Reduced order conformal finite-difference time-domain method and system based on eigenvalue perturbation

Abstract

The application discloses a reduced-order conformal finite-difference time-domain method and system based on eigenvalue perturbation, and relates to the technical field of electromagnetic simulation. The steps are as follows: a discretized C-FDTD time evolution matrix of an electromagnetic target to be simulated is constructed; numerical stability analysis is performed on the discretized C-FDTD time evolution matrix to obtain stability constraint conditions; a transition matrix positive definiteness criterion is established according to the stability constraint conditions; controllable perturbation is applied to eigenvalues that do not satisfy the transition matrix positive definiteness criterion, and the C-FDTD time evolution matrix is reconstructed after correction; a model reduction technique is used to reduce the order of the reconstructed C-FDTD time evolution matrix, and numerical analysis of an electromagnetic field is performed based on the reduced time evolution matrix. The method can be used to perform iterative calculation with a larger time step under the premise of ensuring data accuracy, effectively improves numerical simulation efficiency, and realizes the collaborative optimization of calculation cost and simulation accuracy.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic simulation technology, and in particular to a reduced-order conformal time-domain finite-difference method and system based on eigenvalue perturbation. Background Technology

[0002] Finite-Difference Time-Domain (FDTD) methods are widely used in electromagnetic field numerical analysis. However, for electromagnetic models containing curved boundaries or complex irregular structures, the discretization characteristics of regular Yee meshes introduce significant step approximation errors, making it difficult for the discrete model to accurately represent the actual geometric structure. This leads to geometric modeling errors and further affects the accuracy of electromagnetic field calculation results. To reduce step approximation errors, researchers have developed conformal FDTD (C-FDTD) based on the FDTD method. The basic idea is to introduce discretization corrections consistent with the actual geometric boundaries near the material interface, thereby improving the modeling accuracy of Yee meshes for curved surfaces and complex irregular structures. However, in practical electromagnetic numerical simulations, the low computational efficiency caused by numerical stability constraints remains a significant factor limiting the full computational advantages of the C-FDTD method. With the continuous development of research on unconditionally stable and weakly conditionally stable FDTD methods, highly stable implicit and semi-implicit C-FDTD methods have attracted widespread attention due to their advantages in overcoming time step stability limitations. However, compared to the traditional explicit C-FDTD method, the implicit C-FDTD method has a more complex iterative format and is more difficult to implement numerically. In recent years, Model Order Reduction (MOR) techniques have been widely applied to FDTD methods; however, the MOR-FDTD method is still constrained by the numerical stability limitations of the traditional FDTD method. Its time step is limited by the Courant-Friedrichs-Lewy (CFL) stability condition, and its computational efficiency is still affected by the mesh size. Therefore, for those skilled in the art, optimizing the traditional C-FDTD method to improve electromagnetic simulation efficiency while maintaining numerical accuracy is an urgent problem to be solved. Summary of the Invention

[0003] The purpose of this invention is to provide a reduced-order conformal finite-difference time-domain method and system based on eigenvalue perturbation to solve the problems mentioned in the background art. It can use a larger time step for iterative calculation while ensuring numerical accuracy, effectively improving the efficiency of numerical simulation, realizing the synergistic optimization of computational overhead and simulation accuracy, and ultimately achieving efficient and stable electromagnetic simulation.

[0004] To achieve the above objectives, the present invention provides the following solution: On one hand, it provides a reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation, the specific steps of which include the following:

[0005] Construct the discretized C-FDTD time evolution matrix of the electromagnetic target to be simulated;

[0006] Numerical stability analysis was performed on the discretized C-FDTD time evolution matrix to obtain stability constraints.

[0007] Based on the stability constraints, a positive definiteness criterion for the transition matrix derived from the discretized C-FDTD time evolution matrix is ​​established.

[0008] A controllable perturbation is applied to the eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix, and the C-FDTD time evolution matrix is ​​reconstructed after correction;

[0009] The reconstructed C-FDTD time evolution matrix is ​​reduced in order using model reduction techniques. Electromagnetic field numerical analysis is then performed based on the reduced time evolution matrix, and the electromagnetic field distribution results of the electromagnetic target to be simulated are output.

[0010] Preferably, for the electromagnetic target to be simulated, a differential form of Maxwell's equations is constructed; the Maxwell's equations are discretized using the C-FDTD method to obtain the discretized C-FDTD time evolution matrix.

