CN118152706A - CSIE electromagnetic scattering analysis method based on characteristic basis function - Google Patents

Abstract

The invention discloses a CSIE electromagnetic scattering analysis method based on a characteristic basis function, which aims at the electromagnetic scattering problem of a metal target to establish CSIE for scattering calculation, and performs triangle mesh subdivision on the surface of the target to generate the basis function; partitioning the target, and dividing the basis function into corresponding blocks; generating a characteristic base function in each block as a new base function to replace the original base function; constructing a reduced impedance matrix equation, iteratively solving the matrix equation under the excitation of the first incident angle to obtain an equivalent surface current under the excitation of the first incident angle, carrying out phase correction, then applying the equivalent surface current to the iterative solution of the next incident angle, and carrying out the iterative solution of the subsequent incident angle by analogy, so as to obtain the equivalent surface currents of all angles, and then solving the equivalent surface magnetic current, thereby solving the far field value of the target scattering electric field. The invention obviously reduces the memory and time required for solving CSIE in the single-station problem, and simultaneously maintains good precision and convergence of CSIE matrix equation.

Description

CSIE electromagnetic scattering analysis method based on characteristic basis function

Technical Field

The invention belongs to the technical field of electromagnetic scattering analysis, relates to a high-efficiency method for rapidly analyzing electromagnetic scattering of a metal target, and particularly relates to a CSIE electromagnetic scattering analysis method based on a characteristic basis function.

Background

The electromagnetic scattering problem has been widely focused by scholars at home and abroad. The moment method (Method of Moments, moM) is an effective way to calculate the electromagnetic scattering properties of the target. Within the moment system, there are different integral equations, each of which has different characteristics. For the integral equation used to solve the electromagnetic scattering problem of a metal target, there are two most classical integral equations, one is the electric field integral equation and the other is the magnetic field integral equation. The electric field integral equation has the advantage of high accuracy, but has the disadvantage of slow convergence, which is particularly apparent when analyzing electrically large complex targets, often without a result being obtained because of the inability to converge. In contrast, the convergence of the magnetic field integral equation is very excellent, but its accuracy is very poor with respect to the electric field integral equation. Therefore, it is desirable to improve the electric field integral equation to solve the slow convergence disadvantage; or improving the magnetic field integral equation to solve the defect of poor precision.

Calderon preprocessing is a slow method for improving the convergence of the electric field integral equation, but uses Buffa-CHRISTIANSEN (BC) function, which requires center-of-gravity encryption on the initial grid to generate more and finer small grids, and the encryption is very complex and difficult to realize by computer codes. And the matrix filling time becomes longer as more and finer grids are involved. Based on the BC function, the normal rotation form is used as a detection function of the magnetic field integral equation, so that the accuracy of the magnetic field integral equation can be improved. Similarly, the BC function is also needed, so that the implementation difficulty is high, and the matrix filling time is long.

Combined Source Integral Equation (CSIE) is a method with two characteristics of high precision and good convergence, CSIE can avoid the defects of slow convergence of an electric field integral equation and poor precision of a magnetic field integral equation, CSIE is just required to use a Rao-Wilton-Glisson (RWG) function instead of a BC function, the basic function structure is simple, the filling time of a matrix is not increased, but compared with the electric field integral equation and the magnetic field integral equation, more impedance matrices are introduced, and more memory is required for solving the matrix equation. Because the complexity of the memory grows along with the square of the number of the unknowns, when the number of the unknowns increases, the memory required for solving the matrix equation is larger and larger until the memory exceeds the currently owned computing resources, so that the solution fails.

Disclosure of Invention

Aiming at the problems, the invention provides a CSIE electromagnetic scattering analysis method based on a characteristic basis function, which can analyze the single-station electromagnetic scattering problem of a metal target more efficiently, and the novel efficient and rapid electromagnetic scattering analysis method uses the characteristic basis function based on CSIE, so that the number of unknowns can be greatly reduced, thereby reducing the computer resources required by electromagnetic scattering calculation, and simultaneously, the solution time of the single-station electromagnetic scattering problem can be greatly reduced by using the initial value guess based on the characteristic basis function.

