Electromagnetic simulation system based on three-dimensional multi-symplectic finite difference time domain method

Abstract

The invention discloses an electromagnetic simulation system based on a three-dimensional polysymplectic finite difference time domain method. The electromagnetic simulation system comprises a construction module, a decomposition module and an integration module, the construction module is used for constructing a Dosin form Maxwell equation set of a to-be-simulated electromagnetic field; the decomposition module is used for decomposing the Dosin form Maxwell equation set into a plurality of one-dimensional equations; and the integration module is used for solving the plurality of one-dimensional equations to obtain a plurality of equation solutions, and combining the plurality of equation solutions to complete electromagnetic simulation. The method can adapt to different types of electromagnetic environments, and is suitable for complex geometric structures and multi-scale problems. Meanwhile, flexible parameter setting and optimization schemes are provided, so that the requirements of various engineering applications are met; in addition, according to an existing computing platform and algorithm framework, the method can be easily integrated into an existing simulation tool, and functions and application scenes of the simulation tool are expanded.

Description

Technical Field

[0001] The invention relates to the field of electromagnetic simulation, and in particular to an electromagnetic simulation system based on a three-dimensional multi-symplectic finite difference time domain method. Background Art

[0002] In the construction of mathematical models in disciplines such as celestial mechanics, quantum mechanics, and electromagnetism, it is necessary to maintain as many inherent characteristics and physical properties of the original system as possible. When developing numerical schemes, traditional algorithms are designed without considering these physical characteristics or inherent properties. Scholars have developed a numerical analysis theory for classical Hamiltonian systems, which is qualitative in geometry and quantitative in numerical values. Academician Feng Kang's numerical integration algorithm, which strictly preserves the symplectic geometric structure of the Hamiltonian system while effectively maintaining its physical properties. This high-performance new algorithm is now called the symplectic algorithm. The symplectic algorithm shows unique advantages in the numerical calculation of Hamiltonian systems and has strong capabilities in numerical prediction and numerical tracking. It has also been widely and deeply applied in many other fields of science, technology, and engineering. The symplectic algorithm shows limitations when applied to the discretization of infinite-dimensional Hamiltonian systems, mainly due to the global nature of the conserved quantities. In contrast, the conservation laws in multi-symplectic Hamiltonian systems are local; they are located in any arbitrary space-time region and are independent of boundary conditions. In addition, the phase space of multi-symplectic Hamiltonian systems is finite-dimensional, so they can be analyzed by finite-dimensional theory. Multi-symplectic algorithms, as a generalization of symplecticity, maintain symplectic structure in both space and time. This makes them significantly superior to other methods in terms of long-term computation and numerical stability.

[0003] The characteristic of the multisymplectic scheme is that it not only obeys global energy conservation, but also obeys strict local conservation laws, which are called multisymplectic conservation laws (MSCL). MSCL ensures that the multisymplectic Hamiltonian system obeys both local energy conservation laws and momentum conservation laws. The advantages of the multisymplectic scheme have been successfully demonstrated in many different equations, including the Korteweg-de Vries equation, the Kadomtsev Petviashvili equation, the Zakharov-Kuznetsov equation, the nonlinear Schrödinger equation, etc. Similarly, the Maxwell equations have a multisymplectic structure and have been mathematically verified by many scholars in the field of mathematics.

[0004] The above work mainly focuses on the mathematical analysis and proof of the superior performance of the multi-symplectic scheme form of Maxwell's equations. However, there is no special discussion on the application of multi-symplectic scheme theory in computational electromagnetics to simulate actual engineering problems. The integration of multi-symplectic theory and numerical calculation and its application in solving electromagnetic field problems and solutions have not yet been proposed. Summary of the invention

[0005] The traditional finite-difference time-domain (FDTD) method is prone to numerical dispersion problems in high frequency bands, resulting in large deviations in simulation results. The present invention is based on a multi-symplectic algorithm, which aims to reduce the impact of numerical dispersion and improve the accuracy of solving Maxwell's equations, thereby more accurately simulating the wave propagation characteristics in complex electromagnetic environments.

[0006] To achieve the above object, the present invention provides an electromagnetic simulation system based on a three-dimensional multi-symplectic finite difference time-domain method, comprising: a building module, a decomposition module and an integration module;

[0007] The construction module is used to construct the multi-symplectic Maxwell equations of the electromagnetic field to be simulated;

[0008] The decomposition module is used to decompose the multi-symplectic Maxwell equations into a plurality of one-dimensional equations;

[0009] The integrated module is used to solve several one-dimensional equations, obtain several solutions to the equations, and combine the several solutions to complete electromagnetic simulation.

