Method for solving electromagnetic scattering of metal-dielectric composite structure based on M-HODLR

Abstract

The present application provides a kind of based on M-HODLR solving electromagnetic scattering method of metal medium composite structure, to solve the problem of large memory consumption and low accuracy of electromagnetic scattering result when solving electromagnetic scattering of metal medium composite structure.The implementation steps of the present application are: the surface of metal medium composite structure is completely meshed;Define the base function on each mesh element after meshing and sort the base function;Fill the total impedance matrix;Total impedance matrix is blocked using M-HODLR method;Each sub-matrix block after blocking is compressed using enhanced ACA method;Solve the matrix equation of the generated metal medium composite structure and calculate the electromagnetic scattering result.The present application has the advantages of significantly reducing the calculation memory, reducing the solving time and the high accuracy of the electromagnetic scattering result of the metal medium composite structure calculated.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation technology, and further relates to a method for solving electromagnetic scattering of metallic dielectric composite structures based on the higher-order improved hierarchically off-digonal low-rank algorithm M-HODLR in the field of electromagnetic radiation technology. Background Technology

[0002] Metal-dielectric composite structures refer to structures composed of multiple single metals and dielectric targets with different electromagnetic properties, assembled through processes such as layering, embedding, and encapsulation. Solving the electromagnetic scattering (ESP) of metal-dielectric composite structures involves applying excitation to the structure and calculating the scattered electric and magnetic fields. This method has wide applications in microwave network design, antenna array analysis, and remote sensing. In recent years, due to the increasing complexity of metal-dielectric composite structures, solving their ESP has become increasingly memory-intensive, time-consuming, and demanding in terms of accuracy. This presents a significant challenge to solving the ESP of metal-dielectric composite targets.

[0003] Xi'an University of Electronic Science and Technology disclosed a method for rapidly solving electromagnetic scattering of metallic dielectric composite structures based on impedance matrix partitioning in its patent application, "A Fast Calculation Method for Foil Cloud Scattering Based on Impedance Matrix Blocking" (Application No. CN202110033830.4, Publication No. CN 112733364 A). The method's implementation steps are as follows: 1) Inputting foil cloud parameters; 2) Establishing a uniformly distributed foil cloud within a cube of volume V; 3) Constructing, discretizing, and calculating matrix elements of the integral equation based on the method of moments; 4) Simplifying the impedance matrix based on coupling zone determination, and then calculating the current matrix in parallel by partitioning; 5) Compensating for errors based on the coupling between sub-regions, obtaining the total current matrix, and then calculating the scattering characteristics. While this method effectively improves the speed of solving matrix equations, it still has shortcomings. After partitioning the impedance matrix, it directly compensates for errors in the coupling between sub-regions without determining whether the partitioned sub-matrices can be further subdivided. This can result in some sub-matrix blocks remaining very large.

[0004] Solving a system of matrix equations still requires a large amount of computation, takes a long time, and consumes a lot of memory.

[0005] Nanjing University of Aeronautics and Astronautics disclosed a method for rapidly solving the electromagnetic scattering characteristics of locally varying metallic dielectric composite structures in its patent application, "A Method for Rapidly Solving the Electromagnetic Scattering Characteristics of Locally Varying Targets" (Application No. CN201610248772.6, Authorization Announcement No. CN 105955924 B). The method comprises the following steps: 1) Calculating the total impedance matrix and its inverse matrix of the target, and expressing them in the form of block matrices; 2) Using the block matrix inversion formula, obtaining the relationship between the impedance matrix and its inverse matrix of the remaining unchanged part; 3) Using the matrix inversion theorem SMW (Sherman-Morrison-Woodbury) formula, expressing the induced current matrix of the remaining part in a form related to the inverse matrix of the total impedance matrix and the voltage matrix. Although this method can significantly reduce computation time and memory consumption, it still has shortcomings. When calculating the target total impedance matrix and its inverse matrix, the method only calculates some elements in the sub-matrix block. This will result in the inability to traverse all the principal elements in the total impedance matrix when filling it, causing a part of the solution obtained after solving the system matrix equation to be lost. Therefore, the accuracy of the obtained electromagnetic scattering results is low. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of the existing technology by proposing an M-HODLR-based method for solving electromagnetic scattering of metallic-dielectric composite structures. This method aims to solve the problems of high memory consumption and low accuracy of target electromagnetic scattering results when solving electromagnetic scattering of metallic-dielectric composite structures.

