A residual estimate non-conformal adaptive encryption volume integral equation electromagnetic simulation method

Abstract

The present application belongs to the technical field of electromagnetic simulation, and specifically provides a residual error estimation non-conformal adaptive encryption volume integral equation electromagnetic simulation method to complete high-precision and high-efficiency electromagnetic simulation of complex multi-scale non-uniform targets. The present application proposes a grid error estimator for volume integral equations, performs error analysis on each tetrahedral grid to obtain error distribution of the grid; then sets an error percentage threshold, selects the grid with the largest error according to the error percentage threshold to mark, and performs non-conformal bisection encryption on the marked grid, thereby completing non-conformal adaptive encryption based on grid error. On the premise of increasing a small amount of calculation unknowns, the grid quality is greatly improved, and the calculation precision and calculation efficiency of electromagnetic simulation are improved. In summary, the present application proposes a residual error estimation non-conformal adaptive encryption volume integral equation electromagnetic simulation method, which is especially suitable for electromagnetic characteristic calculation of complex multi-scale non-uniform targets.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation technology, specifically providing an electromagnetic simulation method for nonconformal adaptive encrypted volume fractional equations with residual estimation. Background Technology

[0002] With the increasingly widespread application of multi-scale targets such as unmanned aerial vehicles and swarm targets, the study of the electromagnetic properties of complex non-uniform media targets such as flocks of birds and locusts is becoming more in-depth. Electromagnetic simulation of complex multi-scale non-uniform targets has gradually become an important research direction in the field of modern computational electromagnetics. However, in actual simulation, it is difficult to obtain high-quality meshes for complex multi-scale non-uniform targets. The quality of the mesh directly affects the efficiency and accuracy of the computation. Therefore, how to obtain high-quality electromagnetic computational meshes is the focus of this invention.

[0003] In computational electromagnetics, computational targets mainly include metallic targets and non-uniform dielectric targets. For metallic targets, the method of moments (MoM) based on the area integral equation is mainly used for solving them. Its principle is the principle of surface equivalence. By setting an equivalent current on the target surface to wrap the object, the metallic target is equivalent to a surface problem in free space. Its characteristic is that there are few unknowns. For non-uniform dielectric targets, the method of moments based on the volume integral equation is mainly used for solving them. Its principle is the principle of volume equivalence. By setting an equivalent volume current throughout the entire volume of the target to fill the object, the non-uniform or lossy dielectric target is equivalent to a volume polarization problem in free space. Its characteristic is that it has strong versatility and can handle complex media. For non-uniform media targets, the paper "R. Liu, G. Xiao, S. Huang and Y. Hu, Multibranch Schaubert–Wilton–Glisson Basis Functions for Electromagnetic Scattering Problem, in IEEE Transactions on Antennas and Propagation, vol. 70, no. 4, pp. 3100-3105, April 2022" proposes a volume integral non-conformal mesh calculation method. However, this calculation method adopts traditional mesh processing methods, directly dividing the mesh according to experience, which makes the density distribution of the mesh lack physical characteristics, thus affecting the computational efficiency and accuracy. On the other hand, for metallic targets, the paper "JA Tobon Vasquez, Z. Peng, J.-F. Lee, G. Vecchi and F. Vipiana, Automatic Localized Nonconformal Mesh Refinement for SurfaceIntegral Equations, in IEEE Transactions on Antennas and Propagation, vol.68, no. 2, pp. 967-975, Feb. 2020" discloses a mesh error estimator for surface integrals. This estimator can obtain the error distribution of the mesh under specific electromagnetic environments, thereby enabling adaptive mesh refinement based on the error distribution to obtain a high-quality mesh. However, this error estimator is only applicable to triangular meshes and can only be used to calculate metallic surfaces, failing to address complex non-uniform media. Therefore, this invention proposes a residual estimation-based nonconformal adaptive refinement method for electromagnetic simulation of volume integral equations, enabling high-precision and high-efficiency electromagnetic simulation of complex multi-scale non-uniform targets. Summary of the Invention

[0004] The purpose of this invention is to provide a non-conformal adaptive densification method for electromagnetic simulation of volume fraction equations with residual estimation, which can be used to calculate the electromagnetic properties of complex multi-scale non-uniform targets. This invention proposes a mesh error estimator for volume fraction equations to accurately estimate the mesh error, and then uses a non-conformal densification method to directly densify specific mesh cells locally without affecting the surrounding mesh. Under the premise of increasing a small number of unknowns, it significantly improves the mesh quality and enhances the computational accuracy and efficiency of electromagnetic simulation.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] An electromagnetic simulation method for nonconformal adaptive densified volume integral equations with residual estimation includes the following steps:

[0007] Step 1. Divide the region into regions based on the dielectric constant or scale characteristics of each region in the calculation target, and perform tetrahedral meshing on each sub-region to obtain mesh information;

[0008] Step 2. Construct volume integral equations within the mesh based on the principle of volume equivalence, and define SWG basis functions, MB-SWG basis functions, or Half-SWG basis functions on the mesh according to the mesh information;

[0009] Step 3. Discretize the electric displacement vector within the grid using basis functions, substitute it into the volume integral equation, and obtain the discrete form of the volume integral equation; then perform the Galerkin test on the discrete form of the volume integral equation to obtain a linear system of equations containing N equations, where N is the total number of basis functions;

[0010] Step 4. Solve the linear equation system using the Krylov subspace iterative method to obtain the electric displacement vector coefficient vector. Then, calculate the electric displacement vector distribution of the target based on the electric displacement vector coefficient vector and the basis function, and then calculate the far field.

[0011] Step 5. Based on the solved electric displacement vector coefficients, construct a residual error estimator for the volume integral equation, and perform error analysis on each tetrahedral mesh to obtain the error distribution of the mesh;

[0012] Step 6. Set an error percentage threshold, select the grid with the largest error according to the error percentage threshold and mark it, then refine the marked grid using a non-conformal bisection method;

[0013] Step 7. When the number of adaptive encryption attempts reaches the preset limit, output the far field; otherwise, proceed to step 2 for iteration.

[0014] Furthermore, in step 2, adjacent tetrahedral meshes are combined into standard tetrahedral pairs. For tetrahedrals that fail to pair successfully, they are paired again in a one-to-many manner to form extended tetrahedral pairs. Based on this, SWG basis functions are defined on the standard tetrahedral pairs, MB-SWG basis functions are defined on the extended tetrahedral pairs, and Half-SWG basis functions are defined on individual tetrahedral faces.

[0015] Furthermore, in step 3, the system of linear equations is expressed as follows: ,in, This represents the impedance matrix, specifically an N-dimensional square matrix; This represents the coefficient vector of the electric displacement vector to be solved, specifically an N×1 dimensional column vector; This represents the activation term, specifically an N×1 dimensional column vector.

[0016] Furthermore, in step 5, the specific process of constructing the residual error estimator for the volume integral equation is as follows:

[0017] First, the residual of the volume fraction equation is expressed as:

[0018] ,

[0019] in, Represents the residual. Represents a position vector. Represents the electric displacement vector. Represents the dielectric constant. Represents the imaginary unit. Represents angular frequency. Indicates the magnetic vector position. Represents the gradient operator. Indicates a scalar bit. Indicates the incident field;

[0020] Then, each tetrahedral mesh is refined by dividing it into eight parts, and the basis functions are redefined on the refined meshes. The residuals are projected onto the redefined basis functions to obtain the projection result, which is expressed as: , This represents the redefined basis functions within the subgrid; the projection result is then normalized, and the normalized result is used as the error estimate of the basis functions within the subgrid, expressed as:

[0021] ,

[0022] in, Describing basis functions The error estimate;

[0023] Finally, for each tetrahedral mesh, the error estimates of all sub-mesh are calculated, and the maximum value is taken as the error estimate of the tetrahedral mesh, expressed as:

[0024] ,

[0025] in, This represents the error estimate of the k-th tetrahedral mesh. This represents the error estimate of the basis function within the subgrid of the k-th tetrahedral mesh.

[0026] Furthermore, in step 6, the error percentage threshold is set to 30%~60%.

[0027] Furthermore, in step 6, the specific process of non-conformal bisection encryption is as follows: select the longest edge of the tetrahedron to be encrypted as the edge to be encrypted, take the midpoint of the edge to be encrypted, and connect this midpoint to the vertex of the tetrahedron that is not on the edge to be encrypted.

[0028] Furthermore, in step 7, the preset upper limit for the number of adaptive encryption attempts is 3 to 6.

