CN111640469A - Three-dimensional simulation method for dendrite coarsening - Google Patents
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Abstract
The invention discloses a three-dimensional simulation method for dendrite coarsening, which comprises the following steps: s1, grid division; s2, inputting physical parameters required in the three-dimensional CA-FDM model; s3, introducing the solid phase fraction and the concentration field of the columnar dendrite into the three-dimensional CA-FDM model; s4, searching a solid-liquid interface; s5, calculating a solid phase fraction; s6, calculating a solute concentration field; s7, updating the cell parameters; s8, judging whether the output condition is met; if not, go to step S4; and if so, outputting the result. The method can reproduce the dendritic crystal coarsening process in the three-dimensional space based on the three-dimensional CA-FDM model. The three-dimensional CA-FDM model can be used for quantitatively researching the contribution of each mechanism to dendrite coarsening. In addition, typical dendrite coarsening patterns can also be reproduced, including small dendrite arm melting, solidification growth at the grooves between dendrite arms, and dendrite arm tip merging, among others. The three-dimensional simulation result is closer to the actual experiment, and effective prediction can be carried out.
Description
Technical Field
The invention belongs to the technical field of dendritic crystal coarsening, and relates to a dendritic crystal coarsening three-dimensional numerical simulation method.
Background
Dendrites are one of the most common metal solidification microstructures. Dendrites are the typical multi-dendrite formation of crystals under a large deviation from equilibrium conditions during solidification of a metal material. In most of the medium-low-speed solidification processes or isothermal heat preservation processes, it is inevitable that the dendritic structures coarsen in the mushy zone. Dendrite coarsening refers to a phenomenon in which the dendrite structure in the solid-liquid two-phase region becomes coarser. Dendrite coarsening can have a significant impact on the final solidification structure, microsegregation, and product performance. Because of the importance of dendrite coarsening problems to academic research and practical applications, great attention is paid to both academia and industry. In the actual dendritic crystal coarsening process, solute diffusion, curvature distribution evolution and dendritic crystal morphology evolution processes are carried out in a three-dimensional space.
However, at present, there is no suitable three-dimensional simulation method for dendrite coarsening.
Disclosure of Invention
Based on this, there is a need to provide a three-dimensional simulation method for dendrite coarsening.
A three-dimensional simulation method for dendrite coarsening comprises the following steps:
s1, dividing the three-dimensional simulation area into a plurality of cubic grids;
s2, inputting physical parameters required in the three-dimensional CA-FDM model according to the physical properties of the material;
s3, introducing the solid phase fraction and the concentration field of the columnar dendrite into the three-dimensional CA-FDM model as initial conditions of dendrite coarsening simulation;
s4, searching a solid-liquid interface;
s5, calculating a solid phase fraction;
s6, calculating a solute concentration field;
s7, updating the cell parameters;
s8, judging whether the output condition is met; if not, go to step S4; and if so, outputting the result.
Optionally, in step S2, the physical parameters include initial composition, pure solvent freezing point, liquidus slope, and solute redistribution coefficient.
Alternatively, in step S4, the solid-liquid interface position is found by a neighbor algorithm in CA.
Alternatively, in step S5, the solid phase growth rate at each time step at the solid-liquid interface grid is calculated by formula 1;
wherein, in formula 1, g represents a shape factor;
is the equilibrium component of the liquid phase at the solid-liquid interface.
Alternatively,
the calculation is obtained through formula 2;
wherein, in formula 2, T*Is the solid-liquid interface temperature; m islIn order to be the slope of the liquidus line,
is a primary component of C0The liquidus temperature of the alloy; Gibbs-Thomson coefficient for solid-liquid interface; wmc is the solid-liquid interface weight curvature coupled with the interface energy anisotropy effect.
Optionally, wmc in three-dimensional space is calculated by equation 3:
in equation 3, ξ is the Cahn-Hoffman vector,
is the unit normal vector at the solid-liquid interface;
nx、nyand nzAre respectively as
The components in the three coordinate axes are,
and
wherein
Q is calculated in the manner of
Optionally, in step S6, the concentration field in the region is calculated by formula 4, and the solution method of formula (4) is explicit FDM;
wherein, in formula 4, t is time; c is the average concentration in the grid; d (fs) is the fraction of solid phase in the gridsRelevant diffusion coefficient, p (f)s) Is a transfer function related to the solid fraction of the mesh.
