Semi-implicit and full-implicit CD-FDTD electromagnetic simulation processing method and system

Abstract

The invention discloses a semi-implicit and full-implicit CD-FDTD electromagnetic simulation processing method and system, and belongs to the field of electromagnetic simulation. The method comprises the following steps: constructing a Maxwell frequency domain equation of an electric field and a magnetic field in a PML region, converting from a frequency domain to a time domain, and rewriting into a first-order differential matrix form; a differential operator matrix R in the differential matrix form is divided into two sub-matrixes R1 and R2, and then two sub-time step updating forms are constructed; substituting the stretching function in the time domain form and the auxiliary variable into the updating form of the two sub-time steps to obtain an updating expression of the electric field and the magnetic field under the two sub-time steps, and further deriving to obtain a leapfrog updating equation of the electric field and the magnetic field; through a semi-implicit / full-implicit LCD-FDTD method of the HO-PML technology, the leapfrog update equation of the electric field and the magnetic field is deduced as a semi-implicit and full-implicit update equation of the electric field and the magnetic field.

Description

Technical Field

[0001] The present invention relates to the field of electromagnetic simulation, and in particular to a semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method and system. Background Art

[0002] Finite-Difference Time-Domain (FDTD) is a highly efficient electromagnetic numerical computation method that accurately simulates the complex behavior of electromagnetic waves during propagation, reflection, and scattering. It is widely used in research fields such as circuit device simulation, antenna design, and bioelectromagnetics. However, the time step of traditional FDTD methods is strictly limited to the spatial discretization scale, that is, it is restricted by the CFL (Courant-Friedrichs-Lewy) stability condition, resulting in a significant computational cost when simulating multiscale electromagnetic problems.

[0003] In order to break through the limitations of the CFL stability conditions, a variety of semi-implicit and fully implicit fast calculation methods have been developed to achieve efficient electromagnetic simulation of fine structures at different spatial scales. Semi-implicit FDTD methods such as the HIE-FDTD method and the WCS-FDTD method have more relaxed stability conditions than traditional FDTD methods, especially for fine structures in one-way and two-way dimensions. In addition, fully implicit FDTD methods such as the LOD-FDTD method and the ADI-FDTD method have unconditional stability characteristics. These methods usually use implicit formats to discretize the spatial derivatives in the fine grid direction, and use explicit updates in other directions. By optimizing the time step of iterative calculations, these methods significantly improve computational efficiency and have therefore been widely used in electromagnetic simulations. In addition, the effective combination of CPML and the above methods provides an effective solution for solving complex open-domain electromagnetic problems. Unfortunately, it has been proven that the semi-implicit HIE-FDTD method, WCS-FDTD method, fully implicit ADI-FDTD method and LOD-FDTD method do not obey the divergence characteristics in the passive region and violate Gauss's law, which is naturally satisfied by the traditional FDTD method.

[0004] Furthermore, absorbing boundaries truncate unbounded physical domains into finite computational domains by simulating or approximating radiation boundary conditions, making them crucial for analyzing open-space electromagnetic problems. Since Berenger introduced PML technology, it has continued to develop and be widely used due to its exceptional absorption capabilities. Among the various absorbing boundary conditions, CFS-PML technology is considered the most effective, outperforming PML, SC-PML, and UPML technologies in absorbing evanescent waves and reducing late reflections. To further improve absorption performance, PML technologies with higher-order CFS factors have been developed in recent years. Compared to first-order CFS-PML technology, higher-order PML technology demonstrates significant advantages in enhancing the absorption of reflected electromagnetic waves. With the introduction of recursive convolution methods, the implementation of PML in time-domain numerical methods has been further simplified. Currently, these methods have been widely used in various numerical methods, demonstrating their strong application potential in electromagnetic simulation. Summary of the Invention

[0005] In view of the shortcomings of the existing technology, the present invention proposes a semi-implicit and fully implicit CD-FDTD electromagnetic simulation method and system under a unified framework.

[0006] The purpose of the present invention can be achieved through the following technical solutions:

[0007] A semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method includes the following steps:

[0008] In the PML region, the Maxwell frequency domain equations of the electric and magnetic fields are constructed, converted from the frequency domain to the time domain, and rewritten in the first-order differential matrix form;

[0009] The differential operator matrix R in the differential matrix form is divided into two sub-matrices R1 and R2, and then the form of two sub-time step updates is constructed;

[0010] Substituting the stretching function and auxiliary variables in the time domain into the two sub-time step update form, the updated expressions of the electric field and magnetic field under the two sub-time steps are obtained, and then the leapfrog update equations of the electric field and magnetic field are derived;

[0011] The semi-implicit / fully implicit LCD-FDTD method of HO-PML technology is used to derive the leapfrog update equations of the electric and magnetic fields into semi-implicit and fully implicit update equations of the electric and magnetic fields.

