CN105868468B - Novel neighbor capturing method based on cellular automaton - Google Patents
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Abstract
The invention relates to a novel neighbor capturing method based on a cellular automaton, in particular to a tissue simulation of different crystal orientation grain growth in a welding pool solidification process or a heat affected zone solid phase change process. The method can accurately describe the grain growth orientation angle, improve the calculation precision and the calculation efficiency of computer simulation, and optimize the cellular automata simulation method to a certain extent. The method mainly comprises the steps of dispersing time, space and states, defining model parameters, decomposing growth speed and defining growth distance inheritance rules, then establishing a grain growth model, compiling a computer program based on the model and calculating to obtain grain growth morphology and solute field distribution with different orientation angles. The invention is used for the microstructure simulation process of solidification or solid phase transition.
Description
Technical Field
The invention relates to a novel neighbor capturing method based on a cellular automaton, and belongs to the field of simulation of growth of welding grains.
Background
The traditional welding technology cannot predict and evaluate the performance of the tissue structure of the welded joint before welding and cannot evaluate different welding processes, so that the work is time-consuming and labor-consuming, and major accidents cannot be effectively avoided. The evolution situation of the microstructure crystal grain appearance in a high-temperature welding state is researched by utilizing a computer simulation technology, the size of the crystal grain is predicted, the distribution situation of the crystal grain size can be estimated, the selection and optimization of key parameters of a welding process are further controlled, the complete prediction of the growth of the crystal grain of the joint is realized, a basis is provided for determining the optimal crystal orientation structure, the optimal performance crystal grain size and the like of a base material, and the purposes of predicting, monitoring and improving the quality of the welding joint are finally achieved.
Cellular Automaton (CA) technology is a generic name for a class of discrete models, and was first proposed by Von Neumann as an ideal model of living organisms. In 1986, the Packard firstly uses CA to simulate and research the formation of a crystalline structure in metal solidification, and introduces CA into the field of materials. More material structure evolution models are developed subsequently, and the processes of solidification, eutectic growth, recrystallization, grain growth, phase precipitation and the like are involved.
The cellular automaton is a random and discrete mathematical model and is composed of 4 basic elements, namely a cell, a state space, a neighborhood and a local rule. The existing method for utilizing cellular automata mainly adopts the principle that a CA model related to metallurgy is established directly from a physical metallurgy concept, a two-dimensional plane is divided into identical square cells, the values of the cells form a positive integer set, each number represents a grain orientation, and grains with the same orientation are gathered together to represent a grain. A Moore type neighbor model is adopted, and randomness is introduced into a CA state transition rule so as to reflect the evolution process of a material organization more truly, and the CA is also called as a random cellular automaton. The CA model is established according to a local rule of physical metallurgy principles (mainly energy principles), and the local rule is simple and only depends on basic physical metallurgy principles. However, research shows that the CA with simple local rules can simulate a series of structure evolution processes such as grain growth, recrystallization, structure evolution in the metal forming process, metal solidification and the like, and is a powerful tool for auxiliary material design.
Disclosure of Invention
The invention provides a novel neighbor capturing method based on a cellular automaton, which aims to solve the problem that the cellular automaton method is difficult to randomly orient, and particularly aims at the tissue simulation of the growth of grains with different crystal orientations in the solidification process of a welding pool or the solid phase change process of a heat affected zone.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a novel neighbor capturing method based on cellular automata,
the method comprises the following steps:
the method comprises the following steps: model simplification and calculation domain dispersion;
step two: defining a neighbor and a neighbor capturing mode;
step three: establishing a grain growth model;
step four: analog calculation and result derivation;
and writing a computer program based on the cellular automaton model, and obtaining and deriving the growth and evolution results of the dendritic crystal of the microstructure with different orientation angles.
The specific process of the step one is as follows:
the following simplifications and assumptions were made in building the computer model:
(1) the molten liquid is an incompressible fluid;
(2) the movement of the liquid-solid interface is determined by local temperature, solute diffusion and solute distribution.