The technical scheme of the invention is as follows: the method comprises the following steps:

Step 1: and establishing a surface area fraction equation CSIE for scattering calculation aiming at the electromagnetic scattering problem of the metal target, then meshing the whole target surface to obtain triangular meshes, and generating RWG basis functions on the triangular meshes.

Step 2: and performing block processing on the target, and dividing the basis functions of the surface of the target into different blocks.

In step 2, there is no strict rule on how to block the target, and a suitable block structure is selected according to practical situations, for example, a binary tree structure, a quadtree structure, and an octree structure. A commonly used chunking structure for three-dimensional object problems is the octree structure.

Step 3: a respective feature basis function is generated within each block. These characteristic basis functions will become new basis functions for replacing the original basis functions.

This step can be understood to be actually a sub-problem of the overall problem. Constructing an impedance matrix equation of the block according to a basis function in the block, and irradiating with plane waves with different angles and different polarizations to obtain current solutions under the irradiation; and then singular value decomposition is carried out on the current solutions to remove redundancy, so as to obtain the characteristic base coefficient of unit orthogonality. These feature basis coefficients are the feature basis functions.

Step4: constructing a reduced impedance matrix equation, and iteratively solving the matrix equation under the first personal angle degree excitation to obtain the equivalent surface current under the first personal angle degree excitation.

Step 5: the equivalent surface current under the excitation of the first incident angle is subjected to phase correction and then applied to the iterative solution of the next incident angle. And solving subsequent incident angle excitation by analogy iteration to obtain equivalent surface currents under all the incident angle excitation, and then solving equivalent surface magnetic currents under all the incident angle excitation.

Step 6: and solving far field values of a scattering electric field of the target according to the obtained equivalent surface current solutions and equivalent surface magnetic current solutions under the excitation of all the incidence angles, and further calculating to obtain a radar scattering cross section RCS of the target.

The method is used for carrying out high-efficiency electromagnetic scattering analysis based on a mixed source integral equation (CSIE), a characteristic basis function and initial value guess, is suitable for single-station electromagnetic scattering solving of a metal target, solves far field values of a target scattering electric field, and can be used for stealth design process of military equipment, target detection of a radar system and the like. The method of the invention obviously reduces the calculation memory and time required by the traditional CSIE when solving the single-station electromagnetic scattering, simultaneously maintains good precision and convergence of CSIE matrix equation, and is suitable for the single-station electromagnetic scattering problem of the electric large target.

The invention has the following beneficial effects:

1. Fewer unknowns: the characteristic base function method is utilized, and the characteristic base function is used as a new base function to replace the original RWG base function, so that the number of unknowns is reduced, and the memory consumption is reduced.

2. The precision is high: because the characteristic basis function is a linear combination of RWG basis functions, the mathematical properties related in space of the basis functions or detection functions are not changed, and the dual detection scheme in the process of detecting the integral equation is reserved, so that the invention maintains the original advantage of high precision of CSIE.

3. Good iteration convergence: since the characteristic basis function is a linear combination of RWG basis functions and the equation is CSIE in nature, the invention retains the advantage of good convergence inherent in CSIE.

Drawings

FIG. 1 is a schematic diagram of octree partitioning of a target in an embodiment of the present invention.

Fig. 2 is a graph of the results of a two-station RCS of a regular rectangular pyramid in an embodiment of the present invention.

Fig. 3 is a single-station RCS result graph of a regular rectangular pyramid in an embodiment of the present invention.

Fig. 4 is a convergence graph of a regular rectangular pyramid in an embodiment of the present invention.

Fig. 5 is a single-station iteration step diagram of a regular rectangular pyramid in an embodiment of the present invention.

Detailed Description

In order to clearly illustrate the technical characteristics of the present invention, the technical scheme of the present invention will be described in detail below by means of specific embodiments and with reference to the accompanying drawings.

