Electromagnetic field simulation method and system based on first-order Mur absorption boundary condition

Abstract

The invention discloses an electromagnetic field simulation method and system based on a first-order Mur absorption boundary condition, and relates to the field of electromagnetic field simulation, and the method comprises the following steps: constructing the first-order Mur boundary condition for FCC-FDTD; initializing an electric field parameter and a magnetic field parameter, and defining a medium parameter, a geometric structure and an excitation source of the to-be-simulated region; and setting the first-order Mur boundary condition, and updating the electric field and the magnetic field based on the first-order Mur boundary condition to complete simulation. The method constructs a novel first-order Mur boundary condition by detailedly deriving an iterative formula of an electric field on an absorption boundary, simulates an electromagnetic field through the novel boundary condition, and compared with a traditional FDTD algorithm, the method is simple and easy to implement and convenient to implement, and wide application of the FCC-FDTD method is facilitated.

Description

Technical Field

[0001] The present invention belongs to the technical field of electromagnetic field simulation, and in particular relates to an electromagnetic field simulation method and system based on first-order Mur absorbing boundary conditions. Background Art

[0002] Finite-difference time-domain (FDTD) methods have made important contributions in many electromagnetic applications, such as electromagnetic scattering, microwave circuit design, and electromagnetic compatibility. However, their accuracy is often affected by numerical dispersion errors. To address these dispersion errors, an alternative method has been developed, called the FDTD method on face-centered cubic grids (FCC-FDTD). This method provides an alternative to the traditional Cartesian (Yee) grid. FCC-FDTD is proposed to improve the numerical dispersion characteristics in three-dimensional scalar wave equation simulations. Comparison with Yee-FDTD shows that FCC-FDTD has a more isotropic dispersion relation and more relaxed stability. The standard FDTD method for Maxwell's equations is reformulated on an FCC grid. The dispersion and stability of this method are also demonstrated and compared with Yee-FDTD.

[0003] Since computers cannot store an unlimited amount of data, when using the FDTD method to solve time-domain electromagnetic field equations in unbounded space, it is necessary to simulate the unbounded surrounding environment by truncating the computational grid and using absorbing boundary conditions (ABC) on artificial boundaries. In 1981, Mur introduced the first-order and second-order ABCs in FDTD and widely used them in open domain electromagnetic problems. Compared with the first-order and second-order ABCs, the perfectly matched layer (PML) is more effective in absorbing electromagnetic waves. Various PMLs have been proposed, such as Berenger's perfectly matched layer (BPML), uniaxial perfectly matched layer (UPML), and convolution perfectly matched layer (CPML). Summary of the invention

[0004] The present invention aims to solve the deficiencies of the prior art and provides the following solutions:

[0005] An electromagnetic field simulation method based on a first-order Mur absorbing boundary condition comprises the following steps:

[0006] Construct the first-order Mur boundary conditions for FCC-FDTD;

[0007] Initialize the electric field parameters and magnetic field parameters, define the medium parameters and geometric structure of the area to be simulated, and the excitation source;

[0008] The first-order Mur boundary condition is set, and the electric field and the magnetic field are updated based on the first-order Mur boundary condition to complete the simulation.

[0009] Preferably, the method for constructing the first-order Mur boundary condition includes:

[0010] According to the electric field differential form and combined with Maxwell-Ampere's law, the electric field iterative differential equation of the electric field components in the FCC grid is obtained;

[0011] According to the magnetic field differential form and combined with Maxwell-Faraday law, the magnetic field iterative differential equation of the magnetic field in the FCC grid is obtained;

[0012] The first-order Mur boundary condition is obtained based on the electric field iterative differential equation and the magnetic field iterative differential equation.

[0013] Preferably, the electric field differential form and the magnetic field differential form are respectively:

[0014]

[0015] Among them, E represents the electric field intensity, H represents the magnetic field intensity, ε represents the dielectric constant of the medium, μ represents the magnetic permeability of the medium, t represents time, A and B represent spatial operator matrices, and x, y, and z represent spatial coordinates.

[0016] Preferably, the electric field iterative differential equation is:

[0017]

[0018] in,

[0019]

[0020] The magnetic field iterative differential equation is:

[0021]

[0022] in,

[0023]

[0024] in, represents the grid coordinate, n represents the number of time steps, Δy represents the size of the FCC grid in the y direction, and Δz represents the size of the FCC grid in the z direction.