[0007] The technical approach to achieving the objective of this invention is to use the M-HODLR method to divide the total impedance matrix of a metallic-dielectric composite structure into blocks until all sub-matrix blocks are low-rank sub-matrix blocks. This yields the total impedance matrix represented by the product of all low-rank sub-matrix blocks, significantly reducing the number of unknowns and thus solving the problem of high memory consumption when solving for electromagnetic scattering in metallic-dielectric composite structures. When compressing the sub-matrix blocks, an enhanced ACA method is used to compress each sub-matrix block after division, using all principal element information of each sub-matrix block to represent it. This solves the problem of low accuracy in electromagnetic scattering results for metallic-dielectric composite structures. Compared with existing technologies, this technique significantly reduces the memory consumption required to solve for the electromagnetic properties of metallic-dielectric composite structures and improves the accuracy of the calculation results.

[0008] The specific steps to achieve the objective of this invention include the following:

[0009] Step 1: Use mesh elements that match the shape of the metal-dielectric composite structure to completely partition the surface of the metal-dielectric composite structure;

[0010] Step 2: Define basis functions on each mesh cell after partitioning. Group the basis functions with the same defined region on the mesh cell into a cluster and sort the basis functions in each cluster.

[0011] Step 3: Calculate the coupling between any two basis functions in the basis function sorting, and form the total impedance matrix from all coupling values;

[0012] Step 4: Divide the total impedance matrix into blocks using the M-HODLR method until all sub-matrix blocks are low-rank sub-matrix blocks, and obtain the total impedance matrix represented by the product of all low-rank sub-matrix blocks.

[0013] Step 5: Compress each sub-matrix block after partitioning using the enhanced ACA method, and express the sub-matrix block using all the principal element information of each sub-matrix block;

[0014] Step 6: Obtain the equivalent electromagnetic current of the metal-dielectric composite structure through the generated matrix equation of the metal-dielectric composite structure;

[0015] Step 7: Calculate the electromagnetic scattering parameters.

[0016] Compared with the prior art, the present invention has the following advantages:

[0017] First, because this invention uses the M-HODLR method to divide the total impedance matrix of the metallic dielectric composite structure into blocks, representing the impedance matrix as the product of all low-rank sub-matrix blocks, the number of unknowns is greatly reduced. This overcomes the problem of high computational memory consumption in existing technologies, giving this invention the advantages of significantly reducing computational memory and solution time.

[0018] Secondly, since the present invention uses the enhanced ACA method to compress each sub-matrix block after partitioning, and uses all the principal element information of each sub-matrix block to express the sub-matrix block, it overcomes the problem that the existing technology loses the principal element information of the sub-matrix block, resulting in low accuracy of solving the electromagnetic scattering results of metal-dielectric composite structures. This makes the present invention have the advantage of high accuracy in solving the electromagnetic scattering results of metal-dielectric composite structures. Attached Figure Description

[0019] Figure 1 This is a flowchart illustrating the implementation of the present invention;

[0020] Figure 2 This invention relates to a composite structure of a dielectric sphere and a PEC cube.

[0021] Figure 3 The X component of the near-field scattered electromagnetic field of the composite structure of dielectric sphere and PEC cube in this invention;

[0022] Figure 4This represents the Z component of the near-field scattered electromagnetic field of the composite structure of the dielectric sphere and PEC cube in this invention. Detailed Implementation

[0023] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0024] Reference Figure 1 The steps for implementing the embodiments of the present invention will be further described below.

[0025] Step 1: Mesh the composite target with metallic media.

[0026] When meshing a metal-medium composite target, the surface of the metal-medium composite structure is completely meshed. Complete meshing means that the meshed structure must maintain the integrity of the metal-medium composite structure.