[0011] Preferably, the expression for the discretized C-FDTD time evolution matrix is:

[0012] ;

[0013] Where A is the coefficient matrix. ; ;u n The source vector is the activation vector; Vector x n The size is N = N e + N h N e and N h These represent the quantities of electric and magnetic fields in the simulation space, respectively. ; ; ;l x , l y and l z These represent the ratios of the lengths of the non-PEC portions along the x, y, and z directions in the Yee mesh containing the PEC surface structure to the side lengths of the Yee mesh; S x S y and S zThese represent the ratios of the area of ​​the non-PEC region projected onto the yoz, xoz, and xoy planes to the unit area of ​​the Yee grid, respectively.

[0014] Preferably, the criterion for determining the numerical stability of the discretized C-FDTD time evolution matrix is: when the modulus of the eigenvalues ​​of the coefficient matrix A is not greater than 1, the numerical iteration is stable.

[0015] Preferably, the specific process of applying controllable perturbations to eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix is ​​as follows:

[0016] Determine the transition matrix Q of the coefficient matrix A at the desired extended time step;

[0017] Eigenvalue decomposition is performed on the transition matrix Q to identify unstable eigenvalues ​​whose magnitude is greater than or equal to 1;

[0018] A positive perturbation with controllable amplitude is applied to the unstable eigenvalue to obtain the perturbed eigenvalue;

[0019] The diagonal matrix is ​​reconstructed from the perturbed eigenvalues, and the transition matrix is ​​reconstructed based on the reconstructed diagonal matrix.

[0020] Preferably, the model reduction technique is a Krylov subspace model reduction technique based on the Arnoldi algorithm.

[0021] On the other hand, a reduced-order conformal time-domain finite-difference system based on eigenvalue perturbation is provided, including a time evolution matrix construction module, a stability analysis module, a positive definiteness criterion determination module, an eigenvalue correction module, and an order reduction module; wherein,

[0022] The time evolution matrix construction module is used to construct the discretized C-FDTD time evolution matrix of the electromagnetic target to be simulated;

[0023] The stability analysis module is used to perform numerical stability analysis on the discretized C-FDTD time evolution matrix and obtain stability constraints.

[0024] The positive definiteness criterion determination module is used to establish a positive definiteness criterion for the transition matrix derived from the discretized C-FDTD time evolution matrix based on the stability constraint conditions.

[0025] The eigenvalue correction module is used to apply controllable perturbations to eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix, and then reconstruct the C-FDTD time evolution matrix after correction.

[0026] The order reduction module is used to reduce the order of the reconstructed C-FDTD time evolution matrix using model order reduction technology, perform electromagnetic field numerical analysis based on the reduced time evolution matrix, and output the electromagnetic field distribution results of the electromagnetic target to be simulated.

[0027] According to specific embodiments provided by this invention, the following technical effects are disclosed: Addressing the limitations of numerical stability and high memory overhead in traditional C-FDTD methods, this invention first introduces an eigenvalue perturbation (EP) strategy to correct the eigenvalues ​​of the time evolution matrix of the C-FDTD method, enabling the corrected C-FDTD method to possess unconditional stability and overcoming the limitation of mesh size on time step size. Furthermore, a model order reduction (MOR) technique is employed to reconstruct the time evolution matrix of the C-FDTD method, significantly compressing the system matrix dimension and reducing computational and storage costs during matrix solution. The proposed EP-MOR-C-FDTD method can perform iterative calculations with larger time steps while maintaining numerical accuracy, effectively improving numerical simulation efficiency and achieving synergistic optimization of computational overhead and simulation accuracy. This invention's method demonstrates effectiveness and superiority in solving large-scale complex electromagnetic transient problems, providing a feasible approach for achieving efficient and stable electromagnetic simulation. Attached Figure Description

[0028] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0029] Figure 1 This is a flowchart of the method of the present invention;

[0030] Figure 2 This is a conformal mesh element diagram of the present invention, which includes a partially curved PEC structure;

[0031] Figure 3(a) shows the distribution of eigenvalues ​​of the coefficient matrix using the C-FDTD method when CFLN=0.25;

[0032] Figure 3(b) shows the distribution of eigenvalues ​​of the coefficient matrix when CFLN=1 using the C-FDTD method;

[0033] Figure 3(c) shows the distribution of eigenvalues ​​of the coefficient matrix when CFLN=1 using the method of the present invention;

[0034] Figure 3(d) shows the distribution of eigenvalues ​​of the coefficient matrix when CFLN=4 using the method of the present invention;

[0035] Figure 4 The graph shows the variation of the electric field component (Ex) amplitude at the detection point in a three-dimensional metal resonant cavity model over time, calculated using different methods.

