Three-dimensional biological electromagnetic material electrical characteristic uncertainty quantification method based on S-CDI-FDTD

Abstract

The invention discloses an S-CDI-FDTD-based three-dimensional biological electromagnetic material electrical characteristic uncertainty quantification method, and belongs to the field of biological electromagnetism. The method comprises the following steps: acquiring a CDI-FDTD update equation of an electric field and a magnetic field; introducing random variables and a function of the random variables, and deriving through a Delta method to obtain an expression containing correlation coefficients among the random variables; taking samples of the electric field, the auxiliary electric field, the magnetic field and the auxiliary magnetic field hx, the dielectric constant and the conductivity as random variables to be substituted into the expression; and representing a field update equation of the mean value and the variance of the S-CDI-FDTD through a correlation coefficient between random variables. According to the three-dimensional uncertainty quantification method, by establishing system coupling of the random variable and the CDI-FDTD implicit update equation, on the premise that the divergence characteristic is obeyed, the electromagnetic field statistical characteristic is obtained through single simulation, efficient and accurate analysis can be conducted on a three-dimensional complex biological electromagnetic model, and the time step length constraint of the CFL stability condition is broken through.

Description

Technical Field

[0001] The present invention relates to the field of bioelectromagnetism, and in particular to a method for quantifying uncertainty in the electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD. Background Art

[0002] In bioelectromagnetic research, the electromagnetic parameters (including relative permittivity and conductivity) of complex media within organisms (such as tissues and cells) exhibit significant uncertainty due to environmental conditions. This parameter fluctuation can directly lead to significant shifts in the electromagnetic field distribution, thereby affecting the accuracy of medical diagnostic equipment (such as MRI and electrical stimulation devices). Therefore, how to efficiently quantify the uncertainty in the electrical parameters of three-dimensional bioelectromagnetic materials and their impact on the statistical characteristics of the electromagnetic field has become a key issue in improving the reliability of bioelectromagnetic technology.

[0003] The finite-difference time-domain (FDTD) method, currently a mainstream tool for numerical electromagnetic field analysis, has inherent limitations in modeling random media: traditional algorithms use fixed mean values ​​for the dielectric electrical parameters for deterministic simulation, resulting in calculations that fail to represent the statistical characteristics of the electromagnetic field in real-world scenarios. To overcome this technical bottleneck, a series of improvements have been proposed in recent years, including methods for setting upper and lower limits on dielectric electrical parameters, the polynomial chaos finite-difference time-domain method (PCM-FDTD), the Monte Carlo finite-difference time-domain method (MC-FDTD), and stochastic FDTD (S-FDTD).

[0004] The medium electrical parameter upper and lower limit setting method calculates the field values ​​of different random medium models when the electrical parameters take on maximum, minimum, and average values, thereby determining the fluctuation range of the field values ​​when electromagnetic waves propagate through the random medium. While this method can initially demonstrate the trend of field value changes, due to the complexity of electromagnetic field coupling, the selected example model may not reflect the overall distribution range of the actual field. Furthermore, statistical results such as standard deviation cannot be obtained through limited model simulations, resulting in limitations in the description of field value changes. The PCM-FDTD method combines polynomial chaos theory with FDTD, introducing random factors into the algorithm and converting it into a stochastic differential equation. Mathematical properties such as the orthogonality of the polynomial basis set are used to simplify the equation into a deterministic differential equation that can be numerically solved, thus solving the problem of obtaining statistical results. However, this method is still limited by the number of uncertain input parameters it can handle.

[0005] To address this limitation, the MC-FDTD method randomly generates multiple sets of parameter samples that conform to the statistical characteristics of the medium's electrical parameters. It then performs FDTD field calculations on each set of parameters and calculates the mean and variance. However, this method requires numerous repeated simulations, resulting in lengthy computational times and low efficiency.