Dispersing the time, space and state of grain growth in the welding process simulated by a computer, and setting each unit grain as a cell; defining a time step of 10 during the welding process-110ms, the size of each square unit cell is 1-100 μm, the number of the unit cells is 500 x 500, and each unit cell comprises three states of a parent phase, a child phase and a transition phase.
The specific process of the second step is as follows:
defining the neighbor cells of each cell as four cells closest to the cell through a Moore type neighbor model, and when one cell is converted into a sub-phase from a parent phase, if the neighbor cell is the parent phase, converting the cell into a transition phase of the parent phase and the sub-phase;
the growth of the crystal grains grows along the X, Y axis of the plane coordinate system, and when the growth direction has an orientation angle with the X axis or the Y axis, the growth speed of the crystal grain tips is decomposed according to the X, Y direction. There are 3 calculation methods:
(1) when the product of the growth speed in the X direction and the time step reaches the side length of the cell and is larger than the product of the growth speed in the Y direction and the time step, the cell is converted into a sub-phase cell, the next cell in the X direction is automatically captured as an interface cell, and the interface cell inherits the growth distance in the Y direction of the sub-phase cell and is substituted into the subsequent calculation;
(2) when the product of the Y-direction growth speed and the time step reaches the side length of the cell and is larger than the product of the X-direction growth speed and the time step, the cell is converted into a sub-phase cell, the next cell in the Y direction is automatically captured as an interface cell, and the interface cell inherits the X-direction growth distance of the sub-phase cell and is substituted into the subsequent calculation;
(3) when the product of the growth speed in the X direction and the time step reaches the side length of the cellular cell and is equal to the product of the growth speed in the Y direction and the time step, the cellular cell is converted into a sub-phase cellular cell, the next cellular cell in the X direction and a neighbor cellular cell which is 45 degrees with the sub-phase cellular cell are automatically captured as interface cellular cells, and the X, Y direction growth distances of the interface cellular cells are all 0 and substituted into the subsequent calculation.
The specific process of the third step is as follows:
and defining initial parameters including initial temperature and initial solute fraction, and giving initial state parameters to all regional cellular units.
Considering component supercooling, curvature supercooling and dynamic supercooling, interface tip liquid phase solute fraction in the model
Is given by the following formula:
the formula I is as follows:
wherein
Is the interface tip liquid phase solute fraction, mLIs the slope of the liquidus line, T0Is an initial solute fraction C0The liquidus temperature of time, gamma is the Gibbs Thomson coefficient, kappa is the interfacial curvature, T is the local temperature, V is the growth rate, mukIs the interfacial kinetic coefficient.
Based on the second law of local conservation of solid-liquid interface and Fick diffusion, carrying out finite difference on the model to obtain: the formula II is as follows:
wherein C isE＝(1-fS)CL+fSCS，DE＝(1-fS)DL+fSDS，CEIs the mean solute fraction of the individual cells, DeIs the average solute diffusion coefficient of an individual cell,
and
solid phase and liquid phase solute fractions at the tip of the solid-liquid interface, CSAnd CLSolute fractions in solid and liquid phases, respectively, DSAnd DLSolid phase and liquid phase solute diffusion coefficients, respectively, Δ t is a time step, Δ x and Δ y are unit cell sizes in x and y directions, respectively, i is a horizontal axis coordinate, j is a vertical axis coordinate, f is a vertical axis coordinateSIs the solid phase fraction.
Based on the interfacial solute conservation and differencing it, a tip growth rate can be obtained that is:
the formula III is as follows:
the formula four is as follows:
the specific process of the step four is as follows:
and compiling a computer program based on the cellular automaton model, and performing iterative calculation and derivation to obtain the crystal grain morphology, solute fraction, temperature distribution and dendritic crystal tip growth speed with different orientation angles.
The invention has the beneficial effects that:
the invention provides a novel neighbor capturing method based on different orientation grain growth of a cellular automaton. The method of the invention can simulate the dendritic crystal growth evolution in the solidification process of the welding pool, dynamically reproduce the structural morphology change in the welding cooling process, accurately describe the crystal grain growth process with different orientation angles, simultaneously improve the calculation precision and the calculation efficiency of computer simulation, and optimize the cellular automata simulation method to a certain extent.
Drawings
The invention is further illustrated with reference to the following figures and examples.
