A method for electromagnetic scattering simulation based on skeleton decomposition preconditioning

Abstract

The present invention belongs to electromagnetic simulation technology and specifically provides an electromagnetic scattering simulation method based on skeleton decomposition preconditioning, which is used to complete electromagnetic scattering simulation calculations of electrically large complex targets. While ensuring the accuracy of the simulation calculations, it improves the problem of ill-conditioned impedance matrices generated by complex targets, and at the same time improves the convergence and reliability of the iterative method. In the present invention, the interpolation decomposition technology and the proxy surface method are first used to select the bone element basis functions, and the skeleton decomposition method is used to obtain the approximate inverse matrix of the impedance matrix; then, the approximate inverse matrix is ​​used as a preprocessor, and the current coefficient is obtained by solving the Krylov subspace iteration method. Finally, the electromagnetic scattering of the metal target is calculated based on the current coefficient. The present invention creatively proposes a new preprocessor. Compared with traditional preprocessors, this preprocessor can further improve the matrix properties, greatly reduce the iteration time, and improve the convergence.

Description

Technical Field

[0001] The present invention belongs to electromagnetic simulation technology, and specifically provides an electromagnetic scattering simulation method based on skeleton decomposition preconditioning. Background Art

[0002] The electromagnetic scattering characteristics of complex targets have important research value in the stealth and anti-stealth aspects of the military field. In order to improve the problem of the ill-conditioned impedance matrix generated by complex targets, a fast direct solution method and an iterative solution method based on preconditioning technology (also known as preconditioning) are mainly used; for example, the document "KLHo, L. Ying. Hierarchical interpolative factorization for elliptic operators: Differential equations [J]. Communications on Pure and Applied Mathematics, 2016, 69 (8): 1415-1451" discloses a fast direct solution method, which calculates the inverse matrix of the impedance matrix based on the skeleton decomposition method to solve the equation. However, the fast direct solution method has the problems of long calculation time and high memory consumption, and it is difficult to calculate the electromagnetic scattering characteristics of complex targets with large electrical sizes; therefore, for the simulation of the electromagnetic scattering characteristics of complex targets with large electrical sizes, the iterative solution method based on preconditioning technology (also known as preconditioning) is considered to be a better method. Summary of the Invention

[0003] The purpose of the present invention is to provide an electromagnetic scattering simulation method based on skeleton decomposition preconditioning, which is used to complete the electromagnetic scattering simulation calculation of electrically large complex targets. While ensuring the accuracy of the simulation calculation, it improves the problem of the ill-conditioned impedance matrix generated by the complex targets and improves the convergence and reliability of the iterative method.

[0004] To achieve the above object, the technical solution adopted by the present invention is:

[0005] An electromagnetic scattering simulation method based on skeleton decomposition preconditioning, characterized by comprising the following steps:

[0006] Step 1. Discretize the target surface using a conformal triangle network and define an RWG basis function on each triangle element pair. Then, use the Galerkin method to discretize the EFIE equation to generate a linear equation system Zx=b, where Z is the impedance matrix, b is the excitation term, and x is the current coefficient to be solved.

[0007] Step 2. Group the targets into octrees, establish the correspondence between RWG basis functions and octrees based on the location information, define non-overlapping and non-adjacent basis function groups as far-zone groups, and overlapping or adjacent basis function groups as near-zone groups;

[0008] Step 3. Use the proxy surface method based on Huygens principle to construct the proxy matrix, and perform interpolation decomposition on the proxy matrix to select the bone element basis function of each octree group;

[0009] Step 4. According to the bone element basis function, the skeleton decomposition method is used to obtain the approximate inverse matrix P of the impedance matrix Z -1 ;

[0010] Step 5. Use the approximate inverse matrix P -1 As a preprocessor, construct the linear equation system P -1 Zx=P -1 b, and solve the linear equations by Krylov subspace iteration method to obtain the best approximate solution x opt ;

[0011] Step 6. According to the current coefficient x opt Calculate the electromagnetic scattering of metal targets.

[0012] Furthermore, in step 2, the specific process of octree grouping is as follows: first, use a cube to cover the target grid. At this time, the cube corresponds to the 0th layer of the octree; then, divide the cube into eight small cubes to obtain the first layer of the octree, and divide it layer by layer until the size of the bottom cube is smaller than the preset value, and construct a total of L layers of octree.

