Electrothermal coupling calculation method considering geometry and material uncertainty

Abstract

The invention provides an electrothermal coupling calculation method considering the uncertainty of geometry and material, the uncertainty of material and geometry is considered at the same time, the defect that a traditional S-FDTD algorithm is limited by the size of a grid due to the time step is overcome, the stability and simulation efficiency of an iteration method are improved, and the method has remarkable innovativeness. And geometric randomness and material randomness are simultaneously introduced into electrothermal coupling calculation for the first time, so that the randomness problem of multi-physical simulation in a multi-scale structure can be efficiently solved, and the problem caused by the fact that only material or geometric single randomness is considered when a traditional S-FDTD simulates an electrothermal problem is solved.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-physical field numerical calculation, and in particular to an electrothermal coupling calculation method taking into account geometric and material uncertainties. Background Art

[0002] Electrothermal problems are widely present 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 properties of the structure, and then affects the performance and reliability of the entire system. Therefore, it is of great significance to simulate electromagnetic and thermal fields simultaneously using numerical methods. In addition, there are a large number of random problems in electrothermal simulation, including geometric uncertainty and material uncertainty. In microwave electronic devices, material uncertainty is usually caused by measurement limitations and material aging, while geometric uncertainty is mainly caused by processing errors, material deformation, and environmental factors such as temperature and humidity. Specifically, the biological tissue characteristics and sizes of the human body show considerable differences between different individuals, and even within the same individual, over time, and 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 performs a large number of deterministic simulations using a set of parameters based on statistical properties. This method is widely used to quantify the effects of randomness in numerical models and has been shown to produce accurate results for electromagnetic problems. However, it requires a large number of runs, resulting in large computational times and slow convergence rates. The PCE method accurately obtains the mean and variance of the field by introducing random factors in the finite difference time domain, converting the random problem into a set of stochastic differential equations, maintaining high accuracy when dealing with low-dimensional random variables. However, it is inefficient for large numbers of random inputs.

[0004] Relevant researchers proposed the S-FDTD algorithm, which can calculate the mean and variance of each time and space point through a single simulation. The computational resources and memory of this method are about twice that of traditional FDTD. At present, the S-FDTD method has been widely used in various random medium problems, including plasma, biological tissue, graphene nanostructure, etc. In recent years, S-HIE-FDTD, S-WLP-FDTD and other methods have been proposed on the basis of S-FDTD. These methods have broadened the stability conditions to a certain extent and have high computational efficiency in dealing with multi-scale problems. Recently, S-FDTD has been extended to solve electromagnetic and thermal field problems with material uncertainty. In addition, the S-FDTD method has also been gradually extended to geometric uncertainty problems, including biological tissue, transmission lines and dielectric plates. However, to date, there has been no research on electrothermal problems that simultaneously consider material randomness and geometric randomness. Summary of the invention

[0005] The purpose of the technical solution of the present invention is to set material parameters and grid size as random variables in electrothermal coupling calculations to consider the influence of material uncertainty and geometric uncertainty on field components.

[0006] The technical solution of the present invention provides an electrothermal coupling calculation method taking into account geometric and material uncertainties, comprising the following steps:

[0007] Step 1: Establish the geometric model of the electrothermal coupling system and perform initial meshing;

[0008] Step 2: Discretize the electric and magnetic field components of the Maxwell equations of the electromagnetic grid topology after segmentation by alternating sampling in space and time, split one time step into two sub-time steps using the ADI method, solve the desired equation using detla for leapfrog ADI-FDTD derivation, eliminate the intermediate terms of ADI-FDTD, and obtain the step-by-step electromagnetic field mean iteration equation without sub-time steps;

[0009] Step 3: Take the variance operator on both sides of the step-by-step electromagnetic field mean iteration equation, respectively, and use the DETLA method and probability theory identity to expand the equation. Considering the geometric uncertainty on the right side of the equation, the grid size is set as the first random variable, and considering the material uncertainty, the relative dielectric constant, conductivity, thermal conductivity, density and specific heat capacity of the material are set as the second random variable. According to the relationship between the correlation coefficients between the random variables in the first random variable and the second random variable, the approximate equation is used to simplify and obtain the electromagnetic field standard deviation iteration equation;

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

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

[0012] Step 6: Take the variance operator on both sides of the step-by-step thermal field iteration equation, respectively, use the DETLA method and probability theory identity to expand the equation, and obtain the thermal field standard deviation iteration equation. Whether the maximum temperature difference solved by two adjacent electrothermal couplings is less than the preset value is used to determine whether the thermal steady state is reached; if the thermal steady state is not reached, repeat the electrothermal coupling iteration process of step 2 to step 6; if the thermal steady state is reached, end the electrothermal coupling iteration process;

[0013] Step 7: Post-process the output results.

