Mixing absorption boundary electromagnetic wave numerical simulation method
By setting the Mur absorption boundary on the outermost layer of CPML and combining with the FDTD method, the problem that the Mur absorption boundary cannot be directly applied in the coordinate telescopic media is solved, and a better electromagnetic wave absorption effect is achieved.
Patent Information
- Application Number
- CN202510261437.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-03-06
- Publication Date
- 2025-07-01
AI Technical Summary
Traditional Mur absorption boundary conditions cannot be directly used in coordinate expansion media, resulting in reflection errors in the CPML layer, affecting the absorption effect of electromagnetic waves.
A numerical simulation method for electromagnetic waves in hybrid absorption boundary is proposed, which seamlessly combines the Mur absorption boundary with the CPML absorption boundary. By setting the Mur absorption boundary on the outermost layer of CPML, the electromagnetic field in the coordinate telescopic medium is solved by using the FDTD method to satisfy the differential equation of the Mur absorption boundary in the CPML layer.
It improves the absorption effect of electromagnetic waves, reduces reflection errors, and improves the absorption performance of CPML.
Smart Images
Figure CN120234995A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of computational electromagnetics, and particularly relates to a numerical simulation method for hybrid absorbing boundary electromagnetic waves. Background Art
[0002] The difference equation solver based on the finite-difference time-domain (FDTD) method is one of the most deeply studied and widely used numerical methods in the field of numerical simulation, and plays an extremely important role in the study of electromagnetic wave propagation characteristics. When using FDTD to solve electromagnetic wave propagation, an absorbing boundary technique needs to be adopted to reduce the reflection error caused by the truncated boundary. Currently, the absorbing boundary technique can be roughly divided into two types. One is the absorbing boundary condition based on the traveling wave operator. Engquist and Majda divided the electromagnetic wave propagation into an outward traveling wave and an inward traveling wave, and derived the absorbing boundary condition based on the one-way wave equation. Mur factorized the traveling wave operator on this basis, and used the Taylor series to expand the decomposition factor to obtain Mur absorbing boundary conditions of different orders such as the first order and the second order. In order to eliminate the reflection of any incident wave, Higdon introduced the electromagnetic wave incident angle into the absorbing boundary condition based on the Engquist-Majda absorbing boundary. In 1997, Ramahi used two complementary operators to calculate the boundary electromagnetic field respectively, and then reduced the reflection caused by the truncated boundary by taking the average. In addition, some scholars have also conducted in-depth research on the high-order forms of the above absorbing boundaries, but the high-order forms involve high-order differential calculations, and the required computing resources and computing time also increase accordingly. The other is the perfectly matched layer (PML) based on lossy media. By setting a special medium with a certain thickness in the outer layer of the calculation region, the wave impedance of this medium is completely matched with the wave impedance of the adjacent medium, so that the electromagnetic wave enters this medium without reflection and gradually decays in this medium, thereby reducing the reflection of the electromagnetic wave. The perfectly matched layer (BPML) proposed by Berenger realizes the absorption of electromagnetic waves relying on this idea. Compared with the first absorbing boundary method, BPML can greatly reduce the reflection in FDTD calculations, so it has attracted the strong interest of many scholars, and coordinate stretching perfectly matched layer (SC-PML), anisotropic medium perfectly matched layer (UPML), approximate perfectly matched layer (NPML), convolutional perfectly matched layer (CPML), etc. with better absorption effects have been proposed on the basis of BPML. Among them, CPML combines the cyclic convolution theory with the complex frequency domain tensor shift perfectly matched layer (CFS-PML), improves the absorption of evanescent waves and low-frequency waves, and is currently the best performing, most popular and most widely used absorbing boundary.
