A ceramic grain growth simulation method based on a general theory calculation

Abstract

The application relates to a ceramic grain growth simulation method based on a universal theory calculation, which comprises the following steps: firstly, setting a two-dimensional periodic simulation area and related parameters, generating an initial grain organization and performing periodic connected domain marking; then, calculating an effective step energy for representing a grain boundary nucleation resistance according to a current temperature, and a grain boundary migration rate and a grain boundary energy product item for representing a grain boundary migration ability, and calculating a grain boundary migration rate in combination with an adjacent grain equivalent diameter ratio; and finally, utilizing a cellular automaton to promote active pixels, and cooperating with a self-adaptive time step adjustment and a fixed physical time interval topological update refresh to realize physical consistency and numerical stability. Compared with the prior art, the method can be repeatedly iterated in a continuous temperature rising and holding sintering process until a termination condition is met, and can accurately, stably and efficiently realize dynamic visualization and quantitative numerical simulation of nanometer ceramic grain growth behavior.

Description

Technical Field

[0001] This invention relates to the field of ceramic material microstructure evolution simulation technology, and in particular to a ceramic grain growth simulation method based on universal theoretical calculations. Background Technology

[0002] Grain size, grain size distribution, and grain boundary topology of ceramic materials have a significant impact on their dielectric properties, mechanical properties, breakdown strength, grain boundary barriers, and sintering densification behavior. Therefore, establishing numerical simulation methods that can describe the evolution of ceramic grain structure is of great importance for material processing design and performance control.

[0003] Traditional grain growth theory generally posits that grain growth is primarily controlled by grain boundary curvature and grain boundary mobility. Under normal circumstances, increasing temperature enhances grain boundary mobility, thereby promoting grain growth. However, in nanoceramic systems, grain growth behavior is often more complex, influenced not only by temperature and grain boundary mobility but also by grain boundary structure, effective step energy, grain boundary roughening, the size ratio of adjacent grains, and the local topological environment. Taking SrTiO3 ceramics as an example, under low-temperature, small-size conditions, grain growth may not entirely follow the traditional Arrhenius-type monotonic growth law, and may exhibit anomalous grain growth behavior.

[0004] Existing methods for simulating grain growth include phase-field methods, Monte Carlo methods, and cellular automata methods. Phase-field methods can describe the evolution of continuous interfaces well, but they are computationally expensive in simulating high grain numbers and large-scale nanostructures. Monte Carlo methods can describe grain coarsening processes, but the correlation between the number of simulation steps and actual physical time is weak. Traditional cellular automata methods are computationally efficient, but they often struggle to simultaneously couple temperature-dependent effective step energy, grain boundary mobility, grain boundary energy, grain size ratio, and actual grain boundary topology.

[0005] Patent CN118866192B discloses a simulation method for the change in grain roundness during the growth of ternary cathode materials, belonging to the field of ternary cathode material technology. This simulation method for the change in grain roundness during the growth of ternary cathode materials includes: S1, acquiring SEM images at different initial moments of isothermal temperatures and using them as the initial state of a cellular automaton; S2, calculating the grain growth rate and determining the grain growth distance at each moment, and determining the cell orientation transition moment by comparing it with the cell edge length; S3, calculating the grain boundary energy difference before and after the cell orientation transition to determine the cell orientation transition; S4, further transformation for convex and concave cells; S5, designing cell state transition rules based on grain growth rate, local grain boundary curvature, and the minimum energy principle to establish a two-dimensional cellular automaton model to simulate the change in grain roundness during grain growth. However, relying solely on traditional geometric empirical rules of equivalent radius and curvature lacks a physical and dynamic description of the effective step energy, temperature-driven grain boundary migration ability product, and true topological size ratio during the entire heating and holding sintering process of ceramic materials. Furthermore, without establishing an adaptive step size and dynamic topology refresh mechanism, it is difficult to accurately characterize the anomalous grain growth behavior of nanoceramics caused by step nucleation resistance, and it is also impossible to guarantee the computational efficiency and stability of long-cycle complex simulations. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the existing technology and provide a ceramic grain growth simulation method based on universal theory calculations, which realizes efficient, stable and physically consistent numerical simulation of grain structure evolution during the heating and holding sintering process of nano-ceramics.

