CN111339650A - Electromagnetic basis function construction method and device with reduced orthogonal dimension - Google Patents

Electromagnetic basis function construction method and device with reduced orthogonal dimension Download PDF

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CN111339650A
CN111339650A CN202010107975.XA CN202010107975A CN111339650A CN 111339650 A CN111339650 A CN 111339650A CN 202010107975 A CN202010107975 A CN 202010107975A CN 111339650 A CN111339650 A CN 111339650A
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党训旺
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Beijing Institute of Environmental Features
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Abstract

The invention relates to an electromagnetic basis function construction method and device for orthogonal dimension reduction, a current determination method, computer equipment and a computer readable storage medium, wherein the electromagnetic basis function construction method comprises the steps of obtaining a matrix equation for solving an electromagnetic problem based on a Galerkin method, inputting a set of control parameters, sampling all the control parameters, and solving the matrix equation to obtain a corresponding solution; performing circular screening on the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to a projection error screening control parameter; and orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function. The method can solve the problems of large calculated amount and slow convergence in the construction process of the orthogonal basis function in the conventional moment method, effectively reduces the unknown number of the electromagnetic problem matrix equation and accelerates the solution of the electromagnetic problem matrix equation.

Description

Electromagnetic basis function construction method and device with reduced orthogonal dimension
Technical Field
The invention relates to the technical field of electromagnetic analysis, in particular to an electromagnetic basis function construction method and device with reduced orthogonal dimensions, a current determination method, computer equipment and a computer readable storage medium.
Background
The moment method is a numerical calculation method for solving Maxwell equations corresponding to electromagnetic problems, and converts the Maxwell equations into matrix equations for solving. The moment method needs to represent the current to be solved on the surface of the object for solving the electromagnetic problem as N0A basic function
Figure BDA0002389027680000015
The linear combination of (a) is as follows:
Figure BDA0002389027680000011
wherein J is a vector formed by the expansion coefficients of the current under the basis function, [ J]i＝JiAnd is also the unknown to be solved for by the matrix equation.
The most basic basis function in the moment method is RWG (Rao-Wilton-Glisson) basis function, the quantity to be solved is expressed as the linear combination of the basis functions, and then a matrix equation corresponding to an integral equation is obtained by a Galerkin method. The process needs to use the inner product of the test basis function and the operator equation to obtain the corresponding matrix equation form of
Figure BDA0002389027680000013
Wherein
Figure BDA0002389027680000014
The method comprises the steps of representing an impedance matrix representing the electromagnetic problem, J representing a vector solution formed by expansion coefficients of current of the electromagnetic problem under a basis function, and b representing a vector obtained after discretization of an excitation source of the electromagnetic problem.
In actual use, RWG basis functions can be linearly combined to form a new basis function. Setting new N basic functions vj(r)}j＝1,2,...,NThe form is as follows:
Figure BDA0002389027680000012
wherein, wjiRepresenting the expansion coefficients of the new basis functions under the RWG basis functions. Generally N is less than N0Thus, the number of basis functions is reduced. This method is called dimensionality reduction.
Using new basis functions vj(r)}j＝1,2,...,NSimilarly, by using Galerkin's method, a new matrix equation can be obtained
Figure BDA0002389027680000021
Wherein:
Figure BDA0002389027680000022
Figure BDA0002389027680000023
each column of (a) is the expansion coefficient of the new basis function under the RWG basis function,
Figure BDA0002389027680000024
referred to as the dimension reduction subspace. J. the design is a squarerbRepresenting the expansion coefficients of the current to be solved under the new basis functions. After solving to obtain JrbThen, can pass through
Figure BDA0002389027680000025
The relationship yields the final current J.
If it is not
Figure BDA0002389027680000026
Is orthogonal to each other, it corresponds to a set of orthogonal basis functions. In a linear space, the linear approximation error of the orthogonal basis function is minimum, and the orthogonal basis function is a basis function with very good mathematical properties.