Step 1: the surface area fraction equation CSIE for scatter calculation is established for the electromagnetic scattering problem of metallic targets. And then mesh dissection is carried out on the whole target surface to obtain triangular meshes, and RWG basis functions are generated on the meshes.

CSIE has the expression of

Wherein Z ₀ is the wave impedance of free space, J is the equivalent surface current, M is the equivalent surface magnetic current,For the external normal, S is the target surface, E ^inc is the incident electric field, r is the field point, r' is the source point, and tan represents the tangential component of the surface. L and K _PV are integral operators, I is a unit operator, and the expression is

I{X(r)}＝X(r) (29)

Where j is an imaginary unit, k ₀ is the wavenumber in free space, p.v. represents the cauchy principal value integral, X (r) represents the function at the field point, X (r ') represents the function at the source point, and G (r, r ') is the green's function in free space. Since CSIE has only one equation but two unknowns, a system of equations is usually constructed with a CS condition expressed as

Step 2: the object is partitioned, i.e. the basis functions of the object surface are partitioned into different blocks.

As shown in fig. 1, the target is partitioned according to an octree structure, and RWG basis functions on the surface of the target are partitioned into different blocks according to the positions of the three-dimensional space in which they are located.

Step 3: a respective feature basis function is generated within each block. These characteristic basis functions will become new basis functions for replacing the original basis functions.

In practice this step can be understood as a sub-problem solving the whole problem. According to the basis function in the block, constructing an impedance matrix equation of the block by using a moment method, and irradiating plane waves with different angles and different polarizations to obtain current solutions under the irradiation. For the mth block, its matrix equation as the mth sub-problem is expressed as

Wherein,Is the L operator impedance matrix of the m-th block,/>Is the K operator impedance matrix of the m-th block,/>Is the incident electric field excitation of the mth block, J _m is the matrix of current coefficients to be solved of the mth block,/>And/>Are respectively the m-th block/>A Gram matrix and a RWG-RWG Gram matrix. Directly solving the matrix equation to obtain a current solution J _m, and performing singular value decomposition on the solution to remove redundancy

J_m＝U∑V^H (32)

Where U and V are unitary matrices, and Sigma is a diagonal matrix with diagonal elements being singular values of J _m. The first k columns of unitary matrix U are determined as characteristic base coefficients according to the threshold of singular value decomposition, and the characteristic base coefficients will show characteristic base functions with the expression as follows

C_m＝U_k (33)

The characteristic base function coefficients of all the blocks are arranged diagonally to obtain the characteristic base function of the whole problem, and the characteristic base coefficient matrix is expressed as:

C＝diag(C₁,C₂,…,C_B) (34)

where B represents the number of blocks.

Step 4: and dispersing the surface integral equation by using a moment method, constructing a reduced impedance matrix equation, and iteratively solving the matrix equation under the first angle incidence excitation to obtain the equivalent surface current and the magnetic current under the first angle incidence excitation.

CSIE reduced matrix equation is expressed as

Wherein Z ^L,CBF is the L operator impedance matrix with Characteristic Basis Function (CBF) as basis function and detection function, Z ^K,CBF is the K operator impedance matrix with Characteristic Basis Function (CBF) as basis function and detection function, V ^E,CBF is the incident electric field excitation with Characteristic Basis Function (CBF) as detection function, G ^fnf,CBF and G ^ff,CBF are respectivelyThe Gram matrix and the CBF-CBF Gram matrix, a ^CBF is the matrix of the characteristic base current coefficient to be solved of the whole problem. The operator impedance matrix and Gram matrix are expressed as

Z^L,CBF＝C^HZ^LC (36)

Z^K,CBF＝C^HZ^KC (37)

G^fnf,CBF＝C^HG^fnfC (38)

G^ff,CBF＝C^HG^ffC (39)

