A high-precision electromagnetic calculation method for lossy dispersive medium based on matrix exponent

Abstract

The application discloses a high-precision electromagnetic calculation method of lossy dispersive medium based on matrix exponent, which comprises the following steps: S1, defining a target space and an excitation source, and determining the positions of the excitation source and observation points in the target space; S2, dividing the target space into an FCC grid, and defining electromagnetic parameters of lossy dispersive medium and non-dispersive medium in the target space; S3, initializing electric field and magnetic field parameters of grid points in the FCC grid; and S4, combining the method of matrix exponent, and iteratively calculating electromagnetic field values in a calculation region, namely the target space. The application considers the influence of actual lossy dispersive medium conductivity on electromagnetic field, and can analyze the propagation characteristics of electromagnetic field in the lossy dispersive medium.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic field calculation, and in particular to a high-precision electromagnetic calculation method for lossy dispersive media based on matrix exponents. Background Technology

[0002] Lossy dispersive media are ubiquitous in nature, such as in biological tissues, water, and soil. The relative permittivity of electrically lossy dispersive media is affected by frequency. Lossy dispersive media have wide applications in many fields, including biomedicine, materials science, and radar technology. When the conductivity of a lossy dispersive medium is 0, there is no conductivity loss, and it is called a lossless dispersive medium. In reality, most lossy dispersive media are lossy dispersive media, and their conductivity is not zero. The finite-difference time-domain (FDTD) method is a commonly used method for calculating electromagnetic field propagation in lossy dispersive media. However, the traditional FDTD method divides the geometric object into discrete Yee cells. As the computation time accumulates, the numerical error increases continuously, leading to significant computational errors in the results.

[0003] The finite-difference time-domain method based on face-centered cubic (FCC) grids (FCC-FDTD) records the components of the electric or magnetic field in three directions at each electric or magnetic field node. Compared with the traditional FDTD method, it has better isotropy and higher computational accuracy. However, the finite-difference time-domain method applied to lossless dispersive media is not applicable to lossy dispersive media. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a high-precision electromagnetic calculation method for lossy dispersive media based on matrix exponents. This method considers the influence of the conductivity of the actual lossy dispersive medium on the electromagnetic field and can analyze the propagation characteristics of the electromagnetic field in the lossy dispersive medium.

[0005] The objective of this invention is achieved through the following technical solution: a high-precision electromagnetic calculation method for lossy dispersive media based on matrix exponents, comprising the following steps:

[0006] S1. Define the target space and excitation source, and determine the positions of the excitation source and observation point in the target space;

[0007] S2. Divide the target space into an FCC grid and define the electromagnetic parameters of the lossy dispersive medium and the non-dispersive medium in the target space;

[0008] S3. Initialize the electric and magnetic field parameters of the grid points in the FCC grid;

[0009] S4. Combining the matrix exponentiation method, the electromagnetic field values ​​within the calculation region are iteratively calculated, where the calculation region is the target space.

[0010] The beneficial effects of this invention are: This invention utilizes the matrix exponentiation method to introduce the mathematical model of lossy dispersive media into the FCC-FDTD method. This patent provides an FCC-FDTD method for lossy dispersive media based on matrix exponentiation. This method uses Fourier transform to integrate the electric field value of the lossy dispersive medium within one time step, and further combines it with the FCC grid structure, achieving higher computational accuracy compared to the traditional FDTD method. Attached Figure Description

[0011] Figure 1 This is a flowchart of the method of the present invention;

[0012] Figure 2 This is a schematic diagram of the computational space;

[0013] Figure 3 The electric field waveforms at observation point 1 as a function of time under the three methods are shown.

[0014] Figure 4 The electric field waveforms at observation point 2 are the electric field waveforms over time as shown by the three methods. Detailed Implementation

[0015] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings, but the scope of protection of the present invention is not limited to the following description.

[0016] Considering the influence of electrical losses in lossy dispersive media, the mathematical model of lossy dispersive media is introduced into the FCC-FDTD algorithm based on the matrix exponent method. When the conductivity is set to 0, this method is also applicable to lossless dispersive media. Thus, a more general finite-difference time-domain electromagnetic calculation model based on face-centered cubic is constructed for lossy dispersive media. Compared with other methods, the calculation of the electromagnetic field of lossy dispersive media based on matrix exponent has higher calculation accuracy.