[0027] In the embodiments of the present invention, the metal-dielectric composite target is as follows: Figure 2 The diagram shows a composite structure consisting of a dielectric sphere and a cube of a perfect electric conductor (PEC). This composite structure is placed in air. The radius of the dielectric sphere in this composite structure is 4m, and its electrical parameters are... =2.0, =1.0, a dielectric sphere encloses a PEC cube with a side length of 3m. The centers of the dielectric sphere and the PEC cube coincide, and their centers are both set as the origin of the coordinate system. Plane wave. Incident along the -Z direction, The wavenumber in free space is used, and the near-field observation points are set at (3.0, 4.0, -5.0~5.0) m, with 201 observation points evenly spaced within the range of (-5.0~5.0 m). In this embodiment of the invention, the Gmsh meshing software is used to mesh the surface of the composite structure composed of a dielectric sphere and PEC cubes. In Gmsh, the mesh element type is set to curved quadrilateral. 6648 curved quadrilateral elements are obtained.

[0028] Step 2: Define basis functions on each mesh cell after partitioning. Group the basis functions with the same defined region on the mesh cell into a cluster and sort the basis functions in each cluster.

[0029] The expressions for the basis functions defined on each unit are:

[0030] ;

[0031] in, These represent the coordinates of the vertices of the grid cells. Indicates the first The basis function at the first vertex along the horizontal axis in each grid cell. Indicates the first The basis function at the second vertex along the horizontal axis in each grid cell. Indicates the first The basis function at the third vertex along the horizontal axis in each grid cell. Indicates the first The basis function at the first vertex along the y-axis in each grid cell. Indicates the first The basis function at the second vertex along the y-axis in each grid cell. Indicates the first The basis functions are located at the third vertex along the y-axis in each grid cell. Based on the distance between the center of the basis function cluster and the origin in the coordinate system of the metallic composite structure, all basis function clusters are sorted by power order. The basis functions within each basis function cluster are then sorted by power order according to the order in which they were defined.

[0032] Step 3: Calculate the coupling between any two basis functions in the basis function sorting, and form the total impedance matrix by combining all coupling values.

[0033] If there are N basis functions in the basis function sorting, the size of the total impedance matrix is ​​NxN. The element in the m-th row and n-th column of the total impedance matrix is ​​obtained after the i-th basis function is coupled with the j-th basis function in the basis function sorting. All the coupled values ​​are combined to form the total impedance matrix, and the values ​​of m and i are equal, and the values ​​of n and j are equal.

[0034] The coupling formula for the basis functions is as follows:

[0035] ;

[0036] ;

[0037] ;

[0038] ;

[0039] ;

[0040] ;

[0041] ;

[0042] ;

[0043] ;

[0044] ;

[0045] ;

[0046] in, Representing vectors and The inner product, Representing the basis functions Representing the basis functions and For the integral operator of the inner region, and For the integral operator of the outer region, It is a uniform space Green's function associated with the equivalent source. The imaginary unit, For normalized angular frequency, For the first Wavenumber of a single target and For the first The relative permittivity and permeability of a single target.

[0047] Step 4: Divide the total impedance matrix into blocks using the M-HODLR method until all sub-matrix blocks are low-rank sub-matrix blocks, and obtain the total impedance matrix represented by the product of all low-rank sub-matrix blocks.

[0048] Step 5: Compress each sub-matrix block after partitioning using the enhanced ACA method, and express the sub-matrix block using all the principal element information of each sub-matrix block.

[0049] The steps of the enhanced ACA method are as follows:

[0050] The first step is to temporarily store all the information in the sub-matrix block. ;

[0051] The second step is to record. The maximum value of the elements that have already been traversed and define =0.5 is used as the threshold value. After reading each element of the submatrix block, select... The largest untraversed element value and examine If the condition is met, then the element can be used as the principal element, and the position of the element is recorded. Each principal element and its corresponding principal element position are combined to form the principal element information of the submatrix block.

[0052] Step 6: Obtain the equivalent electromagnetic current of the metal-dielectric composite structure through the generated matrix equation of the metal-dielectric composite structure.

[0053] The matrix equation of the metal-dielectric composite structure is as follows:

[0054] ;

[0055] ;

[0056] Where Z represents the total impedance matrix of the metallic-dielectric composite structure. , Let each represent the electromagnetic current coefficient matrix to be determined. and Let represent the excitation vectors of the electric and magnetic fields of the metallic-dielectric composite structure, respectively. Represents the first in the basis function sorting basis functions Representing the incident electric field and incident magnetic field, respectively. Representing vectors and The inner product, Representing vectors and The inner product of.

