Electromagnetic scattering calculation method and system based on layered Krylov subspace
By employing a hierarchical Krylov subspace electromagnetic scattering calculation method, the target region is divided into blocks and basis functions are constructed. Combined with a sparse model and recovery algorithm, the problem of high computational complexity in electromagnetic scattering of large targets is solved, and efficient electromagnetic scattering solutions are achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-25
- Publication Date
- 2026-03-24
AI Technical Summary
Existing technologies for solving electromagnetic scattering problems of large targets suffer from extremely high computational complexity and memory consumption due to the method of moments. Furthermore, as the target size increases, the number of basis functions grows rapidly, resulting in low computational efficiency and memory utilization.
We employ a hierarchical Krylov subspace electromagnetic scattering calculation method. By dividing the target region into blocks, we construct an initial Krylov subspace basis function and perform projection, compression, and orthogonalization operations to recursively construct a compact global basis function matrix. Combined with the sparse induced current model and the LSMR-Tikhonov recovery algorithm, we improve computational efficiency and memory utilization.
While ensuring computational accuracy, it significantly improves the solution efficiency and numerical stability of electromagnetic scattering problems, and reduces computation time and memory requirements.
Smart Images
Figure CN121723710A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of electromagnetic numerical calculation, in particular to a layered Krylov subspace-based electromagnetic scattering calculation method and system. BACKGROUND
[0002] The method of moments is one of the main numerical methods for analyzing the electromagnetic scattering characteristics of targets. However, when solving the electromagnetic scattering problem of large-size targets, the method of moments needs to construct and solve large-scale dense matrix equations, resulting in extremely high computational complexity and memory consumption. The introduction of compressed sensing technology significantly reduces the complexity of filling the impedance matrix and solving the current coefficient by converting the matrix equation into an underdetermined system and utilizing the sparsity of induced current.
[0003] In the method of moments based on compressed sensing, constructing a sparse basis is the key to realizing sparse representation of current. In the prior art, characteristic basis functions, characteristic mode functions, and Krylov subspace basis functions based on block have been used to construct sparse bases. However, as the size of the target increases, the number of basis functions grows rapidly, making the subsequent sensing matrix large and the computational efficiency and memory utilization low. Therefore, it is severely limited in solving large-scale electromagnetic scattering problems. SUMMARY
[0004] The purpose of the present application is to provide a layered Krylov subspace-based electromagnetic scattering calculation method and system, which uses an algebraic multiscale decomposition framework to recursively construct a compact and well-orthogonal basis function set, significantly improving computational efficiency and memory usage while ensuring accuracy.
[0005] A layered Krylov subspace-based electromagnetic scattering calculation method, comprising: performing block processing on the electromagnetic scattering region of the target to obtain a plurality of blocks; and constructing initial Krylov subspace basis functions for each block to obtain an initial basis function set; performing projection, compression, and orthogonalization operations on the initial basis function set to obtain a final global basis function matrix; constructing an induced current sparse model, and based on the induced current sparse model, combining the final global basis function matrix to obtain a reconstructed induced current.
[0006] Optionally, the projection, compression, and orthogonalization operations on the initial basis function set to obtain a final global basis function matrix comprise: the initial basis function set is taken as the first layer global basis function; the first layer global basis function expression is ; wherein, is the first layer global basis function, is the initial Krylov subspace basis function of the first block, is the initial Krylov subspace basis function of the second block, is the initial Krylov subspace basis function of the p-th block, p is the total number of blocks; projecting the first layer global basis function to obtain the second layer compressed impedance matrix and the second layer compressed excitation vector; the expression of the second layer compressed impedance matrix is: ; wherein, is the second layer compressed impedance matrix, T is the transpose, is the first layer compressed impedance matrix, that is, the initial impedance matrix; the expression of the second layer compressed excitation vector is: ; wherein, is the second layer compressed excitation vector, is the first layer compressed excitation vector, that is, the initial excitation vector; obtaining the second layer initial global basis function according to the second layer compressed impedance matrix and the second layer compressed excitation vector; orthogonalizing the second layer initial global basis function based on singular value decomposition to obtain the second layer global basis function; By analogy, the k-th layer global basis function is obtained, and the first layer global basis function to the k-th layer global basis function are combined into a final global basis function matrix.
