Electromagnetic wave finite-difference time-domain method and system based on anisotropic media conditions

Abstract

The present invention discloses an electromagnetic wave finite-difference time-domain (FDTD) method and system based on anisotropic media conditions, relating to the field of electromagnetic detection. Starting from Maxwell's equations, the method involves the constitutive relationship of the medium in tensor representation, which is given anisotropic properties. The method creatively introduces transverse wave vector parameters into the classic one-dimensional FDTD method, replacing the transverse partial derivatives in Maxwell's equations that do not belong to the propagation direction with the corresponding transverse wave vector parameters. This method can realize a one-dimensional solution for two-dimensional or three-dimensional electromagnetic problems. By setting the corresponding transverse wave vector parameters, the field intensity distribution of the electromagnetic wave can also be calculated under different conditions of normal incidence or oblique incidence.

Description

Technical Field

[0001] The present invention relates to the fields of computational electromagnetics and electromagnetic detection of layered media, and in particular to a quasi-one-dimensional electromagnetic wave finite-difference time-domain method and system under anisotropic media conditions. Background Art

[0002] There are three main categories of methods commonly used to solve electromagnetic field problems, each of which can be further divided into several methods. The first category is analytical methods, with perturbation methods, variational methods, and boundary element methods being common. The second category is semi-analytical numerical methods, such as the semi-analytical method, weighted residual method, and method of moments. The third category is numerical methods. Currently, commonly used numerical methods in computational electromagnetics include the finite difference time domain method (FDTD), the transmission line matrix method (TLM), and the finite element method (FEM). The analytical portion of the method of moments is relatively simple, but its high computational workload makes it very difficult to solve large-scale and complex objects. The finite element method has the advantages of versatility and flexibility. However, for the simulation of complex electromagnetic systems, as the discretization interval decreases, the size of the algebraic matrix equations generated by this method increases, leading to increased errors in the numerical calculation results. Therefore, this method also has limitations. FDTD has always been a research hotspot due to its simplicity and high efficiency. It is one of the core algorithms in computational electromagnetics. Its theoretical system is relatively mature, but there are still many issues that need further research. Summary of the Invention

[0003] The purpose of the present invention is to provide an electromagnetic wave finite-difference time-domain method and system based on anisotropic medium conditions, which can be applied to the electromagnetic wave propagation process and field quantity calculation in layered anisotropic media, and is also applicable to the numerical rapid calculation of normal-incidence and oblique-incidence electromagnetic waves in two-dimensional or three-dimensional situations.

[0004] To achieve the above object, the present invention provides the following solutions:

[0005] The present invention provides an electromagnetic wave finite-difference time-domain method based on anisotropic medium conditions, characterized in that the method includes:

[0006] Based on the Maxwell curl equation for electromagnetic waves, a medium constitutive relation in tensor representation is applied to obtain a Maxwell curl equation with anisotropic properties, and the Maxwell curl equation with anisotropic properties is decomposed to obtain scalar equations for electric field intensity and magnetic field intensity in all directions in a rectangular coordinate system;

[0007] An electromagnetic wave is selected to propagate along a first propagation direction, and the partial derivatives of the remaining propagation directions are replaced by transverse wave vectors in corresponding directions, thereby obtaining a second propagation direction transverse wave vector replacing the second propagation direction partial derivative and a third propagation direction transverse wave vector replacing the third propagation direction partial derivative; when the first propagation direction is the x direction, the second propagation direction and the third propagation direction are respectively the y direction and the z direction; when the first propagation direction is the y direction, the second propagation direction and the third propagation direction are respectively the x direction and the z direction; when the first propagation direction is the z direction, the second propagation direction and the third propagation direction are respectively the x direction and the y direction;

[0008] Setting a space-time grid according to the Yee cell, and adopting a discrete method of alternating sampling in space and time for the electric field component and the magnetic field component in the electromagnetic field according to the scalar equation and the space-time grid, so as to convert the Maxwell curl equation containing the time variable into a time-domain discrete difference equation of the Maxwell curl equation;

[0009] Modifying the time-domain discrete difference equation of the Maxwell curl equation according to the second propagation direction transverse wave vector and the third propagation direction transverse wave vector to obtain a modified Maxwell curl difference equation, and redistributing and defining the curl calculation part in the electromagnetic field update equation corresponding to the modified Maxwell curl difference equation to obtain a first electromagnetic field update equation;

[0010] According to the correspondence between the positions of the electromagnetic field quantity spatial nodes defined in the space-time grid and the grid, averaging the grid-unmatched parts in the first electromagnetic field update equation to obtain a second electromagnetic field update equation;

[0011] The magnetic field and electric field are updated within a preset time step according to the second electromagnetic field update equation to obtain a spatial electromagnetic field.

