Matrix index-based lossy dispersive medium high-precision electromagnetic calculation method

Abstract

The invention discloses a lossy dispersive medium high-precision electromagnetic calculation method based on a matrix index, and 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 FCC grids, and defining electromagnetic parameters of a lossy dispersive medium and a non-dispersive medium in the target space; s3, initializing electric field and magnetic field parameters of grid points in the FCC grid; and S4, in combination with a matrix index method, carrying out iterative calculation on the electromagnetic field value in a calculation region which is a target space. According to the method, the influence of the actual conductivity of the lossy dispersive medium on the electromagnetic field is considered, and the propagation characteristics of the electromagnetic field in the lossy dispersive medium can be analyzed.

Description

Technical Field

[0001] The 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 index. Background Art

[0002] Lossy dissipative media are ubiquitous in nature, such as biological tissues, water, soil, etc. The relative dielectric constant of lossy dissipative media is affected by frequency. Lossy dissipative media are widely used in many fields such as biomedicine, materials science, radar technology, etc. When the conductivity of a lossy dissipative medium is 0, there is no conductivity loss, and it is called a lossless dissipative medium. In practice, most lossy dissipative media are lossy dissipative media, and their conductivity is not 0. The finite-difference time-domain method (FDTD) is a commonly used method for calculating the propagation of electromagnetic fields in lossy dissipative media. The traditional FDTD method divides the geometric object into discrete YEE cells. As the calculation time accumulates, the numerical error will continue to increase, resulting in large calculation errors in the results.

[0003] In the finite-difference time-domain method (FCC-FDTD) based on the face-centered cubic (FCC) grid, each electric field or magnetic field node records the components of the electric field or magnetic field in three directions. Compared with the traditional FDTD method, the finite-difference time-domain method applied to lossless dissipative media is not suitable for lossy dissipative media. Summary of the invention

[0004] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a high-precision electromagnetic calculation method for lossy dissipative media based on matrix index, taking into account the influence of the conductivity of the actual lossy dissipative medium on the electromagnetic field, and being able to analyze the propagation characteristics of the electromagnetic field in the lossy dissipative medium.

[0005] The object of the present invention is achieved through the following technical solution: a high-precision electromagnetic calculation method for lossy dissipative media based on matrix index, 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 FCC grids and define electromagnetic parameters of lossy dispersive media and non-dispersive media in the target space;

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

[0009] S4. Combining the matrix index method, iterative calculation is performed on the electromagnetic field value in the calculation area, and the calculation area is the target space.

[0010] The beneficial effect of the present invention is that the present invention introduces the mathematical model of lossy dispersive media into the FCC-FDTD method by using the matrix index method. This patent provides an FCC-FDTD method for lossy dispersive media based on the matrix index. This method uses Fourier transform to integrate the electric field value of the lossy dispersive medium within one time step, and further combines the FCC grid structure, which has higher calculation accuracy than the traditional FDTD method. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 is a flow chart of the method of the present invention;

[0012] Figure 2 It is a schematic diagram of the calculation space;

[0013] Figure 3 The electric field waveforms of the three methods changing with time at observation point 1;

[0014] Figure 4 The electric field waveforms of the three methods changing with time at observation point 2. DETAILED DESCRIPTION

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

[0016] Considering the influence of electric loss of lossy dissipative medium, the mathematical model of lossy dissipative medium is introduced into the FCC-FDTD algorithm based on the matrix exponential method. When the conductivity is set to 0, this method is also applicable to non-lossy dissipative medium. Therefore, a more general face-centered cubic finite-difference time-domain electromagnetic calculation model for lossy dissipative medium is constructed. Compared with other methods, the electromagnetic field calculation of lossy dissipative medium based on matrix exponential has higher calculation accuracy.

[0017] The present invention uses a matrix index method to introduce a mathematical model of a lossy dispersive medium into the FDTD method, converts a correction equation into a first-order differential equation, and then applies it to the unit cell structure of the FCC to obtain a time domain iteration equation of the lossy dispersive medium suitable for the FCC structure.

