Electromagnetic scattering fast calculation method and device, terminal and storage medium

Abstract

The application provides a kind of electromagnetic scattering fast calculation method, device, terminal and storage medium, belong to electromagnetic field simulation calculation field, wherein the method includes: S1, medium partition is carried out to target and each region is given material parameter;S2, generate table area region set, and the common boundary is classified as medium-medium interface set and ideal conductor surface set;S3, define the Green function kernel corresponding to its wave number;S4, construct unified boundary integral equation set;S5, expand unknown equivalent surface flow, and discrete continuous boundary integral equation set into matrix form linear system;S6, matrix form linear system decomposition and encapsulation;S7, solve to generate global unknown vector;S8, the far field electric field of target and the RCS available result of calculation.The electromagnetic scattering fast calculation method, device, terminal and storage medium of the application can improve the accuracy and calculation efficiency in electromagnetic field simulation calculation.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic field simulation calculation, and in particular relates to a fast electromagnetic scattering calculation method, device, terminal and storage medium. Background Technology

[0002] Numerical simulation of electromagnetic fields of complex targets is mainly used to evaluate observable measurements such as the scattering field and RCS of complex structures such as aircraft, ships and vehicles under electromagnetic wave illumination in a virtual environment. It can provide repeatable "numerical experiments" for engineering problems such as stealth design, antenna layout and electromagnetic compatibility.

[0003] In practical engineering, the method of moments (MoM) is a commonly used method for calculating electromagnetic scattering. However, when dealing with complex targets, since real-world targets often contain both ideal conductors and multiple media, traditional methods require establishing separate equation systems for different regions and then solving them in a coupled manner. This leads to complex modeling and introduces interface errors. Furthermore, the dense matrix generated by the traditional MoM requires large storage space, has high direct solution complexity, and exhibits slow convergence of the iterative algorithm in strongly coupled structures, near resonant frequencies, and wide-band calculations. Therefore, it cannot meet the engineering requirements for electromagnetic simulation of large-scale complex targets. Summary of the Invention

[0004] In view of this, the present invention aims to provide a fast calculation method, device, terminal and storage medium for electromagnetic scattering to solve the above-mentioned technical problems.

[0005] To achieve the above objectives, the technical solution created by this invention is implemented as follows: In a first aspect, embodiments of the present invention provide a fast calculation method for electromagnetic scattering, including: S1. Obtain the three-dimensional geometric model and simulation excitation parameters of the target, divide the target into medium partitions and assign material parameters to each region, and mark the regions that meet the ideal conductor conditions as ideal conductor boundaries; S2. Mesh the three-dimensional geometric model of the target, divide the entire computational domain into multiple surface regions, and generate a set of surface regions; calculate the complex permittivity, wave number, and wave impedance of each surface region based on the material parameters of each region; traverse the common boundary of all mesh elements, classify the common boundary into a set of dielectric-dielectric interfaces and a set of ideal conductor surfaces according to the material differences between the regions on both sides of the common boundary, and record the surface region index pairs and unit normal vectors on both sides of each dielectric-dielectric interface; S3. Define a Green's function kernel corresponding to the wavenumber for each table domain region, and define single-layer operators and two-layer coupled operators based on the Green's function kernel; S4. Based on the set of dielectric-dielectric interfaces and the set of ideal conductor surfaces, a unified set of boundary integral equations is constructed. Among them, the PMCHWT coupled integral equation is established based on the continuity condition at the dielectric-dielectric interface, and the linear combination equation CFIE of EFIE and MFIE is established at the boundary of the ideal conductor surface. S5. On the grid divided in step S2, the unknown equivalent surface flow is expanded using RWG basis functions, and the continuous boundary integral equations in step S4 are discretized into a linear system in matrix form using Galerkin projection. S6. Decompose the linear system in matrix form obtained in step S5 into the sum of the near-field matrix and the far-field matrix, and encapsulate it into a first black-box operator that can be called by the iterative solver. The near-field matrix is ​​calculated directly through explicitly stored matrix blocks, and the far-field matrix is ​​calculated through the matrix free fast algorithm. S7. The generalized minimum residual iteration method is used to solve the linear system in matrix form. During the iteration process, the first black box operator is called to perform matrix-vector multiplication, and the second black box operator corresponding to the preconditioner constructed based on the near field matrix is ​​called to preprocess the iteration residual. Finally, the global unknown vector is generated. S8. Reconstruct the spatial distribution of equivalent surface current and equivalent surface magnetic current based on the global unknown vector, and calculate the target's far-field electric field and RCS.

[0006] Furthermore, the material parameters include dielectric constant, magnetic permeability, and electrical conductivity; In step S2, the complex permittivity, wavenumber, and wave impedance of each surface region are calculated using the following formula: ; ; ; In the above formula, , and These are the respective table regions The dielectric constant, permeability, and conductivity of are . Complex permittivity, For wave number, For wave impedance, Angular frequency, It is the imaginary unit.

[0007] Furthermore, in step S3, the Green's function kernel is defined as: ; Among them Field point position vector, The source point position vector, The distance between two points; The single-layer operator and the two-layer coupled operator defined based on the Green's function kernel are used to express the scattering field within the surface domain, and their mathematical expressions are as follows: ; In the above formula, It is a single-layer operator. It is a two-layer coupled operator. and These respectively represent the fields in the table. The equivalent current density and equivalent magnetic flux density Operator mapping of the generated scattered electric field and scattered magnetic field.

[0008] Furthermore, in step S4, establishing the PMCHWT coupled integral equation at the medium-medium interface based on the continuity condition includes: For media-media interface Introducing surface current With surface magnetic current Based on the physical boundary conditions at the interface constraining the continuity of the electric and magnetic fields, the equivalent current and equivalent magnetic current are expressed by the following formulas: ; in, and For the interface equivalent source in the table domain region The scattered field components generated in the process, and Let the incident electric field and incident magnetic field be respectively. Substituting the above scattering field operator expression into the continuity condition, we can obtain the medium-medium interface. Coupled integral equations: ; The establishment of the linear combination equation CFIE of EFIE and MFIE at the boundary of the ideal conductor surface includes: For the boundary of an ideal conductor surface Retaining only the equivalent surface current position, the calculation employs a linear combination of EFIE and MFIE to form a combined two-field integral equation, with the total tangential electric field being zero. The resulting linear combined equation CFIE is as follows: ; In the above formula, For combination coefficients, It represents the external dielectric wave impedance.