[0006] The S-FDTD method uses the Delta method (based on the principle of Taylor polynomial expansion) to obtain the mean and standard deviation of the field values ​​in a single simulation, effectively solving the problem of low computational efficiency of the MC-FDTD method. However, similar to traditional FDTD, the time step of S-FDTD is determined by the Courant-Friedrichs-Lévy (CFL) condition of the finest spatial discretization size. Therefore, when the time step exceeds the maximum time step determined by the CFL condition, the calculation result may grow infinitely.

[0007] To overcome the CFL condition limitations of traditional explicit FDTD, researchers have developed novel algorithms such as weak conditional stability (WCS-FDTD) and hybrid implicit-explicit (HIE-FDTD). KaiKun Niu's team extended the HIE-FDTD method to random media and proposed the S-HIE-FDTD method, which significantly improved computational efficiency. However, when dealing with models with complex three-dimensional geometries, this efficiency improvement is constrained by the increased spatial discretization. To further improve computational efficiency, researchers have proposed unconditionally stable FDTD algorithms, including the alternating direction implicit (ADI-FDTD) and local one-dimensional (LOD-FDTD) methods. ADI-FDTD uses an alternating direction implicit iterative strategy to update the electromagnetic field, while LOD-FDTD achieves local one-dimensional processing through spatial decoupling, offering higher computational efficiency. However, the traditional ADI / LOD-FDTD methods suffer from the significant flaw of not satisfying Gauss's law during the discretization process, which prevents the strict compliance of charge conservation properties. To address this, researchers have proposed a divergence-obeying implicit finite-difference time-domain (CDI-FDTD) method. This method has the following advantages: unconditional stability, compliance with divergence, no RHS matrix operator, single-step frog leaping scheme, etc. These features make the CDI-FDTD method more accurate, efficient and applicable in electromagnetic simulation, especially when simulating complex lossy media with three-dimensional fine structures.

[0008] Therefore, based on the analysis of the defects of existing technologies, the quantification of the electromagnetic properties of three-dimensional random media still faces three major technical bottlenecks: 1) The explicit algorithm is constrained by the CFL stability condition, resulting in low efficiency in large-scale simulations; 2) The implicit algorithm sacrifices the charge conservation property while improving efficiency; 3) The existing technology has not yet achieved the effective integration of random analysis methods and unconditionally stable FDTD algorithms (especially in three-dimensional bio-electromagnetic modeling scenarios). Summary of the Invention

[0009] In view of the shortcomings of the existing technology, the present invention proposes a method for quantifying the uncertainty of the electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD.

[0010] The purpose of the present invention can be achieved through the following technical solutions:

[0011] A first aspect of the present invention relates to a method for quantifying uncertainty in the electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD, comprising the following steps:

[0012] Obtain the CDI-FDTD update equations for the electric and magnetic fields;

[0013] Introduce random variables and functions of random variables, and derive the expression containing the correlation coefficient between random variables through the Delta method;

[0014] The electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r and conductivity σ e Substitute into the expression as a random variable;

[0015] Through the electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r and conductivity σ e The correlation coefficient between them represents the field update equations of the mean and variance of S-CDI-FDTD.

[0016] Optionally, the CDI-FDTD update equations of the electric field and magnetic field are:

[0017]

[0018] Where ε0 and μ0 are the permittivity and permeability of free space, respectively, and ε r and σ e are the relative permittivity and conductivity, Δt is the time step, Δx, Δy and Δz are the spatial steps, the subscript n and the subscripts i, j, k indicate that the relevant field variables are sampled at time nΔt and position (iΔx, jΔy, kΔz) respectively.

[0019] Optionally, the step of deriving an expression containing correlation coefficients between random variables by using a Delta method includes:

[0020] g(X1,...,X N ) represents random variables X1,...,X N The function of Using the expected value operator The linear property of where a iIs a constant, according to the Delta method, ignoring the high-order terms in the Taylor series, we get:

[0021]

[0022] g(X1,...,X N ) is expressed as:

[0023]

[0024] where σ[·] and σ 2 [·] denotes the standard deviation and variance operators respectively;

[0025] Using the Delta method and following the derivation in , we obtain:

[0026]

[0027] For the variance operation, the following identity holds:

[0028]

[0029] in is the covariance, is a random variable X i With X j The correlation coefficient between .