[0013] Furthermore, in step 3, the decomposition threshold for interpolation decomposition of the proxy matrix is ​​set to 0.01-0.1.

[0014] It should be noted that the core innovation of the present invention is to creatively propose to calculate the approximate inverse matrix (P) of the impedance matrix (Z) obtained by the skeleton decomposition method in the fast and direct solution method. -1 ) is applied to the iterative solution method based on preconditioning technology (also known as preconditioning) as a preprocessor (P -1 ). Based on the preprocessor (P -1) The iterative solution method can quickly and accurately obtain the calculation results at a much lower computational cost than the fast direct solution method; and the specific process of selecting the bone element basis function by the proxy surface method based on Huygens principle can be referred to the literature "Y. Brick, AE Yilmaz. Fast multilevel computation of low-rank representation of H-matrix blocks [J]. IEEE Transactions on Antennas and Propagation, 2016, 64 (12): 5326-5334 and Y. Brick, AE Yilmaz. Rapid rank estimation and low-rank approximation of impedance matrix blocks using proxy grids [J]. IEEE Transactions on Antennas and Propagation, 2018, 66 (10): 5359-5369", and the interpolation decomposition method involved can be referred to the literature "E. Liberty, F. Woolfe, PG Martinsson, et al. Randomized algorithms for the low-rank ap proximation of matrices[J].Proceedings of theNational Academy ofSciences,2007,104(51):20167-20172”, and the above skeleton decomposition method can be referred to the literature “KLHo,L.Ying.Hierarchical interpolative factorization for ellipticoperators:Differential equations[J].Communications on Pure and AppliedMathematics,2016,69(8):1415-1451”, which will not be described in detail in the present invention.

[0015] It is further emphasized that the essence of the present invention is an iterative solution method based on preconditioning technology, and the core innovation is to propose a new preprocessing sub- -1 ); Although, the preprocessor (P -1) is calculated using a skeleton decomposition method, but it is different from the skeleton decomposition method in the fast direct solution method. In order to obtain accurate calculation results, the decomposition threshold of the interpolation decomposition in the skeleton decomposition method of the fast direct solution method must be set to less than 0.001, and the smaller the threshold, the longer the calculation time and the greater the memory consumption. In the present invention, the skeleton decomposition method is used to calculate the preprocessing sub-(P -1 ) process, it is only necessary to set the decomposition threshold of the interpolation decomposition to 0.01~0.1. -1 ) Iterative solution method can quickly and accurately obtain the calculation results; Obviously, the calculation time and memory consumption of the present invention are smaller, and the preprocessing sub-(P -1 ) Compared with traditional preprocessors, it can further improve matrix properties, greatly reduce iteration time and improve convergence.

[0016] In summary, the beneficial effect of the present invention is to propose an electromagnetic scattering simulation method based on skeleton decomposition preconditioning for calculating the electromagnetic scattering of complex special targets; the present invention creatively proposes a new preprocessing sub-(P -1 ), compared with the traditional preprocessor, this preprocessor can further improve the matrix properties of the impedance matrix generated by the electrically large and complex target, greatly reduce the iteration time and improve the convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 is a discrete grid of metal balls in an embodiment of the present invention.

[0018] Figure 2 This is the bone element basis function grid of the metal ball in the embodiment of the present invention.

[0019] Figure 3 2 is a comparison diagram of the HH polarization bistatic RCS results of the metal sphere in an embodiment of the present invention.

[0020] Figure 4 This is an iteration comparison diagram in an embodiment of the present invention. DETAILED DESCRIPTION

[0021] In order to make the purpose, technical solutions and beneficial effects of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments.

[0022] This embodiment provides an electromagnetic scattering simulation method based on skeleton decomposition preconditioning. Taking the electromagnetic scattering of a metal sphere as an example, the method calculates the bistatic radar cross section (RCS). The method specifically includes the following steps:

[0023] Step 1. Figure 1As shown, the radius of the target metal ball is r = 5.0m, the frequency of the incident plane wave is 300MHz, and the incident direction is The electric field polarization direction is along the x-direction. The average mesh size of the triangular grid is set to 0.1λ, and the surface of the metal sphere is discretized using a triangular grid. Basis functions are defined on the grid, and the number of basis functions is 113973. The equivalent current of the surface to be solved is expressed by multiplying the basis functions by the current coefficient to be solved, generating a linear equation system Zx=b, where Z is the impedance matrix, b is the excitation term, and x is the current coefficient to be solved.

