An efficient electromagnetic simulation method for anisotropic media

Abstract

This invention discloses an efficient electromagnetic simulation method for anisotropic media, belonging to the field of electromagnetic simulation technology. The method of this invention applies local time-stepping technology to the DGTD anisotropic media scattering simulation method to efficiently obtain high-precision simulation results. While ensuring accuracy, it adaptively optimizes the time step based on the mesh size, significantly reducing the total number of iterations, thereby further improving computational efficiency compared to existing methods.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation technology, specifically relating to an efficient electromagnetic simulation method for anisotropic media. Background Technology

[0002] In computational electromagnetics, the study of electromagnetic problems in anisotropic media is of great significance. Traditional methods for solving anisotropic media problems include semi-analytical methods and frequency domain methods. Semi-analytical methods can only be applied to simple electromagnetic models, limiting their scope of application. While frequency domain methods, such as VIE, VSIE, and FEM, can be used for complex models, their efficiency is low due to the need to solve large matrix equations, especially when solving broadband electromagnetic problems, where solving at each frequency point further reduces the efficiency of solving broadband problems.

[0003] The novel discontinuous Galerkin time-domain (DGTD) method can be used to analyze electromagnetic scattering problems in fully tensor anisotropic media. Its core advantages include: high versatility, directly calculating tensor wave impedance by solving eigenvalue problems without local coordinate transformations, and uniformly handling lossy / lossless and fully tensor anisotropic media; broadband efficiency, as a time-domain method, obtaining wide-bandgap scattering characteristics in a single simulation; strong geometric adaptability, using tetrahedral mesh discretization to accurately simulate complex targets; high accuracy, introducing second-order nodal basis functions to significantly reduce memory and computation time; and further improvement in modeling simplicity and computational performance by combining incident wave direct loading (IWDL) and PML absorbing boundaries. Numerical examples verify that this method outperforms traditional frequency-domain methods and earlier DGTD implementations in terms of accuracy, stability, and computational efficiency. However, the method uses the traditional leapfrog time step, employing the same time step throughout the iteration process, resulting in relatively low efficiency, especially for multi-scale electromagnetic problems. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide an efficient electromagnetic simulation method for anisotropic media. It applies local time stepping technology to the DGTD anisotropic media scattering simulation method, and adaptively optimizes the time step based on the mesh size while ensuring accuracy, which significantly reduces the total number of iterations and greatly improves the computational efficiency.

[0005] The technical problem addressed by this invention is solved as follows:

[0006] An efficient electromagnetic simulation method for anisotropic media includes the following steps:

[0007] Step 1: Discretize the electromagnetic scattering target and the surrounding half-wavelength space into a three-dimensional mesh to generate a tetrahedral element mesh set. The geometric parameters of each tetrahedral element mesh include volume and surface area, and the corresponding material properties are represented by the dielectric constant tensor and permeability tensor in full tensor form to characterize the anisotropic medium. The wave impedance and waveguide admittance in the anisotropic medium are introduced based on the dielectric constant tensor and permeability tensor.

[0008] Step 2: For each tetrahedral element mesh, calculate the local maximum stable time step that satisfies the numerical stability condition based on its own geometric parameters and material properties.

[0009] Step 3: Starting from Maxwell's equations, derive the DGTD time-domain stepping equations for anisotropic media; use the tetrahedral element node basis functions as the basis functions for the test function and electromagnetic field components, and expand the electric field component, magnetic field component, and test function respectively; substitute the expanded forms of the electric field component, magnetic field component, and test function into the DGTD time-domain stepping equations for anisotropic media, and write the differential with respect to time as a difference scheme to obtain the stepping expressions for the electric field component and magnetic field component;

[0010] Step 4: Set the minimum value of the local maximum stable time step of all tetrahedral element meshes as the global base time step; based on the relationship between the local maximum stable time step of each tetrahedral element mesh and the global base time step, divide the time steps on all tetrahedral elements into three groups, forming three regions with different simulation fineness.

[0011] Step 5: Introduce a local time stepping strategy. Different time steps are used for different regions. Based on the stepping expressions of electric field components and magnetic field components, perform differentiated time stepping updates to achieve asynchronous time stepping.