[0009] Furthermore, in step S5, the unknown equivalent surface flow includes equivalent surface current and equivalent surface magnetic current. Expanding the unknown equivalent surface flow using RWG basis functions on the mesh defined in step S2 includes: Construct a set of vector basis functions on the triangular mesh divided in step S2. The vector basis function set adopts the RWG basis function; the mathematical expressions for the equivalent surface current and equivalent surface magnetic current are as follows: ; In the above formula, and The complex coefficients to be determined; For an unknown current on an ideal conductor surface The same basis function system as the equivalent surface current is used for expansion and incorporated into the same current coefficient vector, thereby defining the global unknown vector: ; The discretization of the continuous boundary integral equations in step S4 into a linear system in matrix form using Galerkin projection includes: A linear system in matrix form is obtained through Galerkin projection. , where the matrix elements are The right-hand vector elements are The inner product is defined as .

[0010] Furthermore, in step S6, the far-field matrix is ​​calculated using a matrix-free fast algorithm, including calculating the far-field matrix using the MLFMA algorithm or the H-matrix algorithm.

[0011] Furthermore, in step S8, the formulas for calculating the far-field electric field and RCS using available results are as follows: ; ; In the above formula, For the far-field electric field, This is the available result for RCS.

[0012] Secondly, embodiments of the present invention also provide a fast electromagnetic scattering calculation device, comprising: The input preprocessing module is used to obtain the three-dimensional geometric model and simulation excitation parameters of the target, divide the target into medium partitions and assign material parameters to each region, and mark the regions that meet the ideal conductor conditions as ideal conductor boundaries; The mesh generation and classification module is used to mesh the 3D geometric model of the target, dividing the entire computational domain into multiple surface regions and generating a set of surface regions. Based on the material parameters of each region, it calculates the complex permittivity, wave number, and wave impedance of each surface region. It traverses the common boundary of all mesh cells and classifies the common boundary into a set of dielectric-dielectric interfaces and a set of ideal conductor surfaces according to the material differences between the regions on both sides of the common boundary. It also records the surface region index pairs and unit normal vectors on both sides of each dielectric-dielectric interface. The kernel and integral definition module is used to define the Green's function kernel corresponding to the wavenumber for each table domain region, and to define single-layer operators and two-layer coupled operators based on the Green's function kernel; The equation construction module is used to construct a unified set of boundary integral equations based on the set of dielectric-dielectric interfaces and the set of ideal conductor surfaces. Specifically, the PMCHWT coupled integral equation is established based on the continuity condition at the dielectric-dielectric interface, and the linear combination equation CFIE of EFIE and MFIE is established at the boundary of the ideal conductor surface. The discretization module is used to expand the unknown equivalent surface flow using RWG basis functions and discretize the continuous boundary integral equations into a linear system in matrix form using Galerkin projection. The matrix computation encapsulation module is used to decompose a linear system in matrix form into the sum of a near-field matrix and a far-field matrix, and encapsulate it into a first black-box operator that can be called by the iterative solver. The near-field matrix is ​​directly calculated through explicitly stored matrix blocks, and the far-field matrix is ​​calculated through a matrix-free fast algorithm. The iterative solution module is used to solve linear systems in matrix form using the generalized minimum residual iterative method. During the iteration process, the first black-box operator is called to perform matrix-vector multiplication, and the second black-box operator corresponding to the preconditioner constructed based on the near-field matrix is ​​called to preprocess the iteration residuals. Finally, the global unknown vector is generated. The output module is used to reconstruct the spatial distribution of the equivalent surface current and the equivalent surface magnetic current based on the global unknown vector, and to calculate the available RCS results of the target.

[0013] Thirdly, embodiments of the present invention also provide a terminal, including: One or more processors; Storage device for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement the fast electromagnetic scattering calculation method provided in the above embodiments.

[0014] Fourthly, embodiments of the present invention also provide a storage medium containing computer-executable instructions, which, when executed by a computer processor, are used to perform the fast electromagnetic scattering calculation method provided in the above embodiments.

[0015] Compared with existing technologies, the electromagnetic scattering fast calculation method, device, terminal, and storage medium described in this invention have the following advantages: This invention provides a fast electromagnetic scattering calculation method, device, terminal, and storage medium. By classifying common boundaries, it transforms the mixed medium distribution in actual simulations into a unified expression of surface parameters and boundary sets, facilitating the simultaneous processing of different types of interfaces in subsequent steps. Furthermore, it expands the equivalent surface current at the medium-medium interface and the unknown current at the ideal conductor surface using RWG basis functions, forming a natural block coupling structure. Galerkin projection is then used to discretize the continuous boundary integral equations into mechanically executable block assembly templates, providing a consistent data layout and interface for subsequent fast matrix-vector multiplication and block preconditioning. Additionally, it decomposes the matrix-form linear system into a fast algorithm of direct near-field explicit sparse matrix multiplication and far-field "matrix-free" operation to reduce complexity. A near-field driven preconditioner constructed from the near-field impedance submatrix is ​​introduced in the GMRES iteration, reducing the number of iteration steps and improving the robustness of strong coupling and convergence across frequency points. Ultimately, it efficiently obtains the far-field electric field and RCS outputs, thus meeting practical application requirements. Attached Figure Description

[0016] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings: Figure 1 A flowchart of the fast electromagnetic scattering calculation method described in Embodiment 1 of this invention is provided; Figure 2 A schematic diagram of the structure of the electromagnetic scattering fast computing device described in Embodiment 2 of the present invention; Figure 3 A schematic diagram of the terminal structure described in Embodiment 3 of the present invention. Detailed Implementation

[0017] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, the accompanying drawings show only the parts relevant to the present invention, and not all of the structures.

[0018] Example 1 Figure 1 The flowchart of the fast electromagnetic scattering calculation method provided in Embodiment 1 of the present invention is shown in the figure. In this embodiment, the fast electromagnetic scattering calculation method specifically includes the following steps: Step S1: Obtain the three-dimensional geometric model and simulation excitation parameters of the target, divide the target into medium partitions and assign material parameters to each region, and mark the regions that meet the ideal conductor conditions as ideal conductor boundaries.