[0003] Based on the above two absorbing boundary conditions, it is natural to think of combining the two absorbing boundaries to further improve the absorption effect. In the field of computational electromagnetics, the Mur absorbing boundary condition has a good absorption effect on perpendicularly incident electromagnetic waves, and thus has been widely studied and applied. Many scholars have explored the Mur absorbing boundary condition under different partial differential equation solvers. The outermost layer of the classical CPML absorbing boundary is usually set as a perfect electric conductor (PEC). Therefore, the perpendicularly incident electromagnetic wave generates reflection at the outermost layer of the CPML, and finally enters the numerical simulation region again, thus bringing reflection error. Combining the Mur absorbing boundary with the CPML is expected to reduce the reflection of the perpendicularly incident electromagnetic wave and further improve the absorption effect of the CPML. However, the traditional Mur absorbing boundary condition does not satisfy the wave equation in the scaled medium. If the traditional Mur absorbing boundary is directly combined with the CPML, it may bring reflection error caused by the mismatch between the two. To combine the Mur and CPML absorbing boundaries, it is necessary to further study the differential equation satisfied by the Mur absorbing boundary in the CPML layer. However, there is no publicly available literature or invention in this regard yet. Summary of the Invention
[0004] In order to overcome the deficiency that the traditional Mur absorbing boundary condition cannot be directly used in the scaled medium, the present invention proposes a numerical simulation method for electromagnetic waves with a hybrid absorbing boundary.
[0005] The technical solution adopted by the present invention to solve its technical problems is:
[0006] A numerical simulation method for electromagnetic waves with a hybrid absorbing boundary, comprising the following steps:
[0007] Step 1, establish a calculation model
[0008] Establish a numerical calculation model for electromagnetic waves with a hybrid absorbing boundary. The principle followed for constructing the calculation model boundary is: the convolutional perfectly matched layer CPML is adjacent to the finite-difference time-domain FDTD calculation region, and the Mur absorbing boundary is located at the outermost boundary of the CPML.
[0009] Step 2, set the parameters of the convolutional perfectly matched layer CPML
[0010] Set the grid size, time step, and parameters of the convolutional perfectly matched layer CPML for the established calculation model.
[0011] Use regular grids to divide the calculation model.
[0012] The convolutional perfectly matched layer CPML boundary is used to calculate the electromagnetic field in the scaled medium. When the electromagnetic wave passes through the boundary between the FDTD region and the CPML region, no reflection occurs.
[0013] Calculate the relative deviation of the electric field at the diagnostic point.
[0014] Step 3, calculate the electromagnetic field distribution inside the model
[0015] Use the FDTD iterative formula and the CPML iterative formula respectively to calculate the electromagnetic field distribution inside the numerical calculation model of the electromagnetic wave with a hybrid absorbing boundary.
[0016] The time steps involved in the FDTD iterative formula and the CPML iterative formula satisfy the CFL condition.
[0017] Step 4, calculate parameter A, parameter B, and parameter C required for the hybrid boundary
[0018] Step 5, calculate the electric field at the outermost boundary of the convolutional perfectly matched layer CPML
[0019] Calculate the electric field at the outermost boundary of the convolutional perfectly matched layer CPML in the x-direction, y-direction, and z-direction respectively.
[0020] Step 6, calculate the magnetic field at the next moment:
[0021] Substitute the electric field value at the outermost boundary of CPML into the FDTD iteration, and use the FDTD method to solve the Maxwell equation to obtain the magnetic field at the next moment.
[0022] Step 7, iteration
[0023] Repeat steps 3 to 6, and perform iterations according to the set number of iteration steps to achieve the numerical simulation of the electromagnetic wave with a hybrid absorbing boundary.
[0024] For the above numerical simulation method of the electromagnetic wave with a hybrid absorbing boundary, in step 1 of establishing the calculation model, in a two-dimensional rectangular coordinate system, the calculation model is a rectangle; in a three-dimensional coordinate system, the calculation model is a cube or a cuboid; the FDTD region is adjacent to the CPML region.