Wherein Z ^L is the L operator impedance matrix, Z ^K is the K operator impedance matrix, and G ^fnf and G ^ff are respectivelyThe Gram matrix and the RWG-RWG Gram matrix, and C ^H represents the conjugate transpose of the characteristic base coefficient matrix; after the characteristic base current solution a ^CBF is obtained, the characteristic base current solution b is obtained according to the following formula ^CBF

G^ff,CBFb^CBF＝Z₀G^fnf,CBFa^CBF (40)

The final current solution is finally given by

a＝Ca^CBF (41)

The final magneto-rheological is given by

b＝Cb^CBF (42)。

Step 5: and (3) carrying out phase correction on the equivalent surface current solution under the excitation of the first incident angle, and then applying the equivalent surface current solution to the iterative solution of the next incident angle. And solving subsequent incident angle excitation by analogy iteration to obtain equivalent surface currents under all the incident angle excitation, and then solving equivalent surface magnetic currents under all the incident angle excitation.

When solving the reduced matrix equation in step 4, when obtaining the equivalent surface current solution under the previous incident angle excitation, firstly restoring the equivalent surface current solution from the coefficient a ^CBF of the characteristic basis function to the coefficient a of the RWG basis function

a＝Ca^CBF (43)

Secondly, carrying out phase correction on the RWG coefficient to obtain the RWG coefficient after the phase correction

Wherein a _m is the mth element of a,Is the modified coefficient/>R _m is the center of the mth RWG basis function, k and k' represent the incident directions of the previous incident angle and the next incident angle, respectively, and e is a natural constant. The corrected RWG coefficients are then projected onto the coefficients of the feature basis functions

I.e. the initial value guess of the next incident angle excitation is solved for iteration.

After the current coefficient and the magnetic current coefficient are obtained by using an iteration solver to solve CSIE matrix equations, the finally obtained equivalent surface current J (r) and magnetic current M (r) can be expressed as follows:

Where N is the number of RWG basis functions, f _n (r) is the nth RWG basis function, and a _n and b _n are the current coefficient and the magnetic current coefficient, respectively, corresponding to the nth RWG basis function.

Step 6: solving far field values of the target scattering electric field according to the obtained equivalent surface current solution and equivalent surface magnetic current solution under the excitation of all incidence angles:

further calculating to obtain a radar scattering cross section RCS of the target, wherein the radar scattering cross section RCS is represented by a symbol sigma, and the calculation expression is as follows:

Wherein the unit of σ is m ².

Example 1

The embodiment verifies the advantages of the invention in terms of memory, precision and iteration convergence by a metal regular rectangular pyramid with the bottom side length of 3m and the height of 6 m.

Fig. 2 shows the results of the two-station RCS of the present invention (only using the part of the mixed source integral equation and the characteristic basis function, CSIE-CBFM for short, the initial guess technique for solving the single-station problem) with the two-station RCS of the different method and the two-station RCS of the Electric Field Integral Equation (EFIE) as the numerical reference solution, and fig. 3 shows the corresponding single-station RCS results, it can be seen that the results of the present invention are more accurate than the Magnetic Field Integral Equation (MFIE). Fig. 4 shows the iterative convergence of the present invention and the iterative convergence of the different methods, and it can be seen that the results of the present invention are more excellent than the Electric Field Integral Equation (EFIE). Fig. 5 shows the number of iteration steps required to reach the convergence threshold in the present invention (CSIE-CBFM) using zero as the initial value (zero initial), the solution of the last incident angle excitation as the initial value (without phase correction), and the solution of the last incident angle excitation plus the phase correction as the initial value (phase correction). It can be seen that using the solution of the last incident angle excitation plus the phase correction as the iteration initial value can effectively reduce the number of the received hesitate to advance further, thereby reducing the convergence time.

Table 1 shows a comparison of the number of unknowns, memory usage, and single-site solution time in solving the single-site problem for conventional CSIE and the present invention (CSIE-CBFM). It can be seen that the present invention has a smaller number of unknowns than conventional CSIE, and therefore requires less memory, embodying a great advantage in terms of memory consumption of the present invention. Because the impedance matrix in the invention has smaller size, the length of the vector product of the matrix in the iterative solution is short, and the number of iterative steps is reduced by combining initial value guessing, the time of the iterative solution is greatly reduced, and the huge advantage of the invention in time consumption is embodied.

