An electromagnetic field simulation method and system based on a first-order mur absorbing boundary condition

Abstract

The application discloses an electromagnetic field simulation method and system based on a first-order Mur absorbing boundary condition, relates to the electromagnetic field simulation neighborhood, and comprises the following steps: constructing a first-order Mur boundary condition for FCC-FDTD; initializing electric field parameters and magnetic field parameters, defining medium parameters and geometric structures of a region to be simulated and an excitation source; setting the first-order Mur boundary condition, updating the electric field and the magnetic field based on the first-order Mur boundary condition, and completing simulation. The application constructs a novel first-order Mur boundary condition by deducing an iteration formula of the electric field on the absorbing boundary in detail, and simulates the electromagnetic field through the novel boundary condition. Compared with a traditional FDTD algorithm, the application is simple and easy to implement, and is helpful to the wide application of the FCC-FDTD method.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of electromagnetic field simulation, and particularly relates to an electromagnetic field simulation method and system based on a first-order Mur absorbing boundary condition. BACKGROUND

[0002] Finite-difference time-domain (FDTD) has made important contributions in many electromagnetic applications, such as electromagnetic scattering, microwave circuit design, and electromagnetic compatibility. However, its accuracy is often affected by numerical dispersion errors. To solve these dispersion errors, an alternative method, called FDTD on face-centered cubic grid (FCC-FDTD), was developed. This method provides a different choice from the traditional Cartesian (Yee) grid. FCC-FDTD was 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 a more relaxed stability. The standard FDTD method for Maxwell's equations was reformulated on FCC grids. The dispersion and stability of this method were also demonstrated and compared with Yee-FDTD.

[0003] Since computers cannot store an infinite amount of data, when solving time-domain electromagnetic field equations using the FDTD method in unbounded space, it is necessary to truncate the computational grid and use absorbing boundary conditions (ABCs) at artificial boundaries to simulate the unbounded surrounding environment. In 1981, Mur introduced first- and second-order ABCs in FDTD and was widely used in open-domain electromagnetic problems. Compared with first- 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 convolutional perfectly matched layer (CPML). SUMMARY

[0004] The application aims to solve the problems of the prior art and provides the following scheme:

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

[0006] A first-order Mur boundary condition for FCC-FDTD is constructed;

[0007] Initialize electric field parameters and magnetic field parameters, define medium parameters and geometric structures of a region to be simulated, and define an excitation source;

[0008] 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.

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

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

[0011] According to the magnetic field differential form, 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] Wherein, E represents electric field intensity, H represents magnetic field intensity, ε represents dielectric constant of medium, μ represents magnetic permeability of medium, t represents time, A and B represent spatial operator matrix, x, y, z represent spatial coordinates.

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

[0017]

[0018] Wherein,

[0019]

[0020] The magnetic field iterative differential equation is:

[0021]

[0022] Wherein,

[0023]

[0024] Wherein, represents grid coordinates, n represents time step number, Δy represents size of the FCC grid in y direction, Δz represents size of the FCC grid in z direction.

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

[0026]

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

[0028] The application further provides an electromagnetic field simulation system based on a first-order Mur absorbing boundary condition, which applies the simulation method in any of the above and comprises a boundary condition construction module, a parameter setting module and a simulation module.

[0029] The boundary condition construction module is used for constructing the first-order Mur boundary condition for the FCC-FDTD.

[0030] The parameter setting module is used for initializing electric field parameters and magnetic field parameters, setting medium parameters and geometric structures of a region to be simulated and an excitation source.

[0031] The simulation module is used for setting the first-order Mur boundary condition, updating electric fields and magnetic fields based on the first-order Mur boundary condition and completing simulation.

[0032] Preferably, the boundary condition construction module comprises an electric field equation construction unit, a magnetic field equation construction unit and a boundary condition construction unit.

[0033] The electric field equation construction unit is used for obtaining an electric field iterative differential equation of electric field components in the FCC grid according to an electric field differential form and in combination with the Maxwell-Ampere law.