In practical problems, batch calculation of control parameters of different values of the electromagnetic model is often required. For matrix equation
Figure BDA0002389027680000027
Assuming that both the matrix Z and the right-hand term b of the equation are related to the parameter μ, the control parameter μ defines a set of matrix equations. In electromagnetic problems μmay be frequency, incident wave direction, scatterer geometry information, etc. The direct solving complexity of the matrix equations is high, and the matrix equations are not suitable for large-scale electromagnetic problems. The basic idea of accelerated solution is to reduce dimensionality by using a new basis function and reduce the number of overall unknowns, and the following two methods are common accelerated solution:
the first method uses a global Basis Function with a certain analytic form, such as a Gaussian Ring Basis Function (Gaussian Ring Basis Function) defined on the whole electromagnetic surface for a disc-type quasi-periodic electromagnetic surface, so that the number of unknowns is effectively reduced, and the solving speed is increased. For an elliptical quasi-periodic electromagnetic surface, a Fourier-Bessel global basis function can be adopted for solving. The method has clear physical connotation, and the orthogonality of the basis functions is determined by an orthogonal mode. The method has the defect of narrow application range, and is only suitable for solving the electromagnetic problem moment method with a simple geometric structure.
The second method uses a numerical method to perform linear combination on the original basis functions, and then obtains new basis functions. A quasi-periodic electromagnetic surface composed of rotating units that produces electromagnetic orbital angular momentum modes can be simulated using the Characteristic Basis Function (CBF) method. The characteristic basis functions extract the information of current distribution formed on the array unit by electromagnetic waves with different incidence angles, so that a characteristic basis function set which can be defined on the whole unit is obtained. The basis of this method is that the electromagnetic wave can be spread out as a superposition of plane waves of different angles. The angle of incidence is chosen sufficiently to represent the effect on the array elements under different electromagnetic field excitations. After the characteristic basis function set is obtained, the equation of the original moment method can be projected on the basis function set, and the number of unknowns is further reduced. The method is suitable for actual quasi-periodic electromagnetic surface simulation by using a Synthetic Basis Function (SBF) method and a certain acceleration method, and corresponding software is written. The method of synthesizing the basis functions uses the scattering currents generated by point sources surrounding the elements in the array as basis functions for the current. Compared with the characteristic basis function, the near-field excitation construction basis function is adopted, the effect of near-field coupling can be represented more effectively, and generally, fewer unknowns can be used than the characteristic basis function. The periodic electromagnetic surface was simulated using Macro Basis Functions (MBF). The macro-basis function extracts the main vector components from the multiply scattered current distribution as new basis functions. In both of these methods, dimension reduction by singular value decomposition is required, and the conventional method is shown in table 1 below.
TABLE 1 orthogonal basis function construction method based on singular value decomposition
Figure BDA0002389027680000031
Wherein the content of the first and second substances,
Figure BDA0002389027680000032
to represent
Figure BDA0002389027680000033
Is determined by the left singular vector of (a),
Figure BDA0002389027680000034
to represent
Figure BDA0002389027680000035
Is determined by the right singular vectors of (a),
Figure BDA0002389027680000036
is a diagonal matrix formed by arranging singular values from large to small. The magnitude of the singular value represents the degree to which the left singular vector contributes to the overall matrix. The smaller the singular value, the smaller the contribution when the corresponding singular vector is represented as a basis function. Since singular value decomposition requires solving matrix eigenvalues, the computation amount of these methods is generally high, and therefore, in order to construct dimension-reduced orthogonal basis functions in an actual electromagnetic problem, it is necessary to study an improved orthogonal basis function construction method.
Disclosure of Invention
The present invention is directed to provide an improved method and apparatus for constructing an electromagnetic basis function with reduced orthogonal dimensions, which solve the problems of large calculation amount, slow convergence, insufficient basis function representativeness, etc. in the conventional method for constructing a basis function with reduced dimensions based on singular value decomposition.
In order to achieve the above object, the present invention provides an orthogonal dimension reduced electromagnetic basis function constructing method, comprising the steps of:
s1, obtaining a matrix equation for solving the electromagnetic problem based on the Galerkin method, inputting a set of control parameters, sampling all the control parameters, and solving the matrix equation to obtain a corresponding solution;
s2, performing circular screening on the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to projection error screening control parameters;
and S3, orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function.
Preferably, when obtaining a matrix equation for solving the electromagnetic problem based on the galois method in step S1, the current to be solved on the target surface is expressed as a linear combination form of RWG basis functions, and the expression of the matrix equation is:
Figure BDA0002389027680000041
wherein, mu represents a control parameter,
Figure BDA0002389027680000042
and J (mu) represents a vector solution formed by the expansion coefficients of the current of the electromagnetic problem under the basis function, and b (mu) represents a vector obtained by discretizing the excitation source of the electromagnetic problem.