[0017] This invention utilizes the matrix exponentiation method to introduce the mathematical model of lossy dispersive media into the FDTD method, transforming the modified equation into a first-order differential equation, and then applying it to the cell structure of FCC to obtain a time-domain iterative equation for lossy dispersive media applicable to the FCC structure.

[0018] First, a mathematical model of the lossy dispersive medium is established, and the frequency domain model of the dielectric constant is as follows:

[0019]

[0020] In the formula, ω is the angular frequency, ε0 ​​is the vacuum permittivity, and ε s ε is the static relative permittivity of the medium. ∞ τ is the relative permittivity at infinite frequency, τ0 is the relaxation time, and σ is the conductivity.

[0021] The constitutive equations for the electric field intensity and electric displacement vector of a lossy dispersive medium are as follows:

[0022] D(ω)=ε0ε r (ω)E(ω) (2)

[0023] Given Maxwell's curl equation in the frequency domain:

[0024]

[0025] Where D is the electric displacement vector, E is the electric field strength, and H is the magnetic field.

[0026] Substituting equations (1) and (2) into equation (3), we get the electric field intensity E in the x-direction. x For example, Maxwell's equations can be rewritten as:

[0027]

[0028] in,

[0029] and The spatial difference of the magnetic field in the y and z directions is expressed as follows in the FCC grid structure:

[0030]

[0031] H1, H2, H3, and H4 are four types of magnetic field nodes in the FCC grid structure.

[0032] To apply the matrix exponent method to the FDTD method for lossy dispersive media, The expression is:

[0033]

[0034] Where κ=ε ∞ , Substituting equation (7) into equation (4),

[0035]

[0036] Ψ Ex1 ,Ψ Ex2 As an auxiliary variable, the expression is:

[0037]

[0038] (8)-(10) are transformed to the time domain, forming matrix equations:

[0039]

[0040] C1 and C2 are coefficient matrices:

[0041]

[0042] (11) can be regarded as a system of first-order differential equations. In the time interval from nΔt to (n+1)Δt, its solution can be expressed as:

[0043]

[0044] in, l1=e -v·Δt l2 = (1 - l1) / v

[0045] Equation (12) is the time-domain iterative equation used in the FCC-FDTD method for calculating the electric field and two auxiliary variables in a lossy dispersive medium.

[0046] This patent thus details the process of introducing the lossy dispersion model into the FDTD method using the matrix exponent method and combining it with the FCC-FDTD method, forming a high-precision calculation method for calculating the propagation of electromagnetic fields in lossy dispersive media. Figure 1 As shown, a high-precision electromagnetic calculation method for lossy dispersive media based on matrix exponents includes the following steps:

[0047] S1. Define the target space and excitation source, and determine the positions of the excitation source and observation point in the target space;

[0048] First, define the target space (geometric target) and the excitation source. This mainly considers the geometric characteristics and size of the target space, creating a three-dimensional or two-dimensional physical model. Define the excitation source type as either a single-frequency excitation source or a broadband excitation source; a single-frequency excitation source uses a sinusoidal signal, while a broadband excitation source uses a Gaussian pulse. Determine the parameters of the excitation source, including frequency, amplitude, and waveform. Finally, determine the positions of the excitation source and the observation point within the target space.

[0049] S2. Divide the target space into an FCC grid and define the electromagnetic parameters of the lossy dispersive medium and the non-dispersive medium in the target space;

[0050] After defining the target space and excitation source, the target is divided into an FCC grid. Based on the frequency domain waveform of the excitation source, the maximum frequency and minimum wavelength of the excitation source are determined. The cell sizes Δx, Δy, and Δz should be less than one-eighth of the minimum wavelength. Considering computational accuracy and efficiency, an appropriate cell size is selected. After determining the cell size, the time step Δt is further set. The time step size should satisfy the following formula to ensure the stability of the calculation method, where c is the speed of light.

[0051]

[0052] After mesh generation, the lossy dispersive medium and the non-dispersive medium in the target space are defined. The size of the space occupied by each medium is determined, and the electromagnetic parameters of the two media are defined. For the non-dispersive medium, the relative permittivity and conductivity need to be determined. The main electromagnetic parameters of the lossy dispersive medium include the relative permittivity at zero frequency, the relative permittivity at infinite frequency, and the relaxation time of the medium.

