Method and apparatus for analyzing solidification of molten pool grains based on continuous growth limiting factor

Abstract

The present application provides a method and device for analyzing the solidification of molten pool grains based on the continuous growth restriction factor, belonging to the field of material processing. The method provided by the present application includes: establishing a first relationship between the continuous growth restriction factor and a model for predicting the continuous growth rate of grains; determining a second relationship between the model for predicting the continuous growth rate of grains and the solidification path according to the first relationship, and as the solidification path increases, the growth curve corresponding to the second relationship shows exponential growth; determining a demarcation point according to the correlation relationship between the solute concentration, welding speed and solidification path; calculating the curvature radius of each point on the growth curve; and determining the demarcation point between planar growth and cellular growth according to the calculated curvature radius. The method and device for analyzing the solidification of molten pool grains based on the continuous growth restriction factor provided by the present application can help to establish a criterion for the conversion of solidification modes near the melting boundary and to master the evolution law of grain size.

Description

Technical Field

[0001] The present application relates to the technical field of material processing, and particularly to a method and device for analyzing the solidification of molten pool grains based on the continuous growth limiting factor. Background Art

[0002] In the field of material processing, especially in the application of alloy materials, the accurate analysis and control of their solidification process are crucial. With the continuous development of industrial technology, the requirements for the performance of alloy materials are getting higher and higher, so it is necessary to deeply study the solidification behavior of alloy materials under various process conditions. Among them, the solidification process of the molten pool is a key link, which involves the growth of grains, the formation of the microstructure, and the final determination of the performance.

[0003] At present, the theoretical research on "the evolution of the solidification structure near the melting boundary of the molten pool" mainly includes numerical calculation methods and analytical methods. Numerical calculation methods can effectively predict the dynamic evolution of the solid-liquid interface during solidification. Compared with numerical calculation methods, analytical methods have the advantages of low calculation cost and high efficiency. However, both of these methods still have deficiencies in predicting the microstructure of high-speed solidification, and cannot reflect the evolution law of the local solidification structure of the molten pool, which limits the research depth of the solidification mode conversion and grain size evolution of the molten pool. Summary of the Invention

[0004] In view of this, the present application provides a method and device for analyzing the solidification of molten pool grains based on the continuous growth limiting factor, which can help to establish a criterion for the solidification mode conversion near the melting boundary and master the evolution law of grain size.

[0005] Specifically, the present application is implemented through the following technical solutions:

[0006] The first aspect of the present application provides a method for analyzing the solidification of molten pool grains based on the continuous growth limiting factor, and the method includes:

[0007] Establish a first relationship between the continuous growth limiting factor and a model for predicting the continuous growth rate of grains, and the first relationship is an inverse relationship;

[0008] Determine a second relationship between the model for predicting the continuous growth rate of grains and the solidification path according to the first relationship. As the solidification path increases, the growth curve corresponding to the second relationship shows exponential growth;

[0009] Determine a demarcation point according to the correlation between the solute concentration, welding speed and the solidification path, and the demarcation point is the point corresponding to the maximum growth rate in the growth curve;

[0010] Calculate the radius of curvature of each point on the growth curve;

[0011] Determine the demarcation point between planar growth and cellular growth based on the calculated radius of curvature.

[0012] The second aspect of the present application provides a molten pool grain solidification analysis device based on a continuous growth limiting factor, and the device includes a establishment module, a determination module, and a calculation module;

[0013] Among them, the establishment module is used to establish a first relationship between the continuous growth limiting factor and the model for predicting the continuous growth rate of grains, and the first relationship is an inverse relationship;

[0014] The determination module is used to determine a second relationship between the model for predicting the continuous growth rate of grains and the solidification path according to the first relationship. As the solidification path increases, the growth curve corresponding to the second relationship shows exponential growth;

[0015] The determination module is further used to determine the demarcation point according to the correlation between the solute concentration, welding speed and the solidification path, and the demarcation point is the point corresponding to the maximum growth rate in the growth curve;

[0016] The calculation module is used to calculate the radius of curvature of each point on the growth curve;

[0017] The determination module is further used to determine the demarcation point between planar growth and cellular growth based on the calculated radius of curvature.

[0018] The method and device for analyzing the solidification of molten pool grains based on the continuous growth limiting factor provided by this application determine the curve reflecting the solidification behavior through the relationship between the solidification process and the continuous growth rate, and then determine the change point of the physical and chemical behavior according to the shape characteristics of the curve, that is, determine the change point of the physical and chemical behavior through the geometric turning change point. On the one hand, it avoids the cumbersome analysis of the physical and chemical processes and the consideration of other physical and chemical characteristics, processes, and interference factors, improving the efficiency of locating the solidification demarcation point; on the other hand, it can accurately calculate the demarcation point through geometric numerical calculations without estimating through other conditions, improving the accuracy of locating the demarcation point. First, by establishing the inverse relationship (the first relationship) between the continuous growth limiting factor and the model for predicting the continuous growth rate of grains, it is possible to more clearly understand the influence mechanism of the limiting factor on the grain growth rate during the solidification of molten pool grains, which helps researchers accurately grasp the limiting factors of grain growth under different conditions and provides a theoretical basis for regulating the solidification process; secondly, determine the second relationship between the model for predicting the continuous growth rate of grains and the solidification path, and it shows exponential growth as the solidification path increases, with the growth rate continuously increasing, which enables the quantitative analysis of the law of the grain growth rate changing with the solidification path during the solidification process and provides a basis for accurately predicting the solidification behavior; further, determine the demarcation point according to the correlation between the solute concentration, welding speed, and solidification path, that is, the point corresponding to the maximum growth rate in the growth curve. This demarcation point can clearly distinguish between two different growth modes of planar growth and cellular growth, which is of great significance for studying the grain characteristics and solidification behavior under different growth modes; in addition, calculate the curvature radius of each point on the growth curve and determine the demarcation point between planar growth and cellular growth again according to the curvature radius, further improving the accuracy and reliability of the demarcation point. The method provided by this application comprehensively considers key factors such as the continuous growth limiting factor, the continuous growth rate of grains, the solidification path, the solute concentration, and the welding speed, and can comprehensively analyze the solidification process of molten pool grains, reflect the evolution law of the local solidification structure of the molten pool, and thus provide help for establishing the criterion for the transformation of the solidification mode near the melting boundary and mastering the law of grain size evolution. Description of the Drawings