[0019] When performing numerical simulations of the electromagnetic field of a target, the first step is to obtain a three-dimensional geometric model of the target (e.g., an aircraft, ship, or vehicle). Example three-dimensional geometric models can be constructed using computer-aided design software (e.g., CAD) or imported from a model library. Subsequently, the geometric model undergoes necessary preprocessing, such as repairing gaps, overlaps, or sharp corners, and simplifying unnecessary fine features to ensure it meets the usability requirements for subsequent mesh generation and topology analysis.

[0020] Furthermore, the simulation excitation parameters in this embodiment should be determined based on the pre-input simulation conditions and preset excitation parameters. The simulation conditions may include, for example, the frequency range of the simulation, the excitation type of the incident electromagnetic wave (e.g., plane wave), the wave polarization (e.g., linear polarization, circular polarization), and the incident direction (azimuth and elevation angles). These parameters can be used in subsequent processing to define, for example, the incident electric field. and incident magnetic field The parameters required for simulation processing.

[0021] After preprocessing the 3D geometric model, since different regions of common complex targets may be made of different materials, this embodiment will divide the target into medium partitions according to the regions and assign material parameters to each region. At the same time, surfaces in the target that meet the conditions of a Perfect Electric Conductor (PEC) (such as metal skin) will be marked as ideal conductor boundaries (i.e., PEC boundaries) for subsequent processing.

[0022] It should be noted that the material parameters in this embodiment may include the dielectric constant. magnetic permeability and conductivity The aforementioned material parameters facilitate the representation of the electromagnetic properties of different materials in subsequent processing, thereby improving the accuracy of subsequent calculations.

[0023] After the above processing is completed, the preprocessed 3D geometric model, the material parameters of each partition (the material parameters can be converted into corresponding material identifiers and added to the partition results), the ideal conductor boundary annotation results, and the simulation excitation parameters can be uniformly packaged into a structured input data package as input for subsequent steps. This completes the transformation from the original geometry to a computable physical model and provides complete initial conditions for the entire simulation process.

[0024] Step S2: Mesh the three-dimensional geometric model of the target, dividing the entire computational domain into multiple surface regions and generating a set of surface regions; calculate the complex permittivity, wave number, and wave impedance of each surface region based on the material parameters of each region; traverse the common boundary of all mesh cells, classify the common boundary into a set of dielectric-dielectric interfaces and a set of ideal conductor surfaces according to the material differences between the regions on both sides of the common boundary, and record the surface region index pairs and unit normal vectors on both sides of each dielectric-dielectric interface.

[0025] After completing step S1, this embodiment will perform surface triangular meshing on the preprocessed 3D geometric model to generate a surface mesh for numerical calculation. Subsequently, the material parameters corresponding to each region in each mesh element are analyzed, and based on the bonding information and material parameters of the geometric model surface, the entire computational domain is divided into multiple surface regions, and a set of surface regions is generated, denoted as: ; in, Indicates the first Each table region, Indicates the number of table ranges.

[0026] Next, at a given simulation angular frequency, and considering the material parameters of each surface region, we calculate its complex permittivity, wave number, and wave impedance. The specific calculation formulas are as follows: ; ; ; In the above formula, , and These are the respective table regions The dielectric constant, permeability, and conductivity of are . Complex permittivity, For wave number, For wave impedance, Angular frequency, The unit is the imaginary unit. Since the complex permittivity, wave number, and wave impedance encapsulate the influence of the material properties in this region on electromagnetic wave propagation, they will serve as the basis for subsequently constructing the kernel function and assembling the system matrix.

[0027] Simultaneously, this embodiment also performs boundary classification processing in step S2. Specifically, during processing, the common boundaries of all mesh cells are traversed, and the material differences are determined based on the differences in material parameters of the surface regions to which the mesh cells on both sides belong. The common boundaries are then divided into a dielectric-dielectric interface set and an ideal conductor surface set according to preset rules, denoted as follows: ; in, This represents the m-th medium-medium interface, where the material parameters of the adjacent surface regions on both sides are different. Let π represent the πth boundary labeled as an ideal conductor (PEC). The above division transforms the mixed medium distribution into an expression of domain parameters + boundary set, which facilitates subsequent unified processing.

[0028] In addition, this embodiment addresses each medium-to-medium interface. It will also record the index pairs of the table regions on both sides: ; in, and These represent the table region numbers corresponding to the positive and negative sides of the interface normal vector, respectively.

[0029] At the same time, for each medium-medium interface It will also define a fixed unit normal vector. Its direction is agreed to be from the region Pointing to area (i.e., satisfying the direction convention) This direction remains unchanged in all subsequent integral calculations to ensure consistency of flux and integral sign.

[0030] Step S3: Define a Green's function kernel corresponding to the wavenumber for each table region, and define single-layer operators and two-layer coupled operators based on the Green's function kernel.

[0031] After obtaining the surface region index topology and the complex permittivity, wavenumber and wave impedance of each surface region, this embodiment will define a unique field propagation kernel and mapping operator for each surface region.

[0032] Specifically, firstly, a Green's function kernel corresponding to the wavenumber is defined for each surface region. This kernel function describes the propagation behavior of the scalar potential generated by the point source in a homogeneous medium with the material properties of that region. It is defined as a free-space scalar kernel, and its mathematical expression is as follows: ; Among them, Field point position vector, The source point position vector, The distance between two points is given. By defining a kernel function for each region, the electromagnetic wave propagation characteristics of different material regions (using...) can be optimized. (For characterization) is directly incorporated into the core parameters of the kernel function.

[0033] To construct the corresponding boundary integral operator for the surface region based on the above kernel function, at least two types of basic operators need to be defined, namely single-layer operators and double-layer coupled operators, in order to express the scattering field within the surface region.

[0034] Among them, the single-layer operator is denoted as The two-layer coupled operator is denoted as Both single-layer and two-layer coupled operators are determined by the kernel function and its spatial derivative. The specific integral expressions of these operators can be determined according to the boundary integral equation formulas used (such as the Stratton-Chu formula), and their function is to map the equivalent surface source (current or magnetic current) defined on the boundary of the region to the field contribution inside the region.

