the position of the group centerlIn which 1 is<l<L；
(2.3) defining the l-1 st layer as the parent group of the l-1 st layer, the l-th layer as the child group of the l-1 st layer, its parent group i being indexed by group ip(ii) a And deleting the empty groups which do not contain the base functions in each layer, and reserving the non-empty groups.
Further, for the two groups with far interaction described in step 2, the group center is used as the sphere center to establish an equivalent spherical surface, specifically as follows:
after octree grouping, each group contains a certain number of RWG basis functions, two groups which take the main role are selected, the center of the group is taken as the sphere center, an equivalent spherical surface with the radius of 3a/2 is generated and taken as a boundary of a far field and a is the side length of a small cube; the uniform sampling is carried out on the equivalent spherical surface, and the direction of each sampling point is set as3 RWG basis functions as basis functions for the remote region of action; the number of sample points is 60.
Further, the adaptive cross approximation method adopted in step 3 compresses a matrix formed by basis functions included in the group and basis functions on the equivalent spherical surface, and determines basis functions having a main effect in the group, specifically as follows:
setting M RWG basis functions in the ith group, setting M RWG basis functions on an equivalent surface of the ith group, and compressing an M multiplied by M impedance matrix formed by the RWG basis functions by an adaptive cross approximation method with the precision of epsilon to obtain:
[Zm,M]≈[Um,r][Vr,M]
[Zm,M]an electric field integral equation impedance matrix formed for M × M RWG basis functions, [ U ]m,r]And [ V ]r,M]Two low-rank matrixes are obtained by compressing through a self-adaptive cross approximation method, and r is an impedance matrix [ Z ] when the compression precision is epsilonm,M]The rank of (d); by a compressive approximation of the above equation, the r principal basis functions in group i are:
Ii＝[i1,i2,i3…ir]
when each group has completed the compressive approximation of the above equation, the group-to-group RWG basis function interactions can be replaced with the interactions of the principal basis functions.
Further, the basis functions with main effects in the determined groups in step 3 are defined as main basis functions, and the interaction between two far interaction groups is calculated as follows:
setting M and N basis functions in the ith and the j groups respectively, wherein the corresponding equivalent basis functions are M and N respectively, and compressing by a self-adaptive cross approximation method to obtain main basis functions in the groups respectively as IiAnd Ij(ii) a The mutual matrix of groups i and j, using the adaptive cross approximation:
[·]-1representing matrix inversion, definitionIn order to receive the matrix, the matrix is,in order to transfer the matrix, the first transfer matrix,for a radiation matrix, the above equation translates to:
[Zm,n]＝[U]i·[D]i,j·[V]j
the above equation is an approximate form of a single-layer adaptive cross approximation method, and for the l-th layer, it is:
wherein L is the number of layers of the octree,is a translation matrix for the l-1 th to l-th layers,for the l to l-1 layer translation matrix,is the transfer matrix of the l-th layer.
Compared with the prior art, the invention has the following remarkable advantages: (1) the programming is simple to realize, the low-frequency numerical value is stable, and the low-rank compression analysis method is irrelevant to the Green function form; (2) the radiation matrix and the receiving matrix of the low-rank decomposition matrix are only related to the current group and are not related to the interaction group, a translation matrix between adjacent layers is introduced, and the low-rank compression decomposition matrix of a high layer can be represented by the lowest-layer nesting, so that the calculation memory and the time are obviously reduced.
Drawings
FIG. 1 is a schematic diagram of a nested cross-approximation method of low frequency electromagnetic property analysis in accordance with the present invention.
FIG. 2 is a schematic diagram of the selection of the main basis functions by group i in the equivalent process according to the present invention.
FIG. 3 shows the placement direction of each equivalent point on the equivalent sphere in the present inventionSchematic of the RWG basis functions of (a).
FIG. 4 is a diagram illustrating a nested cross-approximation process of group i and group j in the present invention.
FIG. 5 is a graph showing the variation of the calculation accuracy and the adaptive cross-approximation truncation accuracy ε of the present invention in example 1.
Fig. 6 is a comparison graph of the results of analyzing the radar scattering cross-sectional area of the metal ball low-frequency electromagnetic scattering characteristics by the method of the invention in example 2 and the conventional adaptive cross-approximation method, respectively.
Detailed Description
The invention is described in further detail below with reference to the figures and specific embodiments.