[0007] The objective of this invention can be achieved through the following technical solutions: A method for simulating ceramic grain growth based on universal theoretical calculations includes the following steps: After setting the two-dimensional periodic simulation region, pixel size, initial number of grains, heating start and end temperatures, heating rate, grain boundary mobility parameters, and effective step energy parameters, an initial grain structure is generated within the two-dimensional periodic simulation region. The initial grain structure is periodically connected to obtain grain numbers, grain areas, and a list of active grain boundary pixels. The equivalent grain diameter is calculated based on the grain area, and the effective step energy, which characterizes grain boundary nucleation resistance, and the product of grain boundary mobility and grain boundary energy, which characterizes grain boundary migration ability, are calculated based on the current temperature. The grain boundary migration rate of each grain is calculated based on the ratio of the equivalent diameters of adjacent grains, the effective step energy, the product of grain boundary mobility and grain boundary energy, and the equivalent grain diameter. The active pixels in the grain boundary active pixel list are updated using a cellular automaton method. When the equivalent diameter of the target grain is greater than that of the adjacent grain, the target grain is designated as a large grain and the adjacent grain as a small grain. This allows the large grain with a positive migration rate to move towards the adjacent small grain region, and the grain area is updated based on the pixel flipping result. The physical time step is adaptively adjusted according to the grain boundary migration rate. When the accumulated physical time reaches a preset fixed physical time interval, the grain topology is updated and refreshed to reconstruct the grain number, grain area, and list of active pixels at grain boundaries. The current temperature is updated according to the heating rate and the start and end temperatures of the heating. As the current temperature increases with the increase of physical time, the calculation of the effective step energy, the product of grain boundary mobility and grain boundary energy, the calculation of grain boundary migration rate, the cellular automaton update, the adaptive adjustment of the physical time step, and the topology update are repeatedly performed to realize the dynamic calculation of the continuous growth and evolution process of ceramic grains until the preset termination temperature, the preset holding time, or the number of grains is lower than the preset termination threshold is reached.

[0008] Furthermore, the specific process of generating the initial grain structure within the two-dimensional periodic simulation region is as follows: Generate uniform seed points for the background within the two-dimensional periodic simulation region; Generate cluster centers, and generate cluster seed points around the cluster centers according to a normal distribution; The background uniform seed points and the cluster seed points are merged as the initial seed points; The initial grain structure is obtained by assigning a grain number to each pixel based on its distance from the nearest initial seed point.

[0009] Furthermore, the effective step energy can be a temperature-dependent function, and its expression is: in, absolute temperature T The effective step energy below, To roughen the reference temperature, The parameter used to control the rate of change of the curve. This is the pre-factor derived from the reference temperature and the reference effective step energy.

[0010] Furthermore, the pre-factor It is determined by the following relationship: in, Reference temperature The effective step energy is given by B, where B is a parameter used to control the rate of change of the effective step energy with temperature.

[0011] Furthermore, the specific process for calculating the grain boundary migration rate of each grain is as follows: Identify adjacent grain pairs and count the grain boundary length between each grain and its adjacent grains; Calculate the equivalent diameter ratio between the target grain and its adjacent grains; When the equivalent diameter of the target grain is larger than that of the adjacent grain, the migration contribution of the grain boundary segment can be calculated based on the equivalent diameter ratio and the effective step energy. The migration contribution of each adjacent grain boundary segment is weighted and summed according to the proportion of the grain boundary length to the total grain boundary length of the grain to obtain the grain-level migration rate.

[0012] Furthermore, grain Relative to adjacent grains The formula for the contribution of grain boundary migration is expressed as: in, Let u be the equivalent diameter of the grain. = / This refers to the grain size ratio. For effective step energy, The current absolute temperature. This is a constant pre-defined based on the Boltzmann constant. For grain boundary mobility, This refers to grain boundary energy.

[0013] Furthermore, the specific process of updating the active pixels in the grain boundary active pixel list using the cellular automata method is as follows: Search for candidate grains with a larger equivalent diameter within the neighborhood of active pixels; The pixel displacement increment that the candidate grain moves toward the current pixel is calculated based on the migration rate corresponding to the candidate grain; The cumulative advance amount is used to determine whether a pixel flip has occurred, and the die number is updated when a pixel flip occurs.