[0035] Using the above operators, the table domain region can be defined. The expression for the scattered field generated by the equivalent source. Introducing the equivalent current density defined on the boundary. With equivalent magnetic flux density Then, the scattered electric field component and the scattered magnetic field component generated in this region can be uniformly expressed as: ; In the above formula, It is a single-layer operator. It is a two-layer coupled operator. and These respectively represent in the table field The equivalent current density and equivalent magnetic flux density Operator mapping of the generated scattered electric field and scattered magnetic field.

[0036] Through the above processing, regional material differences can be represented as... Encapsulated in the form of a regional core and the operators derived from them , This enables subsequent communication at any medium-to-medium interface. When constructing the boundary equations, there is no need to perform separate branch calculations for different materials; instead, only the regions on both sides of the interface need to be indexed. Substitute the operators into the corresponding side table domains respectively and It can calculate the field contribution under different material conditions, thus greatly improving the algorithm's versatility, modularity, and maintainability.

[0037] Step S4: Based on the set of dielectric-dielectric interfaces and the set of ideal conductor surfaces, construct a unified set of boundary integral equations; among which, the PMCHWT coupled integral equation is established at the dielectric-dielectric interface based on the continuity condition, and the linear combination equation CFIE of EFIE and MFIE is established at the boundary of the ideal conductor surface.

[0038] After the above processing is completed, this embodiment will construct a unified set of boundary integral equations based on the dielectric-dielectric interface set and the ideal conductor surface set, so as to facilitate subsequent coupled calculations.

[0039] Specifically, for media-media interfaces Introducing surface current With surface magnetic current Based on the physical boundary conditions at the interface constraining the continuity of the electric and magnetic fields, the equivalent current and equivalent magnetic current are expressed by the following formulas: ; in, and For the interface equivalent source in the table domain region The scattered field components generated in the process, and Let the incident electric field and incident magnetic field be respectively. Substituting the above scattering field operator expression into the continuity condition, we can obtain the medium-medium interface. Coupled integral equations: ; The equation clearly shows that the interface equation is obtained by combining the contributions of the upper and lower region operators using a fixed-symbol template. This structured combination method lays the foundation for the naturally regular block structure that emerges after the system matrix is ​​discretized, which is beneficial for improving the computational and processing efficiency of large-scale problems.

[0040] For the set of ideal conductor surfaces At each boundary in the equation, the total tangential electric field on the surface is zero, and only the equivalent surface current exists. As an unknown, to suppress the instability caused by the internal resonance of the closed conductor structure and to improve the convergence of the iterative solution and the continuity of the cross-frequency calculation, this method adopts the combined field integral equation (CFIE).

[0041] First, the tangential form of its electric field integral equation (EFIE) is as follows: ; Its magnetic field integral equation (MFIE) takes the following form: ; in, and For operators corresponding to free space (or external medium), It represents the identity operator.

[0042] The EFIE and MFIE equations are linearly combined to form the CFIE equations: ; In the above formula, For combination coefficients, The external dielectric impedance is represented by MFIE components. By introducing MFIE components, CFIE effectively eliminates internal resonant modes, making it particularly suitable for solving electromagnetic scattering problems of large-sized and complex-shaped conductors.

[0043] Next, this embodiment will unify and number all PMCHWT equations at dielectric-dielectric interfaces and CFIE equations at ideal conductor boundaries, forming a complete global boundary integral equation system. This represents the complete set of unknowns for the system. Includes all media interfaces For and all PEC surfaces It can be uniformly represented as: ; This entire system can be abstracted and represented in the form of operator equations: ; in, For each region operator and in accordance with The combined global operator mapping, For the entrance field and The global excitation term is formed by tangential projection. The above processing can avoid interface errors caused by multi-system coupling; at the same time, the discretization naturally yields a block structure, providing a direct basis for subsequent fast matrix-vector multiplication and block preconditioning, thus improving the efficiency and robustness of large-scale solutions.

[0044] Step S5: Expand the unknown equivalent surface flow on the mesh divided in step S2 using RWG basis functions, and discretize the continuous boundary integral equations in step S4 into a linear system in matrix form using Galerkin projection.

[0045] In step S5 of this embodiment, the system is transformed into a finite-dimensional linear algebraic system that can be solved by a computer through numerical discretization.

[0046] Specifically, firstly, based on the partitioning results in step S2, a set of Rao-Wilton-Glisson (RWG) vector basis functions should be constructed, denoted as... ,in This represents the total number of basis functions. RWG basis functions are defined on the edges of the mesh and have good tangential continuity and divergence properties, making them very suitable for expanding surface current-like unknowns in surface integral equations.

[0047] For topologically complex structures (such as meshes with three key points or multiple intersecting regions sharing edges), the system can share the support domains and associated degrees of freedom of the basis functions based on a preset strategy. By merging duplicate degrees of freedom and introducing constraints, the consistency of the distribution of unknowns at the interface is ensured, effectively reducing the risk of numerical ill-conditioning or spurious solutions caused by inconsistencies in mesh topology.

[0048] The unknown equivalent surface flow is expanded using the aforementioned RWG basis functions. It should be noted that the unknown equivalent surface flow provided in this embodiment includes the equivalent surface current. and equivalent surface magnetic current Its specific expanded expression is as follows: ; in, and The coefficients to be determined are denoted as .

[0049] For the unknown current at the PEC surface It can be adopted with The same basis function system is expanded and incorporated into the same current coefficient vector, thereby defining a global unknown vector: ; By incorporating the current / magnetic flux degrees of freedom at the dielectric-dielectric interface and the ideal conductor interface into a unified numbering system, a block coupling structure corresponding to the physical equations in step S4 is naturally formed at the matrix level. This provides a consistent and efficient data layout and access interface for the subsequent implementation of fast matrix-vector multiplication and block preconditioning techniques.

[0050] Next, this step will discretize the continuous boundary integral equations using the Galerkin method. The test function and the expansion basis functions are the same, both being... Apply both sides of the global equation to the test function. Perform inner product projection to transform the continuous boundary integral equation system. Consider the following dense linear algebraic system: ; matrix elements Test basis functions Operators acting on source basis functions The inner product is obtained and can be uniformly written as: ; Right-hand vector elements We obtain this from the projection of the incident excitation term: ; The preferred definition of inner product is: ; in This indicates the interface where the corresponding equation is located.