The invention discloses a self-adaptive nested cross approximation method for analyzing low-frequency electromagnetic characteristics, which is used for accelerating the far-field interaction calculation of an integral equation impedance matrix by adopting a self-adaptive nested cross approximation method based on an electric field integral equation aiming at a low-frequency electromagnetic characteristic simulation platform in engineering design, and reducing the consumption of calculation resources required by electromagnetic simulation. With reference to fig. 1, the specific steps are as follows:
step 1, extracting a grid file of an electromagnetic analysis target, and setting incident electromagnetic wave parameters;
step 2, grouping target grids to be analyzed by adopting an octree method according to the average grid number in each group, establishing an octree index, defining discrete grids and the groups where the discrete grids are located, then counting the number of basis functions in each non-empty group, and stopping subdivision of the octree when the average value of the number of the basis functions is smaller than a certain threshold value; establishing a membership relationship between groups of adjacent layers, establishing an electric field integral equation in a discrete grid area, and dividing the discrete grid area into a near field area and a far field area according to the distance between the groups; for two groups with far interaction, the center of the group is used as the sphere center to establish an equivalent spherical surface, and the directions of uniform distribution on the equivalent spherical surface are established3 mutually perpendicular Rao-Wilton-Glisson basis functions.
The specific steps of octree grouping the analyzed low-frequency target grids according to the number of discrete grids in the finest layer group are as follows:
(2.1) defining a cube containing the three-dimensional object under analysis as layer 0, then equally dividing the cube into 8 microcubes, defining as layer 1, and continuing to equally divide each microcube into 8 microcubes until layer L, so that the number of average discrete edges in each group does not exceed 50;
(2.2) the number of groups contained in the first layer is 8lAll groups in each layer are numbered 1 to 8 in sequence according to the position of the group centerlIn which 1 is<l<L；
(2.3) defining the l-1 st layer as the parent group of the l-1 st layer, the l-th layer as the child group of the l-1 st layer, its parent group i being indexed by group ip. And deleting the empty groups which do not contain the base functions in each layer, and reserving the non-empty groups.
The remote interaction group establishes an equivalent surface as follows:
taking two dimensions as an example, combining fig. 2, after octree grouping, each group contains a certain number of RWG basis functions, the invention selects RWG basis functions which take the main role to approximate far-field interaction, takes the group center as the sphere center, automatically generates an equivalent sphere with the radius of 3a/2 through a program, the distance is the boundary of far and near fields, and a is the side length of a small cube. The uniform sampling is carried out on the equivalent spherical surface, and the directions are as follows according to the figure 3, 3 spherical coordinate systems are placed at each sampling pointRWG basis functions, RWG basis functions on the equivalent plane may be equivalent to the basis functions of the far-acting region. Typically 60 samples can satisfy the low frequency calculation accuracy.
And 3, forming a matrix by the basis functions of the groups in the octree and the basis functions on the equivalent spherical surface, compressing the matrix by adopting a self-adaptive cross approximation method and determining the basis functions with main functions in the groups, wherein the basis functions are defined as main basis functions. Thus the interaction between two far interaction groups can be quickly calculated from the principal basis functions in both groups.
The adaptive cross approximation method compresses a matrix formed by basis functions included in the group and basis functions on the equivalent spherical surface, and can select the basis functions having main functions in the group, specifically as follows:
as shown in fig. 2, setting M RWG basis functions in the ith group, and M RWG basis functions on the equivalent surface, the M × M impedance matrix formed by the RWG basis functions is compressed by an adaptive cross-approximation method with the accuracy of ∈, so as to obtain:
[Zm,M]≈[Um,r][Vr,M] (1)
[Zm,M]an electric field integral equation impedance matrix formed for M × M RWG basis functions, [ U ]m,r]And [ V ]r,M]Two low-rank matrixes are obtained by compressing through a self-adaptive cross approximation method, and r is an impedance matrix [ Z ] when the compression precision is epsilonm,M]Is determined. The r principal basis functions in group i are obtained by a compressive approximation of equation (1):
Ii＝[i1,i2,i3…ir] (2)
when each group has completed the compressive approximation of equations (1) and (2), the group-to-group RWG basis function interactions may be replaced with the primary basis function interactions.
