An electro-thermal coupling calculation method considering geometric and material uncertainties

Abstract

The application provides an electro-thermal coupling calculation method considering geometric and material uncertainty, simultaneously considers material and geometric uncertainty, overcomes defects of traditional S-FDTD algorithms limited by time step and grid size, improves stability and simulation efficiency of an iterative method, and has obvious innovation. Moreover, geometric randomness and material randomness are simultaneously introduced into electro-thermal coupling calculation for the first time, so that a multi-physical simulation randomness problem in a multi-scale structure can be solved efficiently, and problems caused by the single randomness of material or geometry in the simulation of electro-thermal problems by traditional S-FDTD are solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of multi-physical field numerical calculation, and particularly relates to an electro-thermal coupling calculation method considering geometric and material uncertainty. BACKGROUND

[0002] Electro-thermal problems exist widely in the fields of packaged circuits, human brain radiation, microwave medical technology, etc. The accumulation of power consumption caused by electromagnetic effects leads to temperature rise, which affects the electromagnetic characteristics of the structure, and further affects the performance and reliability of the whole system. Therefore, it is of great significance to simultaneously simulate electromagnetic field and thermal field by using numerical methods. In addition, there are a large number of random problems in electro-thermal simulation, including geometric uncertainty and material uncertainty. In microwave electronic devices, the uncertainty of materials is usually caused by measurement limitations and material aging, while the uncertainty of geometry is mainly caused by processing errors, material deformation, and environmental factors such as temperature and humidity. Specifically, the biological tissue characteristics and size of the human body show considerable differences between different individuals, even within the same individual, and over time, in different environments.

[0003] FDTD can be combined with various statistical methods to simulate uncertainty problems, including PCE-FDTD, MC-FDTD and S-FDTD. MC-FDTD uses a set of parameters based on statistical characteristics to perform a large number of deterministic simulations. This method is widely used to quantify the impact of randomness in numerical models and has been proven to produce accurate results for electromagnetic problems. However, it requires a large number of runs, resulting in a large amount of computing time and slow convergence speed. The PCE method transforms the random problem into a set of stochastic differential equations by introducing a random factor in the time domain finite difference, thereby accurately obtaining the mean and variance of the field, and maintaining high accuracy when dealing with low-dimensional random variables. However, it is very inefficient for a large number of random inputs.

[0004] Related researchers have proposed the S-FDTD algorithm, which can calculate the mean and variance of each time and spatial point through one simulation. The computational resources and memory of this method are about twice those of traditional FDTD. Currently, the S-FDTD method has been widely applied to various random medium problems, including plasmas, biological tissues, graphene nanostructures, etc. In recent years, based on S-FDTD, methods such as S-HIE-FDTD, S-WLP-FDTD, etc. have been proposed, which have expanded the stability condition to some extent and have high computational efficiency in dealing with multi-scale problems. Recently, S-FDTD has been extended to solve the problems of electromagnetic field and thermal field with material uncertainty. In addition, the S-FDTD method has also been gradually extended to geometric uncertainty problems, including biological tissues, transmission lines, and dielectric plates. However, so far, there is still no research on electro-thermal problems considering both material randomness and geometric randomness. SUMMARY

[0005] The purpose of the technical scheme of the present application is to set material parameters and grid size as random variables in electro-thermal coupling calculation to consider the influence of material uncertainty and geometric uncertainty on field components.

[0006] The technical scheme of the present application provides an electro-thermal coupling calculation method considering geometric and material uncertainty, comprising the following steps:

[0007] Step 1: establishing a geometric model of an electro-thermal coupling system and performing initial mesh partitioning;

[0008] Step 2: alternately sampling and discretizing electric and magnetic field components in space and time on the Maxwell equation of the partitioned electromagnetic mesh topology, splitting one time step into two sub-time steps by using the ADI method, deriving a leapfrog ADI-FDTD by using detla to solve the expected equation, eliminating the intermediate term of ADI-FDTD, and obtaining a step-by-step electromagnetic field mean value iteration equation without sub-time steps;

[0009] Step 3: taking a variance operator on both sides of the step-by-step electromagnetic field mean value iteration equation, expanding the equation by using the detla method and the identity of probability theory, considering geometric uncertainty on the right side of the equation, setting the grid size as the first random variable, considering material uncertainty, setting the relative permittivity, conductivity, thermal conductivity, density and specific heat capacity of the material as the second random variable, simplifying the equation by approximation according to the relationship between the correlation coefficients of the first random variable and the second random variable, and obtaining an electromagnetic field standard deviation iteration equation;