[0025] For the above numerical simulation method of the electromagnetic wave with a hybrid absorbing boundary, step 1 further includes:
[0026] Establish a cube model with a side length of 40 mm, and select the vacuum medium parameters for the model medium parameters. Place an electric dipole at the center of the cube, with the direction of the electric dipole moment being the +z direction, and the coordinates of the diagnostic position M being (2, 20, 20), with the coordinate unit being mm. The electric dipole moment p(t) of the electric dipole is given by the following formula:
[0027]
[0028] In formula (4), T is the radiation source parameter, T = 26.53 ps, and t is the numerical simulation time.
[0029] For the above numerical simulation method of the electromagnetic wave with a hybrid absorbing boundary, step 2 further includes:
[0030] The computational model is meshed with cubes having a side length of 0.5 mm, the time step Δt is 0.48 ps, the number of CPML layers is 10, the stretching factor k max = 1, the complex frequency shift factor α max = 0.2, the polynomial order m = 3, the total number of iteration steps is 4000, and the relative deviation of the z-direction electric field at the diagnostic point is given by the following formula:
[0031]
[0032] In formula (5), Error(t) is the relative deviation of the z-direction electric field at time t, is the diagnostic electric field value of the z-direction electric field at time t with a truncated boundary, is the reference electric field value of the z-direction electric field in free space at time t, is the maximum value of the reference electric field in the z direction of free space.
[0033] For the above numerical simulation method of electromagnetic waves with a hybrid absorbing boundary, step 4 further includes:
[0034] The calculation method of the hybrid boundary related parameters is given by the following formula:
[0035]
[0036] In formula (1), Δx, Δt, and v respectively represent the grid step, the time step, and the propagation speed of electromagnetic waves, k x is the complex frequency shift parameter in the x direction, c x is the attenuation coefficient combination parameter in the x direction, A is calculation parameter 1, B is calculation parameter 2, and C is calculation parameter 3.
[0037] For the above numerical simulation method of electromagnetic waves with a hybrid absorbing boundary, step 5 further includes:
[0038] Calculate the electric field at the outermost boundary of the convolutional perfectly matched layer CPML. For the x direction, the calculation method is as follows:
[0039] The calculation formula for the left boundary is:
[0040]
[0041] The calculation formula for the right boundary is:
[0042]
[0043] In formulas (2) and (3), i, j, and k respectively represent the index values in the x, y, and z directions, and α xare the auxiliary variable of the electric field in the x - direction and the complex frequency shift factor in the x - direction respectively, Δt is the time step, E n+1 (i + 1,j,k) represents the electric field value at the index value (i + 1,j,k) at the (n + 1) - th moment, E n (i + 1,j,k) represents the electric field value at the index value (i + 1,j,k) at the n - th moment, E n+1 (i,j,k) represents the electric field value at the index value (i,j,k) at the (n + 1) - th moment, E n (i,j,k) represents the electric field value at the index value (i,j,k) at the n - th moment.
[0044] For the electric fields at the outermost boundaries of CPML in the y - direction and z - direction, the calculation method is similar to that in the x - direction.
[0045] The beneficial effects of the present invention are:
[0046] A numerical simulation method for electromagnetic waves with a hybrid absorbing boundary. Aiming at the problem that the traditional Mur absorbing boundary condition cannot be directly applied to coordinate - stretching media, the FDTD method is adopted, and the first - order difference form satisfied by the Mur absorbing boundary in coordinate - stretching media is given. Therefore, the hybrid absorbing boundary condition proposed by the present invention can seamlessly combine the Mur absorbing boundary condition and the CPML absorbing boundary, thereby improving the absorption effect.