[0014] Furthermore, the pixel flipping process is combined with a local interface energy criterion determined based on the number of opposite neighbors in the pixel neighborhood. If the local interface energy after pixel flipping is not higher than that before flipping, flipping is allowed. If the local interface after pixel flipping is higher than before flipping, the flipping is accepted according to the hot activation probability.

[0015] Furthermore, the specific process of adaptively adjusting the physical time step is as follows: Screening for grains with positive migration rates; A dynamic size shield is used to exclude grains with an area smaller than a set threshold, and a reference migration rate is determined. The physical time step is calculated based on the reference migration rate, ensuring that the advance distance of the representative fastest die within a single time step does not exceed the preset pixel distance.

[0016] Furthermore, the specific process of updating and refreshing the grain topology is as follows: A lightweight topology update is performed at a first time interval, the lightweight topology update including periodic connected domain relabeling and active grain boundary pixel list reconstruction; A complete topology refresh is performed at the second time interval, which includes periodic connected domain remarking, active grain boundary pixel list reconstruction, and cumulative advance validity verification. When comparing the grain number before and after the update and the target grain number on a pixel-by-pixel basis, if the pixel grain number changes or the target grain number is no longer in the neighborhood of the pixel, the cumulative advance amount and the target grain number of the pixel are cleared; otherwise, the cumulative advance amount of the pixel is retained.

[0017] Compared with the prior art, the present invention has the following beneficial effects: 1. Improve the physical consistency of microstructure evolution: The effective step energy, which characterizes the nucleation resistance of grain boundaries, and the product of grain boundary mobility and grain boundary energy, which characterizes the migration ability of grain boundaries, are calculated in real time based on the current temperature. The grain boundary migration rate of each grain is calculated based on the equivalent diameter ratio of adjacent grains, the effective step energy, the product term, and the equivalent diameter of the grains. This allows the simulation process to reasonably reflect the non-uniform growth phenomenon caused by the nucleation resistance of steps in nanoceramics under low temperature and small size conditions, and improves the accuracy of describing anomalous grain growth behavior.

[0018] 2. Simplified model parameters and calculation process: Grain boundary mobility and grain boundary energy are combined into a single product term as an equivalent characterization of grain boundary mobility, and directly used in cellular automata to solve for grain boundary propagation speed, which reduces the number of independent input parameters and simplifies the speed iteration process in cellular automata.

[0019] 3. Balancing computational efficiency and numerical stability: In cellular automata calculations, only active pixels in the list of active pixels at grain boundaries are updated, and the physical time step is adaptively adjusted based on the grain boundary migration rate after being filtered by a dynamic size shield. This ensures that the advancing distance of the representative fastest grain within a single step does not exceed the preset pixel distance, reducing unnecessary computational overhead across the entire process. At the same time, it avoids non-physical interface flipping or numerical computation instability caused by excessively large step sizes during high-speed evolution.

[0020] 4. Reduce the interference of boundary cutting on evolution calculation: Set up a two-dimensional periodic simulation region, and after generating the initial grain structure, use connected component labeling and disjoint set algorithm to periodically label the cross-boundary grains, unify the numbering of cross-boundary grains and connected regions, reduce the impact of boundary cutting on grain size distribution statistics and grain boundary migration calculation, and maintain the topological coherence of the structure evolution calculation.

[0021] 5. Correcting topology errors that have evolved over a long period of time and maintaining the continuity of low-speed advancement: Lightweight updates and full refreshes of the grain topology are performed at fixed physical time intervals to reconstruct the grain number, grain area, and active pixel list. The validity of the accumulated advancement amount is verified during the full refresh. The accumulated advancement amount is cleared only when the pixel number changes or the advancement target fails. This timely corrects the deviation between active pixels and grain number generated during the evolution process and avoids the problem of interruption of small low-speed grain boundary advancement caused by global forced zeroing.