[0012] Step 6: Repeat the asynchronous time stepping process of Step 5 until the set total simulation time or error meets the set threshold requirements; during or after the time iteration, calculate and output the broadband electromagnetic scattering characteristics of the anisotropic medium target, including the radar cross section, through field transformation from the time domain to the frequency domain.

[0013] Furthermore, in step 1, the average mesh size for three-dimensional mesh discretization is one-tenth of a wavelength.

[0014] Furthermore, the specific process of step 1 is as follows:

[0015] The electromagnetic scattering target to be solved and the surrounding space within a half-wavelength range are taken as the entire solution domain V, which is divided into N tetrahedral element meshes, i.e.:

[0016]

[0017] Where 1≤q≤N, This represents the q-th tetrahedral element mesh;

[0018] For anisotropic media, the electric displacement vector D and the electric field intensity E, and the magnetic induction intensity B and the magnetic field intensity H satisfy the following constitutive relation:

[0019]

[0020]

[0021] in, and , respectively, are the tensor coefficients of the permittivity and permeability. and These are the relative permittivity tensor and the permeability tensor, respectively. and These are the vacuum permittivity and vacuum permeability, respectively, expressed as follows:

[0022] ,

[0023] in, This represents the electric displacement component produced by the electric field in the m direction in the p direction. Let m represent the magnetic induction intensity component produced by the magnetic field in the direction m in the direction p, where 1 ≤ p ≤ 3 and 1 ≤ m ≤ 3.

[0024] Introducing wave impedance and waveguide admittance in anisotropic media:

[0025]

[0026]

[0027] in, and Let these represent the wave impedance and waveguide admittance of the current tetrahedral element e, respectively. and represents the tensor coefficients of the dielectric constant and permeability of the current tetrahedral element e, respectively.

[0028] Furthermore, the specific process of step 2 is as follows:

[0029] For the tetrahedral element e, the local maximum stable time step that satisfies the numerical stability condition is... for:

[0030]

[0031] in, and Let be the volume and total surface area of ​​the current tetrahedral element e, respectively. and represents the average trace of the permeability and permittivity of the current tetrahedral element e, respectively; the subscript e+ indicates the mesh of the adjacent tetrahedral elements of the current tetrahedral element e. and Let represent the average traces of the permeability and dielectric constant of the tetrahedral element e+, respectively, and max denotes the maximum value.

[0032] Furthermore, in step 3, the Maxwell equations are solved using the DGTD method to obtain the following DGTD time-domain step equations for the anisotropic medium:

[0033]

[0034]

[0035]

[0036]

[0037] in, For the test function, and Let these represent the electric and magnetic fields of the current tetrahedral element e, respectively. This indicates taking the derivative with respect to time. To express the curl, It is a volume fraction; and Let e ​​and e+ represent the wave impedances of the tetrahedral elements e and e+, respectively. Let e ​​be the unit outward normal vector pointing from tetrahedral element e to tetrahedral element e+. and These represent the electric and magnetic fields of the tetrahedral element e+, respectively. For the area infinitesimal component, Here is the dissipation coefficient. and Let e ​​and e+ represent the waveguide admittances of the tetrahedral elements e and e+, respectively.

[0038] Electric field of tetrahedral unit e and magnetic field They are represented as follows:

[0039]

[0040] in, , and These represent the unit vectors in the x, y, and z directions, respectively. , and Let x, y, and z represent the components of the electric field, respectively. , and These represent the x-direction, y-direction, and z-direction components of the magnetic field, respectively.

[0041] Furthermore, in step 3, the tetrahedral element nodal basis functions are used as the basis functions for the test function and the electromagnetic field components, and the expansion is performed. For the x-direction component, the expansion is expressed as:

[0042]

[0043]

[0044] in, , and Let x, y, and z represent the electric field components at the j-th node of the tetrahedral element e, respectively. , and Let X represent the x-direction, y-direction, and z-direction magnetic field components of the j-th node on the tetrahedral element e, respectively. and These represent the basis functions for the electromagnetic field components and the basis functions for the test function, respectively.