[0053] S3. Initialize the electric and magnetic field parameters of the grid points in the FCC grid;

[0054] Initialize the electric and magnetic field parameters of the grid points in space. For a grid with coordinates (iΔx, jΔy, kΔz), Δx, Δy, and Δz are the dimensions of the cell in the x, y, and z directions, respectively, and i, j, and k are the number of nodes in the cell in the x, y, and z directions of the computational space. The four types of electric field nodes recorded in the cell are as follows:

[0055] E1=(iΔx,jΔy,kΔz)

[0056] E2=((i+0.5)Δx,(j+0.5)Δy,kΔz)

[0057] E3=((i+0.5)Δx,jΔy,(k+0.5)Δz)

[0058] E4=(iΔx,(j+0.5)Δy,(k+0.5)Δz)

[0059] The coordinates of the four types of magnetic field nodes are:

[0060] H1=((i+0.25)Δx, (j+0.25)Δy, (k+0.25)Δz)

[0061] H2=((i+0.75)Δx, (j+0.75)Δy, (k+0.5)Δz)

[0062] H3=((i+0.75)Δx, (j+0.75)Δy, (k+0.75)Δz)

[0063] H4=((i+0.75)Δx, (j+0.25)Δy, (k+0.75)Δz)

[0064] Before iterating over the electric and magnetic fields, the field values ​​of all grids must be initialized. This means that all four electric and magnetic field values ​​within each grid except the one containing the excitation source are initialized to zero. The observation point is one grid within the FCC grid.

[0065] S4. Combining the matrix exponentiation method, the electromagnetic field values ​​within the calculation region are iteratively calculated, where the calculation region is the target space.

[0066] S401. At half-integer time steps (n+0.5)Δt, update the magnetic field values ​​within the computational domain. The magnetic field update equations are the same for both lossy dispersive and non-dispersive media:

[0067]

[0068] Where s = {1, 2, 3, 4} represents four types of nodes. They surround H in the x, y, and z directions respectively. s The spatial difference of the electric field at (i,j,k) is given by the coefficients of the terms in the iterative equation as follows:

[0069]

[0070] μ x (i,j,k),μ y (i,j,k),μ z (i,j,k) represents the permeability of this node (i,j,k) in the x, y, and z directions.

[0071] For the permeability of this node (i,j,k) in the x, y, and z directions;

[0072] S402. After updating the magnetic field values ​​in the computational domain, add half a time step and update the electric field values ​​of the grid in the space occupied by the nondispersive medium in the computational domain at integer time steps (n+1)Δt:

[0073]

[0074] They orbit E in the x, y, and z directions respectively. s The spatial difference of the magnetic field at (i,j,k) and the coefficients of each term in the iterative equation for the electric field are as follows:

[0075]

[0076]

[0077] ε x (i,j,k),ε y (i,j,k),ε z (i,j,k) is the dielectric constant of node (i,j,k) in the x, y, and z directions.

[0078] The conductivity of this node (i,j,k) in the x, y, and z directions.

[0079] ε x (i,j,k),εy (i,j,k),ε z (i,j,k) is the dielectric constant of node (i,j,k) in the x, y, and z directions.

[0080] The conductivity of this node (i,j,k) in the x, y, and z directions.

[0081] S403. After updating the electric field value of the nondispersive medium, update the electric field value of the lossy dispersive medium mesh region using the matrix exponential method. At this time, it is necessary to calculate the auxiliary variables and electric field according to the matrix exponential method, and then substitute them into the following formula:

[0082]

[0083] At this point, the expressions for each coefficient in the equation are:

[0084]

[0085]

[0086] Where, β x (i,j,k),β y (i,j,k),β z (i,j,k) are introduced intermediate variables, and the expression is:

[0087]

[0088] ε ∞x (i,j,k),ε ∞y (i,j,k),ε ∞z (i,j,k) is the dielectric constant of node (i,j,k) when the frequencies in the x, y, and z directions are infinite. ε sx (i,j,k),ε sy (i,j,k),ε sz (i,j,k) is the dielectric constant of node (i,j,k) when the frequency is 0 in the x, y, and z directions; τ x (i,j,k),τ y (i,j,k),τ z (i,j,k) represents the relaxation time of this node (i,j,k) in the x, y, and z directions.

[0089] S404. The electric and magnetic field values ​​of the entire calculation region have now been updated, and the electric and magnetic field values ​​of the observation point in the x, y, and z directions at the current time have been recorded.

[0090] S405. Determine if the maximum set iteration time step has been reached; if not, return to S401 to update the electric and magnetic fields in the new time step; if the maximum has been reached, stop the calculation.

