CDI-FDTD non-uniform grid modeling method and system

Abstract

The invention discloses a CDI-FDTD non-uniform grid modeling method and system. The method comprises the steps of performing grid division on an area where an electromagnetic target is located by using a non-uniform grid technology; setting parameters of a simulation space and an initial state of an electromagnetic field variable; a CDI-FDTD numerical iteration formula based on the non-uniform grid technology is obtained; performing electromagnetic simulation on the to-be-simulated electromagnetic target according to the method to obtain the electromagnetic characteristics of the to-be-simulated electromagnetic target; the method has the advantages that the electromagnetic target with a fine structure is simulated while the calculation efficiency and precision are guaranteed.

Description

Technical Field

[0001] The present invention relates to the field of efficient numerical simulation of electromagnetic waves, and in particular to a CDI-FDTD non-uniform grid modeling method and system. Background Art

[0002] With the rapid advancement of electronic technology, the simulation models of electronic devices have become increasingly complex. Therefore, how to accurately and efficiently simulate the electromagnetic phenomena of these models has become particularly critical. In this context, the development and application scope of computational electromagnetics are expanding. In addition to the Finite-Difference Time Domain (FDTD) method used in this study, the common numerical methods used in computational electromagnetics include the Method of Moments (MoM) and the Finite Element Method (FEM). Among these methods, the FDTD method has the characteristics of simple numerical implementation, intuitive description of physical processes, easy parallel calculation, and a single calculation can obtain wide-band information. It is gradually used in electromagnetic compatibility problems, microwave circuit simulation and design simulation. However, in the process of using the FDTD method, the main problem faced is that the FDTD method is limited by the CFL (Courant-Friedrichs-Lewy) condition and the traditional Yee cell modeling of electromagnetic models will produce step errors.

[0003] In order to overcome the limitation of the grid size in the computational space on the time step, a series of unconditionally stable and weakly conditionally stable FDTD methods have been proposed and used in electromagnetic simulation. Common implicit unconditionally stable algorithms include the alternating direction implicit FDTD (ADI-FDTD) method, the single-step ADI-FDTD method, the local one-dimensional FDTD method, the FLOD-FDTD method, the hybrid explicit and implicit HIE-FDTD method, and the CN-FDTD method. In addition, there are explicit unconditionally stable algorithms, which are mainly based on the spatial filtering method and the eigenvalue filtering algorithm of the FDTD method. These optimized FDTD methods are not limited by the CFL conditions in the traditional FDTD method and can use a larger time step. Therefore, while keeping the physical simulation time unchanged, a larger time step can reduce the number of iterations, thereby improving the efficiency of numerical calculations.

[0004] In order to solve the step approximation problem in modeling using uniform Yee grids, several improved grid processing technologies have been introduced into the FDTD method, including hybrid subgrid technology, conformal grid technology and non-uniform grid technology. For example, the electromagnetic wave time domain efficient numerical hybrid method based on subgrid technology disclosed in Chinese Patent Publication No. CN111090958A is a hybrid subgrid technology. The hybrid subgrid technology is mainly used in areas with tiny and irregular structures. By using fine grids in these areas and coarse grids in other areas, the need to use fine grids to divide the entire calculation area is avoided, thereby significantly improving the calculation efficiency and reducing memory consumption. However, the subgrid technology requires complex space-time field value exchange at the junction of coarse and fine grids. Simple exchange may cause false reflections at the boundary and affect the accuracy of the results. Complex exchange schemes increase the difficulty and complexity of programming. Conformal grid technology is particularly suitable for modeling surface structures. According to different dielectric material properties, it is divided into technologies for dielectric targets and PEC targets. Especially when processing PEC targets, if the proportion of PEC in the cell is too low, the time step needs to be reduced to ensure the stability of the numerical simulation, which may increase the simulation time.

[0005] In the development of FDTD algorithm, non-uniform grid technology was introduced to reduce the step error generated in the modeling process. In this technical application, the coarse grid covers the background of the problem space, while the fine grid is dedicated to the fine area of ​​the model. By gradually changing the grid size of the coarse and fine grid areas, a smooth transition is achieved and the numerical reflection caused by the grid conversion is minimized. Compared with uniform fine grids, the total number of cells in non-uniform grids is significantly reduced, which can shorten the total simulation time while maintaining the same level of accuracy. Therefore, many scholars have conducted in-depth research on non-uniform grid FDTD methods, including stability analysis, the application of non-uniform cylindrical grids, and the total field and scattered field problems under non-uniform grids. Although non-uniform grids reduce the number of cells, there is still a trade-off between simulation time and computational accuracy under CFL conditions. To address this challenge, many researchers have explored the combination of unconditionally stable methods and non-uniform grid technology, such as ADI-FDTD, frog leapfrog ADI-FDTD, and WCS-FDTD methods. These methods show better computational efficiency than traditional non-uniform grid FDTD under non-uniform grids.

[0006] Recently, a new CDI-FDTD (divergence-compliant implicit finite-difference time-domain) method has been proposed. This method not only has the advantage of unconditional stability, but also maintains the compliant divergence characteristics satisfied by the traditional FDTD method, and its iterative formula is similar to the traditional FDTD method. In terms of simulating the characteristics of complex electromagnetic targets, the CDI-FDTD method shows higher numerical calculation efficiency and accuracy than the traditional non-compliant divergence implicit FDTD algorithm, but this method cannot simulate electromagnetic targets with fine structures. Summary of the invention

[0007] The technical problem to be solved by the present invention is that the modeling method of electronic devices in the prior art is difficult to ensure the calculation efficiency and accuracy while simulating electromagnetic targets with fine structures.