[0029] Based on the above technical solution, the beneficial effects of the present invention are as follows:

[0030] This invention provides a non-conformal adaptive densification method for electromagnetic simulation of volume fractional equations using residual estimation, for calculating the electromagnetic properties of complex multi-scale non-uniform targets. It provides a residual error estimator for the volume fractional equations, enabling accurate error estimation of the mesh. Then, a non-conformal densification method is used to locally densify specific mesh cells without affecting surrounding meshes. This invention is particularly suitable for calculating the electromagnetic properties of complex multi-scale non-uniform targets. The non-conformal adaptive densification significantly improves mesh quality, substantially increasing computational accuracy and efficiency with only a small increase in computational unknowns. Attached Figure Description

[0031] Figure 1 This is a schematic diagram of the mesh partitioning of the target calculation in an embodiment of the present invention.

[0032] Figure 2 This is a residual distribution diagram of the initial grid in an embodiment of the present invention.

[0033] Figure 3 This is an error distribution diagram of the initial grid in an embodiment of the present invention.

[0034] Figure 4 This is a schematic diagram illustrating the principle of non-conformal binary search encryption in an embodiment of the present invention.

[0035] Figure 5The graphs shown are bistatic radar cross section (RCS) curves calculated from the mesh obtained by the fifth non-conformal adaptive encryption and the mesh obtained by direct subdivision in this embodiment of the invention.

[0036] Figure 6 This is a graph showing the root mean square error (RMSE) of the bistatic RCS of the nonconformal adaptive encrypted volume fractional equation electromagnetic simulation method for residual estimation in this embodiment of the invention, as a function of the number of encryption iterations. Detailed Implementation

[0037] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0038] This embodiment provides a non-conformal adaptive encrypted volume integral equation electromagnetic simulation method for residual estimation, which specifically includes the following steps:

[0039] Step 1. Divide the regions according to the dielectric constant or scale characteristics of each region in the calculation target, and perform tetrahedral meshing on each sub-region to obtain mesh information;

[0040] In this embodiment, the calculation target is a non-uniform medium, such as... Figure 1 As shown, the bottom dielectric plate is 1m long and 1m wide, and 0.1m high, with a relative permittivity of 2; the upper cross structure is composed of two cuboids, each 0.5m long, 0.1m wide, and 0.1m high, with a relative permittivity of 10; the incident wave frequency is 100MHz, the incident direction is along the z-direction, the electric field polarization direction is along the x-direction, the partition size of the dielectric plate is 0.3m, and the partition size of the cross structure is 0.1m.

[0041] Step 2. Construct volume integral equations within the mesh based on the principle of volume equivalence. Define SWG basis functions, MB-SWG basis functions, or Half-SWG basis functions on the mesh according to the mesh information. Form standard tetrahedral pairs by connecting adjacent tetrahedral meshes. For tetrahedrals that fail to pair successfully, pair them again in a one-to-many manner to form extended tetrahedral pairs. Based on this, define SWG basis functions on the standard tetrahedral pairs, MB-SWG basis functions on the extended tetrahedral pairs, and Half-SWG basis functions on individual tetrahedral faces.

[0042] Step 3. Discretize the electric displacement vector within the grid using basis functions, substitute it into the volume integral equation, and obtain the discrete form of the volume integral equation; then perform the Galerkin test on the discrete form of the volume integral equation, and simultaneously use N basis functions to form the inner product on both sides of the volume integral equation to expand the original equation into N equations, forming a linear system of N equations.

[0043] Define the coefficient terms on the left-hand side of the linear equation system as the impedance matrix. , The matrix is ​​N-dimensional; the coefficients of the basis function expansion are used as the coefficients to be solved, forming the electric displacement vector coefficient vector to be solved. , The system of linear equations is an N×1 dimensional column vector; the coefficient terms on the right-hand side are defined as excitation terms. , Given an N×1 dimensional column vector, a system of linear equations is derived: ;

[0044] Step 4. Solve the linear equation system using the Krylov subspace iterative method to obtain the electric displacement vector coefficient vector. Then, based on the electric displacement vector coefficient vector... The electric displacement vector distribution of the target is obtained by calculating the basis functions, and then the far-field is calculated.

[0045] Step 5. Based on the solved electric displacement vector coefficients A residual error estimator for the volume fraction equation is constructed, and error analysis is performed on each tetrahedral mesh to obtain the error distribution of the mesh.

[0046] In this embodiment, the construction process of the residual error estimator for the volume fraction equation is as follows:

[0047] The volume fraction equation is expressed as:

[0048] ,

[0049] in, Represents a position vector. Represents the electric displacement vector. Represents the dielectric constant. Represents the imaginary unit. Represents angular frequency. Indicates the magnetic vector position. Represents the gradient operator. Indicates a scalar bit. Indicates the incident field;

[0050] The residual form of the volume integral equation is:

[0051] ,

[0052] in, Represents the residual;

[0053] Due to the characteristics of the Galerkin test method, the result of projecting the residuals onto the currently defined basis functions is zero. Therefore, in order to reveal the residuals and use them to estimate the grid error, it is necessary to construct a new set of basis functions.