[0047] A numerical simulation method for electromagnetic waves with a hybrid absorbing boundary. The outermost layer of the CPML absorbing boundary is set as the Mur absorbing boundary condition, and combined with the CPML boundary, a hybrid absorbing boundary condition CMur - CPML is proposed. The finite - difference time - domain (FDTD) method is used to solve the electromagnetic field in coordinate - stretching media, effectively improving the absorption effect of electromagnetic waves. Brief Description of the Drawings
[0048] Figure 1 is a schematic diagram of a two - dimensional hybrid boundary;
[0049] Figure 2 is a schematic diagram of a three - dimensional calculation model of the electromagnetic field radiated by an electric dipole in Example 1;
[0050] Figure 3 When the hybrid absorbing boundary CMur - CPML and the traditional CPML are used as the absorbing boundaries respectively in Example 1, the relative error comparison of E z at point M;
[0051] Figure 4 When the hybrid absorbing boundary CMur - CPML and the traditional CPML are used as the absorbing boundaries respectively in Example 1, the cumulative relative error comparison of E z at point M. Detailed Embodiments
[0052] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.
[0053] A numerical simulation method for hybrid absorbing boundary electromagnetic waves includes the following steps:
[0054] Step 1: Establish a calculation model. When constructing the model boundary, the principle followed is that the Convolutional Perfectly Matched Layer (CPML) is adjacent to the Finite-Difference Time-Domain (FDTD) calculation region, and the Mur absorbing boundary is located at the outermost boundary of the CPML, as Figure 1 shown.
[0055] In a two-dimensional rectangular coordinate system, it is generally a rectangle, and in a three-dimensional coordinate system, it is generally a cube or a cuboid. The FDTD region is adjacent to the CPML region, and the hybrid absorbing boundary is set at the outermost layer of the CPML.
[0056] Step 2: Set the CPML parameters. Different PML parameters affect the absorption effect of the PML region on electromagnetic waves. The model is meshed using regular grids.
[0057] The CPML boundary is used to calculate the electromagnetic field in the coordinate stretching medium, so that no reflection occurs when the electromagnetic wave passes through the junction of the FDTD region boundary and the CPML region boundary. The CPML is also called the coordinate stretching perfectly matched layer, or the Near PML (NPML).
[0058] Step 3: Calculate the electromagnetic field distribution in the model using the FDTD iteration formula and the CPML iteration formula respectively.
[0059] The time step sizes involved in the FDTD iteration formula and the CPML iteration formula used need to satisfy the CFL condition to ensure the stability of the calculation.
[0060] Step 4: Calculate the parameter A, parameter B, and parameter C required for the hybrid boundary.
[0061] The calculation method of the parameters is given by the following formula:
[0062]
[0063] In the above formula, Δx, Δt, and v respectively represent the grid step size, the time step size, and the propagation speed of the electromagnetic wave, k x is the complex frequency shift parameter in the x direction, c x is the attenuation coefficient combination parameter in the x direction, A is the calculation parameter 1, B is the calculation parameter 2, and C is the calculation parameter 3.
[0064] Step 5: Calculate the electric field at the outermost boundary of CPML using the parameter A (parameter one), parameter B (parameter two), and parameter C (parameter three) obtained in Step 4.
[0065] The method for calculating the electromagnetic field at the outermost boundary of CPML is given by the following formula:
[0066] Calculation method for the left boundary:
[0067]
[0068] Calculation method for the right boundary:
[0069]
[0070] In equations (2) and (3), i, j, and k respectively represent the index values in the x, y, and z directions. and α x are respectively the auxiliary variable of the electric field in the x direction and the complex frequency shift factor in the x direction. Δt is the time step, and E n+1 (i + 1, j, k) represents the electric field value at the index value (i + 1, j, k) at the (n + 1)-th moment, and E n (i + 1, j, k) represents the electric field value at the index value (i + 1, j, k) at the n-th moment.
[0071] E n+1 (i, j, k) represents the electric field value at the index value (i, j, k) at the (n + 1)-th moment, and E n (i, j, k) represents the electric field value at the index value (i, j, k) at the n-th moment.
[0072] Step 6: Substitute the electric field value calculated in Step 5 into the FDTD iteration to calculate the magnetic field at the next moment.
