Array antenna electromagnetic simulation method and system based on non-conformal domain decomposition

Abstract

The application discloses an array antenna electromagnetic simulation method and system based on non-conformal region decomposition. The application allows the sub-domain profiled grid to be completely independently set in the number, shape and node distribution of cells by the non-conformal region decomposition technology, does not need to ensure the grid consistency at the sub-domain boundary, breaks the constraint of the conformal grid, makes the calculation domain be flexibly decomposed into any non-overlapping sub-domain, and greatly reduces the grid generation difficulty under the complex geometric scene. The application guarantees the accurate coupling of the electromagnetic field between the sub-domains under the non-conformal grid by the high-precision interface constraint, and improves the reliability of the simulation result. The application separates the sub-domain independent solving and global assembly, so that the grid generation and electric field solving of each sub-domain can be completely parallelized, significantly shortens the calculation time in large-scale array antenna simulation, and improves the efficiency.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation, and particularly relates to an electromagnetic simulation method and system for array antennas based on non-conformal region decomposition. Background Technology

[0002] The Finite Element Method (FEM), as a mainstream numerical method for solving electromagnetic field problems, is widely used to handle electromagnetic problems with complex geometries and multi-physics coupling. This method transforms continuous electromagnetic field problems into discrete systems of algebraic equations by discretizing the computational domain, thus enabling solutions. However, with the increase in problem scale, especially in high-frequency electromagnetic problems, the FEM faces challenges in terms of computational efficiency and memory consumption. Therefore, improving the computational efficiency of the FEM in large-scale electromagnetic simulations has become one of the current research hotspots.

[0003] In the design and simulation of antenna arrays, array antennas are typically organized in the form of antenna elements. Each antenna element requires a separate mesh generation, and the meshes of all elements are then stitched together. However, as the number of array elements increases, the mesh generation time increases significantly, making the traditional method impractical. This high computational cost limits the rapid simulation and optimization of large antenna arrays. To address this issue, a method has been proposed that meshes only the antenna elements and then constructs the overall antenna array by stitching together the individual element meshes. This method allows for adjustment of the antenna array size, optimization of system performance, and simplification of the mesh generation process; it is called the domain decomposition method. However, this traditional domain decomposition method still faces some problems and challenges.

[0004] Traditional domain decomposition methods require conformal meshes at the boundaries between different subdomains, meaning the meshes must be consistent at these boundaries. This stringent requirement limits their application in complex geometries and irregular regions. Furthermore, mesh generation techniques face numerous challenges in practical applications, especially when dealing with irregular shapes and non-conforming meshes. Optimizing the speed and efficiency of mesh generation while maintaining computational accuracy remains a challenge. When mesh generation does not meet conformal requirements, the continuity of fields on the left and right sides of the boundary becomes a critical issue. Ensuring the continuity of the physical field between regions without altering the mesh structure is a pressing problem. Moreover, traditional domain decomposition methods often rely on a single computational model, making parallel mesh processing difficult. As computational scales increase, efficiently parallelizing the computational tasks of each subdomain becomes a bottleneck for improving electromagnetic simulation efficiency. Therefore, a new method is urgently needed to achieve efficient electromagnetic simulation without relying on conformal meshes. This method should be able to handle complex geometries and irregular regions, optimize the speed and efficiency of mesh generation, ensure the continuity of the physical field between regions, and achieve parallel mesh processing, thereby improving the efficiency and accuracy of electromagnetic simulation. Summary of the Invention

[0005] The purpose of this invention is to overcome the bottleneck between mesh generation and computational efficiency in the prior art, especially for large-scale electromagnetic simulation tasks, by providing an array antenna electromagnetic simulation method and system based on non-conformal region decomposition.

[0006] In a first aspect, the present invention provides an electromagnetic simulation method for array antennas based on non-conformal region decomposition, comprising:

[0007] The overall computational domain is decomposed into several non-overlapping subdomains using a non-conformal region, and the mesh of each subdomain is a non-conformal mesh.

[0008] Lagrange multipliers are defined using interface current traces and interface electric field tangential traces; based on the Lagrange multipliers, symmetric transmission conditions are constructed for adjacent subdomains; all Lagrange multiplier degrees of freedom are mapped to the discrete space of the corresponding subdomains and interfaces to realize the transformation and equilibrium of the physical field, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains and ensuring the correct transmission of the physical field on the domain surface.