[0019] Figure 1 It is a flowchart of the first embodiment of the method for analyzing the solidification of molten pool grains based on the continuous growth limiting factor provided by this application;

[0020] Figure 2 It is a schematic diagram of the second relationship shown by this application;

[0021] Figure 3 It is a schematic diagram of the change of the actual planar growth distance and the distance between the demarcation point and the melting boundary under different liquid solute concentrations shown by this application;

[0022] Figure 4Schematic diagram of the evolution curves of the actual planar growth distance and the distance between the demarcation point and the melting boundary at different welding speeds shown for this application;

[0023] Figure 5 Schematic diagram of the structure of the second embodiment of the molten pool grain solidification analysis device based on the continuous growth limiting factor provided by this application. Detailed implementation manners

[0024] Here, the exemplary embodiments will be described in detail, and the examples are shown in the drawings. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The implementation manners described in the following exemplary embodiments do not represent all implementation manners consistent with this application.

[0025] The terms used in this application are only for the purpose of describing specific embodiments and are not intended to limit this application. The singular forms "a", "the", and "said" used in this application are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term "and / or" used herein refers to and includes any or all possible combinations of one or more of the associated listed items.

[0026] It should be understood that although the terms first, second, third, etc. may be used in this application to describe various information, such information should not be limited to these terms. These terms are only used to distinguish the same type of information from each other. For example, without departing from the scope of this application, the first information may also be referred to as the second information, and similarly, the second information may also be referred to as the first information. Depending on the context, the word "if" as used herein may be interpreted as "when" or "while" or "in response to determining".

[0027] Specific embodiments are given below to introduce the technical solutions of this application in detail.

[0028] Figure 1 Flowchart of the first embodiment of the molten pool grain solidification analysis method based on the continuous growth limiting factor provided by this application. Please refer to Figure 1 , the method provided in this embodiment may include:

[0029] S101. Establish a first relationship between the continuous growth limiting factor and a model for predicting the continuous growth rate of grains, and the first relationship is an inverse relationship.

[0030] It should be noted that the continuous growth limiting factor is a parameter used to describe the degree of restriction on the continuous growth of grains during the solidification process of materials. During the solidification of grains in the molten pool, there are various factors that affect the growth of grains, and the continuous growth limiting factor comprehensively considers the hindering effects of these factors on grain growth. It may be related to factors such as solute concentration, temperature gradient, solidification rate, etc. For example, a higher solute concentration may lead to an increase in the continuous growth limiting factor, thereby restricting the continuous growth rate of grains. Its role is to quantitatively measure the degree of constraint of various factors on grain growth, providing an important indicator for understanding and regulating the solidification process. The model for predicting the continuous growth rate of grains is a method to predict the continuous growth rate of grains during solidification through mathematical expressions or theoretical models. This model usually considers multiple variables, such as temperature, solute concentration, solidification path, continuous growth limiting factor, etc. It establishes the relationship between these variables and the continuous growth rate of grains so that the growth of grains can be predicted based on known process parameters and conditions. For example, the model may be expressed as a functional relationship between the continuous growth rate of grains and temperature, solute concentration, and continuous growth limiting factor. By measuring and analyzing these variables, this model can be used to predict the growth rate of grains under different conditions, providing a basis for optimizing the solidification process of materials and controlling the grain structure.

[0031] Specifically, establishing the first relationship between the continuous growth limiting factor and the model for predicting the continuous growth rate of grains includes:

[0032] (1) Determining the associated common parameters based on the continuous growth limiting factor and the model for predicting the continuous growth rate of grains.

[0033] It should be noted that when establishing the model of the continuous growth limiting factor and the continuous growth rate of grains, it is necessary to determine the associated common parameters. The common parameters reflect the key variables that affect the interaction between the two. These parameters usually include temperature and temperature gradient, solute concentration, liquidus slope, solidification path, partition coefficient, and diffusion coefficient, etc.

[0034] Among them, temperature changes affect the diffusion rate of atoms, thereby affecting the limiting factor. The temperature gradient determines the direction and speed of heat transfer, affecting the rate of the solidification process; solute concentration affects the formation and growth of crystal nuclei, and a higher solute concentration usually increases the value of the limiting factor; the liquidus slope reflects the influence of alloy composition changes on the melting point. During the solidification process, the magnitude of the liquidus slope determines the rate of change of the liquid phase composition with temperature. If the liquidus slope is large, it means that with a slight change in temperature, the liquid phase composition will change significantly, which will affect the distribution of solutes between the liquid and solid phases, and then change the composition and structure at the solid-liquid interface, thereby affecting the continuous growth limiting factor; the distribution coefficient represents the distribution ratio of solutes between the solid and liquid phases. When the distribution coefficient is small, it means that solutes tend to remain in the liquid phase. During the solidification process, this will lead to the enrichment of solutes at the solid-liquid interface, thereby increasing the continuous growth limiting factor; the diffusion coefficient describes the diffusion ability of solute atoms in the liquid and solid phases. If the diffusion coefficient is small, the diffusion of solute atoms near the solid-liquid interface will be relatively slow, which will lead to the accumulation of solutes at the interface, increasing the continuous growth limiting factor.

[0035] (2)Determine the model of the growth limiting factor based on solute concentration, liquidus slope, and distribution coefficient, and optimize the model of the growth limiting factor by combining other parameters except the common parameters among the associated common parameters to obtain the model of the continuous growth limiting factor.

[0036] Combined with the determined common parameters and the research of relevant theories, the model of the growth limiting factor can be determined based on solute concentration, liquidus slope, and distribution coefficient. Specifically, the model of the growth limiting factor is as follows:

[0037] ;

[0038] Among them, Q is the growth limiting factor, m L is the liquidus slope, C 0 is the solute concentration in the liquid, k is the distribution coefficient.

[0039] Furthermore, consider parameters such as the solidification path, welding speed, diffusion coefficient, and the angle between the welding speed direction at a certain point on the solid-liquid interface and the normal of the fusion line to optimize the model of the growth limiting factor to obtain the model of the continuous growth limiting factor. Specifically, the model of the continuous growth limiting factor is as follows:

[0040] ;

[0041] Among them, Q C is the continuous growth limiting factor, mL is the liquidus slope, C 0 is the solute concentration in the liquid, k is the distribution coefficient, v is the welding speed, ξ is the solidification path, α is the angle between the welding speed direction at a point on the solid-liquid interface and the normal of the fusion line, D is the diffusion coefficient of the solute in the liquid.