[0012] During the iteration process, the semi-implicit and fully implicit update equations update the magnetic field h n , then update the electric field auxiliary variables in the HO-PML region; then display the updated equation to update the electric field E n+1 / 2 , update the electric field e by semi-implicit and fully implicit update equations n+1 / 2 ; Then update the auxiliary variables of the magnetic field in the HO-PML region and update the magnetic field H by displaying the update equation n+1 .

[0013] Optionally, a semi-implicit solution is constructed by implicitly processing the spatial partial derivatives of matrices R1 and R2 in one or two directions; a fully implicit solution is constructed by implicitly processing the spatial partial derivatives of matrices R1 and R2 in three directions.

[0014] Alternatively, the Maxwell frequency domain equations can be expressed in the following matrix form:

[0015]

[0016] The cyclic symbols (i, j, k)∈(x, y, z),(y, z, x),(z, x, y), s e and s h They represent the stretching functions corresponding to the electric field and magnetic field respectively. For the Q-order PML, the stretching function is defined as follows:

[0017]

[0018] in are the parameters in the PML region, the subscript u=x, y, z indicates the direction of the electromagnetic field, and q indicates the HO-PML order;

[0019] make Representing the inverse Fourier transform of the stretching function, formula (1) and formula (2) are converted from the frequency domain to the time domain and rewritten as the first-order differential matrix form:

[0020]

[0021] where U=[E x ,E y ,E z ,H x ,H y ,H z ], as well as

[0022]

[0023] Among them, E x ,E y ,E z ,H x ,H y ,H z Represent the components of the electric field and magnetic field in the x, y, and z directions respectively, μ0 represents the magnetic permeability in free space, ε0 represents the relative dielectric constant, D hThe curl operator, D, represents the magnetic field part modified by the stretching function. e represents the curl operator where the electromagnetic part is modified by the stretch function.

[0024] Optionally, the two sub-timestep updates are of the form:

[0025]

[0026] in:

[0027]

[0028] Where I represents the identity matrix.

[0029] Optionally, the calculation method of the leapfrog update equation includes the following steps:

[0030] The time domain expression of the stretching function is expressed as:

[0031]

[0032] Where δ(t) is the unit pulse function, u(t) represents the unit step function, and the auxiliary variable is defined as:

[0033]

[0034] Substituting equations (8)-(10) into equations (5) and (6), we obtain:

[0035]

[0036] in is the HO-PML modified spatial partial derivative matrix, ψ hq =[ψ hxzq ,ψ hxyq ,ψ hyxq ,ψ hyzq ,ψ hzyq ,ψ hzxq ] T , as well as:

[0037]

[0038]

[0039] The following operations are performed on equations (11)-(14) to eliminate the intermediate processes of the electric and magnetic fields, and the updated equations of the electric and magnetic fields in the leapfrog format are obtained:

[0040]

[0041] Alternatively, the semi-implicit and fully implicit update equation calculation methods for the electric and magnetic fields include the following steps: exchanging the explicit and implicit update equations in equations (19) and (20) to obtain:

[0042]

[0043] By introducing auxiliary variables h and e, equations (21) and (22) are further simplified to obtain the following RHS update structure without matrix operators:

[0044]

[0045] where e=[e x ,e y ,e z ] T ,h=[h x ,h y ,h z ] T ;

[0046] The convolution term ψ in equations (24) and (26) is solved by using the circular convolution technique. eq and ψ hq After processing, we get:

[0047]

[0048] in

[0049] Optionally, the y direction is handled implicitly:

[0050]

[0051] (2) Implicit processing of y and z directions:

[0052]

[0053]

[0054] (3) Fully implicit processing:

[0055]

[0056] where the cyclic symbol (w,v)∈(x,y,z),(y,z,x).

[0057] A second aspect of the present invention relates to a semi-implicit and fully implicit CD-FDTD electromagnetic simulation system under a unified framework, comprising:

[0058] The differential matrix construction module is used to construct the Maxwell frequency domain equations of the electric and magnetic fields in the PML region, convert them from the frequency domain to the time domain, and rewrite them into a first-order differential matrix form;

[0059] A differential operator matrix construction module of a time step update form, used for constructing two sub-time step update forms by dividing the differential operator matrix R in the differential matrix form into two sub-matrices R1 and R2;

[0060] The electric and magnetic field leapfrog update module is used to bring the stretching function and auxiliary variables in the time domain form into the form of the two sub-time step updates to obtain the updated expressions of the electric and magnetic fields in the two sub-time steps, and then derive the leapfrog update equations of the electric and magnetic fields;

[0061] and, a semi-implicit and fully implicit update module for electric and magnetic fields, which is used to derive the leapfrog update equations for electric and magnetic fields into semi-implicit and fully implicit update equations for electric and magnetic fields using the semi-implicit / fully implicit LCD-FDTD method of the HO-PML technique;

[0062] Among them, the semi-implicit and fully implicit update equations update the magnetic field h n , then update the electric field auxiliary variables in the HO-PML region; then display the updated equation to update the electric field E n+1 / 2 , update the electric field e by semi-implicit and fully implicit update equations n+1 / 2 ; Then update the auxiliary variables of the magnetic field in the HO-PML region and update the magnetic field H by displaying the update equation n+1 .

