A non-uniform conformal FDTD method based on subgridding technique

Abstract

This invention belongs to the field of numerical simulation in computational electromagnetics, specifically a non-uniform conformal FDTD method based on submesh technology. This invention introduces non-uniform conformal geometric corrections within the submesh to achieve high-precision fitting of curved surface boundaries, reducing step approximation errors and improving the accuracy of field calculations near the boundaries. At the master-submesh interface, a bidirectional coupling transfer is established that simultaneously satisfies line integral conservation and flux conservation, avoiding disruption of the conservation relationships required by the non-uniform conformal FDTD method. This effectively suppresses numerical reflections at the interface, reduces energy non-physical errors, and improves long-term stability. Furthermore, by using fine meshes only in the neighborhood of local microstructures and coordinating with time sub-cycle progression, global mesh refinement and the global time step being limited by the smallest unit are avoided, significantly reducing storage and computation time overhead while maintaining accuracy. This invention balances geometric modeling accuracy, interface stability, and computational efficiency, and has high engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of numerical simulation of computational electromagnetics, specifically a non-uniform conformal FDTD method based on submesh technology, which adopts conserved interface coupling to solve electromagnetic problems of curved structures and multi-scale. Background Technology

[0002] As modern electronic systems evolve towards higher frequencies, greater integration, and increased complexity, computational electromagnetics faces unprecedented challenges. Since its introduction by Kane Yee in 1966, the Finite-Difference Time-Domain (FDTD) method has become one of the mainstream algorithms in this field due to its explicit iteration, natural parallelism, and wide bandwidth solution capabilities. However, when dealing with complex engineering problems involving intricate surface structures and multi-scale features, traditional FDTD methods struggle to balance geometric modeling accuracy with computational efficiency.

[0003] First, the accuracy of geometric modeling is limited by the characteristics of curved surfaces. Traditional FDTD relies on regular Cartesian meshes, which necessitate the use of a "step approximation" when dealing with curved or sloping boundaries. This geometric quantization error not only reduces the solution accuracy of the boundary field to first order but also introduces non-physical spurious scattering, severely affecting the calculation of sensitive parameters such as the radar cross section (RCS). To address this issue, the Local Conformal FDTD method proposed by Dey and Mittra has achieved a breakthrough. This method corrects the update coefficients by accurately calculating the area and edge length of the truncated mesh, successfully maintaining second-order accuracy while significantly reducing the mesh size, becoming an important means of solving geometric modeling errors.

[0004] Secondly, computational efficiency is often constrained by the multi-scale characteristics of engineering models. Actual electromagnetic structures typically contain both large-scale global structures and fine-scale local structures. If the entire computational domain is uniformly refined according to the minimum geometric dimensions to analyze the fine structures, the number of grids will increase cubically with spatial dimensions, leading to a sharp rise in storage and computational overhead. Although non-uniform grids can reduce the number of grids to some extent, they are still constrained by the Cronbach's alpha (CFL) stability condition in explicit FDTD, where the global time step is determined by the minimum spatial step, thus significantly increasing the number of iterations. To balance accuracy and efficiency, subgrid techniques refine only the local regions containing fine structures and use smaller time steps locally in conjunction with time sub-loops. This achieves improved local accuracy and controlled overall computational load without refining the entire domain or spreading the time step constraint corresponding to the minimum grid scale to the entire domain.

[0005] However, existing subgrid methods are all based on the traditional FDTD update framework and lack conformal treatment of surface boundaries. They typically use step approximations, which easily introduce significant step errors, thus affecting computational accuracy. More importantly, the field exchange at the master-subgrid interface often relies on empirical processing such as point value interpolation, extrapolation, or simple averaging. This makes it difficult to simultaneously satisfy the loop integral consistency and surface flux conservation relationship required by non-uniform conformal FDTD methods, easily introducing numerical reflections, non-physical energy gains, and even leading to a decrease in stability at the interface.