[0012] The present invention also provides an electromagnetic wave finite-difference time-domain system based on anisotropic medium conditions, characterized in that the system includes:

[0013] a scalar equation determination module for applying a medium constitutive relation in tensor representation based on the Maxwell curl equation for electromagnetic waves to obtain a Maxwell curl equation with anisotropic properties, and decomposing the Maxwell curl equation with anisotropic properties to obtain scalar equations for electric field intensity and magnetic field intensity in all directions in a rectangular coordinate system;

[0014] A wave vector replacement module is used to select an electromagnetic wave to propagate along a first propagation direction, and replace the partial derivatives of the remaining propagation directions with the transverse wave vectors in the corresponding directions, thereby obtaining a second propagation direction transverse wave vector that replaces the second propagation direction partial derivative and a third propagation direction transverse wave vector that replaces the third propagation direction partial derivative; when the first propagation direction is the x direction, the second propagation direction and the third propagation direction are respectively the y direction and the z direction; when the first propagation direction is the y direction, the second propagation direction and the third propagation direction are respectively the x direction and the z direction; when the first propagation direction is the z direction, the second propagation direction and the third propagation direction are respectively the x direction and the y direction;

[0015] a differential equation determination module for setting a space-time grid based on Yee cells and, based on the scalar equation and the space-time grid, adopting a discrete method of alternating sampling in space and time for the electric field component and the magnetic field component in the electromagnetic field, thereby converting the Maxwell curl equation containing a time variable into a time-domain discrete difference equation of the Maxwell curl equation;

[0016] a first updating module, configured to modify the time-domain discrete difference equation of the Maxwell curl equation according to the second propagation direction transverse wave vector and the third propagation direction transverse wave vector to obtain a modified Maxwell curl difference equation, and redistribute and define a curl calculation portion in the electromagnetic field update equation corresponding to the modified Maxwell curl difference equation to obtain a first electromagnetic field update equation;

[0017] a second updating module, configured to average the grid-unmatched portions of the first electromagnetic field update equation according to the correspondence between the positions of the electromagnetic field quantity spatial nodes defined in the space-time grid and the grid, to obtain a second electromagnetic field update equation;

[0018] The electromagnetic field update calculation module updates the magnetic field and electric field within a preset time step according to the second electromagnetic field update equation to obtain a spatial electromagnetic field.

[0019] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0020] The present invention provides an electromagnetic wave finite-difference time-domain method and system based on anisotropic medium conditions, wherein: 1) the differential formula derivation based on the Maxwell equations involves the medium constitutive relationship in tensor representation, which makes it have anisotropic characteristics and can be applied to the processing of layered media under anisotropic conditions, thereby obtaining numerical solutions of electric field intensity and magnetic field intensity under different media conditions. 2) The transverse wave vector parameter is creatively introduced, that is, the transverse wave vector in each direction independent of the propagation direction is used to replace the partial derivative of the Maxwell equation with respect to that direction, which can achieve a one-dimensional solution to two-dimensional or three-dimensional problems, simplifying formula derivation and code writing. 3) The electromagnetic field update equation is clearly divided into blocks; the curl calculation part is redefined, and the update equation is divided into three parts: isotropic term, anisotropic electric field term, and anisotropic magnetic field term, which are gradually iterated to facilitate code detection and implementation. 4) The quasi-one-dimensional anisotropic time-domain finite-difference algorithm has a wider applicability, laying a solid foundation and practical skills for subsequent processing to achieve electromagnetic calculations under more complex initial conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 This is a flow chart for the derivation and implementation of the quasi-one-dimensional finite-difference time-domain method of the present invention.

[0022] Figure 2 In the present invention, under situation Ⅰ, E z 、H y Field value change effect diagram.

[0023] Figure 3 In the present invention, under situation II, E y 、E z 、H y 、H z Field value change effect diagram.

[0024] Figure 4 In the present invention, under situation III, E y 、E z 、H y 、H z Field value change effect diagram.

[0025] Figure 5 In the present invention under situation IV, E y 、E z 、H y 、H z Field value change effect diagram.

[0026] Figure 6 The present invention is in case V, E y 、E z 、H y 、H z Field value change effect diagram.

[0027] Figure 7 In the present invention, under situation VI, E y 、E z 、H y 、H z Field value change effect diagram.

[0028] Figure 8 In the present invention, under situation VII, E y 、E z 、H y 、H z Field value change effect diagram.

[0029] Figure 9 In the present invention, under situation VIII, E y 、E z 、H y 、H z Field value change effect diagram.

[0030] Figure 10 In the present invention under situation IX, E y 、E z 、H y 、H z Field value change effect diagram. DETAILED DESCRIPTION

[0031] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0032] The basic principles of the quasi-one-dimensional electromagnetic wave finite-difference time-domain method for anisotropic media proposed in this invention are as follows: Based on the Maxwell curl equations, the constitutive relations in tensor representation are applied to impart anisotropic properties. After selecting the propagation direction, a transverse wave vector parameter is creatively introduced, and the partial derivatives in the remaining directions are replaced by the wave vectors in each direction. The electric and magnetic field components in the electromagnetic field are discretized using alternating sampling in space and time. This discretization method transforms the time-dependent Maxwell curl equations into a set of difference equations, and the spatial electromagnetic field is solved step by step along the time axis. Each electric (or magnetic) field component requires the electric (or magnetic) field components in each direction at the previous moment, as well as the curl calculation term associated with the magnetic field (or electric field). In the calculation, the electric field (or magnetic field) at a sample point in space is directly related to the surrounding magnetic field (or electric field). The medium parameters are assigned to each cell in the space. By modifying the medium permittivity and permeability represented by the tensors, the electromagnetic component values ​​in the anisotropic case can be solved.

