CN110765412B - Method for solving electromagnetic scattering of electric large object by centroid segmentation wavelet moment method - Google Patents
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Abstract
The invention discloses a method for solving electromagnetic scattering of an electric large object by a centroid segmentation wavelet moment method, and belongs to the field of calculation electromagnetism in electromagnetic engineering. Aiming at the conditions that the formed matrix is complex, the calculated amount is large and errors are easy to occur when the electromagnetic scattering of the electric large-size conductor target is calculated, firstly, triangular dispersion is carried out on the target surface by utilizing RWG basis functions, then, each original triangle is divided into 9 identical sub-triangles, impedance matrix elements are efficiently filled by adopting a mass center segmentation method, and further, the impedance matrix without singular problem in calculation is obtained, secondly, the sparse matrix generated by discrete wavelet transformation is used for acting on the efficiently filled impedance matrix, and the dense impedance matrix is thinned by utilizing the multi-resolution and vanishing moment characteristics of the wavelet basis, so that the calculation time is reduced. Simulation proves that the method provided by the invention can be used for rapidly calculating the electromagnetic scattering characteristics of the large electric object under the condition of ensuring the calculation accuracy.
Description
Technical Field
The invention belongs to the field of calculation electromagnetism in electromagnetic engineering, and particularly relates to a method for solving electromagnetic scattering of an electric large object by a centroid segmentation wavelet moment method.
Background
As an algorithm for solving the integral equation, the moment method directly carries out integral solving on the source region, does not need to set boundary conditions and does not generate dispersion errors, so that various electromagnetic field problems can be accurately solved, but the algorithm is limited by green functions, and a complex matrix equation needs to be solved, particularly when electromagnetic scattering of an electric large-size conductor target is calculated, a full-rank dense matrix can be formed, the inversion is complex, the calculated amount is larger, and errors are easy to occur. Based on this, there are several documents currently in the study of the rapid solution of electromagnetic scattering properties of electrically large objects.
J.M.Song, C.C.Lu proposes a fast multistage sub-method (Fast Multipole Method, FMM) and a multilayer fast multistage sub-method (Multilevel Fast Multipole Algorithm, MLFMA) for the problem of conductor scattering. These two algorithms are very effective in solving the free space target scattering problem, and can effectively reduce the time for calculating the impedance matrix, but such methods often require knowledge of the green's function of the actual problem when solving. When the green's function expression is not simple and clear enough, the complexity of formula derivation and programming is increased, and the algorithm has the defect of poor portability, and the formula is often required to be derived and programmed again for different problems. Ding, Z.Li, and R.S Chen propose filling of the matrix with higher order basis functions, which although a good description of the physical properties is possible, inevitably lead to multiple integration operations, and Anhui university Zhang Aikui proposes equivalent dipole moment method (EDM) which corresponds to the effect between two intervals as a function of the current basis and the near field generated by the electric dipole moment. The representation of the impedance matrix elements becomes very simple and the elements can be calculated directly, which avoids double integration, but is limited by the size of the triangular faces of the target surface segmentation
In summary, the calculation problem of the electromagnetic scattering characteristics of the electric large object in the existing literature is still to be further explored, and based on the method, the invention provides a rapid method for solving the electromagnetic scattering characteristics of the electric large object based on a wavelet moment method of centroid segmentation.
Disclosure of Invention
The invention aims to provide a high-precision and high-efficiency method for solving electromagnetic scattering of an electric large object by a wavelet moment method of centroid segmentation.
The aim of the invention is achieved by the following technical scheme:
the invention relates to a quick solving problem of electromagnetic scattering characteristics of an ideal electric large object, which is characterized in that an impedance matrix filled by the ideal electric large object is subjected to discrete processing by adopting a wavelet moment method based on a centroid segmentation method, so that the scattering characteristics of the electric large object can be quickly solved, impedance matrix elements are efficiently filled by utilizing the centroid segmentation method, and then an impedance matrix without singular problem in calculation is obtained, then a sparse matrix generated by discrete wavelet transformation is used for acting on the high-efficiency filled impedance matrix, and dense impedance matrix is thinned by utilizing the multi-resolution and vanishing moment characteristics of a wavelet base, so that the calculation time is reduced.
The method for solving electromagnetic scattering of the electric large object by using a centroid segmentation wavelet moment method specifically comprises the following steps:
step (1) obtaining the relation between a scattering electric field and an induced current according to a bit function and a potential function theory;
step (2), the tangential component of the electric field intensity of the conductor surface is zero according to the ideal conductor boundary condition;
step (3) adopts a gamma method, and uses RWG basis functions as weight functions to obtain a discrete electric field integral equation;
performing triangular division on the surface of the conductor by using RWG basis functions to obtain integral of a function g on an original triangle;
step (5) obtaining a matrix equation;
and (6) transforming the matrix equation.