Alternatively, C is calculated by equation 5:
C＝Csfs+Cl(1-fs) Equation 5
Wherein, in formula 5, CsIs a solid phase component of the lattice; clIs the liquid phase component of the grid.
Alternatively, D (f)s) Obtained by calculation of equation 6:
D(fs)＝kpDsfs+Dl(1-fs) Equation 6.
Alternatively, p (f)s) Through a maleEquation 7 is calculated to obtain:
p(fs)＝kpfs+(1-fs) Equation 7.
The dendritic crystal coarsening three-dimensional simulation method provided by the invention simulates the dendritic crystal coarsening phenomenon of the metal material based on the three-dimensional CA-FDM model, and can reproduce the dendritic crystal coarsening process in a three-dimensional space. The established three-dimensional CA-FDM model comprises solidification/melting, solute diffusion, interface curvature evolution and interaction mechanisms thereof, and can be used for quantitatively researching the contribution of the mechanisms to dendrite coarsening. In addition, the three-dimensional simulation method for dendrite coarsening can reproduce a typical dendrite coarsening mode, including small dendrite arm melting, solidification growth at grooves between dendrite arms, dendrite arm tip merging and the like. And the interface curvature distribution and the solid-liquid specific surface area evolution rule in the dendrite coarsening process can be predicted. The three-dimensional simulation result is closer to the actual dendrite coarsening experiment, and the dendrite structure evolution in the isothermal heat preservation process can be effectively predicted.
Drawings
Fig. 1 is a flowchart of a three-dimensional simulation method of dendrite coarsening according to an embodiment of the invention.
FIG. 2 is a dendrite topography at the initial time of isothermal dendrite coarsening simulation.
FIG. 3 is a graph of dendrite coarsening simulation results of a three-dimensional CA-FDM model simulating succinonitrile-2.0 wt.% acetone (SCN-2.0 wt.% ACE) alloy.
FIG. 4 is a diagram of the solid-liquid interface curvature distribution and the interface specific surface area evolution obtained in the three-dimensional dendrite coarsening simulation process.
FIG. 5 is an enlarged view of the coarsening evolution of the dendrite arms in box I of FIG. 3.
Fig. 6 is an enlarged view of the coarsening evolution of the dendrite arms in box II of fig. 3.
FIG. 7 is a graph showing a comparison of simulation results of dendrite coarsening in three dimensions and a two-dimensional slice display.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to specific embodiments. It should be understood that the detailed description and specific examples, while indicating the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention.
Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items.
The invention provides a three-dimensional simulation method for dendrite coarsening, in particular to a three-dimensional numerical simulation method for the dendrite coarsening phenomenon of a metal material in a solid-liquid two-phase region in an isothermal heat preservation process.
Referring to fig. 1, a three-dimensional simulation method of dendrite coarsening includes the following steps:
s1, dividing the three-dimensional simulation area into a plurality of cubic grids;
s2, inputting physical parameters required in the three-dimensional CA-FDM model according to the physical properties of the material;
s3, introducing the solid phase fraction and the concentration field of the columnar dendrite into the three-dimensional CA-FDM model as initial conditions of dendrite coarsening simulation;
s4, searching a solid-liquid interface;
s5, calculating a solid phase fraction;
s6, calculating a solute concentration field;
s7, updating the cell parameters;
s8, judging whether the output condition is met; if not, go to step S4; and if so, outputting the result.
In step S1, the simulation area is divided into a plurality of cubic grids, and those skilled in the art can select and set the grid size according to actual situations.
In step S2, CA in the three-dimensional CA-FDM model is Cellular Automation (CA), and FDM is a Finite Difference Method (FDM), that is, a three-dimensional cellular automation method is used to describe the morphological evolution of the dendrite tissue in the three-dimensional space; meanwhile, a finite difference method is adopted to solve a diffusion equation so as to calculate the concentration field. The three-dimensional CA-FDM model is adopted to simulate the dendritic crystal coarsening phenomenon of a metal material in a three-dimensional space, the solidification and melting processes of dendritic crystal arms can be reproduced, the interaction between the solidification/melting of the dendritic crystal arms and the diffusion and curvature distribution of solutes is described, and the dendritic crystal morphology evolution rule under the isothermal heat preservation condition is predicted.
