Method for predicting the microstructure of steel during heat treatment

Abstract

Providing a method for simulating steel microstructures with higher accuracy using cellular automata. [Solution] This is a method for predicting the microstructure of steel when it is heat-treated in the single-phase region of the austenite phase. The method involves a calculation process in which the space is divided into multiple cells and the calculated grain size, which is the average grain size of the microstructure for given parameters, is calculated at each time step using the cellular automaton method, and a parameter optimization process in which the parameters are optimized using Bayesian optimization on experimental data obtained in advance, and these processes are repeated a predetermined number of times. In the minimization function in Bayesian optimization, the computational cost in the cellular automaton method is given as a penalty term in addition to the error in the calculated grain size.

Description

[Technical Field]

[0001] This invention relates to a method for predicting the microstructure of steel when it is heat-treated in the single-phase region of the austenite phase, and more particularly to a method for predicting the microstructure of steel during heat treatment by simulation using the cellular automaton method. [Background technology]

[0002] Carburizing is a widely applied treatment used to harden the surface of power transmission components such as gears, pulleys, and shafts. However, if heat treatment is performed at high temperatures to accelerate the carburizing process, or if cold forging is performed before carburizing, abnormal grain growth is more likely to occur during the carburizing process. Such abnormal grain growth during carburizing increases the amount of heat treatment strain and can lead to a decrease in the fatigue strength of the component. Therefore, simulating the prediction of such grain growth was considered.

[0003] Several methods have been proposed for simulating grain growth. One such method is the cellular automation method, which divides space into cells and calculates the interactions between adjacent cells. It has been proposed to use this method to introduce the influence of the dispersion state of precipitates into the simulation of austenite grain growth during carburizing, thereby predicting the occurrence of abnormal grain growth (see Non-Patent Literature 1).

[0004] On the other hand, in cellular automaton methods, it is fundamentally impossible to derive the time corresponding to a step or the length of a cell solely through calculation; a process is required to match the time scale and spatial computation parameters with experimental data.

[0005] Here, for example, in Patent Document 1, a material structure calculation method is disclosed in which data of a material structure experimentally obtained is incorporated into a simulation (calculation model) for calculating the temporal change of the material structure of aluminum. Here, regarding the calculation model, it is stated that the parameters included in the calculation model are adjusted so as to reduce the error from the data of the material structure experimentally obtained, and the temporal change of the material structure can be calculated with higher accuracy.

Prior Art Documents

Patent Documents

[0006]

Patent Document 1

Non-Patent Documents

[0007]

Non-Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0008] In Patent Document 1, specific methods such as parameter adjustment for error reduction are not described, but in the above-described steel structure prediction method as well, for example, it is conceivable to conduct experiments and perform optimization in combination with the cellular automaton method by Bayesian estimation. However, if trial and error is repeatedly carried out without restraint so as to match the parameters with the experimental data, the simulation will have to be repeated many times, which requires a great deal of time and is not practical.

[0009] The present invention has been made in view of the above circumstances, and an object thereof is to provide a method for more accurately simulating a steel structure formed when steel is heat-treated in a single-phase region of an austenite phase, using the cellular automaton method.

Means for Solving the Problems

[0010] In other words, the present invention aims to provide a method for efficiently calculating fitting parameters for reducing the error between calculated values and experimental values in the simulation of grain growth in the austenite phase using the cellular automaton method, without spending much time, and for obtaining the optimal values of the optimal fitting parameters for minimizing the error from the experimental values.

[0011] Therefore, the method for predicting the steel structure during heat treatment according to the present invention is a method for predicting the steel structure when steel is heat-treated in a single-phase region of the austenite phase, which includes a calculation process of dividing space into a plurality of cells and calculating the calculated grain size, which is the average grain size of the steel structure with respect to the given parameters using the cellular automaton method, for each time step, and a parameter optimization process of optimizing the parameters using Bayesian optimization for experimental data in which the parameters have been acquired in advance, and in a process of repeating these for a specified number of times, in the minimization function in the Bayesian optimization, in addition to the error of the calculated grain size, the calculation cost in the cellular automaton method is given as a penalty term.

[0012] According to such a feature, it is possible to optimize the parameters while reducing the calculation cost from the minimization function, and as a result, simulate the steel structure with higher accuracy.

[0013] In the above-described invention, the experimental data may be data of the average grain size of the austenite phase with respect to the heat treatment time. According to such a feature, the steel structure can be simulated with even higher accuracy.