The interaction between the two far interaction groups can be quickly calculated by the main basis functions in the two groups, as follows:
as shown in FIG. 4, M and N basis functions are set in the ith and the j groups respectively, corresponding equivalent basis functions are M and N respectively, and main basis functions in the groups are I respectively after compression by a self-adaptive cross approximation methodiAnd Ij. The mutual matrix of groups i and j can be written as:
[·]-1representing matrix inversion, definitionIn order to receive the matrix, the matrix is,in order to transfer the matrix, the first transfer matrix,for a radiation matrix, the formula (3) can be written as:
[Zm,n]＝[U]i·[D]i,j·[V]j (4)
from (4), the receiving matrix [ U ] can be seen]iThe same radiation matrix [ V ] is only related to the basis functions corresponding to group i, equivalent basis functions and main basis functions]jOnly the basis functions corresponding to group j, the equivalent basis functions, and the primary basis functions. However, for the traditional adaptive cross-approximation method, the form is
[Zm,n]＝[U]ij·[V]ij (5)
It can be seen that the receiving matrix and the radiation matrix are simultaneously related to the groups i and j, so the computation efficiency of the adaptive cross approximation method is far lower than that of the adaptive nested cross approximation method proposed by the present invention. Equation (4) is an approximate form of the single-layer adaptive nested cross-approximation method, and can be written as:
matrix arrayAndthe definition of (3) is the same as that of the formula (4), and L is the number of octree layers.Is a translation matrix for the l-1 th to l-th layers,for the l to l-1 layer translation matrix,is the transfer matrix of the l-th layer. The impedance matrix of the L-th layer can be approximated by the radiation and reception matrices of the finest layer L by a translation matrix between adjacent layers.
It can be seen from equations (3) - (6) that the approximation process of the adaptive nested cross approximation method for low-frequency electromagnetic characteristic analysis provided by the invention is numerical operation of the impedance matrix, and is independent of the specific form of the green function, so that the method provided by the invention has the advantages of easy application and simple programming implementation.
And 4, after the fast approximation of the matrix is obtained, the current coefficient of the matrix equation can be solved by using an iterative solver, and the low-frequency electromagnetic characteristic can be solved by using the current coefficient.
Example 1
With reference to fig. 5, the method according to the invention simulates low frequency electromagnetic characteristics. From a cylindrical grid with a radius of 0.5 m and a height of 4 m, 2 far-field interaction groups were taken, the group size being 0.5 m, containing RWG basis functions of 1200 and 1030, respectively, and an incident plane wave frequency of 30 MHz. FIG. 4 shows a two-norm error curve of the calculation results of the adaptive nested cross approximation method and the high-precision moment method as the truncation error ε of the adaptive cross approximation method decreases. The high-precision moment method adopts a Gaussian integration point with an internal and external integration of 61. The moment method in the figure is the result of using the outer integral 3 and the inner integral 7 as the integration point. It can be seen that the calculation accuracy of the adaptive nested cross approximation method gradually improves with the reduction of the truncation error epsilon, when epsilon is 10-4The time calculation precision reaches the calculation precision of a typical moment method, and the calculation precision of low-frequency electromagnetic characteristic analysis can be completely met.
Example 2
Referring to fig. 6, for a radar scattering cross section (RCS) with a radius of 0.5 m, the frequency of an incident plane wave is 600MHz, and the incident angle is vertical incidence according to the method of the present invention and the conventional adaptive cross approximation method. It can be seen that the two groups of results are well matched, and the correctness of the method is further proved. The unknown quantity of the object dispersion is 6660, the number of octree layers is 3, the time and memory calculated by adopting the self-adaptive cross approximation method is 35 minutes and 1.8GB, and the time and memory calculated by adopting the self-adaptive nested cross approximation method of the invention are 8 minutes and 300MB respectively. It can be seen that the calculation efficiency of the low-frequency electromagnetic scattering property is remarkably improved by the method.