[0036] Figure 5 The diagram shows the resonant frequencies of the three-dimensional cavity calculated using different methods in a three-dimensional metal resonant cavity model.

[0037] Figure 6 The graph shows the time-domain waveform of the probe point as a function of time, calculated by different methods on a non-uniform electromagnetic model containing a lossy medium.

[0038] Figure 7 The relative error diagram of the time-domain waveforms of the probe points calculated by the EP-MOR-C-FDTD method and the traditional FDTD method on a non-uniform electromagnetic model containing a lossy medium is shown.

[0039] Figure 8 This is a schematic diagram showing the calculation results of the average electric field on the transmission plane in a three-dimensional waveguide model as a function of time.

[0040] Figure 9 This is a graph showing the transmission coefficient as a function of frequency in a three-dimensional waveguide model.

[0041] Figure 10 This is a graph showing the relative errors of the traditional FDTD method and the EP-MOR-C-FDTD method relative to the C-FDTD method in a three-dimensional waveguide model. Detailed Implementation

[0042] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0043] This invention provides a reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation, such as... Figure 1 As shown, the specific steps include the following:

[0044] S1. Construct the discretized C-FDTD time evolution matrix of the electromagnetic target to be simulated;

[0045] S2. Perform numerical stability analysis on the discretized C-FDTD time evolution matrix to obtain stability constraints;

[0046] S3. Establish a positive definiteness criterion for the transition matrix derived from the discretized C-FDTD time evolution matrix based on stability constraints;

[0047] S4. Apply controllable perturbations to the eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix, and reconstruct the C-FDTD time evolution matrix after correction.

[0048] S5. The reconstructed C-FDTD time evolution matrix is ​​reduced in order using model reduction technology. Electromagnetic field numerical analysis is performed based on the reduced time evolution matrix, and the electromagnetic field distribution results of the electromagnetic target to be simulated are output.

[0049] Furthermore, for the electromagnetic target to be simulated, a differential form of Maxwell's equations is constructed; the C-FDTD method is used to discretize Maxwell's equations, obtaining the discretized C-FDTD time evolution matrix. The specific derivation process is as follows:

[0050] For a general isotropic medium, the differential form of Maxwell's equations can be expressed as:

[0051] (1)

[0052] (2)

[0053] Using the C-FDTD method to numerically discretize (1) and (2), the numerical iterative formulas for E and H can be written in the following matrix form:

[0054] (3)

[0055] in , , , . and It is the identity matrix. , , , It is a diagonal matrix containing the permittivity, permeability, conductivity, and permeability, respectively. n It contains all the excitation source terms. Meanwhile, the expressions for matrices K1 and K2 are as follows:

[0056] (4)

[0057] like Figure 2 As shown, l x , l y and l z These represent the ratios of the lengths of the non-PEC portions along the x, y, and z directions in the Yee mesh containing the PEC surface structure to the side lengths of the Yee mesh, respectively. And S...x S y and S z These represent the ratios of the area of ​​the non-PEC region projected onto the yoz, xoz, and xoy planes to the unit area of ​​the Yee grid, respectively.

[0058] For ease of derivation, (3) is rearranged into the following form:

[0059] (5)

[0060] in , , Vector x n The size is N=N e +N h N e and N h These represent the quantities of electric and magnetic fields in the simulation space, respectively.

[0061] Multiply both sides of (5) by the matrix on the left. The following numerical iterative formulas for the field components in the traditional C-FDTD method can be obtained:

[0062] (6)

[0063] in , .

[0064] Furthermore, according to the stability criteria of numerical methods, when the modulus of the eigenvalues ​​of the coefficient matrix A in (6) does not exceed 1, the constructed numerical method can be considered stable, and the conditions it needs to satisfy are as follows:

[0065] (7)

[0066] Assuming the eigenvectors of matrix A in (7) are α, we can obtain:

[0067] (8)

[0068] Further elaboration of (8) yields:

[0069] (9)

[0070] Let matrix , Add to both sides of (9) And simultaneously multiply by α on the left H Z, we can obtain:

[0071] (10)

[0072] Where, α H Let be the conjugate transpose of α.