[0008] The present invention solves the above technical problems by the following technical means: a CDI-FDTD non-uniform grid modeling method, comprising the following steps:

[0009] Step 1: Use non-uniform grid technology to divide the area where the electromagnetic target is located into grids;

[0010] Step 2: Set the parameters of the simulation space and the initial state of the electromagnetic field variables;

[0011] Step 3, obtaining the CDI-FDTD numerical iteration formula based on non-uniform grid technology;

[0012] Step 4: Perform electromagnetic simulation on the electromagnetic target to be simulated according to the methods of steps 1 to 3 to obtain the electromagnetic characteristics of the electromagnetic target to be simulated.

[0013] The present invention also provides a CDI-FDTD non-uniform grid modeling system, comprising:

[0014] A grid division module is used to divide the area where the electromagnetic target is located into grids using a non-uniform grid technology;

[0015] Initialization module, used to set the parameters of the simulation space and the initial state of the electromagnetic field variables;

[0016] Iterative formula acquisition module, used to obtain CDI-FDTD numerical iterative formula based on non-uniform grid technology;

[0017] The electromagnetic simulation module is used to perform electromagnetic simulation on the electromagnetic target to be simulated according to the method from the grid division module to the iterative formula acquisition module to obtain the electromagnetic characteristics of the electromagnetic target to be simulated.

[0018] The advantages of the present invention are as follows: the present invention optimizes the grid distribution of the electromagnetic model by introducing the non-uniform grid technology within the CDI-FDTD framework, thereby allowing the focus on the key areas of the model through fine grids while applying the coarse grids to the background areas on the basis of maintaining the original unconditional stability and compliant divergence characteristics. This grid configuration method reduces the demand for computing resources and improves the accuracy of numerical simulation. Therefore, the entire scheme combines the non-uniform grid technology with the CDI-FDTD method to simulate electromagnetic targets with fine structures, which not only retains the unconditional stability characteristics and compliant divergence properties of the CDI-FDTD method, but also uses a smaller number of grids to accurately model fine structure targets, ensuring computing efficiency and accuracy while being able to simulate electromagnetic targets with fine structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 A schematic diagram of the distribution of electric and magnetic fields in a non-uniform grid in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention;

[0020] Figure 2 This is a diagram of the modulus value results of the growth factor under different CFLs of a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention; wherein, Figure 2 (a) is the modulus result of the growth factor of the time step of the CDI-FDTD non-uniform grid modeling method under the maximum grid limit of the CFL condition. Figure 2 (b) Figure 2 (c) Figure 2 (d) The modulus results of the growth factor when the time step of the CDI-FDTD non-uniform grid modeling method is 1, 2, and 4 times the CFL condition limit;

[0021] Figure 3 This is the result diagram of the growth factor modulus of the traditional FDTD method based on non-uniform grid technology; Figure 3 (a) is the modulus result of the growth factor of the time step of the traditional FDTD method under the minimum grid limit of the CFL condition. Figure 3 (b) is the result diagram of the modulus of the growth factor of the time step of the traditional FDTD method under the maximum grid limit of the CFL condition;

[0022] Figure 4 Schematic diagram of the divergence of a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention; wherein, Figure 4 (a) is a schematic diagram of the divergence of a uniform grid. Figure 4 (b) Figure 4 (c) Figure 4 (d) Schematic diagram of non-uniform grid divergence when the CFL value is 1, 2, and 4 respectively;

[0023] Figure 5 A schematic diagram of a three-dimensional spatial model of a frequency selective circuit in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention; wherein, Figure 5 (a) is a schematic diagram of the model structure of the frequency selection circuit. Figure 5 (b) is the schematic diagram of PEC structural parameters and the mesh division of the plane;

[0024] Figure 6 A time domain waveform diagram of a CDI-FDTD non-uniform grid modeling method and a comparison method on a transmission plane disclosed in an embodiment of the present invention;

[0025] Figure 7 A schematic diagram of transmission coefficients at different frequencies of a CDI-FDTD non-uniform grid modeling method and a comparison method disclosed in an embodiment of the present invention;

[0026] Figure 8 A schematic diagram of a microstrip rectangular patch antenna model in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention;

[0027] Fig. 9 A time domain waveform of the sampling current at port 1 of a CDI-FDTD non-uniform grid modeling method and a comparative method disclosed in an embodiment of the present invention

[0028] Fig.10 A sampled voltage time domain waveform diagram at port 1 of a CDI-FDTD non-uniform grid modeling method and a comparative method disclosed in an embodiment of the present invention

[0029] Fig.11 A power reflection coefficient (S11) result diagram of a CDI-FDTD non-uniform grid modeling method and a comparative method disclosed in an embodiment of the present invention;

[0030] Fig.12 A schematic diagram of a three-stage microstrip low-pass filter model in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention;

[0031] Fig.13 A schematic diagram of the result of sampling current at port 1 of a three-stage microstrip low-pass filter model in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention in the frequency domain;