TABLE 1

While there have been described what are believed to be the preferred embodiments of the present invention, it will be apparent to those skilled in the art that many more modifications are possible without departing from the principles of the invention.

Claims (7)

1. The CSIE electromagnetic scattering analysis method based on the characteristic basis function is characterized by comprising the following steps of:

Step 1: establishing a surface area component equation CSIE for scattering calculation aiming at the electromagnetic scattering problem of a metal target, then carrying out grid subdivision on the whole target surface to obtain triangular grids, and generating RWG basis functions on the triangular grids;

step 2: partitioning the target, and dividing the basis functions of the surface of the target into different blocks;

step 3: generating a characteristic base function in each block as a new base function to replace the original base function;

Step4: constructing a reduced impedance matrix equation, and iteratively solving the matrix equation under the first personal angle degree excitation to obtain the equivalent surface current under the first personal angle degree excitation;

Step 5: performing phase correction on the equivalent surface current under the excitation of the first incident angle, applying the phase correction to the iteration solution of the next incident angle, performing the similar iterative solution on the subsequent man-angle excitation to obtain the equivalent surface current under the excitation of all the incident angles, and then solving the equivalent surface magnetic current under the excitation of all the incident angles;

Step 6: and solving far field values of a scattering electric field of the target according to the obtained equivalent surface current solutions and equivalent surface magnetic current solutions under the excitation of all the incidence angles, and further calculating to obtain a radar scattering cross section RCS of the target.

2. The method for analyzing CSIE electromagnetic scattering according to claim 1, wherein in step 1, CSIE is expressed as

Wherein Z ₀ is the wave impedance of free space, J is the equivalent surface current, M is the equivalent surface magnetic current,For the external normal, S is the target surface, E ^inc is the incident electric field, r is the field point, r' is the source point, and subscript tan represents the tangential component of the surface; l and K _PV are integral operators, I is a unit operator, and the expression is:

I{X(r)}＝X(r) (4)

Wherein j is an imaginary unit, k ₀ is a wavenumber in free space, p.v. represents a cauchy principal value integral, X (r) represents a function at a field point, X (r ') represents a function at a source point, G (r, r ') is a green's function in free space, and the expression is

Since CSIE has only one equation but two unknowns, it is necessary to construct an equation set with a CS condition expressed as

3. The method for analyzing electromagnetic scattering based on CSIE th step of characteristic basis functions according to claim 1, wherein the step 2 is specifically: the target is segmented according to an octree structure, and RWG basis functions of the surface of the target are divided into different blocks according to the positions of the RWG basis functions in a three-dimensional space.

4. The method for analyzing electromagnetic scattering based on CSIE th step of characteristic basis functions according to claim 1, wherein the step 3 is specifically: constructing an impedance matrix equation of the block by using a moment method according to a basis function in the block, and irradiating with plane waves with different angles and different polarizations to obtain current solutions under the irradiation; for the mth block, its matrix equation as the mth sub-problem is expressed as

Wherein,Is the L operator impedance matrix of the m-th block,/>Is the K operator impedance matrix of the m-th block,/>Is the incident electric field excitation of the mth block, J _m is the matrix of current coefficients to be solved of the mth block,/>And/>Are respectively the m-th block/>A Gram matrix and an RWG-RWG Gram matrix; directly solving the matrix equation to obtain a current solution J _m, and performing singular value decomposition on the solution to remove redundancy

J_m＝UΣV^H (8)

Wherein U and V are unitary matrices, the superscript H represents the conjugate transpose of the matrix, and Sigma is a diagonal matrix whose diagonal elements are the singular values of J _m; the first k columns of unitary matrix U are determined as characteristic base coefficients according to the threshold of singular value decomposition, the characteristic base coefficients are used for representing characteristic base functions, and the expression of the characteristic base function C _m of the mth block is as follows

C_m＝U_k (9)

The characteristic base coefficients of all the blocks are arranged diagonally to obtain the characteristic base function of the whole problem, and the characteristic base coefficient matrix C is expressed as:

C＝diag(C₁,C₂,…,C_B) (10)

where B represents the number of blocks.