[0014] Preferably, the electromagnetic grid topology includes shape function information, unit adjacency relations and boundary conditions.

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

[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] 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 dielectric constant, σ is the conductivity, δ x ,δ y ,δ z are the spatial differential operators in each direction respectively.

[0021] Preferably, when using DETLA to solve the expected equation for leapfrog ADI-FDTD derivation, the average value of the random variable is used to calculate the expectation of the field, using the following identity:

[0022]

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

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

[0025]

[0026] Among them, 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 theory 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] Among them, ρ X,Y Represents the covariance coefficient of the random variable, and its value is between -1 and 1.

[0030] Preferably, one time step is split into three sub-time steps to obtain a first sub-time step, a second sub-time step and a third sub-time step, and tridiagonal matrix solutions are performed respectively to obtain the mean value 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] in:

[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] in:

[0047]

[0048]

[0049] The technical solution of the present invention proposes an electrothermal coupling calculation method that takes into account geometric and material uncertainties. By increasing the value of CFLN, the calculation time of the electrothermal simulation is significantly reduced, and the acceleration of the electrothermal coupling numerical calculation is achieved. The accuracy of the algorithm is guaranteed by comparison with the MC-FDTD and S-FDTD methods. It has a strong advantage in multi-scale electrothermal problems, and takes into account the uncertainties of materials and geometry, effectively solving the limitations of existing methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 The specific process of the algorithm for electrothermal coupling for random problems;

[0051] Figure 2 It is a schematic diagram of a low-pass filter;

[0052] Figure 3 This is the S parameter simulation result of the low-pass filter;

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

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

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

[0056] The embodiment of the present invention provides an electrothermal coupling calculation method that takes into account geometric and material uncertainties, which is used to solve the problem of consuming time and memory when solving the iterative equations of the mean and standard deviation of the electromagnetic field in the electrothermal coupling calculation process under time-domain electromagnetic field excitation and bringing the mean and standard deviation distribution of the thermal field into the heat conduction equation. This method effectively reduces the number of simulation time steps and significantly reduces the simulation time. Compared with the traditional S-FDTD method, the present invention can accurately and efficiently solve multi-scale structures. The method comprises the following steps:

[0057] In electromagnetic solutions, the statistical changes of electrical properties and geometry are directly inserted into the FDTD method as random variables. Based on the leapfrog ADI-FDTD formula, the DETLA method is used to derive the mean and variance of the electromagnetic field at each point. Using the DETLA formula, it can be observed that the mean formula of the electromagnetic field is consistent with the original equation. To solve the standard deviation component of the electromagnetic field using the DETLA formula, it is necessary to calculate the partial derivatives of each component and perform related mathematical approximations to obtain the final iterative equation. The steps are as follows:

[0058] Step 1: Modeling. Establish the geometric model of the electrothermal coupling system, perform initial meshing on it, and obtain the electromagnetic mesh topology, that is, shape function information, unit adjacency relationship, and boundary conditions.

[0059] Step 2: Numerical calculation of the mean value of the electromagnetic field. Load the electromagnetic grid topology, and for the Maxwell equations of the time-domain electromagnetic field, perform differential discretization on the electric and magnetic field components in space and time by alternately sampling them, so as to gradually solve the electromagnetic field in time and space.

[0060]

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

[0062] Using the ADI method, a 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 and magnetic field values, ε r is the relative dielectric constant, σ is the conductivity, δ x ,δ y ,δ z are the spatial differential operators in each direction respectively.

[0068] Then, through algebraic operations, DETLA is used to solve the expected equation to derive the mean equation, which is consistent with the original equation. Following the traditional leapfrog ADI-FDTD derivation method, the intermediate terms of ADI-FDTD are eliminated, and finally the step-by-step electromagnetic field mean iteration equation without sub-time steps is obtained as follows:

[0069]

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

[0071] When approximating the mean and standard deviation equations of S-leapfrog ADI-FDTD based on the delta method, the average value of the random variable can be directly used to calculate the expectation of the field. The standard deviation requires a series of mathematical formulas to derive, and the following identities are required:

[0072]

[0073] where E{·} is the expectation, g(·) is a generic function, and x i (i=1,2,3,…,n) represents a random variable, represents the mean 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 step-by-step electromagnetic field mean iteration equation in step 2, take the variance operator on both sides of the equation, and expand the electromagnetic field equation using the DETLA 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 random variable covariance coefficient, and its value is between -1 and 1. According to the simulation results, using a correlation coefficient of 1 is found to overestimate the variance compared to the MC method.