[0014] In the invention described above, the error may be characterized by the steps of: obtaining a square fitting function obtained by least-squares fitting to the time evolution of the square of the average grain size in the experimental data; and comparing the square fitting function with the square of the calculated grain size relative to the time step obtained in the calculation step. According to this feature, the steel structure can be simulated with higher accuracy in accordance with the experimental data.

[0015] In the invention described above, the calculation cost may be characterized by including at least the number of time steps. With this characteristic, parameters can be obtained to reduce the number of time steps, and the steel structure can be simulated with higher accuracy.

[0016] In the above invention, the minimization function g min This is expressed as the sum of the error term g1 of the calculated particle size and the penalty term g2 (Equation 1),

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[0017] [Figure 1] This flowchart shows an example of a method for predicting the microstructure of steel during heat treatment. [Figure 2] This figure shows the range of other cells around cell x that should be considered when calculating curvature. [Figure 3] This flowchart shows an example of a simulation method using the CA method. [Figure 4] This is an example of the initial tissue used in the CA method. [Figure 5] This graph shows the values ​​of the minimization function when (a) Bayesian optimization is used and (b) when parameters are selected randomly. [Figure 6] This graph compares grain growth using the CA method with grain growth using experimental data, based on the obtained parameter set. [Modes for carrying out the invention]

[0018] First, we will describe the outline of the method for predicting the steel microstructure during heat treatment in this embodiment.

[0019] In this embodiment, the cellular automaton method (sometimes referred to as the CA method) is used to predict the grain growth of austenite grains in the microstructure during heat treatment. The CA method in this embodiment includes three parameters that are artificially determined. These parameters are the maximum value of the random variable (random number), the cell size of the box counting method, and the number of calculation cells. Since these parameters are artificially determined, it is necessary to select the optimal combination of values ​​to perform a simulation that matches the actual microstructure. In other words, unless a combination close to the optimal one can be selected from the many possible parameter combinations, a simulation that matches the actual microstructure cannot be performed.

[0020] Therefore, Bayesian optimization is performed to optimize these three parameters so that the results conform to the experimental data. In Bayesian optimization, it is necessary to predict grain growth as the steel microstructure during heat treatment using the CA method and compare this with the experimental data. In other words, performing Bayesian optimization requires repeatedly using the CA method, which has a relatively high computational cost, and randomly trying combinations of the three parameters is inefficient. Therefore, a penalty term related to computational cost is added to the minimization function (objective function) used in Bayesian optimization to reduce the computational cost.

[0021] Figure 1 outlines the general process for predicting the microstructure of steel during heat treatment.

[0022] First, the initial structure and initial parameters to be used in the CA method are set and input (S1, S2). Of these, the initial parameters include the three artificially determined parameters mentioned above. The initial structure is obtained by dividing space into multiple cells, and can be, for example, a structure obtained by randomly assigning orientations to a two-dimensional Voronoi tessellation structure.

[0023] Based on the input initial structure and initial parameters, the calculated grain size is calculated by repeating the CA method up to a specified time step (S3). This is a calculation process in which the calculated grain size, which is the average grain shape of the steel structure, is calculated at each time step for the parameters input above.

[0024] The system checks whether the number of Bayesian optimization iterations described later has exceeded a specified number. If it has not (S4; No), it refers to the previously entered experimental data (S5) and calculates the value of the minimization function to match the experimental data (S6). These are parameter optimization steps that use Bayesian optimization to optimize the three parameters mentioned above to match the previously acquired experimental data. In particular, the minimization function includes a penalty term representing the computational cost in the CA method, in addition to the error of the calculated particle size relative to the experimental data. In other words, Bayesian optimization optimizes the three parameters to reduce the error relative to the experimental data while keeping the computational cost in the CA method from becoming too high.

[0025] Using the obtained minimization function value, the mean μ and variance σ used in the acquisition function are determined by Gaussian process regression. 2 The result is calculated (S6), and the acquisition function is obtained (S9). In Bayesian optimization, a combination of parameters to be used in the next CA method is selected based on the acquisition function. This process is repeated, and when the specified number of optimization iterations is exceeded (S4; Yes), the process is terminated.

[0026] Next, we will explain cellular automata methods.