[0022] 6. Achieve continuous quantitative calculation of heating and holding processes: Update the current temperature according to the heating rate and the start and end temperatures of heating, and repeatedly execute the calculation of effective step energy and product terms, migration rate calculation, cellular automaton update, adaptive time step adjustment and topology update during the heating or holding process until the preset termination temperature, preset holding time or grain number termination threshold is reached. This realizes continuous quantitative calculation and process control across heating and holding stages, and can output microstructure data and images that evolve with temperature and time. Attached Figure Description

[0023] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0024] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0025] Example 1 This embodiment provides a ceramic grain growth simulation method based on universal theoretical calculations, such as... Figure 1 As shown, the specific process is as follows: After setting the two-dimensional periodic simulation region, pixel size, initial number of grains, heating start and end temperatures, heating rate, grain boundary mobility parameters, and effective step energy parameters, an initial grain structure is generated within the two-dimensional periodic simulation region. Specifically, it is generated using the non-uniform seed density Voronoi method, including: A portion of uniform background seed points are generated within the simulation region to avoid the formation of excessively large void grains; Generate several cluster centers, and generate cluster seed points around the cluster centers according to a normal distribution; The background uniform seed points and the clustered seed points are merged and used as the initial seed points for Voronoi; Grain numbers are assigned based on the distance of each pixel to the nearest seed point, resulting in an initial microstructure with a wide grain size distribution.

[0026] The initial grain structure is periodically connected to form a label, yielding grain numbers, grain areas, and a list of active grain boundary pixels. The equivalent grain diameter is then calculated based on the grain area using the following formula: in, For the first The number of pixels contained in a single grain. This represents the actual length of a single pixel.

[0027] Calculate the effective step energy used to characterize grain boundary nucleation resistance based on the current temperature: in, Let T be the effective step energy, TR be the roughening reference temperature, B be the parameter controlling the rate of change of the curve, and K be the pre-factor derived from the reference temperature and the reference effective step energy. The natural logarithm of the pre-factor K satisfies the following relationship: in, Reference temperature The effective step energy is given by B, where B is a parameter used to control the rate of change of the effective step energy with temperature.

[0028] The product of grain boundary mobility and grain boundary energy, used to characterize grain boundary migration capability, is a term where the grain boundary mobility M and grain boundary energy γ are not calculated as two independent variables, but rather their product Mγ is used as an equivalent characterization of grain boundary migration capability. This approach reduces the number of model parameters and facilitates direct application in cellular automata calculations to solve for grain boundary propagation speed.

[0029] Grain boundary migration is typically controlled by a thermal activation mechanism, and grain boundary mobility can be expressed as an Arrhenius-type temperature function: in, It is the pre-grain boundary mobility factor. For grain boundary energy, Q R is the apparent activation energy for grain boundary migration, and R is the gas constant. This represents the current absolute temperature.

[0030] The grain boundary migration rate of each grain is calculated based on the equivalent diameter ratio of adjacent grains, the effective step energy, the product of the grain boundary mobility and the grain boundary energy, and the equivalent grain diameter. Specifically, this includes: Identify adjacent grain pairs; Calculate the grain boundary length between each grain and its adjacent grains; Calculate the equivalent diameter ratio between the target grain and its adjacent grains; When the target grain is larger than its neighboring smaller grains, the migration contribution of the grain boundary segment can be calculated based on the grain size ratio and the effective step energy. The specific formula is as follows: in, Let u be the equivalent diameter of grain u, and v be the diameter of adjacent grains. = / Where is the grain size ratio, and C is a constant determined by the Boltzmann constant. For effective step energy.

[0031] The grain-level migration rate is obtained by weighting the contributions of all adjacent grain boundary segments according to the grain boundary length. in: In the formula, Let u be the grain-level migration rate. For the set of grains adjacent to grain u, The length of the grain boundary between grain u and grain v. The total grain boundary length of grain u. It contributes to grain boundary migration.

[0032] The active pixels in the grain boundary active pixel list are updated using a cellular automaton method. When the equivalent diameter of the target grain is larger than that of the adjacent grains, the target grain is designated as a large grain, and the adjacent grains are designated as small grains. This allows the large grains with positive migration rates to move towards the adjacent small grain regions, and the grain area is updated based on the pixel flipping results. The physical time step is adaptively adjusted according to the grain boundary migration rate, specifically including: Only active pixels located near grain boundaries are updated; For each active pixel, search for candidate grains with larger grain sizes within its neighborhood; The pixel displacement increment of the candidate die advancing towards the current pixel is calculated based on the migration rate corresponding to the candidate die. Each pixel corresponds to a cumulative advance amount (acc) and its corresponding target die number (acc_target), which is used to record the cumulative advance degree of the target die to the pixel.

