金属过冷熔体凝固过程微观组织及凝固特性的相场法表征
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摘要
大部分材料,尤其是金属材料,在生产制备过程中均要经历由液态到固态的转变,即凝固过程。凝固学的中心任务是探索凝固组织的形成规律和控制方法,固/液界面形态研究是凝固理论研究的重要组成部分它决定了最终的凝固组织和材料性能。由于影响凝固组织形成的因素较多,近年来发展的数值模拟方法为再现金属凝固过程的组织演化,揭示微观组织的形成规律提供的有效手段。作为模拟复杂微观组织方法的一种,相场法已经广为研究者们所接受。
本文从Ginzburg-Landau理论出发,基于熵函数建立了二元合金相场模型,采用均匀网格显示差分法对相场和溶质场控制方程求解。通过研究方程数值解收敛的稳定性条件,确定了空间步长与时间步长的选择与物性参数之间的限制性关系；采用交替方向隐式方法求解温度场控制方程,避免了时间步长的限制。利用VC++操作平台的兼容性,采用C语言实现离散方程的程序编制,利用Tecplot软件实现模拟结果的计算机图像生成及其动态显示；并对凝固过程中枝晶／胞晶尖端生长速度、曲率半径、固相率等进行了计算。研究表明：
Ni-Cu二元合金等温凝固过程中,各向异性系数γ强烈影响界面形态与枝晶尖端稳态行为,对于<100>枝晶生长过程,当γ>1/15时,枝晶生长界面形态呈现不连续的小晶面生长,原始的相场法模型变得不可用,需要对其进行修正计算；对于非典型的<110>枝晶生长,当各向异性系数γ=0.02时,<110>枝晶生长呈现完全对称的“雪花状”晶结构；当γ>0.02时,则[110]方向枝晶主支生长较[101]与[011]方向更为发达,枝晶生长对称形貌被破坏,形成“羽毛状’甚至“针状”结构,且沿[110]方向溶质截留效应加剧。而对于<1]0>枝晶生长,界面动力学β的影响也较为显著,随着β的增加,侧向分支出现大量的合并现象,形成“扇形状”及“盘状”晶,但是晶体生长的对称性并未遭到破坏,溶质截留效应表现明显,当β=0.60,溶质截留水平近似于完全截留。
通过对Ni-Cu二元合金等温模拟与非等温模拟的对比发现：结晶潜热的释放使固相温度升高,而且在固相内二次枝晶生长较为集中的区域温度最高,发生明显的再辉现象。潜热的释放使液相中的过冷度降低,同一凝固时间非等温凝固时的固相率相对较小,并随着时间变化固相率的差距也在增大,枝晶没有等温凝固时的发达。此外,较高的温度利于液相溶质扩散,导致非等温条件下的固／液界面溶质浓度较低,溶质截留效应增强。当存在热扰动时,在较小的界面厚度条件下,热扰动极易被放大,使得枝晶尖端生长稳定性下降,枝晶生长受到抑制;扩散系数DT对枝晶生长的影响主要是热扩散层完成的,当DT增大时,凝固产生的热量能较快的扩散出去,凝固区域的温度上升更加缓慢；溶质梯度系数δ对枝晶形貌几乎没有影响,但随着溶质梯度的增大,固相区溶质分布趋于一致,溶质偏析程度下降。
在二元合金定向凝固过程中,随着界面推进速度的提高,界面形态呈现平-胞-平转变,界面前沿固、液相成分逐渐逼近,从而表现出强的溶质截留效应。在低的凝固速度下,各向异性系数γ对界面形态及尖端生长速度影响明显。当各向异性较弱时,凝固过程晶体生长形态为海藻状组织形貌；随着γ增大,晶体生长形态呈现胞状枝晶及胞晶生长,尖端生长速度增加,稳定性下降。但是,定向凝固的平界面生长过程几乎不受到各向异性的影响。除了界面推进速度,物性参数对溶质截留效应也存在明显影响,随着梯度系数δ及界面动力学系数β的增加,固相溶质浓度显著增加,溶质截留效应增强。
对流对枝晶生长影响明显。引入对流后,枝晶呈现不对称生长,逆流方向枝晶臂生长最发达,顺流枝晶臂生长速度最缓慢,其枝晶臂最短,水平方向枝晶臂生长速度介于前两者之间。通过研究多个晶粒在对流条件下的凝固过程发现,各晶粒之间在相互竞争生长的同时,整体沿着逆流方向生长。通过研究对流作用下各向异性γ对枝晶生长的影响发现,在较小各向异性条件下,各方向枝晶主支生长都比较粗大,随着各向异性增大,各向主枝都逐渐变细,产生侧向分支,发生明显的颈缩效应。当γ增大到0.07时,引发枝晶变异,发生小晶面枝晶生长,界面上出现棱角；顺流枝晶生长产生尖端分裂,枝晶根部形成断裂趋势。对二元合金凝固过程,对流影响也较为明显,随着对流的增强,逆流枝晶生长速度呈现近似线性增加,溶质截留效应明显；而水平尖端生长速度几乎不变。而顺流枝晶尖端生长在弱的对流作用下,随着流速的增加,其现表现为下降趋势；但是对流的进一步增强,则表现为上升趋势,但是顺流枝晶尖端生长速度始终小于逆流方向及水平方向。
The transformation must be happened from liquid to solid in the process of manufacturing for most materials, especially for metal materials. The central task of solidification is the formation mechanism and control method of the microstructure, and studies of the solid/liquid interfacial morphology is an important part of solidification theory, then the interface formation determines the final microstructures and materials properties. Many factors affect the formation of solidification, so the numerical simulation offers researchers a convenient way to visualize the evolution process and reveal the formation mechanism of microstructures during solidification in recent years. As one of the numerical simulation methods to elucidate the complex microstructure evolution, the phase-field method has been widely accepted by researchers.
Based on Ginzburg Landau theory and entropy function, a phase field model of binary alloy is established and an explicit difference method with uniform grid is used to solve the phase field and solute field controlled equation in this paper. Through the studies of stable conditions of convergence in equations'numerical solutions, the choice of the space steps and time step and the restrictive relationship among physical parameters is determined, and the alternating direction implicit method for solving temperature field controlled equation is also employed to avoid the restrictions of time step. Using the compatibility of VC++platform, the C Programming Code is implemented to complete the phase-field simulation. The image generation and dynamic displaying of salutation results are visualized by using Tecplot software, and the tip velocity of dendrite/cell, the curvature radius and the solidifying rate are calculated in the process of solidification. The simulation results consist of following aspects.
The crystalline anisotropy coefficient y is a crucial parameter that determines the interfacial morphology and the tip operating state for an isothermal solidification process of Cu-Ni binary alloy. For the<100> dendrites growth, the slope of the interface has discontinuities and appears faceted when y>1/15, then the initial phase model becomes inapplicable and must to be regularized. For the atypical dendritic growth of <110> directions, the interfacial morphology presents a perfectly symmetrical snowflake structure when γ=0.02. When γ＞0.02, the main branch of [110] is more developed than the [101] and [011] branches, and the symmetry of dendrite morphology is disrupted, and interfacial morphology changes to a feathery shape even a needle shape, then the level of solute trapping is severe along with [110]. The interface dynamic coefficient β also has a significant effect on dendritic growth of <110> dendrites. With increasing of β, the coalescence of some side-branches are observed, and the crystal growth patterns are formed include a sector form and a plate formation, but the symmetry of dendrite morphology is not disrupted, and the solute trapping appears obvious. When β=0.60, the level of solute trapping is almost complete.