[0042] (3) Determine the model for predicting the continuous grain growth rate by combining the KGT model, the common parameters, and the proportional relationship between the model of the growth limiting factor and the model for predicting the continuous grain growth rate.

[0043] It should be noted that in the model for predicting the continuous grain growth rate, the KGT model is a commonly used method that combines factors such as solute distribution, temperature gradient, and crystal growth rate to describe and predict the growth behavior of grains. For the content of the KGT model, please refer to the description of related technologies and will not be elaborated here. By combining the KGT model and the determined growth limiting factor, and analyzing the proportional relationship between the model of the growth limiting factor and the model for predicting the continuous grain growth rate. Specifically, since the continuous grain growth rate is inversely proportional to the growth limiting factor, that is, the larger the growth limiting factor, the smaller the continuous grain growth rate. Thus, a prediction model for the continuous grain growth rate can be further established. Specifically, the model for predicting the continuous grain growth rate is as follows:

[0044] ;

[0045] Among them, v c is the continuous grain growth rate, D is the diffusion coefficient of the solute in the liquid, Γ is the Gibbs-Thomson coefficient, Q is the growth limiting factor, is the degree of supercooling.

[0046] (4) Establish the first relationship based on the model for predicting the continuous grain growth rate and the continuous growth limiting factor.

[0047] Combined with the above description, specifically, replace the growth limiting factor in the model for predicting the continuous grain growth rate with the continuous growth limiting factor to obtain the first relationship, where the first relationship can be expressed as:

[0048] ;

[0049] Among them, v c is the continuous grain growth rate,m L is the liquidus slope, C 0 is the solute concentration in the liquid, k is the partition coefficient, v is the welding speed, ξ is the solidification path, α is the angle between the welding speed direction at a point on the solid-liquid interface and the normal of the fusion line, D is the diffusion coefficient of the solute in the liquid, Q C is the continuous growth limiting factor, Γ is the Gibbs-Thomson coefficient.

[0050] It should be noted that the first relationship characterizes the constraint and influence mechanism of the continuous growth limiting factor on the grain continuous growth rate during the solidification process of the molten pool; the output value of the model of the grain continuous growth rate changes with the continuous growth limiting factor. During the solidification process of the molten pool, the continuous growth limiting factor encompasses various factors affecting the continuous growth of grains, and these factors reflect the constraint on the grain continuous growth rate through this comprehensive index of the continuous growth limiting factor. For example, when the solute concentration involved in the continuous growth limiting factor increases, solute atoms accumulate at the solid-liquid interface, changing the equilibrium state of the interface and increasing the resistance to grain growth. This resistance is represented by the increase in the continuous growth limiting factor, thereby imposing a constraint on the grain continuous growth rate and reducing it.

[0051] When the continuous growth limiting factor changes, it means that the conditions affecting grain growth have changed. If the continuous growth limiting factor increases, it indicates that the factors restricting grain growth have strengthened. For example, it may be that the liquidus slope becomes larger, making the change in the liquid phase composition caused by temperature change more drastic, resulting in more severe segregation of solutes at the interface, or the partition coefficient becomes smaller, making the solute more inclined to remain in the liquid phase and exacerbating the enrichment of solutes at the interface. These changes will all increase the continuous growth limiting factor, thereby causing the value of the model used to predict the grain continuous growth rate to decrease accordingly, that is, the grain continuous growth rate decreases. Conversely, if the continuous growth limiting factor decreases, it means that the restriction on grain growth weakens, and the grain continuous growth rate will increase accordingly. This dynamic change relationship accurately reflects the influence of the complex physical and chemical processes during the solidification process of the molten pool on grain growth.

[0052] S102. Determine the second relationship between the model for predicting the grain continuous growth rate and the solidification path according to the first relationship. As the solidification path increases, the growth curve corresponding to the second relationship shows exponential growth.

[0053] Combined with the above description, it is known that the first relationship is that the grain continuous growth rate (represented by the model for predicting the grain continuous growth rate) is inversely proportional to the continuous growth limiting factor. And the continuous growth limiting factor itself is related to various factors during the solidification process, and the solidification path is one of the key factors. During the solidification process, as the solidification path extends (which can be the advancement of time or the change of spatial position, such as the distance that the molten pool experiences from the starting solidification point to complete solidification), factors such as the solute distribution and temperature gradient will change, and these changes will directly affect the continuous growth limiting factor.

[0054] Figure 2 For the schematic diagram of the second relationship shown in this application, please refer to Figure 2 , as the solidification path increases, the growth curve corresponding to the second relationship shows exponential growth, and the growth rate of the growth curve continuously increases. By changing the input and output variables of the first relationship, the second relationship can be obtained. Specifically, the input and output of the first relationship can be calibrated as Figure 2 the content in, to obtain the second relationship, so there is no need to recalculate. Among them, Figure 2 PG in represents planar growth, CG represents cellular growth, vc represents the growth rate, and rmin represents the minimum curvature radius, ξ S represents the distance from the solidification boundary.

[0055] Please continue to refer to Figure 2 , the second relationship between the model for predicting the grain continuous growth rate and the solidification path shows that as the solidification path increases, the growth rate of the growth curve corresponding to the second relationship changes from small to large, and there is a turning point from slow to fast. According to the change direction and change value of the growth rate, the turning point of the growth rate change can be determined. Specifically, in Figure 2 this turning point is N. The value of the grain continuous growth rate is relatively stable before point N, and the growth rate of the grains is very small at this stage, that is, the solidification mode of the molten pool near the melting boundary can be assumed to be near-planar growth. When the solid-liquid interface gradually approaches point N, the value of the grain continuous growth rate begins to increase and rises rapidly in the later stage, indicating that the phenomenon of planar growth to cellular growth occurs at point N. Among them, planar growth is a growth mode in which the solid-liquid interface basically advances in a planar shape during the solidification process. In this growth mode, the solidification front is relatively smooth, and new crystal layers grow on the existing crystal layers in an approximately parallel manner. Cellular growth refers to a growth mode in which the solidification interface no longer maintains a planar state but forms a series of convex structures similar to cells during the solidification process. These convex cellular structures can be regular hexagons or other shapes, and they grow from the solidification front into the liquid phase.