[0033] FDTD, a numerical method for solving Maxwell's equations, is based on simple formula iteration and does not require complex asymptotic approximations or Green's functions. Although it is a time-domain solution, the frequency-domain response of the solution over a wide frequency band can be obtained through Fourier transforms. Utilizing quasi-one-dimensional techniques and introducing the transverse electromagnetic wave vector parameter, one-dimensional solutions for two- and three-dimensional problems can be obtained, as well as field intensity distributions for normal and oblique incidence. Furthermore, the tensor-based representation of the medium's constitutive relations enables the calculation of electromagnetic field quantities in layered anisotropic media, addressing electromagnetic scattering and radiation from complex-shaped targets and inhomogeneous media. Therefore, the quasi-one-dimensional electromagnetic wave finite-difference time-domain method in anisotropic media has a wider range of applicability. Furthermore, FDTD can conveniently display the temporal evolution of the electromagnetic field as time progresses, which can be displayed in pseudo-color on a computer. This electromagnetic field visualization clearly illustrates the physical processes, facilitating analysis and design.

[0034] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0035] Example 1

[0036] like Figure 1 As shown, this embodiment provides an electromagnetic wave finite-difference time-domain method based on anisotropic medium conditions, the method comprising:

[0037] S1: Based on the Maxwell curl equation for electromagnetic waves, the medium constitutive relation in tensor representation is applied to obtain the Maxwell curl equation with anisotropic properties, and the Maxwell curl equation with anisotropic properties is decomposed to obtain scalar equations of electric field intensity and magnetic field intensity in all directions in the rectangular coordinate system.

[0038] (1) Starting from Maxwell's curl equations and considering the general characteristics of the medium, the Maxwell equations in the lossy medium region can be expressed as follows:

[0039]

[0040]

[0041] where ε=ε0ε r , μ=μ0μ r , ε, μ are the dielectric constant and magnetic permeability of the medium, ε0, μ0 are the dielectric constant and magnetic permeability of the medium in vacuum, ε r , μ r is the dielectric constant and magnetic permeability of the medium, σ e is the conductivity tensor, σ mis the equivalent magnetic loss tensor, expressed as follows:

[0042]

[0043] The constitutive relations expressed as tensors have anisotropic properties. By modifying the values ​​of electromagnetic parameters, numerical solutions calculated using the difference method in isotropic and anisotropic media can be obtained.

[0044] The constitutive relationship expressed in tensor form makes the dielectric constant tensor, magnetic permeability tensor and the related electric and magnetic field update coefficients have anisotropic characteristics. The electromagnetic field quantity calculation under layered anisotropic media can be realized by modifying the dielectric constant tensor and magnetic permeability tensor of the medium in any region.

[0045] S2: Select an electromagnetic wave to propagate along a first propagation direction, and replace the partial derivatives in the remaining propagation directions with the transverse wave vectors in the corresponding directions, obtaining a second propagation direction transverse wave vector that replaces the second propagation direction partial derivative, and a third propagation direction transverse wave vector that replaces the third propagation direction partial derivative; when the first propagation direction is the x direction, the second propagation direction and the third propagation direction are the y direction and the z direction, respectively. When the first propagation direction is the y direction, the second propagation direction and the third propagation direction are the x direction and the z direction, respectively; when the first propagation direction is the z direction, the second propagation direction and the third propagation direction are the x direction and the y direction, respectively.

[0046] For example, in one-dimensional case, the electromagnetic wave is selected to propagate along the x direction, and the spatial sampling of each field component in the grid is: electromagnetic component E x 、H y 、H z In space, it is the position of node i, H x 、E y 、E z In space The node positions make the arrangement of electromagnetic components just conform to the spatial discretization of the curl operator; at the same time, it is agreed that the sampling time of the electric field component is an integer time step, and the sampling time of the magnetic field component is a half-integer time step, so as to realize the alternating sampling of the electric field and magnetic field in time sequence, and the spatial position is suitable for the differential calculation of Maxwell's equations.

[0047] Creative introduction of the transverse wave vector parameter k y 、k z (Taking propagation in the x-direction as an example), replacing the transverse partial derivatives in Maxwell's equations that do not belong to the propagation direction with the related transverse wave vector parameters can achieve a one-dimensional solution to two-dimensional or three-dimensional problems, reducing the number of parameters and simplifying the formula derivation and FDTD update iteration.

[0048] By setting the two transverse wave vector parameters k y 、k z, it is possible to implement one-dimensional solutions for two-dimensional or three-dimensional electromagnetic problems, as well as the field intensity distribution of electromagnetic waves under different conditions of normal and oblique incidence. Furthermore, due to the tensor-represented constitutive relations, the distribution of electromagnetic parameters within an arbitrary one-dimensional region is defined, and the medium parameters are assigned to every cell in space. Therefore, numerical solutions for the electric field intensity E and magnetic field intensity H can be obtained for media with different characteristics.

[0049] S3: A space-time grid is set up according to the Yee cell, and according to the scalar equation and the space-time grid, the electric field component and the magnetic field component in the electromagnetic field are discretely sampled in space and time in an alternating manner, so as to transform the Maxwell curl equation containing the time variable into a time-domain discrete difference equation of the Maxwell curl equation.