Some phenomena have been found by contrasting simulation results of isothermal and non-isothermal solidification process for Ni-Cu alloy. Release of latent heat results in a high temperature distribution in solid area, and the highest temperature appears in the center area of some developed side-branches, that is, recalescence occurs. Release of latent heat reduces supercooling of liquid, so dendritic branches and the solid ratios are comparatively small comparing to isothermal solidification, and the gap becomes bigger according to time up. Moreover, the solute concentration is low in the region of solid/liquid interface at high temperature, and the level of solute trapping is strong. Under the conditions of a small interface thickness, the thermal noise is enlarged so easily, and the tip stability of the dendrite growth is declined, then the crystal growth is restrained. The influence of diffusion coefficient DT to crystal growth is achieved by thermal diffusion layer mainly. With the increment of DT, the latent heat diffuses easily, and the extent of the solid temperature is slight. The solute gradient coefficient δ would affect the dendrite feature hardly, but with elevating of solute gradient, the distribution of solid solute is consistent, and the severity of solute segregation is reduced
During the directional solidification process of binary alloys, with the increment of interface speed Ⅴ, the transition from plane to cells/fine cellular structures, then to planar structures(plane-cell-plane) will happen, and the level of solute trapping is severe following with the concentration of solid/liquid trends to equal in the interfacial region. The effect of crystalline anisotropy on the interfacial morphology and tip velocity is obvious at low interface speed. The solid-liquid interface shape with seaweed microstructure is achieved at a small anisotropy coefficient y. And the interfacial morphology changes through transition from seaweed to cellular dendrite or cell with the increment of y, and cell tip velocity increases correspondingly, but the stability of the cell tip velocity becomes down. Then, the operating behavior for planar growth is hardly any affected by the crystalline anisotropy. In addition to the interface speed, some physical properties have a significant impact on the solute trapping effect. With increasing of the solute concentration gradient δ and the interface dynamic coefficient β, the solute concentration in solid increases correspondingly, thus the level of solute trapping is strong.
The effect of melt convection on the dendrtic growth is obvious. In the presence of fluid flow, the symmetry of dendrite morphology is disrupted. The primary dendrite growing upstream is most luxuriated, and the branches in a horizontal direction normal to the flow come next, then the downstream branch is slowest. Many grains growing along with the preferred orientation morphology under forced flow is studied. It is demonstratcd that the dendrite growth is controlled mainly by flow and the upstream tips are highest, but the grains impact each other when they are closer. The effect of crystalline anisotropy on the interfacial morphology is investigated under forced flow. The simulation results indicate that, the main braches are all obese at different directions at low γ; with increasing of y, they become slender with abundant in side-branches, and the necking is evident. When γ=0.07, the variation of dendrite morphology distortion happens, and the slope of the interface has edges and corners and appears faceted growth; then, the downstream dendrite generates tip-splitting, and the dendritic root trends fracture. Convection affects the interfacial morphology of binary alloys in the process of solidification also. With the enhancement of convection, the upstream tip velocity increases approximately linear, and the level of solute trapping is severe, but the horizontal tips are nearly steady. While the stable growth velocity of downstream tip decreases accordingly at feeble convection, and then increases suddenly when the speed of the fluid flow crosses the critical value.
本文从Ginzburg-Landau理论出发,基于熵函数建立了二元合金相场模型,采用均匀网格显示差分法对相场和溶质场控制方程求解。通过研究方程数值解收敛的稳定性条件,确定了空间步长与时间步长的选择与物性参数之间的限制性关系；采用交替方向隐式方法求解温度场控制方程,避免了时间步长的限制。利用VC++操作平台的兼容性,采用C语言实现离散方程的程序编制,利用Tecplot软件实现模拟结果的计算机图像生成及其动态显示；并对凝固过程中枝晶／胞晶尖端生长速度、曲率半径、固相率等进行了计算。研究表明：
Ni-Cu二元合金等温凝固过程中,各向异性系数γ强烈影响界面形态与枝晶尖端稳态行为,对于<100>枝晶生长过程,当γ>1/15时,枝晶生长界面形态呈现不连续的小晶面生长,原始的相场法模型变得不可用,需要对其进行修正计算；对于非典型的<110>枝晶生长,当各向异性系数γ=0.02时,<110>枝晶生长呈现完全对称的“雪花状”晶结构；当γ>0.02时,则[110]方向枝晶主支生长较[101]与[011]方向更为发达,枝晶生长对称形貌被破坏,形成“羽毛状’甚至“针状”结构,且沿[110]方向溶质截留效应加剧。而对于<1]0>枝晶生长,界面动力学β的影响也较为显著,随着β的增加,侧向分支出现大量的合并现象,形成“扇形状”及“盘状”晶,但是晶体生长的对称性并未遭到破坏,溶质截留效应表现明显,当β=0.60,溶质截留水平近似于完全截留。
通过对Ni-Cu二元合金等温模拟与非等温模拟的对比发现：结晶潜热的释放使固相温度升高,而且在固相内二次枝晶生长较为集中的区域温度最高,发生明显的再辉现象。潜热的释放使液相中的过冷度降低,同一凝固时间非等温凝固时的固相率相对较小,并随着时间变化固相率的差距也在增大,枝晶没有等温凝固时的发达。此外,较高的温度利于液相溶质扩散,导致非等温条件下的固／液界面溶质浓度较低,溶质截留效应增强。当存在热扰动时,在较小的界面厚度条件下,热扰动极易被放大,使得枝晶尖端生长稳定性下降,枝晶生长受到抑制;扩散系数DT对枝晶生长的影响主要是热扩散层完成的,当DT增大时,凝固产生的热量能较快的扩散出去,凝固区域的温度上升更加缓慢；溶质梯度系数δ对枝晶形貌几乎没有影响,但随着溶质梯度的增大,固相区溶质分布趋于一致,溶质偏析程度下降。
在二元合金定向凝固过程中,随着界面推进速度的提高,界面形态呈现平-胞-平转变,界面前沿固、液相成分逐渐逼近,从而表现出强的溶质截留效应。在低的凝固速度下,各向异性系数γ对界面形态及尖端生长速度影响明显。当各向异性较弱时,凝固过程晶体生长形态为海藻状组织形貌；随着γ增大,晶体生长形态呈现胞状枝晶及胞晶生长,尖端生长速度增加,稳定性下降。但是,定向凝固的平界面生长过程几乎不受到各向异性的影响。除了界面推进速度,物性参数对溶质截留效应也存在明显影响,随着梯度系数δ及界面动力学系数β的增加,固相溶质浓度显著增加,溶质截留效应增强。
对流对枝晶生长影响明显。引入对流后,枝晶呈现不对称生长,逆流方向枝晶臂生长最发达,顺流枝晶臂生长速度最缓慢,其枝晶臂最短,水平方向枝晶臂生长速度介于前两者之间。通过研究多个晶粒在对流条件下的凝固过程发现,各晶粒之间在相互竞争生长的同时,整体沿着逆流方向生长。通过研究对流作用下各向异性γ对枝晶生长的影响发现,在较小各向异性条件下,各方向枝晶主支生长都比较粗大,随着各向异性增大,各向主枝都逐渐变细,产生侧向分支,发生明显的颈缩效应。当γ增大到0.07时,引发枝晶变异,发生小晶面枝晶生长,界面上出现棱角；顺流枝晶生长产生尖端分裂,枝晶根部形成断裂趋势。对二元合金凝固过程,对流影响也较为明显,随着对流的增强,逆流枝晶生长速度呈现近似线性增加,溶质截留效应明显；而水平尖端生长速度几乎不变。而顺流枝晶尖端生长在弱的对流作用下,随着流速的增加,其现表现为下降趋势；但是对流的进一步增强,则表现为上升趋势,但是顺流枝晶尖端生长速度始终小于逆流方向及水平方向。
The transformation must be happened from liquid to solid in the process of manufacturing for most materials, especially for metal materials. The central task of solidification is the formation mechanism and control method of the microstructure, and studies of the solid/liquid interfacial morphology is an important part of solidification theory, then the interface formation determines the final microstructures and materials properties. Many factors affect the formation of solidification, so the numerical simulation offers researchers a convenient way to visualize the evolution process and reveal the formation mechanism of microstructures during solidification in recent years. As one of the numerical simulation methods to elucidate the complex microstructure evolution, the phase-field method has been widely accepted by researchers.
Based on Ginzburg Landau theory and entropy function, a phase field model of binary alloy is established and an explicit difference method with uniform grid is used to solve the phase field and solute field controlled equation in this paper. Through the studies of stable conditions of convergence in equations'numerical solutions, the choice of the space steps and time step and the restrictive relationship among physical parameters is determined, and the alternating direction implicit method for solving temperature field controlled equation is also employed to avoid the restrictions of time step. Using the compatibility of VC++platform, the C Programming Code is implemented to complete the phase-field simulation. The image generation and dynamic displaying of salutation results are visualized by using Tecplot software, and the tip velocity of dendrite/cell, the curvature radius and the solidifying rate are calculated in the process of solidification. The simulation results consist of following aspects.