[0056] The method provided by the present invention, wherein the second relationship is the relationship between the distance to the solidification boundary and the grain continuous growth rate, and the second relationship is an exponential growth curve, that is, the absolute value continuously increases, but the growth rate increases from small to large. Determining the demarcation point further includes determining the largest inscribed circle of each position point, obtaining the radius of the largest inscribed circle, and taking the position point corresponding to the minimum value of the radius of the largest inscribed circle as the demarcation point. As an alternative embodiment, since the second relationship is a continuously changing curve, continuous calculation may result in a large amount of calculation. Determining the demarcation point may include: predicting the candidate area of the solidification boundary according to the solidification behavior, where the candidate area is the area within a preset distance from the solidification boundary, predicting the distribution characteristics of the solidification speed according to the solidification behavior, and determining the candidate area of the demarcation point of the second relationship based on the candidate area and the distribution characteristics, and calculating the demarcation point in the candidate area of the demarcation point.

[0057] S103. Determine the demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path, and the demarcation point is the point corresponding to the maximum growth rate in the growth curve.

[0058] It should be noted that during the solidification process of the molten pool, the solute concentration plays a key role. A higher solute concentration will cause the segregation of solutes at the solid-liquid interface, thereby affecting the solidification path. As solidification progresses, the redistribution of solute atoms will change the composition and temperature conditions at the solidification front, thus affecting the grain growth mode and the solidification path. For example, when the solute concentration increases, the possibility of constitutional supercooling increases, and the solidification path may deviate from the planar growth mode and change to a more complex mode such as cellular growth. The welding speed is directly related to the heat input and cooling speed of the molten pool. A faster welding speed means that the molten pool receives relatively less heat input per unit time and the cooling speed increases, which will affect the solidification time and solidification path of the molten pool. For example, during rapid welding, the solidification path of the molten pool may be shortened, and the time for solute diffusion is also correspondingly reduced, making the distribution of solutes at the solid-liquid interface more uneven, thereby affecting grain growth and the solidification mode.

[0059] It should also be noted that the solidification path is the result of the combined action of factors such as solute concentration and welding speed, and it can be reflected by describing the change trajectories of parameters such as temperature and composition during the process of the molten pool changing from liquid to solid. Different combinations of solute concentration and welding speed will result in different solidification paths, and these paths may cause the grain growth to exhibit planar growth, cellular growth or other more complex growth modes.

[0060] Specifically, determining the demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path includes:

[0061] (1) Obtain the grain growth conditions at the alloy melting boundary under different liquid solute concentrations and welding speeds.

[0062] It should be noted that different liquid solute concentrations and welding speed conditions can be simulated by experimental means, such as setting different welding process parameters in the laboratory. For example, alloy materials with different compositions are used to change the solute concentration, and at the same time, the welding speed is controlled by adjusting the parameters of the welding equipment (such as welding current, voltage, etc.).

[0063] During the welding process, observation equipment such as high-speed cameras and microscopes is used to observe the grain growth at the alloy melting boundary. The high-speed camera can capture the dynamic growth process of grains during solidification, while the microscope can provide more detailed information on the grain microstructure, including the size, shape, orientation, etc. of the grains. For each combination of liquid solute concentration and welding speed, the growth state of the grains at different times is recorded in detail. For example, record the time when the grains start to form, the growth direction of the grains, the growth rate of the grains in different directions, etc.

[0064] Based on the above growth situation, determine the curves of the actual planar growth distance and the distance between the demarcation point and the melting boundary varying with the liquid solute concentration at different liquid solute concentrations.

[0065] It should be noted that according to the observed grain growth situation, determine the range of the planar growth stage. The characteristic of the planar growth stage is that the solid-liquid interface advances basically in a planar shape, and at this time, the planar growth distance can be measured. At the same time, by observing the change in the grain growth mode (such as the transition from planar growth to cellular growth), determine the position of the demarcation point, and then measure the distance between the demarcation point and the melting boundary.

[0066] As the liquid solute concentration changes, repeat the above measurement process. Organize the data of the actual planar growth distance and the distance between the demarcation point and the melting boundary at different liquid solute concentrations. Taking the liquid solute concentration as the abscissa and the planar growth distance and the distance between the demarcation point and the melting boundary as the ordinates respectively, plot two curves. Figure 3 The schematic diagram of the changes in the actual planar growth distance and the distance between the demarcation point and the melting boundary at different liquid solute concentrations shown in this application is as follows. Please refer to Figure 3 , and through the curves, the influence of the liquid solute concentration on planar growth and the position of the demarcation point can be intuitively reflected.

[0067] Based on the above growth situation, determine the evolution curves of the actual planar growth distance and the distance between the demarcation point and the melting boundary at different welding speeds.

[0068] Similar to the analysis of the concentration of liquid solutes, for different welding speed conditions, based on the observed grain growth at the alloy melting boundary, determine the planar growth distance and the distance between the demarcation point and the melting boundary. With the change of the welding speed, repeat the measurement and recording. Taking the welding speed as the abscissa and the planar growth distance and the distance between the demarcation point and the melting boundary as the ordinates, plot two evolution curves. Figure 4 Figure 196 shows a schematic diagram of the evolution curves of the actual planar growth distance and the distance between the demarcation point and the melting boundary at different welding speeds shown in this application. Please refer to Figure 4 Figure 197 to show the influence of the welding speed on planar growth and the position of the demarcation point through the evolution curves.

[0069] (4)Based on the above curves and evolution curves, determine the relevant relationship and obtain the demarcation point.

[0070] Through the comprehensive analysis of the above curves related to the liquid solute concentration and the evolution curves of the welding speed, the relevant relationship among the solute concentration, the welding speed, and the solidification path can be determined. For example, it is observed that as the liquid solute concentration increases and the welding speed decreases, the planar growth distance may decrease, and the distance between the demarcation point and the melting boundary may change (such as decrease or increase).

[0071] Based on these curves and evolution curves, the demarcation point can be obtained by finding the characteristic point where the planar growth ends and the growth mode changes (i.e., the demarcation point). This characteristic point may be the point where the slope of the curve changes abruptly, or the point that satisfies certain conditions (such as a specific proportional relationship between the planar growth distance and the distance between the demarcation point and the melting boundary). The demarcation point determined according to these analyses can be used to accurately distinguish different grain growth modes such as planar growth and cellular growth.