[0010] Preferably, the workflow of the construction module includes: starting from the basic form of Maxwell's equations, deriving a multi-symplectic form representation; and constructing a new difference format by introducing the multi-symplectic Preissman scheme to construct the multi-symplectic form of Maxwell's equations.

[0011] Preferably, the Hamiltonian equations of the multisymplectic scheme with m space dimensions include:

[0012]

[0013] Among them, I represents the unit matrix; M and K1, K2, K3 represent antisymmetric matrices, S(z) represents a smooth function, which has been set as Hamiltonian or energy generalization; C1, C2, C3 are the coefficients in K1, K2, K3; Z t Indicates discreteness in the time direction; Z1, Z2, and Z3 indicate discreteness in other directions; represents the Hamiltonian operator.

[0014] Preferably, the building module discretizes the Hamiltonian equations of the multi-symplectic scheme using an implicit midpoint symplectic scheme to obtain the Preissman format in the multi-symplectic algorithm.

[0015]

[0016] Among them, n represents the number of time steps; Δt represents the time step; Δx, Δy, Δz represent the spatial steps of the x, y, and z axes; i, j, and k represent coordinates.

[0017] Preferably, the multi-symplectic Preissman scheme is used for discretization in space and time

[0018] Combining the multi-symplectic Preissman format with Maxwell's equations yields the multi-symplectic Preissman scheme:

[0019]

[0020] Among them, the central discrete operator is as follows:

[0021]

[0022] Where E represents the electric field; H represents the magnetic field; n represents the number of time steps; δ t Indicates time discreteness; δ x Indicates the discreteness in the x-axis direction; δ y Indicates discreteness in the y-axis direction; δ z Indicates discreteness in the z-axis direction; x, y, z indicate coordinate directions; μ indicates magnetic permeability; ε indicates dielectric constant.

[0023] Preferably, the decomposition module decomposes the multi-symplectic Maxwell equations into several one-dimensional sub-problems by introducing a local one-dimensional method:

[0024]

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] The present invention can be applied to complex geometric structures and multi-scale problems by adapting to different types of electromagnetic environments. At the same time, the present invention provides flexible parameter settings and optimization solutions to meet the needs of various engineering applications; in addition, it is compatible with existing computing platforms and algorithm frameworks, and can be easily integrated into existing simulation tools to expand their functions and application scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] In order to more clearly illustrate the technical solution of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0028] Figure 1 A schematic diagram of the system structure of an embodiment of the present invention;

[0029] Figure 2 A schematic diagram of a medium calculation model according to an embodiment of the present invention;

[0030] Figure 3 It is a simulation diagram of numerical results of an embodiment of the present invention;

[0031] Figure 4 Graph showing relative errors of an embodiment of the present invention. DETAILED DESCRIPTION

[0032] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only 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.

[0033] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0034] Embodiment 1

[0035] like Figure 1 As shown, it is a schematic diagram of the system structure of this embodiment, including: a building module, a decomposition module and an integration module; the building module is used to construct the multi-symplectic Maxwell equations of the electromagnetic field to be simulated; the decomposition module is used to decompose the multi-symplectic Maxwell equations into a number of one-dimensional equations; the integration module is used to solve a number of one-dimensional equations, obtain a number of equation solutions, and combine the several equation solutions to complete the electromagnetic simulation.

[0036] The following will describe in detail how the present invention solves the technical problems in practical work in conjunction with this embodiment.

[0037] First, the building module converts the Maxwell equations into a multi-symplectic form and applies the multi-symplectic Preissman scheme, which can effectively reduce numerical dispersion errors when dealing with high-frequency electromagnetic wave problems. Especially in complex medium environments, this method can accurately reproduce the propagation characteristics of electromagnetic waves. The core of this embodiment is to convert the Maxwell equations into a multi-symplectic form. This process includes introducing the Preissman scheme in the multi-symplectic algorithm, which enables it to maintain the energy and momentum conservation of the system in the numerical simulation of the electromagnetic field.

[0038] The specific steps are as follows:

[0039] The Maxwell equations are discretized to form a multi-symplectic matrix structure.