[0050] Among them, a space-time grid is set according to the Yee cell. Specifically, in the space-time grid, the magnetic field component sampling is set to the n+1 / 2 moment in time, and the electric field component sampling is set to the n moment. In space, the electric field component in the first propagation direction, the magnetic field component in the second propagation direction, and the magnetic field component in the third propagation direction are set as the i node position in the grid, and the magnetic field component in the first propagation direction, the electric field component in the second propagation direction, and the electric field component in the third propagation direction are set as the i+1 / 2 node position in the grid.

[0051] (3) The Maxwell curl equation is discretized in time based on the time sampling of the field quantity. The time partial derivative is approximated by the first-order central difference, and the result is:

[0052]

[0053]

[0054] According to the magnetic field component time step sampling The time-step sampling of the electric field component is set at n+1, and the time-domain discrete difference format of Maxwell's curl equation is obtained:

[0055]

[0056]

[0057] The same as described in S1, ε, μ are the dielectric constant and magnetic permeability of the medium, σ e is the conductivity tensor, σ m is the equivalent magnetic loss tensor, and Δt is the time interval.

[0058] That is, finite differences are used to approximate the spatial derivatives and time derivatives in Maxwell's equations, and then a set of equations are constructed to use the instantaneous field value of the previous time step to calculate the instantaneous field value of the next time step, thereby obtaining an algorithm that advances time forward to simulate the process of the electromagnetic field in the time domain and obtain the time domain discrete difference format of Maxwell's curl equations.

[0059] The time and space grids are introduced into Maxwell's curl equations, and the first-order partial derivatives with respect to time and space are approximated by central differences. At the same time, the average approximation is used to achieve grid matching, which makes the FDTD algorithm that advances over time numerically stable and convergent.

[0060] S4: The time-domain discrete difference equation of the Maxwell curl equation is modified according to the second propagation direction transverse wave vector and the third propagation direction transverse wave vector to obtain a modified Maxwell curl difference equation, and the curl calculation part in the electromagnetic field update equation corresponding to the modified Maxwell curl difference equation is redistributed and defined to obtain a first electromagnetic field update equation.

[0061] Since the propagation direction is selected as the x direction, according to the exponential representation of the electric field and the magnetic field, the transverse wave vector parameter is creatively introduced, that is, the partial derivatives about the y and z directions are expressed by the corresponding transverse wave vector components k y 、k z Instead, retain the partial derivative in the x direction and modify the Maxwell curl equation.

[0062] (4) The time domain discrete difference format is processed using a quasi-one-dimensional algorithm, as follows:

[0063] Curl operations in equations The matrix form of is:

[0064]

[0065] The electric and magnetic fields can be expressed as:

[0066] E(r)=E0e -jk·r

[0067] H(r)=H0e -jk·r (9)

[0068] Where E0, H0 are complex vectors, containing the amplitude and phase information of the corresponding components of the electromagnetic field. x ,k y ,k z ) T , called the wave vector r = (x, y, z) T , 0 is called the position vector.

[0069] Since the propagation direction is selected along the x direction, the partial derivatives in the y and z directions are replaced by the transverse wave vectors in their directions:

[0070]

[0071]

[0072] Where j is the imaginary unit, k y , k z is the transverse wave vector in the y and z directions. Modify the C matrix to obtain:

[0073]

[0074] Define equation (6), and the update coefficients in equation (7) are as follows:

[0075]

[0076]

[0077] C ee , C eh , C hh , C he The coefficients are all 3*3 matrices, as shown below:

[0078] At this point, the simplified electromagnetic field update equations in all directions are obtained:

[0079]

[0080]

[0081] Expand it to show:

[0082]

[0083]

[0084] The calculation part of the magnetic field (or electric field) curl required for the electric field (or magnetic field) update is redefined, DH zx It means the partial derivative of the z-direction component of the magnetic field in the x-direction, C Ey The calculation of the curl of the y-direction component of the electric field is:

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097] The curl calculation part in the derived electric and magnetic field update equations is redistributed and defined, that is, y 、k z The term and the partial derivative with respect to x are defined as the curl calculation of the electromagnetic field. The six elements on the right side of the formula are divided into isotropic terms, anisotropic electric field terms, and anisotropic magnetic field terms. This avoids the inconvenience caused by complex formulas and facilitates code modification and testing.

[0098] (5) According to the time step recursion, the propagation process of electromagnetic wave field quantities in all directions in space can be obtained:

[0099]

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] S5: According to the correspondence between the positions of the electromagnetic field quantity space nodes defined in the space-time grid and the grid, average the grid-unmatched parts in the first electromagnetic field update equation to obtain a second electromagnetic field update equation.

[0106] (6) The FDTD method discretizes the one-dimensional geometric space of the problem into spatial geometric points. At these grid points, the electric and magnetic field components are placed at spatially discrete positions. The spatial nodes are mapped to the grid iterations. The i-node position corresponds to the grid iteration as (1:nx). The node corresponds to the grid iteration (1:nx+1), where nx is the number of spatial cells. x 、H y 、H z The iteration on the spatial grid is (1:nx), H x 、E y 、E z The iteration on the spatial grid is (1:nx+1). Since the spatial position is an alternating half grid, the H on the left and right sides x 、E y 、E z The field quantity is not updated and can be directly set to 0 as the boundary of the metal plates on both sides.

[0107] When applying the FDTD algorithm, it is necessary to initialize the constants required for the FDTD algorithm implementation, such as the dielectric constant and magnetic permeability in vacuum; the grid size and number required for the problem space definition; the transverse wave vector parameter k y 、k z settings, etc.