[0045] Substituting the expanded forms of the electric field components, magnetic field components, and the test function into the DGTD time-domain stepping equations for anisotropic media, we obtain the stepping expressions for the electric and magnetic field components. For the x-direction components, the difference scheme expression of the stepping equations is:

[0046]

[0047]

[0048] in, and Let n represent the electric field in the x-direction of the tetrahedral element e at time n and time n+1, respectively. and Let n and n represent the x-direction magnetic fields of the tetrahedral element e at times n-0.5 and n+0.5, respectively.

[0049] mass matrix for:

[0050]

[0051] Stiffness matrices in the x, y, and z directions , and They are respectively:

[0052]

[0053]

[0054]

[0055] in, , and Let x, y, and z represent the partial derivatives, respectively.

[0056] The flux matrix of the current tetrahedral element e and its adjacent faces to the k-th adjacent tetrahedral element e+ and They are represented as follows:

[0057]

[0058]

[0059] Where 1≤k≤4, Let represent the electromagnetic basis function of the j-th node of the k-th adjacent tetrahedral element e+. This represents the adjacent faces of the current tetrahedral element e and the kth adjacent tetrahedral element e+;

[0060] In the step equation difference scheme expression, , , , , , , , , , , , , , , , , and Determined by the mass matrix, stiffness matrix, and flux matrix, specifically expressed as:

[0061]

[0062] in, and Let represent the magnetic fields in the y and z directions of the tetrahedral element e at time n+0.5, respectively;

[0063]

[0064] in, , and Let x, y, and z represent the components of the normal vector between the current tetrahedral element e and its k-th neighboring tetrahedral element, respectively. , and Let x, y, and z represent the magnetic fields in the x-direction, y-direction, and z-direction of the k-th adjacent tetrahedral element e+ at time n+0.5, respectively.

[0065]

[0066] in, , and Let x, y, and z represent the electric fields in the x-direction, y-direction, and z-direction of the k-th adjacent tetrahedral element e+ at time n, respectively. and Let represent the electric fields in the y-direction and z-direction of the tetrahedral element e at time n, respectively;

[0067]

[0068]

[0069]

[0070] in, , and Let x, y, and z represent the magnetic fields in the x, y, and z directions of the k-th adjacent tetrahedral element e+ at time n-0.5, respectively. , and Let x, y, and z represent the magnetic fields of the tetrahedral element e at time n-0.5.

[0071] Furthermore, in step 4, according to The multiple relationship between the tetrahedral elements and the global base time step Δt divides all tetrahedral elements into three regions:

[0072] Region 1: △t e Satisfying △t≤△t e <2△t;

[0073] Region 2: △t e Satisfying 2△t≤△t e <4△t;

[0074] Region 3: △t e Satisfying △t e ≥4△t.

[0075] Furthermore, the specific process of step 5 is as follows:

[0076] The electric and magnetic field components in region 1 are updated once every Δt.

[0077] The electric and magnetic field components in region 2 are updated every 2Δt.

[0078] The electric and magnetic field components in region 3 are updated every 4Δt.

[0079] Furthermore, in step 5, for the electric field component in the x-direction, the step equation difference scheme expression for different regions is as follows:

[0080] Area 1:

[0081]

[0082] Area 2:

[0083]

[0084] Area 3:

[0085]

[0086] The time step length of one overall cycle is 4Δt, and the subscript in the above formula is... , , , , They represent the electric field at , , , , The field values ​​at these five moments.

[0087] The beneficial effects of this invention are:

[0088] This invention provides an efficient simulation method for anisotropic media, which applies local time stepping technology to the DGTD anisotropic media scattering simulation method to efficiently obtain high-precision simulation results. While ensuring accuracy, the time step is adaptively optimized based on the mesh size, which significantly reduces the total number of iterations, thereby further improving the computational efficiency on the basis of the original method. Attached Figure Description

[0089] Figure 1 The diagrams shown are the model diagram and the mesh partition diagram in the method described in the embodiment, where (a) is the front view, (b) is the side view, (c) is the top view, (d) is the bottom view, and (e) is the model mesh.