[0009] Based on the Lagrange multipliers, a bilinear interface condition is defined to allow the electromagnetic response to propagate naturally between subdomains, thus achieving correct coupling of various fields between non-conformal meshes.

[0010] Solve for the electric field in each subdomain;

[0011] The electric field information and interface conditions within each subdomain are assembled into a global matrix, and the electric field information of the complete computational domain is obtained by solving the global matrix.

[0012] Preferably, the non-conformal mesh is configured independently for the number of cells, shape, and node distribution of each subdomain mesh.

[0013] Preferably, the connectivity between subdomains after non-conformal region decomposition is expressed via a Boolean matrix. Definition: For any two adjacent subdomains, there exists a pair of interfaces. and , and They represent the first Subdomain facing the first Interfaces between subdomains, the first Subdomain facing the first The interface between subdomains; and because of the domain surface It is an interface and The overall computational domain decomposed into a skeleton composed of a set of domain surfaces. The set of interfaces that each subdomain faces to other subdomains forms the domain boundary within that subdomain. .

[0014] Preferably, the Lagrange multiplier It only applies to the skeleton after the global computational domain is decomposed. The electric field in the decomposed subdomain is continuous within its own subdomain, but discontinuities are allowed at the interface.

[0015] Preferably, the specific implementation process of constructing symmetric transport conditions for adjacent subfields based on Lagrange multipliers is as follows:

[0016] Apply symmetrical transmission conditions to both sides of each pair of interfaces, i.e., each pair of adjacent subdomains and Symmetry conditions for the tangential components of the electromagnetic field and the tangential components of the magnetic field on both sides of the domain surface.

[0017] Preferably, the process of mapping all Lagrange multiplier degrees of freedom to the discrete space of corresponding subdomains and interfaces to achieve the transformation and equilibrium of the physical field, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains and ensuring the correct transmission of the physical field on the domain surface, is as follows:

[0018] For each pair of adjacent subdomains and Theoretically, two sets of symmetrical interface conditions need to be applied to the domain surface;

[0019] In non-conformal meshes, the degrees of freedom on both sides of the interface are usually different. It is necessary to consider the interface degree of freedom mapping and weights. Therefore, an interface-subdomain projection mapping matrix is ​​introduced. To achieve the transformation and equilibrium of physical fields;

[0020] In the finite element discretization and assembly process, all interface-related degrees of freedom need to be projected onto their respective discrete spaces through the above mapping matrix, so as to achieve accurate conversion and balance of physical information under different subdomains and different grids; in order to avoid the repeated inclusion of the tangential electric field trace, only the master equation on one side needs to be retained.

[0021] Preferably, the bilinear interface conditions include a subdomain-interface coupling term, an interface-subdomain coupling term, and an interface regularization term.

[0022] In a second aspect, the present invention provides an electromagnetic simulation system for an array antenna that implements the above-described method, comprising:

[0023] The subdomain construction module is responsible for decomposing the overall computational domain into several non-overlapping subdomains using a non-conformal region, where the mesh of each subdomain is a non-conformal mesh; defining Lagrange multipliers using interface current traces and interface electric field tangential traces; constructing symmetric transport conditions for adjacent subdomains based on the Lagrange multipliers; mapping all Lagrange multiplier degrees of freedom to the discrete spaces of the corresponding subdomains and interfaces to achieve the transformation and equilibrium of physical fields, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains; defining bilinear interface conditions based on the Lagrange multipliers, so that the electromagnetic response is naturally transmitted between subdomains, achieving correct coupling of various fields between non-conformal meshes;

[0024] The first calculation module is responsible for solving the electric field in each subdomain;

[0025] The second calculation module is responsible for assembling the electric field information and interface conditions in each subdomain into a global matrix, and obtaining the electric field information of the complete calculation domain by solving the global matrix.

[0026] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to perform the above-described method.

[0027] Fourthly, the present invention provides a computing device, including a memory and a processor, wherein the memory stores executable code, and the processor executes the executable code to implement the above-described method.