[0032] Fig.14 A schematic diagram of the result of sampling voltage at port 1 of a three-stage microstrip low-pass filter model in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention in the frequency domain;

[0033] Fig.15 A schematic diagram of S11 and S21 results of a three-stage microstrip low-pass filter model in a CDI-FDTD non-uniform grid modeling method disclosed in an embodiment of the present invention. DETAILED DESCRIPTION

[0034] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in combination with the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0035] Example 1

[0036] Embodiment 1 of the present invention provides a CDI-FDTD non-uniform grid modeling method, comprising the following steps:

[0037] S1. Use non-uniform grid technology to divide the area where the electromagnetic target is located into grids; the specific process is as follows:

[0038] When modeling electromagnetic targets, the choice of mesh unit size is a trade-off between reducing solution time and improving accuracy. For electromagnetic models that do not have fine structures or fine structures that do not affect the simulation results, choosing an appropriate mesh size ensures reasonable simulation time and correct simulation structure. However, when the target model is modeled in a coarse mesh form, some areas of the model are not completely occupied by a mesh. When modeling with Yee mesh, the minimum unit is a mesh, so the resulting model is often not accurate enough. In addition, when the target is modeled with a fine mesh, the fine mesh can better capture details that the coarse mesh cannot model well. However, for most areas, the modeling method of using a fine mesh will lead to oversampling, which will only increase the computer memory usage without having a substantial impact on the simulation results. At the same time, increasing the number of meshes will also increase the time required for simulation. For such models with right-angle structures, non-uniform mesh technology can be selected. This mesh technology can effectively handle complex structures and provide high-precision simulation results.

[0039] First, fine meshes are used to model the fine areas of the model, and coarse meshes are used to model the rest of the area.

[0040] Next, the area between the fine mesh area and the coarse mesh area is gradually changed to achieve a smooth transition between the two and minimize the false reflections generated between the meshes. Figure 1 The distribution of electric and magnetic fields in a one-dimensional non-uniform grid is shown. In the non-uniform grid area, the size of each electric field grid is different, and the grid size in the x direction is Δx.e (1) Δx e (2), ..., Δx e (i) ...; the magnetic field is located between two adjacent electric fields, and the grid size is Δx h Indicates that, where Δx h (i) = Δx e (i-1) / 2+Δx e (i) / 2. Similarly, in three-dimensional space, the electric field grid size and magnetic field grid size in the other two directions are represented by Δy e , Δy h , Δz e and Δz h express.

[0041] S2, setting the parameters of the simulation space and the initial state of the electromagnetic field variables;

[0042] S3. Obtain the Maxwell equations and discretize them using the unconditionally stable divergence-preserving implicit finite-difference time-domain (CDI-FDTD) method to obtain the CDI-FDTD numerical iterative formula based on the non-uniform grid technology. The specific process is as follows:

[0043] First, Maxwell's equations involve only two curl equations when describing an isotropic and homogeneous medium in a passive case:

[0044]

[0045] Where, E is the electric field strength, H is the magnetic field strength, ε and μ are the dielectric constant and magnetic permeability in free space respectively. is the Hamiltonian operator, which can be expressed as where e x ,e y , and e z are the coordinate unit vectors in the rectangular coordinate system, and The partial derivatives in the x, y and z directions respectively. t is the time variable, Represents the time variation of the magnetic field.

[0046] First, the same numerical discretization strategy as the LOD-FDTD method is used to process (1) and (2), and we can obtain:

[0047]

[0048] Among them I n is the n-order unit matrix, u=[E,H] T , the coefficient matrices A and B are

[0049]

[0050] 0 n represents the n-th order zero matrix, It means to find the partial derivative of the electric field in different directions. It means to find the partial derivative of the magnetic field in different directions.

[0051] definition and The forms are:

[0052]

[0053] Among them, u c The matrix form of the electric and magnetic field intensity components, u c =[E xc ,E yc ,E zc ,H xc ,H yc ,H zc ] T Next, by substituting equations (8) and (9) into equations (3) and (4), we can obtain:

[0054]

[0055] In formula (9), c The sampling time of and u is changed from n+1 to n, and it is put into formula (10), and we can get:

[0056]

[0057] E c represents the electric field intensity vector that obeys the divergence. According to formula (12), we can get E c Iteration formula at time n+1 / 2:

[0058]

[0059] Similarly, replace u in formula (8) c The sampling time of and u is changed from n+1 / 2 to n-1 / 2, and is substituted into formula (11) (it should be noted that u in (11) c The sampling time of and u needs to be changed from n+1 to n), and finally, E c The iteration formula at time n-1 / 2 is:

[0060]

[0061] Next, subtract formula (14) from formula (13) to get E c The numerical iteration formula is:

[0062]

[0063] Similarly, H can be derived c The iteration formula is:

[0064]

[0065] Finally, in order to facilitate the numerical solution of equations (15) and (16), the magnetic field auxiliary variable h is defined as c , electric field auxiliary variable e c

[0066]

[0067] Substituting formula (17) and (18) into formula (15) and (16) at the same time, we can get

[0068]

[0069] Therefore, formulas (17)-(20) are the numerical iteration formulas of the CDI-FDTD method, E c and H c are the output electric and magnetic fields respectively.

