A three-dimensional FDTD metal conformal stabilization simulation method

Abstract

The application discloses a three-dimensional FDTD metal conformal stable simulation method, which comprises the following steps: grid construction and boundary identification, effective electromagnetic parameter calculation, stable condition pre-check, field quantity iterative update and boundary condition convergence judgment. Through the integral form of the effective dielectric constant and the magnetic permeability, the post-weighted stabilization strategy and the strict integral update rule of the metal boundary, the high-precision and high-stability electromagnetic simulation of the complex medium boundary and the metal boundary is realized, and the robustness of the algorithm to the time step is improved, and the practical application scene is expanded.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic field and electromagnetic wave numerical simulation technology, and in particular to a three-dimensional FDTD metal conformal stabilization simulation method. Background Technology

[0002] The FDTD (Finite Difference Time Domain) algorithm is one of the core methods for numerical simulation of electromagnetic fields. The traditional FDTD algorithm uses a stepped approximation to handle the medium and metal boundaries. This method will fit irregular curved surfaces and inclined interfaces into a stepped mesh, introducing significant geometric errors, especially in complex boundary scenarios where the simulation accuracy is severely reduced.

[0003] To improve the accuracy of the stepped approximation, the existing technology has developed the FDTD conformal technique. Although it can improve the boundary fitting accuracy by correcting the mesh parameters, it still has many technical shortcomings: First, it is accompanied by a decrease in numerical stability, resulting in insufficient algorithm robustness; second, it imposes strict restrictions on the selection of time steps, which narrows the application scope of the algorithm and increases the computational cost; third, it lacks strict integral form guarantees for the treatment of metal boundaries, which can easily destroy the consistency of boundary conditions and further affect the accuracy of simulation results.

[0004] In summary, existing technologies cannot simultaneously ensure the accuracy and stability of FDTD simulations under complex boundaries. The industry urgently needs a 3D FDTD conformal algorithm that can solve problems such as geometric errors, numerical instability, and imprecise handling of metal boundaries. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a three-dimensional FDTD metal conformal stabilization simulation method. By using the integral form of effective dielectric constant and permeability, a weighted stabilization strategy for the latter term, and a strict integral update rule for metal boundaries, it achieves high-precision and high-stability electromagnetic simulation of complex dielectric and metal boundaries. At the same time, it improves the robustness of the algorithm to time steps and expands its practical application scenarios.

[0006] This invention provides the following solutions:

[0007] This invention provides a three-dimensional FDTD metal conformal stabilization simulation method, the method comprising:

[0008] S1. Mesh Construction and Boundary Identification: Construct a 3D network of the simulation region and identify all conformal boundary cells through the distribution of medium parameters;

[0009] S2. Calculation of effective electromagnetic parameters: For each conformal cell, the effective dielectric constant and effective permeability of the surface are calculated in integral form, and then the effective parameters of the average electric field interval and the effective parameters of the average magnetic field interval are obtained by interval Gauss-Legend numerical integration.

[0010] S3. Stability condition pre-verification: For conformal meshes near metal boundaries, verify whether they meet stability conditions by checking relative length and relative area, and mark unstable cells.

[0011] S4. Field quantity iterative update: The electric field and magnetic field are iterated alternately according to the FDTD time step rule, and the unstable cells are corrected during the iteration process;

[0012] S5. Perform a boundary condition convergence check. If the condition does not meet the criteria, return to step S4 to continue iterating. If the condition meets the criteria, the simulation is complete.

[0013] Furthermore, step S1 includes the following processes:

[0014] S1.1 Construct a three-dimensional cube simulation region, set the mesh step size, including the step size Δx, Δy and Δz in the x-axis, y-axis and z-axis directions; identify all arched boundary cells, record the type of medium and area ratio in the cells, the position and outline of the metal boundary, and the outer region of the conductor;

[0015] S1.2, The scatterer is a spherical dielectric particle, with a diameter d and a dielectric constant ε. r and permeability μ r The particle surface is a complex curved medium boundary;

[0016] S1.3 The simulation region boundary adopts CPML absorbing boundary, and the number of thickness layers is set.

