CN110765412A - Method for solving electromagnetic scattering of large electric object by using 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 computational electromagnetism in electromagnetic engineering. Aiming at the conditions that when electromagnetic scattering of an electrically large-size conductor target is calculated, a formed matrix is complex, the calculated amount is large, and errors are easy to occur, firstly, RWG basis functions are utilized to carry out triangle dispersion on the surface of the target, then each original triangle is divided into 9 identical sub-triangles, a centroid segmentation method is adopted to efficiently fill impedance matrix elements, further, an impedance matrix without singularity problems during calculation is obtained, secondly, a sparse matrix generated by discrete wavelet transformation is utilized to act on the efficiently filled impedance matrix, and the multi-resolution and moment vanishing characteristics of wavelet bases are utilized to enable the dense impedance matrix to be sparse, 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 property of the electric large object under the condition of ensuring the calculation accuracy.
Description
Technical Field
The invention belongs to the field of computational electromagnetism in electromagnetic engineering, and particularly relates to a method for solving electromagnetic scattering of an electric large object by a wavelet moment method of centroid segmentation.
Background
As an algorithm for solving an integral equation, a moment method directly carries out integral solution on a source region, boundary conditions are not required to be set, and dispersion errors are not generated, so that various electromagnetic field problems can be accurately solved. Based on this, some documents are in the research of the fast solving problem of the electromagnetic scattering property of the electrical large object.
Song, c.c.lu, proposed Fast multipolymer (FMM) and multi-layer Fast Multipole Algorithm (MLFMA) for the problem of conductor scattering. The two algorithms are very effective for solving the free space target scattering problem, and can effectively reduce the time for calculating the impedance matrix, but the green function of the actual problem is usually required to be known when the solution is solved by the methods. When the Green function expression is not simple and clear enough, the complexity of formula derivation and programming is increased, the algorithm has the defect of poor portability, and the formula derivation and programming are required to be carried out again for different problems. D.ding, z.li, and r.s Chen proposes to fill the matrix with higher-order basis functions, which, although well described for physical properties, inevitably lead to multiple-integral operations, zhang ajin, university of anhui proposes the equivalent dipole moment method (EDM), which is equivalent to the effect between two intervals as a basis function of the current and the near field generated by the electric dipole moment. The expression of the elements of the impedance matrix becomes very simple and the elements can be calculated directly, which avoids double integration, but it is limited by the size of the triangular faces of the target surface segmentation
In summary, the existing literature has yet to be further explored on the calculation problem of the electromagnetic scattering property of the electrically large object, and based on the method, the invention provides a fast method for solving the electromagnetic scattering property of the electrically large object by using a wavelet moment method based on centroid segmentation.
Disclosure of Invention
The invention aims to provide a method for solving the electromagnetic scattering of an electric large object by a high-precision and high-efficiency centroid segmentation wavelet moment method.
The purpose of the invention is realized by the following technical scheme:
the invention relates to a fast solving problem of electromagnetic scattering characteristics of an ideal electrical large object, which is characterized in that a centroid segmentation method is used as a basis, a filled impedance matrix is subjected to discrete processing by a wavelet moment method, the scattering characteristics of the electrical large object can be further fast solved, the impedance matrix elements are efficiently filled by the centroid segmentation method, an impedance matrix without singularity problem during calculation is obtained, then a 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 a wavelet base, so that the calculation time is reduced.
The method for solving the electromagnetic scattering of the large electric object by the wavelet moment method of centroid segmentation specifically comprises the following steps:
the method comprises the following steps of (1) obtaining the relation between a scattering electric field and induced current according to a bit function and potential function theory;
step (2) deducing that the tangential component of the electric field intensity on the surface of the conductor is zero according to the boundary condition of the ideal conductor;
step (3) adopting a Galois-gold method, and using RWG basis functions as weight functions to obtain a discrete electric field integral equation;
performing triangulation on the surface of the conductor by using the RWG basis function to obtain an 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 following steps:
the relationship between the scattering electric field and the induced current is obtained according to the theory of the bit function and the potential function as follows:
wherein A (r) represents a magnetic vector position,
represents the electric scale position, and the expression of A (r) is as follows:
j (r) represents the equivalent current of the target surface, G (R) represents the Green's function of free space, wherein 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 scattering body.
The step (2) specifically comprises the following steps:
substituting the formulas (2) and (3) into the formula (1)
Where L represents the operator for calculating the electric field radiation from the current, the tangential component of the intensity of the electric field at the surface of the conductor is zero as known from the ideal conductor boundary conditions, i.e. L is the operator for calculating the electric field radiation from the current
wherein EiRepresenting the incident electric field and η representing the wave impedance.