[0108] In order to ensure that the solution of the discretized difference equation is convergent and stable, a stability condition must be set, that is, Where CFL is the stability coefficient, and its value is between 0 and 1.

[0109] Based on the correspondence between spatial nodes and grid iterations, the coefficients and electromagnetic field quantities required for the updated equations are preset and initialized. By modifying the electromagnetic parameters ε and μ, isotropic or anisotropic media can be set.

[0110] S6: updating the magnetic field and electric field within a preset time step according to the second electromagnetic field update equation to obtain a spatial electromagnetic field.

[0111] (7) According to the defined correspondence between the spatial node positions of the electromagnetic field quantities and the grid, the grid-unmatched parts in the initial update equation are averaged to obtain the final update equation.

[0112] The selected magnetic field time step is The electric field time step is at time n+1. Therefore, after entering the FDTD update loop, the magnetic field is iterated first, and then the electric field is calculated. In the isotropic case, each field quantity needs to be updated with the value of the current field quantity at the previous moment and the value of another field at the half grid on the left and right sides. In the anisotropic case, each field quantity needs to be iteratively updated with the value of the current field quantity at the previous moment and the values ​​of the other five field quantities. The six elements on the right are divided into isotropic terms, anisotropic electric field terms, and anisotropic magnetic field terms, as shown in the following formula:

[0113]

[0114]

[0115]

[0116]

[0117]

[0118]

[0119] Thus, the electromagnetic field update equation for the quasi-one-dimensional finite-difference time-domain method for electromagnetic waves in anisotropic media has been derived. This formula can be used to calculate electromagnetic wave field quantities in layered anisotropic media, as well as the field strength at normal and oblique incidence in two-dimensional or three-dimensional scenarios.

[0120] After entering the FDTD loop, the first thing to do is to update the wave source. The defined time-varying Gaussian wave source is applied to the center of the problem space. Then the magnetic field is updated, and then the electric field is updated. During the iteration process, the values ​​of the electromagnetic field quantities are stored for drawing. Perfect metal plates are set on the left and right sides to observe the propagation and reflection of electromagnetic waves between the two plates.

[0121] According to the stability conditions, set the update time step to 2000, the space step to 0.001, and the number of cells to 1000 until the update is completed. Then set different medium conditions and calculate the corresponding electromagnetic field quantities. The electromagnetic field quantity diagrams at time steps of 400, 800, 1200, and 1600 are respectively intercepted. Where data1 represents the metal plates on both sides, and data2 represents E y Component, data3 represents E z Component, data3 represents H y Component, data4 represents H z Quantity.

[0122] This embodiment is not only applicable to the electromagnetic wave propagation process and field quantity calculation in layered anisotropic media, but also further considers the rapid numerical calculation of obliquely incident electromagnetic waves in two-dimensional or three-dimensional situations. The processing process of this method starts from Maxwell's equations, involves the constitutive relationship of the medium in tensor representation, and makes it anisotropic. It defines the initial electromagnetic parameter distribution in an arbitrary one-dimensional region, thereby obtaining numerical solutions for electric and magnetic field intensities in different media situations. The greatest advantage of this invention lies in the creative introduction of transverse wave vector parameters to the classic one-dimensional finite-difference time-domain method. That is, the transverse partial derivatives in Maxwell's equations that do not belong to the propagation direction are replaced with the corresponding transverse wave vector parameters, which can realize a one-dimensional solution for two-dimensional or three-dimensional electromagnetic problems. By setting the corresponding transverse wave vector parameters, the field intensity distribution of electromagnetic waves under different situations of normal incidence or oblique incidence can be calculated. Compared with the classical one-dimensional electromagnetic wave finite-difference time-domain method, the quasi-one-dimensional electromagnetic wave finite-difference time-domain method under anisotropic media conditions has a wider applicability and can lay a solid theoretical foundation and practical skills for handling electromagnetic calculations under more complex initial conditions.

[0123] The following is a verification of the stability and convergence of the results under the quasi-one-dimensional algorithm, including the following steps:

[0124] 1) Set the one-dimensional problem space to a geometry with a distance of 1 meter and an air filling medium. Define the boundaries as parallel perfect conductors. Select a current source with a Gaussian waveform and place it at the center of the problem space. After iterative FDTD updates, obtain the instantaneous electric and magnetic fields in the computational domain.

[0125] 2) In the FDTD method, the electric and magnetic fields are sampled at discrete points in space and time. The sampling period, i.e., the space and time steps in the calculation, must comply with certain restrictions to ensure the stability of the solution. The numerical stability of the FDTD method is determined by the CFL condition, which requires that the time increment relative to the spatial grid is less than a certain value. In the one-dimensional case, that is:

[0126]

[0127] Where c is the maximum speed of light in the computational domain, Δx is the spatial step size, and Δt is the time step size. If the partitioning is non-uniform, the minimum value of Δx in the propagation direction is selected.

[0128] 3) Initialize the electromagnetic field quantities and update coefficients required by the quasi-one-dimensional anisotropic finite-difference time-domain method. x 、H y 、H z and its related C hh 、C he The length of H is nx.x 、E y 、E z and its related C ee 、C eh are both nx+1. As can be deduced from the formula, the update coefficient is related to the medium's dielectric constant, magnetic permeability, electric loss, and magnetic loss tensors. Therefore, by setting the values ​​of the electromagnetic parameters, the medium can be modified to obtain the values ​​of the next-dimensional field quantities for isotropic and anisotropic media.