[0073] Step 7: Repeat Steps 3 to 6 according to the number of iteration steps to achieve the numerical simulation of electromagnetic waves.
[0074] To calculate the magnetic field at the next moment, the FDTD method is used to solve the Maxwell equations, so as to calculate the magnetic field based on the known electric field.
[0075] Example 1
[0076] The present invention provides a method for seamlessly combining the Mur absorption boundary and the CPML absorption boundary, and proposes a new absorption boundary CMur-CPML. To facilitate a clear understanding of the absorption effect of the absorption boundary proposed by the present invention by those skilled in the art, the present invention is further described by taking the electromagnetic field radiated by an electric dipole in three-dimensional cases as the numerical simulation object.
[0077] Step 1: Establish a cube model with a side length of 40 mm, asFigure 2 As shown in the figure, J in the figure z represents the current density in the z direction. The model medium parameters are selected as those of vacuum medium. An electric dipole is placed at the center of the cube, the direction of the electric dipole moment is the +z direction, the coordinates of the diagnostic position M are (2, 20, 20) (the coordinate unit is mm), and the electric dipole moment p(t) of the electric dipole is given by the following formula:
[0078]
[0079] where T = 26.53 ps.
[0080] Step 2, set the grid size, time step, and CPML parameters. The model is meshed with a cube with a side length of 0.5 mm, the time step is 0.48 ps, the number of CPML layers is 10, k max = 1, α max = 0.2, the polynomial order m = 3, the total number of iteration steps is 4000, and the relative deviation of the electric field in the z direction at the diagnostic point is given by the following formula:
[0081]
[0082] Step 3, use the FDTD iteration formula and the CPML iteration formula to calculate the electromagnetic field distribution in the model.
[0083] Step 4, calculate the parameter A, parameter B, and parameter C required for the mixed boundary.
[0084]
[0085] Step 5, calculate the electric field value at the outermost boundary of the CPML. The calculation method is given by the following formula (similar for the y direction and z direction):
[0086] Calculation method for the left boundary:
[0087]
[0088] Calculation method for the right boundary:
[0089]
[0090] Step 6, calculate the magnetic field at the outermost boundary of the convolutional perfectly matched layer CPML
[0091] Step 7, by repeating steps 3 to 6, realize the numerical simulation of the electromagnetic field radiated by the electric dipole.
[0092] Figure 3 and Figure 4 respectively show E at point M within 4000 time steps zThe relative error and cumulative relative error. The results show that within the entire simulation time range, the relative error and cumulative relative error of CMur-CPML are much smaller than those of CPML, which means that after a long-time numerical simulation, the absorbing boundary of CMur-CPML has a much smaller impact on the field in the computational domain, indicating that the present invention has a better absorption effect than CPML.
Claims
1. A method for numerical simulation of hybrid absorbing boundary electromagnetic waves, characterized in that: The following steps are involved: Step 1: Establish a calculation model: A numerical calculation model of electromagnetic waves with hybrid absorption boundary is established. The principle followed in constructing the boundary of the calculation model is that the convolution perfectly matched layer CPML is adjacent to the finite difference time domain FDTD calculation area, and the Mur absorption boundary is located at the outermost boundary of the CPML. Step 2: Set the convolutional perfectly matched layer CPML parameters: Set the grid size, time step, and convolutional perfectly matched layer (CPML) parameters for the established computational model; use a regular grid to divide the computational model; The convolution perfectly matched layer CPML boundary is used to calculate the electromagnetic field in the coordinate telescopic medium; electromagnetic waves do not reflect when they pass through the junction of the FDTD region boundary and the CPML region boundary; Calculate the relative deviation of the electric field at the diagnosis point; Step 3: Calculate the electromagnetic field distribution in the model: The FDTD iteration method and CPML iteration method are used to calculate the electromagnetic field distribution in the hybrid absorbing boundary electromagnetic wave numerical calculation model. The time steps involved in the FDTD iteration and CPML iteration satisfy the CFL condition; Step 4, calculate the parameters A, B and C required for the mixed boundary: Step 5, calculate the electric field at the outermost boundary of the convolution perfectly matched layer CPML: Calculate the electric field at the outermost boundary of the convolution perfectly matched layer CPML in the x, y and z directions respectively; Step 6, calculate the magnetic field at the next moment: The electric field value of the outermost boundary of the CPML is brought into the FDTD iteration, and the Maxwell equation is solved using the FDTD method to obtain the magnetic field at the next moment; Step 7, Iteration: Repeat steps 3 to 6, iterating according to the set number of iteration steps to achieve the numerical simulation of the mixed absorption boundary electromagnetic waves.