[0057] Step 7: Calculate the electromagnetic scattering parameters.

[0058] The equivalent electromagnetic current of the metallic-dielectric composite structure is calculated using the following formula to obtain the electromagnetic scattering parameters of the metallic-dielectric composite structure:

[0059] ;

[0060] in, This represents the scattered electric field of a metallic-dielectric composite structure. The scattered magnetic field of the metallic-dielectric composite structure Represents the imaginary unit. Represents the normalized angular frequency. Indicates wave number, , Representing relative permittivity and permeability, respectively. This represents the equivalent current on the surface of a metallic-dielectric composite structure. It represents the equivalent magnetic current on the surface of a metallic-dielectric composite structure.

[0061] The effects of this invention will be further illustrated below with simulation experiments:

[0062] 1. Simulation experimental conditions:

[0063] The hardware platform for the simulation experiment of this invention is: Dell T7920 workstation, OMP parallel, 10 threads.

[0064] The software platform for the simulation experiment of this invention is: Windows 10 operating system and Visual Studio 2017 + Intel Visual Fortran 2020.

[0065] The simulation object of this invention is a composite structure composed of a dielectric sphere and an ideal electric conductor (PEC) cube, such as... Figure 2 As shown. This composite structure is placed in air. The radius of the dielectric sphere in the composite structure is 4m, and the electrical parameters are... =2.0, =1.0, a dielectric sphere encloses a PEC cube with a side length of 3m. The centers of the dielectric sphere and the PEC cube coincide, and the center points of both are set as the origin of the coordinate system. Plane wave. Incident along the -Z direction, The wavenumber in free space is given. The near-field observation points are (3.0, 4.0, -5.0~5.0) m, and 201 observation points are set at equal intervals within the range of (-5.0~5.0 m).

[0066] 2. Simulation content:

[0067] The simulation experiment of this invention uses the present invention and two existing technologies to solve the electromagnetic scattering parameters of the composite structure composed of a dielectric sphere and an ideal electrical conductor PEC cubic block. The simulation time and memory consumption of the present invention and existing technologies are compared, as shown in Table 1. The accuracy of the obtained electromagnetic scattering results for the composite structure of the dielectric sphere and PEC cubic block is also shown. The components of the electromagnetic scattering results in the X-axis direction are as follows: Figure 3 As shown, the component of the electromagnetic scattering result in the Z-axis direction is as follows: Figure 4 As shown.

[0068] The two existing technologies used in the simulation experiment are:

[0069] Prior art 1 refers to the method for solving electromagnetic scattering problems based on an improved HODLR proposed by Z. Rong et al. in “Fast direct solution of integral equations with modified HODLR structure for analyzing electromagnetic scattering problems”, IEEE Trans. Antennas Propag., vol. 67, no. 5, pp. 3288–3295, 2019.

[0070] Prior art 2 refers to the method proposed by M. Jiang et al. in “Strong admissibility skeletonization factorization for fast direct solution of electromagnetic scattering from conducting objects,” IEEE Trans. Antennas Propag., vol. 69, no. 10, pp. 6607-6617, 2021., which is based on strong admissibility skeletonization factorization to solve electromagnetic scattering problems.

[0071] 3. Simulation Result Analysis:

[0072] Table 1. Simulation calculation parameters of the composite structure of dielectric sphere and PEC cube

[0073] Prior Art 1 This invention Existing technology 2 Number of quadrilateral units 95160 6648 3264 DOFs (J / M) 190320 / 134832 52476 / 37136 58521 / 43041 Fill time (s) 21189.795 1236.56 1701.18 Find the inverse time (s). 673.0566 101.8673 120.0402 Near-field calculation time (s) 18.68 3.52 4.11 Total simulation time (s) 21811.5316 1341.9473 1825.3302 Memory consumption (GBytes) 69.888 18.405 28.818

