A Modeling Method for Wave Propagation in Conductor Dielectrics

Abstract

The present invention discloses a modeling method for wave propagation in a conductor medium, comprising the following steps: step 1: modeling the conductor medium as a dielectric body with frequency dispersion and spatial dispersion in FDTD; step 2: calculating the electromagnetic field using the standard Yee FDTD update equation and the electric flux density and electric field intensity update equation; step 3: calculating the stability condition of the spatial dispersion FDTD method by using the von Neumann method in combination with the Routh-Hurwitz criterion; step 4: introducing a perfectly matched layer and proposing a modified Berenger PML formula. For the first time, the application of spatial dispersion effect in FDTD modeling is considered. By using the modified PML, the conductor medium can be completely matched with the absorption boundary, thereby avoiding the corner scattering of a conductor medium block of limited size in the simulation, greatly improving the convergence performance, and can be used to develop practical applications of conductor media in antenna and microwave engineering.

Description

Technical Field

[0001] The invention relates to the field of electromagnetic wave technology, and in particular to a time-domain finite-difference modeling method for wave propagation in a conductor medium. Background Art

[0002] Existing dispersive FDTD methods can be divided into three types: recursive convolution (RC) method, auxiliary differential equation (ADE) method and Z transform method. The RC method relates the electric flux density to the electric field intensity through the convolution integral, which can be discretized into a cumulative sum. The dispersive FDTD method using the RC scheme is used to simulate different types of dispersive materials. The ADE method introduces additional differential equations to relate the electric field to the current. Through the inverse Fourier transform and discretization scheme, the frequency-dependent material properties are considered in the FDTD simulation. Another dispersive FDTD method is based on the Z transform: the time domain convolution integral is introduced into a multiplication through the Z transform, establishing a recursive relationship between the electric flux density and the electric field.

[0003] After searching, the Chinese patent with the announcement number CN114491987B discloses a modeling method for electromagnetic fields when electromagnetic waves propagate in marine multilayer media. It points out that in marine layered conductive media, the construction of electromagnetic wave propagation models is difficult, and the calculation efficiency and accuracy are low.

[0004] The Chinese patent with announcement number CN111143991B discloses a transverse magnetic wave transmission model of a dielectric wrapped conductor and a construction method thereof, which points out the problem that the existing nonlinear gravity surface wave transmission mathematical model and the cone-shaped electromagnetic wave space transmission model are not applicable to dielectric wrapped conductors.

[0005] However, the simulation of conductor media can be done in two ways: simulating physical structures or through the effective medium method. The second method assumes that the size of the inclusions is very small compared to the operating wavelength. If the conductor medium FDTD modeling is achieved by simulating the physical structure, for accuracy, special processing of the Yee FDTD algorithm (such as conformal FDTD) is usually required, which requires a lot of computing resources. In actual numerical modeling, if complex sources are considered, the simulation effect is not good. Therefore, a modeling method for wave propagation in conductor media is urgently needed. Summary of the invention

[0006] The purpose of the present invention is to solve the defects in the prior art and to propose a modeling method for wave propagation in a conductor medium.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] A method for modeling wave propagation in a conductor medium comprises the following steps:

[0009] Step 1: Model the conductor medium as a dielectric with frequency dispersion and spatial dispersion in FDTD;

[0010] Step 2: Calculate the electromagnetic field using the standard Yee FDTD update equations and the electric flux density and electric field intensity update equations;

[0011] Step 3: Calculate the stability condition of the spatial dispersion FDTD method by using the von Neumann method combined with the Routh-Hurwitz criterion;

[0012] Step 4: Introduce the perfectly matched layer and propose the modified Berenger PML formula.

[0013] Furthermore, in step 1, the geometric shape of the conductor medium is specifically a rectangular grid composed of parallel ideal conductive thin wires.

[0014] Furthermore, in step 1, if the grid period of the conductive medium is much smaller than the wavelength, the conductive medium is regarded as a uniaxial dielectric with a frequency- and space-dependent equivalent dielectric constant:

[0015]

[0016] in, is the equivalent dielectric constant vector, xx,yy,zz are the position vectors, ε(k,q x ) is the equivalent dielectric constant value at the vector position, ε0 is the vacuum dielectric constant value, k0 is the wave number corresponding to the plasma frequency, k is the free space wave number, q x is the component of the wave vector q in the direction of the conductor;

[0017] The plasma frequency depends on the grid periods a and b and the wire radius r:

[0018]

[0019] in, ξ is the independent variable of function F, and n is the period. When a=b, F(1)=0.5275;

[0020] The conductor medium is modeled as a dielectric with frequency dispersion and spatial dispersion in FDTD using equation (1).