[0030] Optionally, the derivation method of the mean field equation includes the following steps:

[0031] Assume that the random function is defined by formula (1); apply formula (2) to formula (1), and we get:

[0032]

[0033]

[0034] in and and They are ε r and σ e The mean of the electric field and magnetic field is the unknown quantity that needs to be updated as time advances.

[0035] Alternatively, assuming that the correlation coefficient of fields of the same type separated by a single time step or a single spatial step in any direction is equal to 1, we obtain:

[0036]

[0037] Optionally, the derivation method of the electric field variance field equation includes the following steps: assuming that the random function g(·) is defined by formula (1a); taking the variance of both sides of formula (1a):

[0038]

[0039] Applying equations (5) and (7) to the left side of equation (9) yields:

[0040]

[0041] for Term, apply formula (4) and formula (7), and get:

[0042]

[0043] For the right side of Equation (9), apply Equation (5) and Equation (7) multiple times and use Equation (4) to obtain:

[0044]

[0045] In formulas (11) and (12):

[0046]

[0047] in, and They are ε r and σ e The standard deviation of ; Formulating on the right side of formula (12) yields:

[0048]

[0049] Let the completed square term be equal to a 2 , the last term is equal to b:

[0050]

[0051]

[0052] Therefore, formula (13) becomes:

[0053] a 2 +b

[0054] Extract common factor a 2 have to:

[0055]

[0056] Take the square root of the resulting equation and use and ignoring the higher-order terms b / 2a, we get:

[0057] Substituting equations (10) and (11) into the left side of equation (9) and equation (14) into the right side, we obtain:

[0058]

[0059] In formula (15), the standard deviation of the electric field and the magnetic field is the unknown quantity to be updated in the time iteration process. Optionally, the magnetic field variance field equation is:

[0060] Assume that the random function g(·) is defined by formula (1); take the variance on both sides of formula (1):

[0061]

[0062] Applying (5) to both sides of (16) and using (7) in the resulting equation yields:

[0063]

[0064] In formula (17):

[0065]

[0066] Where, in (17b), the coefficient and The subscript position indices should be (i,j,k+1 / 2) and (i,j+1,k+1 / 2) respectively.

[0067] The second aspect of the present invention relates to a system for quantifying uncertainty in the electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD, comprising:

[0068] CDI-FDTD update equation construction module for electric and magnetic fields, used to obtain CDI-FDTD update equations for electric and magnetic fields;

[0069] Random function building module is used to introduce random variables and functions of random variables, and derive the expression containing the correlation coefficient between random variables through the Delta method; and the electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r and conductivity σ e Substitute into the expression as a random variable;

[0070] And, the field update equation derivation module of mean and variance is used to pass the electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h xThe sample and dielectric constant ε r and conductivity σ e The correlation coefficient between them represents the field update equations of the mean and variance of S-CDI-FDTD.

[0071] A third aspect of the present invention relates to a computer-readable storage medium storing instructions, which, when executed, can implement the above-mentioned S-CDI-FDTD-based method for quantifying the uncertainty of electrical properties of three-dimensional bioelectromagnetic materials.

[0072] A fourth aspect of the present invention relates to a biomaterial electromagnetic simulation device, comprising the above-mentioned computer-readable storage medium or the above-mentioned three-dimensional bioelectromagnetic material electrical property uncertainty quantification system based on S-CDI-FDTD.