Claims (3)

1. An adaptive nested cross-approximation method for low-frequency electromagnetic property analysis, comprising the steps of:
step 1, extracting a grid file of an electromagnetic analysis target, and setting incident electromagnetic wave parameters;
step 2, grouping target grids to be analyzed by adopting an octree method according to the average grid number in each group, establishing an octree index, defining discrete grids and the groups where the discrete grids are located, then counting the number of basis functions in each non-empty group, and stopping subdivision of the octree when the average value of the number of the basis functions is smaller than a set threshold value; establishing a membership relationship between groups of adjacent layers, establishing an electric field integral equation in a discrete grid area, and dividing the discrete grid area into a near field area and a far field area according to the distance between the groups; for two groups with far interaction, the center of the group is used as the sphere center to establish an equivalent spherical surface, and the directions of uniform distribution on the equivalent spherical surface are established3 mutually perpendicular Rao-Wilton-Glisson basis functions;
step 3, forming a matrix by the basis functions of the groups in the octree and the basis functions on the equivalent spherical surface, compressing the matrix by adopting a self-adaptive cross approximation method, determining the basis functions with main effects in the groups, defining the basis functions as main basis functions, and calculating the interaction between two remote interaction groups;
the method adopts a self-adaptive cross approximation method to compress a matrix formed by the basis functions contained in the group and the basis functions on the equivalent spherical surface and determine the basis functions having main functions in the group, and comprises the following specific steps:
setting M RWG basis functions in the ith group, setting M RWG basis functions on an equivalent surface of the ith group, and compressing an M multiplied by M impedance matrix formed by the RWG basis functions by an adaptive cross approximation method with the precision of epsilon to obtain:
[Zm,M]≈[Um,r][Vr,M] (1)
[Zm,M]an electric field integral equation impedance matrix formed for M × M RWG basis functions, [ U ]m,r]And [ V ]r,M]Two low-rank matrixes are obtained by compressing through a self-adaptive cross approximation method, and r is an impedance matrix [ Z ] when the compression precision is epsilonm,M]The rank of (d); by a compressive approximation of equation (1), the r main basis functions in group i are:
Ii＝[i1,i2,i3…ir] (2)
when each group has completed the compressive approximation of equations (1) and (2), the group-to-group RWG basis function interactions can be replaced with the interactions of the principal basis functions;
the basis functions with main effects in the determined groups are defined as main basis functions, and the interaction between two far interaction groups is calculated, specifically as follows:
setting M and N basis functions in the ith and the j groups respectively, wherein the corresponding equivalent basis functions are M and N respectively, and compressing by a self-adaptive cross approximation method to obtain main basis functions in the groups respectively as IiAnd Ij(ii) a The mutual matrix of groups i and j, using the adaptive cross approximation:
[]-1representing matrix inversion, definitionIn order to receive the matrix, the matrix is,in order to transfer the matrix, the first transfer matrix,for a radiation matrix, equation (3) then translates to:
[Zm,n]＝[U]i·[D]i,j·[V]j (4)
equation (4) is an approximate form of the single-layer adaptive cross-approximation method, and for the l-th layer:
wherein L is the number of layers of the octree,is a translation matrix for the l-1 th to l-th layers,for the l to l-1 layer translation matrix,a transfer matrix being a l-th layer;
and 4, solving a current coefficient by using an iterative solver, and calculating the low-frequency electromagnetic characteristic through the current coefficient.
2. The adaptive nested cross approximation method for low-frequency electromagnetic property analysis according to claim 1, wherein the target meshes to be analyzed in step 2 are grouped by an octree method according to the average number of meshes in each group, specifically as follows:
(2.1) defining a cube containing the three-dimensional object under analysis as layer 0, then equally dividing the cube into 8 microcubes, defining as layer 1, and continuing to equally divide each microcube into 8 microcubes until layer L, so that the number of average discrete edges in each group does not exceed 50;
(2.2) the number of groups contained in the first layer is 8lAll groups in each layer are numbered 1 to 8 in sequence according to the position of the group centerlIn which 1 is<l<L；
(2.3) defining the l-1 st layer as the parent group of the l-1 st layer, the l-th layer as the child group of the l-1 st layer, its parent group i being indexed by group ip(ii) a And deleting the empty groups which do not contain the base functions in each layer, and reserving the non-empty groups.
3. The adaptive nested cross-approximation method for low-frequency electromagnetic property analysis according to claim 1, wherein for the two groups of far interaction described in step 2, an equivalent sphere is established with the group center as the sphere center, specifically as follows:
after octree grouping, each group contains a certain number of RWG basis functions, two groups which take the main role are selected, the center of the group is taken as the sphere center, an equivalent spherical surface with the radius of 3a/2 is generated and taken as a boundary of a far field and a is the side length of a small cube; the uniform sampling is carried out on the equivalent spherical surface, and the direction of each sampling point is set as3 RWG basis functions as basis functions for the remote region of action; the number of sample points is 60.
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