[0073] Due to the matrix and Both are real symmetric matrices. Taking the conjugate transpose of both sides of (10) yields:

[0074] (11)

[0075] Using a similar method to the derivation of (10), the following formula can be obtained from (9):

[0076] (12)

[0077] Taking the conjugate transpose of both sides of (12) yields:

[0078] (13)

[0079] Adding equations (12) and (13), and based on (10) and (11), we obtain the following formula:

[0080] (14)

[0081] and,

[0082] (15)

[0083] Because of the matrix , , and All are non-negative real symmetric matrices, so the matrix in (15) is a positive semi-definite matrix.

[0084] At the same time, in order to ensure the stability of the numerical algorithm, it is necessary to ensure the eigenvalues ​​of the coefficient matrix A. .according to We can obtain:

[0085] (16)

[0086] Based on (14), (15) and (16), we can conclude that:

[0087] (17)

[0088] To further derive the matrix The relationship with the time step Δt is defined by an auxiliary variable P that satisfies the following formula:

[0089] (18)

[0090] in,

[0091] (19)

[0092] The matrix containing conformal parameters in (18) Singular value decomposition yields:

[0093] (20)

[0094] In this matrix, both U and V are orthogonal matrices. It is a matrix singular values ​​s i [i = 1,2,...,min(N e N h The singular matrix formed by )].

[0095] Next, substituting (20) into (18) yields the following formula:

[0096] (twenty one)

[0097] Let matrix And it satisfies the following formula:

[0098] (twenty two)

[0099] According to (22), matrices P1 and P have the same eigenvalues. The eigenvalues ​​of matrix P1 are... As shown below:

[0100] (twenty three)

[0101] Where, n = min(N) e N h ).

[0102] According to (17), (18), (22) and (23), when the matrix When the matrix is ​​a positive semi-definite matrix, the eigenvalues ​​of matrix P1 can be obtained. The conditions that must be met are as follows:

[0103] (twenty four)

[0104] Furthermore, the following formula can be obtained:

[0105] (25)

[0106] Arrange (25) to get:

[0107] (26)

[0108] (26) The stability constraints of the traditional C-FDTD method are given. When the time step of the C-FDTD method does not exceed the stability constraint, the matrix... Preserving the positive semi-definite property, thus ensuring that the eigenvalues ​​of the coefficient matrix A satisfy... Therefore, the constructed time-stepping system can maintain numerical stability.

[0109] Furthermore, when the time step of the traditional C-FDTD method exceeds the stability limit, i.e., the time step satisfies the following formula:

[0110] (27)

[0111] eigenvalues ​​of matrix P1 It will no longer retain its non-negativity property, and the positive definiteness of matrix P1 satisfies the following formula:

[0112] (28)

[0113] Furthermore, based on (18), (22), and (28), the matrix can be obtained. The positive definiteness criterion is as follows:

[0114] (29)

[0115] Let the transition matrix Eigenvalue decomposition of matrix Q yields:

[0116] (30)

[0117] Where Λ is the eigenvalue λ of matrix Q. i (i = 1,2,...,N e + N h The diagonal matrix formed by ) is G, where G is the corresponding eigenvector matrix.

[0118] According to (29), some eigenvalues ​​λ in the diagonal matrix Λ of eigenvalues i < 0, and at the same time, matrix Λ satisfies:

[0119] (31)

[0120] At this point, (16) will no longer hold, and the eigenvalues ​​of the coefficient matrix A cannot be guaranteed. This causes numerical methods to no longer satisfy stability properties, leading to divergent calculation results. Therefore, for simulations requiring high mesh resolution, a small mesh size will severely limit the range of time steps in the C-FDTD method, thus affecting its numerical computation efficiency. Furthermore, when the time step of the traditional C-FDTD method exceeds the stability limit, some eigenvalues ​​in the transition matrix derived from the time evolution matrix will often no longer satisfy the positive definiteness criterion, causing the corresponding modes to continuously grow over time and leading to numerical instability.

[0121] To extend the numerical stability conditions of the C-FDTD method, eigenvalue perturbation (EP) is used to correct the transition matrix Q and its eigenvalue matrix Λ in the C-FDTD method. First, the eigenvalues ​​in matrix Λ that do not satisfy the positive definite condition are subject to controlled perturbation, resulting in the following formula:

[0122] (32)

[0123] in It is a very small positive number.