Preferably, the step S1 further includes dividing the triangular mesh on the target surface before obtaining the matrix equation for solving the electromagnetic problem based on the galois method.
Preferably, in step S1, the expression of the RWG basis function is:
Figure BDA0002389027680000051
wherein n represents the nth side in the triangular mesh, and lnRepresenting the length of the nth side, one side corresponding to two triangles S+、S-Having an area of
Figure BDA0002389027680000052
P represents the position vector r minus the vertex v not on the common edgefreeFormed vector, p ═ r-vfree。
Preferably, the step S2 is based on greedy method to solve the obtained solution { J (mu) }i)}i＝1,2,...,MPerforming the circular screening further comprises:
s2-1, initializing one-dimensional dimension reduction subspace
Figure BDA0002389027680000053
S2-2, setting the maximum cycle number N and the error threshold epsilon, and cycling N from 2 to N to perform the following steps:
calculating a subspace for reducing J (mu) of each solution to the existing dimension
Figure BDA0002389027680000054
Projection error e ofn(μ)：
Figure BDA0002389027680000055
Wherein, JnIs to reduce subspace from J (mu) to existing dimension
Figure BDA0002389027680000056
Projection of (2);
if the absolute value is maximum εn(mu) if the error threshold epsilon is smaller than N, then the loop is exited; otherwise, screening out the epsilon with the maximum absolute valuen(mu) constructing an n-dimensional dimension reduction subspace corresponding to the control parameter mu
Figure BDA0002389027680000057
Figure BDA0002389027680000058
Figure BDA0002389027680000059
S2-3, ending the circulation to obtain the dimensionality reduction subspace of N dimensions
Figure BDA00023890276800000510
Where M denotes the number of input set of control parameters mu and N denotes the dimension of the dimension reduction subspace.
Preferably, the step S3 is implemented by orthogonalizing the constructed dimension reduction subspace using an orthogonal trigonometric decomposition method.
The invention also provides an electromagnetic basis function construction device with reduced orthogonal dimension, which comprises:
the parameter sampling unit is used for obtaining a matrix equation for solving the electromagnetic problem based on a Galerkin method, inputting a set of control parameters, sampling all the control parameters, and solving the matrix equation to obtain a corresponding solution;
the circular screening unit is used for circularly screening the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to the projection error screening control parameters;
and the orthogonal processing unit is used for orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function.
The invention also provides a current determination method, which comprises the following steps:
establishing a current solving model according to the electromagnetic problem analysis;
constructing corresponding orthogonal basis functions by adopting the electromagnetic basis function construction method with the reduced orthogonal dimension as described in any one of the above items;
and substituting the constructed orthogonal basis functions into a current solving model to solve the current.
The invention also provides a computer device comprising a memory storing a computer program and a processor implementing the steps of the orthogonal dimension reduced electromagnetic basis function construction method of any preceding claim when the processor executes the computer program.
The present invention also provides a computer readable storage medium having stored thereon a computer program which, when executed by a processor, performs the steps of any of the orthogonal dimension reduced electromagnetic basis function construction methods described above.
The technical scheme of the invention has the following advantages: the invention provides an orthogonal dimension reduced electromagnetic basis function construction method and device, a current determination method, computer equipment and a computer readable storage medium, wherein the orthogonal dimension reduced electromagnetic basis function construction method comprises three steps of control parameter sampling, dimension reduction and orthogonalization, wherein the dimension reduction step is based on a greedy method and projection error screening, the problems of large calculation amount, slow convergence and insufficient basis function representativeness in the conventional orthogonal dimension reduced electromagnetic basis function construction process are solved, the unknown number of an electromagnetic problem matrix equation is effectively reduced, and the solution is accelerated.
The current determination method provided by the invention determines the electromagnetic basis function required by solving the current by adopting an electromagnetic basis function construction method with reduced orthogonal dimension, and has the advantages of high calculation speed and higher efficiency.