[0051] Because PMCHWT contains at each media-media interface and The coupling terms, the discrete matrix Naturally presents a 2×2 block structure: ; Each block matrix is ​​obtained by combining the operators of the two sides, and the two components of the right-hand vector are obtained by projecting the incident field: ; If the PEC plane CFIE is included, the corresponding row block is obtained by projecting the linear combination of CFIE, EFIE+MFIE, and is in the same block structure as described above. Within the framework assembly, the templated connection method of the upper and lower region contributions in the above steps, after Galerkin projection, is directly transformed into a mechanically executable block assembly template, allowing different materials and different calculation processes to be assembled only through indexes. The corresponding matrix block contribution can be obtained, which significantly reduces the complexity of multi-media systems.

[0052] Step S6: Decompose the linear system in matrix form obtained in step S5 into the sum of the near-field matrix and the far-field matrix, and encapsulate it into a first black-box operator that can be called by the iterative solver. The near-field matrix is ​​calculated directly through explicitly stored matrix blocks, and the far-field matrix is ​​calculated through a matrix-free fast algorithm.

[0053] After completing the aforementioned steps, this step will first decompose the linear system in matrix form according to its near-field and far-field contributions, and then encapsulate the decomposition results into a first black-box operator that can be called by an iterative solver (such as GMRES). This allows the core operator required for solving the linear system to function without generating the full matrix. ; in, As the near-field matrix, it explicitly stores and represents the strong coupling interactions between all spatially close basis function pairs. These interactions cannot be approximated by low-rank and must be calculated precisely. The far-field matrix represents the weak coupling interactions between all the basis function pairs that are far apart. These interactions are compressible and suitable for approximation using fast algorithms. This is the first black-box operator.

[0054] Since iterative solvers only need to rely on a unified interface To obtain The result is such that the solver is transparent, as it is unaware of the storage details of the underlying matrices. Furthermore, this process concentrates the high computational cost on the controllable, precisely processed near-field matrices, while allowing the computationally intensive far-field matrices to be processed through scalable, fast algorithms, thus improving computational efficiency.

[0055] It should be noted that in this embodiment, the near-field contribution calculation can be directly performed by calling the standard Sparse Matrix-Vector Multiplication (SpMV) routine, where the near-field matrix is ​​explicitly assembled and sparsely stored. This calculation logic is deterministic, numerically stable, and can be efficiently reused. The far-field contribution calculation, instead of performing direct double numerical integration on each pair of far-field basis functions, uses matrix-free fast algorithms, such as the MLFMA algorithm or the H-matrix algorithm. The core idea of ​​these algorithms is to utilize the compressibility of far-field interactions to transform dense multiplications between many basis function pairs into a series of low-rank block aggregation-transfer-placement operations, thereby significantly reducing computational complexity.

[0056] Furthermore, to adapt to multi-medium electromagnetic problems, this step explicitly models the medium dependence of the far-field kernel function. Since different surface regions have different wavenumbers, the far-field kernel and translation operator vary with the wavenumber. To avoid the additional overhead caused by repeatedly constructing hierarchical structures for each medium in engineering, this scheme adopts a method of sharing the same geometric tree, configuring kernel-related translation operators according to the medium, and accumulating contributions. The far-field contribution is expressed as the sum of the far-field operators in each region: ; in Using the wavenumber of this region A corresponding set of related operators is used, reusing the same set of spatial hierarchies and box adjacency relationships during computation, only for each Maintain the corresponding translations and complete the aggregation-translation-downlink process within the same framework. Finally, sum the contributions from each region to obtain the total. .

[0057] Step S7: Solve the linear system in matrix form using the generalized minimum residual iteration method. During the iteration process, the first black-box operator is called to perform matrix-vector multiplication, and the second black-box operator corresponding to the preconditioner constructed based on the near-field matrix is ​​called to preprocess the iteration residuals. Finally, the global unknown vector is generated.

[0058] To stably solve linear systems in matrix form, this embodiment introduces a preconditional iterative solution framework. On the one hand, it maintains the matrix-free first black-box operator. To reduce computational complexity, and on the other hand, to construct an efficient approximate inverse operator. And through the second black box operator Preprocessing the residual direction in each iteration significantly improves spectral properties and increases convergence speed, forming... + Two black-box solution chains enable scalable solutions for large-scale dense systems.

[0059] Specifically, the objective equation to be solved in this step is: ; in, For globally unknown vectors, Given the excitation terms. Unlike direct inversion or direct decomposition, this step uses the GMRES class Krylov subspace iteration method: during the iteration process, only repeated calls are needed. To construct Krylov subspace information without explicitly forming it. Simultaneously, a preconditioner is introduced at a key position in each iteration to reduce the number of iterations and enhance robustness to strongly coupled cases. To this end, a preconditioner operator is first constructed. At this point, the preconditioner is not based on complete dense... Instead, it preferentially relies on the more sparse near-field matrix that can be explicitly obtained. To construct an approximate inverse, thus maintaining the preconditioning effect while keeping the construction cost within an acceptable range, a typical implementation path can be adopted in practice, using incomplete decomposition to obtain... Approximate factor, thus The effect can be achieved by back-substituting the triangular factors, and then encapsulating the preconditioner into a second black-box operator. Its definition is: ; in This refers to the residuals generated during the iteration process.

[0060] Using the above framework, two key calculations can be completed in a typical iteration: First, calling... First, obtain the response for the current search direction, which is used to update the Krylov subspace and form a small-scale least squares problem; second, call... The current residual is preprocessed to advance the iteration on the equivalent spectrum after preconditioning, reducing the convergence lag caused by excessively large distribution of extreme eigenvalues. This combination of "matrix-free master operator + near-field driven preconditioner" can significantly reduce the total number of iteration steps without significantly increasing the computational complexity of each step, compared to the unpreprocessed GMRES.

[0061] Step S8: Reconstruct the spatial distribution of equivalent surface current and equivalent surface magnetic current based on the global unknown vector, and calculate the target's far-field electric field and RCS available results.