[0124] Next, the perturbed eigenvalues Reconstructing the diagonal matrix yields:

[0125] (33)

[0126] Finally, the perturbed transition matrix is ​​further reconstructed, as shown in the following expression:

[0127] (34)

[0128] Among them, matrix Given a positive semi-definite matrix, during numerical iteration, the following is used: The original matrix Q is replaced, thus ensuring that the numerical stability of the system is maintained even when the time step exceeds the stability limit.

[0129] Furthermore, according to (6), the Eigenvalue Perturbation C-FDTD (EP-C-FDTD) method can be obtained by performing eigenvalue perturbation processing on the traditional C-FDTD method. The numerical iteration formula of the corresponding electromagnetic field components can be written in the following form:

[0130] (35)

[0131] Because (35) allows for iterative solutions using time steps exceeding the stability constraints of C-FDTD, the EP-C-FDTD method has higher numerical computation efficiency compared to the traditional C-FDTD method. However, the coefficient matrix in (35) and The large dimensionality of the coefficient matrix leads to high computational complexity, which is the main factor restricting further improvement of the computational efficiency of the EP-C-FDTD method. To address this, Model Order Reduction (MOR) is introduced to modify the coefficient matrix in (35). and Dimensionality reduction is performed to reduce the computational complexity of the numerical iteration process, thereby further improving the computational efficiency of the algorithm.

[0132] Furthermore, the model reduction technique is a Krylov subspace model reduction technique based on the Arnoldi algorithm.

[0133] Using the Arnoldi method to generate a matrix The Krylov subspace formed by vector x:

[0134] (36)

[0135] Where m is the order of the Krylov subspace. (36) is a basis R. m =[r1,r2,...,r m [This constitutes (35)] and The projection matrix. Where, matrix R is... m The dimension is (N) e +N h )×m( It should be noted that matrix R... m It is an orthogonal matrix that satisfies H m It is the upper Hessenberg matrix H m+1,m The first m lines, R m With H m The relationship between them can be represented as:

[0136] (37)

[0137] in, Multiply both sides of (37) by the left at the same time. And by simplification we can get

[0138] (38)

[0139] Among them, H mRepresentation matrix The projection onto the Kryloy subspace, where H is... m Representation matrix The reduced-order matrix is ​​obtained by multiplying both sides of equation (35) on the left. At the same time, the properties of orthogonal matrices are used. It can be obtained

[0140] (39)

[0141] (39) can be further derived into the following form:

[0142] (40)

[0143] in , .

[0144] Finally, (40) presents the numerical iterative formula for solving the electromagnetic field components using the method of the present invention, namely the EP-MOR-C-FDTD method. It is worth noting that the correlation coefficient matrix in (40)... and The dimension of m is much smaller than the coefficient matrix in (35). and Dimensions (N = N) e + N h Therefore, compared with the traditional C-FDTD method, the EP-MOR-C-FDTD method can significantly reduce computational complexity and effectively improve the efficiency of electromagnetic field solution.

[0145] To verify the numerical stability of the proposed method, this invention considers a metal resonant cavity model with dimensions of 2m × 2m × 2m, in which a PEC sphere with a radius of 0.25m is placed at the center, and the size of the cubic mesh is ∆x = ∆y = ∆z = 0.05m. The excitation source is a Gaussian pulse.

[0146] (41)

[0147] Where T s = 2.4 ns, t0 = 4 T s .

[0148] The extension factor CFLN for the stability condition of CFL is defined as the difference between the time step ∆t used in the proposed method and the maximum time step ∆t allowed by the stability condition in the traditional FDTD method. max The ratio, i.e., CFLN = ∆t / ∆t max .

[0149] As shown in Figures 3(a)-3(d), in free space, when the stability limit of C-FDTD is exceeded (CFLN=1), the system inherits the unstable eigenvalues ​​in (6), causing some eigenvalues ​​to lie outside the unit circle. When the EP-MOR-C-FDTD method is used to perform controlled perturbations on the eigenvalues ​​that do not meet the positive definite condition, it can be seen that all eigenvalues ​​are strictly located inside or on the unit circle, which shows that the reduced-order model maintains numerical stability in free space.