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FIG. 1 is a schematic diagram of steps of a method for constructing electromagnetic basis functions with reduced orthogonal dimensions according to an embodiment of the present invention;
FIG. 2 is a diagram of RWG basis functions in an embodiment of the invention;
FIG. 3 is a schematic diagram of a square patch grid in an embodiment of the invention;
FIG. 4 is a schematic current diagram of the first 10 reduced-dimension basis functions of a square patch in accordance with an embodiment of the present invention;
FIG. 5 is a schematic diagram of a Malta cross cell grid in an embodiment of the present invention;
FIG. 6 is a schematic current diagram of the first 10 dimensionality reduced basis functions of the Malta cross cell in an embodiment of the present invention
FIG. 7 is a schematic diagram of a mesh of a patch with holes and a variation thereof according to an embodiment of the present invention;
figure 8 is a schematic current diagram of the first 10 reduced-dimension basis functions of a perforated patch in an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, are within the scope of the present invention.
Example one
As shown in fig. 1, a method for constructing electromagnetic basis functions with reduced orthogonal dimensions according to an embodiment of the present invention includes the following steps:
s1, control parameter sampling: a matrix equation for solving the electromagnetic problem is obtained based on a Galerkin method, a set of control parameters is input, all the control parameters are sampled, and the matrix equation is solved to obtain a corresponding solution.
When solving an actual electromagnetic problem, such as scattering of electromagnetic waves by a target, an electromagnetic radiation problem of an antenna, and the like, according to a maxwell equation set, accurate solution can be realized in a form of solving an integral equation. An electromagnetic problem solving system based on a moment method is set up for calculation, a Galerkin method is a representative method in the moment method, control parameters are expressed as a linear combination form of a group of basis functions based on the Galerkin method, and then corresponding matrix equations are obtained, namely a Maxwell equation set is converted into the matrix equations for solving. Preferably, the target surface current can be used as a control parameter to be solved, and after the current is obtained through solving, other information can be obtained through current derivation.
Preferably, when obtaining a matrix equation for solving the electromagnetic problem based on the galois method in step S1, the current to be solved on the target surface is expressed as a linear combination form of RWG basis functions, and the expression of the matrix equation is:
Figure BDA0002389027680000081
wherein, mu represents a control parameter,
Figure BDA0002389027680000082
indicating electromagnetic problemsAnd J (mu) represents a vector solution formed by the expansion coefficients of the current of the electromagnetic problem under the basis function, and b (mu) represents a vector obtained by discretizing the excitation source of the electromagnetic problem. That is, in step S1, a matrix equation for solving the electromagnetic problem is obtained based on the galileon method
Figure BDA0002389027680000083
Input a set of control parameters [ mu ]i}i＝1,2,...,MSampling all control parameters and solving the matrix equation
Figure BDA0002389027680000084
Obtain the corresponding solution { J (mu) }i)}i＝1,2,...,MWhere M represents the number of input set of control parameters μ.
The RWG basis functions are two-dimensional basis functions commonly used in the moment method and are functions defined on a triangular grid. By the method, the RWG basis functions can be linearly combined quickly according to specific electromagnetic problems, and corresponding representative dimensionality reduction basis functions are constructed.
Considering that the geometric triangular meshes required in the simulation should be calculated before establishing the matrix equation, step S1 preferably further includes dividing the triangular meshes on the target surface before obtaining the matrix equation for solving the electromagnetic problem based on the galois method. Further, common meshing tools include GMSH, Hypermesh, etc., which are not further limited herein.
Preferably, as shown in fig. 2, the expression of the original basis function, i.e., the RWG basis function, is:
Figure BDA0002389027680000091
wherein n represents the nth side in the triangular mesh, and lnRepresenting the length of the nth side, one side corresponding to two triangles S+、S-The nth side corresponds to Sn +、Sn -Having an area of
Figure BDA0002389027680000092
P represents the position vector r minus the vertex v not on the common edgefreeFormed vector, p ═ r-vfree. With such a definition, the vertical components of the RWG basis functions on both sides of the common edge can be guaranteed to be continuous.
S2, circular screening based on a greedy method: and circularly screening the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to the projection error screening control parameters.