[0072] It should be noted that determining the demarcation point based on the relevant relationship among the solute concentration, the welding speed, and the solidification path mainly starts from the perspective of macroscopic process parameters, considering the relationship between factors such as the solute concentration and the welding speed, which have an important impact on the solidification process in actual production, and the solidification path, so as to find a demarcation point that can distinguish planar growth and cellular growth. This demarcation point more reflects the influence of external process conditions on the growth mode transition. Among them, this demarcation point is the point corresponding to the maximum growth rate in the growth curve, which represents a key state in the solidification process. Before and after this point, the growth mode and rate of grain growth will change significantly.

[0073] S104. Calculate the radius of curvature of each point on the growth curve.

[0074] It should be noted that the demarcation point between the two growth modes of planar growth and cellular growth can be more accurately determined by calculating the radius of curvature of each point on the growth curve. Specifically, in the planar growth stage, the growth curve is relatively gentle and the radius of curvature is large; while when the growth mode begins to change to cellular growth, the degree of curvature of the curve changes and the radius of curvature becomes smaller. By analyzing the change in the radius of curvature, this change in the growth mode can be captured more acutely.

[0075] When specifically implemented, calculating the radius of curvature of each point on the growth curve includes:

[0076] (1) Determining the curve equation corresponding to the growth curve based on the growth curve.

[0077] It should be noted that data fitting techniques can be used to determine the curve equation corresponding to the growth curve. Specifically, please refer to the description of related technologies and will not be elaborated here.

[0078] (2) Obtaining the coordinates of each point on the growth curve.

[0079] The coordinates of a series of points can be obtained by discretely taking values of the independent variable (such as variables related to the solidification path), and then substituting these values into the curve equation to calculate the corresponding values of the dependent variable (such as the grain growth rate).

[0080] (3) Based on the curve equation and the coordinates of each point, calculating the first derivative and the second derivative corresponding to each point respectively.

[0081] An analytical derivative method (for simple function forms) or a numerical derivative method (for complex or discrete data) can be selected according to the actual situation to calculate the first derivative and the second derivative corresponding to each point respectively.

[0082] (4) Calculating the curvature corresponding to each point based on the curvature formula and the first derivative and the second derivative calculated for each point.

[0083] It should be noted that for a planar curve y = f(x) The curvature k at a certain point on the curve

[0084] ;

[0085] where, is the first derivative of this point, is the second derivative of this point.

[0086] Substituting the first derivative and the second derivative of each point calculated previously into this curvature formula, the curvature of each point can be calculated.

[0087] (5) Calculate the radius of curvature corresponding to each point based on the curvature corresponding to each point.

[0088] It should be noted that the radius of curvature R and the curvature k are related as (when k ≠ 0). Therefore, the specific formula for the radius of curvature is as follows:

[0089] ;

[0090] where is the first derivative of each point calculated, is the second derivative of each point calculated.

[0091] S105. Determine the demarcation point between planar growth and cellular growth according to the calculated radius of curvature.

[0092] It should be noted that the point where the minimum value of the calculated radius of curvature is located is the demarcation point between planar growth and cellular growth. In the growth curve, the curve in the planar growth stage is relatively flat because during planar growth, the change in the grain growth rate is relatively slow, the slope change of the growth curve is small, the corresponding curvature is small, and thus the radius of curvature is large. When the growth mode changes from planar growth to cellular growth, the change in the grain growth rate intensifies. This change is reflected in the growth curve as a sharp increase in the degree of curve bending, an increase in curvature, and a corresponding decrease in the radius of curvature. Therefore, the point with the minimum radius of curvature represents the position where the curve bending degree is the largest, that is, the position where the growth mode changes significantly, and this position can reasonably be considered as the demarcation point between planar growth and cellular growth.

[0093] It should be noted that determining the demarcation point between planar growth and cellular growth according to the calculated radius of curvature starts from the microscopic geometric perspective and determines the demarcation point by analyzing the change in the radius of curvature of the growth curve. This demarcation point focuses more on distinguishing different growth modes from the morphological characteristics of the curve. Combining the determination of the demarcation point from the macroscopic process parameter perspective before and the determination of the demarcation point from the microscopic geometric perspective here, determining the demarcation point from different perspectives helps to more comprehensively and accurately understand and grasp the transition process between planar growth and cellular growth, and provides a more reliable basis for the analysis and control of the molten pool grain solidification process.

[0094] Optionally, after determining the demarcation point between planar growth and cellular growth according to the calculated radius of curvature, it includes:

[0095] (1) Obtain the grain growth rate data of the melting boundary of the alloy material under different process parameter conditions.

[0096] It should be noted that changing process parameters such as welding speed, solute concentration, and temperature gradient will affect the solidification process of alloy materials at the melting boundary. Experiments are carried out under various combinations of different process parameters, and data on the grain growth rate of alloy materials at the melting boundary are collected. Advanced measurement techniques, such as high-speed photography combined with image analysis software, can be used to measure the grain growth rate to ensure the accuracy and reliability of the data.

[0097] (2) Perform phase-field model simulation based on the correlation between the data and the solidification path and obtain the simulation results.

[0098] It should be noted that the phase-field model is a microscopic structure evolution simulation method based on the continuum theory. It describes the transition region between different phases by introducing an order parameter (phase-field variable). During the simulation of grain growth, the phase-field model can consider various physical factors such as solute diffusion, heat diffusion, and interfacial energy. According to the collected grain growth rate data and their correlation with the solidification path, the parameters in the phase-field model (such as interfacial energy coefficient, solute diffusion coefficient, etc.) are reasonably set. These parameter settings are based on experimental data and the physical properties of the material, enabling the phase-field model to accurately reflect the actual solidification process.

[0099] Run the phase-field model for simulation calculations to simulate the solidification process of alloy materials under different process parameters. As the simulation time progresses, the phase-field model can output microscopic structure information at different times, including the shape, size, and growth rate of grains, etc. These simulation results can intuitively display the growth evolution of grains during the solidification process.

[0100] (3) Determine the data points of the grain growth rate varying with the solidification path and the position of the transition point from planar growth to cellular growth based on the simulation results.