[0073] Beneficial effects of the present invention:

[0074] This paper proposes a three-dimensional uncertainty quantification method based on the Stochastic Complying Divergence Implicit FDTD (S-CDI-FDTD) method. By establishing a system coupling between random variables and the CDI-FDTD implicit update equation, the following is achieved while ensuring compliance with the divergence characteristics:

[0075] 1) Obtain the statistical characteristics of the electromagnetic field (mean, variance) through a single simulation;

[0076] 2) Efficient and accurate analysis of complex three-dimensional bio-electromagnetic models;

[0077] 3) Breaking the time step constraint of CFL stability conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] The present invention will be further described below with reference to the accompanying drawings.

[0079] Figure 1 The normalized divergence of the electric flux density on the surface S1 is performed using formula (18) in Example 1 of the present application; (a) S-FLOD-FDTD method; (b) S-CDI-FDTD method; (c) S-FDTD method;

[0080] Figure 2 This is the simulation area for analyzing the propagation of electromagnetic waves in three layers of biological tissue in Example 3 of the present application;

[0081] Figure 3Comparison of the mean time-domain amplitude (a) and variance time-domain amplitude (b) of the electric field x component calculated by S-FDTD, MC-FDTD, and S-CDI-FDTD in Example 3 of the present application at x = 5 mm, y∈[0,500] mm, z = 5 mm;

[0082] Figure 4 This is a comparison of the time-domain amplitude of the electric field x-component variance calculated by S-FDTD, S-FLOD-FDTD, and S-CDI-FDTD in Example 3 of the present application at x=5mm, y∈[0,500]mm, and z=5mm. DETAILED DESCRIPTION

[0083] 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.

[0084] Example 1 Stochastic Coordinate Divergence Implicit Finite-Difference Time-Domain Method (S-CDI-FDTD)

[0085] This embodiment discloses a random coincident divergence implicit finite-difference time-domain method, comprising the following steps:

[0086] Updates to CDI-FDTD x and H x The equation is as follows:

[0087]

[0088] Here, ε0 and μ0 are the permittivity and permeability of free space, respectively, and ε r and σ e are the relative permittivity and conductivity, Δt is the time step, Δx, Δy and Δz are the spatial steps, the subscript n and the subscripts i, j, k indicate that the relevant field variables are sampled at time nΔt and position (iΔx, jΔy, kΔz) respectively.

[0089] It is worth noting that the subscript position index of coefficients b and p should be consistent with the subscript position of the electric field. Specifically, in formula (1a), the subscript position index of coefficients b and p should be (i+1 / 2,j,k). Similarly, in formula (1c), the subscript position index of coefficients b and b should be (i+1 / 2,j,k). + The subscript position indices should be (i,j,k+1 / 2) and (i,j+1,k+1 / 2) respectively.

[0090] A. Random Model

[0091] Let g(X1,...,X N ) represents random variables X1,...,X N The function of Using the expected value operator The linear property of where a i Is a constant, according to the Delta method, ignoring the high-order terms in the Taylor series, we get:

[0092]

[0093] g(X1,...,X N ) is expressed as:

[0094]

[0095] where σ[·] and σ 2 [·] denote the standard deviation and variance operators, respectively. Similarly, using the Delta method and following the derivation in , we obtain:

[0096]

[0097] Furthermore, for the variance operation, the following identity holds:

[0098]

[0099] in is the covariance, It's X i With X j The correlation coefficient between .

[0100] Update equation (1) and use equations (2)-(5) together to derive the update equations of S-CDI-FDTD, namely the mean field equation and the variance field equation.

[0101] The derivation assumption E in this embodiment x 、e x 、H x and h x The sample, that is and Dielectric constant ε r and conductivity σ e is a random variable.

[0102] Furthermore, it is assumed that fields of the same type (electric and magnetic) separated by a single time step or a single space step in any direction (including those multiplied by coefficients p, b) are highly correlated, that is, the correlation coefficient is equal to 1.

[0103]

[0104] Therefore, the proposed field update equations for the mean and variance of S-CDI-FDTD only involve the electric field, magnetic field, auxiliary field, ε r and σ e The correlation coefficient between and ).