[0006] Therefore, there is an urgent need for a finite-difference time-domain (FDTD) method that can maintain conformal geometric accuracy while taking into account computational efficiency, so as to improve the simulation accuracy and computational efficiency of complex surfaces and multi-scale electromagnetic structures and meet the needs of engineering analysis and design. Summary of the Invention

[0007] To address the aforementioned problems and shortcomings, and to resolve the difficulty of balancing conformal geometric accuracy and computational efficiency when dealing with complex targets containing intricate surface structures using traditional FDTD methods, this invention provides a non-uniform conformal FDTD method based on submesh technology. This method employs conserved interface coupling: both the master and submesh utilize non-uniform conformal FDTD to improve the geometric modeling accuracy of the surface structure; the submesh is refined in local regions and employs time sub-loops to enhance multi-scale computational efficiency; and geometrically weighted line integral conservation and flux conservation constraints are established at the master-submesh interface based on conformal equivalent side length / equivalent area, achieving bidirectional consistent transmission of electric and magnetic fields. This suppresses numerical reflection at the interface, avoids non-physical energy gains, and improves algorithm stability, demonstrating high engineering application value.

[0008] The specific technical solution is as follows:

[0009] A non-uniform conformal FDTD method based on submesh technology includes the following steps:

[0010] Step 1: Establish a non-uniform conformal master grid covering the entire computational domain; then select a local sub-region containing fine structures and curved boundaries as the refinement region, and refine the master grid cells of the refinement region according to the refinement factor. Subdivide the mesh to generate sub-mesh. The outer boundary of the sub-mesh is defined as the main-sub-mesh interface, and the main mesh and the sub-mesh exchange field quantities at the interface.

[0011] Both the master and sub-mesh follow the spatiotemporal staggered arrangement mechanism of standard Yee cells, satisfying the following conditions: the electric field is located at the center of the mesh edge and at integer time steps. The magnetic field is located at the center of the grid surface and at half-integer time steps. , where n is an integer greater than or equal to 0.

[0012] Define the electromagnetic field of the main grid as uppercase. Conformal geometric parameters are in uppercase. Subgrids correspond to lowercase. and The main grid uses a non-uniform spatial step size that varies with location. and reference time step If the submesh performs local fine-grained encryption in the encrypted region, then the spatial step size is refined. Refine the time step , Odd numbers are used to ensure that there exists a subgrid time step that is equal to the half-integer time step of the main grid. Alignment allows for the synchronous exchange of magnetic field quantities at the master-submesh interface and avoids time interpolation.

[0013] Based on this, the conformal edge lengths in the master and child meshes are calculated and stored. and conformal area ,in , , The direction is a three-dimensional rectangular coordinate. To be parallel to Conformal edge length of the main mesh in the direction, For respectively located in The conformal area of ​​the main mesh in a plane is calculated, and the same applies to the sub-mesh.

[0014] Step 2: Set the excitation source and boundary conditions, and initialize all electromagnetic field values ​​in the computational domain to zero. Set the total number of time steps to [number missing]. The initial time step n=0.

[0015] Step 3, utilize The main grid electric field at time The magnetic field of the entire main grid, excluding the sub-grid coverage area, is updated according to formulas (1), (2), and (3) to obtain... Magnetic field prediction at time .

[0016] In non-uniform conformal FDTD, the non-uniform conformal mesh used needs to correct the element geometry through conformal processing near the boundary of the ideal conductor PEC to reduce the error caused by the step approximation. This method defines the conformal edge length within the mesh, excluding the portion of the ideal conductor PEC. and conformal area .

[0017] Based on the integral form of Faraday's law, for Magnetic field component The magnetic field update formulas for non-uniform conformal FDTD are as follows:

[0018] (1)

[0019] (2)

[0020] (3)

[0021] Meanwhile, its electric field update formula is:

[0022] (4)

[0023] (5)

[0024] (6)

[0025] in For time step, Where is the dielectric constant. Permeability, Represent Orientation of grid cell index, Represent Grid step size in the direction.