In step S2, the physical parameters include initial composition, pure solvent freezing point, liquidus slope, and solute redistribution coefficient. In one embodiment, the physical parameters required in the three-dimensional CA-FDM model are input based on the physical properties of the SCN-ACE alloy.
Alternatively, in step S4, the solid-liquid interface position is found by a neighbor algorithm in CA.
Alternatively, in step S5, in the three-dimensional CA-FDM model, the migration velocity of the solid-liquid interface is proportional to the difference between the actual concentration and the equilibrium concentration at the interface.
Calculating the solid phase growth rate △ f in each time step △ t at the solid-liquid interface grid by the formula 1s；
Wherein, in formula 1, g represents a shape factor;
is the equilibrium component of the liquid phase at the solid-liquid interface.
Further, in the present invention,
the calculation is obtained through formula 2;
wherein, in formula 2, T*Is the solid-liquid interface temperature; m islIn order to be the slope of the liquidus line,
is a primary component of C0The liquidus temperature of the alloy; Gibbs-Thomson coefficient for solid-liquid interface; wmc is the solid-liquid interface weight curvature coupled with the interface energy anisotropy effect.
Wherein, wmc in the three-dimensional space is obtained by calculation according to formula 3:
in equation 3, ξ is the Cahn-Hoffman vector,
is the unit normal vector at the solid-liquid interface;
nx、nyand nzAre respectively as
The components in the three coordinate axes are,
and
wherein
Q is calculated in the manner of
Optionally, in step S6, the concentration field in the region is calculated by formula 4, and the solution method of formula (4) is explicit FDM;
wherein, in formula 4, t is time; c is the average concentration in the grid; d (fs) is the fraction of solid phase in the gridsRelevant diffusion coefficient, p (f)s) Is a transfer function related to the solid fraction of the mesh.
Alternatively, C is calculated by equation 5:
C＝Csfs+Cl(1-fs), equation 5
Wherein, in formula 5, CsIs a solid phase component of the lattice; clIs the liquid phase component of the grid.
Alternatively, D (f)s) Obtained by calculation of equation 6:
D(fs)＝kpDsfs+Dl(1-fs) Equation 6.
Alternatively, p (f)s) Calculated by equation 7:
p(fs)＝kpfs+(1-fs) Equation 7.
In step S7, parameters such as the lattice state, orientation, and solid fraction of each cell are updated in CA.
The dendritic crystal coarsening simulation method provided by the invention simulates the dendritic crystal coarsening phenomenon of the metal material based on the three-dimensional CA-FDM model, and can reproduce the dendritic crystal coarsening process in a three-dimensional space. The established three-dimensional CA-FDM model comprises solidification/melting, solute diffusion, interface curvature evolution and interaction mechanisms thereof, and can be used for quantitatively researching the contribution of the mechanisms to dendrite coarsening. In addition, the three-dimensional simulation method for dendrite coarsening can reproduce a typical dendrite coarsening mode, including small dendrite arm melting, solidification growth at grooves between dendrite arms, dendrite arm tip merging and the like. And the interface curvature distribution and the solid-liquid specific surface area evolution rule in the dendrite coarsening process can be predicted. The three-dimensional simulation result is closer to the actual dendrite coarsening experiment, and the dendrite structure evolution in the isothermal heat preservation process can be effectively predicted.
The following description will be given by taking the three-dimensional dendrite morphology evolution simulation in the isothermal heat preservation process of SCN-2.0 wt.% ACE alloy as an example.
The established three-dimensional CA-FDM model is adopted to simulate and study the dendritic crystal coarsening phenomenon of the SCN-2.0 wt.% ACE transparent alloy in the heat preservation process at the temperature of 50.83 ℃.
FIG. 2 is a graph of the morphology of columnar dendrites at the initial time of isothermal incubation simulation, with a calculated solid fraction in the region of 21%. The columnar dendrites were obtained by directional solidification simulation of SCN-2.0 wt.% ACE alloy, under directional solidification simulation conditions: the temperature gradient is 7000 ℃/m, the initial supercooling degree is 1 ℃, and the cooling speed is 1 ℃/s.