[0074] Table 1 records some important calculation parameters when solving for electromagnetic scattering in the dielectric sphere and PEC cubic composite structure. The number of quadrilateral elements in Table 1 refers to the number of elements obtained by meshing the surface of the dielectric sphere and PEC cubic composite structure. The DOF (J / M) in Table 1 refers to the number of equivalent currents and equivalent magnetic currents to be calculated in the matrix equations for constructing the dielectric sphere and PEC cubic composite structure. The fill time in Table 1 refers to the time required to fill the total impedance matrix of the dielectric sphere and PEC cubic composite structure. The inversion time in Table 1 refers to the time required to calculate the inverse of the total impedance matrix. The near-field calculation time in Table 1 refers to the time required to calculate the electromagnetic scattering of the dielectric sphere and PEC cubic composite structure at the observation point. The sum of the fill time, inversion time, and near-field calculation time is the total simulation time. The simulation time of prior art 1 is 21811.5316 seconds and the memory consumption is 69.888 GBytes. The simulation time of prior art 2 is 1825.3302 seconds and the memory consumption is 28.818 GBytes. In contrast, the simulation time of this invention is 1341.9473 seconds and the memory consumption is 18.405 GBytes. Comparing the calculation parameters of this invention with those of prior art, it is found that this invention has a shorter simulation time and less memory consumption when calculating the electromagnetic scattering of the composite structure of the dielectric sphere and PEC cube.

[0075] The following is combined with Figure 3 and Figure 4 The simulation results further illustrate the electromagnetic scattering results calculated by this invention.

[0076] Reference Figure 3 The horizontal axis represents the Z-coordinate of the observation point, ranging from -5.0 to 5.0, in meters. (Refer to...) Figure 4 The horizontal axis represents the Z-coordinate of the observation point, ranging from -5.0 to 5.0, in meters. Figure 3 (a) The vertical axis represents the real part of the electromagnetic scattering of the composite structure of the dielectric sphere and PEC cube in the X-axis direction. Figure 3 (b) The vertical axis represents the imaginary part of electromagnetic scattering in the X-axis direction of the composite structure of dielectric sphere and PEC cube. Figure 4 (a) The vertical axis represents the real part of the electromagnetic scattering of the composite structure of dielectric sphere and PEC cube in the Z-axis direction. Figure 4 (b) The vertical axis represents the imaginary part of electromagnetic scattering in the Z-axis direction of the composite structure of dielectric sphere and PEC cube. Figure 3 and Figure 4The 1st-proposed curve represents the electromagnetic scattering result of the dielectric sphere and PEC cubic composite structure calculated by prior art 1, the 2nd-original curve represents the electromagnetic scattering result of the dielectric sphere and PEC cubic composite structure calculated by prior art 2, and the 2nd-proposed curve represents the electromagnetic scattering result of the dielectric sphere and PEC cubic composite structure calculated by the present invention. Comparing the results of the present invention with those of the prior art in the figures, the accuracy of the calculation results of the present invention is superior to that of the prior art.

[0077] This invention presents an M-HODLR-based method for solving electromagnetic scattering in metallic dielectric composite structures. This method overcomes the problems of high memory consumption and low accuracy of electromagnetic scattering results in existing technologies. As a result, this invention has the advantages of significantly reducing computational memory, reducing solution time, and achieving higher accuracy of calculation results.

[0078] The above description of the embodiments is intended to enable those skilled in the art to understand and apply the technology of this application. Those skilled in the art can readily make various modifications to these examples and apply the general principles described herein to other embodiments without creative effort. Therefore, this application is not limited to the above embodiments, and any improvements and modifications made to this application by those skilled in the art based on the disclosure herein should be within the scope of protection of this application.