[0021] Further, in step 2, the steps for calculating the electromagnetic field are as follows:

[0022] In order to consider the dispersion of the material, the relationship between the electric flux density D and the electric field intensity e in the frequency domain is introduced as follows:

[0023]

[0024] Where ω is the frequency and ε0 is the dielectric constant of vacuum;

[0025] The constitutive relation in the space-time domain is obtained from the above formula:

[0026]

[0027] Where C is the speed of light and t is time;

[0028] By rewriting equation (4) and discretizing the rewritten equation, we can get the update formula:

[0029]

[0030] The coefficients are:

[0031]

[0032] Furthermore, in step 4, in the perfectly matched layer in the conductor medium modeling, D is introduced into the update equation, and the two quantities D and H are processed separately:

[0033]

[0034] The updated equation for the H component is consistent with the original Berenger PML formula;

[0035] Wherein: the update formula (8) is extended to PML to adapt the conductor medium and MPML.

[0036] Compared with the prior art, the present invention has the following beneficial effects:

[0037] The present invention considers the application of spatial dispersion effect in FDTD modeling for the first time. Stability analysis shows that if the standard central difference approximation and central average operator are used to discretize the differential equations proposed in the present invention, the traditional Courant stability restrictions will be retained. By using the modified PML, the conductor medium can be fully matched with the absorbing boundary, thereby avoiding the corner scattering of the finite-sized conductor medium block in the simulation and greatly improving the convergence performance. The proposed spatial dispersion FDTD method will show its unique value when using complex sources in simulation, and can be used to develop practical applications of conductor media in antenna and microwave engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0039] Figure 1is a schematic diagram of the geometric shape of the conductor medium in an embodiment of the present invention;

[0040] Figure 2 The figure is a schematic diagram of the implementation logic of the modeling method of wave propagation in a conductor medium in an embodiment of the present invention. DETAILED DESCRIPTION

[0041] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0042] The modeling method of wave propagation in conductor medium includes the following steps:

[0043] Step 1: Model the conductor medium as a dielectric with frequency dispersion and spatial dispersion in FDTD;

[0044] Step 2: Calculate the electromagnetic field using the standard Yee FDTD update equation and the electric flux density and electric field intensity update equation proposed in the present invention;

[0045] Step 3: Calculate the stability condition of the proposed spatial dispersion FDTD method by using the von Neumann method combined with the Routh-Hurwitz criterion. If the standard discretization scheme is used, the traditional Courant stability limit will be retained;

[0046] Step 4: Introduce a perfectly matched layer and propose a modified Berenger PML formula.

[0047] It should be noted that steps 1-2 are specifically a modeling algorithm for the conductor dielectric wave propagation model;

[0048] Step 3 is to propose stability conditions for the modeling algorithm;

[0049] Step 4 introduces a modified PML to make the modeling more complete.

[0050] As the preferred embodiment of this application:

[0051] Step 1: Figure 1 As shown, the conductor medium is an artificial material composed of a regular grid of ideal conductive wires, and the geometric shape of the conductor medium is specifically a rectangular grid composed of parallel ideal conductive thin wires.

[0052] If the grid period is much smaller than the wavelength, the conductor medium can be considered as a uniaxial dielectric with a frequency- and space-dependent effective permittivity:

[0053]

[0054] in is the equivalent dielectric constant vector, xx, yy, zz are the position vectors, ε(k,q x ) is the equivalent dielectric constant value at the vector position, ε0 is the vacuum dielectric constant value, k0 is the wave number corresponding to the plasma frequency, k is the free space wave number, q x is the component of the wave vector q in the wire direction. The plasma frequency depends on the grid periods a and b and the wire radius r:

[0055]

[0056] in, ξ is the independent variable of the function F, and n is the period. When a=b, F(1)=0.5275.

[0057] Using equation (1), the conductor medium can be modeled as a dielectric with frequency dispersion and spatial dispersion in FDTD.

[0058] Step 2: In order to consider the dispersion of the material, the relationship between the electric flux density D and the electric field intensity E in the frequency domain is introduced:

[0059]

[0060] where ω is the frequency and ε0 is the dielectric constant of vacuum.

[0061] The constitutive relation in the space-time domain is obtained from the above formula:

[0062]

[0063] where c is the speed of light and t is time. The above equation only contains the x component of the field. This is because the wire width is considered to be very thin, so the dielectric constant in the y and z directions is the same as that of free space. By introducing the central finite difference operator in time and space and the central average operator in time, (4) will be rewritten as:

[0064]

[0065] Where Δ x and Δ t is the grid edge length and the time step. The time-centered finite difference operator δ t , the spatially centered finite difference operator δ x and the central average operator μ t The definitions of these three operators are as follows:

[0066]

[0067] Where n is an integer, F represents the corresponding field component, (m x , m y , m z) are the corresponding discrete points in FDTD.

[0068] Discretize (5):

[0069]

[0070] Then we can solve the above formula to get E x The update formula is:

[0071]

[0072] The coefficients are:

[0073]

[0074] It is worth noting that from (8), we can see that the calculation of the update equation requires D of the previous two time steps x With E x The stored value of .