[0062] In the final stage of the simulation calculation, this embodiment will reconstruct the global unknown vector to facilitate the calculation of the physical results of the far-field electric field and RCS, so as to meet the needs of practical applications.

[0063] First, based on the discrete expansion relation, the solution vector... Reconstruct the continuous distribution of the equivalent surface flow using RWG basis functions. Expanding on this: ; in Covering the media-media interface and PEC surface, Covering the medium-medium interface. This process transforms the "coefficient solution" into a physical surface source distribution usable for integration calculations. To ensure consistency with the aforementioned multi-medium operator framework, if it is necessary to cover a certain surface region... For in-field evaluation, the aforementioned region operator representation can be directly reused: ; This allows us to obtain the scattered field at any observation point and superimpose it with the incident field to obtain the total field, avoiding the need to re-derive or introduce a post-processing model inconsistent with the main solution.

[0064] Secondly, the far-field and typical scattering observables are calculated, taking the external region as an example, in the far-field direction. Upper structural directional spectrum integral: ; And thus the far-field electric field is obtained. ; ; Accordingly, the available RCS results can be calculated based on this. The calculation formula is as follows: ; After completing the above calculations, the results can be used as the final output, and the simulation calculation is complete.

[0065] This embodiment provides a fast calculation method for electromagnetic scattering. By classifying common boundaries, it transforms the mixed medium distribution in actual simulation into a unified expression of surface parameters and boundary sets, facilitating the simultaneous handling of different types of interfaces in subsequent processing. It also expands the equivalent surface current at the medium-medium interface and the unknown current at the ideal conductor surface using RWG basis functions, forming a natural block coupling structure. Furthermore, Galerkin projection discretizes the continuous boundary integral equations into mechanically executable block assembly templates, providing a consistent data layout and interface for subsequent fast matrix-vector multiplication and block preconditioning. Additionally, it decomposes the matrix-form linear system into a fast algorithm of direct near-field explicit sparse matrix multiplication and far-field "matrix-free" operation to reduce complexity. In the GMRES iteration, a near-field driven preconditioner constructed from the near-field impedance submatrix is ​​introduced to reduce the number of iterations, improve strong coupling and convergence robustness across frequency points, and ultimately efficiently obtain the far-field electric field and RCS outputs, thus meeting practical application requirements.

[0066] Example 2 Figure 2 This is a schematic diagram of the electromagnetic scattering fast calculation device provided in Embodiment 2 of the present invention, as shown below. Figure 2 As shown, the electromagnetic scattering fast calculation device includes: The input preprocessing module 210 is used to obtain the three-dimensional geometric model and simulation excitation parameters of the target, divide the target into medium partitions and assign material parameters to each region, and mark the regions that meet the ideal conductor conditions as ideal conductor boundaries; The mesh generation and classification module 220 is used to mesh the three-dimensional geometric model of the target, divide the entire computational domain into multiple surface regions, and generate a set of surface regions; based on the material parameters of each region, it calculates the complex permittivity, wave number, and wave impedance of each surface region; it traverses the common boundary of all mesh cells, and classifies the common boundary into a set of dielectric-dielectric interfaces and a set of ideal conductor surfaces according to the material differences between the regions on both sides of the common boundary, and records the surface region index pairs and unit normal vectors on both sides of each dielectric-dielectric interface; The kernel and integral definition module 230 is used to define the Green's function kernel corresponding to the wavenumber for each table domain region, and to define single-layer operators and two-layer coupled operators based on the Green's function kernel; The equation construction module 240 is used to construct a unified set of boundary integral equations based on the set of dielectric-dielectric interfaces and the set of ideal conductor surfaces. Among them, the PMCHWT coupled integral equation is established based on the continuity condition at the dielectric-dielectric interface, and the linear combination equation CFIE of EFIE and MFIE is established at the boundary of the ideal conductor surface. Discretization module 250 is used to expand the unknown equivalent surface flow using RWG basis functions and discretize the continuous boundary integral equation system into a linear system in matrix form using Galerkin projection. The matrix calculation encapsulation module 260 is used to decompose a linear system in matrix form into the sum of a near-field matrix and a far-field matrix, and encapsulate it into a first black-box operator that can be called by an iterative solver. The near-field matrix is ​​calculated directly through explicitly stored matrix blocks, and the far-field matrix is ​​calculated through a matrix free fast algorithm. The iterative solution module 270 is used to solve linear systems in matrix form using the generalized minimum residual iterative method. During the iteration process, the first black-box operator is called to perform matrix-vector multiplication, and the second black-box operator corresponding to the preconditioner constructed based on the near-field matrix is ​​called to preprocess the iterative residuals, and finally the global unknown vector is generated. Output module 280 is used to reconstruct the spatial distribution of equivalent surface current and equivalent surface magnetic current based on the global unknown vector, and to calculate the far-field electric field and RCS of the target.

[0067] The electromagnetic scattering fast calculation device provided in this embodiment of the invention can execute the electromagnetic scattering fast calculation method provided in this embodiment of the invention, and has the corresponding functional modules and beneficial effects of the method.

[0068] Example 3 Figure 3 This is a schematic diagram of the structure of a terminal provided in Embodiment 3 of the present invention. Figure 3 A block diagram is shown of an exemplary terminal 12 suitable for implementing embodiments of the present invention. Figure 3 The terminal 12 shown is merely an example and should not impose any limitations on the functionality and scope of use of the embodiments of the present invention.

[0069] like Figure 3 As shown, terminal 12 is presented in the form of a general-purpose computing device. The components of terminal 12 may include, but are not limited to: one or more processors or processing units 16, system memory 28, and bus 18 connecting different system components (including system memory 28 and processing unit 16).

[0070] Bus 18 represents one or more of several bus architectures, including a memory bus or memory controller, a peripheral bus, a graphics acceleration port, a processor, or a local bus using any of the various bus architectures. For example, these architectures include, but are not limited to, the Industry Standard Architecture (ISA) bus, the Micro Channel Architecture (MAC) bus, the Enhanced ISA bus, the Video Electronics Standards Association (VESA) local bus, and the Peripheral Component Interconnect (PCI) bus.