[0033] When the target grain advances the pixel by an increment of Δp within the current time step: In the formula For candidate grain migration rates, This represents the physical time step.

[0034] If the target grain is the same as the previous time step, the cumulative advance is updated as follows: If the target grain changes, the original accumulation process is considered to have failed, and accumulation will restart, i.e., acc_new=Δp.

[0035] When the cumulative advance amount satisfies: acc_new≥1, it means that the target die has completed the advance of one pixel length, and the pixel is immediately flipped to the target die number.

[0036] After the pixel flip is completed, the accumulated advance amount is not immediately cleared to zero. Instead, the remaining advance amount, which exceeds one pixel length, is retained and updated as follows: acc_new = acc_new - 1.

[0037] If the remaining cumulative advance amount is greater than zero, the corresponding target grain number (acc_target) will continue to be maintained; if the remaining cumulative advance amount is less than the set threshold, the cumulative advance amount will be cleared to zero and the target grain number will be reset to empty, so as to wait for the subsequent new grain boundary advance to start accumulating again.

[0038] In addition, when a complete topology refresh causes changes in grain adjacency, in order to avoid mismatch between historical cumulative advance information and the new grain topology, the invalid cumulative advance amount and the corresponding target grain number are cleared; for update processes that only periodically re-mark connected domains and do not change the local grain boundary adjacency, the valid cumulative advance information is retained to ensure the continuity of the grain boundary migration process.

[0039] The dynamic size shield is used to eliminate abnormal control of the time step by grains with excessively small area. The judgment method is as follows: in, Let i be the area of ​​the i-th grain. denoted as the average area of ​​the currently effective grains, and f is the area shield factor.

[0040] The maximum positive migration rate after shield filtering is selected as the reference migration rate. Calculate the theoretical time step: in To preset the maximum single-step advance pixel distance, upper and lower limits are imposed on the step size. .

[0041] The cumulative advance amount is used to determine whether a pixel flip has occurred. The pixel flip process further includes a local interface energy criterion, which statistically analyzes the change in local interface energy before and after the pixel flip. In this embodiment, the number of neighbors with different grain numbers within the eight-neighborhood of a pixel is used to approximate the local interface energy. Specifically, taking the pixel to be flipped as the center, the number of neighbors with different grain numbers from the center pixel within the eight-neighborhood before and after flipping is counted, and these are respectively used as the local interface energy before flipping. and the local interface after flipping The calculation formulas are as follows: in, The grain number of the i-th neighboring pixel; The number of the grain to which the center pixel belonged before flipping; The target grain number is to be flipped; I(·) is an indicator function, which takes the value 1 when the condition is met, and 0 otherwise.

[0042] When pixel flipping causes a decrease in local interface energy, the flip is accepted directly; when local interface energy increases, the hot activation probability is calculated according to the following formula: When ΔE≤0: P=1 When ΔE>0: Where P is the pixel flip acceptance probability; This is a thermal fluctuation parameter used to characterize the effect of thermal disturbance on interface migration.

[0043] This method is used to evaluate the relative change of local interface energy before and after pixel flipping, rather than calculating the absolute value of the actual grain boundary energy of the material, which is consistent with the discretization method of local interface energy in cellular automata grain growth simulation.

[0044] If multiple candidate dies compete for the same pixel, the die number that will ultimately occupy the pixel is determined based on the advance amount.

[0045] When the accumulated physical time reaches a preset fixed physical time interval, the grain topology is updated and refreshed to reconstruct the grain number, grain area, and list of active pixels at grain boundaries; specifically including: Perform periodic connected component relabeling and active grain boundary pixel list reconstruction according to the first time interval; A complete topology refresh is performed at the second time interval. The complete topology refresh includes periodic remarking of connected domains, reconstruction of the active pixel list at grain boundaries, and verification of the validity of the cumulative advance amount. When the pixel grain number changes or the advance target grain is no longer a neighboring grain of the pixel, the cumulative advance amount of the pixel is cleared to zero; otherwise, the cumulative advance amount of the pixel is retained.