FIG. 1 is a schematic diagram of a cell after spatial and state dispersion;
FIG. 2 is a main flow of defining neighbor capture and growth distance inheritance;
FIG. 3(a) is a graph of grain growth with a certain orientation angle;
FIGS. 3(b1) -3(b3) are graphs showing the grain growth process in actual simulation;
FIGS. 3(c1) - (c4) are views showing the grain growth process obtained by decomposing the growth rate by the orientation angle;
FIG. 4 is a flow chart of microscopic grain growth structure simulation;
fig. 5 shows the simulation results of grain growth and solute distribution at different orientation angles.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
aiming at the cellular automata computer simulation of the grain growth process during solidification or solid phase change, a novel neighbor capturing method for grain growth at different orientation angles is provided. The method is characterized in that: decomposing the grain growth speed of the transition phase unit cells between the parent phase and the child phase, converting the transition phase unit cells into the child phase unit cells after the transition conditions are met, simultaneously capturing the interface unit cells in a new step according to the meeting conditions, and adding the unsatisfied conditions into the newly captured interface unit cells as initial conditions, thereby finishing the growth of the grains with a certain orientation angle. And finally, calculating and exporting the result after writing the code. The method comprises the following steps:
the method comprises the following steps: model simplification and computational domain discretization
The following simplifications and assumptions were made in building the computer model:
(1) the molten liquid is an incompressible fluid;
(2) the movement of the liquid-solid interface is determined by local temperature, solute diffusion and solute distribution.
Dispersing the time, space and state of grain growth in the welding process simulated by a computer, and setting each unit grain as a cell; defining a time step of 10 during the welding process-110ms, the size of each square unit cell is 1-100 μm, the number of the unit cells is 500 x 500, each unit cell comprises a mother phase and a child phasePhase and transition phase. Referring to the actual situation of the welding process, a time step of 1ms, a cell size of 5 μm, a cell number of 500 × 500, and a cell state of 3 are used in the example, which are respectively a liquid phase, a solid phase, and a solid-liquid interface, as shown in fig. 1.
Step two: definition of neighbor and neighbor capture mode
The main flow is shown in fig. 2. Defining the neighbor cells of each cell as four cells closest to the cell through a Moore type neighbor model, and when one cell is converted into a sub-phase from a parent phase, if the neighbor cell is the parent phase, converting the cell into a transition phase of the parent phase and the sub-phase;
the actual grain growth process has a fixed growth orientation based on its grain properties of FCC (face centered cubic), BCC (body centered cubic), HCP (hexagonal close packed) such that the FCC grain tip growth directions are perpendicular to each other, and then its grains can grow in X, Y, Z directions in a cartesian coordinate system and in X, Y directions in a two-dimensional form as shown in fig. 3 (a). Fig. 3(b1) -3(b3) show the results of the original grain growth simulation, which shows that the crystal can only be grown in the X, Y direction, and the difference in orientation angle cannot be considered. As shown in fig. 3(c1) - (c4), the growth of the crystal grains grows along the X, Y axis of the planar coordinate system, and when the growth direction has an orientation angle with the X axis or the Y axis, the growth rate of the crystal grain tips is decomposed in the X, Y direction. There are 3 calculation methods:
(1) when the product of the growth speed in the X direction and the time step reaches the side length of the cell and is larger than the product of the growth speed in the Y direction and the time step, the cell is converted into a sub-phase cell, the next cell in the X direction is automatically captured as an interface cell, and the interface cell inherits the growth distance in the Y direction of the sub-phase cell and is substituted into the subsequent calculation;
(2) when the product of the Y-direction growth speed and the time step reaches the side length of the cell and is larger than the product of the X-direction growth speed and the time step, the cell is converted into a sub-phase cell, the next cell in the Y direction is automatically captured as an interface cell, and the interface cell inherits the X-direction growth distance of the sub-phase cell and is substituted into the subsequent calculation;
(3) when the product of the growth speed in the X direction and the time step reaches the side length of the cellular cell and is equal to the product of the growth speed in the Y direction and the time step, the cellular cell is converted into a sub-phase cellular cell, the next cellular cell in the X direction and a neighbor cellular cell which is 45 degrees with the sub-phase cellular cell are automatically captured as interface cellular cells, and the X, Y direction growth distances of the interface cellular cells are all 0 and substituted into the subsequent calculation. By adopting the method, the grain growth process of any orientation angle can be calculated.