[0101] Extract the data of the grain growth rate varying with the solidification path from the simulation results of the phase-field model. These data form a series of data points, depicting the dynamic process of grain growth under the simulation conditions. By observing the changes in the growth morphology of grains during the simulation, determine the transition point from planar growth to cellular growth. In the phase-field model, this transition point can be judged based on characteristics such as the flatness of the grain interface and the change in growth direction. For example, when the grain interface starts to show protrusions from a relatively smooth plane, forming a cellular-like structure, it can be determined as the transition point.

[0102] (4) Compare the positions of the data points with the position of the point corresponding to the maximum growth curvature, and compare the position of the transition point from planar growth to cellular growth with the position point corresponding to the point of the maximum growth curvature on the solidification path to determine the validity of the simulation results and the second relationship.

[0103] It should be noted that the point corresponding to the maximum growth curvature is the theoretical boundary point between planar growth and cellular growth obtained from the previous analysis of the growth curve. By comparing the data points and transition points obtained from the phase-field model simulation with it, it can be verified whether the simulation results are consistent with the theoretical analysis. If the data points and transition points in the simulation results are close to the point corresponding to the maximum growth curvature theoretically, it indicates that the phase-field model can accurately simulate the grain growth process, and at the same time proves that the second relationship (the relationship between grain growth rate and solidification path) obtained based on the growth curve analysis is effective.

[0104] In specific implementation, for the data points, their positions on the solidification path coordinate axis and the values of the grain growth rate can be compared. If the data points obtained from the simulation are near the same solidification path position as the point corresponding to the maximum growth curvature theoretically, and the change trends of the growth rates are also consistent, then it indicates that the simulation results are consistent with the theoretical analysis. For the planar growth to cellular growth transition point, compare its position on the solidification path. If the position of the transition point obtained from the simulation is close to the position of the point corresponding to the maximum growth curvature theoretically on the solidification path, it indicates that the phase-field model has successfully captured the transition of the growth mode, further verifying the effectiveness of the simulation results and the second relationship.

[0105] Optionally, determine the boundary point between planar growth and cellular growth according to the calculated curvature radius, and then, it includes:

[0106] (1) Divide the molten pool solidification process into different stages based on the boundary point, and the stages include a planar growth stage and a cellular growth stage.

[0107] It should be noted that when the boundary point between planar growth and cellular growth is determined by calculating the curvature radius, the two main stages in the molten pool solidification process can be clearly distinguished. The interval from the start of solidification to the boundary point is the planar growth stage, and its characteristic is that the solid-liquid interface basically advances forward in a planar shape. And after the boundary point is the cellular growth stage, at this time, a series of convex structures similar to cells are formed on the solidification interface and grow into the liquid phase. Such a division helps to study the molten pool solidification process more carefully. Different growth stages are dominated by different factors and have different effects on the final microstructure and properties of the material. For example, the grain structure formed in the planar growth stage is relatively regular and smooth, while the cellular growth stage will introduce complex microstructural changes.

[0108] (2) Determine the influence degree of the liquid solute concentration and welding speed in each stage on the solidification result.

[0109] It should be noted that during the planar growth stage, a lower liquid solute concentration is conducive to maintaining the planar growth mode because the low solute concentration reduces the enrichment of solutes at the solid-liquid interface and avoids constitutional supercooling. If the solute concentration increases slightly, planar growth may still be maintained within a certain range, but it will reduce the stability of planar growth. For example, when the solute concentration increases, the temperature gradient at the solidification front may be required to be more stringent to suppress the generation of constitutional supercooling, otherwise, the cellular growth stage may be entered prematurely. At this time, a faster welding speed will increase the cooling rate of the molten pool, and the temperature gradient at the solidification front can be increased during this stage. A larger temperature gradient helps to maintain planar growth because it can more effectively conduct heat away from the solidification front and suppress local supercooling. However, if the welding speed is too fast, it may lead to insufficient solute diffusion and accumulation in local areas, which will also have an adverse effect on planar growth.

[0110] During the cellular growth stage, a higher liquid solute concentration is an important factor promoting cellular growth. The enrichment of solutes at the solid-liquid interface changes the liquid phase composition at the interface and reduces the melting point of the liquid phase. This causes the liquid in local areas to start solidifying at a temperature slightly higher than the equilibrium solidification temperature, forming cellular protrusions. As the solute concentration further increases, the cellular structure may become more complex, such as smaller cellular sizes and more cellular branches. At this time, a slower welding speed during the cellular growth stage will slow down the cooling rate of the molten pool, providing more time for solute diffusion and the growth of the cellular structure. At the same time, due to the slow cooling rate, the temperature gradient is relatively small, and constitutional supercooling is easier to maintain, which is conducive to the continuous progress of cellular growth. If the welding speed increases, cellular growth may be inhibited because the heat dissipation is too fast, and the growing cellular structure may be surrounded by the rapidly solidifying liquid phase before it has time to fully develop.

[0111] (3)Based on the degree of influence, determine the relationship between the planar growth distance and the average grain size during the solidification process of the molten pool.

[0112] It should be noted that during the planar growth stage, the length of the planar growth distance directly affects the initial grain size. A longer planar growth distance means that the grains have more growth time in the planar direction, which will increase the grain size during the planar growth stage. For example, under the same liquid solute concentration and welding speed conditions, the grain size at the end of the planar growth stage of a molten pool with a longer planar growth distance is relatively larger.

[0113] It should also be noted that the cellular growth stage is carried out on the basis of the planar growth stage, and the planar growth distance will affect the initial state of cellular growth. If the planar growth distance is short, cellular growth may start earlier, and due to the small initial grain size, the development of the cellular structure may be restricted to a certain extent. While a longer planar growth distance provides a better basis for cellular growth, which may make the cellular structure more regular and developed. From the perspective of the average grain size, the final average grain size of the molten pool with a long planar growth distance and sufficient cellular growth may be larger because the grains have sufficient growth space and time during both the planar growth stage and the cellular growth stage. At the same time, the interaction between the planar growth distance and the cellular growth stage also affects the distribution of grain size.

[0114] Optionally, determine the demarcation point between planar growth and cellular growth according to the calculated radius of curvature. After that, it further includes:

[0115] (1) Measure the grain size of the local area of the molten pool at the alloy melting boundary to obtain the average grain size data.