[0063] A third aspect of the present invention relates to a computer-readable storage medium storing instructions, which, when executed, implement the semi-implicit and fully implicit CD-FDTD electromagnetic simulation methods under the above-mentioned unified framework.

[0064] A fourth aspect of the present invention relates to an electromagnetic wave simulation device, comprising the above-mentioned computer-readable storage medium.

[0065] Beneficial effects of the present invention:

[0066] The simulation method presented in this application and its practical application demonstrate that the proposed CFS-PML technique, combined with semi-implicit and fully implicit LCD-FDTD methods, can maintain the original stability conditions. When applied to electromagnetic simulations of a human brain model, the simulation method demonstrates not only excellent absorption performance but also unconditional stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] The present invention will be further described below with reference to the accompanying drawings.

[0068] Figure 1 This is a flow chart of the semi-implicit / fully implicit LCD-FDTD method in Example 1 of the present application.

[0069] Figure 2 The eigenvalues ​​of the matrix M are amplified by the CD-HIE-FDTD method combined with the HO-PML technology in Example 2 of the present application, (a) first-order PML, (b) second-order PML.

[0070] Figure 3 The eigenvalues ​​of the matrix M are amplified by the CD-WCS-FDTD method combined with the HO-PML technology in Example 2 of the present application, (a) first-order PML, (b) second-order PML.

[0071] Figure 4 The eigenvalues ​​of the matrix M are amplified by the LCDI-FDTD method combined with the HO-PML technology in Example 2 of the present application, (a) first-order PML, (b) second-order PML.

[0072] Figure 5 The geometric structure of the low-pass filter in Example 3 of the present application is: W = 20.32 mm, L = 19.4718 mm, L1 = 4.233 mm, L2 = 2.5398 mm, L3 = 8.46 mm, W1 = 5.6896 mm, W2 = 2.4384 mm, h = 0.794 mm.

[0073] Figure 6 This is the relationship between the PML relative reflection error and the time step in the LCD-HIE-FDTD method in Example 3 of the present application.

[0074] Figure 7 These are the S parameters (a) S11 and (b) S21 of the low-pass filter in Example 3 of the present application.

[0075] Figure 8 This is a schematic diagram of the dual-band T-type antenna structure in Example 3 of this application.

[0076] Figure 9 This is the relationship between the PML relative reflection error and the time step in the LCD-WCS-FDTD method in Example 3 of the present application.

[0077] Figure 10 This is the relationship between the PML relative reflection error and the time step in the LCD-WCS-FDTD method in Example 3 of the present application.

[0078] Figure 11 This is the three-dimensional human brain geometric model in Example 3 of this application.

[0079] Figure 123. The relationship between the PML relative reflection error and the time step in the LCDI-FDTD method in Example 3 of the present application: (a) CFLN=1; (b) CFLN=4, (c) CFLN=8.

[0080] Figure 13 Figure 3 shows the specific absorption rate (SAR) results calculated using different methods in Example 3 of this application. (a) LCDI-FDTD (CFLN = 1) method with a relatively large simulation space; (b) CDI-FDTD (CFLN = 1) method based on HO-PML technology; (c) CDI-FDTD (CFLN = 4) method based on HO-PML technology; (d) CDI-FDTD (CFLN = 8) method based on HO-PML technology. DETAILED DESCRIPTION

[0081] 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 any creative efforts shall fall within the scope of protection of the present invention.

[0082] Example 1: High-order PML implementation of semi-implicit / fully implicit LCD-FDTD method

[0083] In the PML region, the Maxwell frequency domain equation including the scaling factor correction can be expressed in the following matrix form:

[0084]

[0085] where the cyclic symbols (i,j,k)∈(x,y,z),(y,z,x),(z,x,y).s e and s h They represent the stretching functions corresponding to the electric field and magnetic field respectively. For the Q-order PML, the stretching function is defined as follows:

[0086]

[0087] in are parameters in the PML region, the subscript u = x, y, z represents the direction of the electromagnetic field, and q represents the HO-PML order.

[0088] make Denote the inverse Fourier transform of the stretching function, and then convert (1) and (2) from the frequency domain to the time domain and rewrite them into a compact first-order differential matrix form, we get:

[0089]

[0090] where U=[E x ,E y ,E z ,H x ,H y ,H z ],

[0091] Based on the Peaceman-Rachford discrete strategy, within a time step Δt, the spatial partial differential operator matrix R in equation (4) can be divided into two sub-matrices R1 and R2, and then the form of two sub-time step updates is constructed, namely:

[0092]

[0093] in:

[0094]

[0095] Where I represents the 6×6 identity matrix. The matrix forms of R1 and R2 vary, and their choice determines the implicit treatment of the update equation. To illustrate this more clearly, Table 1 lists the construction of matrices R1 and R2 when establishing implicit equations in the y-direction, the y and z-directions, and the x, y, and z-directions, respectively.