[0091] The embodiments will now be described in detail, with examples illustrated in the accompanying drawings.

[0092] Figure 2 This diagram illustrates the computational space, which measures 2.4m × 2.4m × 2.4m. The central region is filled with air, measuring 0.2m × 0.1m × 0.1m, while the remaining area is filled with a lossy dispersive medium. The dielectric constant of this lossy dispersive medium at infinite frequency is ε. ∞ =7, static dielectric constant is ε s =10, relaxation time is τ1 = 0.7 ns. Consider using an FCC mesh size of 0.05m × 0.05m × 0.05m to partition the computational space. Select an electric dipole radiation-modulated Gaussian pulse as the excitation source to calculate the propagation of the electromagnetic field in a lossy dispersive medium. The specific steps of the high-precision electromagnetic calculation method for lossy dispersive media based on matrix exponents provided in this application are as follows:

[0093] Step 1: Define the computation space size and define the excitation source. The excitation source expression is as follows:

[0094]

[0095] Where, ω=2π×0.3×10 9 rad / s, t0=9π / 2ω, t1=4.3ns.

[0096] Step 2: Use the FCC mesh to partition the target, determine the location of the excitation source, and place the electric dipole at the center of the computational region. At the same time, define the locations of the observation points: the coordinates of observation point 1 are (0.7, 1.2, 1.2), and the coordinates of observation point 2 are (1.2, 0.7, 1.2).

[0097] Step 3: In the FCC grid, the electromagnetic field of the air region is calculated using the traditional FCC-FDTD method, and the electromagnetic field of the lossy dispersive medium region is updated using the matrix exponent-based FCC-FDTD method, finally obtaining the distribution of the electromagnetic field in the entire calculation region.

[0098] Figure 3 The electric field waveforms at observation point 1 are the electric field waveforms that change over time using the three methods. The calculation results of the commercial software CST are selected as a reference. The mean square error of the ME-FCC-FDTD method proposed in this patent is 1.22, and the mean square error of the ME-FDTD method is 1.55. Figure 4The figures show the electric field waveforms at observation point 2 as a function of time using the three methods. The mean square error (MSE) of the ME-FCC-FDTD method proposed in this patent is 1.67, and the MSE of the ME-FDTD method is 3.12. The calculation results show that the method proposed in this patent significantly improves the calculation accuracy of electromagnetic field propagation in lossy dispersive media.

[0099] The above describes the specific embodiments and simulation verification of the present invention. It should be noted that those skilled in the art will clearly understand that the above embodiments and simulations are only for illustrating and verifying the rationality and feasibility of the method, and are not intended to limit the method of the present invention. Although the embodiments effectively illustrate and describe the present invention, many variations exist without departing from the spirit of the present invention. Without departing from the spirit and essence of the method of the present invention, those skilled in the art can make various corresponding changes or modifications according to the method of the present invention, but these corresponding changes or modifications all fall within the protection scope claimed by the method of the present invention.

Claims

1. A high-precision electromagnetic calculation method for lossy dispersive media based on matrix exponentials, characterized in that: The method comprises the following steps: S1. defining a target space and an excitation source, and determining the positions of the excitation source and an observation point in the target space; S2. dividing the target space into an FCC grid, and defining electromagnetic parameters of a lossy dispersive medium and a non-dispersive medium in the target space; S3. initializing electromagnetic field parameters of grid points in the FCC grid; S4. iteratively calculating electromagnetic field values in a calculation region, i.e., the target space, by combining a matrix exponential method; The step S4 comprises: S401. updating the magnetic field values in the calculation region at a half-integer time step (n+0.5)Δt, and the magnetic field update equations of the lossy dispersive medium and the non-dispersive medium are the same: ； where represents four kinds of nodes, respectively, the spatial difference of the electric field in the x, y, z three directions around the iteration equation, the coefficient of each term is as follows: ； ； ； for the node magnetic permeability in x, y, z directions, For this node Permeability in x, y, z directions; S402. after updating the magnetic field values of the calculation region, increasing a half time step, and updating the electric field values of the grid in the space occupied by the non-dispersive medium in the calculation region at an integer time step (n+1)Δt: ； respectively the spatial differentiation of the magnetic field in the x, y, z directions The coefficients in the electric field iteration equation are as follows: ； ； ； for this node dielectric constant in x, y, z directions, for this node conductivity in x, y, z directions; for the node dielectric constant in x, y, z directions, for the node conductivity in x, y, z directions; S403. after updating the electric field values of the non-dispersive medium, updating the electric field values of the grid region of the lossy dispersive medium by using the matrix exponential method: At this time, the expression of each coefficient in the equation is: ； ； wherein is an introduced intermediate variable, the expression is: ； for this node dielectric constant at infinite frequency in the x, y, z directions, for this node dielectric constant at zero frequency in the x, y, z directions; for this node relaxation time in the x, y, z directions; S404. after updating the electric field values and the magnetic field values of the entire calculation region, recording the electric field values and the magnetic field values of the observation point in the current time x, y and z directions; S405. determining whether the maximum number of iteration time steps is reached; if not, returning to S401 to update the electric field and the magnetic field in a new time step; if the maximum number of iteration time steps is reached, stopping the calculation.