[0025] Preferably, the first-order Mur boundary condition is:

[0026]

[0027] Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

[0028] The present invention also provides an electromagnetic field simulation system based on the first-order Mur absorption boundary condition, wherein the simulation system applies any of the above-mentioned simulation methods, including: a boundary condition construction module, a parameter setting module and a simulation module;

[0029] The boundary condition building module is used to build the first-order Mur boundary condition for FCC-FDTD;

[0030] The parameter setting module is used to initialize electric field parameters and magnetic field parameters, and to set medium parameters and geometric structure and excitation source of the area to be simulated;

[0031] The simulation module is used to set the first-order Mur boundary condition, update the electric field and the magnetic field based on the first-order Mur boundary condition, and complete the simulation.

[0032] Preferably, the boundary condition building module includes: an electric field equation building unit, a magnetic field equation building unit and a boundary condition building unit;

[0033] The electric field equation construction unit is used to obtain the electric field iterative differential equation of the electric field components in the FCC grid according to the electric field differential form in combination with Maxwell-Ampere's law;

[0034] The magnetic field equation construction unit is used to obtain a magnetic field iterative differential equation of the magnetic field in the FCC grid according to the magnetic field differential form in combination with the Maxwell-Faraday law;

[0035] The boundary condition construction unit obtains the first-order Mur boundary condition based on the electric field iterative differential equation and the magnetic field iterative differential equation.

[0036] Preferably, the electric field differential form and the magnetic field differential form are respectively:

[0037]

[0038]

[0039] Among them, E represents the electric field intensity, H represents the magnetic field intensity, ε represents the dielectric constant of the medium, μ represents the magnetic permeability of the medium, t represents time, A and B represent spatial operator matrices, and x, y, and z represent spatial coordinates.

[0040] Preferably, the electric field iterative differential equation is:

[0041]

[0042] in,

[0043]

[0044]

[0045] The magnetic field iterative differential equation is:

[0046]

[0047] in,

[0048]

[0049]

[0050] in, represents the grid coordinate, n represents the number of time steps, Δy represents the size of the FCC grid in the y direction, and Δz represents the size of the FCC grid in the z direction.

[0051] Preferably, the first-order Mur boundary condition is:

[0052]

[0053] Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

[0054] Compared with the prior art, the present invention has the following beneficial effects:

[0055] The present invention constructs a new first-order Mur boundary condition by deriving the iterative formula of the electric field on the absorbing boundary in detail, and simulates the electromagnetic field through the new boundary condition. Compared with the traditional FDTD algorithm, the present invention is simple and easy to implement, which is conducive to the wide application of the FCC-FDTD method. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] 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.

[0057] Figure 1 A schematic diagram of a method flow of an embodiment of the present invention;

[0058] Figure 2 A schematic diagram showing only the electric field structure in space and time near the left end of a grid according to an embodiment of the present invention;

[0059] Figure 3 In the embodiment of the present invention, E z1 、E z4 , and Schematic diagram of distribution in the yz plane;

[0060] Figure 4 The electric field E in the embodiment of the present invention is z1 Distribution diagram at different times, where Figure 4 (a) shows the electric field intensity distribution at 0.2048ns, Figure 4 (b) shows the electric field intensity distribution at 0.2501ns, Figure 4 (c) shows the electric field intensity distribution at 0.2918ns, Figure 4 (d) represents the electric field intensity distribution at 0.3335ns, Figure 4 (e) shows the electric field intensity distribution at 0.3752ns, Figure 4 (f) represents the electric field intensity distribution at 0.4169ns;

[0061] Figure 5 is a schematic diagram of a free space containing a dielectric block in an embodiment of the present invention;

[0062] Figure 6 The embodiment of the present invention uses FCC-FDTD and Yee-FDTD to calculate the E of the observation point z and E z1 Comparison chart;

[0063] Figure 7 A schematic diagram of the FSS calculation domain according to an embodiment of the present invention;

[0064] Figure 8 is the electric field E of the observation point obtained by Yee-FDTD and FCC-FDTD in the embodiment of the present invention z and E z1 +E z3 Schematic diagram of the results;

[0065] Fig. 9 is the S calculated by Yee-FDTD and FCC-FDTD in the embodiment of the present invention. 11 | and |S 21 |Schematic diagram of. DETAILED DESCRIPTION

[0066] 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.

[0067] 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.

[0068] Embodiment 1

[0069] In this embodiment, if Figure 1 As shown, an electromagnetic field simulation method based on a first-order Mur absorbing boundary condition comprises the following steps:

[0070] S1. Construct the first-order Mur boundary condition for FCC-FDTD.