[0007] Optionally, the expression of the induced current sparse model is: ; wherein, is the final global basis function matrix, is the reconstructed induced current, is the sparse coefficient vector of the reconstructed induced current on .
[0008] Optionally, the method for constructing the induced current sparse model, based on the induced current sparse model, and combining the final global basis function matrix to obtain the reconstructed induced current, comprises: constructing the induced current sparse model and the compressed sensing calculation model; the expression of the compressed sensing calculation model is: ; wherein, is a measurement matrix, which is obtained according to an initial impedance matrix, is the final global basis function matrix, is a sensing matrix, , To measure the excitation vector, it is obtained from the initial excitation vector. To rebuild the induced current in sparse coefficient vector on; Based on the LSMR-Tikhonov recovery algorithm, the current coefficients are analyzed. Solving this problem yields the optimal sparse coefficient vector; the expression is: ; in, For regularization parameters; The optimal sparse coefficient vector and the final global basis function matrix are substituted into the sparse model of the induced current to obtain the reconstructed induced current.
[0009] This invention also provides an electromagnetic scattering computation system based on hierarchical Krylov subspaces, comprising: The block construction module is used to divide the target electromagnetic scattering region into blocks to obtain several blocks; and to construct the initial Krylov subspace basis functions for each block to obtain the initial basis function set. The function processing module is used to perform projection, compression, and orthogonalization operations on the initial basis function set to obtain the final global basis function matrix. The induced current module is used to construct a sparse model of induced current. Based on the sparse model of induced current and combined with the final global basis function matrix, the reconstructed induced current is obtained.
[0010] Optionally, the function processing module specifically comprises: The initial set of basis functions is used as the first-level global basis functions; the expression for the first-level global basis functions is: ; in, For the first level of global basis functions, Let be the initial Krylov subspace basis functions for the first block. The initial Krylov subspace basis functions for the second block are... Let be the initial Krylov subspace basis function for the p-th block, where p is the total number of blocks; Projecting the first-layer global basis functions yields the second-layer compressed impedance matrix and the second-layer compressed excitation vector; the expression for the second-layer compressed impedance matrix is: ; in, This is the second-level compression impedance matrix, where T is the transpose. This is the first-level compression impedance matrix, i.e., the initial impedance matrix; The expression for the second-layer compression excitation vector is: ; in, This is the compression excitation vector for the second layer. This is the first layer compression excitation vector, i.e., the initial excitation vector; The initial global basis functions of the second layer are obtained based on the second layer compression impedance matrix and the second layer compression excitation vector; The initial global basis functions of the second layer are orthogonalized based on singular value decomposition to obtain the global basis functions of the second layer. By analogy, the global basis functions of the kth layer are obtained, and the global basis functions of the 1st layer to the kth layer are combined into the final global basis function matrix.
[0011] Optionally, the sparse expression for the induced current sparse model is: ; in, This is the final global basis function matrix. To reconstruct the induced current, To rebuild the induced current in A sparse coefficient vector on.
[0012] Optionally, the sensing current module specifically comprises: Construct the sparse model of the induced current and the compressed sensing computation model; the expression of the compressed sensing computation model is: ; in, The measurement matrix is obtained from the initial impedance matrix. This is the final global basis function matrix. For the sensing matrix, , To measure the excitation vector, it is obtained from the initial excitation vector. To rebuild the induced current in sparse coefficient vector on; Based on the LSMR-Tikhonov recovery algorithm, the current coefficients are analyzed. Solving this problem yields the optimal sparse coefficient vector; the expression is: ; in, For regularization parameters; The optimal sparse coefficient vector and the final global basis function matrix are substituted into the sparse model of the induced current to obtain the reconstructed induced current.