2. The method of claim 1, wherein the method is characterized by: The step S1 comprises: defining a target according to the geometric characteristics of the target space and the size of the target space, and creating a three-dimensional or two-dimensional physical model; defining the type of the excitation source as a single-frequency excitation source or a wideband excitation source; wherein the single-frequency excitation source adopts a sinusoidal signal, and the wideband excitation source adopts a Gaussian pulse; determining the parameters of the excitation source, including frequency, amplitude and waveform; determining the positions of the excitation source and the observation point in the target space.

3. The method of claim 1, wherein: The step S2 divides the target space into an FCC grid, and needs to determine the cell size and the time step, including: Based on the frequency domain waveform of the excitation source, the maximum frequency and minimum wavelength of the excitation source are determined, and the size of the unit cell should be less than one-eighth of the minimum wavelength; After the cell size is determined, the time step is set The time step size should satisfy the following equation to ensure the stability of the calculation method: ； where c is the speed of light.

4. The method of claim 1, wherein: In the step S2, the electromagnetic parameters of the lossy dispersive medium and the non-dispersive medium in the target space are defined, the electromagnetic parameters of the non-dispersive medium include the relative permittivity and the conductivity, and the electromagnetic parameters of the lossy dispersive medium include the relative permittivity at zero frequency, the relative permittivity at infinite frequency and the relaxation time of the medium.

5. The method of claim 1, wherein: The FCC grid, i.e., the face-centered grid, has four types of electric field nodes and four types of magnetic field nodes in each grid, and the distribution form is like the unit cell structure of SiC in chemistry; the observation point is a grid in the FCC grid; For coordinates The grid, Let be the dimensions of the cell in the x, y, and z directions, and let i, j, and k be the number of nodes in the cell in the x, y, and z directions of the computation space. The four types of electric field nodes recorded in the cell are as follows: ； The coordinates of the four types of magnetic field nodes are: ； In the step S3, the electric field and the magnetic field parameters of the grid points in the FCC grid are initialized, i.e., the four types of electric field values and the magnetic field values in the other grids except the grid at the position of the excitation source are initialized to zero.

6. The method of claim 1, wherein: When the matrix exponential method is used to update the electric field values of the grid region of the lossy dispersive medium, the principle of the matrix exponential method is as follows: A1, a mathematical model of the lossy dispersive medium is established, i.e., a frequency domain model of the permittivity is as follows: ； wherein is the angular frequency, is the vacuum permittivity, is the static relative permittivity of the medium, is the relative permittivity at infinite frequency, is the relaxation time, is the conductivity; The constitutive equation of the electric field intensity and the electric displacement vector of the lossy dispersive medium is as follows: ； A2, Maxwell curl equation in the frequency domain: ； wherein is the electric displacement vector, is the electric field intensity, is the magnetic field; electric field intensity in the x direction The Maxwell equations are rewritten as ； wherein ; and For the spatial difference of the magnetic field in the y and z directions, the expression in the FCC lattice structure is: ； ； For the four kinds of magnetic field nodes in the FCC lattice structure; A3, the matrix exponential method is introduced, The expression is: ； wherein ; Thus we get, ； For the auxiliary variable, the expression is: ； ； A4, Transform to time domain, form matrix equation: ； For the coefficient matrix: ， ； The matrix equation is considered as a system of first order differential equations with the auxiliary variables initialized to 0, in to a time period, whose solution is expressed as: ； wherein , , , , ; Thus the time domain iterative equations of the electric field and two auxiliary variables of the lossy dispersive medium are calculated.

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