[0096] Table 1 Construction of R1 and R2 in semi-implicit / fully implicit schemes

[0097]

[0098] According to the different implicit processing methods in Table 1, the method of implicit processing in a single direction or two directions is called a semi-implicit scheme; if implicit processing is used for the spatial partial derivatives in all three directions, it is called a fully implicit scheme.

[0099] According to equation (3), the time domain expressions of the stretching functions in equations (5) and (6) can be expressed as:

[0100]

[0101] Where δ(t) is the unit pulse function, u(t) represents the unit step function, and the auxiliary variable is defined as:

[0102]

[0103] also, It can also be obtained in the same way.

[0104] Substituting equations (8)-(10) into equations (5) and (6), we obtain:

[0105]

[0106] in is the HO-PML modified spatial partial derivative matrix, ψ hq =[ψ hxzq ,ψ hxyq ,ψ hyxq ,ψ hyzq ,ψ hzyq ,ψ hzxq ] T , as well as:

[0107]

[0108] Next, we perform the following operations on Equations (11)-(14) to eliminate the intermediate processes of the electric and magnetic fields, and obtain the updated equations of the electric and magnetic fields in the leapfrog format.

[0109] Substitute equation (14) into equation (13) and eliminate H n+1 / 2 , and step back one sub-time step, we get:

[0110]

[0111] Then, substitute equation (12) into equation (11) and eliminate H n+1 / 2 ,get:

[0112]

[0113] Subtracting Equation (17) from Equation (18) yields the leapfrog update equation for the electric field from time n-1 / 2 to time n+1 / 2:

[0114]

[0115] Similarly, the following steps can be performed to derive the updated equations for the magnetic field from time n to time n+1:

[0116] Step 1) Substitute equation (11) into equation (12) and eliminate E n ;

[0117] Step 2) Substitute (13) into equation (14) and eliminate E n+1 ;

[0118] Step 3) Subtract the two expressions obtained in Steps 1 and 2. The magnetic field leapfrog update equation is as follows:

[0119]

[0120] Equations (19) and (20) form a leapfrog scheme for the electric and magnetic fields, so that the electromagnetic field only needs to be updated once in a sub-time step.

[0121] To further derive the semi-implicit / fully implicit LCD-FDTD method combined with the HO-PML technique, it is necessary to exchange the explicit and implicit update equations in equations (19) and (20), thus obtaining:

[0122]

[0123] By introducing auxiliary variables h and e, equations (21) and (22) are further simplified to obtain the following RHS update structure without matrix operators:

[0124]

[0125] where e=[e x ,e y ,e z ] T ,h=[h x ,h y ,h z ] T .

[0126] The convolution term ψ in equations (24) and (26) is solved by using the circular convolution technique. eq and ψ hq After processing, we get:

[0127]

[0128] in

[0129] Equations (23)-(28) form the update formulas for the semi-implicit / fully implicit LCD-FDTD method combined with the HO-PML technique. This method has a leapfrog time-stepping form, satisfies the divergence characteristics of the electromagnetic field, and has an update structure with no matrix operators on the RHS.

[0130] Applying the matrix operators for constructing semi-implicit / fully implicit equations introduced in Table 1 to equations (23) and (25), we obtain different forms of auxiliary variable iteration formulas, as follows:

[0131] (1) Implicit processing of the y direction:

[0132]

[0133] (2) Implicit processing of y and z directions:

[0134]

[0135]

[0136] (3) Fully implicit processing:

[0137]

[0138] where the cyclic symbol (w,v)∈(x,y,z),(y,z,x).

[0139] Furthermore, substituting the matrix operators for constructing semi-implicit / fully implicit equations introduced in Table 1 into Equations (24) and (26), we obtain the iterative equations for the electric and magnetic fields:

[0140]

[0141] Equations (27)-(33) give the updated equations of the electromagnetic field components of the semi-implicit / fully implicit LCD-FDTD method in the HO-PML region. Figure 1 Schematic diagram showing the execution flow of the semi-implicit / fully implicit LCD-FDTD method.

[0142] Example 2

[0143] This embodiment specifically analyzes the stability and compliance divergence in embodiment 1.

[0144] A. Stability Analysis

[0145] In this embodiment, the von Neumann method is used to analyze the numerical stability of the semi-implicit and fully implicit LCD-FDTD methods combined with high-order PML technology. Taking the second-order PML as an example, according to equations (27)-(33), its update formula can be expressed in the following matrix form:

[0146]

[0147] Among them U pml =[E,H,ψ e1 ,ψ e2 ,ψ h1 ,ψ h2 ] T ,M is the magnification matrix, coefficient matrix M L and M R for:

[0148]

[0149] in

[0150]

[0151] In addition, according to equations (29)-(31), when different implicit schemes are implemented, the relationship between the auxiliary variables e, h and the electromagnetic field components E, H is different, resulting in the matrix in equation (34) Next, we conduct an in-depth analysis of the numerical stability of the three different implicit forms of LCD-FDTD algorithms in the PML region.