Further, step S2 is based on the solution { J (μ) obtained in the round-robin filtering step S1 of the greedy methodi)}i＝1,2,...,MAnd further comprising:
s2-1, initializing one-dimensional dimension reduction subspace
Figure BDA0002389027680000093
S2-2, looping N from 2 to N, given a maximum number of loops N and an error threshold epsilon (where N can theoretically be taken as the dimension of vector J at maximum), the following steps are performed:
calculating a subspace for reducing J (mu) of each solution to the existing dimension
Figure BDA0002389027680000094
Projection error e ofn(μ)：
Figure BDA0002389027680000095
Wherein, JnIs to reduce subspace from J (mu) to existing dimension
Figure BDA0002389027680000096
Projection of (2);
if the absolute value is maximum εn(μ) is smaller than the error threshold epsilon, then N equals N, the loop exits, i.e. the value is assigned again to N and the process goes to step S2-3;
otherwise, screening out the epsilon with the maximum absolute valuen(mu) constructing an n-dimensional dimension reduction subspace corresponding to the control parameter mu
Figure BDA0002389027680000097
Figure BDA0002389027680000098
Figure BDA0002389027680000099
S2-3, ending the circulation to obtain the dimensionality reduction subspace of N dimensions
Figure BDA00023890276800000910
Where M denotes the number of input set of control parameters mu and N denotes the dimension of the dimension reduction subspace.
And S3, orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function.
Preferably, after the loop is finished, the dimension of the structure is reduced by adopting an orthogonal triangular decomposition (QR) method
Figure BDA0002389027680000101
Orthogonalizing and outputting an orthogonalized N-dimensional reduced subspace
Figure BDA0002389027680000102
The invention solves the problems of large calculated amount and slow convergence in the construction process of the electromagnetic basis function with reduced orthogonal dimension, effectively reduces the unknown number of the electromagnetic problem matrix equation, accelerates the solution of the electromagnetic problem matrix equation and simultaneously improves the representativeness of the basis function.
Example two
The second embodiment provides an electromagnetic basis function constructing apparatus with reduced orthogonal dimension, which includes: the device comprises a parameter sampling unit, a circular screening unit and an orthogonal processing unit.
The parameter sampling unit is used for obtaining a matrix equation for solving the electromagnetic problem based on a Galerkin method, inputting a set of control parameters, sampling all the control parameters, and solving the matrix equation to obtain a corresponding solution;
the circular screening unit is used for circularly screening the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to the projection error screening control parameters;
and the orthogonal processing unit is used for orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function.
EXAMPLE III
The third embodiment of the invention provides a current determination method, which comprises the following steps:
and establishing a current solving model according to the electromagnetic problem analysis.
Constructing corresponding orthogonal basis functions using an orthogonal dimensionality reduced electromagnetic basis function construction method as described in any one of the preceding claims.
And substituting the constructed orthogonal basis functions into a current solving model to solve the current.
Example four
The fourth embodiment is basically the same as the third embodiment, and the same parts are not described again, except that:
referring to fig. 3 and 4, fig. 3 is a grid schematic diagram of a square patch for solving current distribution, fig. 4 shows currents corresponding to 10 dimensionality reduction basis functions in the front of the square patch obtained by solving, for the square patch shown in fig. 3, selecting a side length from 1m to 2m, uniformly taking 31 points, an incident plane wave pitch angle theta from 0 to 90 degrees taking 21 points, and an azimuth angle phi from 0 to 180 degrees taking 41 points, so that 31 × 21 × 41-26691 current distribution vectors can be obtained.
EXAMPLE five
The fifth embodiment is basically the same as the fourth embodiment, and the same parts are not described again, except that:
referring to fig. 5 and 6, fig. 5 is a schematic grid diagram of a Malta cross unit of current distribution to be solved, fig. 6 shows currents corresponding to 10 dimensionality reduction basis functions in front of the solved Malta cross unit, and the Malta cross unit in fig. 5 is a common quasi-periodic electromagnetic surface unit. The same element shape and incident wave parameters as those in the previous example are selected, and a dimension reduction subspace with N-10 dimensions is constructed by using the same singular value decomposition, so as to make a current distribution diagram, as shown in fig. 6. It can be seen that as the number increases, the oscillation of the current distribution also increases, corresponding to higher order modes.
EXAMPLE six
Sixth embodiment is basically the same as fifth embodiment, and the same parts are not described again, but the differences are:
referring to fig. 7 and 8, fig. 7(a) is a grid diagram of a perforated patch for solving current distribution, and fig. 7(b) is an alternate diagram of the perforated patch. The transformation relationship is as follows:
Figure BDA0002389027680000111
Figure BDA0002389027680000112
where x, y are the original coordinates of the mesh of the perforated patch, L is half of the total side length of the perforated patch, a0Is half the original edge length of the hole, x ', y' are transformed coordinates, and a is half the edge length of the transformed hole.