[0017] Furthermore, the effective dielectric constant of the surface mentioned in step S2 is calculated using the following formula:

[0018] ,

[0019] Where, ε eff (x) is the effective dielectric constant of the cross-section x, A is the area of ​​the integral surface of the cross-section x, M is the number of dielectrics contained in the cross-section x, and ε m (x) is the dielectric constant of the m-th medium;

[0020] The effective permeability of a surface is calculated using the following formula:

[0021] ,

[0022] in, Let μ be the effective permeability of the tangential surface x. m (x) is the first The permeability of a medium, S m (x) is the first The area ratio of each medium;

[0023] For the interval [xi x i+1 ], x i and x i+1 Let i and i+1 represent the effective parameters of the average magnetic field interval for the i-th and (i+1)-th tangents, respectively. It is calculated using the following formula:

[0024] ,

[0025] in, ;

[0026] Effective parameters of average electric field range It is calculated using the following formula:

[0027] .

[0028] Furthermore, step S3 is validated in the following cases:

[0029] A. If the simulation scenario involves magnetic materials, the following process is included:

[0030] For a conformal metal mesh near the surface of a spherical dielectric particle, the relative lengths of the electric field integral lines in the y and z directions are calculated using the following formula. and and relative area :

[0031] ,

[0032] ,

[0033] Among them, l y and l z These represent the effective lengths of the electric field integral lines on the y-axis and z-axis, respectively. , , and These represent the relative lengths of the electric field integral lines along the positive z-axis, negative z-axis, positive y-axis, and negative y-axis, respectively.

[0034] Strict verification ≥0.05 and ≤12, mark the edge cells that do not meet the stability condition;

[0035] B. If the simulation scenario involves non-magnetic materials, the following process is included:

[0036] Calculate the following correction parameters:

[0037] ,

[0038] ,

[0039] in, , , and Let be the relative lengths of the electric field integral lines of the tangent x along the positive z-axis, negative z-axis, positive y-axis, and negative y-axis, respectively. Let x be the relative area of ​​the cross section;

[0040] Strict verification Mark the edge cells that do not meet the stability conditions.

[0041] Furthermore, step S4 includes the following processes:

[0042] S4.1, for the interval [x i x i+1 For cells within a given area, the electric field is updated in the following cases:

[0043] ,

[0044] in, and These are the positions with x-axis index i. Time and The electric field value at time t. For time step, , , and These represent the positive z-axis direction, negative z-axis direction, positive y-axis direction, and negative y-axis direction of the x-axis section, respectively. Magnetic field components at different times; effective parameters of the average electric field interval The values ​​calculated in step S2 are used in the case of conformal dielectric, and then corrected near the metal boundary using the following formula:

[0045] ,

[0046] Among them, l x For the interval [x i x i+1 The effective length outside an ideal conductor;

[0047] S4.2, for the interval [x i x i+1 The cells within the [ ] are updated with magnetic fields in the following cases:

[0048] A. If the simulation scenario involves magnetic materials, the magnetic field will be updated in the following scenarios:

[0049] a. Conformal dielectric:

[0050] ,

[0051] in, and At position i along the x-axis. Time and The magnetic field value at time [time]. , , and These represent the positive z-axis direction, negative z-axis direction, positive y-axis direction, and negative y-axis direction of the x-axis section, respectively. The electric field components at time;

[0052] b. Metal boundary:

[0053] ;

[0054] in, and Represents the cross section x Time and The magnetic field value at time [time]. and denoted as the effective length of the electric field integral line in the positive and negative z-axis directions;

[0055] B. If the simulation scenario uses non-magnetic materials, then the magnetic field is updated using the following equation:

[0056] ,

[0057] Where, μ0= .

[0058] Furthermore, for unstable cells in magnetic material scenarios, the magnetic field is updated again using the following formula:

[0059] .