[0071] Terminal 12 typically includes a variety of computer system readable media. These media can be any available media that can be accessed by terminal 12, including volatile and non-volatile media, removable and non-removable media.

[0072] System memory 28 may include computer system readable media in the form of volatile memory, such as random access memory (RAM) 30 and / or cache memory 32. Terminal 12 may further include other removable / non-removable, volatile / non-volatile computer system storage media. By way of example only, storage system 34 may be used to read and write non-removable, non-volatile magnetic media (… Figure 3 Not shown; usually referred to as a "hard drive"). Although Figure 3 Not shown, a disk drive for reading and writing to a removable non-volatile disk (e.g., a "floppy disk") and an optical disk drive for reading and writing to a removable non-volatile optical disk (e.g., a CD-ROM, DVD-ROM, or other optical media) may be provided. In these cases, each drive may be connected to bus 18 via one or more data media interfaces. Memory 28 may include at least one program product having a set (e.g., at least one) of program modules configured to perform the functions of the embodiments of the present invention.

[0073] A program / utility 40 having a set (at least one) of program modules 42 may be stored, for example, in memory 28. Such program modules 42 include, but are not limited to, an operating system, one or more application programs, other program modules, and program data. Each or some combination of these examples may include an implementation of a network environment. Program modules 42 typically perform the functions and / or methods described in the embodiments of the present invention.

[0074] Terminal 12 can also communicate with one or more external devices 14 (e.g., keyboard, pointing device, display 24, etc.), and with one or more devices that enable a user to interact with terminal 12, and / or with any device that enables terminal 12 to communicate with one or more other computing devices (e.g., network card, modem, etc.). This communication can be performed via input / output (I / O) interface 22. Furthermore, terminal 12 can also communicate with one or more networks (e.g., local area network (LAN), wide area network (WAN), and / or public networks, such as the Internet) via network adapter 20. As shown, network adapter 20 communicates with other modules of terminal 12 via bus 18. It should be understood that, although not shown in the figures, other hardware and / or software modules can be used in conjunction with terminal 12, including but not limited to: microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data backup storage systems.

[0075] The processing unit 16 executes various functional applications and data processing by running programs stored in the system memory 28, such as implementing the fast electromagnetic scattering calculation method provided in the embodiments of the present invention.

[0076] Example 4 Embodiment 4 of the present invention also provides a storage medium containing computer-executable instructions, which, when executed by a computer processor, are used to perform the fast electromagnetic scattering calculation method as described in any of the above embodiments.

[0077] The computer storage medium of this invention can be any combination of one or more computer-readable media. A computer-readable medium can be a computer-readable signal medium or a computer-readable storage medium. A computer-readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of computer-readable storage media (a non-exhaustive list) include: an electrical connection having one or more wires, a portable computer disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination thereof. In this document, a computer-readable storage medium can be any tangible medium that contains or stores a program that can be used by or in conjunction with an instruction execution system, apparatus, or device.

[0078] Computer-readable signal media may include data signals propagated in baseband or as part of a carrier wave, carrying computer-readable program code. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. Computer-readable signal media may also be any computer-readable medium other than computer-readable storage media, capable of sending, propagating, or transmitting programs for use by or in connection with an instruction execution system, apparatus, or device.

[0079] Program code contained on a computer-readable medium may be transmitted using any suitable medium, including, but not limited to, wireless, wire, optical fiber, RF, etc., or any suitable combination thereof.

[0080] Computer program code for performing the operations of this invention can be written in one or more programming languages ​​or a combination thereof, including object-oriented programming languages ​​such as Java, Smalltalk, and C++, as well as conventional procedural programming languages ​​such as "C" or similar programming languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN)—or can be connected to an external computer (e.g., via the Internet using an Internet service provider).

[0081] Note that the above description is merely a preferred embodiment of the present invention and the technical principles employed. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and various obvious changes, readjustments, and substitutions can be made without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in detail through the above embodiments, the present invention is not limited to the above embodiments, and may include many other equivalent embodiments without departing from the concept of the present invention, the scope of which is determined by the scope of the appended claims.

Claims

1. A fast calculation method for electromagnetic scattering, characterized in that... include: S1. Obtain the three-dimensional geometric model and simulation excitation parameters of the target, divide the target into medium partitions and assign material parameters to each region, and mark the regions that meet the ideal conductor conditions as ideal conductor boundaries; S2. Mesh the three-dimensional geometric model of the target, divide the entire computational domain into multiple surface regions, and generate a set of surface regions; calculate the complex permittivity, wave number, and wave impedance of each surface region based on the material parameters of each region; traverse the common boundary of all mesh elements, classify the common boundary into a set of dielectric-dielectric interfaces and a set of ideal conductor surfaces according to the material differences between the regions on both sides of the common boundary, and record the surface region index pairs and unit normal vectors on both sides of each dielectric-dielectric interface; S3. Define a Green's function kernel corresponding to the wavenumber for each table domain region, and define single-layer operators and two-layer coupled operators based on the Green's function kernel; S4. Based on the set of dielectric-dielectric interfaces and the set of ideal conductor surfaces, a unified set of boundary integral equations is constructed. Among them, the PMCHWT coupled integral equation is established based on the continuity condition at the dielectric-dielectric interface, and the linear combination equation CFIE of EFIE and MFIE is established at the boundary of the ideal conductor surface. S5. On the grid divided in step S2, the unknown equivalent surface flow is expanded using RWG basis functions, and the continuous boundary integral equations in step S4 are discretized into a linear system in matrix form using Galerkin projection. S6. Decompose the linear system in matrix form obtained in step S5 into the sum of the near-field matrix and the far-field matrix, and encapsulate it into a first black-box operator that can be called by the iterative solver. The near-field matrix is ​​calculated directly through explicitly stored matrix blocks, and the far-field matrix is ​​calculated through the matrix free fast algorithm. S7. The generalized minimum residual iteration method is used to solve the linear system in matrix form. During the iteration process, the first black box operator is called to perform matrix-vector multiplication, and the second black box operator corresponding to the preconditioner constructed based on the near field matrix is ​​called to preprocess the iteration residual. Finally, the global unknown vector is generated. S8. Reconstruct the spatial distribution of equivalent surface current and equivalent surface magnetic current based on the global unknown vector, and calculate the target's far-field electric field and RCS.