[0105] B. Mean Field Equations

[0106] Assume that the function g(.) is defined by equation (1). Applying equation (2) to equation (1), we can obtain:

[0107]

[0108] in and and They are ε r and σ e Comparing Equation (1) with Equation (8), we can see that the update of the mean field equations is the same as the update of the "classical" field equations, except that the variables are replaced by the means. In Equation (8), the means of the electric and magnetic fields are the unknowns that need to be updated as time advances.

[0109] C. Variance Field Equation

[0110] Assume that the function g(·) is defined by equation (1a). Derive E x The first step in the variance field equation is to take the variance of both sides of equation (1a):

[0111]

[0112] Note that b and p are related to the random variable ε r and σ e function, so the left side of formula (9) can be applied with formula (5) and formula (7), and we can get:

[0113]

[0114] for Term, apply formula (4) and formula (7), we can get:

[0115]

[0116] For the right side of Equation (9), applying Equations (5) and (7) multiple times and using Equation (4) yields:

[0117]

[0118] In formulas (11) and (12),

[0119]

[0120] in, and They are ε r and σ e The standard deviation of . Formulating the right side of equation (12) yields:

[0121] Let the completed square term be equal to a 2 , the last term is equal to b:

[0122]

[0123]

[0124] Therefore, formula (13) becomes:

[0125] a 2 +b

[0126] Extract common factor a 2 have to:

[0127]

[0128] Take the square root of the resulting equation and use and ignoring the higher-order terms b / 2a, we get:

[0129]

[0130] Substituting equations (10) and (11) into the left side of equation (9) and equation (14) into the right side, we obtain

[0131]

[0132] In (15), the standard deviations of the electric and magnetic fields are unknown quantities to be updated during the time iteration process.

[0133] This embodiment uses similar steps to derive e x 、h x and H x Variance field equation. Let the function g(·) be defined by equation (1). The first step is to take the variance of both sides of equation (1):

[0134]

[0135] Applying (5) to both sides of (16) and using (7) in the resulting equation, we obtain:

[0136]

[0137] In formula (17):

[0138]

[0139] Where, in (17b), the coefficient and The subscript position indices should be (i,j,k+1 / 2) and (i,j+1,k+1 / 2) respectively.

[0140] Example 2. Analysis of the divergence characteristics of the S-CDI-FDTD method

[0141] In this embodiment, in order to verify the divergence-compliant characteristics of the S-CDI-FDTD method proposed in the above embodiment, the electric flux density divergence is numerically calculated and converted into a decibel scale with a unit of 1 coulomb per meter cubed (C / m):

[0142]

[0143] We compared the divergence of the S-FDTD method, the S-FLOD-FDTD method and the S-CDI-FDTD method. The following observation surface (unit: mm) is selected

[0144] s1:x=15,0≤y≤30,0≤z≤30

[0145] The normalized divergence values ​​on the S1 surface are calculated by S-CDI-FDTD, S-FLOD-FDTD (described later) and S-FDTD using different random methods, such as Figure 1 As shown in Figure 3, the numerical divergence of the proposed S-CDI-FDTD is comparable to that of the explicit S-FDTD, and both are much smaller than the divergence of S-FLOD-FDTD in the passive region of the S1 surface, indicating that the proposed method is consistent with the divergence.