The crystalline anisotropy coefficient y is a crucial parameter that determines the interfacial morphology and the tip operating state for an isothermal solidification process of Cu-Ni binary alloy. For the<100> dendrites growth, the slope of the interface has discontinuities and appears faceted when y>1/15, then the initial phase model becomes inapplicable and must to be regularized. For the atypical dendritic growth of <110> directions, the interfacial morphology presents a perfectly symmetrical snowflake structure when γ=0.02. When γ＞0.02, the main branch of [110] is more developed than the [101] and [011] branches, and the symmetry of dendrite morphology is disrupted, and interfacial morphology changes to a feathery shape even a needle shape, then the level of solute trapping is severe along with [110]. The interface dynamic coefficient β also has a significant effect on dendritic growth of <110> dendrites. With increasing of β, the coalescence of some side-branches are observed, and the crystal growth patterns are formed include a sector form and a plate formation, but the symmetry of dendrite morphology is not disrupted, and the solute trapping appears obvious. When β=0.60, the level of solute trapping is almost complete.
Some phenomena have been found by contrasting simulation results of isothermal and non-isothermal solidification process for Ni-Cu alloy. Release of latent heat results in a high temperature distribution in solid area, and the highest temperature appears in the center area of some developed side-branches, that is, recalescence occurs. Release of latent heat reduces supercooling of liquid, so dendritic branches and the solid ratios are comparatively small comparing to isothermal solidification, and the gap becomes bigger according to time up. Moreover, the solute concentration is low in the region of solid/liquid interface at high temperature, and the level of solute trapping is strong. Under the conditions of a small interface thickness, the thermal noise is enlarged so easily, and the tip stability of the dendrite growth is declined, then the crystal growth is restrained. The influence of diffusion coefficient DT to crystal growth is achieved by thermal diffusion layer mainly. With the increment of DT, the latent heat diffuses easily, and the extent of the solid temperature is slight. The solute gradient coefficient δ would affect the dendrite feature hardly, but with elevating of solute gradient, the distribution of solid solute is consistent, and the severity of solute segregation is reduced
During the directional solidification process of binary alloys, with the increment of interface speed Ⅴ, the transition from plane to cells/fine cellular structures, then to planar structures(plane-cell-plane) will happen, and the level of solute trapping is severe following with the concentration of solid/liquid trends to equal in the interfacial region. The effect of crystalline anisotropy on the interfacial morphology and tip velocity is obvious at low interface speed. The solid-liquid interface shape with seaweed microstructure is achieved at a small anisotropy coefficient y. And the interfacial morphology changes through transition from seaweed to cellular dendrite or cell with the increment of y, and cell tip velocity increases correspondingly, but the stability of the cell tip velocity becomes down. Then, the operating behavior for planar growth is hardly any affected by the crystalline anisotropy. In addition to the interface speed, some physical properties have a significant impact on the solute trapping effect. With increasing of the solute concentration gradient δ and the interface dynamic coefficient β, the solute concentration in solid increases correspondingly, thus the level of solute trapping is strong.
The effect of melt convection on the dendrtic growth is obvious. In the presence of fluid flow, the symmetry of dendrite morphology is disrupted. The primary dendrite growing upstream is most luxuriated, and the branches in a horizontal direction normal to the flow come next, then the downstream branch is slowest. Many grains growing along with the preferred orientation morphology under forced flow is studied. It is demonstratcd that the dendrite growth is controlled mainly by flow and the upstream tips are highest, but the grains impact each other when they are closer. The effect of crystalline anisotropy on the interfacial morphology is investigated under forced flow. The simulation results indicate that, the main braches are all obese at different directions at low γ; with increasing of y, they become slender with abundant in side-branches, and the necking is evident. When γ=0.07, the variation of dendrite morphology distortion happens, and the slope of the interface has edges and corners and appears faceted growth; then, the downstream dendrite generates tip-splitting, and the dendritic root trends fracture. Convection affects the interfacial morphology of binary alloys in the process of solidification also. With the enhancement of convection, the upstream tip velocity increases approximately linear, and the level of solute trapping is severe, but the horizontal tips are nearly steady. While the stable growth velocity of downstream tip decreases accordingly at feeble convection, and then increases suddenly when the speed of the fluid flow crosses the critical value.
引文
[1]王文清,李魁盛.铸造工艺学[M].机械工业出版社,2002.10
[2]柳百成,荆涛.铸造工程的模拟仿真与质量控制[M].北京：机械工业出版社,2001
[3]周尧和,胡壮麒等.凝固技术[M].北京：机械工业出版社,1998
[4]Boettinger W J, Coriell S R, Greer A L et al. Solidification microstructures:recent developments, future directions[J]. Acta Mater., 2000,48:43-70
[5]柳百成.铸件充型凝固过程国内外研究进展[J].铸造,1999,48(8)：78-81
[6]Liu B C. Progress in Solidification Modeling of Cast Iron in China[J]. International J. of Cast Metals Research,1999,11(5),259-266
[7]王延露.基于MAGMAsoft的铸造充型凝固过程分析与研究[D].兰州理工大学硕士学位论文.兰州,2006,6
[8]刘小刚.Al-Cu合金等温凝固的相场法模拟[D].沈阳工业学院硕士学位论文.沈阳,2003.3
[9]王君.纯物质凝固过程枝晶生长现象的研究和模拟[D].昆明理工大学博士学位论文.昆明,2002.6
[10]D.罗伯.计算材料学[M].北京：化学工业出版社.2003
[11]Zhu P P, Smith R W. Dynamic simulation of crystal growth by Monte Carlo method-I:Model description and kinetics[J]. Acta Metall.,1992, 40(2):350-361
[12]Wolfram S. Statistical mechanics of cellular automata[J]. Rev. of Modern phys.,1983,55:601-644.
[13]Wolfram S. Universality and complexity in cellular automata[J]. Physica D,1984,10:1-35.
[14]Wolfram S. Cellular automata as models of complexity [J], Nature, 1984:311:419-448.
[15]Rappaz M, Gandin C A. Probabilostic Modelling of Microstructure Formation in Solidification Processes[J]. Acta Metall.,1993,41(2): 350-360
[16]Gandin C A, Rappaz M. Coupled Finite Element-Cellular Automation Model for The Prediction of Dendritic Grain Structures in Solidification Processes[J]. Acta Metall.,1994,42(7):2233-2246
[17]Brown S G R. Simulation of diffusional composite growth using the cellular automaton finite difference(CAFD) method[J]. J. of Mater. Sci. 1998,33(19):4769-4773
[18]Jarvis D J, Brown S G R. Modelling of non-equilibrium solidification in ternary alloys:comparison of 1D,2D, and 3D cellular automaton finite difference simulations[J]. Mater. Sci. and Techn.,2000,16: 1420-1424
[19]Oldfield W.A Quantitative Approach to casting Solidification[J]. Freezing of Cast Iron. ASM Trans.1966,59(2):945-960
[20]Hunt J D. Steady state columnar and equiaxed growth of dendrites and eutectic[J]. Mater. Sci. Eng.,1984,65:75-89
[21]Wang C Y, Beckermann C. Equiaxed Dendritic Solidification with Convection-I:Multiscale/Multiphase Modeling[J]. Metall. Trans. A, 1996,27:2754-2764
[22]Wang C Y, Beckermann C. Equiaxed Dendritic Solidification with Convection-Ⅱ:Numerical Simulation for an Al-4wt%Cu Alloy[J]. Metall. Trans. A,1996,27:2765-2783
[23]Wang C Y, Beckermann C. Equiaxed Dendritic Solidification with Convection-Ⅲ:Comparisons with NH4Cl-H2O Experiments[J]. Metall. Trans. A,1996,27:2784-2795
[24]Karma A, Rappel W J. Phase-field Method for Computationally Efficient Modeling of Solidification with Arbitrary Interface Kinetics[J]. Phys. Rev. E,1996,53(4):3017-3020
[25]Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth[J]. Phys. Rev. Lett.,1996,77(10):4050-4053
[26]Karma A, Rappel W J. Quantitative phase-field modeling of dendritic growth in two and three dimensions[J]. Phys. Rev. E,1998,57(4): 4323-4349
[27]于艳梅.过冷熔体枝晶生长的相场法模拟[D].西北工业大学博士毕业论文.西安,2002.6
[28]于艳梅,杨根仓,赵达文等.凝固微观组织数值模拟的研究进展[J].特种铸造及有色合金,2002,(3)：30-33
[29]Asta M, Beckermann C, Karma A et al. Solidification microstructures and solid-state parallels:recent developments, future directions[J]. Acta Mater.,2009,57:941-971
[30]Jeong J H, Goldenfeld N, Dantzig J A. Phase field model for three-dimensional dendritic growth with fluid flow[J]. Phys. Rev. E,2001,64:041602
[31]Provatas N, Goldenfeld N, Dantzig J. Efficient computation of dendritic microstructures using adaptive mesh refinement[J]. Phys. Rev. Lett.,1998,80:3308
[32]Lan C W, Hsu C M, Liu C C. Efficient adaptive phase field simulation of dendritic growth in a forced flow at low suppercooling[J]. J. Cryst. Growth,2002,241:379-386
[33]Lan C W, Hsu C M, Liu C C et al. Adaptive phase field simulation of dendritic growth in a forced flow at various supercoolings[J]. Phys. Rev. E,2002,65:061601.