2. The fast calculation method for electromagnetic scattering according to claim 1, characterized in that: The material parameters include dielectric constant, magnetic permeability, and electrical conductivity; In step S2, the complex permittivity, wavenumber, and wave impedance of each surface region are calculated using the following formula: ； ； ； In the above formula, , and These are the respective table regions The dielectric constant, permeability, and conductivity of are . Complex permittivity, For wave number, For wave impedance, Angular frequency, It is the imaginary unit.

3. The fast calculation method for electromagnetic scattering according to claim 2, characterized in that: In step S3, the Green's function kernel is defined as: ； Among them Field point position vector, The source point position vector, The distance between two points; The single-layer operator and the two-layer coupled operator defined based on the Green's function kernel are used to express the scattering field within the surface domain, and their mathematical expressions are as follows: ； In the above formula, It is a single-layer operator. It is a two-layer coupled operator. and These respectively represent in the table field The equivalent current density and equivalent magnetic flux density Operator mapping of the generated scattered electric field and scattered magnetic field.

4. The fast calculation method for electromagnetic scattering according to claim 3, characterized in that: In step S4, establishing the PMCHWT coupled integral equation based on the continuity condition at the medium-medium interface includes: For media-media interface Introducing surface current With surface magnetic current Based on the physical boundary conditions at the interface constraining the continuity of the electric and magnetic fields, the equivalent current and equivalent magnetic current are expressed by the following formulas: ； in, and For the interface equivalent source in the table domain region The scattered field components generated in the process, and Let the incident electric field and incident magnetic field be respectively. Substituting the above scattering field operator expression into the continuity condition, we can obtain the medium-medium interface. Coupled integral equations: ； The establishment of the linear combination equation CFIE of EFIE and MFIE at the boundary of the ideal conductor surface includes: For the boundary of an ideal conductor surface Retaining only the equivalent surface current position, the calculation employs a linear combination of EFIE and MFIE to form a combined two-field integral equation, with the total tangential electric field being zero. The resulting linear combined equation CFIE is as follows: ； In the above formula, For combination coefficients, It represents the external dielectric wave impedance.

5. The fast calculation method for electromagnetic scattering according to claim 1, characterized in that: In step S5, the unknown equivalent surface flow includes equivalent surface current and equivalent surface magnetic current. Expanding the unknown equivalent surface flow using RWG basis functions on the mesh defined in step S2 includes: Construct a set of vector basis functions on the triangular mesh divided in step S2. The vector basis function set adopts the RWG basis function; the mathematical expressions for the equivalent surface current and equivalent surface magnetic current are as follows: ； In the above formula, and The complex coefficients to be determined; For an unknown current on an ideal conductor surface The same basis function system as the equivalent surface current is used for expansion and incorporated into the same current coefficient vector, thereby defining the global unknown vector: ； The discretization of the continuous boundary integral equations in step S4 into a linear system in matrix form using Galerkin projection includes: A linear system in matrix form is obtained through Galerkin projection. , where the matrix elements are The right-hand vector elements are The inner product is defined as .

6. The fast calculation method for electromagnetic scattering according to claim 1, characterized in that: In step S6, the far-field matrix is ​​calculated using a matrix-free fast algorithm, including calculating the far-field matrix using the MLFMA algorithm or the H-matrix algorithm.

7. The fast calculation method for electromagnetic scattering according to claim 1, characterized in that: In step S8, the formulas for calculating the far-field electric field and RCS using available results are as follows: ； ； In the above formula, For the far-field electric field, This is the available result for RCS.

8. A fast electromagnetic scattering calculation device, characterized in that, include: The input preprocessing module is used to obtain the three-dimensional geometric model and simulation excitation parameters of the target, divide the target into medium partitions and assign material parameters to each region, and mark the regions that meet the ideal conductor conditions as ideal conductor boundaries; The mesh generation and classification module is used to mesh the 3D geometric model of the target, dividing the entire computational domain into multiple surface regions and generating a set of surface regions. Based on the material parameters of each region, it calculates the complex permittivity, wave number, and wave impedance of each surface region. It traverses the common boundary of all mesh cells and classifies the common boundary into a set of dielectric-dielectric interfaces and a set of ideal conductor surfaces according to the material differences between the regions on both sides of the common boundary. It also records the surface region index pairs and unit normal vectors on both sides of each dielectric-dielectric interface. The kernel and integral definition module is used to define the Green's function kernel corresponding to the wavenumber for each table domain region, and to define single-layer operators and two-layer coupled operators based on the Green's function kernel; The equation construction module is used to construct a unified set of boundary integral equations based on the set of dielectric-dielectric interfaces and the set of ideal conductor surfaces. Specifically, the PMCHWT coupled integral equation is established based on the continuity condition at the dielectric-dielectric interface, and the linear combination equation CFIE of EFIE and MFIE is established at the boundary of the ideal conductor surface. The discretization module is used to expand the unknown equivalent surface flow using RWG basis functions and discretize the continuous boundary integral equations into a linear system in matrix form using Galerkin projection. The matrix computation encapsulation module is used to decompose a linear system in matrix form into the sum of a near-field matrix and a far-field matrix, and encapsulate it into a first black-box operator that can be called by the iterative solver. The near-field matrix is ​​directly calculated through explicitly stored matrix blocks, and the far-field matrix is ​​calculated through a matrix-free fast algorithm. The iterative solution module is used to solve linear systems in matrix form using the generalized minimum residual iterative method. During the iteration process, the first black-box operator is called to perform matrix-vector multiplication, and the second black-box operator corresponding to the preconditioner constructed based on the near-field matrix is ​​called to preprocess the iteration residuals. Finally, the global unknown vector is generated. The output module is used to reconstruct the spatial distribution of the equivalent surface current and the equivalent surface magnetic current based on the global unknown vector, and to calculate the far-field electric field and RCS of the target.

9. A terminal, characterized in that, The terminal includes: One or more processors; Storage device for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement the fast electromagnetic scattering calculation method as described in any one of claims 1-7.

10. A storage medium containing computer-executable instructions, which, when executed by a computer processor, are used to perform the fast electromagnetic scattering calculation method as described in any one of claims 1-7.

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