[0010] Step 4: calculating the mean value and standard deviation of electromagnetic field dissipation power density according to the mean value and standard deviation components of the step-by-step electromagnetic field mean value iteration equation and the electromagnetic field standard deviation iteration equation, and substituting them into the heat conduction equation as the heat source of the heat conduction thermal field mean value and standard deviation;

[0011] Step 5: constructing a boundary value problem of the heat conduction equation of the heat source, splitting one time step into three sub-time steps by using the ADI method and using the alternating direction implicit method, and performing tridiagonal matrix solution respectively to obtain the mean value distribution of the thermal field and a step-by-step thermal field iteration equation;

[0012] Step 6: taking a variance operator on both sides of the step-by-step thermal field iteration equation, expanding the equation by using the detla method and the identity of probability theory, obtaining a thermal field standard deviation iteration equation, and judging whether the thermal steady state is reached according to whether the maximum temperature difference obtained by two adjacent electro-thermal coupling solutions is less than a preset value; if the thermal steady state is not reached, repeating the electro-thermal coupling iteration process of steps 2-6; if the thermal steady state is reached, ending the electro-thermal coupling iteration process.

[0013] Step 7: Post-processing output results.

[0014] Preferably, the electromagnetic mesh topology includes shape function information, element adjacency relationship and boundary conditions.

[0015] Preferably, the ADI method is used to split one time step into two sub-time steps, a first sub-time step and a second sub-time step, according to the following formula:

[0016] The first sub-time step is from time n to time n+1 / 2:

[0017]

[0018] The second sub-time step is from time n+1 / 2 to time n+1:

[0019]

[0020] Wherein, p = 1 + σΔt / 4ε0ε r , q = 1 - σΔt / 4ε0ε r , Δt is the time step, E and H are the electric field and magnetic field values respectively, ε r is the relative permittivity, σ is the conductivity, δ x , δ y , δ z are the spatial differential operators in each direction.

[0021] Preferably, when deriving the leapfrog ADI-FDTD using detla to solve the desired equation, the mean value of the random variable is used to calculate the expectation of the field, and the following identity is used:

[0022]

[0023] Wherein, E{·} is the expectation, g(·) is the generic function, x i (i = 1, 2, 3, …, n) represents a random variable, represents the mean value of the random variable.

[0024] Preferably, when deriving the electromagnetic field standard deviation iterative equation, the following identity is used:

[0025]

[0026] Wherein, σ 2 {·} represents the variance of the random variable, and σ{·} represents the standard deviation of the random variable.

[0027] Preferably, when deriving the electromagnetic field standard deviation iterative equation, the following probability identity is used:

[0028] σ 2 {aX±bY}=a 2 σ 2 {X}+b 2 σ 2 {Y}±2abσ{X}σ{Y}ρ X,Y

[0029] where ρ X,Y represents the covariance coefficient of random variables, whose value is between -1 and 1.

[0030] Preferably, the one time step is split into three sub-time steps, i.e., a first sub-time step, a second sub-time step and a third sub-time step, and three diagonal matrix solutions are performed respectively to obtain the mean formula of the thermal field as follows:

[0031] The first sub-time step is from time n to time n+1 / 3:

[0032]

[0033] The second sub-time step is from time n+1 / 3 to time n+2 / 3:

[0034]

[0035] The third sub-time step is from time n+2 / 3 to time n+1:

[0036]

[0037] wherein:

[0038]

[0039] The standard deviation of the thermal field is as follows:

[0040] The first sub-time step is from time n to time n+1 / 3:

[0041]

[0042] The second sub-time step is from time n+1 / 3 to time n+2 / 3:

[0043]

[0044] The third sub-time step is from time n+2 / 3 to time n+1:

[0045]

[0046] wherein:

[0047]

[0048]

[0049] The technical scheme of the present application provides an electro-thermal coupling calculation method considering geometric and material uncertainties, which significantly reduces the calculation time of electro-thermal simulation by increasing the value of CFLN, realizes the acceleration of electro-thermal coupling numerical calculation, and is compared with the MC-FDTD and S-FDTD methods to ensure the accuracy of the algorithm, has a strong advantage in multi-scale electro-thermal problems, and considers the uncertainties of materials and geometry, effectively solving the limitations of existing methods. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 The algorithm flow of the electro-thermal coupling of the random problem is shown in the figure;