[0018] First, a mathematical model of dissipative medium is established. The frequency domain model of dielectric constant is as follows:

[0019]

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

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

[0022] D(ω)=ε0ε r (ω)E(ω) (2)

[0023] Maxwell's curl equation in the known 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 can get the electric field strength E in the x direction: x For example, Maxwell's equations can be rewritten as:

[0027]

[0028] in,

[0029] and is the spatial difference of the magnetic field in the y and z directions, and its expression in the FCC grid structure is:

[0030]

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

[0032] In order to apply the matrix exponential method to the FDTD method with dissipative media, The expression is:

[0033]

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

[0035]

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

[0037]

[0038] (8)-(10) are converted to the time domain to form the matrix equation:

[0039]

[0040] C1, C2 are coefficient matrices:

[0041]

[0042] (11) can be regarded as a first-order differential equation system. Its solution in the time period from nΔt to (n+1)Δt can be expressed as:

[0043]

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

[0045] Equation (12) is used to calculate the time domain iterative equations of the electric field and two auxiliary variables in the FCC-FDTD method for lossy dissipative media.

[0046] So far, this patent has demonstrated in detail the process of introducing the lossy dispersive model into the FDTD method using the matrix index method and combining it with the FCC-FDTD method to form a high-precision calculation method for calculating the electromagnetic field propagation in lossy dispersive media. Figure 1 As shown, a high-precision electromagnetic calculation method for lossy dissipative media based on matrix index 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. Mainly consider the geometric characteristics of the target space, the size in the target space, and create a three-dimensional or two-dimensional physical model; define the excitation source type as a single-frequency excitation source or a broadband excitation source; among them, the single-frequency excitation source uses a sine signal, and the broadband excitation source uses a Gaussian pulse; and determine the parameters of the excitation source, including frequency, amplitude, waveform, etc.; finally, determine the position of the excitation source and the observation point in the target space.

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

[0050] After defining the target space and excitation source, divide the target into FCC grids. According to the excitation source frequency domain waveform, determine the maximum frequency and minimum wavelength of the excitation source. The cell size Δx, Δy, Δz should be less than one eighth of the minimum wavelength. Considering the calculation accuracy and efficiency, select the appropriate cell size. After determining the cell size, further set the time step Δt. 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 meshing, define the lossy dispersive medium and non-dispersive medium in the target space. Determine the size of the space occupied by the two media and define the electromagnetic parameters of the two media. The relative dielectric constant and conductivity of the non-dispersive medium need to be determined. The main electromagnetic parameters of the lossy dispersive medium include the relative dielectric constant at zero frequency, the relative dielectric constant at infinite frequency, and the relaxation time of the medium.

[0053] S3. Initialize the electric field 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, Δz are the dimensions of the cell in the x, y, and z directions, i, j, k are the number of nodes of the cell in the x, y, and z directions of the computational space, and 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 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 the electric field and magnetic field are iterated, the field values ​​of all grids must be initialized, that is, the four electric field values ​​and magnetic field values ​​in the grids other than the grid where the excitation source is located are initialized to zero. The observation point is a grid in the FCC grid;

[0065] S4. Combining the matrix index method, iterative calculation is performed on the electromagnetic field value in the calculation area, and the calculation area is the target space.

[0066] S401. At the time of half-integer time steps (n+0.5)Δt, update the magnetic field value in the calculation area. The magnetic field update equations for lossy dispersive media and non-dispersive media are the same:

[0067]

[0068] Among them, s = {1, 2, 3, 4} represents four types of nodes. Around H in the x, y, and z directions respectively s The spatial difference of the electric field at (i, j, k), the coefficients of each term in the iterative equation are as follows:

[0069]

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

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

[0072] S402. After updating the magnetic field value in the calculation area, add half a time step, and update the electric field value of the grid in the space occupied by the non-dispersive medium in the calculation area in an integer number of time steps (n+1)Δt:

[0073]

[0074] Around E in the x, y, and z directions respectively s The spatial difference of the magnetic field at (i, j, k), the coefficients of each term in the electric field iteration equation 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 this 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 this 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 non-dispersive medium, the electric field value of the lossy dispersive medium grid area is updated using the matrix index method. At this time, the auxiliary variables and the electric field need to be calculated according to the matrix index method, and then brought into the following formula:

[0082]

[0083] At this time, the expressions of the coefficients in the equation are:

[0084]

[0085]

[0086] Among them, β x (i,j,k),β y (i,j,k),β z (i, j, k) are the 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 this node (i,j,k) when the frequency is infinite in the x, y, and z directions, ε sx (i,j,k),ε sy (i,j,k),ε sz (i,j,k) is the dielectric constant of this 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) is the relaxation time of this node (i,j,k) in the x, y, and z directions;

[0089] S404. So far, the electric field value and magnetic field value of the entire calculation area are updated, and the electric field value and magnetic field value of the observation point in the three directions of x, y, and z at the current time are recorded;

[0090] S405. Determine whether the maximum value of the set iterative time step number has been reached; if not, return to S401 to update the electric field and magnetic field in a new time step; if the maximum value has been reached, the calculation stops.