[0060] The beneficial effects of this invention based on its technical solution are as follows:

[0061] (1) The simulation accuracy is greatly improved: by using the integral form of effective dielectric constant and effective permeability, the actual spatial distribution of complex medium boundary is accurately fitted, eliminating the geometric error caused by the traditional step approximation from the root; the strict integral update rule of metal boundary ensures the consistency of boundary conditions and further improves the accuracy of simulation results.

[0062] (2) Significantly enhanced numerical stability: The weighted stabilization strategy can effectively suppress the numerical oscillation problem caused by conformal meshes and solve the defect of poor stability of existing conformal techniques; the clear mesh stability conditions greatly improve the robustness of the algorithm to time steps, allow for a larger time step selection, reduce the number of iterations, and reduce computational costs.

[0063] (3) Wide range of applications and strong versatility: The algorithm supports high-precision and stable processing of complex medium boundaries (curved surfaces, inclined interfaces, etc.) and metal boundaries. There is no need to design separate algorithms for different boundary types. It can be widely used in various electromagnetic field numerical simulation scenarios such as microwave device design, antenna simulation, and electromagnetic compatibility analysis.

[0064] (4) Simple to implement and easy to engineer: The algorithm is optimized based on the classic FDTD framework. The core update equation is highly compatible with the traditional FDTD algorithm. It can be applied in engineering by simply adding effective parameter calculation, stability condition verification and subsequent weighted correction modules to the original simulation program. The modification cost is low. Attached Figure Description

[0065] To more clearly illustrate the technical solutions in the embodiments of this specification or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0066] Figure 1 This is a flowchart illustrating a three-dimensional FDTD metal conformal stabilization simulation method provided by the present invention.

[0067] Figure 2 This is a schematic diagram of the Yee grid index.

[0068] Figure 3 This is a schematic diagram of the cross-section x of the effective volume of Ex.

[0069] Figure 4 This is a schematic diagram of the effective volume section x of Hx.

[0070] Figure 5 This is a schematic diagram of conformal design for Hx metals.

[0071] Figure 6 This is a schematic diagram of a scatterer.

[0072] Figure 7 This is a schematic diagram of the Mie scattering cross section of a spherical medium particle.

[0073] Figure 8 This is a schematic diagram of the relative error of spherical medium particles. Detailed Implementation

[0074] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention are within the protection scope of the embodiments of the present invention.

[0075] First, the principles of conformal updates of electric and magnetic fields will be explained using a 3D Yee mesh as an example:

[0076] Figure 2 It displays the node distribution locations of electric and magnetic field components, providing coordinate references for defining conformal mesh parameters. Figure 3 E was shown x The effective volume cross-section x of the surface element A, and the mesh size. and magnetic field component H y and H z The positive and negative positions of the symbols indicate the integral region of the conformal update of the electric field. Figure 4 H was marked x The effective volume cross-section x of the surface element A, and the mesh size. and electric field component E y and E z The positive and negative positions of the numbers indicate the integral region of the conformal update of the magnetic field. Figure 5 The location of the metal boundary, the effective area A(x) of the conformal mesh outside the metal, and the effective length l of the electric field edge are labeled. y and l z This demonstrates the effective integration region for conformal magnetic field under metallic boundaries.

[0077] In FDTD technology, a cell is the most basic spatial discrete unit, that is, a smallest cube (3D) or square (2D) in a Yee mesh. The tangent x of a cell refers to the part of a cell that intersects with the YOZ plane at x=x0. A 3D cell is a face, and a 2D cell is a line. No face elements are required.

[0078] I. Conformal update of electric field (taking the electric field Ex in the x-axis direction as an example):

[0079] 1. The theoretical basis is Maxwell's curl equation:

[0080] ,

[0081] in, Here, H is the differential operator, D is the magnetic field strength, and t is the electric displacement vector.