Claims

1. A method for solving electromagnetic scattering in metallic dielectric composite structures based on M-HODLR, characterized in that, The total impedance matrix of the metallic-dielectric composite structure is divided into blocks using the M-HODLR method, and each off-diagonal sub-matrix block is compressed using the enhanced ACA method. The steps of this electromagnetic scattering solution method are as follows: Step 1: Use mesh elements that match the shape of the metal-dielectric composite structure to completely partition the surface of the metal-dielectric composite structure; Step 2: Define basis functions on each mesh cell after partitioning. Group the basis functions with the same defined region on the mesh cell into a cluster, and sort the basis functions in each basis function cluster. Step 3: Calculate the coupling between any two basis functions in the basis function sorting, and form the total impedance matrix from all coupling values; Step 4: Divide the total impedance matrix into blocks using the M-HODLR method until all sub-matrix blocks are low-rank sub-matrix blocks, and obtain the total impedance matrix represented by the product of all low-rank sub-matrix blocks. Step 5: Compress each sub-matrix block after partitioning using the enhanced ACA method, and express the sub-matrix block using all the principal element information of each sub-matrix block; The steps of the enhanced ACA method are as follows: The first step is to temporarily store all the information in the sub-matrix block. ; The second step is to record. The maximum value of the elements that have already been traversed and define =0.5 is used as the threshold value. After reading each element of the submatrix block, select... The largest untraversed element value and examine If the condition is met, the element is taken as the pivot and its position is recorded. Each pivot and its corresponding pivot position are combined to form the pivot information of the sub-matrix block. Step 6: Obtain the equivalent electromagnetic current of the metal-dielectric composite structure through the generated matrix equation of the metal-dielectric composite structure; Step 7, calculate electromagnetic scattering parameters: ； in, This represents the scattered electric field of a metallic-dielectric composite structure. The scattered magnetic field of the metallic-dielectric composite structure The symbol representing the imaginary unit. Represents the normalized angular frequency. Represents the vacuum wavenumber. , Representing relative permittivity and permeability, respectively. This represents the equivalent current on the surface of a metallic-dielectric composite structure. It represents the equivalent magnetic current on the surface of a metallic-dielectric composite structure.

2. The electromagnetic scattering method for solving metallic-dielectric composite structures based on M-HODLR according to claim 1, characterized in that, The complete meshing mentioned in step 1 means that the meshed mesh must maintain the integrity of the metallic dielectric composite structure.

3. The electromagnetic scattering method for solving metallic-dielectric composite structures based on M-HODLR according to claim 1, characterized in that, The basis functions mentioned in step 2 are as follows: ； in, These represent the coordinates of the vertices of the grid cells. Indicates the first The basis function at the first vertex along the horizontal axis in each grid cell. Indicates the first The basis function at the second vertex along the horizontal axis in each grid cell. Indicates the first The basis function at the third vertex along the horizontal axis in each grid cell. Indicates the first The basis function at the first vertex along the y-axis in each grid cell. Indicates the first The basis function at the second vertex along the y-axis in each grid cell. Indicates the first The basis function at the third vertex along the y-axis in each grid cell.

4. The electromagnetic scattering method for solving metallic-dielectric composite structures based on M-HODLR according to claim 1, characterized in that, The sorting of basis functions in each basis function cluster mentioned in step 2 refers to sorting all basis function clusters by power order based on the distance between the center of the basis function cluster and the origin in the coordinate system of the metallic dielectric composite structure, and sorting the basis functions in each basis function cluster by power order based on the order in which the basis functions were defined.

5. The electromagnetic scattering method for solving metallic-dielectric composite structures based on M-HODLR according to claim 1, characterized in that, The coupling described in step 3 is shown in the following equation. ； ； ； ； ； ； ； ； ； ； ； in, Representing vectors and The inner product, Indicates the first basis functions Indicates the first basis functions and The integral operator representing the inner region, and Indicates the integral operator for the outer region. Represents the uniform space Green's function associated with the equivalent source. The symbol representing the imaginary unit. Represents the normalized angular frequency. Indicates the first Wavenumber of a single target and They represent the first The relative permittivity and permeability of a single target.

6. The electromagnetic scattering method for solving metallic-dielectric composite structures based on M-HODLR according to claim 1, characterized in that, The matrix equation of the metal-dielectric composite structure described in step 6 is as follows: ； ； Where Z represents the total impedance matrix of the metallic-dielectric composite structure. , Let f(x) represent the equivalent electromagnetic current matrices of the metallic-dielectric composite structure to be determined. and Let represent the excitation vectors of the electric and magnetic fields of the metallic-dielectric composite structure, respectively. Represents the first in the basis function sorting basis functions Representing the incident electric field and incident magnetic field, respectively. Representing vectors and The inner product of.

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