[0027] In the CA method, it is necessary to define the rules for moving to the next time step. In this example, the grain growth behavior of austenite grains during heat treatment is reproduced using a grain growth model based on the CA method. Applying the general theory of grain growth to the CA method, the driving force of grain growth per unit volume is expressed using the radius of curvature as shown in (Equation 4). Here, Rg is the radius of curvature, σ is the grain boundary energy, and x and y are the two-dimensional coordinates of the cell.

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[0028] Also, the mobility of grain boundaries is considered to be temperature-dependent and is expressed as in (Equation 5).

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[0029] Here, M0 is the coefficient of mobility, Q is the activation energy, R is the gas constant, and T is the temperature. Regarding the curvature, by using the Box Count method by Pierre et al., the influence of the curvature on the driving force of grain growth was introduced. In the Box Count method, the radius of curvature R g represents [x, y].

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[0030] [[ID=2B]] Here, b represents the cell range to be considered around the cell x, y as shown in FIG. 2. For example, if b = 5, the curvature is calculated considering up to the second-nearest cells. δ[x, y][x + i, y + j] is 1 when the cell x, y and the cell x + i, y + j are different grains and 0 if they are the same grain.

[0031] The handling of grains will be described. In this example, each grain was given a crystal orientation in the same manner as the actual structure. In so doing, for convenience of calculation, quaternions representing the orientation were used. Representing the orientation of grains using quaternions has been proposed, for example, by Takahashi et al. For example, if the quaternions representing the orientations of cubic crystals a and b are q a , q b respectively, then this orientation difference θ a,b is easily expressed as in (Equation 7).

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[0032] Here, e iThese are 24 quaternions representing symmetry operations in a cubic crystal. (p,q) represents the inner product of quaternions p and q. Using Euler angles or lattice vectors requires more complex operations compared to using quaternions. In other words, using quaternions allows for the calculation of orientation differences with lower computational cost. Furthermore, the four real parts of a quaternion can be treated equivalently. Therefore, random orientations can be easily obtained by simply assigning uniform random numbers to the four real parts and normalizing them.

[0033] Next, we will discuss the treatment of grain boundary energy. We applied the Read-Shockrey equation (8) to the grain boundary energy, and incorporated the orientation dependence of the grain boundary energy.

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[0034] The grain boundary energy σ[x,y] of each cell is expressed as the sum of the grain boundary energies with the surrounding cells, as shown in (Equation 9).

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[0035] Regarding the mobility of grain boundaries, we introduced the orientation dependence of mobility by applying Humphry et al.'s equation (10).

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[0036] The grain growth rate v is expressed as shown in (Equation 11).

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[0037] In this embodiment, as shown in (Equation 12), when ΔG decreases, if the grain growth rate v is greater than or equal to a certain value z given by a random number, grain growth is reproduced according to the rule shown in the equation, which states that the adjacent crystal grains are replaced and become the same phase so that ΔG is lowest.

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[0038] Here, ΔG after [x,y]-[x+i,y+j] This is ΔG at cell x,y when only cell [x,y] is replaced by the crystal grain of cell [x+i,y+j]. z is the maximum value of z max These are uniformly random numbers with a minimum value of 0, and are updated for each cell and step.

[0039] As shown in Figure 3, an example of a combination of processing steps for the method described above is shown in a flowchart.

[0040] First, the initial structure and constants are set (S11: Initialization step). The model space consisting of austenite grains is divided into multiple cells, and the temperature T and the maximum value of the uniform random number z are set. max Then, a crystal orientation is determined for each cell.

[0041] Next, the time step counting begins (S12). The following processing is performed after each time step count.

[0042] Next, the driving force ΔG is calculated for all cells (S13: Driving force calculation step). That is, the radius of curvature R of the grain boundary between the cell and the surrounding cells is calculated using (Equation 6). g [x,y] is calculated, and the grain boundary energy σ calculated from the crystal orientation is used, along with the radius of curvature R of the grain boundary. g The driving force ΔG for grain growth is calculated using [x,y] in equation (4).

[0043] Furthermore, the driving force when a cell is replaced by an adjacent cell is also calculated for all cell combinations (S14). Here, the driving force when a cell is replaced by an adjacent cell is calculated in advance in order to detect the cell that will result in the minimum driving force, as will be described later.

[0044] Next, the mobility M is calculated for all cells (S15: Mobility Calculation Step). Here, the mobility M is calculated from the crystal orientation and temperature T using (Equation 10).