2. The hybrid absorbing boundary electromagnetic wave numerical simulation method according to claim 1, characterized in that: The step 1 establishes a calculation model. In a two-dimensional rectangular coordinate system, the calculation model is a rectangle; in a three-dimensional coordinate system, the calculation model is a cube or a cuboid; the FDTD area is adjacent to the CPML area.
3. The hybrid absorbing boundary electromagnetic wave numerical simulation method according to claim 1, characterized in that: The step 1 further comprises: Establish a cube model with a side length of 40 mm, and select the vacuum medium parameters as the model medium parameters; place the electric dipole at the center of the cube, with the electric dipole moment direction in the +z direction, and the coordinates of the diagnostic position M are (2,20,20), with the coordinate unit in mm. The electric dipole moment p(t) of the electric dipole is given by the following formula: In formula (4), T is the radiation source parameter, T = 26.53 ps, and t is the numerical simulation time.
4. The method for numerical simulation of hybrid absorbing boundary electromagnetic waves according to claim 3, characterized in that: Step 2 further comprises: The computational model was meshed using a cube with a side length of 0.5 mm, a time step Δt of 0.48 ps, 10 CPML layers, and a stretch factor k. max =1, complex frequency shift factor α max = 0.2, the polynomial order m = 3, the total number of iterations is 4000, and the relative deviation of the electric field in the z direction at the diagnosis point is given by the following formula: In formula (5), Error(t) is the relative deviation of the electric field in the z direction at time t, is the diagnostic electric field value of the electric field in the z direction at time t with a truncation boundary, is the reference electric field value of the electric field in the z direction in free space at time t, is the maximum value of the reference electric field in the z direction of free space.
5. The hybrid absorbing boundary electromagnetic wave numerical simulation method according to claim 1, characterized in that: The step 4 further comprises: The calculation method of the mixed boundary related parameters is given by the following formula: In formula (1), Δx, Δt and v represent the grid step, time step and propagation speed of electromagnetic waves respectively, k x is the complex frequency shift parameter in the x direction, c x is the combined parameter of the attenuation coefficient in the x direction, A is the calculation parameter one, B is the calculation parameter two, and C is the calculation parameter three.
6. The hybrid absorbing boundary electromagnetic wave numerical simulation method according to claim 1, characterized in that: The step 5 further comprises: Calculate the electric field at the outermost boundary of the convolution perfectly matched layer CPML. For the x direction, the calculation method is as follows: The calculation method for the left boundary is: Right boundary calculation method: In formulas (2) and (3), i, j, and k represent the index values in the x, y, and z directions respectively. and α x are the auxiliary variables of the electric field in the x direction and the complex frequency shift factor in the x direction, Δt is the time step, E n+1 (i+1,j,k) represents the electric field value at time n+1 with index (i+1,j,k), E n (i+1,j,k) represents the electric field value at time n with index value (i+1,j,k), E n+1 (i,j,k) represents the electric field value at time n+1 with index (i,j,k), E n (i,j,k) represents the electric field value at time n with index value (i,j,k); For the electric field at the outermost boundary of the CPML in the y and z directions, the calculation method is similar to that in the x direction.