[0070] Finally, the specific iteration formula of the non-uniform grid CDI-FDTD method is given, and the iteration formula is shown as:

[0071]

[0072]

[0073] Among them, n represents the nth moment, express Time coordinates The x-direction magnetic field intensity at the position, Δt represents the time step used in numerical discretization, μ represents the magnetic permeability coefficient in free space, Represents the coordinates at time n The auxiliary variable of the electric field in the y direction of the position, Represents the coordinates at time n The z-direction electric field auxiliary variable of the position, Δz e (k) represents the electric field grid size between k+1 and k in the z direction, Δy e (j) represents the electric field grid size between j+1 and j in the y direction; express Time coordinates The magnetic field strength in the y direction of the position, Represents the coordinates at time n The x-direction electric field auxiliary variable of the position, Δxe (i) represents the electric field grid size between i+1 and i in the x direction; express Time coordinates The z-direction magnetic field strength of the position; Indicates the coordinates at time n+1 The electric field strength in the x direction at the position, ε represents the dielectric constant in free space, Δy h (j) represents the y direction and The magnetic field grid size between h (j) represents the z direction and The magnetic field grid size between ; Indicates the coordinates at time n+1 The electric field strength in the y direction at the position, Δz h (k) represents the z direction and The magnetic field grid size between h (i) indicates the x direction and The magnetic field grid size between ; Indicates the coordinates at time n+1 The electric field strength in the z direction at the position.

[0074] The implicit iteration formula is:

[0075]

[0076] in, express Time coordinates Auxiliary variable of magnetic field strength in the x direction, express Time coordinates Auxiliary variable of magnetic field strength in the y direction of position, express Time coordinates Auxiliary variable of magnetic field strength in the z direction; They represent the first magnetic field coefficient in the x direction, the second magnetic field coefficient in the x direction, and the third magnetic field coefficient in the x direction at the coordinate (i, j, k), respectively. They represent the first magnetic field coefficient in the y direction, the second magnetic field coefficient in the y direction, and the third magnetic field coefficient in the y direction at the coordinate (i, j, k), respectively. represent the first magnetic field coefficient in the z direction, the second magnetic field coefficient in the z direction, and the third magnetic field coefficient in the z direction at the coordinate (i, j, k), respectively, and

[0077]

[0078] For the auxiliary variable e c The iterative formula in each direction at time n is:

[0079]

[0080] in, They represent the first electric field auxiliary variable coefficient in the x direction, the second electric field auxiliary variable coefficient in the x direction, and the third electric field auxiliary variable coefficient in the x direction at the coordinate position (i, j, k), respectively. They represent the first electric field auxiliary variable coefficient in the y direction, the second electric field auxiliary variable coefficient in the y direction, and the third electric field auxiliary variable coefficient in the y direction at the coordinate (i, j, k), respectively. They represent the first electric field auxiliary variable coefficient in the z direction, the second electric field auxiliary variable coefficient in the z direction, and the third electric field auxiliary variable coefficient in the z direction at the coordinate (i, j, k), respectively, and

[0081]

[0082]

[0083] Formulas (21)-(32) constitute the numerical iteration formula of the non-uniform grid CDI-FDTD method. It can be found that the iteration formula of the explicit part is similar to that of the non-uniform grid FDTD method, and the iteration formula of the implicit part has a fundamental form without right-hand-sides (RHS) operators. Therefore, the numerical implementation of the non-uniform grid CDI-FDTD method is relatively easy.

[0084] S4. According to the von Neumann method and Gaussian divergence theorem, the unconditional stability and divergence characteristics of the proposed method are verified; the specific process is as follows:

[0085] S41. Stability analysis of CDI-FDTD method based on non-uniform grid technology:

[0086] First, the numerical iteration formula of the above CDI-FDTD method is written in the following matrix form:

[0087] M L P n+1 =M R P n (33)

[0088] Where P = [E x ,E y ,E z ,H x ,H y ,H z ]T , with E x For example, E x =[E x (1,1,1),E x (1,1,2),…,E x (m,n+1,q+1)] T , the other field components are expressed in a similar way. At the same time, m, n, q are the maximum index values ​​of the field components in the x, y, and z directions, respectively. Here

[0089]

[0090] The field component of electromagnetic waves in the space domain is defined as:

[0091]

[0092] Where, φ=E,H, represents the amplitude of the field component. ζ represents the time growth factor, k p (p = x, y, z) represents the Fourier wave number in the p direction. The subscripts i, j, k represent the node positions of the Yee grid, and Δp (p = x, y, z) represents the discretized grid size.

[0093] The spatial first-order partial derivative is approximated by the second-order central difference discretization, and we get

[0094]

[0095] in, (p = x, y, z). Substituting equation (29) into equation (27), we obtain

[0096] MP n =(ζM L -M R ) n =0 (36)

[0097] In order to ensure that there is a non-zero solution to equation (36), the value of the determinant of the coefficient matrix M should be 0. At the same time, in order to ensure that the numerical method remains stable during the iteration process, the modulus of the growth factor ζ must be less than or equal to 1. The expression of ζ can be obtained by solving the determinant of the matrix M. However, the expression of ζ is too cumbersome, so for the convenience of calculation, the parameters of the non-uniform grid area in S41 are substituted into the matrix M to determine the modulus of the growth factor ζ. It is worth noting that the wave number k is not involved in S41. p The definition of wave number k p is assigned a random array.