[0045] Here, a designated cell to emphasize the ease of movement is selected in advance, and it is determined whether the cell from which the ease of movement was calculated is one of the designated cells (S16). If it is a designated cell to emphasize the ease of movement (S16: Yes), the ease of movement is multiplied by a predetermined emphasis coefficient to make it an emphasized ease of movement (S17). If it is not a designated cell to emphasize the ease of movement (S16: No), the ease of movement remains unchanged.

[0046] Then, the grain growth rate v is calculated when the cell is replaced with the one that minimizes the driving force (S18). Here, first, for one cell, the driving force ΔG when it is replaced with the adjacent cell calculated above is calculated. after [x,y]-[x+i,y+j] The adjacent cell with the smallest value is extracted as a replacement candidate cell. Then, the driving force ΔG when the replacement candidate cell replaces the original cell is calculated. after [x,y]-[x+i,y+j] If the value is small, perform the substitution check.

[0047] In determining substitution, the grain growth rate v of the candidate cell to be substituted is compared with a uniform random number z (S19). In detail, the driving force ΔG after [x,y]-[x+i,y+j] Multiplying by mobility M gives the grain growth rate v, and the generated uniform random number z (0 ≤ z ≤ z) max) is compared with the uniform random number z. If the grain growth rate v is greater than the uniform random number z (S19: Yes), it is determined that the cell will be replaced by the replacement candidate cell. In other words, it is determined that the crystal grains corresponding to the original cell and the crystal grains corresponding to the replacement candidate cell are in the same phase and grain growth will occur. In this case, the original cell and the replacement candidate cell are replaced as a single cell (S20: Replacement step). On the other hand, if the grain growth rate is not greater than the uniform random number z (S19: No), it is determined that no grain growth will occur and no cell replacement is performed.

[0048] Perform the above process for all cells. Furthermore, to incorporate the effect of precipitates acting as pinning particles, a process that takes into account the growth of precipitates by Ostwald growth may be introduced.

[0049] Finally, the process is repeated (S21) by returning to the start of the time step (S12) until a predetermined time step is reached. In other words, a time corresponding to one count of the time step is set, and the count is repeated until the time to observe grain growth through simulation is reached. Note that the order of calculating the driving force (S13 and S14) and calculating the mobility (S15) can be reversed, and the calculation of mobility can be performed first.

[0050] Thus, the CA method allows for the simulation of the grain growth process, and the calculated grain size, which is the average of the crystal grain sizes obtained by the CA method, can be calculated. However, as mentioned above, the CA method repeats calculations for each cell and each time step, so at least the number of time steps and the number of cells contribute to high computational costs.

[0051] Next, we will explain an example of optimizing three parameters used in the CA method described above, which can be determined artificially, using Bayesian optimization.

[0052] As shown in Figure 4, the initial structure used in the CA method was a structure obtained by randomly assigning orientations to a two-dimensional Voronoi tessellation structure. To ensure that the initial grain size and average grain size in the experimental results shown later were the same, the calculation range was a square with sides of 3.83 mm. The physical parameters used in the CA method were the values ​​shown in Table 1. Of these, the three parameters Z were artificially assigned. max , b, X max (Y max We will optimize the following using Bayesian optimization.

[0053] [Table 1]

[0054] Next, I will explain how the experimental data was obtained.

[0055] First, a 50 kg ingot was melted by vacuum induction melting to a mass% composition of 0.18C-0.08Si-0.60Mn-1.2Cr-Fe, forged to a diameter of 42 mm, soaked at 1573 K, and then annealed at 1373 K for 1 hour and 1173 K for 1 hour. Subsequently, to simulate cold forging, the ingot was machined to φ15 × 22.5 mm L and subjected to 30% cold working using a press. Furthermore, heat treatment simulating carburizing was performed at a holding temperature of 1273 K for four different holding times: 0 minutes, 20 minutes, 40 minutes, and 80 minutes, followed by water quenching. The resulting cross-section of the sample was then mirror-polished, etched with a grain revealing solution using picric acid aqueous solution, and the prior γ grains were observed and photographed using an optical microscope. Finally, the grain size was measured according to the method described in ISO 643:1983. In other words, the experimental data represents the average grain size of the austenite phase against the heat treatment time.

[0056] Next, we will explain the Bayesian optimization method.