[0040] First, starting from the basic form of Maxwell's equations, its multi-symplectic form is derived. By introducing the multi-symplectic Preissman scheme, a new difference format is constructed, which can maintain the energy conservation and momentum conservation of the system. The Hamiltonian equation of the multi-symplectic scheme with m space dimensions is:

[0041]

[0042] Among them, I represents the unit matrix; M and K1, K2, K3 represent antisymmetric matrices, S(z) represents a smooth function, which has been set as Hamiltonian or energy generalization; C1, C2, C3 are the coefficients in K1, K2, K3; Z t Indicates discreteness in the time direction; Z1, Z2, and Z3 indicate discreteness in other directions; represents the Hamiltonian operator.

[0043] The implicit midpoint symplectic scheme is used to discretize the above formula and obtain the Preissman format in the multi-symplectic algorithm:

[0044]

[0045] Among them, n represents the number of time steps; Δt represents the time step; Δx, Δy, Δz represent the spatial steps of the x, y, and z axes; i, j, and k represent coordinates.

[0046] (2) The numerical format is adjusted through the multi-symplectic Preissman scheme to maintain the long-term stability of physical conservation quantities.

[0047] In the numerical solution process, the multi-symplectic Preissman scheme is used for spatial and temporal discretization. The characteristic of this scheme is that it can effectively control numerical dispersion, so that the accuracy of the solution is guaranteed in long-term simulation.

[0048] Combining the multi-symplectic Preissman scheme with Maxwell's equations yields the multi-symplectic Preissman scheme:

[0049]

[0050] Among them, the central discrete operator is as follows:

[0051]

[0052] Among them, E represents the electric field; H represents the magnetic field; n represents the number of time steps; Δt represents the time step; Δx, Δy, Δz represent the spatial steps of the x, y, z axes; i, j, k represent coordinates.

[0053] This embodiment uses a dispersion relation contour map using kx and ky as coordinate axes at a specific frequency. In addition, due to the symmetric nature of the numerical dispersion relation, the contour figures projected onto the (kx, kz) plane and the (ky, kz) plane are omitted because they are the same as the contour figures of the (kx, ky) plane. For smaller k values, the numerical contour is almost circular, and the dispersion performance of the conventional method shows distortion in the edge area and high-frequency area, while the method of this embodiment shows excellent dispersion performance.

[0054] In order to reduce the computational complexity, this embodiment introduces a local one-dimensional method, which significantly reduces the consumption of computational resources while maintaining the accuracy of the results by decomposing the three-dimensional problem into multiple one-dimensional sub-problems for solution.

[0055] The multi-symplectic Hamiltonian system is decomposed into three local one-dimensional subsystems using the decomposition module, thereby obtaining a local one-dimensional multi-symplectic Preissman scheme that allows the Maxwell system, which is expressed as follows:

[0056]

[0057] Finally, the integrated module solves each sub-problem separately to reduce the overall computational effort; and through appropriate boundary conditions and interface matching technology, it ensures that the solutions between sub-problems can be seamlessly combined to form a complete three-dimensional solution.

[0058] This embodiment also verifies the effectiveness of the present invention through a series of numerical experiments, and theoretically derives that the system has stability and good dispersion properties, and finally verifies through numerical examples that the computational efficiency is significantly improved compared to traditional methods.

[0059] The stability analysis of this embodiment is performed using the Fourier method. Using the plane wave solution of Maxwell's equations, the matrix between the nth time step and the n+1th time step in the spatial spectrum domain is rewritten:

[0060]

[0061] Among them, A, A1, A2, A3, A4 represent the matrix composed of coefficients; B represents the matrix composed of coefficients; ψ represents the matrix composed of electric field and magnetic field; n represents the number of time steps;

[0062] The stability analysis of the multi-symplectic Preissman format is performed, the coefficients are matrixed, and then the plane wave solution of Maxwell's equations is substituted into the matrix to obtain the following matrix:

[0063]

[0064] The solved eigenvalues ​​are:

[0065] λ1=λ2=1

[0066]

[0067] in

[0068]

[0069] w x =csin(k xΔx / 2)Δt / Δx

[0070] w y =csin(k y Δy / 2)Δt / Δy

[0071] w z =csin(k z Δz / 2)Δt / Δz

[0072] In the formula, w x 、w y 、w z Indicates the intermediate transition amount; c indicates; k x , k y , k z represents the wave number in different directions; Δz represents the spatial discretization in the z direction; λ1, λ2, λ3, λ4, λ5, λ6 represent the eigenvalues;

[0073] It can be clearly seen that the modulus of the eigenvalues ​​is 1, thus proving that the scheme of the method of the present invention is unconditionally stable. The unconditional stability of the method of the present invention is also verified in the above numerical examples.