[0077] Expand the left side of the second step equation and get:

[0078]

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

[0080]

[0081] The above equation can be approximated by And ignoring the higher-order terms, we get:

[0082]

[0083] In addition, for the right side of the equation in step 2, the number of variables is very large, including both material and geometric randomness. First, considering the geometric uncertainty, the grid sizes Δx, Δy and Δz can be set as random variables. Secondly, due to the uncertainty of the material, the relative dielectric constant, 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, when solving the formula according to the DETLA method, Taylor formula expansion is performed, and then a series of mathematical derivations are performed to obtain the standard deviation formulas from n to n+1 / 2 and n+1 / 2 to n+1. Due to the large number of variables, the relationship between the correlation coefficients of each variable should be considered to simplify the equation. It is sorted out through a series of approximate formula algebraic algorithms, eliminating the intermediate terms of the iterative formula, ignoring other high-order terms, and solving six implicit equations to obtain the final iterative formula.

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

[0085]

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

[0087]

[0088] in

[0089]

[0090] Then, following the traditional derivation steps of leapfrog ADI-FDTD, the iterative equation of the standard deviation of the electromagnetic field is obtained.

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

[0092]

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

[0094]

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

[0096] Step 4: Electrothermal conversion. Based on the mean and standard deviation components of the electromagnetic field, calculate the mean and standard deviation of the electromagnetic field dissipated power density and substitute them into the heat conduction equation as the heat source for the mean and standard deviation of the heat conduction field.

[0097] Calculate the mean and standard deviation components of the electromagnetic field dissipated power as the heat source Q and σ{Q} of the thermal field mean and standard deviation. The specific formula is as follows:

[0098] Q=σ|E 2

[0099] σ{Q}=σ{σ}|E 2

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

[0101]

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

[0103] Using the ADI method and the alternating direction implicit method, a time step is divided into three sub-time steps, and three tridiagonal matrix solutions are performed respectively. In each sub-time step, only one direction of heat conduction is considered, only one direction is implicit, and the other two directions are explicit. Implicit solutions are performed in each direction. Then, each time step only requires solving a set of tridiagonal matrix equations to obtain the thermal field mean iteration equation. That is, heat is transferred from the node (i, j, k) to (i+1, j+1, k+1) along the x, y, and z directions at the same time, and in each sub-time step, we only consider heat conduction in one direction. According to the delta method, the mean 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 the thermal field standard deviation component, the random changes in geometry and materials should be taken into account. According to the discretized thermal field equation in step 5, the final thermal field standard deviation equation is solved by the iterative formula for solving the electromagnetic field standard deviation as follows:

[0114] (1) The 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] Whether the thermal steady state is reached is determined by whether the maximum temperature difference solved by two adjacent electrothermal couplings is less than a preset value; if the thermal steady state is not reached, the electrothermal coupling iteration process from steps two to six is ​​repeated; if the thermal steady state is reached, the electrothermal coupling iteration process is terminated.

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

[0124] by Figure 2 The filter is used as an example. The method proposed in the embodiment of the present invention is used to perform electrothermal coupling simulation. A sinusoidal excitation source with a center frequency of 4 GHz is added at one end of the filter feeder, and the model is truncated with a convolution perfectly matched layer (CPML) absorption boundary condition. The ambient temperature is set to 300 K, and the convection boundary condition is used for thermal analysis. The convection coefficient h is set to 5 W / (m2·K). In the simulation, we choose Δx=0.4064 mm, Δy=0.4233 mm, Δz=0.265 mm, so the calculation domain contains 90×110×26 grids.

[0125] Considering the uncertainty of the filter geometry, the filter dimensions in each direction (Δx, Δy, Δz) are assumed to vary uniformly by 1%. 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 are shown in Table 1. They are consistent 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 electric field on the filter surface obtained by MC-FDTD (a), S-FDTD (b) and S-leapfrog ADI-FDTD (c) respectively. Figure 5 The mean and standard deviation distribution of the cross-sectional temperature are given. The results show that the MC-FDTD (a), S-FDTD (b) and S-leapfrog ADI-FDTD (c) methods have significant consistency in the mean and variance of the transient temperature and variance distribution, thus 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 S-leapfrog ADI-FDTD method runs 3 times faster than S-FDTD and 634 times faster than the MC-FDTD method, while the memory usage is 68.6% less than the MC-FDTD method.