[0034] The magnetic field equation construction unit is used for obtaining a magnetic field iterative differential equation of magnetic fields in the FCC grid according to a magnetic field differential form and 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 as follows:

[0037]

[0038]

[0039] wherein E represents electric field intensity, H represents magnetic field intensity, ε represents dielectric constant of a medium, μ represents 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 as follows:

[0041]

[0042] wherein,

[0043]

[0044]

[0045] The iterative differential equation for the magnetic field is:

[0046]

[0047] in,

[0048]

[0049]

[0050] in, Δy represents the grid coordinates, n represents the time step, 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 beneficial effects of the present invention are as follows:

[0055] This invention constructs a novel first-order Mur boundary condition by deriving the iterative formula for the electric field on the absorbing boundary in detail. The electromagnetic field is simulated using this novel boundary condition. Compared with the traditional FDTD algorithm, this invention is simple, easy to implement, and conducive to the widespread application of the FCC-FDTD method. Attached Figure Description

[0056] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are 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.

[0057] Figure 1 This is a schematic diagram of the method flow according to an embodiment of the present invention;

[0058] Figure 2 This is a schematic diagram showing only the electric field structure in the spacetime near the left end of the grid in an embodiment of the present invention;

[0059] Figure 3 E in the embodiments of the present invention z1 E z4 , and A schematic diagram of the distribution in the yz plane;

[0060] Figure 4 Electric field E for the embodiment of the present application z1 Distribution diagram at different time, wherein, Figure 4 (a) represents the electric field intensity distribution at 0.2048 ns, Figure 4 (b) represents the electric field intensity distribution at 0.2501 ns, Figure 4 (c) represents the electric field intensity distribution at 0.2918 ns, Figure 4 (d) represents the electric field intensity distribution at 0.3335 ns, Figure 4 (e) represents the electric field intensity distribution at 0.3752 ns, Figure 4 (f) represents the electric field intensity distribution at 0.4169 ns;

[0061] Figure 5 Free space diagram containing dielectric block for the embodiment of the present application;

[0062] Figure 6 E for the observation point calculated by FCC-FDTD and Yee-FDTD for the embodiment of the present application z and E z1 Comparison diagram;

[0063] Figure 7 FSS calculation domain diagram for the embodiment of the present application;

[0064] Figure 8 Electric field E for the observation point obtained by Yee-FDTD and FCC-FDTD for the embodiment of the present application z and E z1 +E z3 Result diagram of E

[0065] Figure 9 |S 11 | and |S 21 | calculated by Yee-FDTD and FCC-FDTD for the embodiment of the present application. DETAILED DESCRIPTION

[0066] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.

[0067] In order to make the above objectives, characteristics and advantages of the present application more obvious and easy to understand, the present application will be further described in detail below with reference to the drawings and specific embodiments.

[0068] Embodiment one

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

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

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

[0073]

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

[0075]

[0076] Wherein,

[0077]

[0078]

[0079] The magnetic field iterative differential equation is:

[0080]

[0081] Wherein,

[0082]

[0083]

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

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

[0086] ​

[0087] where Ψ = x, y, z, ξ = x, y, z, and Ψ ≠ ξ.

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

[0089]

[0090]

[0091] Let the nodes in the yoz plane be and The update equation needs to be solved for the nodes on the boundary. As shown in Figure 2 , 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 for solving the boundary condition includes: by setting a virtual electric field and discretizing , the time derivative of the electric field can be approximated using the following finite difference expression:

[0093]

[0094] By time-averaging the spatial finite difference at the appropriate position, we obtain:

[0095]

[0096] where E z2 , E z3 each represent an 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; ultimately we obtain:

[0097]

[0098] where 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, the iterative formula for the boundary value can be derived. In the yoz plane, and E z1 , E z4The relationship is shown as Figure 3 It can be seen that by linear interpolation, we can get Figure 3

[0099]

[0100] For the calculation of E z1 , it 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 , which does not belong to type I, type II and type III, it can be obtained from the electric field calculation in the xoy plane. Similarly, the iterative formula of the other five absorbing boundary electric fields can be derived.