[0013] The effects of this invention are as follows: The application is based on a layered Krylov subspace electromagnetic scattering calculation method, which first carries out block processing on a target electromagnetic scattering region to obtain a plurality of blocks, and constructs initial Krylov subspace base functions of each block to obtain an initial base function set; then carries out projection, compression and orthogonalization operations on the initial base function set to obtain a final global base function matrix; finally, a sparse model of induced current is constructed, and based on the sparse model of induced current, the final global base function matrix is combined to obtain reconstructed induced current; an algebraic multi-scale decomposition framework is adopted to recursively construct a compact and well-orthogonal base function set, and a more stable LSMR-Tikhonov algorithm is adopted for current recovery, which significantly improves the calculation efficiency and memory usage while ensuring accuracy. BRIEF DESCRIPTION OF DRAWINGS
[0014] Figure 1 is a flowchart of the layered Krylov subspace electromagnetic scattering calculation method of the application; Figure 2 is a result schematic diagram of the method of the application. DETAILED DESCRIPTION
[0015] Hereinafter, embodiments of the application will be described with reference to the accompanying drawings.
[0016] Figure 1 is a flowchart of the layered Krylov subspace electromagnetic scattering calculation method of the application, as shown in Figure 1 the application provides a layered Krylov subspace electromagnetic scattering calculation method, which comprises: S1, carrying out block processing on a target electromagnetic scattering region to obtain a plurality of blocks; and constructing initial Krylov subspace base functions of each block to obtain an initial base function set.
[0017] Specifically, the target electromagnetic scattering region is divided into p blocks. The matrix equation in the method of moments: ; can be converted into the following block form: ; Wherein, Z is the initial impedance matrix, I is the reconstructed induced current, i.e. the induced current vector to be solved, V is the initial excitation vector. is the self-impedance of the ith block, is the mutual impedance between the ith block and the jth block, is the induced current of the ith block, is the excitation vector of the ith block.
[0018] In order to eliminate the discontinuity of the current between the blocks, the of each block is extended, is the wavelength of the incident plane wave; after expansion, the self-impedance of the ith block is , and the excitation vector is , For each expanded block, construct its a-th Krylov subspace as follows: ; Generate a set of orthonormal bases in this subspace by Arnoldi iteration ; ; where is the a-th basis function of the ith block.
[0019] After removing the numerical values of the expansion region, the initial basis function of the ith block is , that is, the initial Krylov subspace basis function of the ith block.
[0020] S2, project, compress and orthogonalize the initial basis function set to obtain the final global basis function matrix.
[0021] Further, S2 includes: S21, take the initial basis function set as the first layer global basis function. The expression of the first layer global basis function is ; where is the first layer global basis function, is the initial Krylov subspace basis function of the first block, is the initial Krylov subspace basis function of the second block, is the initial Krylov subspace basis function of the p-th block, and p is the total number of blocks.
[0022] S22, project the first layer global basis function to obtain the second layer compressed impedance matrix and the second layer compressed excitation vector. The expression of the second layer compressed impedance matrix is ; where is the second layer compressed impedance matrix, T is the transpose, is the first layer compressed impedance matrix, that is, the initial impedance matrix.
[0023] The expression of the second layer compressed excitation vector is ; where is the second layer compressed excitation vector, is the first layer compressed excitation vector, that is, the initial excitation vector.
[0024] S23, obtaining the initial global basis function of the second layer according to the compressed impedance matrix of the second layer and the compressed excitation vector of the second layer.
[0025] S24, orthogonalizing the initial global basis function of the second layer based on singular value decomposition to obtain the global basis function of the second layer.
[0026] Specifically, the initial global basis function of the second layer is generated by Arnoldi iteration, and then the initial global basis function of the second layer is orthogonalized by singular value decomposition (SVD) to obtain the global basis function of the second layer; the expression is: ; Wherein, U and W are unitary matrices generated after SVD, is a diagonal matrix of singular values.
[0027] The first column vectors in U are reserved as the global basis function of the second layer .