[0051] Figure 2 The schematic diagram of the low-pass filter is shown in the figure;

[0052] Figure 3 The simulation results of the S parameters of the low-pass filter are shown in the figure;

[0053] Figure 4 The cross-sectional distribution of the mean and variance of the electric field of the low-pass filter at the simulation time t = 0.3 ns is shown in the figure;

[0054] Figure 5 The cross-sectional distribution of the mean and variance of the temperature of the low-pass filter at the simulation time t = 0.3 ns is shown in the figure. DETAILED DESCRIPTION

[0055] The present application will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present application and not to limit the scope of the present application. In addition, it should be understood that after reading the content taught by the present application, those skilled in the art can make various modifications or changes to the present application, and these equivalent forms also fall within the scope of the claims attached to the present application.

[0056] The embodiment of the present application provides an electro-thermal coupling calculation method considering geometric and material uncertainties, which is used to solve the problem that the mean and standard deviation distribution of the thermal field obtained by bringing the heat conduction equation into the electromagnetic field mean and standard deviation iterative equation in the electro-thermal coupling calculation process under the excitation of the time-domain electromagnetic field, resulting in time consumption and memory problems. The method effectively reduces the time step of the simulation, and significantly reduces the simulation time. Compared with the traditional S-FDTD method, the present application can accurately and efficiently solve the multi-scale structure. The method comprises the following steps:

[0057] In the electromagnetic solution, the statistical variation of electrical performance and geometry is directly inserted into the FDTD method as a random variable. Based on the leapfrog ADI-FDTD formula, the direct estimation of the mean and variance of the electromagnetic field at each point can be obtained according to the detla method. The mean formula of the electromagnetic field can be observed to be consistent with the original equation by using the detla formula. The standard deviation component of the electromagnetic field is solved by using the detla formula, which requires calculating the partial derivative of each component and performing relevant mathematical approximation and simplification, so as to obtain the final iterative equation. The steps are as follows:

[0058] Step 1: Modeling. The geometric model of the electro-thermal coupling system is established, and initial mesh division is performed to obtain the electromagnetic mesh topology, i.e. shape function information, element adjacency relationship and boundary conditions.

[0059] Step 2: Numerical calculation of electromagnetic field mean. Load the electromagnetic mesh topology, and for the Maxwell equation of the time domain electromagnetic field, the electromagnetic field is gradually solved in time and space by taking the alternative sampling discretization of the electric and magnetic field components in space and time.

[0060]

[0061] where E is the electric field intensity, H is the magnetic field intensity, J is the current density, and J m is the magnetic current density.

[0062] By using the ADI method, one time step is divided into two sub-time steps, and the following formula is obtained (taking Ex and Hz components as an example):

[0063] (1) The first sub-time step: from time n to time n+1 / 2:

[0064]

[0065] (2) The second sub-time step: from time n+1 / 2 to time n+1:

[0066]

[0067] where p=1+σΔt / 4ε0ε r , q=1-σΔt / 4ε0ε r , Δt is the time step, E and H are the electric field and magnetic field values respectively, ε r is the relative permittivity, σ is the conductivity, δ x , δ y , δ z are the spatial differential operators in each direction.

[0068] Then, the mean equation is derived by algebraic operation using delta, which is consistent with the original equation and follows the traditional leapfrog ADI-FDTD derivation method. The intermediate term of ADI-FDTD is eliminated, and the final mean value iteration equation of the split-step electromagnetic field without sub-time step is as follows:

[0069]

[0070] Compared with the ADI-FDTD equation, only six three-diagonal matrices need to be solved, which improves the calculation efficiency.

[0071] When the mean value and standard deviation equations of S-leapfrog ADI-FDTD are derived based on the delta method approximation, the expectation of the field can be calculated directly using the mean value of the random variable, and the standard deviation needs to be derived by a series of mathematical formulas, which requires the use of the following identities:

[0072]

[0073] where E{·} is the expectation, g(·) is the generic function, x i (i = 1, 2, 3, …, n) represents the random variable, represents the mean value of the random variable, σ 2 {·} represents the variance of the random variable, and σ{·} represents the standard deviation of the random variable.