Step three: establishment of grain growth model
The model building process is shown in fig. 4. The dendritic crystal growth process is mainly controlled by temperature distribution and solute diffusion during solidification, and the solute fraction gradient is the driving force for solid-phase and liquid-phase solute diffusion, so the temperature and solute fraction need to be calculated firstly. And defining initial parameters including initial temperature and initial solute fraction, and assigning initial state parameters to the unit cells in all the regions.
Considering component supercooling, curvature supercooling and dynamic supercooling, interface tip liquid phase solute fraction in the model
Is given by the following formula:
the formula I is as follows:
wherein
Is the interface tip liquid phase solute fraction, mLIs the slope of the liquidus line, T0Is an initial solute fraction C0The liquidus temperature of time, gamma is the Gibbs Thomson coefficient, kappa is the interfacial curvature, T is the local temperature, V is the growth rate, mukIs the interfacial kinetic coefficient.
Based on the second law of local conservation of solid-liquid interface and Fick diffusion, carrying out finite difference on the model to obtain: the formula II is as follows:
wherein C isE＝(1-fS)CL+fSCS，DE＝(1-fS)DL+fSDS，CEIs the mean solute fraction of the individual cells, DeIs the average solute diffusion coefficient of an individual cell,
and
solid phase and liquid phase solute fractions at the tip of the solid-liquid interface, CSAnd CLSolute fractions in solid and liquid phases, respectively, DSAnd DLSolid phase and liquid phase solute diffusion coefficients, respectively, Δ t is a time step, Δ x and Δ y are unit cell sizes in x and y directions, respectively, i is a horizontal axis coordinate, j is a vertical axis coordinate, f is a vertical axis coordinateSIs the solid phase fraction.
Based on the interfacial solute conservation and differencing it, a tip growth rate can be obtained that is:
the formula III is as follows:
the formula four is as follows:
step four: analog computation and derivation of results
And compiling a computer program based on the cellular automaton model, and performing iterative calculation and derivation to obtain the crystal grain morphology, solute fraction, temperature distribution and dendritic crystal tip growth speed with different orientation angles.
The microstructure simulation of the Al-3% Cu alloy welding solidification process is used for analysis.
The method comprises the following steps: model simplification and computational domain discretization
Referring to the actual situation of the welding process, the following simplifications and assumptions are made in establishing the computer model: (1) the molten liquid is an incompressible fluid; (2) the movement of the liquid-solid interface is determined by local temperature, solute diffusion and solute distribution.
And (5) discretizing the time, space and state of the computer simulation. Defining the time step of the simulation as 10-1ms), the size of each square unit cell is 5 μm and the number of unit cells is 500 × 500, the number of states per unit cell (generally including 3 states in total of a parent phase, a child phase, and a transition phase).
Step two: definition of neighbor and neighbor capture mode
A Moore-type neighbor model is used, i.e., the neighbors of each cell are defined as the four cells closest to the cell. When a unit cell changes from a parent phase to a child phase, if a neighboring unit cell is the parent phase, the unit cell changes to a transition phase (interface) between the parent phase and the child phase.