[0090] Figure 2This is a comparison chart of the RCS calculation results of the method described in the embodiment and the prior art. Detailed Implementation

[0091] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0092] This embodiment provides an efficient electromagnetic simulation method for anisotropic media. Taking the electromagnetic scattering of an amygdala coating model as an example, the schematic diagram of the model is shown below. Figure 1 As shown, (a) is the front view, (b) is the side view, (c) is the top view, (d) is the bottom view, and (e) is the model mesh, including the following steps:

[0093] Step 1: Discretize the electromagnetic scattering target and its surrounding space (at a distance of half a wavelength from the target edge) into a three-dimensional mesh to generate a tetrahedral element mesh set with an average mesh size of one-tenth of a wavelength.

[0094] The geometric parameters of each tetrahedral element mesh include volume and surface area, and the corresponding material properties are represented by dielectric constant tensors and permeability tensors in full tensor form to characterize anisotropic media.

[0095] Furthermore, the specific process of step 1 is as follows:

[0096] The entire solution domain V is divided into N grids. ,Right now:

[0097]

[0098] For anisotropic media, the electric displacement vector D and the electric field strength E, and the magnetic induction intensity B and the magnetic field strength H will no longer have a simple linear proportional relationship, but will become the following constitutive relationship:

[0099]

[0100]

[0101] in, and These are the tensor coefficients of the dielectric constant and permeability. and These are the relative permittivity tensor and the permeability tensor. and These are the vacuum permittivity and vacuum permeability, and their specific expressions are as follows:

[0102] ,

[0103] in, This represents the electric displacement component produced by the electric field in the m direction in the p direction. Let p represent the magnetic induction intensity component produced by a magnetic field in the m direction in the p direction, where 1 ≤ p ≤ 3 and 1 ≤ m ≤ 3; where diagonal elements (p = m) represent responses in the same direction and off-diagonal elements (p ≠ m) represent cross-coupling between different directions.

[0104] Introducing wave impedance and waveguide admittance in anisotropic media:

[0105]

[0106]

[0107] in, and Let these represent the wave impedance and waveguide admittance of the current tetrahedral element e, respectively. and represents the tensor coefficients of the dielectric constant and permeability of the current tetrahedral element e, respectively.

[0108] Step 2: For each discrete tetrahedral element mesh generated in Step 1, calculate the local maximum stable time step that satisfies the numerical stability condition based on its own geometric parameters and material properties.

[0109] Furthermore, the specific process of step 2 is as follows:

[0110] In the traditional frog-leap strategy's DGTD method, the time step of each discrete tetrahedral cell is... Determined by the following formula:

[0111]

[0112] in, and Let be the volume and total surface area of ​​the current tetrahedral element e, respectively. and represents the average trace of the permeability and permittivity of the current tetrahedral element e, respectively; the subscript e+ indicates the mesh of the adjacent tetrahedral elements of the current tetrahedral element e. and Let represent the average traces of the permeability and dielectric constant of the tetrahedral element e+, respectively, and max denotes the maximum value.

[0113] Step 3: Starting from Maxwell's equations, derive the DGTD time-domain step formula for anisotropic media;

[0114] Furthermore, the specific process of step 3 is as follows:

[0115] Solving Maxwell's equations using the DGTD method yields the following step equations:

[0116]

[0117]

[0118]

[0119]

[0120] in, For the test function, and Let these represent the electric and magnetic fields of the current tetrahedral element e, respectively. This indicates taking the derivative with respect to time. To express the curl, It is a volume fraction; and Let e ​​and e+ represent the wave impedances of the tetrahedral elements e and e+, respectively. Let e ​​be the unit outward normal vector pointing from tetrahedral element e to tetrahedral element e+. and These represent the electric and magnetic fields of the tetrahedral element e+, respectively. For the area infinitesimal component, Here is the dissipation coefficient. and Let e ​​and e+ represent the waveguide admittance of the tetrahedral elements e and e+, respectively.

[0121] Electric field of tetrahedral unit e and magnetic field They are represented as follows:

[0122]

[0123] in, , and These represent the unit vectors in the x, y, and z directions, respectively. , and Let x, y, and z represent the components of the electric field, respectively. , and These represent the x-direction, y-direction, and z-direction components of the magnetic field, respectively.