The step (3) specifically comprises the following steps: the discrete electric field integral equation can be obtained by adopting a Galois method and using RWG basis functions as weight functions
The step (4) specifically comprises the following steps: when using centroid-cut distribution, the RWG basis function is used to triangulate the conductor surface, and the integral of the function g over the original triangle can be expressed as
So that the impedance matrix elements can be rewritten to
The step (5) specifically comprises the following steps: the matrix equation can be obtained as follows
ZI＝V (11)
Z, I and V respectively represent an impedance matrix, an unknown vector and an excitation vector, and U is assumed to be a non-singular matrix of size and a wavelet transformation matrix.
The step (6) specifically comprises the following steps: the wavelet transformation matrix U is introduced, and the matrix equation can be transformed:
Z'I'＝V' (12)
Z'＝UZUTI'＝(UT)-1I V'＝UV (13)
t denotes the transpose of the matrix.
The invention has the beneficial effects that:
based on a centroid subdivision method, the filled impedance matrix is subjected to discrete processing by a wavelet moment method, so that the scattering characteristics of the electrically large object can be rapidly solved, and the characteristics of multi-resolution and moment of disappearance of the wavelet basis are utilized to thin and sparse the dense impedance matrix, thereby reducing the calculation time.
Drawings
FIG. 1 is a schematic diagram of triangle centroid segmentation;
FIG. 2 is a cut rectangular plate;
FIG. 3 is a schematic view of an ideal conductor ball after being split;
FIG. 4 is a two station RCS of a rectangular plate;
FIG. 5 is a two station RCS of an ideal conductor ball;
FIG. 6 is a graph of the impedance matrix fill time and the calculated total time for a rectangular plate;
fig. 7 shows the impedance matrix fill time and the calculated total time for an ideal conductor ball.
Detailed Description
The following further describes embodiments of the present invention with reference to the accompanying drawings:
the first embodiment is as follows:
to calculate the electromagnetic scattering of an electrical large object, we first obtain the relationship between the scattering electric field and the induced current according to the bit function theory as follows
Wherein A (r) represents a magnetic vector position,
indicating the electric scale position, A (r)
J (r) represents the equivalent current of the target surface, G (R) represents the Green's function of free space, wherein 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 where the source current is located, i.e., the surface of the scattering body, the substitution of equations (15) and (16) into equation (14) yields:
l represents the operator for calculating the electric field radiation from the current, and the tangential component of the electric field intensity on the surface of the conductor is zero as known from the ideal conductor boundary conditions, i.e.:
Eirepresenting the incident electric field, η representing the wave impedance the discrete electric field integral equation is obtained using the Galois gold method using the RWG basis functions as the weight functions:
with centroid-cut distribution, the RWG basis functions are used to triangulate the conductor surface, with each divided triangle being subdivided into nine identical small triangles, as shown in FIG. 1. The black point is located at the centroid position of the triangle. The large black point represents the position of the field point, the small black point represents the position of the source point, and we can clearly see that the situation that the field point and the far point are overlapped does not occur at the moment, namely the singularity problem occurring in the calculation of the impedance element is avoided, and at the moment, the integral 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:
denotes the center of each sub-corner, AmRepresenting the area of the original triangle, the magnetic and electric scale bits being rewritable
So that the impedance matrix elements can be rewritten to
The centroid segmentation method can solve the problem of singularity and simultaneously realize efficient filling of the impedance matrix, but has no influence on calculation of an impedance matrix equation. The fundamental reason is that the moment method introduces a green function as an integral kernel function when calculating the scattering problem. Although it can accurately describe the propagation process of the electromagnetic field, the matrix generated by the method is a full-rank dense matrix and is complex to calculate. Combining wavelet transform with the 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, and 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 an impedance matrix, an unknown vector and an excitation vector, and U is assumed to be a non-singular matrix of size and a wavelet transformation matrix. For any vector V, because the wavelet basis has multiresolution and vanishing moment properties. Therefore, U · V corresponds to a decomposition vector. The wavelet transformation matrix U is introduced into the traditional MoM, and the matrix equation can be transformed.
Z'I'＝V' (25)
Z'＝UZUTI'＝(UT)-1I V'＝UV (26)
T represents the transposition of the matrix, U serves as a sparse matrix, and for the matrix Z, the wavelet transformed matrix is UZUTDue to (Z)-1)'＝UZ-1UTIs a sparse matrix, therefore, solving for I ═ UT(Z')-1V' ratio solution I ═ Z-1V saves computation time, reducing the total computation time to compute the unknown quantity.
The method of the invention is utilized to respectively calculate the two-station RCS of a two-dimensional rectangular plate and a three-dimensional conductor ball, and FIGS. 2 and 3 are respectively a schematic structural diagram after triangulation by adopting Matlab function delaunay, wherein the plate size is 6 lambda multiplied by 6 lambda, the radius of the conductor ball is 1 lambda, the incident plane wave is 300MHz, and the incident angle is
The dual station RCS observation angle is
Fig. 4 and fig. 5 show simulation results, which show that the method of the present invention is well matched with the conventional method, and fig. 6 and fig. 7 show the 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, which show that the method of the present invention can significantly reduce the filling time and the total calculation time of the elements of the impedance matrix.