[0074] Embodiment 2

[0075] In conjunction with this embodiment, the following will describe in detail how the present invention can be put into practical application, its application scenarios and other related contents.

[0076] The present invention realizes long-term stable simulation of electromagnetic systems through the energy and momentum conservation characteristics of the multi-symplectic algorithm. During long-term simulation, the traditional finite difference time domain (FDTD) method often causes system instability due to accumulated errors, while the method of the present invention can maintain the stability of the numerical solution and reduce the error accumulation effect. Verification shows that the simulation time can still remain stable when extended to thousands of time steps.

[0077] In both the method of the present invention and the FDTD method simulation, the following formula is used as the excitation source:

[0078] E(t)=E0*exp[-(t-t0) / τ) 2 ]

[0079] Wherein, E(t) represents the electric field strength at time t; E0 represents the initial electric field strength of the Gaussian pulse; t represents the time variable; t0 represents the center time of the Gaussian pulse; τ represents the pulse width parameter, which determines the duration of the Gaussian pulse; in this embodiment, τ is set to 5.333e -9 s;t0=2.13e -8 s.

[0080] The spatial step size is set to 0.1m. The Gaussian source is excited at the 50th element and propagates in both directions in free space. The two algorithms propagate 5000 grids (1e -6 The waveform generated after s) is shown in the figure below. It is worth noting that the time step used by the method of the present invention is 3e -10 s (CFL = 1.27), while the time step used by the FDTD method is 1e -10 s (CFL = 0.42). By using different CFL conditions and keeping the same actual physical time, the unconditional stability of the method of the present invention is effectively proved. The waveform generated by the FDTD method has obvious hysteresis, while the waveform generated by the present invention is very consistent with the analytical solution. The method of the present invention has greater stability than the FDTD method.

[0081] When other conditions of the above settings remain unchanged, t0 is changed to 1.333e -8 s, there will be a small mutation in the waveform at the beginning of the Gaussian pulse. Experiments show that after the waveform is transmitted for a certain distance, the waveform of the FDTD scheme will have obvious jitter. In addition to a small mutation on the entire waveform, there is an obvious periodic jitter between the two waveforms. The waveform of the scheme of the present invention is highly consistent with the waveform of the source, and the mutation generated by the initial waveform of the Gaussian pulse source remains consistent in the scheme of the present invention, without additional mutation. Moreover, the error of the FDTD scheme increases with time, while the error of the scheme of the present invention does not change basically. From this numerical example, it can be understood that the present invention can highly maintain the state of the original waveform and can maintain it for a long time.

[0082] By converting the Maxwell equations into multi-symplectic form and applying the multi-symplectic Preissman scheme, the present invention can effectively reduce the numerical dispersion error when dealing with high-frequency electromagnetic wave problems. Numerical simulation results show that under the same grid density, the calculation accuracy of the method of the present invention is improved by about 30% compared with the traditional FDTD method. Especially in complex medium environments, the method can accurately reproduce the propagation characteristics of electromagnetic waves.

[0083] The local one-dimensional method is used to reduce the computational complexity of complex three-dimensional problems. By decomposing the problem into one-dimensional sub-problems, the overall computing resource consumption is reduced and the simulation speed is increased by about 50%. This processing method significantly shortens the calculation time without affecting the accuracy of the results, and is particularly suitable for large-scale electromagnetic simulations.

[0084] In this embodiment, the cavity size is set to 0.02m×0.02m, while the CFL condition is kept unchanged for multi-mode analysis (TE12, TE22, TE23, TE33). It can be seen from the experiment that the numerical results of all four modes in the scheme of the present invention exceed the numerical results of the FDTD method. Specifically, the numerical error of the FDTD method in the TE33 mode is 0.68%, while the error of the scheme of the present invention is only 0.005%. Table 1 gives the detailed numerical results of the four cavity modes, indicating that the results of the scheme of the present invention are closer to the analytical solution, with an error of less than 0.02%, highlighting its high accuracy.