[0124] The tetrahedral element nodal basis functions are used as the basis functions for both the test function and the electromagnetic field components for expansion. Taking the x-direction component as an example, the expansion is as follows:

[0125]

[0126]

[0127] in, , and Let x, y, and z represent the electric field components at the j-th node of the tetrahedral element e, respectively. , and Let X represent the x-direction, y-direction, and z-direction magnetic field components of the j-th node on the tetrahedral element e, respectively. and These represent the basis functions of the electromagnetic field components and the basis functions of the test function, respectively.

[0128] Substituting the expanded forms of the test function and the electric field basis function into the electric and magnetic field step equations, and writing the differential with respect to time in a difference scheme, we obtain the expressions corresponding to the three components of the electromagnetic field.

[0129] Taking the x-direction component as an example, the corresponding step equation difference scheme expression is:

[0130]

[0131]

[0132] in, and Let n represent the electric field in the x-direction of the tetrahedral element e at the current time n and the next time n+1, respectively. and Let n represent the x-direction magnetic field of tetrahedral element e at the current time n-0.5 and the next time n+0.5, respectively;

[0133] mass matrix for:

[0134]

[0135] Stiffness matrices in the x, y, and z directions , and They are respectively:

[0136]

[0137]

[0138]

[0139] in, , and Let x, y, and z represent the partial derivatives, respectively.

[0140] The flux matrix of the current tetrahedral element e and its adjacent faces to the k-th adjacent tetrahedral element e+ and They are represented as follows:

[0141]

[0142]

[0143] Where 1≤k≤4, Let represent the electromagnetic basis function of the j-th node of the k-th adjacent tetrahedral element e+. This represents the adjacent faces of the current tetrahedral element e and the kth adjacent tetrahedral element e+;

[0144] In the step equation difference scheme expression, , , , , , , , , , , , , , , , , and Determined by the mass matrix, stiffness matrix, and flux matrix, specifically expressed as:

[0145]

[0146] in, and These represent the magnetic fields in the y and z directions of the tetrahedral element e at the current time n+0.5, respectively.

[0147]

[0148] in, , and Let x, y, and z represent the components of the normal vector between the current tetrahedral element e and its k-th neighboring tetrahedral element, respectively. , and These represent the magnetic fields in the x, y, and z directions of the kth adjacent tetrahedral element e+ at the current time n+0.5, respectively.

[0149]

[0150] in, , and Let x, y, and z represent the electric fields in the x-direction, y-direction, and z-direction of the k-th adjacent tetrahedral element e+ at the current time n, respectively. and Let represent the electric fields in the y-direction and z-direction of the tetrahedral element e at the current time n, respectively;

[0151]

[0152]

[0153]

[0154] in, , and Let x, y, and z represent the magnetic fields in the x, y, and z directions of the k-th adjacent tetrahedral element e+ at the previous time n-0.5, respectively. , and Let x, y, and z represent the magnetic fields of the tetrahedral element e at the previous time n-0.5, respectively.

[0155] As can be seen from the above equation, the three components of the electromagnetic field will no longer be determined solely by the original components; the mass matrix and stiffness matrix must be correlated with... and When multiplying, the terms corresponding to the flux matrix must be multiplied by the tensor coefficients. , , and Multiply.

[0156] Step 4: Set a global base time step (usually the minimum of all local maximum time steps), and divide the time steps on all tetrahedral elements into several groups (e.g., Δt, 2Δt, 4Δt). Use the base time step on finer elements, and use several times the base time step on larger elements.

[0157] Furthermore, the specific process of step 4 is as follows:

[0158] Calculate the maximum allowable stable step size Δt for each tetrahedral element. e According to △t e Based on the multiple relationship with the reference step size Δt, all elements are divided into three regions:

[0159] Region 1 (Finest): Contains cells that can only withstand very small step sizes, where Δt e Satisfying △t≤△t e<2△t. These are typically meshes located at sharp edges and in areas with fine structures.

[0160] Region 2 (Medium): Contains units that can withstand slightly larger step sizes, with Δt e Satisfying 2△t≤△t e <4△t.