[0075] The magnetic field H and electric flux D are calculated by the standard Yee FDTD update equations, then E x It is calculated by the algorithm proposed by the present invention, that is, formula (8), and it is obvious that:

[0076] In step 3: In order to analyze the numerical stability of the proposed spatial dispersion FDTD method, the relationship derived from Faraday and Ampere's laws is considered:

[0077]

[0078] where μ0 is the magnetic permeability of vacuum, is a vector differential operator.

[0079] Discretizing the matrix form of (10) and the matrix form of the constitutive relation (5) by the central difference operator yields:

[0080]

[0081] in,

[0082] Substitute the following solution into the discretized equation:

[0083]

[0084] where F is the field component, is the complex amplitude, is the numerical wave vector of the discrete mode. Z is a complex variable that can represent the error growth in one time step, expressed as

[0085]

[0086] After a series of calculations, the stability polynomial is as follows:

[0087]

[0088] in α=x,y,z,Δ α is the grid side length, is the numerical wave vector of the discrete mode. For the purpose of simplification, the terms leading to the stability condition of free space (ie the conventional stability condition) are omitted from the stability polynomial.

[0089] In order to avoid the numerical root search in obtaining the stability condition, the stability polynomial is expressed in the complex plane according to equation (13):

[0090]

[0091] By building a Routh table for (14), we can obtain the following stability conditions:

[0092]

[0093] In order to satisfy the stability condition in (16), it is sufficient to satisfy the conventional Courant stability condition, namely:

[0094]

[0095] Therefore, the traditional Courant stability condition is also applicable to the modeling of conductor media, and no additional conditions are required. However, if the discretization scheme proposed in the present invention is not used, for example, the central average operator is not used in the discretization of the constitutive relation (4) This may cause the discretized formula to become unstable.

[0096] Step 4: Generally, the PML needs to be placed near the conductor medium at half the wavelength. The present invention uses a modified Berenger PML formula, namely MPML. In the perfectly matched layer in the conductor medium modeling, D is introduced into the update equation, and the two quantities D and H are processed separately:

[0097]

[0098] The updated equation for the H component is consistent with the original Berenger PML formula.

[0099] It is worth noting that the update formula (8) proposed in the present invention also needs to be extended to PML to adapt to the conductor medium and MPML.

[0100] like Figure 2 As shown in the figure, in the specific implementation, the modeling process is as follows:

[0101] By (1) modeling the conductor medium as a dielectric with frequency dispersion and spatial dispersion;

[0102] Calculate the modeling time step using the conventional Courant stability condition (16) or the stability condition calculated by the present invention (15);

[0103] The modeled electromagnetic field components are updated using the traditional Yee FDTD update equation and the update formula (8) proposed by the present invention;

[0104] MPML is introduced to update the electromagnetic field components according to the original Berenger PML formula and formula (17). And the update formula (8) is also extended to PML.

[0105] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A modeling method for wave propagation in a conductor medium, characterized in that: The following steps are involved: Step 1: Model the conductor medium as a dielectric with frequency dispersion and spatial dispersion in FDTD; Step 2: Calculate the electromagnetic field using the standard Yee FDTD update equations and the electric flux density and electric field intensity update equations; Step 3: Calculate the stability condition of the spatial dispersion FDTD method; Step 4: Introduce the perfectly matched layer and propose the modified Berenger PML formula; In step 1, the conductor medium is specifically a rectangular grid composed of parallel ideal conductive thin wires; In step 2, if the grid period of the conducting medium is smaller than the wavelength, the conducting medium is considered as a uniaxial dielectric with a frequency- and space-dependent equivalent dielectric constant: in is the equivalent dielectric constant vector, xx, yy, zz are the position vectors, ε(k,q x ) is the equivalent dielectric constant value at the vector position, ε0 is the vacuum dielectric constant value, k0 is the wave number corresponding to the plasma frequency, k is the free space wave number, q x is the component of the wave vector q in the direction of the conductor; The plasma frequency depends on the grid periods a and b and the wire radius r: in, ξ is the independent variable of function F, n is the period, when a=b, F(1)=0.5275; The conductor medium is modeled as a dielectric with frequency dispersion and spatial dispersion in FDTD using equation (1).

2. The modeling method for conductor dielectric wave propagation according to claim 1, characterized in that: In step 2, the steps to calculate the electromagnetic field are as follows: In order to consider the dispersion of the material, the relationship between the electric flux density D and the electric field intensity E in the frequency domain is introduced as follows: D x (ω)=ε(ω,q x )E x (oh) Where ω is the frequency and ε0 is the dielectric constant of vacuum; The constitutive relation in the space-time domain is obtained from the above formula: Where c is the speed of light and t is time; By rewriting equation (4) and discretizing the rewritten equation, we can get the update formula: The coefficients of the above formula are: Where Δ t is the time step, Δ x is the spatial step length.

3. The modeling method for conductor dielectric wave propagation according to claim 2, characterized in that: In step 4, in the perfectly matched layer in the conductor medium modeling, D is introduced into the update equation, and the two quantities D and H are processed separately: The updated equation for the H component is consistent with the original Berenger PML formula; Wherein: the update formula (8) is extended to PML to adapt the conductor medium and MPML.

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