[0152] (1) CD-HIE-FDTD method combined with HO-PML technology

[0153] According to the relationship between the auxiliary variables e, h and the electromagnetic field components E, H in equation (36), the matrix can be determined The specific form is:

[0154]

[0155]

[0156] where δ u Represents the first-order spatial partial derivative operator. To achieve numerical discretization, the derivative is approximated by central difference, so δ u =2jsin(k u Δu / 2) / Δu,k u represents the wave number in the u direction,

[0157] (2) CD-WCS-FDTD method combined with HO-PML technology

[0158] According to the relationship between the auxiliary variables e, h and the electromagnetic field components E, H in equation (37), the matrix can be determined The specific form is

[0159]

[0160] (3) CDI-FDTD method in the HO-PML region

[0161] According to the relationship between the auxiliary variables e, h and the electromagnetic field components E, H in equation (38), the matrix can be determined The specific form is

[0162]

[0163]

[0164] According to the von Neumann stability analysis method, all eigenvalues ​​of the magnified matrix M in Equation (34) must be no greater than 1 to ensure numerical stability during the algorithm's iterations. Due to the large dimensions of the magnified matrix, commonly used computing software such as Mathematica and Maple have difficulty directly determining the range of its eigenvalues. Therefore, this paper numerically solves the eigenvalues ​​by substituting specific values.

[0165] For the case where the update equations are implicit in the y direction and explicit in the x and z directions, the allowed time step size for the LCD-HIE-FDTD method is: Δt HIE ≤min[Δx / c0,Δz / c0]. For the case where the update equations are implicit in the y and z directions and explicit in the x direction, the time step allowed by the LCD-WCS-FDTD method is: Δt HIE ≤Δx / c0. For the case where the update equation is fully implicit, the LCDI-FDTD method is unconditionally stable. Therefore, in the stability analysis of the first-order and second-order PML, the grid size and time step of the three methods are set as:

[0166] (1) LCD-HIE-FDTD method: Δx=0.4064mm, Δy=0.0794mm, Δz=0.4233mm, Δt=Δx / c0;

[0167] (2) LCD-WCS-FDTD method: Δx=0.75mm, Δy=0.25mm, Δz=0.2mm, Δt=Δx / c0;

[0168] (3) LCDI-FDTD method: Δx=2mm, Δy=2mm, Δz=2mm, Δt=CFLN×Δt FDTD , where CFLN is set to 8, is the maximum time step allowed by the traditional FDTD method.

[0169] Furthermore, in the PML region, the stretching function defined in equation (3) satisfies the following conditions: That is, satisfying κ u ≥1,α u ≥0, and σ u ≥ 0. In the first-order PML, the relevant parameters are defined as follows: In the second-order PML, the relevant parameters are defined as follows: In order to further analyze its stability, the parameter range of the first-order scaling factor is set to: m∈(1,6),k max ∈(1,30),α max ∈(0,0.25),σ facter ∈(0.5,5). The parameter range of the second-order scaling factor is set to: m1, m2, m3∈(1,6), k max ∈(1,30),σ facter1 and σ facter2 ∈(0.1,5),α0∈(0,0.5).

[0170] Finally, these parameters are substituted into Eq. (34) and a large set of random wave numbers k u Calculate the growth factor above. Figure 2-4 The eigenvalues ​​of the amplified matrix M using three LCD-FDTD methods combined with the HO-PML technique are demonstrated. It can be seen that all eigenvalues ​​lie within the unit circle of the complex plane, indicating that the modulus of the eigenvalues ​​of this method, |ζ|, is ≤ 1, thus verifying the numerical stability of the proposed method. In summary, the stability analysis above demonstrates that the proposed CFS-PML technique, combined with the semi-implicit and fully implicit LCD-FDTD methods, can maintain the original stability conditions.

[0171] B. Subject to divergence analysis

[0172] The Gaussian divergence theorem is not only a key property in electromagnetic theory, but also an important component of Maxwell's equations. Therefore, maintaining the divergence property is essential in the numerical discretization process.

[0173] According to equation (32), the updated equation of the electric field is:

[0174]

[0175] in:

[0176]

[0177] In the PML region, the divergence relationship of the electric flux density can be expressed as:

[0178]

[0179] Substituting equation (35) into equation (37), we obtain:

[0180]

[0181] in According to equation (36), we get:

[0182]

[0183] According to the vector identity, Therefore, equation (38) can be written as:

[0184]

[0185] Equation (40) shows that in the passive region, the electric flux density diverges to 0 with numerical iteration. This shows that the semi-implicit and fully implicit LCD-FDTD methods combined with the HO-PML technique satisfy the Gaussian divergence theorem.