[0116] It should be noted that an optical microscope or an electron microscope can be used to observe the microstructure of the molten pool at the alloy melting boundary. By taking pictures of the grains in the local area and then using image analysis software to measure the size of each grain.

[0117] (2) Analyze the average grain size data to form an average grain size evolution curve.

[0118] Taking relevant parameters such as solidification time or solidification path as the abscissa and the average grain size as the ordinate, plot the average grain size data measured at different times or different solidification stages in a coordinate system, and connect these data points with a curve to form an average grain size evolution curve. This curve intuitively shows the change trend of grain size with time or solidification path during the entire solidification process of the molten pool. By observing the shape of the curve, the stage characteristics of grain growth can be initially judged. For example, the stage with a larger curve slope may indicate a faster grain growth rate, while the stage with a smaller slope may indicate a slower growth rate or inhibited growth.

[0119] (3) Analyze the variation relationship of the average grain size under different solute concentrations and welding speeds based on the evolution curve.

[0120] It should be noted that when the solute concentration changes, the change in the average grain size evolution curve is observed. At a lower solute concentration, the planar growth stage may be longer, and the average grain size may gradually and steadily increase during the planar growth stage because the low solute concentration is conducive to planar growth, enabling the grains to grow under relatively stable conditions. As the solute concentration increases, a change in the growth mode may occur earlier, transitioning from planar growth to cellular growth. During the cellular growth stage, due to solute segregation and complex growth patterns, the growth trend of the average grain size may change, and there may be a slowdown in growth, fluctuations, or grain refinement because the enrichment of solute atoms at the solid-liquid interface changes the growth conditions, affecting the driving force and growth direction of grain growth.

[0121] Regarding the welding speed, a faster welding speed will accelerate the cooling rate of the molten pool. On the average grain size evolution curve, this is manifested as a possible shortening of the planar growth stage, a reduction in the time for grain size growth during the planar growth stage, resulting in a relatively smaller average grain size at the end of the planar growth stage. During the cellular growth stage, rapid cooling may inhibit the full growth of the cellular structure, restricting the growth of the average grain size. Conversely, at a slower welding speed, the molten pool cools more slowly, the planar growth stage may be extended, which is conducive to grain growth during the planar growth stage. Moreover, during the cellular growth stage, the slower cooling rate provides more favorable conditions for the growth of the cellular structure, which may result in a larger increase in the average grain size, but may also lead to an increase in the non-uniformity of the grain size. Through such analysis, the specific impact of the welding speed on grain size changes can be clarified, providing a basis for controlling the welding process to obtain the desired grain size.

[0122] The method provided by this application, by determining the demarcation point between planar growth and cellular growth, on the one hand, can clearly divide the solidification process of the molten pool into different stages. This division helps to deeply understand the physical mechanisms and growth characteristics of each stage. For example, the grain growth during the planar growth stage is relatively regular and stable, while the cellular growth stage shows the growth of a complex cellular structure. This makes the analysis of the solidification process of the molten pool more accurate from the whole to the local. During the analysis process, factors such as solute concentration and welding speed that affect the solidification process are comprehensively considered. The roles of these factors are different in different growth stages. By studying their effects in detail, the microscopic structure evolution during the solidification process of the molten pool can be grasped more comprehensively. For example, the solute concentration affects the growth stability during the planar growth stage and the formation of the cellular structure during the cellular growth stage; the welding speed affects the time and morphology of grain growth by changing the cooling rate of the molten pool.

[0123] In the second aspect, a first relationship (inverse relationship) between the continuous growth limiting factor and the model for predicting the continuous grain growth rate is established, and a second relationship between the model for predicting the continuous grain growth rate and the solidification path is established (the growth curve shows exponential growth with an increasing growth rate). These relationships quantitatively describe the interaction between the grain growth rate and other factors through a mathematical model, making the prediction of the grain growth rate more accurate.

[0124] In the third aspect, the effectiveness of the model and simulation results is verified in various ways. For example, after determining the demarcation point, the grain growth rate data under different process parameter conditions are obtained, phase field model simulations are carried out, and the simulation results are compared with the points corresponding to the maximum growth curvature obtained based on theoretical analysis. This multi-dimensional verification method can ensure that the model and simulation results can accurately reflect the actual situation of the molten pool solidification process and improve the reliability of the analysis method.

[0125] Corresponding to the foregoing embodiment of a method for analyzing the solidification of molten pool grains based on the continuous growth limiting factor, the present application also provides an embodiment of an apparatus for analyzing the solidification of molten pool grains based on the continuous growth limiting factor.

[0126] Figure 5 It is a schematic structural diagram of Embodiment 2 of the apparatus for analyzing the solidification of molten pool grains based on the continuous growth limiting factor provided by the present application. Please refer to Figure 5 The apparatus provided in this embodiment includes a building module 510, a determination module 520, and a calculation module 530;

[0127] Among them, the building module 510 is used to establish a first relationship between the continuous growth limiting factor and the model for predicting the continuous grain growth rate, and the first relationship is an inverse relationship;

[0128] The determination module 520 is used to determine a second relationship between the model for predicting the continuous grain growth rate and the solidification path according to the first relationship. As the solidification path increases, the growth curve corresponding to the second relationship shows exponential growth;

[0129] The determination module 520 is further used to determine the demarcation point according to the correlation between the solute concentration, welding speed and the solidification path, and the demarcation point is the point corresponding to the maximum growth rate in the growth curve;

[0130] The calculation module 530 is used to calculate the curvature radius of each point on the growth curve;

[0131] The determination module 520 is further used to determine the demarcation point between planar growth and cellular growth according to the calculated curvature radius.

[0132] The apparatus of this embodiment can be used to execute Figure 1The steps of the method embodiments shown are similar in specific implementation principles and implementation processes, and will not be elaborated here.

[0133] For the implementation processes of the functions and roles of each unit in the above device, please refer to the implementation processes of the corresponding steps in the above method for details, and will not be elaborated here.

[0134] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to the partial descriptions of the method embodiments. The device embodiments described above are only illustrative. The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed to multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the solution of this application. Those of ordinary skill in the art can understand and implement it without creative efforts.

[0135] The above are only the preferred embodiments of this application and are not intended to limit this application. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of this application shall be included within the scope of protection of this application.