[0161] Region 3 (largest): Contains elements that can withstand larger step sizes, whose Δt e Satisfying △t e ≥4△t. These are typically meshes located in smooth, large-volume regions.

[0162] Step 5: Introduce a local time stepping strategy, perform differentiated time stepping updates, and realize asynchronous time stepping;

[0163] Furthermore, the specific process of step 5 is as follows:

[0164] Perform differentiated time-step updates

[0165] This step is the "execution phase" of the local time stepping strategy, which optimizes the computational load.

[0166] Assigning step size: Each region uses a fixed step size that matches the grouping for field updates.

[0167] The field (E1 / H1) in region 1 is updated once every Δt.

[0168] The field (E2 / H2) in region 2 is updated every 2Δt.

[0169] The field (E3 / H3) in region 3 is updated every 4Δt.

[0170] Taking the electric field x-component as an example, the step equation difference scheme expression for different regions is as follows:

[0171] Area 1:

[0172]

[0173] Area 2:

[0174]

[0175] Area 3:

[0176]

[0177] The expressions for the y and z components are similar to those for the x component.

[0178] The time step length of one overall cycle is 4Δt, and the subscript in the above formula is... , , , , They represent the electric field at , , , , The field values ​​at these five moments.

[0179] Coordination of rhythm: As shown in Table 1, this creates a "rhythm difference" in updates. Within a cycle of one update (4Δt) for region 3, region 2 updates twice, and region 1 updates four times. This directly reduces the total number of field updates for regions 2 and 3, thus saving a significant amount of computation.

[0180] Table 1. Coordination rhythm of the method described in this embodiment.

[0181]

[0182] Step 6: Repeat the asynchronous time progression process of Step 5 until the set total simulation time or error meets the set threshold requirement. During or after the time iteration, calculate and output the target's broadband electromagnetic scattering characteristics, such as the radar cross section, through a field transformation from the time domain to the frequency domain.

[0183] Furthermore, the specific process of step 6 is as follows:

[0184] This step ensures that calculations proceed within a consistent timeframe and provides results for subsequent analysis.

[0185] Synchronization Time: Global time is advanced in units of a complete maximum step size cycle (4△t). That is, after each complete cycle containing four updates of region 1, two updates of region 2, and one update of region 3, the current time t is updated to t+4△t.

[0186] Record the results: At this synchronized time point, record the electromagnetic field coefficients for all regions. Since this is a time-domain simulation, these field values ​​will be used to perform a Fourier transform to obtain the frequency-domain scattering characteristics (such as RCS), for example... Figure 2 As shown.

[0187] To demonstrate the effectiveness of this invention, the amygdala coating RCS (red line) calculated in step 6 is compared with the calculation results of the commercial electromagnetic simulation software FEKO (circle) and the traditional frog-jump time-step DGTD method. Figure 2 As shown in the figure, the present invention also has high calculation accuracy for more complex targets.

[0188] The method described in this embodiment was used to calculate the locally coated spherical target, and the calculation time was statistically analyzed and compared with that of existing methods. The comparison results are shown in Table 2. As can be seen from the table, the calculation time required by the present invention is significantly reduced compared with traditional algorithms and existing electromagnetic simulation software FEKO.

[0189] Table 2 Comparison of Calculation Time for Different Methods

[0190]

[0191] The above description is merely a specific embodiment of the present invention. Any feature disclosed in this specification may be replaced by other equivalent or similar features unless otherwise specified. All disclosed features, or steps in all methods or processes, may be combined in any way except for mutually exclusive features and / or steps.