5. The method for electromagnetic scattering analysis of CSIE based on characteristic basis functions as set forth in claim 4, wherein step 4 is specifically: dispersing a surface integral equation by using a moment method, constructing a reduced impedance matrix equation, and iteratively solving the matrix equation under the first angle incidence excitation to obtain an equivalent surface current under the first angle incidence excitation;

CSIE reduced matrix equation is expressed as

Wherein Z ^L,CBF is the L operator impedance matrix with characteristic basis function CBF as basis function and detection function, Z ^K,CBF is the K operator impedance matrix with characteristic basis function CBF as basis function and detection function, V ^E,CBF is the incident electric field excitation with characteristic basis function CBF as detection function, G ^fnf,CBF and G ^ff,CBF are respectivelyThe matrix and the CBF-CBF Gram matrix, a ^CBF is a matrix of the characteristic base current coefficient to be solved of the whole problem; the operator impedance matrix and Gram matrix are expressed as

Z^L,CBF＝C^HZ^LC (12)

Z^K,CBF＝C^HZ^KC (13)

G^fnf,CBF＝C^HG^fnfC (14)

G^ff,cBF＝C^HG^ffC (15)

Wherein Z ^L is the L operator impedance matrix, Z ^K is the K operator impedance matrix, and G ^fnf and G ^ff are respectivelyThe matrix and RWG-RWG Gram matrix, C ^H represents the conjugate transpose of the characteristic base coefficient matrix; after the characteristic base current solution a ^CBF is obtained, the characteristic base current solution b is obtained according to the following formula ^CBF

G^ff,CBFb^CBF＝Z₀G^fnf,CBFa^CBF (16)

The final solution a is finally given by

a＝Ca^CBF (17)

The magneto-rheological b is given by

b＝Cb^CBF (18)。

6. The method for electromagnetic scattering analysis of CSIE based on characteristic basis functions of claim 5, wherein step 5 is specifically: when solving the reduced matrix equation in step 4, when obtaining the equivalent surface current solution under the previous incident angle excitation, firstly restoring the equivalent surface current solution from the coefficient a ^CBF of the characteristic basis function to the coefficient a of the RWG basis function

a＝Ca^CBF (19)

Secondly, carrying out phase correction on the RWG coefficient to obtain the RWG coefficient after the phase correction

Wherein a _m is the mth element of a,Is the modified coefficient/>R _m is the center of the mth RWG basis function, k and k' represent the incident directions of the previous angle and the next angle, respectively, and e is a natural constant;

The corrected RWG coefficients are then used to calculate Projection onto coefficients of characteristic basis functions

Namely, the initial value guess of the excitation of the next incident angle is obtained by iteration;

After the current coefficient and the magnetic current coefficient are obtained by using an iteration solver to solve CSIE matrix equation, the finally obtained equivalent surface current J (r) and magnetic current M (r) are expressed as follows:

Where N is the number of RWG basis functions, f _n (r) is the nth RWG basis function, and a _n and b _n are the current coefficient and the magnetic current coefficient, respectively, corresponding to the nth RWG basis function.

7. The method for electromagnetic scattering analysis of CSIE based on characteristic basis functions as set forth in claim 6, wherein the step 6 is specifically: according to the obtained equivalent surface current solution and equivalent surface magnetic current solution under the excitation of all incidence angles, solving a far field value E ^sca of a target scattering electric field:

further calculating to obtain a radar scattering cross section RCS of the target, wherein the radar scattering cross section RCS is represented by a symbol sigma, and the calculation expression is as follows:

Wherein the unit of σ is m ².