[0091] DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings.

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

[0093] Step 1: Define the size of the calculation space and 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 grid to divide the target, determine the location of the excitation source, and place the electric dipole at the center of the calculation area. At the same time, define the location of the observation point, 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 traditional FCC-FDTD method is used to calculate the electromagnetic field in the air area, and the FCC-FDTD method based on the matrix index is used to update the electromagnetic field in the lossy dissipative medium area, and finally the distribution of the electromagnetic field in the entire calculation area is obtained.

[0098] Figure 3 It is the electric field waveform of the three methods changing with time at observation point 1. 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 electric field waveforms of the three methods at observation point 2 that change with time, the mean square error of the ME-FCC-FDTD method proposed in this patent is 1.67, and the mean square error of the ME-FDTD method is 3.12. From the calculation results, it can be seen that the method proposed in this patent greatly improves the calculation accuracy of electromagnetic field propagation in lossy dissipative media.

[0099] The above are specific implementation methods and simulation verifications of the present invention. It should be pointed out that those skilled in the art can clearly understand that the above embodiments and simulations cited by the present invention are only used to illustrate and verify the rationality and feasibility of the method, and are not used to limit the method of the present invention. Although the present invention can be effectively illustrated and described by the embodiments, there are many changes in the present invention 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 deformations according to the method of the present invention, but these corresponding changes or deformations all belong to the protection scope required by the method of the present invention.

Claims

1. A high-precision electromagnetic calculation method for lossy dissipative media based on matrix index, characterized by: The following steps are involved: S1. Define the target space and excitation source, and determine the positions of the excitation source and observation point in the target space; S2. Divide the target space into FCC grids and define electromagnetic parameters of lossy dispersive media and non-dispersive media in the target space; S3. Initialize the electric field and magnetic field parameters of the grid points in the FCC grid; S4. Combining the matrix index method, iterative calculation is performed on the electromagnetic field value in the calculation area, and the calculation area is the target space.

2. According to claim 1, a high-precision electromagnetic calculation method for lossy dissipative media based on matrix index is characterized by: The step S1 comprises: When defining the target, a three-dimensional or two-dimensional physical model is created based on the geometric characteristics of the target space and the size in the target space; Define the excitation source type as single-frequency excitation source or broadband excitation source; the single-frequency excitation source uses a sine signal, and the broadband excitation source uses a Gaussian pulse; Determine the parameters of the excitation source, including frequency, amplitude and waveform; Determine the locations of the excitation source and observation point in the target space.

3. The high-precision electromagnetic calculation method for lossy dissipative media based on matrix index according to claim 1, characterized in that: The step S2 divides the target space into FCC grids, and needs to determine the cell size and time step, including: According to the frequency domain waveform of the excitation source, determine the maximum frequency and minimum wavelength of the excitation source. The cell size Δx, Δy, Δz should be less than one eighth of the minimum wavelength. After determining the cell size, set the time step Δt. The time step size should satisfy the following formula to ensure the stability of the calculation method: where c is the speed of light.

4. The high-precision electromagnetic calculation method for lossy dissipative media based on matrix index according to claim 1, characterized in that: In 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 relative permittivity and conductivity; the electromagnetic parameters of the lossy dispersive medium include relative permittivity at zero frequency, relative permittivity at infinite frequency, and relaxation time of the medium.

5. The high-precision electromagnetic calculation method for lossy dissipative media based on matrix index according to claim 1, characterized in that: The FCC grid, i.e., the face center grid, has four types of electric field nodes and four types of magnetic field nodes in each grid, and the distribution form is similar to the unit cell structure of SiC in chemistry; the observation point is a grid in the FCC grid; For a grid with coordinates (iΔx, jΔy, kΔz), Δx, Δy, Δz are the dimensions of the cell in the x, y, and z directions, i, j, k are the number of nodes of the cell in the x, y, and z directions of the computational space, and the four types of electric field nodes recorded in the cell are as follows: E1=(iΔx,jΔy,kΔz) E2=((i+0.5)Δx,(j+0.5)Δy,kΔz) E3=((i+0.5)Δx,jΔy,(k+0.5)Δz) E4=(iΔx,(j+0.5)Δy,(k+0.5)Δz) The coordinates of the four magnetic field nodes are: H1=((i+0.25)Δx, (j+0.25)Δy, (k+0.25)Δz) H2=((i+0.75)Δx, (j+0.75)Δy, (k+0.5)Δz) H3=((i+0.75)Δx, (j+0.75)Δy, (k+0.75)Δz) H4=((i+0.75)Δx, (j+0.25)Δy, (k+0.75)Δz) In step S3, the electric field and magnetic field parameters of the grid points in the FCC grid are initialized, that is, the four electric field values ​​and magnetic field values ​​inside the grids other than the grid where the excitation source is located are initialized to zero.