Since the Al-Cu alloy is a face-centered cubic crystal, the FCC grain tip growth directions are perpendicular to each other, and grow in the X, Y direction in a microscopic two-dimensional simulation. As shown in fig. 3(c1) - (c4), the growth of the crystal grains grows along the X, Y axis of the plane coordinate system, and when the growth direction has an orientation angle with the X axis or the Y axis, the growth rate of the crystal grain tips is decomposed in the X, Y direction. There are 3 calculation methods: (1) when the product of the growth speed in the X direction and the time step reaches the side length of the cell and is larger than the product of the growth speed in the Y direction and the time step, the cell is converted into a sub-phase cell, the next cell in the X direction is automatically captured as an interface cell, and the interface cell inherits the growth distance in the Y direction of the sub-phase cell and is substituted into the subsequent calculation; (2) when the product of the growth speed in the Y direction and the time step reaches the side length of the cell and is larger than the product of the growth speed in the X direction and the time step, the cell is converted into a sub-phase cell, the next cell in the Y direction is automatically captured as an interface cell, and the interface cell inherits the growth distance in the X direction of the sub-phase cell and is substituted into the subsequent calculation; (3) when the product of the growth speed in the X direction and the time step reaches the side length of the cellular cell and is equal to the product of the growth speed in the Y direction and the time step, the cellular cell is converted into a sub-phase cellular cell, the next cellular cell in the X direction and a neighbor cellular cell which is 45 degrees with the sub-phase cellular cell are automatically captured as interface cellular cells, and the X, Y direction growth distances of the interface cellular cells are all 0 and substituted into the subsequent calculation. By adopting the method, the grain growth process of any orientation angle can be calculated.
Step three: establishment of grain growth model
The dendritic crystal growth process is mainly controlled by temperature distribution and solute diffusion during solidification, and the solute fraction gradient is the driving force of solid-phase and liquid-phase solute diffusion, so the temperature and solute fraction need to be calculated firstly. And defining initial parameters including initial temperature and initial solute fraction, and assigning initial state parameters to the cellular units in all the regions. The material parameters and simulation parameters used in the micro-model are shown in table 1.
TABLE 1 Material parameters and simulation parameters used in the micro-model
Considering component supercooling, curvature supercooling and dynamic supercooling, interface tip liquid phase solute fraction in the model
Is given by the following formula:
the formula I is as follows:
wherein
Is the interface tip liquid phase solute fraction, mLIs the slope of the liquidus line, T0Is an initial solute fraction C0The liquidus temperature of time, Г is the Gibbs Thomson coefficient, κ is the interfacial curvature, T is the local temperature, V is the growth rate, μkIs the interfacial kinetic coefficient.
Based on the second law of local conservation of solid-liquid interface and Fick diffusion, carrying out finite difference on the model to obtain:
the formula II is as follows:
wherein C isE＝(1-fS)CL+fSCS，DE＝(1-fS)DL+fSDS，CEIs the mean solute fraction of the individual cells, DeIs the average solute diffusion coefficient of an individual cell, C
And
solid phase and liquid phase solute fractions at the tip of the solid-liquid interface, CSAnd CLSolute fractions in solid and liquid phases, respectively, DSAnd DLSolid phase and liquid phase solute diffusion coefficients, respectively, Δ t is a time step, Δ x and Δ y are unit cell sizes in x and y directions, respectively, i is a horizontal axis coordinate, j is a vertical axis coordinate, f is a vertical axis coordinateSIs the solid phase fraction.
Based on the interfacial solute conservation and differencing it, a tip growth rate can be obtained that is:
the formula III is as follows:
the formula four is as follows:
step four: analog computation and derivation of results
According to the temperature and solute diffusion in the welding solidification process, grain growth evolution calculation of different orientation angles is carried out, during simulation, the grain growth direction has randomness and forms a certain included angle with the X-axis direction, the growth process of any grain is the growth process of the grain of any angle, the grain of any orientation can be normally grown and evolved by setting the orientation angle of the grain, and finally, the grain appearance, the solute fraction, the temperature distribution and the dendritic crystal tip growth speed are obtained and derived.
In light of the foregoing description of the preferred embodiment of the present invention, it is to be understood that various changes and modifications may be made by those skilled in the art without departing from the spirit and scope of the invention. The technical scope of the present invention is not limited to the content of the specification, and must be determined according to the scope of the claims.