[0046] The current temperature is updated according to the stated heating rate and the start and end temperatures of the heating process. As the current temperature increases with physical time, the heating formula is followed. The current temperature is updated, and the calculation of the effective step energy, grain boundary mobility, and grain boundary energy product term, grain boundary migration rate calculation, cellular automaton update, physical time step adaptive adjustment, and topology update refresh are repeatedly performed to realize the dynamic calculation of the continuous growth and evolution process of ceramic grains until the preset termination temperature, preset holding time, or grain number is lower than the preset termination threshold is reached. At this time, the simulation snapshot is automatically saved. The snapshot includes grain number matrix, grain area, active grain boundary pixels, current temperature, physical time, and cumulative progress status.

[0047] The method further includes a thermal insulation calculation step, which includes: The system reads the status file saved during the heating simulation and restores the grain number matrix, grain area array, cumulative propulsion amount, target grain number, grain boundary active pixel list, current temperature, and cumulative physical time. If the system is set to hold at the same temperature, the holding temperature is taken from the current temperature recorded in the status file; if the system is set to hold at a specified temperature, the holding temperature is taken from the user-defined value. During the holding process, the temperature remains constant, and the effective step energy... The product of grain boundary mobility and grain boundary energy The calculations were performed at this holding temperature and used for subsequent grain-level migration rates and cellular automata propulsion.

[0048] During the heat preservation process, physical time continues to advance, but the temperature no longer changes according to the heating rate. The system saves microstructure snapshots according to the preset total heat preservation time, heat preservation snapshot interval, and grain number reduction ratio. The heat preservation simulation ends when the preset heat preservation time is reached or the effective grain number falls below a preset threshold. Through this heat preservation continuation calculation mechanism, the grain growth, grain co-occurrence, and average grain size changes of the microstructure at the same heating endpoint under different heat preservation times can be analyzed.

[0049] Example 2 This embodiment uses two-dimensional ceramic grain structure as a simulation object to illustrate the specific implementation method of this embodiment in the process of low-temperature small-size ceramic grain growth.

[0050] First, the two-dimensional simulation region is set as a square grid with a grid size of 3200. 3200, the actual length corresponding to a single pixel is: = That is, 25nm. The initial grain number is set to 820,000, so that the initial average grain size is approximately on the order of 100nm. The initial heating temperature is set to: The temperature to be stopped during heating is set as follows: The heating rate is set as follows: R=10˚C / min This parameter combination is used to simulate the microstructure evolution of ceramic grains during the heating process under low temperature and small size conditions.

[0051] In the initial tissue generation stage, the ratio of uniform seed points in the background is set to [value missing]. The number of cluster centers is set to The diffusion radius of the cluster normal distribution is set to Each pixel. After merging to generate the initial Voronoi organization, periodic connected component labeling is performed to obtain the initial state.

[0052] During the physical parameter calculation phase, the roughening reference temperature is set to... The effective step energy curve control parameters are set as follows: The reference temperature is set to The reference effective step energy can be set as Therefore, the prefactor is determined. And calculate in real time at different temperatures At the same time, and As a uniform constant given in the model, the current absolute temperature is solved. Below .

[0053] During the propulsion and time step control phase, a representative maximum propulsion distance for a single step is set. Pixel, single-step local pixel flip hard limit is Pixel. The area shield factor in time step control is set to The maximum time step is limited to The minimum time step is limited to .

[0054] During the topology refresh phase, the time interval for lightweight topology updates is set to every [time value missing]. The process is executed once per physical time interval; the time interval for a complete topology refresh and the validity check of its cumulative advance is every [time period missing]. It is executed once per physical time.

[0055] Regarding process preservation, it is set that when the temperature increases... Or the number of grains has decreased by more than [amount missing] compared to the last storage. At that time, a complete snapshot file is automatically saved, including the grain number matrix, grain area, cumulative advance, target grain number, current temperature, and cumulative physics time. The simulation continues to iterate until the current temperature reaches... Or the number of grains decreases to a preset termination threshold.