The second embodiment is as follows:
the invention provides a calculation method for efficient filling and sparseness of an impedance matrix, aiming at the problems that the impedance matrix formed when the electromagnetic scattering characteristics of an electrically large object are solved is complex and is not easy to solve directly.
Establishing an electric field integral equation based on an ideal conductor:
(1) by the theory of potential function, we can obtain the following relationship between the scattering electric field and the induced current:
wherein, the centroid cutting method is adopted to write the magnetic vector position and the electric mark position into
Substituting equations (28) and (29) into (27) can give
(2) The boundary conditions of an ideal conductor surface require that the tangential component of the electric field at the conductor surface be zero, i.e. zero
(3) By adopting a Galois method, an impedance matrix equation can be obtained by taking RWG basis functions as weight functions and dispersing an electric field integral equation
Z·I＝V (32)
Wherein the impedance matrix
Excitation vector
Applying a sparse matrix generated by discrete wavelet transform to the impedance matrix Z in (7)mnMake it sparse, i.e.
Z'＝T·Z·T' I'＝T·I V'＝T·V (35)
Then
Z'·I'＝V' (36)
And (5) carrying out simulation analysis. As shown in fig. 1 and 2, the schematic diagrams after triangulation of two-dimensional and three-dimensional electrical large object models are shown, fig. 3 and 4 are radar scattering cross-section schematic diagrams, matrix filling time and total calculation time under two conditions are obtained through simulation, and accuracy and high efficiency of the method are proved.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. The method for solving the electromagnetic scattering of the large electric object by the wavelet moment method of centroid segmentation is characterized by comprising the following steps of:
the method comprises the following steps of (1) obtaining the relation between a scattering electric field and induced current according to a bit function and potential function theory;
step (2) deducing that the tangential component of the electric field intensity on the surface of the conductor is zero according to the boundary condition of the ideal conductor;
step (3) adopting a Galois-gold method, and using RWG basis functions as weight functions to obtain a discrete electric field integral equation;
performing triangulation on the surface of the conductor by using the RWG basis function to obtain an integral of a function g on an original triangle;
step (5) obtaining a matrix equation;
and (6) transforming the matrix equation.
2. The method for solving the electromagnetic scattering of the electrically large object by the wavelet moment method of centroid segmentation as claimed in claim 1, wherein the step (1) specifically comprises:
the relationship between the scattering electric field and the induced current is obtained according to the theory of the bit function and the potential function as follows:
wherein A (r) represents a magnetic vector position,
represents the electric scale position, and the expression of A (r) is as follows:
j (r) represents the equivalent current of the target surface, G (R) represents the Green's function of free space, wherein 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 scattering body.
3. The method for solving the electromagnetic scattering of the electrically large object by the wavelet moment method of centroid splitting according to claim 1, wherein the step (2) specifically comprises:
substituting the formulas (2) and (3) into the formula (1)
Where L represents the operator for calculating the electric field radiation from the current, the tangential component of the electric field strength at the surface of the conductor is zero as known from the ideal conductor boundary conditions, i.e. L is the product of the calculation of the electric field radiation from the current
wherein EiRepresenting the incident electric field and η representing the wave impedance.
4. The method for solving the electromagnetic scattering of the electrically large object by the wavelet moment method of centroid splitting according to claim 1, wherein the step (3) specifically comprises: the discrete electric field integral equation can be obtained by adopting a Galois method and using RWG basis functions as weight functions
5. The method for solving the electromagnetic scattering of the electrically large object by the wavelet moment method of centroid splitting according to claim 1, wherein the step (4) specifically comprises: when using centroid-cut distribution, the RWG basis function is used to triangulate the conductor surface, and the integral of the function g over the original triangle can be expressed as
So that the impedance matrix elements can be rewritten to
6. The method for solving the electromagnetic scattering of the electrically large object by the wavelet moment method of centroid splitting according to claim 1, wherein the step (5) specifically comprises: the matrix equation can be obtained as follows
ZI＝V (11)
Z, I and V respectively represent an impedance matrix, an unknown vector and an excitation vector, and U is assumed to be a non-singular matrix of size and a wavelet transformation matrix.
7. The method for solving the electromagnetic scattering of the electrically large object by the wavelet moment method of centroid splitting according to claim 1, wherein the step (6) specifically comprises: the wavelet transformation matrix U is introduced, and the matrix equation can be transformed:
Z'I'＝V' (12)
Z'＝UZUTI'＝(UT)-1I V'＝UV (13)
t denotes the transpose of the matrix.
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| CN114491992A (en) * | 2022-01-11 | 2022-05-13 | 电子科技大学 | Efficient electromagnetic scattering method based on equivalent dipole moment and physical optical method |
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