[0129] 4) According to the above formula, the updated equation of the electromagnetic field of the quasi-one-dimensional time-domain finite-difference method is derived, and the six electromagnetic components of the redistributed electric field (magnetic field) isotropic term, anisotropic electric field term, and anisotropic magnetic field term are gradually iterated. During the process, the grid consistency of the tensor coefficient, field quantity, and curl calculation part must be ensured.

[0130] 5) Since the sampling time step of the selected magnetic field component is The electric field component is sampled at time step n+1, so upon entering the loop, the magnetic field is updated first, followed by the electric field iteration. Within the set time step, the electric and magnetic fields are updated alternately to complete the field propagation. The field values ​​at each time step are recorded and plotted.

[0131] 6) Set the two sides of the one-dimensional problem space as metal plate boundaries. Therefore, the tangential electric field component will disappear on the metal plate. In order to meet this condition, the specific implementation method is to The field quantities at the nodes are forced to 0 at the first and last grid cells, which allows us to see the correct behavior of the field propagation in the problem space and the incidence and reflection on the metal plate surface.

[0132] Then the quasi-one-dimensional finite-difference time-domain method of electromagnetic waves is applied to calculate under different media conditions; by modifying k y 、k z The value of is used to calculate the field value under normal incidence and oblique incidence, and to modify the dielectric constant tensor and permeability tensor to realize the field value calculation in any area and any medium type:

[0133] Case I: Set ε and μ to: k y =0,k z =0,

[0134] The results are as follows Figure 2As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, its normal incidence is defined to be perpendicular to the x axis and parallel to the metal plates on both sides, the medium in the space is set to air, and the source term is incorporated into the z-direction component of the electric field. In this case, there are only variables in the z-direction of the electric field and the y-direction of the magnetic field. The wave propagates from the middle to both sides, and after passing through the metal plate, it is reflected, the electric field is reversed, and the magnetic field remains unchanged, and it changes from propagating to both sides to propagating to the center.

[0135] In case II, set ε and μ to be: k y =10,k z =0,

[0136] The results are as follows Figure 3 As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vector in the y direction changes, which causes the source term to change, and the incident wave changes from normal incidence to oblique incidence. When the space medium is still set to air, the electromagnetic field value changes, and variables in the y direction of the electric field and the z direction of the magnetic field appear. The wave propagates from the middle to both sides, passes through the metal plate, and is reflected. The direction of the electric field is reversed, and the direction of the magnetic field remains unchanged.

[0137] Case III: Set ε and μ to: k y =0,k z =10,

[0138] The results are as follows Figure 4 As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vector in the z direction changes, which causes the source term to change, and the incident wave changes from normal incidence to oblique incidence. When the space medium is still set to air, the value of the electromagnetic field changes, and variables in the y direction of the electric field and the z direction of the magnetic field appear. Due to the duality relationship between the field quantities, this situation is consistent with Case III. The wave propagates from the middle to both sides, passes through the metal plate, and is reflected. The direction of the electric field is reversed, and the direction of the magnetic field remains unchanged.

[0139] Case IV sets ε and μ to be: k y =10,k z =10,

[0140] The results are as follows Figure 5As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vectors in the y and z directions change, which causes the source term to change, and the incident wave changes from normal incidence to a more complex oblique incidence. When the space medium is still set to air, the change in the electromagnetic field value is more obvious than that in Case II and Case III, and variables in the y direction of the electric field and the z direction of the magnetic field appear. The wave propagates from the middle to both sides, passes through the metal plate, and is reflected. The direction of the electric field is reversed, and the direction of the magnetic field remains unchanged.

[0141] Case V Set ε, μ to be: k y =10,k z =10,

[0142] The results are as follows Figure 6 As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vectors in the y and z directions change, and the incident wave setting of Case IV is used, that is, in the case of oblique incidence, the dielectric constant in the entire space is modified to obtain anisotropic media, the electromagnetic field value changes, the wave propagation mode remains unchanged, and it still propagates from the middle to both sides, but the propagation speed changes. After passing through the metal plates on both sides, reflection occurs, the electric field direction is reversed, and the magnetic field direction remains unchanged.

[0143] Case VI: Set ε and μ to: y =3,k z =5

[0144] set up Within range In the rest of the range

[0145]

[0146] The results are as follows Figure 7 As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vectors in the y and z directions change, the oblique incidence situation is still selected, the spatial medium type is modified, and the medium in the (250~750) region is set to have symmetrical anisotropic characteristics. When the wave passes through this region during propagation, the propagation speed slows down, and the field quantity is reflected after reaching and passing through the medium. The remaining part is transmitted to the metal plate and then reflected. The direction of the electric field is reversed, and the direction of the magnetic field remains unchanged.

[0147] Case VII: Set ε and μ to: k y =3,k z =5;

[0148] set up Within range In the rest of the range

[0149]

[0150] The results are as follows Figure 8 As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vectors in the y and z directions change, the oblique incidence case is still selected, the spatial medium type is modified, and the spatial step is set in the (500~750) region. Compared with case VII, the medium has arbitrary anisotropic properties. When the wave passes through this region during propagation, the propagation speed slows down, and the field quantity is reflected after reaching and passing through the medium. The remaining part is transmitted to the metal plate and then reflected. The direction of the electric field is reversed, and the direction of the magnetic field remains unchanged.