The step (1) specifically comprises:
the relationship between the scattered electric field and the induced current obtained according to the bit function and the potential function theory is as follows:
wherein A (r) represents the magnetic sagittal position,
and (3) representing an electric sign bit, wherein the expression of A (r) is as follows:
j (R) represents the equivalent current of the target surface, G (R) represents the free-space Green's function, where the Green's function is
η is the wave impedance in free space, k is the propagation constant in free space, R represents the distance from the field point to the origin, and the integration region is the surface on which the source current is located, i.e. the surface of the diffuser.
The step (2) specifically comprises:
substituting the formula (2) and the formula (3) into the formula (1) to obtain
Wherein L represents an operator for calculating electric field radiation from current, and the tangential component of the electric field intensity of the conductor surface is zero as known from ideal conductor boundary conditions, namely
wherein Ei Represents the incident electric field and η represents the wave impedance.
The step (3) specifically comprises: adopts a gamma method, uses RWG basis function as weight function, and can obtain discrete electric field integral equation
The step (4) specifically comprises: when using centroid distribution, the conductor surface is triangulated using RWG basis functions, and the integral of the function g over the original triangle can be expressed as
The impedance matrix elements can thus be rewritten into
The step (5) specifically comprises: the matrix equation can be obtained as follows
ZI＝V (11)
Z, I and V respectively represent impedance matrixes, unknown vectors and excitation vectors, and U is a wavelet transformation matrix on the assumption that U is a non-singular matrix with a size.
The step (6) specifically comprises: the wavelet transformation matrix U is introduced, and a matrix equation can be transformed:
Z'I'＝V' (12)
Z'＝UZU T I'＝(U T ) -1 I V'＝UV (13)
t represents the transpose of the matrix.
The invention has the beneficial effects that:
the impedance matrix based on the centroid segmentation method and filled with the impedance matrix is subjected to discrete processing by adopting a wavelet moment method, so that the scattering characteristic of an electric main body can be rapidly solved, and the dense impedance matrix is thinned and thinned by utilizing the multi-resolution and vanishing moment characteristics of the wavelet matrix, so that the calculation time is reduced.
Drawings
FIG. 1 is a schematic diagram of a triangle centroid cut;
FIG. 2 is a rectangular flat plate after being split;
FIG. 3 is an ideal conductor sphere after dissection;
FIG. 4 is a double station RCS of rectangular flat panel;
FIG. 5 is a double station RCS for an ideal conductor sphere;
FIG. 6 is an impedance matrix fill time and calculated total time for a rectangular flat panel;
fig. 7 is an impedance matrix fill time and calculated total time for an ideal conductor sphere.
Detailed Description
The following is a further description of embodiments of the invention, taken in conjunction with the accompanying drawings:
first embodiment:
the electromagnetic scattering of an electric large object is calculated, and the relation between a scattering electric field and an induced current is firstly obtained according to a bit function theory as follows
Wherein A (r) represents the magnetic sagittal position,
representing the electrical index, the expression of A (r)
J (R) represents the equivalent current of the target surface, G (R) represents the free-space Green's function, where the Green's function is
Eta isThe wave impedance in free space, k is the propagation constant in free space, R represents the distance from the field point to the origin, and the integration region is the surface on which the source current is located, i.e. the surface of the scatterer. Substituting the formulas (15) and (16) into (14) can obtain:
l represents an operator for calculating electric field radiation by current, and tangential component of electric field intensity of the surface of the conductor is zero according to ideal conductor boundary conditions, namely:
E i represents the incident electric field and η represents the wave impedance. By adopting a gamma method and using RWG basis functions as weight functions, a discrete electric field integral equation can be obtained:
with centroid distribution, the conductor surface is triangulated using RWG basis functions, each of which is subdivided into nine identical small triangles, as shown in fig. 1. The black dot is located at the centroid of the triangle. The large black points represent the positions of the field points, the small black points represent the positions of the source points, and the situation that the field points and the far points are coincident at the moment can be clearly seen, namely the problem of singularity in the process of calculating impedance elements is avoided, and at the moment, the integration can be approximated by the weighted summation of the central values on each sub-triangle. Thus, the integral of the function g over the original triangle can be expressed as:
The impedance matrix elements can thus be rewritten into
The mass center segmentation method can realize the efficient filling of the impedance matrix while solving the problem of singularity, but has no influence on the calculation of an impedance matrix equation. The root cause is that the moment method introduces a green's function as an integral kernel in calculating the scattering problem. Although it can accurately describe the propagation process of electromagnetic fields, the matrix it produces is a full rank dense matrix, which is computationally complex. Combining wavelet transforms with a moment method can effectively solve this problem. Since the complexity of solving the linear equation is related to the number of non-zero elements in the coefficient matrix, the more zero elements, the more sparse the coefficient matrix, the lower the computational complexity. The electric field integral equation can be solved by a moment method, and the unknown current is also expanded into a superposition form of RWG basis functions. The matrix equation can be obtained as follows
ZI＝V (24)
Z, I and V respectively represent impedance matrixes, unknown vectors and excitation vectors, and U is a wavelet transformation matrix on the assumption that U is a non-singular matrix with a size. For any vector V, the wavelet basis has multi-resolution and vanishing moment characteristics. Therefore, u·v corresponds to a decomposition vector. The wavelet transformation matrix U is introduced into a conventional MoM, and the matrix equation can be transformed.