[0150] To verify the correctness and effectiveness of the EP-MOR-C-FDTD method proposed in this invention, a three-dimensional metal resonant cavity model is used as an example for numerical simulation in this embodiment, and the calculation results are analyzed. The resonant cavity model occupies 40×40×40 grid cells, and the center of the model contains a PEC sphere with a radius of 0.25m. The spatial grid size in the simulation is set to 0.05m. A Gaussian pulse excitation source is set in the simulation region, with parameters T. w =2.4ns, t0=4T w For this model, the coefficient matrix constructed using the traditional C-FDTD method has a dimension of 398520×398520, while the coefficient matrix of the EP-MOR-C-FDTD method proposed in this invention has a dimension of only 3600×3600, significantly reducing the dimensionality of the computational equations. In numerical simulations, the traditional FDTD method, the C-FDTD method, and the proposed EP-MOR-C-FDTD method are compared and studied. The time step used by the C-FDTD method satisfies CFLN=0.25, while the time step used by the traditional FDTD method and the EP-MOR-C-FDTD method corresponds to CFLN=1, which exceeds the stability condition range allowed by the C-FDTD method.

[0151] Figure 4 and Figure 5 The reference solution and the time-domain response and frequency-domain characteristics of the resonant cavity at the probe point calculated by the three methods are presented. Comparison with the reference solution shows that both the C-FDTD and EP-MOR-C-FDTD methods achieve high computational accuracy; however, the traditional FDTD method, due to its limited ability to characterize bending boundaries, shows a significant deviation from the reference solution. Notably, with a larger time step (CFLN=1), the results obtained by the EP-MOR-C-FDTD method are in excellent agreement with the results obtained by the C-FDTD method under the condition of CFLN=0.25. The EP-MOR-C-FDTD method maintains high numerical accuracy and good stability even with a larger time step.

[0152] In addition, regarding Figure 4The results show that the relative errors of the traditional FDTD method and the EP-MOR-C-FDTD method relative to the C-FDTD method were calculated according to Equation (42). The results show that the maximum relative error of the EP-MOR-C-FDTD method is only -63 dB throughout the simulation, demonstrating high computational accuracy. Compared with the traditional FDTD method, the EP-MOR-C-FDTD method can not only effectively reconstruct the system response, but also maintain the original numerical accuracy well. At the same time, the error analysis results further illustrate that the introduction of eigenvalue perturbation and model order reduction techniques did not significantly affect the accuracy of the C-FDTD method.

[0153] Table 1 compares the simulation time and memory usage of the traditional FDTD method, the C-FDTD method, and the EP-MOR-C-FDTD method. It can be seen that the EP-MOR-C-FDTD method has a certain advantage in memory usage compared to the traditional FDTD and C-FDTD methods. Furthermore, because the EP-MOR-C-FDTD method overcomes the stability limitation of the time step, it can use a larger time step, thus significantly improving numerical computation efficiency and achieving a marked improvement in runtime compared to the C-FDTD method.

[0154] (42)

[0155] Table 1. Simulation time and memory usage of the three methods

[0156] Method Simulation Time (s) Memory Cost (MB) Traditional FDTD 42.31 64.82 C-FDTD 167.48 70.93 EP-MOR-C-FDTD 12.11 30.55

[0157] To further verify the applicability of the method of this invention in complex media environments, simulation verification was performed on a non-uniform electromagnetic model containing lossy media. Spatial discretization used a uniform mesh with a mesh size of Δx=Δy=Δz=Δ=0.1m. A Gaussian pulse was used as the excitation source in the simulation, with parameters T. w =5ns, t0=4T w For this model, the system coefficient matrix constructed based on the traditional C-FDTD method has a dimension of 742955×742955, while the coefficient matrix corresponding to the EP-MOR-C-FDTD method has a dimension of only 4000×4000, thus significantly reducing the system size. Figure 6 The variation of the electric field amplitude at the detection point over time is presented. The results show that the time-domain response of the EP-MOR-C-FDTD method is highly consistent with that of the C-FDTD method, while the results of the traditional FDTD method show a significant deviation from the reference solution. Figure 7The relative errors of the traditional FDTD method and the EP-MOR-C-FDTD method compared to the C-FDTD method are further demonstrated. It is evident that the error of the EP-MOR-C-FDTD method does not exceed that of the traditional FDTD method throughout the entire simulation time range. This indicates that the proposed method can maintain high computational accuracy in electromagnetic simulations involving complex lossy dielectric structures, thus verifying the correctness and reliability of the proposed method.