[0057] Bayesian optimization is a machine learning technique aimed at minimizing an unknown function f(x). It is used for searching for hyperparameters in computationally expensive calculations. A key feature of Bayesian optimization is that the minimization function (objective function) is treated as a statistical model. From this model, the parameter x of f, which will be determined in the next step, is obtained, resulting in the function f(x). This process is repeated to optimize the function. In many cases, including this example, a Gaussian process is used as this statistical model. A Gaussian process is a type of stochastic process; given a function f and parameter x, a Gaussian distribution is obtained. The mean μ(x) and variance σ of this Gaussian distribution are... 2 (x) can be expressed using the kernel function k(x, x') as follows:

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[0058] Here, k(x) = (k(x,x1), ..., k(x,x n )) T , k i,j =k(i,j), where y is the actual value of the function to be minimized and n is the number of data points used in the Gaussian process regression.

[0059] In this example, the three parameters described above were optimized for experimental data using Bayesian optimization with a Gaussian process. The kernel function of the Gaussian process was given by (Equation 15) below. Depending on the data used, a maternal kernel may also be used as the kernel function.

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[0060] Here, θ0, θ1, and θ2 are optimized to fit the experimental data using Gaussian process regression. The acquisition function used is the Upper Confidence Band (UCB), given by the following equation, with a β value of 0.8. Provability Improvement (PI) or Expected Improvement (EI) may also be used as the acquisition function.

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[0061] In the example, the parameter to be optimized by Bayesian optimization is X max (Y max ), Z max These are the three simulation parameters: b,

[0062] To perform the initial Gaussian process regression, 27 initial parameter sets were prepared, and simulations using the CA method were performed for all of these sets. The 27 initial parameter sets consist of 3 × 3 × 3 = 27 combinations, each of the three variables as shown in Table 2. In this case, the CA simulations were performed using the same parameters with five different random number seeds, and the results of the minimization function calculations for the five different random number seeds were averaged. For the subsequent Gaussian process regression, the parameter sets were standardized to have a mean of 0 and a variance of 1, and the regression was performed. [Table 2]

[0063] Next, in order to optimize, we first interpret the experimental results in real time. Generally, in grain growth, the square of the grain size is proportional to time (square law), and this can be expressed using the radius corresponding to the grain size as shown in (Equation 17) below.

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[0064] Here, R(t) is the average radius of the crystal grain at time t, R0 is the average radius of the initial crystal grain, and α and t are the coefficient and time, respectively. Then, the particle size data from the experimental results in real time was fitted to this equation using the least squares method. In other words, a square-fit function was obtained by least squares fitting the time evolution of the square of the average particle size from the experimental data.

[0065] Next, we adjust the real time to match the time step of the CA method. If Δt is the real time per time step, then we should calculate Δt such that h shown in (Equation 18) below is minimized.

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[0066] Here, R ca 2 [s] is the square of the average radius of the crystal grain at time step s, which is determined by the CA method. To find the Δt that minimizes h, the point where dh / d(Δt)=0 was determined analytically.

[0067] Next, we define the minimization function for Bayesian optimization as shown in (Equations 19) to (21) below.

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[0068] Here, S is the number of calculation steps, S max R(t) is the number of steps corresponding to the longest retention time of the experimental data (i.e., 80 minutes), R(t) is half the grain size (radius) at time t corresponding to each calculation step of the square fitting function fitted to the experimental data using the square law, R ex max This is the average radius of the crystal grains at the longest retention time in the experimental data. Also, R ca (S) and R ca (S max ) are calculation steps S and S respectively. maxThis is half the calculated particle size (radius). The first term g1 of the minimization function represents the error rate of the calculated particle size by the CA method compared to the average particle size within the time range of the experimental data. Here, from (Equation 17), R = (αt + R0 2 ) 1 / 2 The second term, g2, is a penalty term imposed when the Δt calculated by the formula is too small and does not exceed the actual experimental time even with the maximum number of steps, which is 3000 steps. In other words, it includes the computational cost due to the number of time steps as a penalty term.

[0069] Using this regression equation, μ and σ were determined for all 27 possible parameter sets shown in Table 2, and the acquisition function was obtained from these values. Furthermore, from the 57,400 parameter sets shown in Table 3, parameters were selected to minimize the value of the minimization function, and simulations were performed using the CA method. In other words, three parameters were selected to reduce the error between the calculated particle size and the average particle size from experimental data, while also keeping the computational cost of the CA method low. This operation was repeated 30 times to optimize the parameters. These simulations and Bayesian optimizations were performed using Python code and C language code.

[0070] [Table 3]

[0071] [Experimental Results] Table 4 shows the grain size measurements of the samples used in the experimental data obtained above, expressed as the average radius of the grains. These results closely matched the square law described above. Based on these results, the slope α that minimizes the error was determined using (Equation 17), resulting in α = 5.25 μm. 2 / s was obtained.