[0028] S25, similarly, the global basis function of the kth layer is obtained, and the global basis function matrix of the first layer to the global basis function of the kth layer is combined into the final global basis function matrix. Specifically, the expression of the initial global basis function of the kth layer is: ; Wherein, is the initial global basis function of the kth layer, is the compressed impedance matrix of the kth layer, is the compressed excitation vector of the kth layer, and m is the order of the Krylov subspace of the kth layer.
[0029] S3, constructing a sparse model of induced current, and based on the sparse model of induced current, combining the final global basis function matrix to obtain the reconstructed induced current.
[0030] Specifically, the sparse expression of the sparse model of induced current is: ; Wherein, is the final global basis function matrix, is the reconstructed induced current, is the sparse coefficient vector of the reconstructed induced current on .
[0031] S3 includes: S31, constructing a sparse model of induced current and a compressed sensing calculation model. The expression of the compressed sensing calculation model is: ; Wherein, To measure the matrix, the initial impedance matrix is obtained, To the final global basis function matrix, To the sensing matrix, , To the measurement excitation vector, the initial excitation vector is obtained, To the sparse coefficient vector of the reconstructed induced current on .
[0032] S32, the current coefficient is solved based on the LSMR-Tikhonov recovery algorithm, and the optimal sparse coefficient vector is obtained; the expression is: ; Wherein, The regularization parameter.
[0033] Specifically, the LSMR-Tikhonov recovery algorithm is the combination of LSMR algorithm and Tikhonov regularization.
[0034] S33, the optimal sparse coefficient vector and the final global basis function matrix are brought into the induced current sparse model to obtain the reconstructed induced current.
[0035] Take the bistatic radar scattering cross section RCS of the almond model with a side length of 1m as an example. The incident excitation is a plane wave with a frequency of 3.5GHz. According to The surface of the almond is divided into 28416 unknowns. The target is divided into 30 blocks. The present method, the traditional block method and the traditional MoM method are used respectively. In the present method, the number of reserved basis functions is 351, which is much smaller than the number of reserved basis functions 10500 of the traditional block method. The present method uses LSMR-Tikhonov algorithm for solving, and the extraction interval is 10. The traditional block method uses least square method for solving, and the extraction interval is 4.
[0036] As shown in Figure 2 , the calculation result of the present method is in good agreement with the traditional MoM, and has high calculation accuracy. Table 1 gives the comparison of calculation time of the present method and the traditional block method. It can be seen that the time of filling matrix and solving is reduced by 19.6% and 87.8% respectively. Although the construction process of basis function is slightly time-consuming, but this also lays a foundation for the stability of subsequent solving. Therefore, the total time is reduced by 14.3%. The example verifies the accuracy and efficiency of the present method.
[0037] Table 1 Comparison of calculation time
[0038] In summary, the application constructs compact basis functions by hierarchical projection, combines the LSMR-Tikhonov recovery algorithm, and improves the solving efficiency and numerical stability of the electromagnetic scattering problem under the premise of ensuring the calculation accuracy. The application has high calculation efficiency and accuracy in solving the bistatic RCS of a three-dimensional electrically large conductor target.
[0039] The application further provides an electromagnetic scattering calculation system based on hierarchical Krylov subspaces, which comprises: The block construction module is configured to perform block processing on the electromagnetic scattering region of the target to obtain a plurality of blocks, and construct initial Krylov subspace basis functions of the blocks to obtain an initial basis function set.
[0040] The function processing module is configured to perform projection, compression and orthogonalization operations on the initial basis function set to obtain a final global basis function matrix.
[0041] The induced current module is configured to construct an induced current sparse model, and obtain reconstructed induced current based on the induced current sparse model and the final global basis function matrix.
[0042] Optionally, the function processing module specifically comprises: The initial basis function set is taken as the first layer global basis function, and the expression of the first layer global basis function is: ; Wherein, is the first layer global basis function, is the initial Krylov subspace basis function of the first block, is the initial Krylov subspace basis function of the second block, is the initial Krylov subspace basis function of the pth block, and p is the total number of blocks.