[0074] Step 3: Numerical calculation of electromagnetic field standard deviation. According to the split-step electromagnetic field mean value iteration equation in step 2, the variance operator is taken on both sides of the equation, and the electromagnetic field equation is expanded using the delta method and probability theory identities. The probability theory identities are as follows:

[0075] σ 2 {aX±bY}=a 2 σ 2 {X}+b 2 σ 2 {Y}±2abσ{X}σ{Y}ρ X,Y

[0076] where ρ X,Y represents the covariance coefficient of random variables, and its value is between -1 and 1. According to the simulation results, compared with the MC method, using the correlation coefficient 1 is found to overestimate the variance.

[0077] The expansion of the left side term of the second step equation is carried out, and by rearranging, we get:

[0078]

[0079] Then, the Using the delta approximation method, we get:

[0080]

[0081] The above equation is approximated by the equation and the high order terms are ignored to get

[0082]

[0083] In addition, for the right side of the step two equation, the number of variables is very large, and it contains material and geometric randomness. First, considering the geometric uncertainty, the grid size Δx, Δy and Δz can be set as random variables. Second, due to the uncertainty of the material, the relative permittivity, conductivity, thermal conductivity, density and specific heat capacity of the material are set as random variables, so the electromagnetic field and thermal field are also random variables. In the derivation process, according to the formula of the detla method, the Taylor formula is expanded, and then a series of mathematical derivations are carried out to get the standard deviation formula from n to n+1 / 2 and n+1 / 2 to n+1. Since the number of variables is large, the relationship between the correlation coefficients of each variable needs to be considered to simplify the equation. Through a series of approximate formula algebraic algorithms, the intermediate terms of the iteration formula are eliminated, and other high order terms are ignored, and the final iteration formula is obtained by solving six implicit equations.

[0084] By extracting common factors and also ignoring other terms, it can be expanded as:

[0085]

[0086] The standard deviation iteration formula of each sub-time step is obtained by rearrangement:

[0087]

[0088] Where

[0089]

[0090] Then follow the traditional derivation steps of leapfrog ADI-FDTD to get the iteration equation of the standard deviation of the electromagnetic field.

[0091] Where the standard deviation of E x is:

[0092]

[0093] Similarly, the standard deviation of H z can be obtained:

[0094]

[0095] For other electromagnetic field components, their update equations can be obtained by similar methods.

[0096] Step 4: Electric-thermal conversion. According to the mean and standard deviation components of the electromagnetic field, the mean and standard deviation of the electromagnetic field dissipation power density are calculated and substituted into the heat conduction equation as the heat source of the mean and standard deviation of the heat conduction thermal field.

[0097] The mean and standard deviation components of the electromagnetic field dissipation power are calculated as the heat source Q and σ{Q} of the mean and standard deviation of the thermal field, and the specific formula is as follows:

[0098] Q = σ|E 2

[0099] σ{Q} = σ{σ}|E 2

[0100] Step 5: Mean value calculation of thermal field. Load the network topology, and construct the boundary value problem of heat conduction for the following heat conduction equation. The heat conduction equation is solved implicitly in each direction, and finally the mean value distribution of the thermal field is obtained.

[0101]

[0102] where T is the temperature, k is the thermal conductivity, ρ m is the density, c ρ is the specific heat capacity.

[0103] Using the ADI method and using the alternating direction implicit method, a time step is divided into three sub-time steps, and three tri-diagonal matrix solutions are performed respectively. In each sub-time step, only one direction of heat conduction is considered, and only one direction is implicit, while the other two directions are explicit. Implicit solution is performed in each direction, and then only one set of tri-diagonal matrix equations needs to be solved in each time step to obtain the heat field mean iteration equation. That is, heat is transmitted from node (i, j, k) to (i+1, j+1, k+1) along x, y, z directions, and in each sub-time step, we only consider one direction of heat conduction. According to the delta method, the mean value of the thermal field can be directly solved.