Z'I'＝V' (25)
Z'＝UZU T I'＝(U T ) -1 I V'＝UV (26)
T represents the transpose of the matrix, the effect of U is a sparse matrix, and for matrix Z, the matrix after wavelet transformation is UZU T Due to (Z -1 )'＝UZ -1 U T As a sparse matrix, therefore, solve for i=u T (Z') -1 V' is greater than solving for i=z -1 V saves computation time, reducing the total computation time to compute an unknown quantity.
The method of the invention is used to calculate the two-dimensional rectangular plate and the two-dimensional RCS of the three-dimensional conductor sphere respectively, and the structure diagram after adopting Matlab function delaunay triangulation is shown in figures 2 and 3, the plate size is 6λ×6λ, the radius of the conductor sphere is 1λ, the incident plane wave is 300MHz, and the incident angle is
The observation angle of the double-station RCS is
Fig. 4 and fig. 5 show simulation results, and it can be seen that the method of the present invention is well matched with the conventional method, and fig. 6 and fig. 7 show comparison between the filling time and the total calculation time of the impedance matrix of the method of the present invention and the conventional method, and it can be seen that the filling time and the total calculation time of the impedance matrix elements of the method proposed by the present invention can be significantly reduced.
Specific embodiment II:
aiming at the problems that an impedance matrix formed when solving the electromagnetic scattering characteristic of an electric large object is complex and difficult to solve directly, the invention provides a calculation method for efficient filling and sparsification of the impedance matrix.
Establishing an electric field integral equation based on an ideal conductor:
(1) By potential function theory, we can get the relationship between the scattering electric field and the induced current as follows:
wherein, by adopting a mass center segmentation method, the magnetic vector position and the electric mark position can be written into
Substituting equations (28) and (29) into (27) to obtain
(2) The ideal conductor surface boundary condition requires zero tangential component of the electric field at the conductor surface, i.e
(3) Adopting a gamma method, taking RWG basis function as a weight function, and dispersing an electric field integral equation to obtain an impedance matrix equation
Z·I＝V (32)
Wherein the impedance matrix
Excitation vector
The sparse matrix generated by discrete wavelet transformation acts on the impedance matrix Z in (7) mn Thinning it, i.e
Z'＝T·Z·T' I'＝T·I V'＝T·V (35)
Then
Z'·I'＝V' (36)
And (5) performing simulation analysis. Fig. 1 and 2 are schematic diagrams after triangulation of two-dimensional and three-dimensional electric large object models, fig. 3 and 4 are schematic diagrams of radar scattering cross sections, and matrix filling time and total calculation time under two conditions are obtained through simulation, so that accuracy and high efficiency of the method are proved.
The above description is only of the preferred embodiments of the present invention and is not intended to limit the present invention, but various modifications and variations can be made to the present invention by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (1)
1. The method for solving electromagnetic scattering of the electric large object by using a centroid segmentation wavelet moment method is characterized by comprising the following steps of:
(1) Establishing an electric field integral equation based on an ideal conductor;
through potential function theory, the relation between the scattered electric field and the induced current is:
wherein A (r) represents the magnetic vector position,
the electrical index is represented by the number of bits,
j (r) represents an equivalent current of the target surface; g (R) represents a green's function in free space,
k is a propagation constant in free space; r represents the distance from the field point to the origin; a (r), ->
Is introduced into the expression of (2), resulting in:
where η is the wave impedance in free space;
from the ideal conductor boundary conditions, the tangential component of the conductor surface electric field strength is zero, i.e
Adopting a gamma gold method, using RWG basis functions as weight functions, and an electric field integral equation based on an ideal conductor is as follows:
(2) Triangulating the conductor surface using RWG basis functions using a centroid segmentation method, each divided triangle being subdivided into nine identical small triangles, the integration being approximated by a weighted summation of the central values over each sub-triangle, the integral of the function g over the original triangle being expressed as:
wherein ,
representing the center of each sub-angle shape; a is that m Representing the area of the original triangle;
the magnetic vector and electrical label bits are rewritten as:
the impedance matrix element is therefore rewritten as:
(3) Solving an electric field integral equation by a moment method, and expanding the unknown current into a superposition form of RWG basis functions;
matrix equation:
ZI＝V
wherein Z, I and V respectively represent an impedance matrix, an unknown vector and an excitation vector; introducing a wavelet transformation matrix U, and transforming a matrix equation:
Z′I′＝V′
Z′＝UZU T I′＝(U T ) -1 I V′＝UV
the wavelet transformation matrix U is a sparse matrix, and for the matrix Z, the matrix after wavelet transformation is UZU T The method comprises the steps of carrying out a first treatment on the surface of the Due to (Z -1 )'＝UZ -1 U T As a sparse matrix, therefore solve for i=u T (Z') -1 V' is greater than solving for i=z -1 And V saves the calculation time and reduces the total calculation time for calculating the unknown quantity.
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