[0026] The electromagnetic field follows the spatiotemporal staggered arrangement mechanism of the standard Yee cell, with the electric field located at the center of the grid edges and integer time steps. The magnetic field is located at the center of the grid surface and at half-integer time steps. Specifically, it is expressed as: , , , , and .in, Representing along Grid cell indices in three coordinate directions, Indicates that it is located at At this location, the time step is The x-axis magnetic field component at time t; Indicates that it is located at At this point, the time step is The y-axis magnetic field component at time t; Indicates that it is located at At this location, the time step is The z-axis magnetic field component at time t; correspondingly, Indicates that it is located at At this location, the time step is The x-axis electric field component at time t; Indicates that it is located at At this point, the time step is The y-axis electric field component at time t; Indicates that it is located at At this point, the time step is The z-axis electric field component at time t.

[0027] Step 4: Refine the time step for the sub-mesh. conduct The next iteration will change the sub-time from Advance to In each sub-loop, the tangential electric field on the edge of the main mesh is set along its conformal edge length. The electric field integral is conformal to the edge length of its corresponding subgrid. The tangential electric field integrals are equal to ensure that the tangential electric field at the subgrid interface satisfies the line integral conservation.

[0028] Based on the spatial correspondence of the electric fields at the interface between the main and sub-grids, the electric field of the main grid is interpolated using formula (7), and the interpolation result is assigned to the electric field of the sub-grid.

[0029] (7)

[0030] for The parameters for direction are: and for Two adjacent main grids at time Towards electric field and , and For the corresponding conformal edge lengths, the electric fields of these two adjacent principal grids pass through... After refinement, it will correspond to Subgrid Towards electric field , for This moment Subgrid Towards the electric field, This represents the corresponding conformal side length. and For value index, and Combining them can achieve this Subgrid Towards electric field , The index for the summation of the conformal side lengths of the submesh. The subscript of the combination parameter in equation (7). Change to and Then it represents the corresponding and Relevant parameters in the direction.

[0031] The electric field of all corresponding main grids on the main-sub grid interface is traversed, and the electric field of the sub grid boundary is assigned according to Equation (7).

[0032] Then, the magnetic and electric fields inside the subgrid are updated alternately using the magnetic and electric field update formulas (1) to (6) from step 3. The next iteration indicates that both the magnetic and electric fields are updated. Second-rate.

[0033] Step 5: When the subgrid time advances to synchronization with the main grid magnetic field At that time, set the conformal area of ​​the main mesh. The magnetic flux on the surface and the conformal area of ​​its corresponding subgrid The magnetic fluxes on the grid are equal to satisfy the flux-weighted conservation. The equivalent magnetic field fed back to the main grid is calculated using formula (8). .

[0034] (8)

[0035] for The parameters for direction are: for Real-time feedback to the main grid Towards the magnetic field, This represents the corresponding conformal area. A principal grid magnetic field passes through... After refinement, it will correspond to Subgrid magnetic field , for Moment Subgrid Towards the magnetic field, For the corresponding conformal area, For summation index. The subscript of the combined parameters in equation (8). Change to and Then it represents the corresponding and Relevant parameters in the direction.

[0036] The feedback magnetic field of the main grid is obtained by summing the magnetic fields of the sub-grids by area weighting according to equation (8). The above relationship refers to a main grid on the interface. During the calculation, all corresponding main grids on the interface between the main grid and the sub-grid are traversed one by one.

[0037] Subsequently, the equivalent magnetic field of feedback was used. Correct the main grid magnetic field estimate obtained in step 3. This allows the physical information of the subgrid structure to be transmitted back to the main grid in a conserved manner.

[0038] Step 6: Use the corrected magnetic field estimate from Step 5. The main grid electric field is updated according to formulas (4) to (6), resulting in... electric field at time Complete from arrive This is a time step update.

[0039] Step 7, for Repeat steps 3 through 6 in sequence until... Finish This process involves updating the data at each time step, thus completing the full-time progression and obtaining the final desired time-domain electromagnetic field result. In this way, the non-uniform conformal FDTD method of subgrid technology is realized through conserved interface coupling.

[0040] Furthermore, , Taking values ​​that are too large can lead to a significant increase in the amount of computation.