[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 invention provides an electrothermal coupling calculation method that takes into account geometric and material uncertainties, proves unconditional stability through the von Neumann method, and can take a larger time step under the same simulation time as the prior art, thereby reducing the simulation calculation time, solving the problem that the stability condition of the CFL of the traditional S-FDTD is limited by the spatial step grid, and verifies the accuracy and effectiveness of the method through numerical simulation. The embodiment of the present invention also takes into account that as the CFLN increases, the numerical dispersion of the S-Leapfrog ADI-FDTD increases, and provides a design that balances the relationship between calculation efficiency and accuracy.

Claims

1. An electrothermal coupling calculation method taking into account geometric and material uncertainties, characterized in that: The following steps are involved: Step 1: Establish the geometric model of the electrothermal coupling system and perform initial meshing; Step 2: Discretize the electric and magnetic field components of the Maxwell equations of the electromagnetic grid topology after segmentation by alternating sampling in space and time, split one time step into two sub-time steps using the ADI method, solve the desired equation using detla for leapfrog ADI-FDTD derivation, eliminate the intermediate terms of ADI-FDTD, and obtain the step-by-step electromagnetic field mean iteration equation without sub-time steps; Step 3: Take the variance operator on both sides of the step-by-step electromagnetic field mean iteration equation, respectively, and use the DETLA method and probability theory identity to expand the equation. Considering the geometric uncertainty on the right side of the equation, the grid size is set as the first random variable, and considering the material uncertainty, the relative dielectric constant, conductivity, thermal conductivity, density and specific heat capacity of the material are set as the second random variable. According to the relationship between the correlation coefficients between the random variables in the first random variable and the second random variable, the approximate equation is used to simplify and obtain the electromagnetic field standard deviation iteration equation; Step 4: Calculate the mean and standard deviation of the electromagnetic field dissipated power density according to the mean and standard deviation components of the step-by-step electromagnetic field mean iteration equation and the electromagnetic field standard deviation iteration equation, and substitute them into the heat conduction equation as the heat source of the heat conduction heat field mean and standard deviation; Step 5: Construct the boundary value problem of the heat conduction equation of the heat source, use the ADI method and the alternating direction implicit method to split one time step into three sub-time steps, and solve the tridiagonal matrix respectively to obtain the mean distribution of the thermal field and the distributed thermal field iterative equation; Step 6: Take the variance operator on both sides of the step-by-step thermal field iteration equation, expand the equation using the DETLA method and probability theory identities, and obtain the thermal field standard deviation iteration equation. Whether the maximum temperature difference solved by two adjacent electrothermal couplings is less than the preset value is used to determine whether thermal steady state is reached; If thermal steady state is not reached, repeat the electrothermal coupling iteration process from step 2 to step 6; If thermal steady state is reached, the electrothermal coupling iteration process ends; Step 7: Post-process the output results.

2. The electrothermal coupling calculation method taking into account geometric and material uncertainties according to claim 1, characterized in that: The electromagnetic grid topology includes shape function information, unit adjacency relations, and boundary conditions.

3. The electrothermal coupling calculation method taking into account geometric and material uncertainties according to claim 1, characterized in that: The ADI method is used to split a time step into two sub-time steps to obtain the first sub-time step and the second sub-time step. The formula is as follows: The first sub-time step is from time n to time n+1 / 2: The second sub-time step is from time n+1 / 2 to time n+1: 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 dielectric constant, σ is the conductivity, δ x ,δ y ,δ z are the spatial differential operators in each direction respectively.

4. The electrothermal coupling calculation method taking into account geometric and material uncertainties according to claim 1, characterized in that: When the leapfrog ADI-FDTD derivation is performed by using DETLA to solve the expected equation, 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, and x i (i=1,2,3,…,n) represents a random variable, Represents the mean of the random variable.

5. The electrothermal coupling calculation method taking into account geometric and material uncertainties according to claim 1, characterized in that: When deriving the electromagnetic field standard deviation iteration equation, the following identity is used: Among them, σ 2 {·} represents the variance of the random variable, and σ{·} represents the standard deviation of the random variable.

6. The electrothermal coupling calculation method taking into account geometric and material uncertainties according to claim 1, characterized in that: When deriving the electromagnetic field standard deviation iteration equation, the following probability theory identity is used: s 2 {aX±bY}=a 2 s 2 {X}+b 2 s 2 {Y}±2abσ{X}σ{Y}ρ X,Y Among them, ρ X,Y Represents the covariance coefficient of the random variable, and its value is between -1 and 1.

7. The electrothermal coupling calculation method taking into account geometric and material uncertainties as claimed in claim 3, characterized in that: The time step is split into three sub-time steps to obtain the first sub-time step, the second sub-time step and the third sub-time step, and the tridiagonal matrix is ​​solved respectively to obtain the mean value formula of the thermal field 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: in: 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: in:

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