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

[0109] In this embodiment, the size of the simulation region is 50mmx50mmx50mm, and the calculation domain is surrounded by the proposed absorbing boundary condition. In order to evaluate the correctness of the absorbing boundary condition of FCC-FDTD, a Gaussian pulse is selected as the excitation source, which is located at the center of the calculation domain. The spatial step is set to Δx=Δy=Δz=0.5mm, and the time step is Δt=0.83391ps. The mathematical expression of the Gaussian pulse is:

[0110]

[0111] where E Z (t) represents the Gaussian pulse, t0 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 electric field intensity distribution at different time points, which illustrates the propagation and absorption process of electromagnetic waves. Among them, Figure 4 ​(a) represents the electric field intensity distribution at 0.2048 ns, Figure 4 (b) represents the electric field intensity distribution at 0.2501 ns, Figure 4 (c) represents the electric field intensity distribution at 0.2918 ns, Figure 4 (d) represents the electric field intensity distribution at 0.3335 ns, Figure 4 (e) represents the electric field intensity distribution at 0.3752 ns, Figure 4 (f) represents the electric field intensity distribution at 0.4169 ns, and Figure 4 In (a), the electromagnetic wave starts to radiate outward from the source point, initiating the propagation process. As time progresses, the wavefront continues to expand and spread throughout the grid. By the time of Figure 5 (f), the electromagnetic wave has propagated to the outer boundary of the simulation domain, where the proposed absorbing boundary condition effectively absorbs. 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 the accuracy of the simulation.

[0114] Example Two

[0115] In this example, the 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 6 , where Lx= Ly= Lz= 0.1 m, and the geometric parameters of the dielectric blocks are: al= 0.08 m, bl= 0.05 m, cl= 0.03 m, a2= 0.02 m, b2= 0.05 m, c2= 0.035 m, a3= 0.025 m, b3= 0.05 m, c3= 0.025 m, respectively. The relative permittivities of the three dielectric blocks ε r are 5, 10, and 15, respectively. The spatial step, time step, and excitation source are the same as in Example One. The excitation point is set at the center of the simulation region, and the observation point is located at (0.053 m, 0.053 m, 0.05 m). As Figure 7 shown, the electric fields calculated by FCC-FDTD and Yee-FDTD are compared, and the results show that the proposed absorbing boundary condition has good accuracy.

[0116] Example Three

[0117] In this example, a frequency selective structure (FSS) is studied to verify the accuracy of the proposed method. The FSS is composed of a perfect electric conductor (PEC) plate with an array of slits, and the geometric dimensions of the unit cell are Figure 8The FSS is simulated using periodic boundary conditions in the x and z directions. In the y direction, an absorbing boundary condition is used to simulate an infinite region. The spatial step is set as Dx=Dy=Dz=1.2mm, and the time step is At=1.96ps. A modulated Gaussian pulse source is used as the excitation source, and its mathematical expression is:

[0118]

[0119] where f c represents the carrier frequency, f c =1.25GHz, τ=0.39ns, and t0=4.5τ.

[0120] In the Yee-FDTD method, the source is introduced into the calculation region through the total field / scattered field (TF / SF) technique. In the FCC-FDTD method, a transparent source is set in the calculation region. 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 the structure, and the incident waveform is recorded. Then the incident waveform can be subtracted from the sum of the incident and scattered waveforms, thereby obtaining the scattered waveform. The distance between the FSS and the observation point A is 0.078m. At the observation point A, the scattered field Ezobtained by the Yee-FDTD method and the scattered field E z1 + E z3 As Figure 9 shown. ​ The |S11| and |S21| values calculated by the Yee-FDTD and FCC-FDTD are shown. It can be seen that the results obtained by the FCC-FDTD are consistent with those of the Yee-FDTD, and at the same time, the correctness of the proposed absorbing boundary condition is also proved.

[0121] Embodiment Four

[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 construction module is used to construct the first-order Mur boundary condition for the FCC-FDTD.