[0056] The rest is the same as in Example 1.

[0057] Example 3 This embodiment illustrates the specific implementation method of continuing the isothermal grain growth simulation after the heating simulation ends or a certain intermediate temperature is reached.

[0058] First, the complete state file saved during the heating simulation is read. This complete state file includes a grain number matrix, a cumulative advance matrix, a target grain number matrix, a grain area array, grain boundary active pixel row and column indices, the number of active pixels, the current temperature, the cumulative physical time, and the accumulated holding time. By reading this complete state file, the grain structure state at the time the state file was saved is restored, enabling continuation of the calculation from the breakpoint.

[0059] Choose between constant temperature insulation or specified temperature insulation mode. When set to constant temperature insulation, the insulation temperature is... Retrieve the current temperature recorded in the status file; when the insulation temperature is set to a specified temperature, the insulation temperature is... Use the user-preset value. In this embodiment, the final temperature of the heating process is used. Perform constant temperature insulation, and set the total insulation time to [value missing]. .

[0060] During the constant temperature insulation process, the absolute temperature is maintained at [value missing]. Unchanged. Calculate the constant effective step energy at this constant temperature. and constant product terms (in and (These are constants in the equivalent model).

[0061] In each physical time step of heat preservation, based on the real-time area of ​​each grain Update equivalent diameter And re-extract the true grain boundary topology and calculate the size ratio. With weighted grain-level migration rate Cellular automata propulsion continues to use the cumulative propulsion criterion and the Metropolis local interface energy criterion. Time step control continues to use the area shield factor. This constrains the theoretical time step to arrive between.

[0062] During the insulation period, every Perform lightweight topology updates every Perform a full topology refresh and cumulative advance validity check to maintain the continuity of low-speed advance. Set the heat preservation snapshot interval to every heat preservation. Or when the number of grains has decreased by more than [a certain percentage] compared to the last storage. A snapshot is triggered and saved, preserving the complete status file and statistical results (including number-average grain size and area-weighted average grain size). When the cumulative holding time reaches... The constant temperature insulation simulation ends when the number of effective grains is lower than the preset threshold.

[0063] The rest is the same as in Example 1.

[0064] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for simulating ceramic grain growth based on universal theoretical calculations, characterized in that, Includes the following steps: After setting the two-dimensional periodic simulation region, pixel size, initial number of grains, heating start and end temperatures, heating rate, grain boundary mobility parameters, and effective step energy parameters, an initial grain structure is generated within the two-dimensional periodic simulation region. The initial grain structure is periodically connected to obtain grain numbers, grain areas, and a list of active grain boundary pixels. The equivalent grain diameter is calculated based on the grain area, and the effective step energy, which characterizes grain boundary nucleation resistance, and the product of grain boundary mobility and grain boundary energy, which characterizes grain boundary migration ability, are calculated based on the current temperature. The grain boundary migration rate of each grain is calculated based on the ratio of the equivalent diameters of adjacent grains, the effective step energy, the product of grain boundary mobility and grain boundary energy, and the equivalent grain diameter. The active pixels in the grain boundary active pixel list are updated using a cellular automaton method. When the equivalent diameter of the target grain is greater than that of the adjacent grain, the target grain is designated as a large grain and the adjacent grain as a small grain. This allows the large grain with a positive migration rate to move towards the adjacent small grain region, and the grain area is updated based on the pixel flipping result. The physical time step is adaptively adjusted according to the grain boundary migration rate. When the accumulated physical time reaches a preset fixed physical time interval, the grain topology is updated and refreshed to reconstruct the grain number, grain area, and list of active pixels at grain boundaries. The current temperature is updated according to the heating rate and the start and end temperatures of the heating. As the current temperature increases with the increase of physical time, the calculation of the effective step energy, the product of grain boundary mobility and grain boundary energy, the calculation of grain boundary migration rate, the cellular automaton update, the adaptive adjustment of the physical time step, and the topology update are repeatedly performed to realize the dynamic calculation of the continuous growth and evolution process of ceramic grains until the preset termination temperature, the preset holding time, or the number of grains is lower than the preset termination threshold is reached.