[0186] Example 3

[0187] To comprehensively evaluate the applicability and effectiveness of semi-implicit and fully implicit LCD-FDTD methods in conjunction with high-order PML (HO-PML) absorbing boundary conditions, this example conducts simulation studies on a typical open-domain electromagnetic problem with multi-scale fine structures. For electromagnetic simulation models with fine structures in one direction, two directions, and three directions, the corresponding LCD-FDTD algorithms are used for modeling and calculation. This systematically evaluates the applicability and accuracy of the method described in Example 1 under various complex scenarios.

[0188] A. High-order PML implementation of LCD-HIE-FDTD method

[0189] In the first case, in order to verify the effectiveness of the high-order PML technology in the LCD-HIE-FDTD method, a unidirectional microstrip filter with a fine structure is used for electromagnetic simulation analysis. Figure 5 As shown in the figure, the low-pass filter is composed of a microstrip patch attached to a dielectric substrate with a dielectric constant of 2.2. The structure is a two-port circuit, one end of which is connected to a voltage source with an internal resistance of 50Ω. The other end is connected to a 50 ohm resistor. The entire computational domain is 80 × 70 × 76 Yee cells, containing 10 layers of PML. The spatial grid size is Δx = 0.4064 mm, Δy = 0.0794 mm, Δz = 0.4233 mm. The time step is Δt HIE =Δx / c0.

[0190] In order to effectively truncate the problem space, this embodiment provides details of the optimized PML parameter selection: (1) For the first-order PML, k max =7,α max =0.05,σ facter =1.3.(2) For high-order PML, the parameters are selected as follows: k max,2 =15, m1=6, m2=3, m3=2, α0=0.09, σ facter1 =0.175,σ facter2=2.75. In addition, to further quantify the absorption performance of PML, this paper defines the relative reflection error (RE), which is calculated as follows:

[0191]

[0192] Where E pml represents the electric field value at the position (31,21,11), which is located at the center of the xoz plane where the voltage source is located and is only one Yee cell away from the PML boundary. In addition, E ref It is a reference value for the electric field in a fairly large simulation domain.

[0193] Table 2THE CPU TIME FOR COMPUTATIONS USING VARIOUS METHODS

[0194]

[0195] Figure 6 The relative reflection error at the detection point under the 10-layer (First order-PML) FO-PML, 8-layer and 10-layer (Highorder-PML) HO-PML absorption boundaries in the CD-HIE-FDTD algorithm is shown. The results show that the high-order PML has a better absorption performance. Even when only 8 layers are used, its absorption effect is comparable to that of the 10-layer first-order PML, or even slightly better. In addition, Table 2 shows the performance of the above three PML configurations in terms of maximum reflection error, CPU time and memory usage. The results show that compared with the 10-layer FO-PML absorption boundary, the 8-layer HO-PML boundary memory and calculation time are reduced by 9.5% and 10.6% respectively. Finally, Figure 7 (a) and (b) show the S11 and S21 curves, respectively, calculated using the LCD-HIE-FDTD method under 10-layer FO-PML and HO-PML boundary conditions and a relatively large simulation volume. As can be seen, the results calculated using the LCD-HIE-FDTD method under the 10-layer HO-PML boundary condition are highly consistent with the reference solution, demonstrating that the LCD-HIE-FDTD method combined with the HO-PML technique can effectively simulate open-domain problems.

[0196] B. High-order PML implementation of LCD-WCS-FDTD method

[0197] Next, to verify the effectiveness of the high-order PML technology in the LCD-HIE-FDTD method, a dual-T antenna with fine structures in two directions was used for electromagnetic simulation analysis. Figure 8As shown, it consists of a double T-shaped patch attached to a dielectric substrate with a dielectric constant of 4.4. Its specific geometric dimensions are listed in Table 2. The entire computational domain occupies 96 × 251 × 55 Yee grids in the x, y, and z directions, including 10 layers of PML absorbing boundaries. To obtain the lowest reflection, the details of the PML parameters are as follows:

[0198] 1) For SC-PML, the parameter is set to k max =1,σ facter =1.5.

[0199] 2) For the first-order PML, the parameter is set to k max =7,α max =0.05,σ facter =1.5.

[0200] 3) For the second-order PML parameters, set to k max,2 =7, m1=5.2, m2=6, m3=2.8, α0=0.05, σ facter1 =0.001,σ facter2 =2.75. In addition, the spatial grid is set to Δx = 0.75 mm, Δy = 0.25 mm, Δz = 0.2 mm. The time step is set to Δt WCS =Δx / c0. The antenna is excited by a voltage source with an internal resistance of 50Ω, and the excitation signal is a Gaussian pulse. Its parameter is τ = 5.6289e -11 s,t0=4.5τ.