[0028] The beneficial effects of the present invention include at least the following:

[0029] This invention uses non-conformal region decomposition technology to allow the mesh of each subdomain to be set completely independently in terms of the number of elements, shape, and node distribution, without having to ensure mesh consistency at the boundaries of subdomains. This breaks the constraints of conformal meshes, allowing the computational domain to be flexibly decomposed into any non-overlapping subdomains, significantly reducing the difficulty of mesh generation in complex geometric scenarios and broadening the applicability of electromagnetic simulation methods.

[0030] This invention utilizes interface current traces and electric field tangential traces to define Lagrange multipliers, which act only on the skeleton after subdomain decomposition. This allows for electric field continuity within subdomains but discontinuities at interfaces, achieving "weak continuity" through constraints. Symmetrical transmission conditions are constructed for adjacent subdomains, forcing matching of electromagnetic physical quantities on both sides of the interface through the symmetric equations of the electromagnetic field tangential components. A projection mapping matrix is ​​introduced to map the Lagrange multiplier degrees of freedom to the discrete space of the subdomains, resolving the degree-of-freedom mismatch problem on both sides of the interface under non-conformal meshes and ensuring the correct transmission of physical fields across the domain surface. These measures guarantee accurate coupling of electromagnetic fields between subdomains under non-conformal meshes, avoiding the accuracy loss caused by mesh mismatch in traditional methods and improving the reliability of simulation results.

[0031] This invention separates independent subdomain solution from global assembly, enabling fully parallel processing of mesh generation and electric field solution for each subdomain. This significantly reduces computation time and improves efficiency in large-scale array antenna simulation. Attached Figure Description

[0032] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0033] Figure 1 A flowchart of an electromagnetic simulation method provided in an embodiment of this application.

[0034] Figure 2 The computational domain in the electromagnetic simulation method provided in an embodiment of this application A schematic diagram of the topological quantities resulting from the partitioning, where (a) is the neighborhood set. (b) Interfaces generated by adjacent domains (c) Internal domain boundaries (d) Decomposed skeleton .

[0035] Figure 3 The computational domain in the electromagnetic simulation method provided in an embodiment of this application A schematic diagram of the domain decomposition.

[0036] Figure 4An embodiment of this application provides a cube decomposed into multiple subdomains of different colors using the FETI-2λ method of the present invention. Each subdomain is decomposed using an unstructured mesh, which is a second-order finite element mesh with a specific element size. exist arrive between.

[0037] Figure 5 This is a comparison diagram of the error convergence of the present invention's method FETI-2λ and the existing method FEM in a tangentially continuous LM space, provided as an embodiment of this application. Detailed Implementation

[0038] The present invention will be further analyzed below with reference to specific embodiments.

[0039] The core objective of this invention is to improve computational efficiency by addressing the generation and optimization of non-conformal meshes, enabling electromagnetic simulations to be performed in more complex geometries without being strictly limited by mesh generation methods. This improvement is particularly evident in large-scale problems and high-frequency electromagnetic simulations. With the continuous improvement of computing power, traditional finite element methods are no longer effective in handling large-scale electromagnetic simulation tasks, especially in the case of multiple subdomains and complex topologies. Computational efficiency and memory consumption have become pressing technical challenges.

[0040] The technical problems addressed by this invention involve multiple aspects. First, regarding the handling of non-conformal meshes, this invention proposes a novel method capable of processing meshes that do not meet conformal requirements, thereby overcoming the limitations of traditional Domain Decomposition (DDM) methods in terms of boundary consistency and adapting to various complex geometries. Second, considering the computational complexity of electromagnetic simulation tasks, this invention implements a parallel computing framework, enabling the computational tasks of each subdomain to be performed in parallel on multiple computing nodes, greatly improving computational efficiency and shortening the simulation time. Furthermore, the method of this invention possesses good scalability and topology independence; computational performance is unaffected by the number of domain partitions and the topology, thus ensuring an efficient computational process.

[0041] To achieve this technical goal, this invention employs a complete solution comprised of a series of innovative technologies. The first is the generation and processing of non-conformal meshes. Traditional finite element methods require meshes to be completely uniform at interfaces; however, this requirement is often impractical for complex geometries and irregular regions. To address this issue, this invention proposes an efficient method for generating non-conformal meshes, which divides the computational domain into multiple subdomains and generates irregular meshes for each subdomain. This method not only handles meshes that do not meet conformality requirements but also overcomes the limitations imposed by mesh consistency in traditional methods through optimization of the mesh in each subdomain.