Claims

1. A highly efficient electromagnetic simulation method for anisotropic media, characterized in that, Includes the following steps: Step 1: Discretize the electromagnetic scattering target and the surrounding half-wavelength space into a three-dimensional mesh to generate a tetrahedral element mesh set. The geometric parameters of each tetrahedral element mesh include volume and surface area, and the corresponding material properties are represented by the dielectric constant tensor and permeability tensor in full tensor form to characterize the anisotropic medium. The wave impedance and waveguide admittance in the anisotropic medium are introduced based on the dielectric constant tensor and permeability tensor. Step 2: For each tetrahedral element mesh, calculate the local maximum stable time step that satisfies the numerical stability condition based on its own geometric parameters and material properties. Step 3: Starting from Maxwell's equations, derive the DGTD time-domain stepping equations for anisotropic media; use tetrahedral element node basis functions as basis functions for the test function and electromagnetic field components, and expand the electric field component, magnetic field component, and test function respectively. Substituting the expansion forms of the electric field components, magnetic field components, and test function into the DGTD time-domain stepping equation for anisotropic media, and writing the differential with respect to time in a difference scheme, we obtain the stepping expressions for the electric field components and magnetic field components. Step 4: Set the minimum value of the local maximum stable time step of all tetrahedral element meshes as the global base time step; based on the relationship between the local maximum stable time step of each tetrahedral element mesh and the global base time step, divide the time steps on all tetrahedral elements into three groups, forming three regions with different simulation fineness. Step 5: Introduce a local time stepping strategy. Different time steps are used for different regions. Based on the stepping expressions of electric field components and magnetic field components, perform differentiated time stepping updates to achieve asynchronous time stepping. Step 6: Repeat the asynchronous time stepping process of Step 5 until the set total simulation time or error meets the set threshold requirements; during or after the time iteration, calculate and output the broadband electromagnetic scattering characteristics of the anisotropic medium target, including the radar cross section, through field transformation from the time domain to the frequency domain.

2. The efficient electromagnetic simulation method for anisotropic media according to claim 1, characterized in that, In step 1, the average mesh size for three-dimensional mesh discretization is one-tenth of a wavelength.

3. The efficient electromagnetic simulation method for anisotropic media according to claim 1, characterized in that, The specific process of step 1 is as follows: The electromagnetic scattering target to be solved and the surrounding space within a half-wavelength range are taken as the entire solution domain V, which is divided into N tetrahedral element meshes, i.e.: Where 1≤q≤N, This represents the q-th tetrahedral element mesh; For anisotropic media, the electric displacement vector D and the electric field intensity E, and the magnetic induction intensity B and the magnetic field intensity H satisfy the following constitutive relation: in, and , respectively, are the tensor coefficients of the permittivity and permeability. and These are the relative permittivity tensor and the permeability tensor, respectively. and These are the vacuum permittivity and vacuum permeability, respectively, expressed as follows: ， in, This represents the electric displacement component produced by the electric field in the m direction in the p direction. Let m represent the magnetic induction intensity component produced by the magnetic field in the direction m in the direction p, where 1 ≤ p ≤ 3 and 1 ≤ m ≤ 3. Introducing wave impedance and waveguide admittance in anisotropic media: in, and Let these represent the wave impedance and waveguide admittance of the current tetrahedral element e, respectively. and represents the tensor coefficients of the dielectric constant and permeability of the current tetrahedral element e, respectively.

4. The efficient electromagnetic simulation method for anisotropic media according to claim 1, characterized in that, The specific process of step 2 is as follows: For the tetrahedral element e, the local maximum stable time step that satisfies the numerical stability condition is... for: in, and Let be the volume and total surface area of ​​the current tetrahedral element e, respectively. and represents the average trace of the permeability and permittivity of the current tetrahedral element e, respectively; the subscript e+ indicates the adjacent tetrahedral element mesh of the current tetrahedral element e. and Let represent the average traces of the permeability and dielectric constant of the tetrahedral element e+, respectively, and max denotes the maximum value.

5. The efficient electromagnetic simulation method for anisotropic media according to claim 1, characterized in that, In step 3, the Maxwell equations are solved using the DGTD method to obtain the following DGTD time-domain step equations for anisotropic media: in, For the test function, and Let these represent the electric and magnetic fields of the current tetrahedral element e, respectively. This indicates taking the derivative with respect to time. To express the curl, It is a volume fraction; and Let e ​​and e+ represent the wave impedances of the tetrahedral elements e and e+, respectively. Let e ​​be the unit outward normal vector pointing from tetrahedral element e to tetrahedral element e+. and These represent the electric and magnetic fields of the tetrahedral element e+, respectively. For the area infinitesimal component, Here is the dissipation coefficient. and Let e ​​and e+ represent the waveguide admittances of the tetrahedral elements e and e+, respectively. Electric field of tetrahedral unit e and magnetic field They are represented as follows: in, , and These represent the unit vectors in the x, y, and z directions, respectively. , and Let x, y, and z represent the components of the electric field, respectively. , and These represent the x-direction, y-direction, and z-direction components of the magnetic field, respectively.