[0201] Table 3DIMENSIONS OF PRINTED DUAL-BAND DOUBLE-T MONOPOLE ANTENNA(UNIT:mm)

[0202]

[0203] Figure 9 The relative reflection error of the LCD-WCS-FDTD method under 10-layer SC-PML, FO-PML and HO-PML boundary conditions is shown. It can be seen that compared with SC-PML and first-order PML, high-order PML (HO-PML) has more obvious advantages in absorption performance and can significantly reduce reflection error. In addition, Figure 10The S11 results of the LCD-WCS-FDTD method are presented for a 10-layer FO-PML and HO-PML boundary conditions over a relatively large computational domain. The results of the LCD-WCS-FDTD method combined with the HO-PML technique are consistent with those of the reference solution. These numerical results further validate the effectiveness of the high-order HO-PML implementation in the LCD-WCS-FDTD method.

[0204] C. High-order PML implementation of LCDI-FDTD method

[0205] Finally, to verify the effectiveness of the high-order PML technology in the LCDI-FDTD method, this example performs electromagnetic simulation analysis on a human brain model with fine structures in three directions. The simulation environment is as follows: Figure 11 As shown. The human brain model comes from the IEEE database. In view of the complex composition of human brain tissue, eight typical tissues such as skin, fat, blood, cerebrospinal fluid, white matter, dura mater, gray matter and skull are selected for modeling. The electromagnetic parameters of each tissue are set as follows: the relative dielectric constants are 20, 3, 8, 1, 8, 22, 1, and 4 respectively. The conductivity is 0.40, 0.026, 1.5, 2.2, 0.3, 0.62, 0.5, and 0.065s / m respectively. In the simulation, the entire calculation domain occupies 155×180×130 Yee grids in the x, y, and z directions. The size of the Yee grid is 2mm×2mm×2mm, and 10 layers of PML absorption boundaries are set at the boundaries. In order to obtain the lowest reflection, the details of the PML parameters are as follows:

[0206] 1) For SC-PML, the parameter is set to k max =1,σ facter =1.1.

[0207] 2) For the first-order PML, the parameter is set to k max =7,α max =0.05,σ facter =1.3.

[0208] 3) For the second-order PML parameters, set to k max,2 =14, m1=2, m2=4, m3=2, α0=0.06, σ facter1 =0.075,σ facter2 =4.

[0209] A plane wave Gaussian pulse with a center frequency of 1 GHz is introduced into the computational domain using the total field / scattered field (TF / SF) technique. The detection point is located above the human brain (in the scattered field region), two grid cells away from the innermost PML boundary. Figure 12Figures (a)-(c) show the relative reflection errors relative to the reference solution for the LCDI-FDTD method (CFLN = 1, 4, and 8) using SC-PML boundaries, first-order PML boundaries, and higher-order PML boundaries. It can be seen that compared to SC-PML boundaries, first-order PML boundaries significantly improve the suppression of late reflections. Furthermore, higher-order PML boundaries further enhance the absorption performance.

[0210] Then, based on the LCDI-FDTD method with high-order PML (CFLN=1, 4, 8) and the reference method, the electric field components were calculated by expanding the original simulation space by 75 grids in the x, y, and z directions and using the LCDI-FDTD method with 20 layers of HO-PML as the absorbing boundary (CFLN=1). The specific absorption rate (SAR) was calculated by substituting them into the following formula:

[0211]

[0212] Where ρ is the mass density. Figure 13 It can be seen that under different CFLN conditions, the SAR distributions of the LCDI-FDTD method based on HO-PML technology are highly consistent with those of the analytical method, which fully verifies that the high-order PML implementation of the LCDI-FDTD method not only has excellent absorption performance but also has unconditional stability.

[0213] Throughout this specification, references to terms such as "one embodiment," "example," or "specific example" indicate that the specific features, structures, materials, or characteristics described in conjunction with that embodiment or example are included in at least one embodiment or example of the present invention. In this specification, schematic representations of these terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0214] The basic principles, main features, and advantages of the present invention are shown and described above. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are merely illustrative of the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention, and such changes and modifications fall within the scope of the invention as claimed.

Claims

1. A semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method, characterized in that: The following steps are involved: In the PML region, the Maxwell frequency domain equations of the electric and magnetic fields are constructed, converted from the frequency domain to the time domain, and rewritten in the form of a first-order differential matrix; The differential operator matrix R in the differential matrix form is divided into two sub-matrices R1 and R2, and then the form of two sub-time step updates is constructed; Substituting the stretching function and auxiliary variables in the time domain into the two sub-time step update form, the updated expressions of the electric field and magnetic field under the two sub-time steps are obtained, and then the leapfrog update equations of the electric field and magnetic field are derived; The semi-implicit / fully implicit LCD-FDTD method of HO-PML technology is used to derive the leapfrog update equations of the electric and magnetic fields into semi-implicit and fully implicit update equations of the electric and magnetic fields. During the iteration process, the semi-implicit and fully implicit update equations update the magnetic field h n , then update the electric field auxiliary variables in the HO-PML region; then display the updated equation to update the electric field E n+1 / 2 , update the electric field e by semi-implicit and fully implicit update equations n+1 / 2 ; Then update the auxiliary variables of the magnetic field in the HO-PML region and update the magnetic field H by displaying the update equation n+1 .