[0041] Furthermore, the total number of time steps set in step 2 It is a positive integer not less than 1, so that steps 3 to 6 are executed at least once, thereby completing one main time step update.

[0042] Furthermore, the total number of time steps set in step 2 For a preset finite positive integer, generally speaking, the total number of time steps The initial empirical setting value should not exceed 10,000, and can be adjusted according to simulation results and computational costs. Although a value that is too large does not change the basic process of this invention, it will lead to an unnecessary increase in computational costs.

[0043] The significant advantages of this invention compared to existing technologies are:

[0044] (1) High accuracy of surface geometry modeling: Non-uniform conformal geometry correction is introduced inside the sub-mesh, which can fit the surface boundary with high accuracy, reduce the step approximation error and improve the accuracy of the boundary field.

[0045] (2) Interface reflection suppression and stability improvement: The interface coupling needs to satisfy both line integral conservation and flux conservation to avoid disrupting the conservation relationship required by the non-uniform conformal FDTD method, thereby significantly reducing numerical reflection at the master-sub-mesh interface and improving long-term stability.

[0046] (3) High computational efficiency: Fine meshes are used only near local microstructures and time sub-cycles are performed to avoid global mesh refinement and global time steps being limited to the smallest unit, thus significantly reducing memory and computation time overhead while ensuring accuracy. Attached Figure Description

[0047] Figure 1This is a flowchart of the present invention;

[0048] Figure 2 This is a model diagram of a dipole antenna array as an example.

[0049] Figure 3 A schematic diagram of a conformal mesh;

[0050] Figure 4 A schematic diagram of the tangential electric field at the main-sub-grid interface;

[0051] Figure 5 A schematic diagram of the tangential magnetic field at the interface between the master and subgrids;

[0052] Figure 6 A 1D polar coordinate diagram of a dipole antenna array for an example embodiment ( );

[0053] Figure 7 A 1D polar coordinate diagram of a dipole antenna array for an example embodiment ( ). Detailed Implementation

[0054] The present invention will now be described in further detail with reference to the embodiments and accompanying drawings.

[0055] To verify the effectiveness of the non-uniform conformal FDTD method using the conserved interface coupled subgrid technique proposed in this invention, this embodiment selects a dipole antenna array for simulation verification. The simulation frequency range is set to 25~35GHz, and the calculation region boundary adopts a 10-layer PML (perfectly matched layer) absorbing boundary condition.

[0056] A non-uniform conformal FDTD method based on submesh technology (e.g.) Figure 1 (As shown), the specific steps are as follows:

[0057] Step 1, construct as follows Figure 2 The dipole antenna array model shown is compared with the global coarse grid (41×41×75), the global fine grid (123×123×225), and the sub-grid scheme of the present invention. In the scheme of the present invention, the background main grid (coarse grid) sets the spatial step size according to the highest frequency of 35GHz. and the time step that satisfies the stability condition Meanwhile, a subgrid with a mesh size of (30×30×10⁵) is created in a local region surrounding the dipole array, and a refinement factor is set. Employing independent refinement space step size And refine the time step Calculate and store the conformal edge lengths in the main mesh and sub-mesh. and conformal area ,in , , The direction is a three-dimensional rectangular coordinate. To be parallel to Conformal edge length of the main mesh in the direction, For respectively located in The conformal area of ​​the main mesh in a plane is calculated, and the same applies to the sub-mesh.

[0058] Step 2: Use discrete-port impedance excitation with an impedance value of 65 ohms. Apply a Gaussian pulse signal to simultaneously excite all four ports, using a 10-layer PML absorbing boundary condition, and initialize the electric and magnetic fields across the entire computational domain to zero. Set the total number of time steps. .

[0059] Step 3, utilize The main grid electric field at time The magnetic field of the entire main grid, excluding the sub-grid coverage area, is updated according to formulas (1), (2), and (3) to obtain... Magnetic field prediction at time .