[0042] In terms of parallel computing, this invention introduces a parallel computing framework that distributes mesh generation and computation across multiple computing nodes for parallel processing. This allows the simulation task to distribute the load across multiple processing units, significantly improving computational efficiency. Especially in large-scale electromagnetic simulation tasks, traditional serial computing methods are insufficient, while the parallel computing framework of this invention enables efficient simulation, meeting the demands of high-performance computing. Furthermore, the method's parallel computing capabilities significantly reduce computation time and improve simulation efficiency when dealing with large-scale antenna arrays or complex electromagnetic environments.

[0043] This invention also pays particular attention to the topology-independent property of domain partitioning. Traditional domain decomposition methods are typically highly dependent on the number of meshes and the topology, limiting their applicability to different simulation tasks. In contrast, the method of this invention can adapt to different numbers of domains, and its computational performance is unaffected by changes in topology. This means that whether dealing with regular geometry or irregular or complex geometries, the method of this invention maintains efficient computation, ensuring computational accuracy while optimizing the speed and efficiency of mesh generation.

[0044] Furthermore, this invention further improves computational efficiency by handling the independence of the number of iterations from the topology. Unlike traditional methods, the number of iterations in this invention is independent of the number of region partitions and the topology. This characteristic makes the simulation process more stable and ensures consistent computational performance when facing computational tasks of different scales and structures.

[0045] This embodiment provides an electromagnetic simulation method for array antennas based on non-conformal region decomposition. See Appendix. Figure 1 This includes the following steps:

[0046] Step S1: Define the overall array antenna as the computational domain. It is then decomposed into M non-overlapping subdomains for each antenna element. See appendix Figure 2 In (a), each antenna element is a subdomain, where M > 1, i ∈ [1, M]. There are no restrictions during the decomposition process. The mesh for each subdomain after decomposition is a non-conformal mesh, where the number of elements, shape, and node distribution of each subdomain's mesh are independently set. In other words, each subdomain after decomposition... The information on both sides of the interface does not need to correspond one-to-one or overlap. This decomposition method is called non-conformal region decomposition, and the resulting mesh is called a non-conformal mesh. See Appendix. Figure 3 .

[0047]

[0048] Within each subdomain Ωᵢ, the generated mesh divides the continuous computational domain into finite cells and approximates the variation of the electromagnetic field on each cell.

[0049] Connectivity between subdomains is achieved through a Boolean matrix. Definition, where Boolean matrix The element in the i-th row and j-th column:

[0050]

[0051] Each subdomain obtained from the decomposition is non-overlapping, therefore the union of any two subdomains is... Represented as:

[0052]

[0053] in, Defined as a domain-face, representing the first... Subdomains and the first A surface shared between subdomains;

[0054] because It also represents a subdomain. and A spatially unified region. When two subdomains are adjacent, they are separated by a domain surface. Achieving boundary coupling is the foundation for realizing physical field continuity and parallel assembly in the domain decomposition finite element method; if they are not adjacent, there is no physical interaction, and there are no coupling terms in the numerical system.

[0055] At the same time, define the computational domain. Other topologies in:

[0056] See appendix Figure 2 In section (a), define a pair of interfaces for any two adjacent subdomains. and , and They represent the first Subdomain facing the first Interfaces between subdomains, the first Subdomain facing the first Interfaces between subdomains;

[0057] And because of the domain It is an interface and Composition, it can be known .

[0058] The set of domains forms the framework of domain decomposition. See appendix Figure 2 Middle (d):

[0059]

[0060] No. The set of interfaces that each subdomain points to from other subdomains forms the domain boundary within that subdomain. See appendix Figure 2 (c)

[0061]

[0062] Step S2: To adapt to subdomain coupling under non-conformal meshes, utilize interface current traces. and interface electric field tangential trace In the skeleton Upper definition of Lagrange multipliers The role of Lagrange multipliers is to ensure the continuity of the electromagnetic field between subdomains at the interface, especially to ensure the transmission of the tangential field. By defining the Lagrange multipliers, the solutions of each subdomain are constrained, thus maintaining the continuity of the physical field at the interface.