[0151] Case VIII: Set ε and μ to: k y =0,k z =5;

[0152] set up Within range set up Within range In the rest of the range

[0153] The results are as follows Figure 9 As shown, the propagation direction is selected as the x-direction, the excitation current screen is placed at the center of the problem space, the transverse wave vector in the z-direction changes, a relatively simple oblique incidence case is set, and more complex spatial medium conditions are set. The spatial step is modified to have different types of spatial media in the (500-750) and (1-500) regions, each with arbitrary anisotropy. As a result, the propagation and reflection of the wave from the center to the sides change. The propagation speed within the medium region is slow, and the field is reflected after passing through the medium. The remaining part is transmitted to the metal plate and reflected again, causing the electric field direction to flip, while the magnetic field direction remains unchanged. The more pronounced the anisotropy, the more dramatic the field changes.

[0154] Case IX: Set ε and μ to be: k y =5,k z =7;

[0155] set up Within range set up Within range In the rest of the range

[0156] The results are as follows Figure 10As shown, the propagation direction is selected as along the x direction, the excitation current screen is placed in the center of the problem space, the transverse wave vectors in the y and z directions change, a more complex oblique incidence situation is set, and the medium in the space step area of ​​(500~750) is set to have a superposition of arbitrary anisotropic properties, that is, the dielectric constant and magnetic permeability parameters in the area are set at the same time. The propagation speed of the wave in this medium is extremely slow, and the field quantity is reflected after passing through the medium, and the propagation speed is restored. The remaining part is transmitted to the metal plate and then reflected again, the direction of the electric field is reversed, and the direction of the magnetic field remains unchanged.

[0157] Example 2

[0158] This embodiment provides an electromagnetic wave finite-difference time-domain system based on anisotropic media conditions, the system comprising:

[0159] The scalar equation determination module is used to apply the medium constitutive relationship in tensor representation based on the Maxwell curl equation of electromagnetic waves to obtain the Maxwell curl equation with anisotropic properties, and decompose the Maxwell curl equation with anisotropic properties to obtain scalar equations of electric field intensity and magnetic field intensity in all directions in a rectangular coordinate system.

[0160] The wave vector replacement module is used to select an electromagnetic wave to propagate along a first propagation direction, and replace the derivatives of the remaining propagation directions with the transverse wave vectors in the corresponding directions, so as to obtain the second propagation direction transverse wave vector replacing the second propagation direction derivative and the third propagation direction transverse wave vector replacing the third propagation direction derivative; when the first propagation direction is the x direction, the second propagation direction and the third propagation direction are respectively the y direction and the z direction; when the first propagation direction is the y direction, the second propagation direction and the third propagation direction are respectively the x direction and the z direction; when the first propagation direction is the z direction, the second propagation direction and the third propagation direction are respectively the x direction and the z direction.

[0161] The differential equation determination module is used to set a space-time grid according to the Yee cell, and adopt a discrete method of alternating sampling in space and time for the electric field component and the magnetic field component in the electromagnetic field according to the scalar equation and the space-time grid, so as to convert the Maxwell curl equation containing the time variable into a time-domain discrete difference equation of the Maxwell curl equation.

[0162] The first update module is used to correct the time-domain discrete difference equation of the Maxwell curl equation according to the second propagation direction transverse wave vector and the third propagation direction transverse wave vector to obtain a corrected Maxwell curl difference equation, and redistribute and define the curl calculation part in the electromagnetic field update equation corresponding to the corrected Maxwell curl difference equation to obtain a first electromagnetic field update equation.

[0163] The second updating module is used to average the grid-unmatched parts in the first electromagnetic field update equation according to the correspondence between the spatial node positions of the electromagnetic field quantities defined in the space-time grid and the grid, so as to obtain the second electromagnetic field update equation.

[0164] The electromagnetic field update calculation module updates the magnetic field and electric field within a preset time step according to the second electromagnetic field update equation to obtain a spatial electromagnetic field.

[0165] Example 3

[0166] This embodiment provides an electronic device, including a memory and a processor, wherein the memory is used to store a computer program, and the processor runs the computer program to enable the electronic device to perform the electromagnetic wave finite-difference time-domain method based on anisotropic medium conditions in embodiment 1.

[0167] Optionally, the above-mentioned electronic device may be a server.

[0168] In addition, an embodiment of the present invention further provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, the electromagnetic wave finite-difference time-domain method based on anisotropic medium conditions of the first embodiment is implemented.