[0043] The first layer global basis function is projected to obtain a second layer compressed impedance matrix and a second layer compressed excitation vector. The expression of the second layer compressed impedance matrix is: ; Wherein, is the second layer compressed impedance matrix, and T is the transpose, is the first layer compressed impedance matrix, that is, the initial impedance matrix.
[0044] The expression of the second layer compressed excitation vector is: ; Wherein, is the second layer compressed excitation vector, is the first layer compressed excitation vector, that is, the initial excitation vector.
[0045] The initial global basis functions of the second layer are obtained based on the second layer compression impedance matrix and the second layer compression excitation vector.
[0046] The initial global basis functions of the second layer are orthogonalized based on singular value decomposition to obtain the global basis functions of the second layer.
[0047] By analogy, the global basis functions of the kth layer are obtained, and the global basis functions of the 1st layer to the kth layer are combined into the final global basis function matrix.
[0048] Optionally, the sparse expression for the induced current sparse model is: ; in, This is the final global basis function matrix. To reconstruct the induced current, To reconstruct the induced current in A sparse coefficient vector on.
[0049] Optionally, the induced current module specifically comprises: A sparse model of the induced current and a compressed sensing computational model are constructed. The expression of the compressed sensing computational model is as follows: ; in, The measurement matrix is obtained from the initial impedance matrix. This is the final global basis function matrix. For the sensing matrix, , To measure the excitation vector, it is obtained from the initial excitation vector. To reconstruct the induced current in A sparse coefficient vector on.
[0050] Based on the LSMR-Tikhonov recovery algorithm, the current coefficients are analyzed. The optimal sparse coefficient vector is obtained by solving the problem. The expression is: ; in, For regularization parameters; By substituting the optimal sparse coefficient vector and the final global basis function matrix into the sparse model of the induced current, the reconstructed induced current is obtained.
[0051] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.
Claims
1. A method for calculating electromagnetic scattering based on hierarchical Krylov subspaces, characterized in that, It includes: The target electromagnetic scattering region is divided into blocks to obtain several blocks; And construct the initial Krylov subspace basis functions for each of the aforementioned blocks to obtain the initial basis function set; The initial basis function set is projected, compressed, and orthogonalized to obtain the final global basis function matrix; A sparse model of induced current is constructed, and the reconstructed induced current is obtained based on the sparse model of induced current and the final global basis function matrix.
2. The electromagnetic scattering calculation method based on hierarchical Krylov subspaces according to claim 1, characterized in that, The projection, compression, and orthogonalization operations performed on the initial basis function set to obtain the final global basis function matrix include: The initial set of basis functions is used as the first-level global basis functions; the expression for the first-level global basis functions is: ； in, For the first level of global basis functions, Let be the initial Krylov subspace basis functions for the first block. The initial Krylov subspace basis functions for the second block are... Let be the initial Krylov subspace basis function for the p-th block, where p is the total number of blocks; Projecting the first-layer global basis functions yields the second-layer compressed impedance matrix and the second-layer compressed excitation vector; the expression for the second-layer compressed impedance matrix is: ； in, This is the second-level compression impedance matrix, where T is the transpose. This is the first-level compression impedance matrix, i.e., the initial impedance matrix; The expression for the second-layer compression excitation vector is: ; in, This is the compression excitation vector for the second layer. This is the first layer compression excitation vector, i.e., the initial excitation vector; The initial global basis functions of the second layer are obtained based on the second layer compression impedance matrix and the second layer compression excitation vector; The initial global basis functions of the second layer are orthogonalized based on singular value decomposition to obtain the global basis functions of the second layer. By analogy, the global basis functions of the kth layer are obtained, and the global basis functions of the 1st layer to the kth layer are combined into the final global basis function matrix.
3. The electromagnetic scattering calculation method based on hierarchical Krylov subspaces according to claim 2, characterized in that, The sparse expression for the induced current sparse model is: ； in, This is the final global basis function matrix. To reconstruct the induced current, To reconstruct the induced current in A sparse coefficient vector on.