[0071] The method for constructing the first-order Mur boundary condition includes: obtaining the electric field iterative differential equation of the electric field component in the FCC grid according to the differential form of the electric field combined with the Maxwell-Ampere law; obtaining the magnetic field iterative differential equation of the magnetic field in the FCC grid according to the differential form of the magnetic field combined with the Maxwell-Faraday law; and obtaining the first-order Mur boundary condition based on the electric field iterative differential equation and the magnetic field iterative differential equation.

[0072] The differential forms of the electric field and the magnetic field are:

[0073]

[0074] Among them, E represents the electric field intensity, H represents the magnetic field intensity, ε represents the dielectric constant of the medium, μ represents the magnetic permeability of the medium, t represents time, A and B represent the spatial operator matrix, and x, y, and z represent the spatial coordinates. The iterative differential equation of the electric field is:

[0075]

[0076] in,

[0077]

[0078]

[0079] The iterative differential equation of the magnetic field is:

[0080]

[0081] in,

[0082]

[0083]

[0084] in, represents the grid coordinate, n represents the number of time steps, Δy represents the size of the FCC grid in the y direction, and Δz represents the size of the FCC grid in the z direction.

[0085] The first-order Mur boundary condition is:

[0086]

[0087] Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

[0088] In this embodiment, after introducing the iterative form of the electric field and magnetic field in the FCC grid, the absorbing boundary conditions of the FCC grid can be derived based on the iterative differential equations of the electric field and the iterative differential equations of the magnetic field, combined with the advection equation. If the result of any of these operators acting on the field alone is zero, then the advection equation automatically holds, including:

[0089]

[0090]

[0091] Let the nodes on the yoz plane be and is a node on the boundary, and the update equation needs to be solved for it. Figure 2 As shown, if the above formula is expanded at the spatial point A and the time point (q+1 / 2)Δt, a stable absorbing boundary condition can be obtained.

[0092] The method of solving the boundary condition includes: setting a virtual electric field and Discretized, the time derivative of the electric field can be approximated using the following finite difference expression:

[0093]

[0094] By performing time averaging of the spatial finite differences at appropriate locations, we can obtain:

[0095]

[0096] in, E z2 、E z3 All represent the electric field value, represents the electric field value at time q+1, represents the electric field value at time q, m + represents the next time step, m - represents the previous time step; finally we get:

[0097]

[0098] Here, Δx represents the size of the FCC grid in the x direction and Δt represents the time step. Similarly, by setting a virtual electric field at point B and discretizing it, the boundary value can be derived: Iterative formula. In the yoz plane, and E z1 、E z4The relationship is as Figure 3 shown; from Figure 3 It can be seen that through linear interpolation we can get:

[0099]

[0100] For E z1 The calculation can be divided into three types: Type I, Type II and Type III.

[0101] For Type I:

[0102]

[0103] For Type II:

[0104]

[0105] For Type III:

[0106]

[0107] for Figure 3 The cases that do not belong to type I, II, or III can be obtained from the electric field calculation of the xoy plane. Similarly, the iterative formulas for the electric fields of the other five absorbing boundaries can be derived.

[0108] S2. Initialize the electric field parameters and magnetic field parameters, define the medium parameters and geometric structure of the area to be simulated, and the excitation source.

[0109] In this example, the size of the simulation area is 50 mm × 50 mm × 50 mm, and the computational domain is surrounded by the proposed absorbing boundary conditions. In order to evaluate the correctness of the absorbing boundary conditions of FCC-FDTD, a Gaussian pulse is selected as the excitation source, located at the center of the computational domain. The spatial step size is set to Δx = Δy = Δz = 0.5 mm, and the time step size is Δt = 0.83391 ps. The mathematical expression of the Gaussian pulse is:

[0110]

[0111] Among them, E Z (t) represents Gaussian pulse, t 0 represents the time delay of the pulse, and τ represents the width parameter of the pulse.

[0112] S3. Set the first-order Mur boundary condition, update the electric field and magnetic field based on the first-order Mur boundary condition, and complete the simulation.

[0113] Figure 4 (a)-(f) show the distribution of electric field intensity at different time points, illustrating the propagation and absorption process of electromagnetic waves. Figure 4(a) shows the electric field intensity distribution at 0.2048ns, Figure 4 (b) shows the electric field intensity distribution at 0.2501ns, Figure 4 (c) shows the electric field intensity distribution at 0.2918ns, Figure 4 (d) represents the electric field intensity distribution at 0.3335ns, Figure 4 (e) shows the electric field intensity distribution at 0.3752ns, Figure 4 (f) represents the electric field intensity distribution at 0.4169ns. Figure 4 In (a), the electromagnetic wave begins to radiate outward from the source, initiating the propagation process. As time goes by, the wavefront continues to expand and spreads across the entire grid. Figure 4 (f), the electromagnetic wave has propagated to the outer boundary of the simulation domain, where it is effectively absorbed by the proposed absorbing boundary condition. This absorption prevents the wave from reflecting back into the computational domain, demonstrating the effectiveness of the absorbing boundary condition in reducing reflected waves and ensuring simulation accuracy.