[0060] In non-uniform conformal FDTD, the non-uniform conformal mesh used needs to be corrected for element geometry through conformal processing near the boundary of the ideal electrical conductor PEC to reduce errors caused by the step approximation. Specifically, such as... Figure 3 As shown, this method defines the conformal edge length within the mesh, excluding the ideal electrical conductor PEC portion. , and conformal area .

[0061] Based on the integral form of Faraday's law, with Taking the example of a non-uniform conformal FDTD, the magnetic field update formula is as follows:

[0062] (3)

[0063] Meanwhile, its electric field update formula is:

[0064] (6)

[0065] in For time step, Where is the dielectric constant. Permeability, Representing along , , Orientation of grid cell index, and Represent and Directional step size. For example... Indicates that it is located at of At the center of the face, the time step is time Towards the magnetic field component, Indicates that it is located at of Towards the center of the edge, the time step is time Towards the electric field component.

[0066] Step 4: Refine the time step using sub-mesh. Perform 3 iterations to change the sub-time from Advance to In each sub-loop, the tangential electric field on the edge of the main mesh is set along its conformal edge length. The electric field integral is conformal to the edge length of its corresponding subgrid. The tangential electric field integrals are equal to ensure that the tangential electric field at the subgrid interface satisfies the line integral conservation.

[0067] according to Figure 4 The spatial correspondence of the electric field at the master-sub-mesh interface is shown. Figure 4 (Time stamps have been omitted from the variables in the text) Taking Xiang as an example, the electric field of the main grid is interpolated using formula (9), and the interpolation result is assigned to the electric field of the sub-grid.

[0068] (9)

[0069] in and for Two adjacent main grids at time Towards electric field and , and These are the corresponding conformal edge lengths. After being refined by a factor of 3, the electric fields of these two adjacent main grids will correspond to 12 sub-grids. Towards electric field , for These 12 subgrids of time Towards the electric field, This represents the corresponding conformal side length. For value index, and These 12 subgrids can be retrieved. Towards electric field , This is the index for the summation of the conformal side lengths of the submesh.

[0070] The above relationship is based on two adjacent master grids at the interface. Taking the electric field as an example, in actual calculation, it is necessary to traverse the electric field of all corresponding main grids on the main-sub-grid interface and complete the assignment of the sub-grid boundary electric field according to equation (9). and The same applies to electric fields.

[0071] Then, the magnetic field and electric field inside the subgrid are updated alternately using the magnetic field and electric field update formulas (1) to (6) from step 3. Three iterations indicate that the magnetic field and electric field are updated three times.

[0072] Step 5: When the subgrid time advances to synchronization with the main grid magnetic field At that time, set the conformal area of ​​the main mesh. The magnetic flux on the surface and the conformal area of ​​its corresponding subgrid The magnetic fluxes on the surface are equal to satisfy the flux weighted conservation.

[0073] like Figure 5 As shown, with Taking Xiang as an example, the equivalent magnetic field fed back to the main grid is calculated using formula (10). .

[0074] (10)

[0075] in for Real-time feedback to the main grid Towards the magnetic field, This represents the corresponding conformal area. A main grid magnetic field, after being refined by a factor of 3, will correspond to 9 sub-grid magnetic fields. , for Nine subgrids of time Towards the magnetic field, For the corresponding conformal area, For summation indexing. The feedback magnetic field of the main grid is obtained by summing the sub-grid magnetic fields by area weighting using equation (10). The above relationship takes a main grid on the interface as an example. In actual calculation, it is necessary to traverse and calculate all corresponding main grids on the main-sub-grid interface one by one. Similarly, and The formula for calculating the equivalent magnetic field is also given in formula (10).

[0076] Subsequently, the equivalent magnetic field of feedback was used. Correct the main grid magnetic field estimate obtained in step 3. This allows the physical information of the subgrid structure to be transmitted back to the main grid in a conserved manner.

[0077] Step 6: Use the corrected magnetic field estimate from Step 5. The main grid electric field is updated according to formulas (4) to (6), resulting in... electric field at time Complete from arrive This is a time step update.