[0104] The formula is as follows:

[0105] (1) The first sub-time step: from time n to time n+1 / 3:

[0106]

[0107] (2) The second sub-time step: from time n+1 / 3 to time n+2 / 3:

[0108]

[0109] (3) The third sub-time step: from time n+2 / 3 to time n+1:

[0110]

[0111] in:

[0112]

[0113] Step 6: Numerical Calculation of Thermal Field Standard Deviation. When solving for the standard deviation components of the thermal field, the random variations in geometry and materials should be considered simultaneously. Based on the discretized thermal field equations from Step 5, the final standard deviation equation for the thermal field is obtained by using an iterative formula for solving the electromagnetic field standard deviation, as follows:

[0114] (1) First sub-time step: from time n to time n+1 / 3:

[0115]

[0116] (2) The second sub-time step: from time n+1 / 3 to time n+2 / 3:

[0117]

[0118] (3) The third sub-time step: from time n+2 / 3 to time n+1:

[0119]

[0120] in:

[0121]

[0122] Determine whether thermal steady state has been reached by checking whether the maximum temperature difference obtained from two consecutive electrothermal coupling solutions is less than a preset value. If thermal steady state has not been reached, repeat the electrothermal coupling iteration process from step two to step six. If thermal steady state has been reached, end the electrothermal coupling iteration process.

[0123] Step 7: Post-process the output results.

[0124] by Figure 2 Using a filter as an example, electrothermal coupling simulations were performed using the method proposed in this embodiment of the invention. A sinusoidal excitation source with a center frequency of 4 GHz was added to one end of the filter feed line, and the model was truncated using a convolutionally perfectly matched layer (CPML) with absorbing boundary conditions. The ambient temperature was set to 300 K, and convective boundary conditions were used for thermal analysis, with the convection coefficient h set to 5 W / (m²·K). In the simulation, we selected Δx = 0.4064 mm, Δy = 0.4233 mm, and Δz = 0.265 mm, therefore the computational domain contained a 90 × 110 × 26 grid.

[0125] Considering the uncertainty of the filter geometry, it is assumed that the filter size varies uniformly by 1% in each direction (Δx, Δy, Δz). In addition, the material uncertainty of the substrate is also considered, as shown in Table 1.

[0126] Table 1 Substrate material parameters

[0127]

[0128] Figure 3 The S-parameter simulation results of the filter. It can be seen that they are in good agreement with the results obtained by S-FDTD (CFLN = 1) and MC-FDTD (CFLN = 1).

[0129] Figure 4 The two-dimensional distribution of the mean and variance of the surface electric field of the filter obtained by MC-FDTD (a), S-FDTD (b) and S-leapfrog ADI-FDTD (c), respectively. At the same time, Figure 5 The mean and standard deviation distribution of the cross-sectional temperature is given. The results show that the MC-FDTD (a), S-FDTD (b) and S-leapfrog ADI-FDTD (c) method has significant consistency in the mean and variance of the transient temperature and variance distribution, thereby verifying the accuracy of the method. Table 2 further shows the comparison of MC-FDTD, S-FDTD and S-leapfrog ADI-FDTD in terms of computing resources. When CFLN = 5, the running speed of the S-leapfrog ADI-FDTD method is 3 times faster than that of S-FDTD, and 634 times faster than that of MC-FDTD, and the memory occupation is 68.6% less than that of MC-FDTD.

[0130] Table 2 Comparison of MC-FDTD, S-FDTD and S-leapfrog ADI-FDTD in terms of computing resources

[0131]

[0132] The embodiment of the present application provides an electro-thermal coupling calculation method considering geometric and material uncertainty, which proves unconditional stability by von Neumann method, can take larger time step under the same simulation time as the prior art, so as to reduce the calculation time of simulation, solves the stability condition of CFL of the traditional S-FDTD which is limited by the space step grid, and verifies the accuracy and effectiveness of the method through numerical simulation. The embodiment of the present application also considers that the numerical dispersion of S-Leapfrog ADI-FDTD is larger as CFLN is larger, and provides a design corresponding to the balance of calculation efficiency and accuracy.