[0098] Figure 2The values ​​of the growth factor ζ corresponding to different CFL conditions are shown in the CDI-FDTD method. (CFLN = Δt CDI-FDTD / Δt,Δt CDI-FDTD is the time step of the CDI-FDTD method, N represents a multiple, CFLN represents N times of CFL, for example, CFL2 represents 2 times of CFL). Figure 2 The horizontal and vertical coordinates represent the real and imaginary parts of the growth factor ζ, respectively. It can be found that the modulus of all growth factors ζ is not greater than 1, which shows that the CDI-FDTD method has unconditional stability when calculating non-uniform grids. Figure 2 (a) The time step is stable under the maximum grid limit of CFL conditions; Figure 2 (b) Figure 2 (c) Figure 2 (d) When the time step is 1, 2, and 4 times the CFL condition limit, it is stable. In addition, the non-uniform grid parameters of S41 are also used to calculate the modulus of the growth factor ζ of the FDTD method under different Δt. Figure 3 As shown in (a), When , the modulus of all growth factors ζ is less than 1, the FDTD method is stable. Figure 3 The results of (b) show that When , the modulus of some growth factors ζ is greater than 1, which will cause the FDTD method to be unstable. This is because the FDTD method is restricted by the CFL condition when dealing with non-uniform grids. However, for the CDI-FDTD method, it has the characteristic of unconditional stability, so it is not restricted by the CFL condition when dealing with spaces containing non-uniform grids.

[0099] S42. Analysis of Gaussian divergence characteristics

[0100] The traditional FDTD method and CDI-FDTD method both satisfy the divergence property. This step further studies whether the non-uniform grid technology will affect the divergence property of the FDTD method and CDI-FDTD method.

[0101] According to the differential Gauss law:

[0102]

[0103] Where D is the electric displacement vector, ρ v is the charge density. Formula (37) shows that in the passive region, the divergence of the electric displacement vector should be zero. Therefore, the divergence of D in the numerical method can be solved to verify whether the FDTD method and the CDI-FDTD method also satisfy the compliance divergence property. The divergence of the electric displacement vector D can be expressed as

[0104]

[0105] Figure 4 The result of the divergence of D on the sampling surface in the CDI-FDTD method is shown. Specifically, Figure 4 (a) uses a uniform grid, while Figure 4 The computational space in (b)-(d) contains a non-uniform grid, and the CFL values ​​are 1, 2, and 4, respectively. Figure 4 From the results (b)-(d), we can see that for the passive area, the results of the non-uniform grid area and the uniform grid area fluctuate around -500dB, and there is no obvious increase. Moreover, the results of each grid are similar to Figure 4 The results of (a) are similar, which shows that the divergence property of the CDI-FDTD method is not affected when the non-uniform grid area is calculated using the CDI-FDTD method. Therefore, the non-uniform grid technology used in the present invention will not affect the divergence property of the numerical method.

[0106] S5. According to the CDI-FDTD method based on the non-uniform grid technology, electromagnetic simulation is performed on the electromagnetic target to be simulated to obtain the electromagnetic characteristics of the electromagnetic target to be simulated. In this embodiment, electromagnetic simulation is performed on the frequency selection circuit, the microstrip patch antenna and the low-pass filter. The specific process is as follows:

[0107] S51. Frequency Selective Surface

[0108] The model of this example (frequency selective surface) is as follows Figure 5 As shown in (a), the size of the simulation space is 35mm×35mm×70mm. In the z direction, the simulation space is truncated by using the CPML method, while the boundaries in the x and y directions are periodic boundaries. The location of the I-shaped structure in the model represents a dielectric plate with a relative dielectric constant of 2.2. The I-shaped structure on the dielectric plate is made of PEC material. The specific PEC structural parameters and the mesh division of the plane are shown in Figure 5 (b) is shown. Figure 5 In (b), the background space is a coarse grid area with a spatial step of 1×10-3m. The PEC structure is modeled using a fine grid with a spatial step of 0.5×10-3m. A smooth transition is achieved between the coarse and fine grids by gradually changing the grid size. This example uses a plane wave, which is a differential Gaussian pulse, where t0=4τ, τ=0.1593ns. The fine grid FDTD method, the coarse grid FDTD method, and the non-uniform grid CDI-FDTD method are used to perform electromagnetic simulation of the model. At the same time, the results obtained by the fine grid FDTD method are used as the reference solution in the example. The transmission plane is used as the detection point, Figure 6For the fine grid FDTD method, the coarse grid FDTD and the non-uniform grid CDI-FDTD method (CFLN = 1, 2, 4) the time domain waveform is obtained in the transmission plane. Figure 7 The transmission coefficient results of the fine grid FDTD method, coarse grid FDTD and non-uniform grid CDI-FDTD method (CFLN = 1, 2, 4) are given as they vary with frequency. Numerical results show that the results obtained by the non-uniform grid CDI-FDTD method are consistent with those of the fine grid FDTD method. In addition, the results obtained by the non-uniform grid CDI-FDTD method at CFLN = 4 are still more accurate than those of the coarse grid FDTD method, which verifies the accuracy of the non-uniform grid CDI-FDTD method.