[34]Lan C W, Liu C C, Hsu C M. An adaptive finite volume method forincompressible heat flow problems in solidification[J]. J. Comput. Phys,2002,178:464-497.
[35]熊玉华,刘林.铸件凝固组织形成的计算机模拟.材料导报,1999,13(3)：20-21
[36]Gandin C A, Jalanti T, Rappaz M. Modeling of Dendritic Grain Structure. Modeling of Casting, Welding and Advanced Solidification Processes Ⅷ.1998:363-374
[37]Zhu M F, Hong C P. A modified cellular automaton model for the simulation of dendritic growth in solidification of alloys[J]. ISIJ Int. 2001,41(5):436-445
[38]Zhu M F, Kim J M, Hong C P. Modeling of globular and dendritic structure evolution in solidificat ion of an Al-7mass%Si alloy[J]. ISIJ Int.,2001,41(9):992-998
[39]Cho S H. Contribution of nucleation process to grain formation in calculating solidification microstructure by CA-DFD[J]. Current Advance in Materials and Process,2002,15, (1):46-50
[40]李殿中,苏仕方,徐雪华等.镍基合金叶片凝固过程微观组织模拟及工艺优化研究[J].铸造,1997,46(8)：1-7
[41]王同敏,金俊泽,郑贤淑.金属凝固过程微观模拟的两个辅助算法[J].铸造,2000,49(8)：484-488
[42]许庆彦,柳百成.采用Cellular Aut omaton法模拟铝合金的微观组 织[J].中国机械工程.2001,12(3)：328-331
[43]Spittle J A, Brown S G R. Computer Simulation of the Effects of Alloy Variables on the Grain Structures of Castings[J]. Acta Metall.,1989, 37(7):1803-1810
[44]Spittle J A, Brown S G R. A Computer Simulation of the Influence of Processing Conditions on the cast Grain Structures[J]. J. Mater. Sci., 1989,23:1777-1781
[45]Zhu P P, Smith R W. Dynamic Simulation of Crystal Growth by Monte Carlo Method Ⅱ. Ingot Microstructures[J]. Acta Metall.,1992, 40(12):3369-3379
[46]金俊泽,王宗廷,郑贤淑等.金属凝固组织形成的仿真研究[J].金属学报,1998,34(9)：928-932
[47]丁雨田,王海南,许广济等.定向凝固微观组织的Monte Carlo模拟[J].甘肃工业大学学报,2001,27(9)：15-18
[48]丁雨田.热型连铸凝固过程微观组织形成的数值模拟[D].兰州理工大学博士学位论文.兰州,2005.6
[49]Maxwell I, Hellawell A. A simple model for grain refinement during solidification[J]. Acta Metall.,1975,23:229-237
[50]Thevoz P H, Desbilles J L, Rappaz M. Modeling of equiaxed microstructure formation in casting[J]. Metall. Trans. A,1989,20(2): 311-322
[51]Saito Y, Goldbeck-Wood G, Muller-Krumbhaar H. Numerical simulation of dendritic growth[J]. Phys. Rev. A,1988,38(4): 2148-2153.
[52]Fix G J. Phase field for free boundary problems. In:Fasono A, Primicerio M, Pitman eds. Free Boundary Problems:Theory and Applications[J]. London,1983:580-589
[53]Collins J B, Levine H. Diffuse interface model of diffusion limited crystal growth[J]. Phys. Rev. B,1985,31(9):6119-6122
[54]Langer J S. Directions in condensed physics, eds. Grinstein G and Maxenko G (World Science,1986):164-168
[55]Caginalp G, Fife P C. Phase field model for interfacial boundaries[J]. Phys. Rev. B,1986,34(7):4940-4943
[56]Caginalp G, Fife P C. Dynamics of layered interfaces arising from phase boundaries[J]. SIAM J. of Appl. Math.,1988,48(3):506-510
[57]Fife P C, Gill S G. The phase-field description of mushy zones[J]. Phys. D,1989,35:267-275
[58]Fife P C, Gill S G. Phase transition mechanisms for the phase-field model under internal heating[J]. Phys. Rev. A,1991,43(2):843-851
[59]Penrose O, Fife P C. Thermodynamically Consistent Models of Phase-field Type for the Kinetics of Phase Transitions[J]. Phys. D,1990, 43:44-62
[60]Kobayashi R. Modeling and Numerical Simulation of Dendritic Crystal Growth[J]. Phys. D,1993,63:410-423
[61]Wang S L, Sekerka R F, Wheeler A A et al. Thermodynamically-Consistent Phase-Field Models for Solidification[J]. Phys. D,1993,69:189-200
[62]Wheeler A A, Murrary B T, Schaefer R J. Computation of dendrites using a phase field model[J]. Phys. D,1993,66(10):243-262
[63]Murrary B T, Wheeler A A, Glicksman M E. Simulation of experimentally observed dendritic growth behavior using a phase-field model[J]. J. Cryst. Growth,1995,154:386-400
[64]Karma A, Rappel W J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics[J]. Phys. Rev. E,1996,53(4):3017-3020
[65]Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth[J]. Phys. Rev. Lett.,1996,77(10):4050-4053
[66]Karma A, Rappel W J. Phase-field simulation of three-dimensional dendrites:is microscopic solvability theory correct[J]. J. Cryst. Growth. 1997,174:54-64
[67]Karma A, Rappel W J. Quantitative phase-field modeling of dendritic growth in two and three dimensions[J]. Phys. Rev. E,1998,57(4): 4323-4349
[68]张玉妥,李殿中,李依依等.用相场方法模拟纯物质等轴枝晶生长[J].金属学报,2000,36(6)：589-591
[69]张玉妥,李殿中,庞维诚等.用相场方法模拟镍的枝晶生长[J].材料研究学报,2002,16(5)：541-543
[70]于艳梅,杨根仓,赵达文等.相场法模拟过冷熔体枝晶生长的界面厚度参数的取值[J].自然科学进展,2001,11(11)：1192-1197
[71]于艳梅,杨根仓,赵达文等.过冷熔体中枝晶生长的相场法数值模 拟[J].物理学报,2001,50(12)：2423-2428
[72]李俊杰,王锦程,杨根仓.相场法模拟界面能各向异性对枝晶生长行为的影响[J].自然科学进展,2005,15(11)：1312-1317
[73]赵代平,荆涛.用捕获液态改进的相场方法模拟三维枝晶生长[J].金属学报,2002,38(12)：1238-1240
[74]Zhu Changsheng, Liu Baicheng, Jing Tao et al. Dependence of dendritic side-branches on parameters in phase-field simulation[J]. Mater. Trans.,2005,46(1):15-19
[75]朱昌盛.相场法凝固微观组织计算机模拟仿真系统的研究与实现[D].兰州理工大学博士学位论文.兰州,2006.11
[76]Wheeler A A, Boettinger W J, McFadden G B. Phase-field model for isothermal phase transition in binary alloy[J]. Phys. Rev. E,1992,45(10): 7424-7439
[77]Warren J A, Boettinger W J. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method[J]. Acta metal. Mater,1995,43(2):689-703
[78]Boettinger W J, Warren J A. The Phase field method:simulation of alloy dendritic solidification during recalescence[J]. Metall. Mater. Trans. A,1996,27A (3):657-668
[79]Wheeler A A, Boettinger W J, McFadden G B. Phase field model of trapping during solidification[J]. Phys. Rev. E,1993,47(4): 1893-1909
[80]Boettinger W J, Wheeler A A, Murray B T, McFadden G B. Prediction of solute trapping at high solidification rates using a diffuse interface phase field theory of alloy solidification[J]. Mater. Sci. and Eng.,1994, 178 A (10):217-223
[81]Conti M. Solidification of binary alloys:Thermal effects studied with the phase-field model[J]. Phys. Rev. E,1997,55(1):765-771
[82]Conti M. Phase and solute fields across the solid-liquid interface of a binary alloy[J]. Phys. Rev. E,1999,55(1):1913-1920
[83]Kim S G, Kim W T, Suzuki T. Interfacial compositions of solid and liquid in a phase-field Model with Finite interface Thickness for Isothermal Solidification in Binary Alloys[J]. Phys. Rev. E,1998,58(3): 3316-3323
[84]Kim S G, Kim W T, Suzuki T. Phase-field model for binary alloys[J]. Phys. Rev. E,1999,60(6):7186-7197
[85]Suzuki T, Ode M, Kim S G, et al. Phase-field model of dendritic growth[J]. J. Crystal Growth,2002,237-239:125-131
[86]Lee J S, Suzuki T. Numerical simulation of isothermal dendritic growth by phase-field model[J]. ISIJ Int.,1999,39(3):246-252