[0082] The local surface element x, which is the tangent to the effective volume of the electric field Ex along the x-axis. By integrating and applying Stokes' theorem, the surface integral on the left side is transformed into a line integral, yielding the integral relationship between the magnetic field and the electric displacement vector:

[0083] ,

[0084] in, The magnetic field strength, It is the electric displacement vector. For an integral surface, For surface element vectors, Let t be the boundary loop of the integral surface, and t be time.

[0085] 2. Definition of Field Quantities and Parameters:

[0086] cross section of a cell When the medium parameter ε(x) (which varies with the x-direction) changes within the interval, the average magnetic field within the tangent is defined. , The grid size satisfies:

[0087] ,

[0088] ,

[0089] .

[0090] Assume there are M types of media on the cross-section x of the cell. For the first The dielectric constant of a medium, Its area percentage within the surface element ( According to the boundary condition that the tangential electric fields are equal at the dielectric interface, we have E xm (x) is the electric field in the m-th medium on the cell section at position x. It is the electric field at the cell section at position x.

[0091] 3. Derivation of effective dielectric constant:

[0092] The effective dielectric constant of the surface is derived using integral form. The electric displacement vector is decomposed into the sum of its dielectric components:

[0093] ,

[0094] Introducing the effective dielectric constant of the surface:

[0095] ,

[0096] The integral can then be simplified to:

[0097] .

[0098] 4. Derivation of the electric field renewal equation:

[0099] Substituting the effective dielectric constant into the integral relationship, and combining it with the difference scheme, we obtain the difference equation for the electric displacement vector, and then substituting it into... ,get Instantaneous update formula:

[0100] ,

[0101] exist We define the average electric field and the effective permittivity of the interval by taking the average value of the electric field integral over the interval:

[0102] ,

[0103] ,

[0104] Analytical calculation Since it depends on the geometry of the medium, there is no universal analytical expression. This method uses Gauss-Legend numerical integration for calculation. ,

[0105] ,

[0106] in: .

[0107] Using the K-point Gauss-Legend integral, let the Gauss point be t. k The corresponding weight is A k ,get:

[0108] ,

[0109] The physical coordinates corresponding to the Gaussian point are denoted as: .

[0110] Finally, the electric field update equation based on dielectric conformality is obtained:

[0111] .

[0112] 5. Electric field correction near the metal boundary:

[0113] when The interval has a length l x Outside the ideal conductor, the interval is corrected by integration, the effective dielectric constant is recalculated, and the form of the electric field update equation remains unchanged, ensuring the consistency of the electric field update near the metal boundary.

[0114] ,

[0115] ,

[0116] .

[0117] II. Conformal Update of Magnetic Field (Taking the magnetic field Hx in the x-axis direction as an example):

[0118] 1. The theoretical basis is Faraday's law of electromagnetic induction:

[0119] ,

[0120] Where E is the electric field strength, B is the magnetic flux density, and t is time.

[0121] Integrating over the tangent x of the effective volume Hx, we transform the surface integral into the line integral of the electric field:

[0122] ,

[0123] 2. Definition of Field Quantities and Parameters:

[0124] Define the average electric field within a surface element. , The grid size satisfies:

[0125] ,

[0126] ,

[0127] ,

[0128] The range of values ​​is .

[0129] Assume that there exists within the surface element a medium, For the first The permeability of the medium, Assuming its area percentage, the effective permeability of the surface is introduced:

[0130] .

[0131] 3. Magnetic field renewal equation:

[0132] Combination The derivation yields Instantaneous update formula:

[0133] ,

[0134] To ensure the conservation of the magnetic field integral, we take the average magnetic field and the effective permeability of the interval as the values:

[0135] ,

[0136] ,

[0137] ,

[0138] Finally, the magnetic field update equation based on the conformal nature of the medium is obtained:

[0139] .