[0158] Table 2 presents a comparison of simulation time and memory usage between the traditional FDTD method, the C-FDTD method, and the EP-MOR-C-FDTD method for this simulation environment. The results show that the EP-MOR-C-FDTD method not only optimizes memory usage compared to the traditional FDTD and C-FDTD methods, but also significantly reduces computation time. This indicates that using a larger time step combined with matrix reduction processing can effectively improve the overall computational efficiency of numerical algorithms.

[0159] Table 2 Simulation time and memory usage of the three methods

[0160] Method Simulation Time (s) Memory Cost (MB) Traditional FDTD 18.49 122.27 C-FDTD 45.59 133.61 EP-MOR-C-FDTD 4.82 43.98

[0161] Furthermore, a three-dimensional waveguide model was selected to verify the method of this invention and further evaluate its performance. The physical dimensions of the three-dimensional waveguide model are 1m × 1.5m × 4m, and the spatial discretization uses a uniform grid with a grid size of Δx = Δy = Δz = Δ = 0.05m. In the simulation, a Gaussian pulse plane wave with a maximum frequency of 0.3GHz was used as the excitation source. For this model, the system coefficient matrix constructed based on the traditional C-FDTD method has a dimension of 301930 × 301930, while the coefficient matrix of the EP-MOR-C-FDTD method has a dimension of only 3000 × 3000. Figure 8 The calculated results of the time-varying average electric field (Ex) on the transmission plane are presented. The results show that the EP-MOR-C-FDTD method and the traditional C-FDTD method exhibit high consistency in the time-domain electric field response, while maintaining good fitting accuracy with the reference solution.

[0162] also, Figure 9 The curves showing the transmission coefficient (S21) versus frequency are presented. The results demonstrate a high degree of consistency between the frequency domain responses obtained by the EP-MOR-C-FDTD method and the C-FDTD method. Regarding... Figure 8 The results shown Figure 10The relative errors of the traditional FDTD method and the EP-MOR-C-FDTD method compared to the C-FDTD method are further presented. It can be seen that the maximum relative error of the EP-MOR-C-FDTD method is less than -60 dB, and the overall error level is lower than that of the traditional FDTD method, verifying the accuracy of the proposed method in waveguide electromagnetic simulation. Furthermore, Table 3 lists the comparison results of the three methods in terms of simulation time and memory usage. The EP-MOR-C-FDTD method significantly outperforms the traditional FDTD method and the C-FDTD method in terms of computation time, while also reducing memory consumption. These results demonstrate that the proposed method can effectively improve computational efficiency while maintaining computational accuracy, thus exhibiting good application potential in large-scale electromagnetic simulations.

[0163] Table 3 Simulation time and memory usage of the three methods

[0164] Method Simulation Time (s) Memory Cost (MB) Traditional FDTD 30.65 44.92 C-FDTD 63.19 50.17 EP-MOR-C-FDTD 8.21 21.90

[0165] This invention proposes an EP-MOR-C-FDTD algorithm based on eigenvalue perturbation and model order reduction. This method effectively extends the stability limit of the time step by applying controllable perturbations to unstable eigenvalues ​​in the time-progressing system matrix that exceed the stability range. Simultaneously, it reduces the system dimensionality by combining structure-preserving model order reduction techniques, thereby significantly reducing computational complexity. Numerical results show that the proposed method achieves high computational accuracy and significantly improves simulation efficiency while reducing memory overhead, while maintaining the structural characteristics of the C-FDTD method. These results verify the effectiveness and superiority of this method in solving large-scale complex electromagnetic transient problems, providing a feasible approach for achieving efficient and stable electromagnetic simulation.

[0166] On the other hand, this invention provides a reduced-order conformal time-domain finite-difference system based on eigenvalue perturbation, including a time evolution matrix construction module, a stability analysis module, a positive definiteness criterion determination module, an eigenvalue correction module, and an order reduction module; wherein...

[0167] The time evolution matrix construction module is used to construct the discretized C-FDTD time evolution matrix of the electromagnetic target to be simulated;

[0168] The stability analysis module is used to perform numerical stability analysis on the discretized C-FDTD time evolution matrix and obtain stability constraints.

[0169] The positive definiteness criterion determination module is used to establish a positive definiteness criterion for the transition matrix derived from the discretized C-FDTD time evolution matrix based on stability constraints.