[0063]

[0064] Lagrange multipliers It only acts on the skeleton Above, among which Represents the interface current trace; Represents the tangential trace of the interface electric field; and These are the interface curl and tangential operators, respectively. To adjust the complex parameters for convergence (theoretical analysis can take...) ); for the computational subdomains obtained from the decomposition The electric field solution. The electric field in the computational domain after the entire decomposition. It represents the set of unknown electric field quantities for all subdomains within the entire partition. They are contiguous within their respective subdomains, but discontinuousness is allowed at the interface. Represents relative permeability. Represents the curl operator, It is a Lagrange multiplier space.

[0065] Apply symmetrical transmission conditions to both sides of each pair of interfaces, i.e., each pair of adjacent subdomains and The symmetry conditions for the tangential components of the electromagnetic field and the tangential components of the magnetic field on both sides of the domain surface are expressed as:

[0066]

[0067] Where x Indicates in the interface The x values ​​on both sides, x Indicates in the interface The x-values ​​on both sides.

[0068] Neighboring subfields and Weak continuity of the tangential field in the non-conformal case is achieved by symmetric conditional coupling.

[0069] Using Lagrange multipliers Rewriting formula (7), we get:

[0070]

[0071] Further analysis of the assembly process under this constraint:

[0072] For each pair of adjacent subdomains and Theoretically, two sets of symmetrical interface conditions need to be applied to the domain surface.

[0073] exist Perspective (acting on) ):

[0074]

[0075] exist Perspective (acting on) ):

[0076]

[0077] in, and These respectively represent the interfaces and The degrees of freedom of the Lagrange multipliers on both sides, Indicates in subdomain The tangential trace of the electric field on one side, Indicates in subdomain The tangential trace of the electric field on one side.

[0078] If we simply add formulas (9) and (10), we get formula (11):

[0079]

[0080] However, in physical calculations, the tangential electric field trace It should only be counted once; otherwise, energy or interface information will be counted repeatedly, violating the law of conservation of energy and physical symmetry.

[0081] In non-conformal meshes, the degrees of freedom on both sides of the interface are usually different. It is necessary to consider the interface degree of freedom mapping and weights. Therefore, an interface-subdomain projection mapping matrix is ​​introduced. To achieve the transformation and equilibrium of physical fields: This indicates that the Lagrange multiplier degrees of freedom of the interface are projected onto the subdomain. Degrees of freedom; This indicates that the Lagrange multiplier degrees of freedom of the interface are projected onto the subdomain. Freedom space.

[0082] In the finite element discretization and assembly process, all interface-related degrees of freedom need to be projected onto their respective discrete spaces through the aforementioned mapping matrix, thereby achieving accurate conversion and balance of physical information under different subdomains and meshes. To avoid the repeated inclusion of tangential electric field traces, only one side needs to be retained in the system (e.g., The master equation is:

[0083]

[0084] in, It is aimed at The symmetry correction term on both sides of the interface, due to The current trace and the electric field tangential trace are already included. If the constraints on both sides are directly assembled separately, it will lead to... The contribution is repeatedly counted in the global energy. This is achieved by introducing a symmetry correction term, which retains only one side of the contribution. This ensures energy conservation and numerical symmetry of the system under non-conformal grids, achieving strict physical constraints and optimal information transfer.

[0085] This step maps the original symmetric relationship of physical quantities in their respective continuous spaces to the linear combination constraint of Lagrange multipliers in discrete space, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains, thus ensuring the correct transmission of physical fields at the interface.

[0086] Step 3: Define bilinear interface conditions, which include subdomain-interface coupling terms, interface-subdomain coupling terms, and interface regularization terms. The regularization terms numerically stabilize the interface degrees of freedom, preventing "zero-mode drift" caused by excessive or unevenly distributed interface degrees of freedom. This ensures the invertibility and solution stability of the global matrix, significantly improving the robustness of the numerical system in non-conformal scenarios and avoiding the coupling instability problem in traditional methods.

[0087] In order to achieve tangential continuity of the interface and global field coupling, and All The test function on, For Lagrange multiplier space The test function within.

[0088] Define subdomain-interface coupling items:

[0089]

[0090] in, express skeleton of all interfaces Tangential trace on, integral sign Indicated throughout the skeleton Perform integral calculations on top. Representing a surface infinitesimal element, The expression represents the vector inner product integration over all interfaces. The term couples the electric field solution within the subdomain with the Lagrange multiplier at the interface, ensuring weak continuity between electric field information in different subdomains.