[0140] III. Conformal update of magnetic field at metallic boundaries:

[0141] 1. Metal boundary correction rules

[0142] When a metallic boundary exists, the line integral of the loop electric field is corrected, taking into account the effective area of ​​the Yee mesh outside the metallic conductor. And the effective length of the edge corresponding to the electric field node outside the conductor. , (In regions where the electric field is not zero), the derivation of the region below the metal boundary is obtained. The updated version:

[0143] ,

[0144] 2. Conformal mesh stability conditions

[0145] Normalization is performed by introducing relative length and relative area:

[0146] ,

[0147] ,

[0148] The conformally processed mesh must satisfy the following two stability conditions; otherwise, the numerical stability of the algorithm will deteriorate:

[0149] Conformal mesh effective area ratio ≥ 5% ;

[0150] The ratio of the longest relative side length to the relative area is ≤12: .

[0151] Substituting the relative parameters, we obtain the normalized updated equation for the metallic boundary magnetic field:

[0152] .

[0153] IV. Weighted stabilization of the latter term:

[0154] 1. Stabilization and correction strategy:

[0155] When the conformal mesh does not satisfy the above stability conditions, a weighted average strategy is used to correct the magnetic field value and suppress numerical oscillations. For example, the corrected formula is:

[0156] ,

[0157] This strategy constrains unstable field values, improves the overall stability of the algorithm, and relaxes the restrictions on the selection of time step.

[0158] 2. Simplified formulas for non-magnetic materials

[0159] If the simulation scenario involves non-magnetic materials ( (vacuum permeability), define the correction factor:

[0160] ,

[0161] ,

[0162] The magnetic field update equation simplifies to:

[0163] ,

[0164] The stability condition then simplifies to:

[0165] .

[0166] Based on the above derivation, this embodiment provides a three-dimensional FDTD metal conformal stabilization simulation method, referring to... Figure 1 The method includes:

[0167] S1. Mesh Construction and Boundary Identification: Construct a 3D network for the simulation region and identify all conformal boundary cells through the distribution of medium parameters. Specifically, this includes the following steps: constructing a 3D cubic simulation region, setting the mesh step size... ;reference Figure 6 The scatterer is a spherical medium particle with a diameter of dielectric constant of the medium magnetic permeability The particle surface is a complex curved medium boundary; the boundary of the simulation region adopts CPML absorbing boundary with a thickness of 8 mesh layers to avoid boundary reflection from interfering with the scattering results.

[0168] S2. Calculation of Effective Electromagnetic Parameters: For each conformal cell, identify the conformal mesh cells on the surface of the spherical particle, and accurately calculate the area ratio of the medium (spherical particle) to the air within each cell. Substituting the formulas for effective surface permittivity and effective surface permeability, we obtain the properties of each conformal cell. , Then, the effective parameters corresponding to the average electric / magnetic field are obtained through interval integration. Specifically, the following formula is used for calculation:

[0169] The effective dielectric constant of the surface is calculated using the following formula:

[0170] ,

[0171] Where, ε eff (x) is the effective dielectric constant of the cross-section x, A is the area of ​​the integral surface of the cross-section x, M is the number of dielectrics contained in the cross-section x, and ε m (x) is the dielectric constant of the m-th medium;

[0172] The effective permeability of a surface is calculated using the following formula:

[0173] ,

[0174] in, Let μ be the effective permeability of the tangential surface x. m (x) is the first The permeability of a medium, S m (x) is the first The area ratio of each medium;

[0175] For the interval [x i x i+1 ], x i and x i+1 Let i and i+1 represent the effective parameters of the average magnetic field interval for the i-th and (i+1)-th tangents, respectively. It is calculated using the following formula:

[0176] ,

[0177] in, ;

[0178] Effective parameters of average electric field range It is calculated using the following formula:

[0179] .

[0180] The parameters for the y-axis and z-axis can be obtained similarly.

[0181] S3. Stability condition pre-verification: This simulation scenario uses non-magnetic materials ( (vacuum permeability), for a conformal metal mesh near the surface of a spherical dielectric particle, the relative effective lengths of the electric field edges in the y and z directions are calculated using the following formula. and and relative area :

[0182] ,

[0183] ,

[0184] Among them, l y and l z These are the effective lengths of the electric field edge in the y and z directions, respectively; , , and , respectively, are the relative effective lengths of the electric field integral lines, and A1 is the area outside the metal in the cross-section;

[0185] Define the conformal correction factor for metals:

[0186] ,

[0187] ,

[0188] The magnetic field update equation simplifies to:

[0189] ,

[0190] The stability condition then simplifies to:

[0191] .