[0170] The eigenvalue correction module is used to apply controllable perturbations to eigenvalues ​​that do not meet the positive definiteness criterion of the transition matrix, and then reconstruct the C-FDTD time evolution matrix after correction.

[0171] The order reduction module is used to reduce the order of the reconstructed C-FDTD time evolution matrix using model order reduction technology, perform electromagnetic field numerical analysis based on the reduced time evolution matrix, and output the electromagnetic field distribution results of the electromagnetic target to be simulated.

[0172] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation, characterized in that, The specific steps include the following: Construct the discretized C-FDTD time evolution matrix of the electromagnetic target to be simulated; Numerical stability analysis was performed on the discretized C-FDTD time evolution matrix to obtain stability constraints. Based on the stability constraints, a positive definiteness criterion for the transition matrix derived from the discretized C-FDTD time evolution matrix is ​​established. A controllable perturbation is applied to the eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix, and the C-FDTD time evolution matrix is ​​reconstructed after correction; The reconstructed C-FDTD time evolution matrix is ​​reduced in order using model reduction techniques. Electromagnetic field numerical analysis is then performed based on the reduced time evolution matrix, and the electromagnetic field distribution results of the electromagnetic target to be simulated are output.

2. The reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation according to claim 1, characterized in that, For the electromagnetic target to be simulated, a differential form of Maxwell's equations is constructed; the C-FDTD method is used to discretize the Maxwell's equations to obtain the discretized C-FDTD time evolution matrix.

3. The reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation according to claim 1, characterized in that, The expression for the discretized C-FDTD time evolution matrix is ​​as follows: ； Where A is the coefficient matrix. ; ;u n The source vector is the activation vector; Vector x n The size is N = N e + N h N e and N h These represent the quantities of electric and magnetic fields in the simulation space, respectively. ; ; ;l x , l y and l z These represent the ratios of the lengths of the non-PEC portions along the x, y, and z directions in the Yee mesh containing the PEC surface structure to the side lengths of the Yee mesh; S x S y and S z These represent the ratios of the area of ​​the non-PEC region projected onto the yoz, xoz, and xoy planes to the unit area of ​​the Yee grid, respectively.

4. The reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation according to claim 3, characterized in that, The criterion for determining the numerical stability of the discretized C-FDTD time evolution matrix is: when the modulus of the eigenvalues ​​of the coefficient matrix A is not greater than 1, the numerical iteration is stable.

5. The reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation according to claim 3, characterized in that, The specific process of applying a controllable perturbation to the eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix is ​​as follows: Determine the transition matrix Q of the coefficient matrix A at the desired extended time step; Eigenvalue decomposition is performed on the transition matrix Q to identify unstable eigenvalues ​​whose magnitude is greater than or equal to 1; A positive perturbation with controllable amplitude is applied to the unstable eigenvalue to obtain the perturbed eigenvalue; The diagonal matrix is ​​reconstructed from the perturbed eigenvalues, and the transition matrix is ​​reconstructed based on the reconstructed diagonal matrix.

6. The reduced-order conformal time-domain finite-difference method based on eigenvalue perturbation according to claim 1, characterized in that, The model reduction technique is a Krylov subspace model reduction technique based on the Arnoldi algorithm.

7. A reduced-order conformal time-domain finite-difference system based on eigenvalue perturbation, characterized in that, It includes modules for constructing the time evolution matrix, stability analysis, determining the positive definiteness criterion, eigenvalue correction, and order reduction; among which, The time evolution matrix construction module is used to construct the discretized C-FDTD time evolution matrix of the electromagnetic target to be simulated; The stability analysis module is used to perform numerical stability analysis on the discretized C-FDTD time evolution matrix and obtain stability constraints. The positive definiteness criterion determination module is used to establish a positive definiteness criterion for the transition matrix derived from the discretized C-FDTD time evolution matrix based on the stability constraint conditions. The eigenvalue correction module is used to apply controllable perturbations to eigenvalues ​​that do not satisfy the positive definiteness criterion of the transition matrix, and then reconstruct the C-FDTD time evolution matrix after correction. The order reduction module is used to reduce the order of the reconstructed C-FDTD time evolution matrix using model order reduction technology, perform electromagnetic field numerical analysis based on the reduced time evolution matrix, and output the electromagnetic field distribution results of the electromagnetic target to be simulated.

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