[87]Kim S G, Kim W T, Lee J S, et al. Large scale simulation of dendritic growth in pure undercooled melt by phase-field model[J]. ISIJ Int.,1999, 39 (4):335-340
[88]Loginova Ⅰ, Amberg G and Agern J. Phase-field simulation of non-isothermal binary alloy solidification[J]. Acta mater.,2001,49: 573-581
[89]Gonzalez-Cinca R, Ramirez-Piscina L, Casademunt J et al. Phase-field simulations and experiments of faceted growth in liquid crystals[J]. Phys. D,1996,99:359-368
[90]Mcfadden G B, Coriell S R, Sekerka R F. Effect of surface free energy anisotropy on dendrite tip shape[J]. Acta Mater.,2000,48:3177-3181
[91]Boettinger W J, Coriell S R, Greer A L et al. Solidification microstructures:recent developments, future directions[J]. Acta Mater. 2000,48:43-70
[92]Eggleston J J, Mcfadden G B, Voorhees P W. A phase-field model for highly anisotropic interfacial energy[J]. Physica D,2001,150:91-103
[93]Debierre J M, Karma A, Celestini F et al. Phase-field approach for faceted solidification[J]. Phys. Rev. E,2003,68(4):1604-1611
[94]Uehara T, Sekerka R F. Phase field simulations of faceted growth for strong anisotropy of kinetic coefficient[J]. J. Cryst. Growth,2003,254: 251-261
[95]Mullis A M. Quantification of mesh induced anisotropy effects in the phase-field method[J]. Comput. Mater. Sci.,2006,36:345-353
[96]Phase field modeling of dendritic growth with high anisotropy[J]. J. Cryst. Growth,2005,275:355-360
[97]Suzuki T, Kim S G, Kim W T. Two-dimensional facet crystal growth of silicon from undercooled melt of Si-Ni alloy[J]. Mater. Sci. Enging. A, 2007,449:99-104
[98]Suwa Y, Saito Y, Onodera H. Three-dimensional phase field simulation of the anisotropy in grain-boundary mobility on growth kinetics and morphology of grain structure[J]. Comput. Mater. Sci.,2007, 40:40-50
[99]Haxhimali T, Karma A, Gonzales F et al. Orientation selection in dendritic evolution[J]. Nature Materials,2006,5:660-664
[100]Clyne T W. Heat flow in controlled directional solidification of metals[J]. J. Cryst. Growth,1980,50:684-690
[101]Sakane J, Lib Q, Evans J W. Mathematical modeling of meniscus profile and melt flow in electromagnetic casters[J]. Metall. Trans. B, 1988,19:397-408
[102]陶文铨.数值传热学[M].西安：西安交通大学出版社,1988
[103]Conti M. Solute trapping in directional solidification at high speed:A one-dimensional study with the phase-field model[J]. Phys. Rev. E,1997, 56:3717-3720
[104]Conti M. Oscillatory instabilities in rapid directional solidification[J]. Phys. Rev. E,1997,56:6267-6270
[105]Boettinger W J, Warren J A. Simulation of the cell to plane front transition during directional solidification at high velocity[J]. J. Cryst. Growth,1999,200:583-591
[106]Mullins W W, Sakerka R F. Stability of a Planar Interface During Solidification of a dilute Alloy[J]. J. Applied Phys.,1964,35:444-451
[107]Kim S G, Kim W T. Phase-field modeling of rapid solidification[J]. Mater. Sci. Eng. A,2001,304:281-286
[108]Diepers H J, Ma D, Steinbach I. History effects during the selection of primary dendrite spacing. Comparison of phase-field simulations with experimental observations[J]. J. Cryst. Growth,2002,237:149-153
[109]Takaki T, Fukuoka T, Tomita Y. Phase-field simulation during directional solidification of a binary alloy using adaptive finite element method[J]. J. Cryst. Growth,2005,283:263-278
[110]Filho R N, Kosterlitz J M, Granato E. Pattern selection in a phase field model for directional solidification[J]. Physica A,2005,354: 333-343
[111]Plapp M. Three-dimensional phase-field simulations of directional solidification[J]. J. Cryst. Growth,2007,303:49-57
[112]Steinbach I. Effect of interface anisotropy on spacing selection in constrained dendrite growth[J]. Acta Mater.,2008,56:4965-4971
[113]Amoorezaei M, Gurevich S, Provatas N. Spacing characterization in Al-Cu alloys directionally solidified under transient growth conditions[J]. Acta Mater.,2010,58:6115-6124
[114]Amoorezaei M, Gurevich S, Provatas N. Orientation selection in solidification patterning[J]. Acta Mater.,2012,60:657-663
[115]于艳梅.过冷熔体枝晶生长的相场法模拟[D].西北工业大学博士学位论文.西安,2002.6
[116]Yu Yan-Mei, Yang Gen-Cang, Zhao Da-Wen et al. Influence of isothermal approximation on the phase-field simulation of directional growth in undercooled melt[J]. Chinese Physics,2003,12(02):211-217
[117]Guo Jingjie, Li Xinzhong, Su Yanqing et al. Phase-field simulation of structure evolution at high growth velocities during directional solidification of Ti55Al45 alloy[J]. Intermetallics,2005,13:275-279
[118]李新中,苏彦庆,郭景杰等Ti-45%Al合金界面形态及微观结构演化的相场模拟[J].中南大学学报,2006,37(5)：856-861
[119]李梅娥,杨根仓,周尧和.二元合金高速定向凝固过程的相场法数值模拟[J].物理学报,2005,54(1)：454-458
[120]王锦程,李俊杰,杨玉娟等.定向凝固界面形态演化及其稳定性的相场法研究[J].中国科学E,2008,38(1)：16-23
[121]Li Junjie, Wang Zhijun, Wang Yaqin, Wang Jincheng. Phase-field study of competitive dendritic growth of converging grains during directional solidification[J]. Acta Mater.,2011,60(4):1478-1493
[122]王雅琴,王锦程,李俊杰.定向倾斜枝晶生长规律及竞争行为的相场法研究[J].物理学报,2012,61(11)：118103
[123]庄建平,龙文元,方立高等.铝合金定向凝固过程枝晶生长的相场法模拟[J].南昌航空工业学院学报,2005,19(3)：4-9
[124]庄建平,龙文元,方立高等.过冷熔体定向凝固过程枝晶生长的相场法模拟[J].热加工工艺,2005,(12)：5-7
[125]Ode M, LEE J S, Kim S G et al. Phase-field Model for Solidification of Ternary Alloys[J]. ISIJ International,2000,40(9):870-874
[126]Kobayashi H, Ode M, Kim S G et al. Phase-field model for solidification of ternary alloys coupled with thermodynamic database[J]. Acta Mater.,2003,48:689-694
[127]Qin R S, Wallach E R, Thomoson R C. A Phase-field Model for the Solidification of Multicomponent and Multiphase alloys[J]. J. of Cryst. Growth,2005,279(8):163-169
[128]Alexandre F F, Leonardo O F. Microsegregation in Fe-C-P Ternary Alloys Using a Phase-Field Model[J]. J. of the Braz. Soc. of Mech. Sci. & Eng.,2009,16(3):173-180
[129]Garcke H, Stinner B. Multicomponent alloy solidification: Phase-field modeling and simulations[J]. Phys. Rev. E,2005,71: 041609(06)
[130]Cha P R, Yeon D H, Yoon J K. Phase-field model for multicomponent alloy solidification[J]. J. of Cryst. Growth,2005,274:281-293
[131]Zhang R J, He Z, Wang X Y et al. A Simple Model for Dendrite Arm Coarsening During Solidification in Multicomponent Alloys. J. of Mater. Sci.,2004,39(20):6379-6381
[132]Zhang R J, Jing T, Jie W Q. Phase-field Simulation of Solidification in Multicomponent Alloys Coupled with thermodynamic and diffusion mobility databases[J]. Acta Mater.,2006,54(8):8055-8061
[133]Zhang R J, Li M, Allison J. Phase field study for the influence of solute interactions on solidification process in multicomponent alloys[J]. Comput. Mater. Sci.,2010,47:832-837.