Claims (3)
1. A novel neighbor capturing method based on cellular automata is characterized by comprising the following steps:
the method comprises the following steps: model simplification and computational domain discretization
Step two: definition of neighbor and neighbor capture mode
Step three: establishment of grain growth model
Step four: analog computation and derivation of results
Writing a computer program based on the cellular automaton model, and obtaining and deriving the growth and evolution results of the dendritic crystal of the microstructure with different orientation angles:
the specific process of the second step is as follows:
defining the neighbor cells of each cell as four cells closest to the cell through a Moore type neighbor model, and when one cell is converted into a sub-phase from a parent phase, if the neighbor cell is the parent phase, converting the cell into a transition phase of the parent phase and the sub-phase;
the growth of the crystal grains grows along the X, Y axes of the plane coordinate system, when the growth direction has an orientation angle with the X axis or the Y axis, the growth speed of the crystal grain tips is decomposed according to the X, Y direction, and there are 3 calculation methods:
(1) when the product of the growth speed in the X direction and the time step reaches the side length of the cell and is larger than the product of the growth speed in the Y direction and the time step, the cell is converted into a sub-phase cell, the next cell in the X direction is automatically captured as an interface cell, and the interface cell inherits the growth distance in the Y direction of the sub-phase cell and is substituted into the subsequent calculation;
(2) when the product of the growth speed in the Y direction and the time step reaches the side length of the cell and is larger than the product of the growth speed in the X direction and the time step, the cell is converted into a sub-phase cell, the next cell in the Y direction is automatically captured as an interface cell, and the interface cell inherits the growth distance in the X direction of the sub-phase cell and is substituted into the subsequent calculation;
(3) when the product of the growth speed in the X direction and the time step reaches the side length of the cellular cell and is equal to the product of the growth speed in the Y direction and the time step, the cellular cell is converted into a sub-phase cellular cell, the next cellular cell in the X direction and a neighbor cellular cell which is 45 degrees with the sub-phase cellular cell are automatically captured as interface cellular cells, and the X, Y direction growth distances of the interface cellular cells are all 0 and substituted into the subsequent calculation;
the specific process of the third step is as follows:
defining initial parameters including initial temperature and initial solute fraction, assigning initial state parameters to all regional cellular units,
considering component supercooling, curvature supercooling and dynamic supercooling, interface tip liquid phase solute fraction in the model
Is given by the following formula:
the formula I is as follows:
wherein
Is the interface tip liquid phase solute fraction, mLIs the slope of the liquidus line, T0Is an initial solute fraction C0The liquidus temperature of time, gamma is the Gibbs Thomson coefficient, kappa is the interfacial curvature, T is the local temperature, V is the growth rate, mukAs a coefficient of the interface dynamics, the coefficient of the interface dynamics,
based on the second law of local conservation of solid-liquid interface and Fick diffusion, carrying out finite difference on the model to obtain:
the formula II is as follows:
wherein C isE＝(1-fS)CL+fSCS，DE＝(1-fS)DL+fSDS，CEIs the average solute fraction of the individual cells, DeIs the average solute diffusion coefficient of an individual cell,
and
solid phase and liquid phase solute fractions at the tip of the solid-liquid interface, CSAnd CLSolute fractions in solid and liquid phases, respectively, DSAnd DLRespectively solid phase and liquid phase solute diffusion coefficients, delta t is a time step, delta x and delta y are unit cell sizes in x and y directions, respectively, i is a horizontal axis coordinate, j is a vertical axis coordinate, fSIs the fraction of the solid phase,
based on the interfacial solute conservation and differencing it, a tip growth rate can be obtained that is:
the formula III is as follows:
the formula four is as follows:
wherein k in formula three and formula four is defined as: the solute distribution coefficient.
2. The novel neighbor capturing method based on cellular automata as claimed in claim 1, wherein the specific process of the first step is as follows:
the following simplifications and assumptions were made in building the computer model:
(1) the molten liquid is an incompressible fluid;
(2) the movement of the liquid-solid interface is determined by local temperature, solute diffusion and solute distribution;
dispersing the time, space and state of grain growth in the welding process simulated by a computer, and setting each unit grain as a cell; defining a time step of 10 during the welding process-110ms, the size of each square unit cell is 1-100 μm, the number of the unit cells is 500 x 500, and each unit cell comprises three states of a parent phase, a child phase and an excess phase.
3. The novel neighbor capturing method based on cellular automata according to claim 1, wherein the specific process of the fourth step is as follows:
and compiling a computer program based on the cellular automaton model, and performing iterative calculation and derivation to obtain the crystal grain morphology, solute fraction, temperature distribution and dendritic crystal tip growth speed with different orientation angles.
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