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

[0125] The electric field differential form and the magnetic field differential form are respectively:

[0126]

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

[0128]

[0129] wherein,

[0130]

[0131]

[0132] The magnetic field iterative differential equation is:

[0133]

[0134] wherein,

[0135]

[0136]

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

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

[0139]

[0140] wherein Ψ=x, y, z, ξ=x, y, z, and Ψ≠ξ.

[0141] The parameter setting module is configured to initialize electric field parameters and magnetic field parameters, set medium parameters and geometric structures of a region to be simulated and an excitation source.

[0142] The simulation module is configured to set a 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.

[0143] The above-described embodiments are merely intended to describe the preferred modes of the present application, and are not intended to limit the scope of the present application. Various modifications and improvements to the technical solutions of the present application made by those skilled in the art without departing from the design spirit of the present application shall fall within the protection scope of the present application as defined by the claims.

Claims

1. An electromagnetic field simulation method based on first-order Mur absorption boundary conditions, characterized in that, Includes the following steps: Construct first-order Mur boundary conditions for FCC-FDTD; Initialize the electric and magnetic field parameters, and define the medium parameters, geometry, and excitation source of the region to be simulated. The first-order Mur boundary conditions are set, and the electric and magnetic fields are updated based on the first-order Mur boundary conditions to complete the simulation. The method for constructing the first-order Mur boundary conditions includes: Based on the differential form of the electric field and combined with Maxwell-Ampere's law, the iterative differential equation of the electric field components in the FCC grid is obtained; Based on the differential form of the magnetic field and combined with Maxwell-Faraday's law, the iterative differential equation of the magnetic field in the FCC grid is obtained; The first-order Mur boundary conditions are obtained based on the iterative differential equations of the electric field and the magnetic field. The differential forms of the electric field and the magnetic field are respectively: Where E represents the electric field strength, H represents the magnetic field strength, ε represents the dielectric constant of the medium, μ represents the permeability of the medium, t represents time, A and B represent spatial operator matrices, and x, y, z represent spatial coordinates; The iterative differential equation for the electric field is: in, The iterative differential equation for the magnetic field is: in, in, Δy represents the grid coordinates, n represents the time step, 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. The first-order Mur boundary condition is: Among them, Ψ=x, y, z, ξ=x, y, z, Ψ≠ξ.

2. An electromagnetic field simulation system based on a first-order Mur absorbing boundary condition, wherein the simulation system applies the simulation method described in claim 1, characterized in that, include: Boundary condition construction module, parameter setting module, and simulation module; The boundary condition construction module is used to construct first-order Mur boundary conditions for FCC-FDTD; The parameter setting module is used to initialize electric field parameters and magnetic field parameters, and to set the medium parameters, geometry, and excitation source of the region to be simulated. The simulation module is used to set the first-order Mur boundary conditions, update the electric and magnetic fields based on the first-order Mur boundary conditions, and complete the simulation.

3. The electromagnetic field simulation system based on first-order Mur absorption boundary conditions according to claim 2, characterized in that, 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 building unit is used to obtain the iterative differential equation of the electric field components in the FCC grid based on the differential form of the electric field and the Maxwell-Ampère law. The magnetic field equation building unit is used to obtain the magnetic field iterative differential equation of the magnetic field in the FCC grid based on the differential form of the magnetic field and combined with Maxwell-Faraday's 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.

4. The electromagnetic field simulation system based on first-order Mur absorption boundary conditions according to claim 3, characterized in that, The differential forms of the electric field and the magnetic field are respectively: Where E represents the electric field strength, H represents the magnetic field strength, ε represents the dielectric constant of the medium, μ represents the permeability of the medium, t represents time, A and B represent spatial operator matrices, and x, y, and z represent spatial coordinates.

5. The electromagnetic field simulation system based on first-order Mur absorption boundary conditions according to claim 4, characterized in that, The iterative differential equation for the electric field is: in, The iterative differential equation for the magnetic field is: in, in, Δy represents the grid coordinates, n represents the time step, 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.

6. The electromagnetic field simulation system based on first-order Mur absorption boundary conditions according to claim 5, 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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