[134]龙文元,蔡启舟,魏伯康等.相场法模拟多元合金过冷熔体中的枝晶生长[J].物理学报,2006,55(3)：1341-1345
[135]Long Y Q, Liu P, Liu Y, Zhang W M, Pan J S. Simulation of Recrystallization Grain Growth During Re-aging Process in the Cu-Ni-Si Alloy Based on Phase Field Model[J]. Materials Letters,2008, 62:3039-3042
[136]袁栋世,王锦程,杨根仓.非对角扩散项对三元合金枝晶生长影响的相场法研究[J].铸造技术,2008,29(4)：461-465
[137]崔海霞,张玉妥,孙强等Fe-Mn-C三元合金等温凝固过程枝晶生长的相场法模拟[J].铸造设备研究,2008,(6)：39-41
[138]Wang Gang, Zeng De-chang, Liu Zhong-wu. Phase field calculation of interface mobility in a ternary alloy[J]. Trans. Nonferrous Met. Soc. China,2012,22:1711-1716
[139]Diepers H J et al. Solidification Processing[M]. Sheffield, Ranmoon House,1997
[140]Diepers H J, Beckermann C, Steinbach D M. Simulation of Convection and Ripening in a Binary Alloy Mush Using the Phase-field Method[J]. Acta Mater.,1999,47(13):3663-3678
[141]Diepers H J, Ma D, Steinbach D M. History effects during the selection of primary dendrite spacing. Comparison of phase-field simulations with experimental observations[J]. J. Crystal Growth,2002, 237:149-153
[142]Tonhardt R, Amberg G. Phase-field simulation of dendritic growth in a shear flow[J]. J. Cryst. Growth,1998,194(7):406-425
[143]Tonhardt R. Convective Effects on Dendritic Solidification[D]. Royal Institute of Technology, Sweden,1999
[144]Tonhardt R, Amberg G. Dendritic Growth of Randomly Oriented Nuclei in a Shear Flow[J]. J Cryst. Growth.2000,213:161-187
[145]Beckermann C, Diepers H J, Steinbach I et al. Modeling Melt Convection in Phase-Field Simulations of Solidification[J]. J. of Comput. Phys.,1999,154:468-496
[146]Tong X, Beckermann C, Karma A. Velocity and Shape Selection of Dentritic Crystals in a Forced Flow. Phys. Rev. E,2000,61:49-53
[147]Tong X, Beckermann C, Karma A et al. Phase-filed Simulations of Dendritic Crystal Growth in Forced Flow. Phys. Rev. E,2001,63: 061601(16)
[148]Anderson D M, Mcfadden G B, Wheeler A A. A phase-field model of solidification with convection[J]. Physica D,2000,135:175-194
[149]Anderson D M, Mcfadden G B, Wheeler A A. A phase-field model with convection:Sharp-Interface Asymptotics[J]. Physica D,2001,151: 305-331
[150]Li Q, Beckermann C. Modeling of free dendritic growth of succinonitrile-acetone alloys with thermosolutal melt convection[J]. J Cryst. Growth.2002,236:482-498
[151]Jeong J H, Goldenfeld N, Dantzig J A. Phase Field Model for Three Dimensional Dendritic Growth with Fluid Flow[J], Phys. Rev. E,2001, 64:041602(14)
[152]Jeong J H, Goldenfeld N, Dantzig J A. Phase Field Model for Three-dimension Dendritic Growth with Fluid Flow[J]. Metall. Mater. Transit. A 2003,34:459-470
[153]Lan C W, Hsu C M, Liu C C. Efficient Adaptive Phase Field Simulation of Dendritic Growth in a Forced Flow[J]. J. Cryst. Growth, 2002,241:379-386
[154]Lan C W, Hsu C M, Liu C C et al. Adaptive Phase Field Simulation of Dendritic Growth in a Forced Flow at Various Supercoolings[J]. Phys. Rev. E,2002,65:061601(12)
[155]Lan C W, Liu C C, Hsu C M. An Adaptive Finite Volume Method for Incompressible Heat Flow Problems in Solidification[J]. J. Comput. Phys.,2002,178:464-491
[156]Lan C W, Lee M H, Chuang M H et al. Phase field modeling of convective and morphological instability during directional solidification of an alloy[J]. J. Cryst. Growth,2006,295:202-208
[157]陈玉娟,陈长乐.相场法模拟对流速度对上游枝晶生长的影响[J].物理学报,2008,57(7)：4585-4589
[158]陈玉娟,陈长乐,邱文旭等.强迫对流下各相场参数对枝晶生长的影响[J].特种铸造及有色合金,2008,28(7)：513-515
[158]袁训锋,丁雨田.强制对流作用下纯Ni枝晶生长的相场法研究[J].铸造技术,2012,33(8)：913-917
[159]袁训锋,丁雨田.强制对流作用下多枝晶生长的相场法研究[J].材料工程,2011,(10)：5-10
[160]袁训锋,丁雨田,郭廷彪.强制对流作用下镁合金枝晶生长的相场法数值模拟[J].中国有色金属学报,2010,20(8)：1474-1480
[161]陈志,陈长乐,郝丽梅.有/无强制流动下定向凝固界面形貌的数值模拟研究[J].稀有金属材料与工程,2010,39(12)：2117-2121
[162]徐祖耀.相变原理[M].科学出版社,1988
[163]田卫星.纯金属凝固过程枝晶生长的相场法研究[D].山东大学博士毕业论文.济南,2007.6
[164]Mullins W W. Morphological Stability of a Particle Growing by Diffusion of Heat Flow[J]. App. Phys. Lett.,1963,34:323-329
[165]常国威,王建中.金属凝固过程中的晶体生长与控制[M].冶金工业出版社,2002
[166]王献孚,熊鳌魁.高等流体力学[M].武汉：华中科技大学出版社,2003
[166]刘全坤.材料成型基本原理[M].北京：机械工业出版社,2004
[167]栾贻国.材料加工中的计算机应用基础[M].哈尔滨：哈尔滨工业大学出版社,2005
[168]陈铁.科学数据图显分析软件Tecplot[J].软件世界,1995,(06)： 28-28
[169]田薇,孙秦,贺海鸥.基于Tecplot的飞行器有限元图形处理技术[J].机械设计与制造,2006,(2)：125-126
[170]Aoyama T, Kuribayashi K. Influence of undercooling on solid/liquid interface morphology in semiconductors[J]. Acta Mater.,2000,48(14): 3739-3744
[171]Henry S, Jarry P, Jouneau P H, Rappaz M. Electron backscattered diffraction investigation of the texture of feathery crystals in aluminum alloys[J]. Metall. Mater. Trans. A,1997,28:207-213
[172]Henry S, Jarry P, Rappaz M.<110> dendrite growth morphologies in aluminum alloys[J]. Metall. Mater. Trans. A,1998,29:2807-2817
[173]Henry S, Minghetti T, Rappaz M. Dendrite growth morphologies in aluminum alloys[J]. Acta Mater.,1998,46:6431-6443
[174]Ahmad N A, Wheeler A A, Boettinger W J et al. Solute trapping and solute drag in a phase-field model of rapid solidification[J]. Phys. Rev. E,1998,58(3):3436-3450
[2]柳百成,荆涛.铸造工程的模拟仿真与质量控制[M].北京：机械工业出版社,2001
[3]周尧和,胡壮麒等.凝固技术[M].北京：机械工业出版社,1998
[4]Boettinger W J, Coriell S R, Greer A L et al. Solidification microstructures:recent developments, future directions[J]. Acta Mater., 2000,48:43-70
[5]柳百成.铸件充型凝固过程国内外研究进展[J].铸造,1999,48(8)：78-81
[6]Liu B C. Progress in Solidification Modeling of Cast Iron in China[J]. International J. of Cast Metals Research,1999,11(5),259-266
[7]王延露.基于MAGMAsoft的铸造充型凝固过程分析与研究[D].兰州理工大学硕士学位论文.兰州,2006,6
[8]刘小刚.Al-Cu合金等温凝固的相场法模拟[D].沈阳工业学院硕士学位论文.沈阳,2003.3
[9]王君.纯物质凝固过程枝晶生长现象的研究和模拟[D].昆明理工大学博士学位论文.昆明,2002.6
[10]D.罗伯.计算材料学[M].北京：化学工业出版社.2003
[11]Zhu P P, Smith R W. Dynamic simulation of crystal growth by Monte Carlo method-I:Model description and kinetics[J]. Acta Metall.,1992, 40(2):350-361
[12]Wolfram S. Statistical mechanics of cellular automata[J]. Rev. of Modern phys.,1983,55:601-644.