Figure 8 shows the currents for the first 10 dimensionally reduced basis functions of the resulting perforated patch solved. The total side length of the perforated patch is 3m, the side length of the middle hole is changed from 1m to 2m, the hole side length of the reference unit is 1m, and as shown in FIG. 7(a), the case that the hole side length is 2m can be changed into the case that the hole side length is 2m in FIG. 7 (b). The same element shape and incident wave parameters as those in the previous example are selected, and a dimensionality reduction subspace with N-10 dimensions is constructed by using the same singular value decomposition, so as to make corresponding current distribution, as shown in fig. 8. It is also found that as the order increases, the oscillation of the current distribution also increases, corresponding to higher order modes.
From the above examples, it can be seen that the larger the singular value of the dimensionality reduction basis function extracted on the quasi-periodic electromagnetic surface unit by the method of the present invention is, the more gradual the change of the dimensionality reduction basis function is, and the corresponding mode is more basic. Therefore, the electromagnetic basis function construction method and the current determination method with reduced orthogonal dimensions can better embody the physical characteristics of current distribution on the unit and can retain the main information of the current distribution.
In particular, in some preferred embodiments of the present invention, there is also provided a computer device comprising a memory storing a computer program and a processor implementing the steps of the orthogonal dimension reduced electromagnetic basis function construction method of any of the above embodiments when the processor executes the computer program.
In other preferred embodiments of the present invention, a computer-readable storage medium is further provided, on which a computer program is stored, which, when being executed by a processor, carries out the steps of the orthogonal dimension reduced electromagnetic basis function construction method described in any one of the above embodiments.
It will be understood by those skilled in the art that all or part of the processes of the methods of the embodiments described above can be implemented by hardware instructions of a computer program, which can be stored in a non-volatile computer-readable storage medium, and when the computer program is executed, the processes of the embodiments of the methods described above can be included, and will not be repeated here.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.

Claims (10)

1. A method of constructing electromagnetic basis functions of reduced orthogonal dimensions, comprising the steps of:
s1, obtaining a matrix equation for solving the electromagnetic problem based on the Galerkin method, inputting a set of control parameters, sampling all the control parameters, and solving the matrix equation to obtain a corresponding solution;
s2, performing circular screening on the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to projection error screening control parameters;
and S3, orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function.
2. The method of claim 1, wherein:
when a matrix equation for solving the electromagnetic problem is obtained based on the galileo method in step S1, the current to be solved on the target surface is expressed as a linear combination form of RWG basis functions, and the expression of the matrix equation is:
Figure FDA0002389027670000011
wherein, mu represents a control parameter,
Figure FDA0002389027670000012
and J (mu) represents a vector solution formed by the expansion coefficients of the current of the electromagnetic problem under the basis function, and b (mu) represents a vector obtained by discretizing the excitation source of the electromagnetic problem.
3. The method of claim 2, wherein:
the step S1 further includes dividing a triangular mesh on the target surface before obtaining a matrix equation for solving the electromagnetic problem based on the galois method.
4. The method of claim 3,
in step S1, the expression of the RWG basis function is:
Figure FDA0002389027670000013
wherein n represents the nth side in the triangular mesh, and lnRepresenting the length of the nth side, one side corresponding to two triangles S+、S-Having an area of
Figure FDA0002389027670000021
P represents the position vector r minus the vertex v not on the common edgefreeFormed vector, p ═ r-vfree。
5. The method of claim 2,
the step S2 is to solve the obtained solution { J (mu) based on a greedy methodi)}i＝1,2,...,MPerforming the circular screening further comprises:
s2-1, initializing one-dimensional dimension reduction subspace
Figure FDA0002389027670000022
S2-2, setting the maximum cycle number N and the error threshold epsilon, and cycling N from 2 to N to perform the following steps:
calculating a subspace for reducing J (mu) of each solution to the existing dimension
Figure FDA0002389027670000023
Projection error e ofn(μ)：
Figure FDA0002389027670000024
Wherein, JnIs to reduce subspace from J (mu) to existing dimension
Figure FDA0002389027670000025
Projection of；
If the absolute value is maximum εn(mu) if the error threshold epsilon is smaller than N, then the loop is exited; otherwise, screening out the epsilon with the maximum absolute valuen(mu) constructing an n-dimensional dimension reduction subspace corresponding to the control parameter mu
Figure FDA0002389027670000026
Figure FDA0002389027670000027
Figure FDA0002389027670000028
S2-3, ending the circulation to obtain the dimensionality reduction subspace of N dimensions
Figure FDA0002389027670000029
Where M denotes the number of input set of control parameters mu and N denotes the dimension of the dimension reduction subspace.