[0091] Define interface-subdomain coupling items:

[0092]

[0093] in, express skeleton of all interfaces Tangential trace on, Represents numerical weights, derived from hybrid physical boundary processing based on transmission conditions. Integral symbol. Indicated throughout the skeleton Perform integral calculations on top. Representing a surface infinitesimal element, The expression represents a weighted average of the degrees of freedom across all interfaces. This term is a "weight exchange term" between the interface variables and the subdomain fields, playing a crucial role in improving the convergence and numerical robustness of the FETI-2λ algorithm.

[0094] Define the regularization term:

[0095]

[0096] in, The tangential trace operator represents the vector field defined over the entire subdomain. and The tangential components are extracted into the skeleton of the set of all interfaces. Above. Integral symbol Indicated throughout the skeleton Perform integral calculations on top. Representing a surface infinitesimal element, The expression representing the area-weighted integral over all interface points originates from the variable transformation of the Lagrange multipliers. Stability provides additional constraints to prevent numerical divergence or loss of degrees of freedom due to variable transformations in hybrid finite element methods.

[0097] Get the interface regularization term:

[0098]

[0099] The integral symbol Indicated throughout the skeleton Perform integral calculations on top. Representing a surface infinitesimal element, The expression represents the accumulation of the inner product of all interface degrees of freedom. The term is the mass matrix or self-coupling term of the interface Lagrange multipliers, used to regularize the interface degrees of freedom, ensuring that the physical quantities on each interface participate in global coupling while also possessing numerical stability. This prevents "zero-mode drift" or "singular matrices" when there are too many interface degrees of freedom. It plays an irreplaceable role in achieving weak continuity, numerical convergence, and uniqueness.

[0100] This completes the constraints for electromagnetic information transfer between subdomains, enabling the correct transfer of electromagnetic information between subdomains and achieving correct coupling of various fields between non-conformal meshes.

[0101] Step S4: Solve for the electric field in each subdomain. Inside, electric field The time-harmonic Maxwell equations for the electromagnetic field must be satisfied, using the variational form:

[0102]

[0103] in, Indicates in subdomain The unknown quantity of the electric field vector on; Relative permeability; It is the relative permittivity; The incident wave number; It represents the curl of the electric field, reflecting the distribution of the magnetic field.

[0104] The current source is expressed as follows:

[0105]

[0106] in, This refers to the applied source current density. Free-space wave impedance (constant); It is the product of the imaginary unit and the wave number, reflecting the time-harmonic characteristics of electromagnetic waves.

[0107] By assembling the electric field information and interface conditions within each subdomain into a global matrix, the following global block sparse system is obtained:

[0108]

[0109] in, subdomain Independent stiffness matrix, by Generate independently; and Interface-Subdomain The generated coupling matrix reflects the information exchange between each subdomain and the interface, and is achieved through a mapping matrix; The interface regularization matrix is ​​represented by... Generate variables to ensure their self-consistency and numerical stability. subdomain A collection of tangential electric field traces; The set of all interface Lagrange multipliers; subdomain The source item.

[0110] Solve for the global matrix (19):

[0111] (1) The decomposition matrix system consists of two sets of equations.

[0112] Subdomain equations (main block):

[0113]

[0114] Interface equations (boundary / constraints):

[0115]

[0116] (2) Construct the Schur complement system.

[0117] In formula (20) Consider it as the main unknown quantity. As the coupling variables, we can transform formula (20) to obtain:

[0118]

[0119] Substituting formula (22) into formula (21) yields the following: The equation:

[0120]

[0121] Further analysis revealed:

[0122]

[0123] make

[0124]

[0125] Where Y represents the constructed Schur complement matrix.

[0126]

[0127] The Schur complement system equations are obtained:

[0128]

[0129] (3) Obtain the solution vector using the direct method or iterative method. This represents the solution for all interface Lagrange multipliers. The solution obtained... Independent back-substitution within each subdomain, i.e.:

[0130]

[0131] Obtain the electric field degree of freedom coefficients for all subdomains .