[13]Wolfram S. Universality and complexity in cellular automata[J]. Physica D,1984,10:1-35.
[14]Wolfram S. Cellular automata as models of complexity [J], Nature, 1984:311:419-448.
[15]Rappaz M, Gandin C A. Probabilostic Modelling of Microstructure Formation in Solidification Processes[J]. Acta Metall.,1993,41(2): 350-360
[16]Gandin C A, Rappaz M. Coupled Finite Element-Cellular Automation Model for The Prediction of Dendritic Grain Structures in Solidification Processes[J]. Acta Metall.,1994,42(7):2233-2246
[17]Brown S G R. Simulation of diffusional composite growth using the cellular automaton finite difference(CAFD) method[J]. J. of Mater. Sci. 1998,33(19):4769-4773
[18]Jarvis D J, Brown S G R. Modelling of non-equilibrium solidification in ternary alloys:comparison of 1D,2D, and 3D cellular automaton finite difference simulations[J]. Mater. Sci. and Techn.,2000,16: 1420-1424
[19]Oldfield W.A Quantitative Approach to casting Solidification[J]. Freezing of Cast Iron. ASM Trans.1966,59(2):945-960
[20]Hunt J D. Steady state columnar and equiaxed growth of dendrites and eutectic[J]. Mater. Sci. Eng.,1984,65:75-89
[21]Wang C Y, Beckermann C. Equiaxed Dendritic Solidification with Convection-I:Multiscale/Multiphase Modeling[J]. Metall. Trans. A, 1996,27:2754-2764
[22]Wang C Y, Beckermann C. Equiaxed Dendritic Solidification with Convection-Ⅱ:Numerical Simulation for an Al-4wt%Cu Alloy[J]. Metall. Trans. A,1996,27:2765-2783
[23]Wang C Y, Beckermann C. Equiaxed Dendritic Solidification with Convection-Ⅲ:Comparisons with NH4Cl-H2O Experiments[J]. Metall. Trans. A,1996,27:2784-2795
[24]Karma A, Rappel W J. Phase-field Method for Computationally Efficient Modeling of Solidification with Arbitrary Interface Kinetics[J]. Phys. Rev. E,1996,53(4):3017-3020
[25]Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth[J]. Phys. Rev. Lett.,1996,77(10):4050-4053
[26]Karma A, Rappel W J. Quantitative phase-field modeling of dendritic growth in two and three dimensions[J]. Phys. Rev. E,1998,57(4): 4323-4349
[27]于艳梅.过冷熔体枝晶生长的相场法模拟[D].西北工业大学博士毕业论文.西安,2002.6
[28]于艳梅,杨根仓,赵达文等.凝固微观组织数值模拟的研究进展[J].特种铸造及有色合金,2002,(3)：30-33
[29]Asta M, Beckermann C, Karma A et al. Solidification microstructures and solid-state parallels:recent developments, future directions[J]. Acta Mater.,2009,57:941-971
[30]Jeong J H, Goldenfeld N, Dantzig J A. Phase field model for three-dimensional dendritic growth with fluid flow[J]. Phys. Rev. E,2001,64:041602
[31]Provatas N, Goldenfeld N, Dantzig J. Efficient computation of dendritic microstructures using adaptive mesh refinement[J]. Phys. Rev. Lett.,1998,80:3308
[32]Lan C W, Hsu C M, Liu C C. Efficient adaptive phase field simulation of dendritic growth in a forced flow at low suppercooling[J]. J. Cryst. Growth,2002,241:379-386
[33]Lan C W, Hsu C M, Liu C C et al. Adaptive phase field simulation of dendritic growth in a forced flow at various supercoolings[J]. Phys. Rev. E,2002,65:061601.
[34]Lan C W, Liu C C, Hsu C M. An adaptive finite volume method forincompressible heat flow problems in solidification[J]. J. Comput. Phys,2002,178:464-497.
[35]熊玉华,刘林.铸件凝固组织形成的计算机模拟.材料导报,1999,13(3)：20-21
[36]Gandin C A, Jalanti T, Rappaz M. Modeling of Dendritic Grain Structure. Modeling of Casting, Welding and Advanced Solidification Processes Ⅷ.1998:363-374
[37]Zhu M F, Hong C P. A modified cellular automaton model for the simulation of dendritic growth in solidification of alloys[J]. ISIJ Int. 2001,41(5):436-445
[38]Zhu M F, Kim J M, Hong C P. Modeling of globular and dendritic structure evolution in solidificat ion of an Al-7mass%Si alloy[J]. ISIJ Int.,2001,41(9):992-998
[39]Cho S H. Contribution of nucleation process to grain formation in calculating solidification microstructure by CA-DFD[J]. Current Advance in Materials and Process,2002,15, (1):46-50
[40]李殿中,苏仕方,徐雪华等.镍基合金叶片凝固过程微观组织模拟及工艺优化研究[J].铸造,1997,46(8)：1-7
[41]王同敏,金俊泽,郑贤淑.金属凝固过程微观模拟的两个辅助算法[J].铸造,2000,49(8)：484-488
[42]许庆彦,柳百成.采用Cellular Aut omaton法模拟铝合金的微观组 织[J].中国机械工程.2001,12(3)：328-331
[43]Spittle J A, Brown S G R. Computer Simulation of the Effects of Alloy Variables on the Grain Structures of Castings[J]. Acta Metall.,1989, 37(7):1803-1810
[44]Spittle J A, Brown S G R. A Computer Simulation of the Influence of Processing Conditions on the cast Grain Structures[J]. J. Mater. Sci., 1989,23:1777-1781
[45]Zhu P P, Smith R W. Dynamic Simulation of Crystal Growth by Monte Carlo Method Ⅱ. Ingot Microstructures[J]. Acta Metall.,1992, 40(12):3369-3379
[46]金俊泽,王宗廷,郑贤淑等.金属凝固组织形成的仿真研究[J].金属学报,1998,34(9)：928-932
[47]丁雨田,王海南,许广济等.定向凝固微观组织的Monte Carlo模拟[J].甘肃工业大学学报,2001,27(9)：15-18
[48]丁雨田.热型连铸凝固过程微观组织形成的数值模拟[D].兰州理工大学博士学位论文.兰州,2005.6
[49]Maxwell I, Hellawell A. A simple model for grain refinement during solidification[J]. Acta Metall.,1975,23:229-237
[50]Thevoz P H, Desbilles J L, Rappaz M. Modeling of equiaxed microstructure formation in casting[J]. Metall. Trans. A,1989,20(2): 311-322
[51]Saito Y, Goldbeck-Wood G, Muller-Krumbhaar H. Numerical simulation of dendritic growth[J]. Phys. Rev. A,1988,38(4): 2148-2153.