[0085] Table 1

[0086]

[0087] 3D Case Analysis

[0088] The computational area is a 130×15×15 grid, and the spatial step size is uniformly set to 6e. -5 m. The time increment of the method in this embodiment is set to 2e -13 s (CFL = 1.73), and the time increment of the FDTD method is set to 2e -14 s (CFL = 0.173). The Gaussian source propagates along the x-axis, while the newly developed periodic boundary conditions of the invention are applied in the other two directions. The newly developed periodic boundary conditions require multiple parameter adjustments of the tridiagonal matrix coefficients according to the magnetic and electric field components required in the iterative process. The dimensions of the medium are all set to 3e -3 m, the specific modeling diagram is as follows Figure 2 As shown. Figure 3 As shown in Figure 2, the numerical results of the solution of the present invention are very close to those of the FDTD method and the rigorous coupled wave analysis (RCWA), among which the numerical results of the method of the present invention are slightly better. In order to more clearly illustrate the difference between the method of the present invention and the FDTD method, this example generates a relative error Figure 4 Obviously, within the calculated frequency range, the accuracy of the proposed method is always higher than that of the FDTD method. The proposed method adopts a more relaxed CFL condition, which makes the proposed method have the characteristic of large amount of calculation, but the calculation time of the proposed method is 73.25s, which is better than the 94.41s of the FDTD method.

[0089] The embodiments described above are only descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. An electromagnetic simulation system based on a three-dimensional multi-symplectic finite difference time-domain method, characterized in that: include: Building blocks, decomposing blocks, and integrating blocks; The construction module is used to construct the multi-symplectic Maxwell equations of the electromagnetic field to be simulated; The decomposition module is used to decompose the multi-symplectic Maxwell equations into a plurality of one-dimensional equations; The integrated module is used to solve several one-dimensional equations, obtain several solutions to the equations, and combine the several solutions to complete electromagnetic simulation.

2. The electromagnetic simulation system based on the three-dimensional multi-symplectic finite difference time-domain method according to claim 1 is characterized in that: The workflow of the construction module includes: starting from the basic form of Maxwell's equations, deriving the multi-symplectic form representation; and constructing a new difference format by introducing the multi-symplectic Preissman scheme to construct the multi-symplectic form of Maxwell's equations.

3. The electromagnetic simulation system based on three-dimensional multi-symplectic finite difference time-domain method according to claim 2 is characterized in that: The Hamiltonian equations for the multisymplectic scheme with m space dimensions are: MZ t +K1Z1+K2Z2+K3Z3=▽ Z S(z) Among them, I represents the unit matrix; M and K1, K2, K3 represent antisymmetric matrices, S(z) represents a smooth function, which has been set as Hamiltonian or energy generalization; C1, C2, C3 are the coefficients in K1, K2, K3; Z t Indicates discreteness in the time direction; Z1, Z2, and Z3 indicate discreteness in other directions; represents the Hamiltonian operator.

4. The electromagnetic simulation system based on three-dimensional multi-symplectic finite difference time-domain method according to claim 3 is characterized in that: The building block uses an implicit midpoint symplectic scheme to discretize the Hamiltonian equations of the multi-symplectic scheme, and obtains the Preissman format in the multi-symplectic algorithm: Among them, n represents the number of time steps; Δt represents the time step; Δx, Δy, Δz represent the spatial steps of the x, y, and z axes; i, j, and k represent coordinates.

5. The electromagnetic simulation system based on three-dimensional multi-symplectic finite difference time-domain method according to claim 4, characterized in that: The multi-symplectic Preissman scheme is used for discretization in space and time. The multi-symplectic Preissman scheme is combined with the Maxwell equations to obtain the multi-symplectic Preissman scheme: Among them, the central discrete operator is as follows: Where E represents the electric field; H represents the magnetic field; n represents the number of time steps; δ t Indicates time discreteness; δ x Indicates the discreteness in the x-axis direction; δ y Indicates discreteness in the y-axis direction; δ z Indicates discreteness in the z-axis direction; x, y, z indicate coordinate directions; μ indicates magnetic permeability; ε indicates dielectric constant.

6. The electromagnetic simulation system based on three-dimensional multi-symplectic finite difference time-domain method according to claim 5, characterized in that: The decomposition module decomposes the multi-symplectic Maxwell equations into several one-dimensional sub-problems by introducing a local one-dimensional method:

Images