[0109] S52. Microstrip patch antenna

[0110] In order to further verify the applicability and accuracy of the non-uniform grid CDI-FDTD method, this example performs electromagnetic simulation on a microstrip rectangular patch antenna. The model of the example is as follows Figure 8 As shown, the relative dielectric constant ε of the dielectric region r The value is 2.2. Fine grid FDTD method, coarse grid FDTD method and non-uniform grid CDI-FDTD method are used to perform electromagnetic simulation on the model. The Δx and Δy of the fine grid FDTD method are 1 mm, while the Δx and Δy of the coarse grid FDTD method are 2 mm. In the non-uniform grid simulation, the grid size used in the background space is the same as that of the coarse grid FDTD, and the grid size around the port and the area around the patch is 1 mm. This is because using a finer grid can improve the accuracy of the input impedance and the calculation of the return loss. Figure 8 The grid division of the microstrip antenna under non-uniform grid is shown. A voltage source is added to the port of the microstrip rectangular patch antenna. The time domain form of the voltage source is a Gaussian pulse, where τ = 3.336 × 10 -11 s, T0=5τ, the resistance is 50Ω.

[0111] Fig. 9 and Fig.10 The time domain waveforms of the sampled current and sampled voltage obtained by different methods in the first 6ns at port 1 are shown. For the results obtained by different methods, the simulation results obtained by the fine grid FDTD method are used as the reference solution and compared with the results of the coarse grid FDTD method and the non-uniform grid CDI-FDTD method. Fig. 9 and Fig.10From the results, we can see that in the subsequent oscillation of 2ns, the results of the coarse-grid FDTD method are significantly different from those of other methods. In contrast, the results obtained by the non-uniform-grid CDI-FDTD method are more consistent with the fine-grid method. At the same time, the results obtained by the non-uniform-grid CDI-FDTD method when CFLN=4 are still closer to the results obtained by the fine-grid FDTD method than the coarse-grid FDTD method. Fig.11 The power reflection coefficient (S11) results obtained by each method at this port are given. Numerical results show that the results obtained by the non-uniform grid CDI-FDTD method are closer to the fine grid FDTD method. This further verifies the accuracy of the non-uniform grid CDI-FDTD method.

[0112] S53. Microstrip low-pass filter

[0113] Finally, the accuracy of the non-uniform grid CDI-FDTD method is verified by simulating a three-stage microstrip low-pass filter. The model diagram and parameters of the low-pass filter are shown in Figure 2. Fig.12 As shown in the figure, the key part of the filter is the microstrip line with a width of 0.15mm in the middle, which is connected to the impedance matching section in the center and the wide feed lines on both sides. The relative dielectric constant of the medium is 10.8. For the non-uniform grid, the microstrip line area is modeled with a fine grid, and other areas are modeled with a coarse grid. The grid size of the coarse grid is 0.1mm, and the grid size of the fine grid is 0.05mm. Different from the above two examples, the simulation space of this example is truncated by PEC. A voltage source with a resistance of 50Ω is set at port 1.

[0114] This example uses the coarse-grid FDTD method, the fine-grid FDTD method, and the non-uniform-grid CDI-FDTD method for electromagnetic simulation, and the results obtained by the fine-grid FDTD method are used as the reference solution. Fig.13 The results of the sampling current in the frequency domain obtained at port 1 by different methods are given and normalized. Fig.14 The result of the sampling voltage of port 1 in the frequency domain is also normalized. The data results show that due to the inaccurate modeling of the coarse grid method, there is a significant difference between the results obtained by the coarse grid FDTD method and the results obtained by the fine grid FDTD method. The non-uniform grid technology can well describe the characteristics of the filter, so the results obtained by the non-uniform grid CDI-FDTD method (CFLN = 1, 2, 4) are consistent with the fine grid FDTD method. Finally Fig.15 The S11 and S21 (forward transmission coefficient) result diagrams of the low-pass filter simulated using different methods are given. Fig.15 The comparison results of various methods in Fig.13 and Fig.14 This further verifies the accuracy of the method proposed in this chapter.

[0115] Through the above technical scheme, the present invention optimizes the grid distribution of the electromagnetic model by introducing non-uniform grid technology within the CDI-FDTD framework, thereby allowing the focus on the key areas of the model through fine grids while applying coarse grids to the background areas while maintaining the original unconditional stability and compliant divergence characteristics. This grid configuration method reduces the demand for computing resources and improves the accuracy of numerical simulation. Therefore, the entire scheme combines non-uniform grid technology with the CDI-FDTD method to simulate electromagnetic targets with fine structures, which not only retains the unconditional stability characteristics and compliant divergence properties of the CDI-FDTD method, but also uses a smaller number of grids to accurately model fine structure targets, ensuring computational efficiency and accuracy while being able to simulate electromagnetic targets with fine structures.

[0116] Example 2

[0117] Based on Example 1, Example 2 of the present invention further provides a CDI-FDTD non-uniform grid modeling system, including:

[0118] A grid division module is used to divide the area where the electromagnetic target is located into grids using a non-uniform grid technology;

[0119] Initialization module, used to set the parameters of the simulation space and the initial state of the electromagnetic field variables;

[0120] Iterative formula acquisition module, used to obtain CDI-FDTD numerical iterative formula based on non-uniform grid technology;

[0121] The electromagnetic simulation module is used to perform electromagnetic simulation on the electromagnetic target to be simulated according to the method from the grid division module to the iterative formula acquisition module to obtain the electromagnetic characteristics of the electromagnetic target to be simulated.