[0146] Example 3. Numerical example

[0147] In this embodiment, the accuracy and efficiency of S-CDI-FDTD are compared with those of the traditional Monte Carlo (MC-FDTD) method and the S-FDTD method for numerical examples. For this example, the absorbing boundary is a coordinate-scaled perfectly matched layer (CPML). All calculations are performed on a desktop computer with 32GB RAM and a 4.9GHz Intel Core i7-12700 processor. This embodiment uses the method proposed in this article to analyze the propagation of electromagnetic waves in human tissue consisting of three layers of skin, fat, and muscle. ( Figure 2The mean and standard deviation of the conductivity and dielectric constant of each layer are shown in Table 1. The simulation area used in this problem is as follows Figure 2 As shown. The size of the simulation area is x∈[0,10]mm, y∈[0,10]mm, z∈[0,500]mm. Periodic boundary conditions (PBCs) are used in the y and x directions, and 10 layers of CPML absorbing boundaries are used in the z direction. A plane wave with an x-polarization and an electric field amplitude of 1V / M and a frequency of 2GHz is incident on the tissue layer. S-CDI-FDTD and S-FDTD were simulated for ρ=1 and ρ=0.38 respectively. Here ρ is the correlation coefficient between the dielectric constant / conductivity and the field value of the electromagnetic field. In this experiment, the thickness of the three layers of tissue is 54mm. The size of the spatial grid is set to Δx=Δy=Δz=1mm, and the time step of S-FDTD and MC-FDTD is selected as Courant time step Δt CFL . The S-CDI-FDTD method can overcome the stability limit and thus allow a larger time step, so CFLN (the ratio of the selected time step to the Courant time step) is selected as 3. The Monte Carlo (MC-FDTD) method is 1000 time-domain finite-difference method (FDTD) simulations. In each simulation, the dielectric constant and conductivity of the tissue layer are independently selected from the Gaussian distribution with the mean and standard deviation listed in Table 1. S-CDI-FDTD, S-FDTD and Monte Carlo methods are used to calculate the mean and variance of the x component of the electric field at x = 5 mm, y = 5 mm and z∈[0,500] mm, and the relevant results are shown in Figure 3 (a) Figure 3 (b) and Figure 4 middle.

[0148] like Figure 4 As shown, this embodiment selects the relatively popular implicit algorithm FLOD-FDTD method and the S-FLOD-FDTD method for comparison. The accuracy of the variance calculated by the FLOD-FDTD method in the range of 225mm-280mm is far inferior to that of the S-FDTD method and the S-CDI-FDTD method. This is because the FLOD-FDTD method itself does not satisfy the Gaussian law (which has been compared in the section on numerical divergence verification). Therefore, this method is no longer considered. Figure 3 As shown in (a), at any location in the simulated area, the means produced by the three methods are almost the same, which shows that the assumption of high correlation between electric field samples and between magnetic field samples (see formula (7)) is accurate. Figure 3(b) shows the time-domain variance magnitude as a function of z. For both values ​​of ρ, the results obtained by S-CDI-FDTD and S-FDTD agree very well with each other. However, the results for ρ = 0.38 are more consistent with those obtained by MC-FDTD. As shown in Table 2, the S-CDI-FDTD method (CFLN = 3) is 40% faster than the S-FDTD method and significantly less than the MC-FDTD method.

[0149] Table 1 Biological three-layer tissue parameters

[0150]

[0151] Table 2 CPU time used by various methods

[0152]

[0153] 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.

[0154] 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 method for quantifying uncertainty in the electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD, characterized by: The following steps are involved: Obtain the CDI-FDTD update equations for the electric and magnetic fields; Introduce random variables and functions of random variables, and derive the expression containing the correlation coefficient between random variables through the Delta method; The electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r and conductivity σ e Substitute into the expression as a random variable; Through the electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r The correlation coefficient between them represents the field update equations of the mean and variance of S-CDI-FDTD.

2. The S-CDI-FDTD-based uncertainty quantification method for electrical properties of three-dimensional bioelectromagnetic materials according to claim 1, characterized in that: The CDI-FDTD update equations for the electric and magnetic fields are: Where ε0 and μ0 are the permittivity and permeability of free space, respectively, and ε r and σ e are the relative permittivity and conductivity, Δt is the time step, Δx, Δy and Δz are the spatial steps, the subscript n and the subscripts i, j, k indicate that the relevant field variables are sampled at time nΔt and position (iΔx, jΔy, kΔz) respectively.