[0078] Figure 6 and Figure 7 This paper presents a comparison between the far-field radiation pattern of the dipole antenna array calculated using this invention and both a globally fine-grid scheme (reference value) and a globally coarse-grid scheme. From... Figure 6 and Figure 7 As can be seen, the calculation results of this invention (red dashed line) are in high agreement with the high-precision results of the global fine mesh (blue solid line), with the curves almost completely overlapping. In contrast, the results of the global coarse mesh (green dashed line) show significant deviations. Furthermore, compared to the global fine mesh scheme, this invention reduces the required memory from 2849.2 Mb to 153.4 Mb (a reduction of approximately 94.6%) and the computation time from 2909.4 s to 131.9 s (an improvement of approximately 22 times) while maintaining the same computational accuracy. This result strongly verifies that the method of this invention can effectively reduce the step approximation error and maintain the stability of the computation process when dealing with multi-scale microstructure problems, and significantly reduce storage requirements and computation time overhead while ensuring computational accuracy.

[0079] As demonstrated by the above embodiments, this invention proposes a non-uniform conformal FDTD method based on a sub-mesh technique with conserved interface coupling. By introducing non-uniform conformal geometric corrections within the sub-mesh, high-precision fitting of surface boundaries is achieved, reducing step approximation errors and improving the accuracy of field calculations near the boundaries. A bidirectional coupling transfer mechanism is established at the master-sub-mesh interface, simultaneously satisfying line integral conservation and flux conservation, avoiding disruption of the conservation relationships required by the non-uniform conformal FDTD method. This effectively suppresses interface numerical reflections, reduces energy non-physical errors, and improves long-term stability. Furthermore, by using fine meshes only in the local microstructure neighborhood and coordinating with time sub-cycle progression, global mesh refinement and the global time step being limited by the smallest unit are avoided, significantly reducing storage and computation time overhead while maintaining accuracy. Therefore, this method balances geometric modeling accuracy, interface stability, and computational efficiency, possessing high engineering application value.