[0122] Specifically, the grid division module is also used for:

[0123] Define the size of the fine grid and the coarse grid, and define the areas corresponding to the fine grid and the coarse grid, divide the area where the electromagnetic target is located into grids, and design a transition method from the fine grid to the coarse grid to make a smooth transition between the two.

[0124] More specifically, the iterative formula acquisition module is also used for:

[0125] Maxwell's equations are constructed, which are satisfied by electromagnetic field quantities in both fine and coarse grids. The Maxwell equations are discretized using the unconditionally stable divergence-preserving implicit finite-difference time-domain method, and the CDI-FDTD numerical iterative formula based on non-uniform grid technology is obtained.

[0126] More specifically, the CDI-FDTD numerical iteration formula based on the non-uniform grid technology includes a display iteration formula, which is:

[0127]

[0128] Among them, n represents the nth moment, express Time coordinates The x-direction magnetic field intensity at the position, Δt represents the time step used in numerical discretization, μ represents the magnetic permeability coefficient in free space, Represents the coordinates at time n The auxiliary variable of the electric field in the y direction of the position, Represents the coordinates at time n The z-direction electric field auxiliary variable of the position, Δz e (k) represents the electric field grid size with a length of k in the z direction, Δy e (j) represents the electric field grid size with a length of j in the y direction;

[0129]

[0130] in, express Time coordinates The magnetic field strength in the y direction of the position, Represents the coordinates at time n The x-direction electric field auxiliary variable of the position, Δx e (i) represents the electric field grid size with length i in the x direction;

[0131]

[0132] in, express Time coordinates The z-direction magnetic field strength of the position;

[0133]

[0134] in, Indicates the coordinates at time n+1 The electric field strength in the x direction at the position, ε represents the dielectric constant in free space, Δy h (j) represents the magnetic field grid size with a length of j in the y direction, Δz h (j) represents the magnetic field grid size with a length of j in the z direction;

[0135]

[0136] in, Indicates the coordinates at time n+1 The electric field strength in the y direction at the position, Δz h (k) represents the magnetic field grid size with a length of k in the z direction, Δx h (i) represents the magnetic field grid size with length i in the x direction;

[0137]

[0138] in, Indicates the coordinates at time n+1 The electric field strength in the z direction at the position.

[0139] More specifically, the CDI-FDTD numerical iteration formula based on the non-uniform grid technology also includes an implicit iteration formula, which is:

[0140]

[0141]

[0142] in, express Time coordinates Auxiliary variable of magnetic field strength in the x direction, express Time coordinates Auxiliary variable of magnetic field strength in the y direction of position, express Time coordinates Auxiliary variable of magnetic field strength in the z direction; They represent the first magnetic field coefficient in the x direction, the second magnetic field coefficient in the x direction, and the third magnetic field coefficient in the x direction at the coordinate (i, j, k), respectively. They represent the first magnetic field coefficient in the y direction, the second magnetic field coefficient in the y direction, and the third magnetic field coefficient in the y direction at the coordinate (i, j, k), respectively. represent the first magnetic field coefficient in the z direction, the second magnetic field coefficient in the z direction, and the third magnetic field coefficient in the z direction at the coordinate (i, j, k), respectively, and

[0143]

[0144] More specifically, the iterative formula of the electric field auxiliary variable in each direction at time n is:

[0145]

[0146]

[0147] in, They represent the first electric field auxiliary variable coefficient in the x direction, the second electric field auxiliary variable coefficient in the x direction, and the third electric field auxiliary variable coefficient in the x direction at the coordinate position (i, j, k), respectively. They represent the first electric field auxiliary variable coefficient in the y direction, the second electric field auxiliary variable coefficient in the y direction, and the third electric field auxiliary variable coefficient in the y direction at the coordinate (i, j, k), respectively. They represent the first electric field auxiliary variable coefficient in the z direction, the second electric field auxiliary variable coefficient in the z direction, and the third electric field auxiliary variable coefficient in the z direction at the coordinate (i, j, k), respectively, and

[0148]

[0149] More specifically, the electromagnetic simulation module is also used for:

[0150] The CDI-FDTD numerical iterative formula based on non-uniform grid technology is used to solve the electric field and magnetic field information of the electromagnetic target to be simulated in all directions, thereby realizing electromagnetic simulation and obtaining the electromagnetic characteristics of the electromagnetic target to be simulated, wherein the electromagnetic characteristics include electric field and magnetic field information.

[0151] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A CDI-FDTD non-uniform grid modeling method, characterized in that: The following steps are involved: Step 1: Use non-uniform grid technology to divide the area where the electromagnetic target is located into grids; Step 2: Set the parameters of the simulation space and the initial state of the electromagnetic field variables; Step 3, obtaining the CDI-FDTD numerical iteration formula based on non-uniform grid technology; Step 4: Perform electromagnetic simulation on the electromagnetic target to be simulated according to the methods of steps 1 to 3 to obtain the electromagnetic characteristics of the electromagnetic target to be simulated.

2. A CDI-FDTD non-uniform grid modeling method according to claim 1, characterized in that: The step one comprises: Define the size of fine grid and coarse grid, define the area corresponding to the fine grid and coarse grid, divide the area where the electromagnetic target is located into grids, and design the transition method from fine grid to coarse grid to make a smooth transition between the two.