2. The ceramic grain growth simulation method based on universal theoretical calculations according to claim 1, characterized in that, The specific process for generating the initial grain structure within the two-dimensional periodic simulation region is as follows: Generate uniform seed points for the background within the two-dimensional periodic simulation region; Generate cluster centers, and generate cluster seed points around the cluster centers according to a normal distribution; The background uniform seed points and the cluster seed points are merged as the initial seed points; The initial grain structure is obtained by assigning a grain number to each pixel based on its distance from the nearest initial seed point.

3. The ceramic grain growth simulation method based on universal theory calculation according to claim 1, characterized in that, The effective step energy is a temperature-dependent function, and its expression is: in, absolute temperature T The effective step energy below, To roughen the reference temperature, The parameter used to control the rate of change of the curve. This is the pre-factor derived from the reference temperature and the reference effective step energy.

4. The ceramic grain growth simulation method based on universal theory calculation according to claim 3, characterized in that, The pre-factor It is determined by the following relationship: in, Reference temperature The effective step energy is given by B, where B is a parameter used to control the rate of change of the effective step energy with temperature.

5. The ceramic grain growth simulation method based on universal theoretical calculations according to claim 1, characterized in that, The specific process for calculating the grain boundary migration rate of each grain is as follows: Identify adjacent grain pairs and count the grain boundary length between each grain and its adjacent grains; Calculate the equivalent diameter ratio between the target grain and its adjacent grains; When the equivalent diameter of the target grain is larger than that of the adjacent grain, the migration contribution of the grain boundary segment can be calculated based on the equivalent diameter ratio and the effective step energy. The migration contribution of each adjacent grain boundary segment is weighted and summed according to the proportion of the grain boundary length to the total grain boundary length of the grain to obtain the grain-level migration rate.

6. The ceramic grain growth simulation method based on universal theory calculation according to claim 5, characterized in that, grains Relative to adjacent grains The formula for the contribution of grain boundary migration is expressed as: in, Let u be the equivalent diameter of the grain. = / This refers to the grain size ratio. For effective step energy, The current absolute temperature. This is a constant pre-defined based on the Boltzmann constant. For grain boundary mobility, This refers to grain boundary energy.

7. The ceramic grain growth simulation method based on universal theory calculation according to claim 1, characterized in that, The specific process of updating the active pixels in the grain boundary active pixel list using the cellular automata method is as follows: Search for candidate grains with a larger equivalent diameter within the neighborhood of active pixels; The pixel displacement increment that the candidate grain moves toward the current pixel is calculated based on the migration rate corresponding to the candidate grain; The cumulative advance amount is used to determine whether a pixel flip has occurred, and the die number is updated when a pixel flip occurs.

8. The ceramic grain growth simulation method based on universal theory calculation according to claim 7, characterized in that, The pixel flipping process is combined with the local interface energy criterion determined based on the number of opposite neighbors in the pixel neighborhood. If the local interface energy after pixel flipping is not higher than that before flipping, flipping is allowed. If the local interface after pixel flipping is higher than before flipping, the flipping is accepted according to the hot activation probability.

9. The ceramic grain growth simulation method based on universal theory calculation according to claim 1, characterized in that, The specific process of adaptively adjusting the physical time step is as follows: Screening for grains with positive migration rates; A dynamic size shield is used to exclude grains with an area smaller than a set threshold, and a reference migration rate is determined. The physical time step is calculated based on the reference migration rate, ensuring that the advance distance of the representative fastest die within a single time step does not exceed the preset pixel distance.

10. The ceramic grain growth simulation method based on universal theory calculation according to claim 1, characterized in that, The specific process of updating and refreshing the grain topology is as follows: A lightweight topology update is performed at a first time interval, the lightweight topology update including periodic connected domain relabeling and active grain boundary pixel list reconstruction; A complete topology refresh is performed at the second time interval, which includes periodic connected domain remarking, active grain boundary pixel list reconstruction, and cumulative advance validity verification. When comparing the grain number before and after the update and the target grain number on a pixel-by-pixel basis, if the pixel grain number changes or the target grain number is no longer in the neighborhood of the pixel, the cumulative advance amount and the target grain number of the pixel are cleared; otherwise, the cumulative advance amount of the pixel is retained.

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