[0132] (4) Use interpolation to globally stitch together the entire computational domain The global electromagnetic field distribution over the domain. For any point within the domain... Determine which subdomain it belongs to. The electric field value is reconstructed using interpolation formulas:

[0133] (29)

[0134] in, The electric field strength at that point is... For the selected basis functions, The first degree of freedom of the electric field in the subdomain One portion, This represents the number of local degrees of freedom for that subdomain. By performing the above interpolation and concatenation on all subdomains, the complete electromagnetic field distribution of the entire computational domain is obtained.

[0135] This completes the entire electromagnetic simulation process.

[0136] This embodiment also compares the error convergence of the FETI-2λ method of the present invention with that of the existing FEM method in the tangentially continuous LM space. The results are respectively referred to [reference needed]. Figure 5 . Figure 4 The method of this invention decomposes a cube into multiple subdomains of different colors, where each subdomain is decomposed using an unstructured mesh. The mesh is a second-order finite element mesh with a specific element size. exist arrive Between. Existing FEM methods use a single global mesh, i.e., the standard finite element method without subdomain decomposition. According to Figure 5Analysis shows that dividing the tangentially continuous LM space into more subdomains (high parallelism) does not result in a loss of accuracy, indicating that the FETI-2λ method is highly scalable and suitable for ultra-large-scale electromagnetic simulations.

[0137] This embodiment also provides an electromagnetic simulation system for an array antenna, the system comprising:

[0138] The subdomain construction module is responsible for decomposing the overall computational domain into several non-overlapping subdomains using a non-conformal region, where the mesh of each subdomain is a non-conformal mesh; defining Lagrange multipliers using interface current traces and interface electric field tangential traces; constructing symmetric transport conditions for adjacent subdomains based on the Lagrange multipliers; mapping all Lagrange multiplier degrees of freedom to the discrete spaces of the corresponding subdomains and interfaces to achieve the transformation and equilibrium of physical fields, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains; defining bilinear interface conditions based on the Lagrange multipliers, so that the electromagnetic response is naturally transmitted between subdomains, achieving correct coupling of various fields between non-conformal meshes;

[0139] The first calculation module is responsible for solving the electric field in each subdomain;

[0140] The second calculation module is responsible for assembling the electric field information and interface conditions in each subdomain into a global matrix, and obtaining the electric field information of the complete calculation domain by solving the global matrix.

[0141] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0142] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0143] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0144] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0145] As described above, this invention first overcomes the strict requirements of traditional domain decomposition methods on conformal meshes by processing non-conformal meshes, enabling computation to handle more complex geometries and greatly improving the applicability and flexibility of the method. When processing irregular meshes, it is no longer necessary to forcibly match the mesh structure at mesh boundaries, thereby reducing computational errors caused by inconsistent mesh partitioning and ensuring computational accuracy. Secondly, this invention significantly improves the computational efficiency of electromagnetic simulation tasks. By adopting a parallel computing framework and a scalable method structure, simulation tasks can be distributed across multiple computing nodes, greatly reducing computation time. The improvement in computational efficiency is particularly prominent in large-scale electromagnetic simulation tasks, meeting the dual requirements of simulation accuracy and computational speed in practical engineering.

[0146] Furthermore, the method of this invention exhibits excellent scalability and topology independence, adapting to different numbers of region divisions and topological structures. This makes the invention applicable not only to regular geometries but also to irregular or complex geometries, demonstrating greater versatility and adaptability. Simultaneously, the method's flexibility and scalability enable its wide application in various electromagnetic simulation tasks, providing a stable and efficient solution.

[0147] In summary, this invention provides an innovative parallel computing method for non-conformal meshes, solving the bottleneck problems of mesh generation and computational efficiency in existing technologies, and demonstrating significant advantages, especially in handling large-scale electromagnetic simulation tasks. By optimizing mesh generation, introducing a parallel computing framework, and addressing the issues of region partitioning and topology independence, this invention not only improves computational efficiency and accuracy but also expands the applicability of electromagnetic simulation methods, thus promoting the development of this field.