3. A CDI-FDTD non-uniform grid modeling method according to claim 2, characterized in that: The step three comprises: Maxwell's equations that are satisfied by electromagnetic field quantities in both fine and coarse grids are constructed. The Maxwell equations are discretized using an unconditionally stable divergence-preserving implicit finite-difference time-domain method, and the CDI-FDTD numerical iterative formula based on non-uniform grid technology is obtained.

4. A CDI-FDTD non-uniform grid modeling method according to claim 3, characterized in that: The CDI-FDTD numerical iteration formula based on the non-uniform grid technology includes a display iteration formula, which is: Among them, n represents the nth moment, express Time coordinates The x-direction magnetic field intensity at the position, Δt represents the time step used in numerical discretization, μ represents the magnetic permeability coefficient in free space, Represents the coordinates at time n The auxiliary variable of the electric field in the y direction of the position, Represents the coordinates at time n The z-direction electric field auxiliary variable of the position, Δz e (k) represents the electric field grid size with a length of k in the z direction, Δy e (j) represents the electric field grid size with a length of j in the y direction; in, express Time coordinates The magnetic field strength in the y direction of the position, Represents the coordinates at time n The x-direction electric field auxiliary variable of the position, Δx e (i) represents the electric field grid size with length i in the x direction; in, express Time coordinates The z-direction magnetic field strength of the position; in, Indicates the coordinates at time n+1 The electric field strength in the x direction at the position, ε represents the dielectric constant in free space, Δy h (j) represents the magnetic field grid size with a length of j in the y direction, Δz h (j) represents the magnetic field grid size with a length of j in the z direction; in, Indicates the coordinates at time n+1 The electric field strength in the y direction at the position, Δz h (k) represents the magnetic field grid size with a length of k in the z direction, Δx h (i) represents the magnetic field grid size with length i in the x direction; in, Indicates the coordinates at time n+1 The electric field strength in the z direction at the position.

5. A CDI-FDTD non-uniform grid modeling method according to claim 4, characterized in that: The CDI-FDTD numerical iteration formula based on the non-uniform grid technology also includes an implicit iteration formula, which is: in, express Time coordinates Auxiliary variable of magnetic field strength in the x direction, express Time coordinates Auxiliary variable of magnetic field strength in the y direction of position, express Time coordinates Auxiliary variable of magnetic field strength in the z direction; They represent the first magnetic field coefficient in the x direction, the second magnetic field coefficient in the x direction, and the third magnetic field coefficient in the x direction at the coordinate (i, j, k), respectively. They represent the first magnetic field coefficient in the y direction, the second magnetic field coefficient in the y direction, and the third magnetic field coefficient in the y direction at the coordinate (i, j, k), respectively. represent the first magnetic field coefficient in the z direction, the second magnetic field coefficient in the z direction, and the third magnetic field coefficient in the z direction at the coordinate (i, j, k), respectively, and 6. A CDI-FDTD non-uniform grid modeling method according to claim 5, characterized in that: The iterative formula of the electric field auxiliary variable in each direction at time n is: in, They represent the first electric field auxiliary variable coefficient in the x direction, the second electric field auxiliary variable coefficient in the x direction, and the third electric field auxiliary variable coefficient in the x direction at the coordinate position (i, j, k), respectively. They represent the first electric field auxiliary variable coefficient in the y direction, the second electric field auxiliary variable coefficient in the y direction, and the third electric field auxiliary variable coefficient in the y direction at the coordinate (i, j, k), respectively. They represent the first electric field auxiliary variable coefficient in the z direction, the second electric field auxiliary variable coefficient in the z direction, and the third electric field auxiliary variable coefficient in the z direction at the coordinate (i, j, k), respectively, and 7. A CDI-FDTD non-uniform grid modeling method according to claim 6, characterized in that: The fourth step comprises: The CDI-FDTD numerical iterative formula based on non-uniform grid technology is used to solve the electric field and magnetic field information of the electromagnetic target to be simulated in all directions, thereby realizing electromagnetic simulation and obtaining the electromagnetic characteristics of the electromagnetic target to be simulated, wherein the electromagnetic characteristics include electric field and magnetic field information.

8. A CDI-FDTD non-uniform grid modeling system, characterized in that: include: A grid division module is used to divide the area where the electromagnetic target is located into grids using a non-uniform grid technology; Initialization module, used to set the parameters of the simulation space and the initial state of the electromagnetic field variables; Iterative formula acquisition module, used to obtain CDI-FDTD numerical iterative formula based on non-uniform grid technology; The electromagnetic simulation module is used to perform electromagnetic simulation on the electromagnetic target to be simulated according to the method from the grid division module to the iterative formula acquisition module to obtain the electromagnetic characteristics of the electromagnetic target to be simulated.

9. A CDI-FDTD non-uniform grid modeling system according to claim 8, characterized in that: The grid partitioning module is also used for: Define the size of fine grid and coarse grid, define the area corresponding to the fine grid and coarse grid, divide the area where the electromagnetic target is located into grids, and design the transition method from fine grid to coarse grid to make a smooth transition between the two.

10. A CDI-FDTD non-uniform grid modeling system according to claim 9, characterized in that: The iterative formula acquisition module is also used for: Maxwell's equations that are satisfied by electromagnetic field quantities in both fine and coarse grids are constructed. The Maxwell equations are discretized using an unconditionally stable divergence-preserving implicit finite-difference time-domain method, and the CDI-FDTD numerical iterative formula based on non-uniform grid technology is obtained.

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