6. The high-precision electromagnetic calculation method for lossy dissipative media based on matrix index according to claim 1, characterized in that: The step S4 comprises: S401. At the time of half-integer time steps (n+0.5)Δt, update the magnetic field value in the calculation area. The magnetic field update equations for lossy dispersive media and non-dispersive media are the same: Among them, s = {1, 2, 3, 4} represents four types of nodes. Around H in the x, y, and z directions respectively s The spatial difference of the electric field at (i, j, k), the coefficients of each term in the iterative equation are as follows: μ x (i,j,k),μ y (i,j,k),μ z (i,j,k) is the magnetic permeability coefficient of this node (i,j,k) in the x, y, and z directions. The magnetic permeability of this node (i, j, k) in the x, y, and z directions; S402. After updating the magnetic field value in the calculation area, add half a time step, and update the electric field value of the grid in the space occupied by the non-dispersive medium in the calculation area in an integer number of time steps (n+1)Δt: Around E in the x, y, and z directions respectively s The spatial difference of the magnetic field at (i, j, k), the coefficients of each term in the electric field iteration equation are as follows: ε x (i,j,k),ε y (i,j,k),ε z (i,j,k) is the dielectric constant of this node (i,j,k) in the x, y, and z directions. The conductivity of this node (i, j, k) in the x, y, and z directions. ε x (i,j,k),ε y (i,j,k),ε z (i,j,k) is the dielectric constant of this node (i,j,k) in the x, y, and z directions. The conductivity of this node (i, j, k) in the x, y, and z directions. S403. After updating the electric field value of the non-dispersive medium, the electric field value of the lossy dispersive medium grid area is updated using the matrix index method: At this time, the expressions of the coefficients in the equation are: Among them, β x (i,j,k),β y (i,j,k),β z (i, j, k) are the introduced intermediate variables, and the expression is: ε ∞x (i,j,k),ε ∞y (i,j,k),ε ∞z (i,j,k) is the dielectric constant of this node (i,j,k) when the frequency is infinite in the x, y, and z directions, ε sx (i,j,k),ε sy (i,j,k),ε sz (i,j,k) is the dielectric constant of this 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) is the relaxation time of this node (i,j,k) in the x, y, and z directions; S404. So far, the electric field value and magnetic field value of the entire calculation area are updated, and the electric field value and magnetic field value of the observation point in the three directions of x, y, and z at the current time are recorded; S405. Determine whether the maximum value of the set iterative time step number has been reached; if not, return to S401 to update the electric field and magnetic field in a new time step; if the maximum value has been reached, the calculation stops.

7. The high-precision electromagnetic calculation method for lossy dissipative media based on matrix index according to claim 6, characterized in that: When the matrix index method is used to update the electric field value of the grid region with lossy dissipative medium, the principle of the matrix index method is as follows: A1. Establish a mathematical model of dissipative medium, that is, the frequency domain model of dielectric constant is as follows: Where ω is the angular frequency, ε0 ​​is the dielectric constant of vacuum, and ε s is the static relative dielectric constant of the medium, ε ∞ is the relative dielectric constant at infinite frequency, τ0 is the relaxation time, and σ is the conductivity; The constitutive equations of the electric field intensity and electric displacement vector of a lossy dissipative medium are as follows: D(ω)=ε0ε r (ω)E(ω) A2. Maxwell curl equation in the known frequency domain: Where D is the electric displacement vector, E is the electric field intensity, and H is the magnetic field; The electric field strength E in the x direction x , Maxwell's equations can be rewritten as: in, and is the spatial difference of the magnetic field in the y and z directions, and its expression in the FCC grid structure is: H1, H2, H3, H4 are four magnetic field nodes in the FCC grid structure; A3, introduce the matrix index method, The expression is: in, From this we get, Ψ Ex1 ,Ψ Ex2 is an auxiliary variable, and the expression is: A4. Convert to time domain and form the matrix equation: C1, C2 are coefficient matrices: The matrix equation is regarded as a first-order differential equation system, and the auxiliary variables are initialized to 0. In the time period from nΔt to (n+1)Δt, its solution is expressed as: in, l1=e -v·Δt , l2=(1-l1) / v, Thus, the electric field in the lossy dissipative medium and the time domain iterative equations of the two auxiliary variables are calculated.

Images