[0054] Each tetrahedral mesh is divided into eight parts for refinement. The basis functions are then redefined on the refined meshes in the same way as in step 2, thus obtaining a more complete set of basis functions.

[0055] Projecting the residuals onto the newly constructed basis functions, i.e., taking the inner product of the residuals and the basis functions, is expressed as: , This represents the redefined basis function;

[0056] The residual can then be expressed using the redefined basis functions as follows: , This represents the total number of redefined basis functions; the larger the value of the inner product, the larger the component of the residual in that basis function, and the larger the error of the grid corresponding to that basis function.

[0057] Therefore, it can be achieved through the inner product. To achieve error estimation of the mesh, and to ensure the comparability of error estimates calculated for meshes of different sizes and media, it is necessary to optimize the inner product. After normalization, it is represented as: This eliminates the influence of grid size on the error estimate;

[0058] Since the overall dimensions of the residuals are consistent with those of the electric field, the error estimate is not affected by the dielectric constant. Therefore, the residuals of the basis functions are ultimately defined as:

[0059] ,

[0060] in, Describing basis functions The error estimate;

[0061] For each tetrahedral mesh, calculate the error estimates of its eight sub-mesh, and take the maximum value as the error estimate of the tetrahedral mesh, expressed as:

[0062] ,

[0063] Wherein, the superscript k represents the number of the tetrahedral mesh, and the subscript m represents the number of the basis function defined within the submesh;

[0064] In this embodiment, the residual distribution of the initial mesh is as follows: Figure 2 As shown, the error distribution of the initial grid is as follows: Figure 3 As shown;

[0065] Step 6. Set the error percentage threshold The percentage is 30%, based on the error percentage threshold. Select the grid with the largest error for marking, and then refine the marked grid using a non-conformal bisection method;

[0066] In this embodiment, non-conformal binary search encryption is as follows: Figure 4 As shown, the specific process is as follows: Select the longest edge of the tetrahedron to be encrypted as the edge to be encrypted, take the midpoint of the edge to be encrypted, and connect this midpoint to the vertex of the tetrahedron that is not on the edge to be encrypted.

[0067] While keeping the adjacent mesh structure unchanged, the non-conformal bisection method refinement divides the original tetrahedral mesh into two parts, thus forming a non-conformal mesh interface. On this non-conformal interface, the MB-SWG basis function can be defined.

[0068] Step 7. When the number of adaptive encryption attempts reaches the preset limit, output the far field; otherwise, proceed to step 2 for iteration.

[0069] In this embodiment, the preset upper limit for the number of adaptive encryption iterations is set to 5. The initial total number of basis functions N is 741. After five iterations of non-conformal adaptive encryption, the total number of basis functions increases to 935, 1201, 1520, 1818, and 2455 respectively. The bistatic radar cross section (RCS) calculated from the mesh obtained by the fifth non-conformal adaptive encryption and the mesh obtained by direct subdivision is as follows: Figure 5 As shown in the figure, the RCS of the two stations is almost completely identical. The unknown quantity after five non-conformal adaptive refinements in this embodiment is 2455, while the unknown quantity of the mesh obtained by direct subdivision is 5592. The unknown quantity is reduced by almost half. This shows that the present invention significantly improves the mesh quality and significantly improves the overall calculation accuracy and efficiency while increasing the number of unknown quantities by a small amount.

[0070] Furthermore, the non-conformal adaptive encryption method in this embodiment is compared with the conformal adaptive encryption method and the global encryption method. The bistation RCS value calculated after each mesh encryption is recorded sequentially. The bistation RCS calculated by the last global encryption is taken as the reference solution. Then, the root mean square error (RMSE) is calculated between the bistation RCS value calculated after each mesh encryption and the reference bistation RCS. The results are as follows. Figure 6 As shown in the figure, the non-conformal adaptive encryption method proposed in this invention can calculate more accurate results with fewer unknowns, thus proving the reliability of the residual error estimator proposed in this invention, and also proving the effectiveness of the residual estimation non-conformal adaptive encryption volume integral equation electromagnetic simulation method proposed in this invention.