[0114] Embodiment 2

[0115] In this example, a free space containing three dielectric blocks with different relative permittivities is considered to verify the accuracy of the proposed method. The geometry of the simulation model is shown in Figure 5 As shown, Lx = Ly = Lz = 0.1m, and the geometric parameters of the dielectric block are: a 1 =0.08m, b 1 =0.05m,c 1 =0.03m, a 2 =0.02m, b 2 =0.05m,c 2 =0.035m, a 3 =0.025m, b 3 =0.05m,c 3 =0.025m. The relative dielectric constants ε of the three dielectric blocks r are 5, 10 and 15 respectively. The spatial step, time step and excitation source are the same as those in Example 1. The excitation point is set at the center of the simulation area, and the observation point is located at (0.053m, 0.053m, 0.05m). Figure 6 As shown in Figure 3, the electric fields calculated by FCC-FDTD and Yee-FDTD are compared, and the results show that the proposed absorbing boundary conditions have good accuracy.

[0116] Embodiment 3

[0117] In this example, a frequency selective structure (FSS) is studied to verify the accuracy of the proposed method. The FSS consists of a perfect electric conductor (PEC) plate with a slit array. The geometric dimensions of the unit cell are as follows: Figure 7 As shown. In the x-axis and z-axis directions, periodic boundary conditions are used to simulate FSS. In the y-axis direction, absorbing boundary conditions are used to simulate infinite regions. The spatial step size is set to Δx = Δy = Δz = 1.2 mm, and the time step size is Δt = 1.96 ps. A modulated Gaussian pulse source is used as the excitation source, and its mathematical expression is:

[0118]

[0119] Among them, f c Indicates the carrier frequency, f c =1.25GHz,τ=0.39ns,t 0 =4.5τ.

[0120] In the Yee-FDTD method, the source is introduced into the calculation area through the full field / scattered field (TF / SF) technique. In the FCC-FDTD method, a transparent source is set in the calculation area. The transparent source calculates the sum of the incident waveform and the scattered waveform. In order to obtain the incident waveform, the calculation is first performed without a structure and the incident waveform is recorded. The incident waveform can then be subtracted from the sum of the incident and scattered waveforms to obtain the scattered waveform. The distance between the FSS and the observation point A is 0.078m. At the observation point A, the scattered field Ez obtained by the Yee-FDTD method and the scattered field E obtained by the FCC-FDTD method are z1 +E z3 like Figure 8 shown. Fig. 9 The |S11| and |S21| values ​​calculated by Yee-FDTD and FCC-FDTD are shown. It can be seen that the results obtained by FCC-FDTD are consistent with those of Yee-FDTD, and the correctness of the proposed absorbing boundary conditions is also proved.

[0121] Embodiment 4

[0122] In this embodiment, an electromagnetic field simulation system based on a first-order Mur absorbing boundary condition includes: a boundary condition construction module, a parameter setting module and a simulation module.

[0123] The boundary condition building module is used to build the first-order Mur boundary conditions for FCC-FDTD.

[0124] The boundary condition construction module includes: an electric field equation construction unit, a magnetic field equation construction unit and a boundary condition construction unit. The electric field equation construction unit is used to obtain the electric field iterative differential equation of the electric field component in the FCC grid according to the differential form of the electric field combined with the Maxwell-Ampere law; the magnetic field equation construction unit is used to obtain the magnetic field iterative differential equation of the magnetic field in the FCC grid according to the differential form of the magnetic field combined with the Maxwell-Faraday law; the boundary condition construction unit obtains the first-order Mur boundary condition based on the electric field iterative differential equation and the magnetic field iterative differential equation.

[0125] The differential forms of the electric field and the magnetic field are:

[0126]

[0127] Among them, E represents the electric field intensity, H represents the magnetic field intensity, ε represents the dielectric constant of the medium, μ represents the magnetic permeability of the medium, t represents time, A and B represent the spatial operator matrix, and x, y, and z represent the spatial coordinates. The iterative differential equation of the electric field is:

[0128]

[0129] in,

[0130]

[0131]

[0132] The iterative differential equation for the magnetic field is:

[0133]

[0134] in,

[0135]

[0136]

[0137] in, represents the grid coordinate, n represents the number of time steps, Δy represents the size of the FCC grid in the y direction, and Δz represents the size of the FCC grid in the z direction.