FIG. 3 is a graph of dendrite coarsening of SCN-2.0 wt.% ACE alloy obtained by three-dimensional CA-FDM modeling during 50.83 deg.C soak. As can be seen from fig. 3, at the initial 0s, the columnar dendrites are branched clearly, the secondary dendrite arms are developed, and the number of the tertiary dendrite arms is also large. As the hold time was 20s and 600s, the primary and secondary dendrite arms became coarser, while the number of smaller tertiary dendrite arms decreased or even disappeared. In general, the dendritic crystal structure becomes more "rounded" and the phenomenon of coarsening of the dendritic crystal is obvious. The two boxes in the figure show a typical dendrite coarsening pattern, which mainly consists of small dendrite arm melting, solidification growth at the inter-dendrite arm groove, and dendrite arm tip merging.
Fig. 4(a) and (b) respectively show the solid-liquid interface curvature distribution and the interface specific surface area evolution condition obtained in the three-dimensional dendrite coarsening simulation process. As can be seen from fig. 4(a), the average curvature (the lowermost curve) distribution was broad and approximately normal in the initial isothermal incubation period (1.1 s). At this time, the curvature value of the maximum probability density in the calculation region is positive and the absolute value is large, indicating that there are a large number of small arms in the dendrite structure. As the isothermal incubation proceeded, the mean curvature profile narrowed, with the peak moving toward zero, indicating that the overall mean curvature gradually tended to a smaller positive absolute value. In addition, the probability of negative mean curvature gradually decreases during isothermal incubation. After isothermal heat preservation for about 600s, most of the weight curvature of the solid-liquid interface is distributed in a positive value with a small absolute value. Average curve with isothermal incubationThe distribution probability of the rate value in a positive value interval close to zero is gradually increased, which shows that the solid-liquid interface gradually tends to be flatter and straighter in the process of isothermal heat preservation dendritic crystal coarsening. As can be seen from FIG. 4(b), the specific surface area S of the solid-liquid interface increases with the holding timeVsAnd is continuously decreased. In addition, the specific surface area S of the solid-liquid interface at the early stage of the heat preservationVsThe falling speed of the temperature-keeping agent is fast and tends to be gentle in the later period of the temperature keeping.
FIG. 5 is an enlarged view of the coarsening evolution of dendrite arms in box I of FIG. 3, wherein FIGS. 5(a), (b), and (c) show the simulation results in terms of solid fraction, actual composition, and equilibrium composition, respectively. As shown in FIG. 5(a), the grooves between the dendrite arms are continuously filled with solid phase, and the tips of the secondary dendrite arms melt slightly. The numbers marked in fig. 5(b) and (c) are the actual composition and equilibrium concentration values for the local site. It can be seen that when the actual composition at a local site is higher than the equilibrium composition, the site melts; otherwise, solidification occurs. The simulation results of fig. 5 demonstrate a typical dendrite coarsening mechanism of small dendrite arm melting and solidification growth at the grooves between dendrite arms.
FIG. 6 is an enlarged view of the coarsening evolution of dendrite arms in box II of FIG. 3, wherein FIGS. 6(a), (b), and (c) show the simulation results in terms of solid fraction, actual composition, and equilibrium composition, respectively. As shown in fig. 6(a), the tips of the adjacent secondary dendrite arms merge, and the liquid phase trapped in the middle gradually takes on a circular shape and is filled with the solid phase. In addition, the trench gaps between the dendrite arms also solidify, resulting in significant coarsening of the primary dendrite arms. The numbers marked in fig. 6(b) and (c) are the actual composition and equilibrium concentration values for the local site. Similar to the simulation results of FIG. 5, a local site melts when its actual composition is higher than its equilibrium composition; otherwise, solidification occurs. The simulation results of fig. 6 demonstrate a typical dendrite coarsening mechanism of dendrite arm tip merging and solidification growth at the inter-dendrite arm groove.
FIG. 7 is a graph showing a comparison of simulation results of dendrite coarsening in three dimensions and a two-dimensional slice display.
In conclusion, the dendrite coarsening simulation method provided by the invention simulates the dendrite coarsening phenomenon of the metal material based on the three-dimensional CA-FDM model, and can reproduce the dendrite coarsening process in a three-dimensional space. The established three-dimensional CA-FDM model comprises solidification/melting, solute diffusion, interface curvature evolution and interaction mechanisms thereof, and can be used for quantitatively researching the contribution of the mechanisms to dendrite coarsening. In addition, the three-dimensional simulation method for dendrite coarsening can reproduce a typical dendrite coarsening mode, including small dendrite arm melting, solidification growth at grooves between dendrite arms, dendrite arm tip merging and the like. And the interface curvature distribution and the solid-liquid specific surface area evolution rule in the dendrite coarsening process can be predicted. The three-dimensional simulation result is closer to the actual dendrite coarsening experiment, and the dendrite structure evolution in the isothermal heat preservation process can be effectively predicted.