Claims

1. An electro-thermal coupling calculation method taking into account geometrical and material uncertainties, characterized in that, The method comprises the following steps: Step 1: a geometric model of an electro-thermal coupling system is established, and initial mesh partitioning is performed; Step 2: the Maxwell equations of the partitioned electromagnetic mesh topology are discretized by alternately sampling electric and magnetic field components in space and time, one time step is split into two sub-time steps by using an ADI method, a leapfrog ADI-FDTD derivation is performed by using a detla to solve the expected equation, the intermediate term of the ADI-FDTD is eliminated, and a step-by-step electromagnetic field mean value iterative equation without a sub-time step is obtained; Step 3: variance operators are taken on both sides of the step-by-step electromagnetic field mean value iterative equation, the equation is expanded by using a detla method and a probability theory identity, in view of geometric uncertainty, the grid size is set as a first random variable, and in view of material uncertainty, the relative permittivity, conductivity, thermal conductivity, density and specific heat capacity of the material are set as second random variables, the relationship between the correlation coefficients of the first random variable and the second random variable is approximated, the equation is simplified, and an electromagnetic field standard deviation iterative equation is obtained; Step 4: the mean value and the standard deviation of the electromagnetic field dissipation power density are calculated according to the mean value and the standard deviation components of the step-by-step electromagnetic field mean value iterative equation and the electromagnetic field standard deviation iterative equation, and the mean value and the standard deviation of the heat conduction equation are substituted into the heat conduction equation as a heat source of the heat conduction thermal field mean value and the standard deviation; Step 5: a boundary value problem of the heat source of the heat conduction equation is constructed, one time step is split into three sub-time steps by using an ADI method and an alternating direction implicit method, and three diagonal matrix solutions are performed, so that the mean value distribution of the thermal field and a distribution thermal field iterative equation are obtained; Step 6: variance operators are taken on both sides of the step-by-step thermal field iterative equation, the equation is expanded by using a detla method and a probability theory identity, a thermal field standard deviation iterative equation is obtained, and whether the maximum temperature difference obtained by two adjacent electro-thermal coupling solutions is less than a preset value is determined to determine whether a thermal steady state is reached; If the thermal steady state is not reached, the electro-thermal coupling iterative process of steps 2-6 is repeated; If the thermal steady state is reached, the electro-thermal coupling iterative process is ended; Step 7: post-processing output results.

2. The electro-thermal coupling calculation method considering geometric and material uncertainties according to claim 1, characterized in that, The electromagnetic mesh topology comprises shape function information, element adjacency relationships and boundary conditions.

3. The electro-thermal coupling calculation method considering geometric and material uncertainties according to claim 1, characterized in that, The ADI method is used to split one time step into a first sub-time step and a second sub-time step, and the formulas are as follows: The first sub-time step is from the n time to the n+1 / 2 time: The second sub-time step is from the n+1 / 2 time to the n+1 time: where p = 1 + σΔt / 4ε0ε r , q = 1 - σΔt / 4ε0ε r , Δt is the time step, E and H are the electric and magnetic field values, ε r is the relative permittivity, σ is the conductivity, δ x , δ y , δ z are the spatial differential operators in each direction.

4. The electro-thermal coupling calculation method considering geometric and material uncertainties according to claim 1, characterized in that, When the detla is used to solve the expected equation to perform the leapfrog ADI-FDTD derivation, the average value of the random variable is used to calculate the expectation of the field; where E{·} is the expectation, g(·) is a generic function, x i (i = 1, 2, 3,..., n) represent random variables, represent the mean of the random variables.

5. The electro-thermal coupling calculation method considering geometric and material uncertainties according to claim 1, characterized in that, When the electromagnetic field standard deviation iterative equation is derived, the following identity is used: where σ 2 {·} denotes the variance of a random variable, σ{·} denotes the standard deviation of a random variable.

6. The electro-thermal coupling calculation method considering geometric and material uncertainties of claim 1, wherein, When the electromagnetic field standard deviation iterative equation is derived, the following probability theory identity is used: σ 2 {aX±bY} = a 2 σ 2 {X}+b 2 σ 2 {Y}±2abσ{X}σ{Y}ρ X,Y where p X,Y represents the random variable covariance coefficient, whose value is between -1 and 1.

7. The method of claim 3, wherein the geometric and material uncertainties are considered. When one time step is split into a first sub-time step, a second sub-time step and a third sub-time step, and three diagonal matrix solutions are performed, the mean value formula of the thermal field is as follows: The first sub-time step is from time n to time n+1 / 3: The second sub-time step is from time n+1 / 3 to time n+2 / 3: The third sub-time step is from time n+2 / 3 to time n+1: wherein: The standard deviation of the thermal field is as follows: The first sub-time step is from time n to time n+1 / 3: The second sub-time step is from time n+1 / 3 to time n+2 / 3: The third sub-time step is from time n+2 / 3 to time n+1: wherein: The standard deviation of the thermal field is as follows:

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