[0138] The first-order Mur boundary condition is:

[0139]

[0140] Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

[0141] The parameter setting module is used to initialize the electric field parameters and magnetic field parameters, and to set the medium parameters and geometric structure of the area to be simulated and the excitation source.

[0142] The simulation module is used to set the first-order Mur boundary conditions, update the electric field and magnetic field based on the first-order Mur boundary conditions, and complete the simulation.

[0143] 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 field simulation method based on first-order Mur absorbing boundary conditions, characterized in that: The following steps are involved: Constructing first-order Mur boundary conditions for FCC-FDTD; Initialize the electric field parameters and magnetic field parameters, define the medium parameters and geometric structure of the area to be simulated, and the excitation source; The first-order Mur boundary condition is set, and the electric field and the magnetic field are updated based on the first-order Mur boundary condition to complete the simulation.

2. The electromagnetic field simulation method based on the first-order Mur absorbing boundary condition according to claim 1 is characterized in that: The method of constructing the first-order Mur boundary condition includes: According to the electric field differential form and combined with Maxwell-Ampere's law, the electric field iterative differential equation of the electric field components in the FCC grid is obtained; According to the magnetic field differential form and combined with Maxwell-Faraday law, the magnetic field iterative differential equation of the magnetic field in the FCC grid is obtained; The first-order Mur boundary condition is obtained based on the electric field iterative differential equation and the magnetic field iterative differential equation.

3. The electromagnetic field simulation method based on the first-order Mur absorbing boundary condition according to claim 2 is characterized in that: The electric field differential form and the magnetic field differential form are respectively: Among them, E represents the electric field intensity, H represents the magnetic field intensity, ε represents the dielectric constant of the medium, μ represents the magnetic permeability of the medium, t represents time, A and B represent spatial operator matrices, and x, y, and z represent spatial coordinates.

4. The electromagnetic field simulation method based on the first-order Mur absorbing boundary condition according to claim 3 is characterized in that: The electric field iterative differential equation is: in, The magnetic field iterative differential equation is: in, in, represents the grid coordinate, n represents the number of time steps, Δy represents the size of the FCC grid in the y direction, and Δz represents the size of the FCC grid in the z direction.

5. The electromagnetic field simulation method based on the first-order Mur absorbing boundary condition according to claim 4 is characterized in that: The first-order Mur boundary condition is: Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

6. An electromagnetic field simulation system based on first-order Mur absorption boundary conditions, the simulation system applying the simulation method according to any one of claims 1 to 5, characterized in that: include: Boundary condition building module, parameter setting module and simulation module; The boundary condition building module is used to build the first-order Mur boundary condition for FCC-FDTD; The parameter setting module is used to initialize electric field parameters and magnetic field parameters, and to set medium parameters and geometric structure and excitation source of the area to be simulated; The simulation module is used to set the first-order Mur boundary condition, update the electric field and the magnetic field based on the first-order Mur boundary condition, and complete the simulation.

7. The electromagnetic field simulation system based on the first-order Mur absorbing boundary condition according to claim 6, characterized in that: The boundary condition building module includes: an electric field equation building unit, a magnetic field equation building unit and a boundary condition building unit; The electric field equation construction unit is used to obtain the electric field iterative differential equation of the electric field components in the FCC grid according to the electric field differential form in combination with Maxwell-Ampere's law; The magnetic field equation construction unit is used to obtain a magnetic field iterative differential equation of the magnetic field in the FCC grid according to the magnetic field differential form in combination with the Maxwell-Faraday law; The boundary condition construction unit obtains the first-order Mur boundary condition based on the electric field iterative differential equation and the magnetic field iterative differential equation.

8. The electromagnetic field simulation system based on the first-order Mur absorbing boundary condition according to claim 7, characterized in that: The electric field differential form and the magnetic field differential form are respectively: Among them, E represents the electric field intensity, H represents the magnetic field intensity, ε represents the dielectric constant of the medium, μ represents the magnetic permeability of the medium, t represents time, A and B represent spatial operator matrices, and x, y, and z represent spatial coordinates.

9. The electromagnetic field simulation system based on the first-order Mur absorbing boundary condition according to claim 8, characterized in that: The electric field iterative differential equation is: in, The magnetic field iterative differential equation is: in, in, represents the grid coordinate, n represents the number of time steps, Δy represents the size of the FCC grid in the y direction, and Δz represents the size of the FCC grid in the z direction.

10. The electromagnetic field simulation system based on the first-order Mur absorbing boundary condition according to claim 9, characterized in that: The first-order Mur boundary condition is: Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

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