The technical features of the embodiments described above may be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the embodiments described above are not described, but should be considered as being within the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express several embodiments of the present invention, and the description thereof is more specific and detailed, but not construed as limiting the scope of the invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the inventive concept, which falls within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.
Claims (10)
1. A three-dimensional simulation method for dendrite coarsening is characterized by comprising the following steps:
s1, dividing the three-dimensional simulation area into a plurality of cubic grids;
s2, inputting physical parameters required in the three-dimensional CA-FDM model according to the physical properties of the material;
s3, introducing the solid phase fraction and the concentration field of the columnar dendrite into the three-dimensional CA-FDM model as initial conditions of dendrite coarsening simulation;
s4, searching a solid-liquid interface;
s5, calculating a solid phase fraction;
s6, calculating a solute concentration field;
s7, updating the cell parameters;
s8, judging whether the output condition is met; if not, go to step S4; and if so, outputting the result.
2. The three-dimensional simulation method of dendrite coarsening according to claim 1 wherein in step S2 the physical parameters include initial composition, pure solvent freezing point, liquidus slope, and solute redistribution coefficient.
3. The three-dimensional simulation method of dendrite coarsening according to claim 1 wherein in step S4 the solid-liquid interface position is found by a nearest neighbor algorithm in CA.
4. The three-dimensional simulation method of dendrite coarsening according to claim 1, wherein in step S5, a solid phase growth rate at each time step at the solid-liquid interface grid is calculated by formula 1;
wherein, in formula 1, g represents a shape factor;
is the equilibrium component of the liquid phase at the solid-liquid interface.
5. The three-dimensional simulation method of dendrite coarsening according to claim 4,
the calculation is obtained through formula 2;
wherein, in formula 2, T*Is the solid-liquid interface temperature; m islIs the slope of the liquidus line, Tl eqIs a primary component of C0The liquidus temperature of the alloy; Gibbs-Thomson coefficient for solid-liquid interface; wmc is the solid-liquid interface weight curvature coupled with the interface energy anisotropy effect.
6. The method for three-dimensional simulation of dendrite coarsening according to claim 5 wherein wmc in three-dimensional space is calculated by equation 3:
in equation 3, ξ is the Cahn-Hoffman vector,
is the unit normal vector at the solid-liquid interface;
nx、nyand nzAre respectively as
The components in the three coordinate axes are,
and
wherein
Q is calculated in the manner of
7. The method for three-dimensional simulation of dendrite coarsening according to claim 1 wherein in step S6 the concentration field in the region is calculated by formula 4, the solution method of formula 4 is explicit FDM;
wherein, in formula 4, t is time; c is the average concentration in the grid; d (fs) is the fraction of solid phase in the gridsRelevant diffusion coefficient, p (f)s) Is a transfer function related to the solid fraction of the mesh.
8. The method for three-dimensional simulation of dendrite coarsening according to claim 7 wherein C is calculated by equation 5:
C＝Csfs+Cl(1-fs) Equation 5;
wherein, in formula 5, CsIs a solid phase component of the lattice; clIs the liquid phase component of the grid.
9. The method for three-dimensional simulation of dendrite coarsening according to claim 7 wherein D (f)s) Obtained by calculation of equation 6:
D(fs)＝kpDsfs+Dl(1-fs) Equation 6.
10. The method for three-dimensional simulation of dendrite coarsening according to claim 7 wherein p (f)s) Calculated by equation 7:
p(fs)＝kpfs+(1-fs) Equation 7.
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| 骈松;张照;包羽冲;刘林;李日;: "基于三维LBM-CA模型模拟Al-4.7%Cu合金的枝晶形貌和成分分布", 材料导报 * |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113139254A (en) * | 2021-05-14 | 2021-07-20 | 苏州大学 | Method for describing columnar crystal morphology in 6xxx series aluminum alloy melting welding pool |
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