[0024] Step 2. Group the target using an octree. Cover the target grid with a large cube. Set the size of the bottom cube to 0.4λ. Build an octree structure with a total of 5 layers to define the far group and the near group.

[0025] Step 3. Use the proxy surface method based on Huygens principle to obtain the proxy matrix, perform interpolation decomposition on the proxy matrix, select the bone element basis function to compress the matrix calculation, and the selected bone element basis function is as follows: Figure 2 As shown in the figure, the number of bone element basis functions finally selected is 17901;

[0026] The specific process of interpolation decomposition is: for the basis function groups i and j that are mutually far-zone groups, the coupling matrix formed is Z ij , which can be expressed as the multiplication of three low-rank matrices:

[0027] Z ij ≈L i ·S ij ·R j

[0028] Among them, S ij Represents the bone element basis function impedance matrix selected from basis function groups i and j, which is also the original matrix Z ij Li and Rj represent projection matrices and are only related to basis function groups i and j respectively; the dimensions of the three low-rank matrices are n i ×k i 、k i ×k j and k j ×n j , n i and n j are the number of unknowns in basis function groups i and j, respectively, and k i and k j are the numbers of selected osteoblasts;

[0029] Interpolation and decomposition of the global impedance matrix will bring a huge amount of calculation. To accelerate the selection of bone bases, the proxy surface method based on the Huygens principle is used. The impedance matrix between the basis function group and its far-zone group can be equivalently replaced by the impedance matrix between the basis function group and a virtual surface. A virtual sphere is used to surround the entire basis function group and constructed at the boundary between the near group and the far-zone group. A series of evenly distributed equivalent points are set on the proxy surface, and two mutually orthogonal constant vector basis functions are defined at each equivalent point. The impedance matrix of the basis function group and the proxy surface can be calculated using the defined constant basis functions. The bone bases can be selected by interpolation and decomposition of the established proxy impedance matrix.

[0030] Step 4. According to the bone element basis function, the skeleton decomposition method is used to obtain the approximate inverse matrix P of the impedance matrix Z -1 ;

[0031] The construction of the multi-layer skeleton matrix structure is as follows:

[0032] After applying the interpolation decomposition plus proxy surface method to the far zone group impedance matrix, the system matrix can be further divided into:

[0033]

[0034] in, The impedance matrix of the neighboring group at the bottom layer of the system matrix, L L and R L represents the projection matrix, Z L-1 Represents the impedance matrix between the bone element basis functions of each basis function group;

[0035] Further reorganization to a coarser layer can obtain a multi-layer skeletonized matrix structure, which is expressed as follows:

[0036]

[0037] Among them, the matrix The impedance matrix of the nearby group not in the bottom layer is reorganized from the bone element basis function in the previous layer;

[0038] The skeleton decomposition method is as follows:

[0039] Starting from the basis function group i of the bottom layer L, the bone element basis functions in the group have been selected by interpolation decomposition; accordingly, the impedance matrix of the system can be used using the appropriate permutation matrix P L,i Decompose into matrix block form:

[0040]

[0041] Among them, i r and i sThey represent the bone element basis function and redundant basis function in basis function group i (all basis functions except bone element basis function are called redundant basis function), and their number is k respectively. i and (n i -k i );i n and i f Represent the nearby group and the distant group of the basis function group i respectively; the interpolation matrix T obtained by interpolation decomposition is i , the impedance matrix between the redundant basis functions and the far zone group can be expressed by the bone element basis function as:

[0042]

[0043] In order to introduce zero matrices in off-diagonal blocks, a diagonal cancellation matrix is ​​defined:

[0044]

[0045]

[0046] Define two triangle elimination matrices:

[0047]

[0048]

[0049]

[0050]

[0051] X represents Schurcomplement, whose value is:

[0052]

[0053]

[0054] It can be seen that the impedance matrix between the redundant basis function and other basis functions is completely eliminated, while the impedance matrix of the bone element basis function is retained; the above process can be simplified as follows:

[0055]

[0056] in:

[0057]

[0058] At this point, the skeleton decomposition of basis function group i is completed, and the remaining matrix We can continue to decompose the next group; after decomposing all the basis function groups of this layer, we can get the decomposition form of this layer as follows:

[0059]

[0060] The final skeleton decomposition preprocessor P -1 The form is

[0061]

[0062] Step 5. Use the approximate inverse matrix P -1 As a preprocessor, construct the linear equation system P -1 Zx=P -1 b, and solve the linear equations by Krylov subspace iteration method to obtain the best approximate solution x opt ;

[0063] Step 6. According to the current coefficient x opt The electromagnetic scattering radar cross-sectional area of ​​the metal target is calculated, such as Figure 3 shown.