[0192] Strict verification Mark the edge cells that do not meet the stability conditions.

[0193] S4. Field quantity iterative update: Set the incident electromagnetic wave to a plane wave, and the excitation source frequency... The corresponding time step Following the FDTD process of alternating electric and magnetic fields, the field values ​​are solved by sequentially substituting the conformal electric and magnetic field update equations of the medium; for the marked unstable cells, a weighted strategy for the latter term is used for correction. Field values ​​are used to ensure iterative stability. This includes the following processes:

[0194] S4.1, for the interval [x i x i+1 For cells within a given area, the electric field is updated in the following cases:

[0195] ,

[0196] in, and These are the positions with x-axis index i. Time and The electric field value at time t, where Δt is the time step. , , and These are the magnetic field components that need to be linearly integrated; and the effective parameters of the average electric field interval. The values ​​calculated in step S2 are used in the case of conformal dielectric, and then corrected near the metal boundary using the following formula:

[0197] ,

[0198] Among them, l x For the interval [x i x i+1 The length outside the ideal conductor;

[0199] S4.2, for the interval [x i x i+1 Within the cells, in this embodiment, the simulation scenario is a magnetic material, and the magnetic field is updated in the following scenarios:

[0200] a. Conformal dielectric:

[0201] ,

[0202] in, and At positions with x-axis index i respectively. Time and The magnetic field value at time [time]. , , and These are the electric field components that need to be integrated by line, namely the electric field components in the positive z-axis direction, negative z-axis direction, positive y-axis direction, and negative y-axis direction of the tangent x.

[0203] b. Metal boundary:

[0204] ;

[0205] in, and Represents the cross section x Time and The magnetic field value at time [time]. and Let be the effective length of the electric field integral around the tangent plane x.

[0206] For unstable cells in magnetic material scenarios, the magnetic field is updated and then corrected again using the following formula:

[0207] .

[0208] The updates for the electric and magnetic fields in the y-axis and z-axis directions can be obtained similarly.

[0209] S5. Perform a boundary condition convergence check. If the boundary condition is not met, return to step S4 to continue iteration. If it is met, the simulation is complete. After 20,000 iterations, the electromagnetic field converges. The simulation yields the Mie scattering cross section and relative error of the spherical medium particles as follows: Figure 7 and Figure 8 As shown in the figure. The results show that the absolute error of the algorithm of the present invention is reduced by 0.174241 dB compared with the traditional volume averaging method, which verifies the high accuracy and high stability of the algorithm.