[0071] The above description is merely a specific embodiment of the present invention. Any feature disclosed in this specification may be replaced by other equivalent or similar features unless otherwise specified. All disclosed features, or steps in all methods or processes, may be combined in any way except for mutually exclusive features and / or steps.

Claims

1. A non-conformal adaptive encrypted volume integral equation electromagnetic simulation method for residual estimation, characterized in that, Includes the following steps: Step 1. Divide the region into regions based on the dielectric constant or scale characteristics of each region in the calculation target, and perform tetrahedral meshing on each sub-region to obtain mesh information; Step 2. Construct volume integral equations within the mesh based on the principle of volume equivalence, and define SWG basis functions, MB-SWG basis functions, or Half-SWG basis functions on the mesh according to the mesh information; Step 3. Discretize the electric displacement vector within the grid using basis functions, substitute it into the volume integral equation, and obtain the discrete form of the volume integral equation; then perform the Galerkin test on the discrete form of the volume integral equation to obtain the linear equation system. Step 4. Solve the linear equation system using the Krylov subspace iterative method to obtain the electric displacement vector coefficient vector. Then, calculate the electric displacement vector distribution of the target based on the electric displacement vector coefficient vector and the basis function, and then calculate the far field. Step 5. Based on the solved electric displacement vector coefficients, construct a residual error estimator for the volume integral equation, and perform error analysis on each tetrahedral mesh to obtain the error distribution of the mesh; Step 6. Set an error percentage threshold, select the grid with the largest error according to the error percentage threshold and mark it, then refine the marked grid using a non-conformal bisection method; Step 7. When the number of adaptive encryption attempts reaches the preset limit, output the far field; otherwise, proceed to step 2 for iteration.

2. The residual estimation non-conformal adaptive encrypted volume fractional equation electromagnetic simulation method according to claim 1, characterized in that, In step 2, adjacent tetrahedral meshes are paired into standard tetrahedral pairs. For tetrahedrals that fail to pair successfully, they are paired again in a one-to-many manner to form extended tetrahedral pairs. Based on this, SWG basis functions are defined on the standard tetrahedral pairs, MB-SWG basis functions are defined on the extended tetrahedral pairs, and Half-SWG basis functions are defined on individual tetrahedral faces.

3. The electromagnetic simulation method for non-conformal adaptive densified volume fractional equations of residual estimation according to claim 1, characterized in that, In step 3, the system of linear equations is expressed as follows: ,in, Represents the impedance matrix. This represents the coefficient vector of the electric displacement vector to be solved. Indicates the incentive term.

4. The electromagnetic simulation method for non-conformal adaptive densified volume fractional equations of residual estimation according to claim 1, characterized in that, In step 5, the specific process of constructing the residual error estimator for the volume integral equation is as follows: First, the residual of the volume fraction equation is expressed as: ， in, Represents the residual. Represents a position vector. Represents the electric displacement vector. Represents the dielectric constant. Represents the imaginary unit. Represents angular frequency. Indicates the magnetic vector position. Represents the gradient operator. Indicates a scalar bit. Indicates the incident field; Then, each tetrahedral mesh is refined by dividing it into eight parts, and the basis functions are redefined on the refined meshes. The residuals are projected onto the redefined basis functions to obtain the projection result, which is expressed as: , This represents the redefined basis functions within the subgrid; the projection result is then normalized, and the normalized result is used as the error estimate of the basis functions within the subgrid, expressed as: ， in, Describing basis functions The error estimate; Finally, for each tetrahedral mesh, the error estimates of all sub-mesh are calculated, and the maximum value is taken as the error estimate of the tetrahedral mesh, expressed as: ， in, This represents the error estimate of the k-th tetrahedral mesh. This represents the error estimate of the basis function within the subgrid of the k-th tetrahedral mesh.

5. The electromagnetic simulation method for non-conformal adaptive densified volume fractional equations of residual estimation according to claim 1, characterized in that, In step 6, the error percentage threshold is set to 30%~60%.

6. The electromagnetic simulation method for non-conformal adaptive densified volume fractional equations of residual estimation according to claim 1, characterized in that, In step 6, the specific process of non-conformal bisection encryption is as follows: select the longest edge of the tetrahedron to be encrypted as the edge to be encrypted, take the midpoint of the edge to be encrypted, and connect this midpoint to the vertex of the tetrahedron that is not on the edge to be encrypted.

7. The electromagnetic simulation method for non-conformal adaptive densified volume fractional equations of residual estimation according to claim 1, characterized in that, In step 7, the preset upper limit for the number of adaptive encryption attempts is 3 to 6.

Images