[0064] The electromagnetic simulation method based on the skeleton decomposition preprocessor is compared with the electromagnetic simulation method without the preprocessor. In this embodiment, the electromagnetic simulation method based on the skeleton decomposition preprocessor reaches the expected convergence after only 7 Krylov subspace iterations, and the residual is less than 0.001, while the electromagnetic simulation method without the preprocessor can hardly reach the expected convergence. Figure 4 As shown; This shows that, compared with the traditional preprocessor, the preprocessor provided by the present invention can further improve the matrix properties, greatly reduce the iteration time, improve the convergence, and be more targeted for complex and special electromagnetic targets, showing strong robustness. In addition, this embodiment discretizes the metal ball with a triangular grid, generating a total of 113,973 unknowns, but the number of bone element basis functions selected by the proxy surface method combined with the interpolation decomposition is only 17,901. It can be seen that the skeletonization method can effectively coarsen the original dense grid, which also greatly saves the time and computing memory used to construct the preprocessor; Finally, the RCS calculated by the electromagnetic simulation method with the skeleton decomposition preprocessor is compared with the RCS calculated by the analytical MIE series, as shown Figure 3 As shown in FIG, the results of the two methods are in good agreement with each other, which shows that the electromagnetic simulation method based on skeleton decomposition preconditioning proposed in the present invention has high accuracy.

[0065] The above description is only a specific embodiment of the present invention. Unless otherwise stated, any feature disclosed in this specification can be replaced by other equivalent or alternative features with similar purposes.

Claims

1. An electromagnetic scattering simulation method based on skeleton decomposition preconditioning, characterized in that: The following steps are involved: Step 1. Discretize the target surface using a conformal triangle network, define the RWG basis function on each triangle element pair, and use the Galerkin method to discretize the EFIE equation to generate a linear equation system; Step 2. Group the targets into octrees, establish the correspondence between RWG basis functions and octrees based on the location information, define non-overlapping and non-adjacent basis function groups as far-zone groups, and overlapping or adjacent basis function groups as near-zone groups; Step 3. Use the proxy surface method based on the Huygens principle to construct a proxy matrix, perform interpolation decomposition on the proxy matrix to select the bone element basis function of each octree group; the decomposition threshold of the interpolation decomposition of the proxy matrix is ​​set to 0.01~0.1; Step 4. According to the osteon basis function, the skeleton decomposition method is used to obtain the approximate inverse matrix of the impedance matrix; Step 5. Use the approximate inverse matrix as a preprocessor to construct a linear equation system, and solve the linear equation system through the Krylov subspace iteration method to obtain the best approximate solution; Step 6. Calculate the electromagnetic scattering of the metal target based on the best approximate solution of the current coefficient.

2. The electromagnetic scattering simulation method based on skeleton decomposition preconditioning according to claim 1, characterized in that: In step 1, the linear equation group is specifically: Zx=b, where Z is the impedance matrix, b is the excitation term, and x is the current coefficient to be solved.

3. The electromagnetic scattering simulation method based on skeleton decomposition preconditioning according to claim 1, characterized in that: In step 2, the specific process of octree grouping is as follows: first, use a cube to cover the target grid. At this time, the cube corresponds to the 0th layer of the octree; then, divide the cube into eight small cubes to obtain the first layer of the octree. Divide layer by layer until the size of the bottom cube is smaller than the preset value, and construct a total of L layers of octree.

4. The electromagnetic scattering simulation method based on skeleton decomposition preconditioning according to claim 1, characterized in that: In step 5, the linear equations are: P -1 Zx=P -1 b, P -1 is the approximate inverse matrix of the impedance matrix, Z is the impedance matrix, b is the excitation term, and x is the current coefficient to be solved.

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