[0210] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0211] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0212] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A three-dimensional FDTD metal conformal stabilization simulation method, characterized in that, The method includes: S1. Mesh Construction and Boundary Identification: Construct a 3D mesh for the simulation region and identify all conformal boundary cells through the distribution of medium parameters; this includes the following processes: S1.1 Construct a three-dimensional cubic simulation region and set the mesh step size, including the step size Δ in the x-axis, y-axis, and z-axis directions. x Δ y and Δ z Identify all arched boundary cells, record the type of medium and area ratio within the cells, the location and outline of the metal boundary, and the outer region of the conductor; S1.2, The scatterer is a spherical medium particle, with a set diameter. d Dielectric constant of the medium ε r and permeability μ r The particle surface is a complex curved medium boundary; S1.3 The simulation region boundary adopts CPML absorbing boundary, and the number of thickness layers is set; S2. Calculation of Effective Electromagnetic Parameters: For each conformal cell, the effective surface permittivity and effective surface permeability are calculated in integral form. Then, the effective parameters of the average electric field interval and the effective parameters of the average magnetic field interval are obtained through interval Gauss-Gande numerical integration. The effective surface permittivity is calculated using the following formula: ， in, ε eff ( x ) is a cross-section x The effective dielectric constant of the surface, A For cross-section x The area of ​​the integral surface, M For cross-section x There are several types of memory media. ε m ( x ) is the first m The dielectric constant of the medium; The effective permeability of a surface is calculated using the following formula: ， in, μ eff ( x ) is a cross-section x Effective permeability of the surface, μ m ( x ) is the first m The permeability of the medium, S m ( x ) is the first m The area ratio of each medium; For the interval [ x i , x i+1 ], x i and x i+1 They represent the first i Cross-section and the first i +1 section, effective parameters of average magnetic field range It is calculated using the following formula: ， in, ; Effective parameters of average electric field range It is calculated using the following formula: ； S3. Stability condition pre-verification: For conformal meshes near metal boundaries, verify whether they meet stability conditions by checking relative length and relative area, and mark unstable cells. S4. Field quantity iterative update: The electric field and magnetic field are iterated alternately according to the FDTD time step rule, and the unstable cells are corrected during the iteration process; S5. Perform a boundary condition convergence check. If the condition does not meet the criteria, return to step S4 to continue iterating. If the condition meets the criteria, the simulation is complete.

2. The three-dimensional FDTD metal conformal stabilization simulation method according to claim 1, characterized in that: Step S3 is verified in the following cases: A. If the simulation scenario involves magnetic materials, the following process is included: For a conformal metal mesh near the surface of a spherical dielectric particle, the relative lengths of the electric field integral lines in the y and z directions are calculated using the following formula. and and relative area : ， ， in, l y and l z These represent the effective lengths of the electric field integral lines on the y-axis and z-axis, respectively. , , and These represent the relative lengths of the electric field integral lines along the positive z-axis, negative z-axis, positive y-axis, and negative y-axis, respectively. A 1 is the area outside the metal within the cut surface; Strict verification and Mark the edge cells that do not meet the stability condition; B. If the simulation scenario involves non-magnetic materials, the following process is included: Calculate the following correction parameters: ， ， in, , , and Cross-sections x The relative lengths of the electric field integral lines along the positive z-axis, negative z-axis, positive y-axis, and negative y-axis. For cross-section x The relative area; Strict verification Mark the edge cells that do not meet the stability conditions.

3. The three-dimensional FDTD metal conformal stabilization simulation method according to claim 2, characterized in that: Step S4 include The following process: S4.1, for the interval [ x i , x i+1 For cells within a given area, the electric field is updated in the following cases: ， in, and They are respectively x Axis direction index is i At the location, Time and The electric field value at time t. For time step, , , and Cross-sections x The positive z-axis direction, negative z-axis direction, positive y-axis direction, and negative y-axis direction Magnetic field components at different times; effective parameters of the average electric field interval The values ​​calculated in step S2 are used in the case of conformal dielectric, and then corrected near the metal boundary using the following formula: ， in, l x For the interval [ x i , x i+1 The effective length outside an ideal conductor; S4.2, for the interval [ x i , x i+1 The cells within the [ ] are updated with magnetic fields in the following cases: A. If the simulation scenario involves magnetic materials, the magnetic field will be updated in the following scenarios: a. Conformal dielectric: ， in, and respectively x Axis direction index is i At the location, Time and The magnetic field value at time [time]. , , and Cross-sections x The positive z-axis direction, negative z-axis direction, positive y-axis direction, and negative y-axis direction The electric field components at time; b. Metal boundary: ； in, and Indicates cross-section x of Time and The magnetic field value at time [time]. and denoted as the effective length of the electric field integral line in the positive and negative z-axis directions; B. If the simulation scenario uses non-magnetic materials, then the magnetic field is updated using the following equation: ， in, μ 0= μ eff ( x ).

4. The three-dimensional FDTD metal conformal stabilization simulation method according to claim 3, characterized in that: For unstable cells in magnetic material scenarios, the magnetic field is updated and then corrected again using the following formula: 。

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