Claims

1. A method for analyzing molten pool grain solidification based on continuous growth limiting factor, characterized in that: The method comprises: Establishing a first relationship between the continuous growth limiting factor and a model for predicting the continuous growth rate of grains, wherein the first relationship is an inverse relationship; Determining a second relationship between the model for predicting the continuous growth rate of grains and the solidification path according to the first relationship, wherein as the solidification path increases, a growth curve corresponding to the second relationship increases exponentially; Determine a demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path, the demarcation point being the point corresponding to the maximum growth rate in the growth curve; determine the demarcation point including: estimating a candidate region of a solidification boundary according to the solidification behavior, the candidate region being a region within a preset distance from the solidification boundary, estimating a distribution characteristic of the solidification speed according to the solidification behavior, determining a candidate region of a demarcation point of the second relationship based on the candidate region and the distribution characteristic, and calculating a demarcation point in the candidate region of the demarcation point; Calculating the radius of curvature of each point on the growth curve; The dividing point between planar growth and cellular growth is determined based on the calculated radius of curvature; The demarcation point is determined by combining the demarcation point determined in the step of determining the demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path and the demarcation point determined in the step of determining the demarcation point between planar growth and cellular growth according to the calculated curvature radius.

2. The method according to claim 1, characterized in that The input and output variables of the first relationship are changed to obtain the second relationship; as the solidification path increases, the growth rate of the growth curve corresponding to the second relationship changes from small to large, and the turning point of the growth rate change is determined according to the change direction and change value of the growth rate.

3. The method according to claim 1, characterized in that Calculating the curvature radius of each point on the growth curve includes: Determining a curve equation corresponding to the growth curve based on the growth curve; Obtaining coordinates of each point on the growth curve; Based on the curve equation and the coordinates of each point, respectively calculate the first-order derivative and the second-order derivative corresponding to each point; Calculate the curvature corresponding to each point based on the curvature formula and the calculated first-order derivative and second-order derivative corresponding to each point; The curvature radius corresponding to each point is calculated based on the curvature corresponding to each point.

4. The method according to claim 1, characterized in that: The step of establishing a first relationship between the continuous growth limiting factor and a model for predicting the continuous growth rate of grains comprises: determining a common parameter associated with a model for predicting a continuous growth rate of grains based on the continuous growth limiting factor; Determining a growth limiting factor model based on solute concentration, liquidus slope and partition coefficient, and optimizing the growth limiting factor model by combining other parameters in the associated common parameters except the common parameters to obtain a continuous growth limiting factor model; Determine a model for predicting the continuous growth rate of grains by combining the KGT model, the common parameters, and the proportional relationship between the model of the growth limiting factor and the model for predicting the continuous growth rate of grains; The first relationship is established based on the model for predicting the continuous growth rate of grains and the continuous growth limiting factor.

5. The method according to claim 1, characterized in that The dividing point between the planar growth and the cellular growth is determined according to the calculated curvature radius, and then, the method comprises: Obtain the grain growth rate data of the melting boundary of the alloy material under different process parameters; Perform phase field model simulation based on the correlation between the data and the solidification path and obtain simulation results; Determine, based on the simulation results, data points of grain growth rate changes with solidification path and the position of the transition point from planar growth to cellular growth; The data point is compared with the position of the point corresponding to the maximum value of the growth curvature, and the position of the transition point from planar growth to cellular growth is compared with the position point corresponding to the point corresponding to the maximum value of the growth curvature on the solidification path to determine the validity of the simulation results and the second relationship.

6. The method according to claim 1, characterized in that The step of determining the demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path includes: Obtain the grain growth at the alloy melting boundary under different liquid solute concentrations and welding speeds; Based on the growth conditions, determining a curve showing the actual plane growth distance and the distance between the dividing point and the melting boundary changing with the liquid solute concentration at different liquid solute concentrations; Determine, based on the growth condition, an evolution curve of the actual plane growth distance and the distance between the dividing point and the melting boundary at different welding speeds; The correlation is determined based on the curve and the evolution curve and a demarcation point is obtained.

7. The method according to claim 1, characterized in that The dividing point between the planar growth and the cellular growth is determined according to the calculated curvature radius, and then, the method comprises: Dividing the molten pool solidification process into different stages based on the dividing point, the stages include a planar growth stage and a cellular growth stage; Determine the influence of liquid solute concentration and welding speed on solidification results at each stage; Based on the degree of influence, the relationship between the plane growth distance and the average grain size during the molten pool solidification process is determined.

8. The method according to claim 1, characterized in that The dividing point between the planar growth and the cellular growth is determined according to the calculated curvature radius, and then, the method comprises: Measure the grain size of the local area of ​​the molten pool at the melting boundary of the alloy to obtain the average grain size data; Analyzing the average grain size data to form an average grain size evolution curve; Based on the evolution curve, the variation relationship of the average grain size under different solute concentrations and welding speeds is analyzed respectively.

9. A molten pool grain solidification analysis device based on continuous growth limiting factor, characterized in that: The device includes an establishment module, a determination module and a calculation module; Wherein, the establishing module is used to establish a first relationship between the continuous growth limiting factor and the model for predicting the continuous growth rate of grains, and the first relationship is an inverse relationship; The determining module is used to determine a second relationship between the model for predicting the continuous growth rate of grains and the solidification path according to the first relationship, and as the solidification path increases, a growth curve corresponding to the second relationship increases exponentially; The determination module is further used to determine a demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path, the demarcation point being the point corresponding to the maximum growth rate in the growth curve; determining the demarcation point comprises: estimating a candidate region of a solidification boundary according to the solidification behavior, the candidate region being a region within a preset distance from the solidification boundary, estimating a distribution characteristic of the solidification speed according to the solidification behavior, determining a candidate region of a demarcation point of the second relationship based on the candidate region and the distribution characteristic, and calculating a demarcation point in the candidate region of the demarcation point; The calculation module is used to calculate the curvature radius of each point on the growth curve; The determination module is further used to determine the boundary point between planar growth and cellular growth according to the calculated radius of curvature; The demarcation point is determined by combining the demarcation point determined in the step of determining the demarcation point according to the correlation between the solute concentration, the welding speed and the solidification path and the demarcation point determined in the step of determining the demarcation point between planar growth and cellular growth according to the calculated curvature radius.

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