6. The efficient electromagnetic simulation method for anisotropic media according to claim 5, characterized in that, In step 3, the tetrahedral element nodal basis functions are used as the basis functions for the test function and the electromagnetic field components, and the expansion is performed. For the x-direction component, the expansion is expressed as: in, , and Let x, y, and z represent the electric field components at the j-th node of the tetrahedral element e, respectively. , and Let X represent the x-direction, y-direction, and z-direction magnetic field components of the j-th node on the tetrahedral element e, respectively. and These represent the basis functions for the electromagnetic field components and the basis functions for the test function, respectively. Substituting the expanded forms of the electric field components, magnetic field components, and the test function into the DGTD time-domain stepping equations for anisotropic media, we obtain the stepping expressions for the electric and magnetic field components. For the x-direction components, the difference scheme expression of the stepping equations is: in, and Let n represent the electric field in the x-direction of the tetrahedral element e at time n and time n+1, respectively. and Let n and n represent the x-direction magnetic fields of the tetrahedral element e at times n-0.5 and n+0.5, respectively. mass matrix for: Stiffness matrices in the x, y, and z directions , and They are respectively: in, , and Let x, y, and z represent the partial derivatives, respectively. The flux matrix of the current tetrahedral element e and its adjacent faces to the k-th adjacent tetrahedral element e+ and They are represented as follows: Where 1≤k≤4, Let represent the electromagnetic basis function of the j-th node of the k-th adjacent tetrahedral element e+. This represents the adjacent faces of the current tetrahedral element e and the kth adjacent tetrahedral element e+; In the step equation difference scheme expression, , , , , , , , , , , , , , , , , and Determined by the mass matrix, stiffness matrix, and flux matrix, specifically expressed as: in, and Let represent the magnetic fields in the y and z directions of the tetrahedral element e at time n+0.5, respectively; in, , and Let x, y, and z represent the components of the normal vector between the current tetrahedral element e and its k-th neighboring tetrahedral element, respectively. , and Let x, y, and z represent the magnetic fields in the x-direction, y-direction, and z-direction of the k-th adjacent tetrahedral element e+ at time n+0.5, respectively. in, , and Let x, y, and z represent the electric fields in the x-direction, y-direction, and z-direction of the k-th adjacent tetrahedral element e+ at time n, respectively. and Let represent the electric fields in the y-direction and z-direction of the tetrahedral element e at time n, respectively; in, , and Let x, y, and z represent the magnetic fields in the x-direction, y-direction, and z-direction of the k-th adjacent tetrahedral element e+ at time n-0.5, respectively. , and Let x, y, and z represent the magnetic fields of the tetrahedral element e at time n-0.

5.

7. The efficient electromagnetic simulation method for anisotropic media according to claim 4, characterized in that, In step 4, according to The multiple relationship between the tetrahedral elements and the global base time step Δt divides all tetrahedral elements into three regions: Region 1: △t e Satisfying △t≤△t e <2△t; Region 2: △t e Satisfying 2△t≤△t e <4△t; Region 3: △t e Satisfying △t e ≥4△t.

8. The efficient electromagnetic simulation method for anisotropic media according to claim 7, characterized in that, The specific process of step 5 is as follows: The electric and magnetic field components in region 1 are updated once every Δt. The electric and magnetic field components in region 2 are updated every 2Δt. The electric and magnetic field components in region 3 are updated every 4Δt.

9. The efficient electromagnetic simulation method for anisotropic media according to claim 6, characterized in that, In step 5, for the electric field component in the x-direction, the step equation difference scheme expression for different regions is as follows: Area 1: Area 2: Area 3: The time step length of one overall cycle is 4Δt, and the subscript in the above formula is... , , , , They represent the electric field at , , , , The field values ​​at these five moments.

Images