Claims

1. An electromagnetic wave finite-difference time-domain method based on anisotropic media conditions, characterized in that: The method comprises: Based on the Maxwell curl equation for electromagnetic waves, a medium constitutive relation in tensor representation is applied to obtain a Maxwell curl equation with anisotropic properties, and the Maxwell curl equation with anisotropic properties is decomposed to obtain scalar equations for electric field intensity and magnetic field intensity in all directions in a rectangular coordinate system; An electromagnetic wave is selected to propagate along a first propagation direction, and the partial derivatives of the remaining propagation directions are replaced by transverse wave vectors in corresponding directions, thereby obtaining a second propagation direction transverse wave vector replacing the second propagation direction partial derivative and a third propagation direction transverse wave vector replacing the third propagation direction partial derivative; when the first propagation direction is the x direction, the second propagation direction and the third propagation direction are respectively the y direction and the z direction; when the first propagation direction is the y direction, the second propagation direction and the third propagation direction are respectively the x direction and the z direction; when the first propagation direction is the z direction, the second propagation direction and the third propagation direction are respectively the x direction and the y direction; Setting a space-time grid according to the Yee cell, and adopting a discrete method of alternating sampling in space and time for the electric field component and the magnetic field component in the electromagnetic field according to the scalar equation and the space-time grid, so as to convert the Maxwell curl equation containing the time variable into a time-domain discrete difference equation of the Maxwell curl equation; Modifying the time-domain discrete difference equation of the Maxwell curl equation according to the second propagation direction transverse wave vector and the third propagation direction transverse wave vector to obtain a modified Maxwell curl difference equation, and redistributing and defining the curl calculation part in the electromagnetic field update equation corresponding to the modified Maxwell curl difference equation to obtain a first electromagnetic field update equation; According to the correspondence between the positions of the electromagnetic field quantity spatial nodes defined in the space-time grid and the grid, averaging the grid-unmatched parts in the first electromagnetic field update equation to obtain a second electromagnetic field update equation; The magnetic field and electric field are updated within a preset time step according to the second electromagnetic field update equation to obtain a spatial electromagnetic field.

2. The method according to claim 1, characterized in that Setting up a space-time grid based on Yee cells includes: In the space-time grid, the magnetic field component sampling is set to n+1 / 2 moment in time, and the electric field component sampling is set to n moment. In space, the electric field component in the first propagation direction, the magnetic field component in the second propagation direction, and the magnetic field component in the third propagation direction are the i node position in the grid, and the magnetic field component in the first propagation direction, the electric field component in the second propagation direction, and the electric field component in the third propagation direction are the i+1 / 2 node position in the grid.

3. The method according to claim 2, characterized in that The scalar equation is: Where ε=ε0ε r , μ=μ0μ r , ε, μ are the dielectric constant and magnetic permeability of the medium, ε0, μ0 are the dielectric constant and magnetic permeability of the medium in vacuum, ε r , μ r is the dielectric constant and magnetic permeability of the medium, σ e is the conductivity tensor, σ m is the equivalent magnetic loss tensor; ▽× represents the curl operation.

4. The method according to claim 3, characterized in that The time domain discrete difference equation of the Maxwell curl equation is: Where Δt is the time interval.

5. The method according to claim 3, characterized in that When the first propagation direction is the x direction, the curl calculation part in the modified Maxwell curl difference equation is: Where j represents the imaginary part; k=(k x ,k y ,k z ) T , called the wave vector.

6. The method according to claim 5, characterized in that The second electromagnetic field update equation is: in, Where C ee,11 to C ee,33 、C eh,11 to C eh,33 、C hh,11 to C hh,33 、C he,11 to C he,33 Corresponding to C ee , C eh , C hh , C he The elements in C Ey represents the curl of the y-component of the electric field.

7. An electromagnetic wave finite-difference time-domain system based on anisotropic media conditions, characterized in that: The system comprises: a scalar equation determination module for applying a medium constitutive relation in tensor representation based on the Maxwell curl equation for electromagnetic waves to obtain a Maxwell curl equation with anisotropic properties, and decomposing the Maxwell curl equation with anisotropic properties to obtain scalar equations for electric field intensity and magnetic field intensity in all directions in a rectangular coordinate system; A wave vector replacement module is used to select an electromagnetic wave to propagate along a first propagation direction, and replace the partial derivatives of the remaining propagation directions with the transverse wave vectors in the corresponding directions, thereby obtaining a second propagation direction transverse wave vector that replaces the second propagation direction partial derivative and a third propagation direction transverse wave vector that replaces the third propagation direction partial derivative; when the first propagation direction is the x direction, the second propagation direction and the third propagation direction are respectively the y direction and the z direction; when the first propagation direction is the y direction, the second propagation direction and the third propagation direction are respectively the x direction and the z direction; when the first propagation direction is the z direction, the second propagation direction and the third propagation direction are respectively the x direction and the y direction; a differential equation determination module for setting a space-time grid based on Yee cells and, based on the scalar equation and the space-time grid, adopting a discrete method of alternating sampling in space and time for the electric field component and the magnetic field component in the electromagnetic field, thereby converting the Maxwell curl equation containing a time variable into a time-domain discrete difference equation of the Maxwell curl equation; a first updating module, configured to modify the time-domain discrete difference equation of the Maxwell curl equation according to the second propagation direction transverse wave vector and the third propagation direction transverse wave vector to obtain a modified Maxwell curl difference equation, and redistribute and define a curl calculation portion in the electromagnetic field update equation corresponding to the modified Maxwell curl difference equation to obtain a first electromagnetic field update equation; a second updating module, configured to average the grid-unmatched portions of the first electromagnetic field update equation according to the correspondence between the positions of the electromagnetic field quantity spatial nodes defined in the space-time grid and the grid, to obtain a second electromagnetic field update equation; The electromagnetic field update calculation module updates the magnetic field and electric field within a preset time step according to the second electromagnetic field update equation to obtain a spatial electromagnetic field.

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