6. The method of claim 1, wherein: in step S3, the dimension reduction subspace of the structure is orthogonalized by an orthogonal trigonometric decomposition method.
7. An apparatus for constructing electromagnetic basis functions with reduced orthogonal dimensions, comprising:
the parameter sampling unit is used for obtaining a matrix equation for solving the electromagnetic problem based on a Galerkin method, inputting a set of control parameters, sampling all the control parameters, and solving the matrix equation to obtain a corresponding solution;
the circular screening unit is used for circularly screening the obtained solution based on a greedy method, and constructing a corresponding dimension reduction subspace according to the projection error screening control parameters;
and the orthogonal processing unit is used for orthogonalizing and outputting the constructed dimension reduction subspace to obtain an orthogonal basis function.
8. A method of determining current, comprising the steps of:
establishing a current solving model according to the electromagnetic problem analysis;
constructing corresponding orthogonal basis functions using the orthogonal dimensionality reduced electromagnetic basis function construction method of any one of claims 1 to 6;
and substituting the constructed orthogonal basis functions into a current solving model to solve the current.
9. A computer device comprising a memory and a processor, the memory storing a computer program, characterized in that the processor, when executing the computer program, implements the steps of the orthogonal dimension reduced electromagnetic basis function construction method of any of claims 1 to 6.
10. A computer-readable storage medium, on which a computer program is stored which, when being executed by a processor, carries out the steps of the orthogonal dimensionality reduced electromagnetic basis function construction method of any one of claims 1 to 6.
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Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110251832A1 (en) * 2010-04-13 2011-10-13 CST-Computer Simulation Technology AG Method, device and computer program product for determining an electromagnetic near-field of a field excitation source of an electrical system
CN104714929A (en) * 2015-02-09 2015-06-17 中国人民解放军理工大学 Method for achieving parallel solution according to orders for AH-FDTD algorithm
CN105814489A (en) * 2013-09-09 2016-07-27 Asml荷兰有限公司 Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures
CN108445303A (en) * 2018-03-08 2018-08-24 北京环境特性研究所 A kind of Electromagnetic scatter simulated behavior method
US20190011600A1 (en) * 2016-01-04 2019-01-10 Schlumberger Technology Corporation Effective Permeability Upscaling for A Discrete Fracture Network
US10282500B1 (en) * 2016-10-17 2019-05-07 Northrop Grumman Systems Corporation Efficient modeling of electromagnetic properties of an assembly
CN110069863A (en) * 2019-04-24 2019-07-30 安徽理工大学 A kind of condensation matrix building method for accelerating Characteristic Basis Function Method iterative solution

Patent Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110251832A1 (en) * 2010-04-13 2011-10-13 CST-Computer Simulation Technology AG Method, device and computer program product for determining an electromagnetic near-field of a field excitation source of an electrical system
CN105814489A (en) * 2013-09-09 2016-07-27 Asml荷兰有限公司 Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures
CN104714929A (en) * 2015-02-09 2015-06-17 中国人民解放军理工大学 Method for achieving parallel solution according to orders for AH-FDTD algorithm
US20190011600A1 (en) * 2016-01-04 2019-01-10 Schlumberger Technology Corporation Effective Permeability Upscaling for A Discrete Fracture Network
US10282500B1 (en) * 2016-10-17 2019-05-07 Northrop Grumman Systems Corporation Efficient modeling of electromagnetic properties of an assembly
CN108445303A (en) * 2018-03-08 2018-08-24 北京环境特性研究所 A kind of Electromagnetic scatter simulated behavior method
CN110069863A (en) * 2019-04-24 2019-07-30 安徽理工大学 A kind of condensation matrix building method for accelerating Characteristic Basis Function Method iterative solution

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
党训旺，等: "等效原理算法在不同等效面下的误差分析" *
王全全，等: "改进的特征基函数法分析电磁散射问题" *

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