[0148] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. An electromagnetic simulation method for array antennas based on non-conformal region decomposition, characterized in that, The method includes: The overall computational domain is decomposed into several non-overlapping subdomains using a non-conformal region, and the mesh of each subdomain is a non-conformal mesh. Lagrange multipliers are defined using interface current traces and interface electric field tangential traces; based on the Lagrange multipliers, symmetric transmission conditions are constructed for adjacent subdomains; all Lagrange multiplier degrees of freedom are mapped to the discrete space of the corresponding subdomains and interfaces to realize the transformation and equilibrium of the physical field, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains and ensuring the correct transmission of the physical field on the domain surface. Based on the Lagrange multipliers, a bilinear interface condition is defined to allow the electromagnetic response to propagate naturally between subdomains, thus achieving correct coupling of various fields between non-conformal meshes. Solve for the electric field in each subdomain; The electric field information and interface conditions in each subdomain are assembled into a global matrix, and the electric field information of the complete computational domain is obtained by solving the global matrix. The process of mapping all Lagrange multiplier degrees of freedom to the discrete space of corresponding subdomains and interfaces to achieve the transformation and equilibrium of the physical field, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains and ensuring the correct transmission of the physical field on the domain surface, is as follows: For each pair of adjacent subdomains and Theoretically, two sets of symmetrical interface conditions need to be applied to the domain surface; In non-conformal meshes, the degrees of freedom on both sides of the interface are different, so the interface degree of freedom mapping and weights need to be considered. Therefore, an interface-subdomain projection mapping matrix is ​​introduced. To achieve the transformation and equilibrium of physical fields; In the finite element discretization and assembly process, all interface-related degrees of freedom need to be projected onto their respective discrete spaces through the above mapping matrix, so as to achieve accurate conversion and balance of physical information under different subdomains and different grids; in order to avoid the repeated inclusion of the tangential electric field trace, only the master equation on one side needs to be retained.

2. The method according to claim 1, characterized in that, The non-conformal mesh is defined by independently setting the number of elements, shape, and node distribution of each subdomain mesh.

3. The method according to claim 1, characterized in that, The connectivity between subdomains after non-conformal domain decomposition is expressed by a Boolean matrix. Definition: For any two adjacent subdomains, there exists a pair of interfaces. and , and They represent the first Subdomain facing the first Interfaces between subdomains, the first Subdomain facing the first The interface between subdomains; and because of the domain surface It is an interface and The overall computational domain decomposed into a skeleton composed of a set of domain surfaces. The set of interfaces that each subdomain faces to other subdomains forms the domain boundary within that subdomain. .

4. The method according to claim 3, characterized in that, The Lagrange multipliers It only applies to the skeleton after the global computational domain is decomposed. The electric field within the decomposed subdomain is continuous within its own subdomain, but discontinuities are allowed at the interface.

5. The method according to claim 1, characterized in that, The specific implementation process of constructing symmetric transmission conditions for adjacent subfields based on Lagrange multipliers is as follows: Apply symmetrical transmission conditions to both sides of each pair of interfaces, i.e., each pair of adjacent subdomains and Symmetry conditions for the tangential components of the electromagnetic field and the tangential components of the magnetic field on both sides of the domain surface.

6. The method according to claim 1, characterized in that, The bilinear interface conditions include subdomain-interface coupling terms, interface-subdomain coupling terms, and interface regularization terms.

7. An electromagnetic simulation system for an array antenna that implements the method of any one of claims 1-6, characterized in that, The system includes: The subdomain construction module is responsible for decomposing the overall computational domain into several non-overlapping subdomains using a non-conformal region, where the mesh of each subdomain is a non-conformal mesh; defining Lagrange multipliers using interface current traces and interface electric field tangential traces; constructing symmetric transport conditions for adjacent subdomains based on the Lagrange multipliers; mapping all Lagrange multiplier degrees of freedom to the discrete spaces of the corresponding subdomains and interfaces to achieve the transformation and equilibrium of physical fields, thereby adapting to the flexible assembly of non-conformal meshes and multiple subdomains; defining bilinear interface conditions based on the Lagrange multipliers, so that the electromagnetic response is naturally transmitted between subdomains, achieving correct coupling of various fields between non-conformal meshes; The first calculation module is responsible for solving the electric field in each subdomain; The second calculation module is responsible for assembling the electric field information and interface conditions in each subdomain into a global matrix, and obtaining the electric field information of the complete calculation domain by solving the global matrix.

8. A computer-readable storage medium having a computer program stored thereon, which, when executed in a computer, causes the computer to perform the method of any one of claims 1-6.

9. A computing device comprising a memory and a processor, wherein the memory stores executable code, and the processor, when executing the executable code, implements the method of any one of claims 1-6.

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