[0072] [Table 4]

[0073] Figure 5(a) shows the results of optimization using Bayesian optimization. The vertical axis represents the minimization function g. min The horizontal axis represents the number of calculations for optimization. The line that decreases in a step-like manner from the upper left to the lower right represents the smallest g calculated so far. min This shows that the minimum g was found on the 40th iteration (13th iteration in the Bayesian optimization calculation), including 27 calculations using 27 sets of parameters. min This means we have obtained the following parameter set. Also, increasing the number of data points as we increase the number of calculations reduces the uncertainty of the Gaussian process regression. Therefore, g is greater than the 40th time mentioned above. min It appears that searching for a set of parameters that minimizes the problem is becoming increasingly difficult. In other words, with only 40 iterations, we are reducing uncertainty and getting closer to finding the optimal set of parameters among 57,400 candidates.

[0074] Figure (b) shows the result of randomly selecting a parameter set and g min The results of the calculation are shown. In the figure, the dashed line represents the minimum g obtained by the Bayesian optimization described above. min This shows that, when a random parameter set is selected, g is faster than when using Bayesian optimization, at least for up to 40 calculations. min The result was significant. In other words, using Bayesian optimization is a method that can find an appropriate set of parameters more quickly than randomly selecting a set of parameters.

[0075] Figure 6 shows the results of simulating grain growth behavior using the CA method with the parameter set obtained by the Bayesian optimization described above, along with the experimental results. The simulation results closely resembled the points obtained from the experimental results and the straight line obtained by the least squares method based on them.

[0076] As described above, by optimizing the parameters of the CA method using Bayesian optimization, it was possible to reproduce the grain growth process of austenite grains. By combining this with Bayesian optimization, it was possible to quickly select a set of CA method parameters with relatively low computational cost from among the countless possible combinations of parameters. In other words, parameter optimization became possible without relying on experience or intuition. Furthermore, because it was compared with experimental data, it became possible to perform calculations that are closer to practical applications, matching the spatial and temporal scales.

[0077] Although representative embodiments of the present invention and modifications thereof have been described above, the present invention is not necessarily limited thereto, and those skilled in the art will be able to find various alternative embodiments and modifications without departing from the spirit of the present invention or the scope of the attached claims.

Claims

1. A method for predicting the microstructure of steel when it is heat-treated in the single-phase region of the austenite phase, A calculation process that divides space into multiple cells and uses the cellular automaton method to calculate the calculated grain size, which is the average grain size of the steel structure for given parameters, at each time step, In a process that repeats a predetermined number of times the parameter optimization process, which involves optimizing the aforementioned parameters using Bayesian optimization on experimental data obtained in advance, A method for predicting the microstructure of a steel during heat treatment, characterized in that the minimization function in the Bayesian optimization includes, in addition to the error in the calculated grain size, the computational cost in the cellular automaton method as a penalty term.

2. The method for predicting the microstructure of steel during heat treatment according to claim 1, characterized in that the experimental data is data on the average grain size of the austenite phase with respect to the heat treatment time.

3. The aforementioned error is, The process involves obtaining a square fitting function obtained by least-squares fitting to the time course of the square of the average particle size in the experimental data, A method for predicting the structure of a steel during heat treatment according to claim 2, characterized by being obtained from a step of comparing the square fitting function with the square of the calculated grain size relative to the time step obtained in the calculation step.

4. The method for predicting the structure of steel during heat treatment according to claim 3, characterized in that the calculation cost includes at least the number of time steps.

5. The minimization function g ｍｉｎ This is the term g of the error in the calculated particle size. １ And, the aforementioned penalty term g ２ It is expressed as the sum of (Equation 1), [Math 1] In the cellular automaton method described above, S: Number of calculation steps S ｍａｘ : Number of calculation steps corresponding to the longest retention time in the experimental data, R: Half of the crystal grain size (radius) at time t corresponding to each calculation step of the square fitting function obtained from experimental data. Rex max : Half of the average particle size (radius) at the longest retention time in the experimental data. Rca(S), Rca(S ｍａｘ ): Calculation steps S and S ｍａｘ Half of the calculated particle size (radius), So, the error term g １ and the penalty term g ２ are each [Math 2] [Math 3] A method for predicting the microstructure of steel during heat treatment according to claim 4, characterized by being represented as follows.

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