4. The electromagnetic scattering calculation method based on hierarchical Krylov subspaces according to claim 1, characterized in that, The construction of the sparse model of the induced current, and the obtaining of the reconstructed induced current based on the sparse model of the induced current and the final global basis function matrix, includes: Construct the sparse model of the induced current and the compressed sensing computation model; the expression of the compressed sensing computation model is: ； in, The measurement matrix is obtained from the initial impedance matrix. This is the final global basis function matrix. For the sensing matrix, , To measure the excitation vector, it is obtained from the initial excitation vector. To reconstruct the induced current in sparse coefficient vector on; Based on the LSMR-Tikhonov recovery algorithm, the current coefficients are analyzed. Solving this problem yields the optimal sparse coefficient vector; the expression is: ； in, For regularization parameters; The optimal sparse coefficient vector and the final global basis function matrix are substituted into the sparse model of the induced current to obtain the reconstructed induced current.
5. An electromagnetic scattering calculation system based on hierarchical Krylov subspaces, characterized in that, It includes: The block construction module is used to divide the target electromagnetic scattering region into blocks to obtain several blocks; And construct the initial Krylov subspace basis functions for each of the aforementioned blocks to obtain the initial basis function set; The function processing module is used to perform projection, compression, and orthogonalization operations on the initial basis function set to obtain the final global basis function matrix. The induced current module is used to construct a sparse model of induced current. Based on the sparse model of induced current and combined with the final global basis function matrix, the reconstructed induced current is obtained.
6. The electromagnetic scattering calculation system based on hierarchical Krylov subspaces according to claim 5, characterized in that, The function processing module is specifically as follows: The initial set of basis functions is used as the first-level global basis functions; the expression for the first-level global basis functions is: ； in, For the first level of global basis functions, Let be the initial Krylov subspace basis functions for the first block. The initial Krylov subspace basis functions for the second block are... Let be the initial Krylov subspace basis function for the p-th block, where p is the total number of blocks; Projecting the first-layer global basis functions yields the second-layer compressed impedance matrix and the second-layer compressed excitation vector; the expression for the second-layer compressed impedance matrix is: ； in, This is the second-level compression impedance matrix, where T is the transpose. This is the first-level compression impedance matrix, i.e., the initial impedance matrix; The expression for the second-layer compression excitation vector is: ; in, This is the compression excitation vector for the second layer. This is the first layer compression excitation vector, i.e., the initial excitation vector; The initial global basis functions of the second layer are obtained based on the second layer compression impedance matrix and the second layer compression excitation vector; The initial global basis functions of the second layer are orthogonalized based on singular value decomposition to obtain the global basis functions of the second layer. By analogy, the global basis functions of the kth layer are obtained, and the global basis functions of the 1st layer to the kth layer are combined into the final global basis function matrix.
7. The electromagnetic scattering calculation system based on hierarchical Krylov subspaces according to claim 6, characterized in that, The sparse expression for the induced current sparse model is: ； in, This is the final global basis function matrix. To reconstruct the induced current, To reconstruct the induced current in A sparse coefficient vector on.
8. The electromagnetic scattering calculation system based on hierarchical Krylov subspaces according to claim 5, characterized in that, The inductive current module is specifically: Construct the sparse model of the induced current and the compressed sensing computation model; the expression of the compressed sensing computation model is: ； in, The measurement matrix is obtained from the initial impedance matrix. This is the final global basis function matrix. For the sensing matrix, , To measure the excitation vector, it is obtained from the initial excitation vector. To reconstruct the induced current in sparse coefficient vector on; Based on the LSMR-Tikhonov recovery algorithm, the current coefficients are analyzed. Solving this problem yields the optimal sparse coefficient vector; the expression is: ； in, For regularization parameters; The optimal sparse coefficient vector and the final global basis function matrix are substituted into the sparse model of the induced current to obtain the reconstructed induced current.