2. The semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to claim 1 is characterized in that: A semi-implicit solution is constructed by implicitly processing the spatial partial derivatives of matrices R1 and R2 in one or two directions; a fully implicit solution is constructed by implicitly processing the spatial partial derivatives of matrices R1 and R2 in three directions.

3. The semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to claim 1 is characterized in that: Maxwell frequency domain equation, expressed in the following matrix form: The cyclic symbols (i, j, k)∈(x, y, z),(y, z, x),(z, x, y), s e and s h They represent the stretching functions corresponding to the electric field and magnetic field respectively. For the Q-order PML, the stretching function is defined as follows: in are the parameters in the PML region, the subscript u=x, y, z indicates the direction of the electromagnetic field, and q indicates the HO-PML order; make Representing the inverse Fourier transform of the stretching function, formula (1) and formula (2) are converted from the frequency domain to the time domain and rewritten as the first-order differential matrix form: where U = [E x , E y , E z , H x , H y , H z , Among them, E x ,Ey,E z ,H x ,Hy,H z Represent the components of the electric field and magnetic field in the x, y, and z directions respectively, μ0 represents the magnetic permeability in free space, ε0 represents the relative dielectric constant, D h The curl operator, D, represents the magnetic field part modified by the stretching function. e represents the curl operator where the electromagnetic part is modified by the stretch function.

4. The semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to claim 1 is characterized in that: The two sub-time steps are updated in the following forms: in: Where I represents the identity matrix.

5. The semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to claim 4 is characterized in that: The calculation method of the leapfrog update equation comprises the following steps: Obtain the time domain expression of the stretching function: Where δ(t) is the unit impulse function, u(t) represents the unit step function, and the auxiliary variable is defined as: Substituting equations (8)-(10) into equations (5) and (6), we obtain: in is the HO-PML modified spatial partial derivative matrix, ψ hq =[ψ hxzq ,ψ hxyq ,ψ hyxq ,ψ hyzq ,ψ hzyq ,ψ hzxq ] T , as well as: The following operations are performed on equations (11)-(14) to eliminate the intermediate processes of the electric and magnetic fields, and the updated equations of the electric and magnetic fields in the leapfrog format are obtained:

6. The semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to claim 5, characterized in that: The method for calculating the semi-implicit and fully implicit update equations for the electric and magnetic fields comprises the following steps: exchanging the explicit and implicit update equations in equations (19) and (20) to obtain: By introducing auxiliary variables h and e, equations (21) and (22) are further simplified to obtain the following RHS update structure without matrix operators: where \(e = [e x , e y , e z T , \(h = [h x , h y , h z T ;\​​ The convolution term ψ in equations (24) and (26) is solved by using the circular convolution technique. eq and ψ hq After processing, we get: in 7. The semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to claim 1 is characterized in that: The implicit processing methods in the y direction include: Implicit processing methods for the y and z directions include: The fully implicit processing methods include: where the cyclic symbol (w,v)∈(x,y,z),(y,z,x).

8. A semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing system, characterized by: include: The differential matrix construction module is used to construct the Maxwell frequency domain equations of the electric and magnetic fields in the PML region, convert them from the frequency domain to the time domain, and rewrite them into a first-order differential matrix form; A differential operator matrix construction module of a time step update form, used for constructing two sub-time step update forms by dividing the differential operator matrix R in the differential matrix form into two sub-matrices R1 and R2; The electric and magnetic field leapfrog update module is used to bring the stretching function and auxiliary variables in the time domain form into the form of the two sub-time step updates to obtain the updated expressions of the electric and magnetic fields in the two sub-time steps, and then derive the leapfrog update equations of the electric and magnetic fields; and, a semi-implicit and fully implicit update module for electric and magnetic fields, which is used to derive the leapfrog update equations for electric and magnetic fields into semi-implicit and fully implicit update equations for electric and magnetic fields using the semi-implicit / fully implicit LCD-FDTD method of the HO-PML technique; Among them, the semi-implicit and fully implicit update equations update the magnetic field h n , then update the electric field auxiliary variables in the HO-PML region; then display the updated equation to update the electric field E n+1 / 2 , update the electric field e by semi-implicit and fully implicit update equations n+1 / 2 ; Then update the auxiliary variables of the magnetic field in the HO-PML region and update the magnetic field H by displaying the update equation n+1 .

9. A computer-readable storage medium storing instructions, characterized in that: When the instructions are executed, the semi-implicit and fully implicit CD-FDTD electromagnetic simulation processing method according to any one of claims 1 to 7 is implemented.

10. An electromagnetic wave simulation device, characterized in that: The computer-readable storage medium of claim 9 is included.

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