Claims

1. A non-uniform conformal FDTD method based on sub-mesh technology, characterized in that, Includes the following steps: Step 1: Establish a non-uniform conformal master grid covering the entire computational domain; then select a local sub-region containing fine structures and curved boundaries as the refinement region, and refine the master grid cells of the refinement region according to the refinement factor. Subdivide the mesh to generate sub-mesh. The outer boundary of the sub-mesh is defined as the main-sub-mesh interface; Both the master and sub-mesh follow the spatiotemporal staggered arrangement mechanism of standard Yee cells, satisfying the following conditions: the electric field is located at the center of the mesh edge and at integer time steps. The magnetic field is located at the center of the grid surface and at half-integer time steps. n is an integer greater than or equal to 0; Define the electromagnetic field of the main grid as uppercase. Conformal geometric parameters are in uppercase. Subgrids correspond to lowercase. and The main grid uses a non-uniform spatial step size that varies with location. and reference time step ; Subgrids, on the other hand, refine the spatial step , the temporal step , Take odd numbers; Based on this, the conformal edge lengths in the master and child meshes are calculated and stored. and conformal area ,in , , The direction is a three-dimensional rectangular coordinate. To be parallel to Conformal edge length of the main mesh in the direction, For respectively located in The conformal area of ​​the main mesh in a plane is calculated, and the same applies to the sub-mesh. Step 2: Set the excitation source and boundary conditions, and set the initial values ​​of all electromagnetic fields in the computational domain to zero. Set the total number of time steps to [value missing]. Initial time step n=0; Step 3, utilize The main grid electric field at time The magnetic field of the entire main grid, excluding the sub-grid coverage area, is updated according to formulas (1), (2), and (3) to obtain... Magnetic field prediction at time ; In non-uniform conformal FDTD, the non-uniform conformal mesh used needs to correct the element geometry through conformal processing near the boundary of the ideal conductor PEC; this method defines the conformal edge length within the mesh excluding the portion of the ideal conductor PEC. and conformal area ; Based on the integral form of Faraday's law, for Magnetic field component The magnetic field update formulas for non-uniform conformal FDTD are as follows: ； ； ； Meanwhile, its electric field update formula is: ； ； ； in For time step, Where is the dielectric constant. Permeability; Represent Orientation of grid cell index, Represent Grid step size in the direction; The electromagnetic field follows the spatiotemporal staggered arrangement mechanism of the standard Yee cell, with the electric field located at the center of the grid edges and integer time steps. The magnetic field is located at the center of the grid surface and at half-integer time steps. Specifically, it is expressed as: , , , , and ;in, Representing along Grid cell indices in three coordinate directions, Indicates that it is located at At this location, the time step is The x-axis magnetic field component at time t; Indicates that it is located at At this location, the time step is The y-axis magnetic field component at time t; Indicates that it is located at At this location, the time step is The z-axis magnetic field component at time t; correspondingly, Indicates that it is located at At this location, the time step is The x-axis electric field component at time t; Indicates that it is located at At this location, the time step is The y-axis magnetic field component at time t; Indicates that it is located at At this location, the time step is The z-axis magnetic field component at time t; Step 4: Refine the time step for the sub-mesh. conduct The next iteration will change the sub-time from Advance to In each sub-loop, the tangential electric field on the edge of the main mesh is set along its conformal edge length. The electric field integral is conformal to the edge length of its corresponding subgrid. The integrals of the tangential electric field on the surface are equal; Based on the spatial correspondence of the electric fields at the interface between the main and sub-grids, the electric field of the main grid is interpolated using formula (7), and the interpolation result is assigned to the electric field of the sub-grid. ； for The parameters for direction are: and for Two adjacent main grids at time Towards electric field and , and For the corresponding conformal edge lengths, the electric fields of these two adjacent principal grids pass through After refinement, it will correspond to Subgrid Towards electric field , for This moment Subgrid Towards the electric field, The corresponding conformal side length; and For value index, and Combining them can achieve this Subgrid Towards electric field , The index for the summation of the conformal side lengths of the submesh; the subscript of the combination parameter in equation (7). Change to and Then it represents the corresponding and Relevant parameters in the direction; The electric field of all corresponding main grids on the main-sub grid interface is traversed, and the electric field of the sub grid boundary is assigned according to Equation (7); Then, the magnetic and electric fields inside the subgrid are updated alternately using the magnetic and electric field update formulas (1) to (6) from step 3. The next iteration indicates that both the magnetic and electric fields are updated. Second-rate; Step 5: When the subgrid time advances to synchronization with the main grid magnetic field At that time, set the conformal area of ​​the main mesh. The magnetic flux on the surface and the conformal area of ​​its corresponding subgrid The magnetic flux on the grid is equal; the equivalent magnetic field fed back to the main grid is calculated using formula (8). ; ； for The parameters for direction are: for Real-time feedback to the main grid Towards the magnetic field, For the corresponding conformal area; a principal grid magnetic field passes through After refinement, it will correspond to Subgrid magnetic field , for Moment Subgrid Towards the magnetic field, For the corresponding conformal area, For summation index; subscripts of the combined parameters in equation (8) Change to and Then it represents the corresponding and Relevant parameters in the direction; The main grid feedback magnetic field is obtained by summing the sub-grid magnetic fields by area weighting in equation (8). During the calculation, all corresponding main grids on the main-sub-grid interface are traversed one by one. Subsequently, the equivalent magnetic field is fed back The main grid magnetic field estimate obtained in step 3 is corrected ; Step 6, the magnetic field estimation in step 5 is corrected , the main grid electric field is updated according to formulas (4)-(6) to obtain the electric field at the moment , one time step update from to is completed; Step 7, for Repeat steps 3 through 6 in sequence until... Finish The process is updated step by step, thus completing the full-time progression and obtaining the final desired time-domain electromagnetic field result.

2. The non-uniformly conformal FDTD method based on subgrids of claim 1, wherein: The .

3. The non-uniformly conformal FDTD method based on subgrids of claim 1, wherein: The total number of time steps set in step 2 It is a positive integer not less than 1.

4. The non-uniformly conformal FDTD method based on subgrids of claim 1, wherein: the total number of time steps set in step 2 not more than 10000.

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