3. The S-CDI-FDTD-based uncertainty quantification method for electrical properties of three-dimensional bioelectromagnetic materials according to claim 2, characterized in that: The step of deriving an expression containing correlation coefficients between random variables by the Delta method includes: g(X1,...,X N ) represents random variables X1,...,X N The function of Using the expected value operator The linear property of where a i Is a constant, according to the Delta method, ignoring the high-order terms in the Taylor series, we get: g(X1,...,X N ) is expressed as: where σ[·] and σ 2 [·] denotes the standard deviation and variance operators respectively; Using the Delta method and following the derivation in , we obtain: For the variance operation, the following identity holds: in is the covariance, is a random variable X i With X j The correlation coefficient between .

4. The S-CDI-FDTD-based uncertainty quantification method for electrical properties of three-dimensional bioelectromagnetic materials according to claim 3, characterized in that: The derivation method of the mean field equation includes the following steps: Assume that the random function is defined by formula (1); apply formula (2) to formula (1), and we get: in and and They are ε r and σ e The mean of the electric field and magnetic field is the unknown quantity that needs to be updated as time advances.

5. The S-CDI-FDTD-based uncertainty quantification method for electrical properties of three-dimensional bioelectromagnetic materials according to claim 3, characterized in that: The correlation coefficient of fields of the same type separated by a single time step or a single space step in any direction is equal to 1, giving:

6. The S-CDI-FDTD-based uncertainty quantification method for electrical properties of three-dimensional bioelectromagnetic materials according to claim 5, characterized in that: The derivation method of the electric field variance field equation comprises the following steps: Assume that the random function g(·) is defined by equation (1a); take the variance on both sides of equation (1a): Applying equations (5) and (7) to the left side of equation (9) yields: for Term, apply formula (4) and formula (7), and get: For the right side of Equation (9), apply Equation (5) and Equation (7) multiple times and use Equation (4) to obtain: In formulas (11) and (12): in, and They are ε r and σ e The standard deviation of ; Formulating on the right side of formula (12) yields: Let the completed square term be equal to a 2 , the last term is equal to b: Therefore, formula (13) becomes: a 2 +b Extract common factor a 2 have to: Take the square root of the resulting equation and use and ignoring the higher-order terms b / 2a, we get: Substituting equations (10) and (11) into the left side of equation (9) and equation (14) into the right side, we obtain: In formula (15), the standard deviations of the electric and magnetic fields are unknown quantities to be updated during the time iteration process.

7. The S-CDI-FDTD-based uncertainty quantification method for electrical properties of three-dimensional bioelectromagnetic materials according to claim 5, characterized in that: The magnetic field variance field equation derivation method includes: Assume that the random function g(·) is defined by formula (1); take the variance on both sides of formula (1): Applying (5) to both sides of (16) and using (7) in the resulting equation yields: In formula (17): Where, in (17b), the coefficient and The subscript position indices should be (i,j,k+1 / 2) and (i,j+1,k+1 / 2) respectively.

8. A system for quantifying uncertainty in the electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD, characterized by: include: CDI-FDTD update equation construction module for electric and magnetic fields, used to obtain CDI-FDTD update equations for electric and magnetic fields; Random function building module is used to introduce random variables and functions of random variables, and derive the expression containing the correlation coefficient between random variables through the Delta method; and the electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r and conductivity σ e Substitute into the expression as a random variable; And, the field update equation derivation module of mean and variance is used to pass the electric field E x , auxiliary electric field e x , magnetic field H x and auxiliary magnetic field h x The sample and dielectric constant ε r The correlation coefficient between them represents the field update equations of the mean and variance of S-CDI-FDTD.

9. A computer-readable storage medium storing instructions, characterized in that: When the instructions are executed, the uncertainty quantification method of electrical properties of three-dimensional bioelectromagnetic materials based on S-CDI-FDTD as described in any one of claims 1 to 7 can be implemented.

10. A biomaterial electromagnetic simulation device, comprising the computer-readable storage medium according to claim 9 or the three-dimensional bioelectromagnetic material electrical property uncertainty quantification system based on S-CDI-FDTD according to claim 8.

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