[52]Fix G J. Phase field for free boundary problems. In:Fasono A, Primicerio M, Pitman eds. Free Boundary Problems:Theory and Applications[J]. London,1983:580-589
[53]Collins J B, Levine H. Diffuse interface model of diffusion limited crystal growth[J]. Phys. Rev. B,1985,31(9):6119-6122
[54]Langer J S. Directions in condensed physics, eds. Grinstein G and Maxenko G (World Science,1986):164-168
[55]Caginalp G, Fife P C. Phase field model for interfacial boundaries[J]. Phys. Rev. B,1986,34(7):4940-4943
[56]Caginalp G, Fife P C. Dynamics of layered interfaces arising from phase boundaries[J]. SIAM J. of Appl. Math.,1988,48(3):506-510
[57]Fife P C, Gill S G. The phase-field description of mushy zones[J]. Phys. D,1989,35:267-275
[58]Fife P C, Gill S G. Phase transition mechanisms for the phase-field model under internal heating[J]. Phys. Rev. A,1991,43(2):843-851
[59]Penrose O, Fife P C. Thermodynamically Consistent Models of Phase-field Type for the Kinetics of Phase Transitions[J]. Phys. D,1990, 43:44-62
[60]Kobayashi R. Modeling and Numerical Simulation of Dendritic Crystal Growth[J]. Phys. D,1993,63:410-423
[61]Wang S L, Sekerka R F, Wheeler A A et al. Thermodynamically-Consistent Phase-Field Models for Solidification[J]. Phys. D,1993,69:189-200
[62]Wheeler A A, Murrary B T, Schaefer R J. Computation of dendrites using a phase field model[J]. Phys. D,1993,66(10):243-262
[63]Murrary B T, Wheeler A A, Glicksman M E. Simulation of experimentally observed dendritic growth behavior using a phase-field model[J]. J. Cryst. Growth,1995,154:386-400
[64]Karma A, Rappel W J. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics[J]. Phys. Rev. E,1996,53(4):3017-3020
[65]Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth[J]. Phys. Rev. Lett.,1996,77(10):4050-4053
[66]Karma A, Rappel W J. Phase-field simulation of three-dimensional dendrites:is microscopic solvability theory correct[J]. J. Cryst. Growth. 1997,174:54-64
[67]Karma A, Rappel W J. Quantitative phase-field modeling of dendritic growth in two and three dimensions[J]. Phys. Rev. E,1998,57(4): 4323-4349
[68]张玉妥,李殿中,李依依等.用相场方法模拟纯物质等轴枝晶生长[J].金属学报,2000,36(6)：589-591
[69]张玉妥,李殿中,庞维诚等.用相场方法模拟镍的枝晶生长[J].材料研究学报,2002,16(5)：541-543
[70]于艳梅,杨根仓,赵达文等.相场法模拟过冷熔体枝晶生长的界面厚度参数的取值[J].自然科学进展,2001,11(11)：1192-1197
[71]于艳梅,杨根仓,赵达文等.过冷熔体中枝晶生长的相场法数值模 拟[J].物理学报,2001,50(12)：2423-2428
[72]李俊杰,王锦程,杨根仓.相场法模拟界面能各向异性对枝晶生长行为的影响[J].自然科学进展,2005,15(11)：1312-1317
[73]赵代平,荆涛.用捕获液态改进的相场方法模拟三维枝晶生长[J].金属学报,2002,38(12)：1238-1240
[74]Zhu Changsheng, Liu Baicheng, Jing Tao et al. Dependence of dendritic side-branches on parameters in phase-field simulation[J]. Mater. Trans.,2005,46(1):15-19
[75]朱昌盛.相场法凝固微观组织计算机模拟仿真系统的研究与实现[D].兰州理工大学博士学位论文.兰州,2006.11
[76]Wheeler A A, Boettinger W J, McFadden G B. Phase-field model for isothermal phase transition in binary alloy[J]. Phys. Rev. E,1992,45(10): 7424-7439
[77]Warren J A, Boettinger W J. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method[J]. Acta metal. Mater,1995,43(2):689-703
[78]Boettinger W J, Warren J A. The Phase field method:simulation of alloy dendritic solidification during recalescence[J]. Metall. Mater. Trans. A,1996,27A (3):657-668
[79]Wheeler A A, Boettinger W J, McFadden G B. Phase field model of trapping during solidification[J]. Phys. Rev. E,1993,47(4): 1893-1909
[80]Boettinger W J, Wheeler A A, Murray B T, McFadden G B. Prediction of solute trapping at high solidification rates using a diffuse interface phase field theory of alloy solidification[J]. Mater. Sci. and Eng.,1994, 178 A (10):217-223
[81]Conti M. Solidification of binary alloys:Thermal effects studied with the phase-field model[J]. Phys. Rev. E,1997,55(1):765-771
[82]Conti M. Phase and solute fields across the solid-liquid interface of a binary alloy[J]. Phys. Rev. E,1999,55(1):1913-1920
[83]Kim S G, Kim W T, Suzuki T. Interfacial compositions of solid and liquid in a phase-field Model with Finite interface Thickness for Isothermal Solidification in Binary Alloys[J]. Phys. Rev. E,1998,58(3): 3316-3323
[84]Kim S G, Kim W T, Suzuki T. Phase-field model for binary alloys[J]. Phys. Rev. E,1999,60(6):7186-7197
[85]Suzuki T, Ode M, Kim S G, et al. Phase-field model of dendritic growth[J]. J. Crystal Growth,2002,237-239:125-131
[86]Lee J S, Suzuki T. Numerical simulation of isothermal dendritic growth by phase-field model[J]. ISIJ Int.,1999,39(3):246-252
[87]Kim S G, Kim W T, Lee J S, et al. Large scale simulation of dendritic growth in pure undercooled melt by phase-field model[J]. ISIJ Int.,1999, 39 (4):335-340
[88]Loginova Ⅰ, Amberg G and Agern J. Phase-field simulation of non-isothermal binary alloy solidification[J]. Acta mater.,2001,49: 573-581
[89]Gonzalez-Cinca R, Ramirez-Piscina L, Casademunt J et al. Phase-field simulations and experiments of faceted growth in liquid crystals[J]. Phys. D,1996,99:359-368
[90]Mcfadden G B, Coriell S R, Sekerka R F. Effect of surface free energy anisotropy on dendrite tip shape[J]. Acta Mater.,2000,48:3177-3181
[91]Boettinger W J, Coriell S R, Greer A L et al. Solidification microstructures:recent developments, future directions[J]. Acta Mater. 2000,48:43-70
[92]Eggleston J J, Mcfadden G B, Voorhees P W. A phase-field model for highly anisotropic interfacial energy[J]. Physica D,2001,150:91-103
[93]Debierre J M, Karma A, Celestini F et al. Phase-field approach for faceted solidification[J]. Phys. Rev. E,2003,68(4):1604-1611
[94]Uehara T, Sekerka R F. Phase field simulations of faceted growth for strong anisotropy of kinetic coefficient[J]. J. Cryst. Growth,2003,254: 251-261
[95]Mullis A M. Quantification of mesh induced anisotropy effects in the phase-field method[J]. Comput. Mater. Sci.,2006,36:345-353
[96]Phase field modeling of dendritic growth with high anisotropy[J]. J. Cryst. Growth,2005,275:355-360
[97]Suzuki T, Kim S G, Kim W T. Two-dimensional facet crystal growth of silicon from undercooled melt of Si-Ni alloy[J]. Mater. Sci. Enging. A, 2007,449:99-104
[98]Suwa Y, Saito Y, Onodera H. Three-dimensional phase field simulation of the anisotropy in grain-boundary mobility on growth kinetics and morphology of grain structure[J]. Comput. Mater. Sci.,2007, 40:40-50
[99]Haxhimali T, Karma A, Gonzales F et al. Orientation selection in dendritic evolution[J]. Nature Materials,2006,5:660-664
[100]Clyne T W. Heat flow in controlled directional solidification of metals[J]. J. Cryst. Growth,1980,50:684-690
[101]Sakane J, Lib Q, Evans J W. Mathematical modeling of meniscus profile and melt flow in electromagnetic casters[J]. Metall. Trans. B, 1988,19:397-408
[102]陶文铨.数值传热学[M].西安：西安交通大学出版社,1988
[103]Conti M. Solute trapping in directional solidification at high speed:A one-dimensional study with the phase-field model[J]. Phys. Rev. E,1997, 56:3717-3720
[104]Conti M. Oscillatory instabilities in rapid directional solidification[J]. Phys. Rev. E,1997,56:6267-6270
[105]Boettinger W J, Warren J A. Simulation of the cell to plane front transition during directional solidification at high velocity[J]. J. Cryst. Growth,1999,200:583-591
[106]Mullins W W, Sakerka R F. Stability of a Planar Interface During Solidification of a dilute Alloy[J]. J. Applied Phys.,1964,35:444-451
[107]Kim S G, Kim W T. Phase-field modeling of rapid solidification[J]. Mater. Sci. Eng. A,2001,304:281-286
[108]Diepers H J, Ma D, Steinbach I. History effects during the selection of primary dendrite spacing. Comparison of phase-field simulations with experimental observations[J]. J. Cryst. Growth,2002,237:149-153
[109]Takaki T, Fukuoka T, Tomita Y. Phase-field simulation during directional solidification of a binary alloy using adaptive finite element method[J]. J. Cryst. Growth,2005,283:263-278
[110]Filho R N, Kosterlitz J M, Granato E. Pattern selection in a phase field model for directional solidification[J]. Physica A,2005,354: 333-343
[111]Plapp M. Three-dimensional phase-field simulations of directional solidification[J]. J. Cryst. Growth,2007,303:49-57
[112]Steinbach I. Effect of interface anisotropy on spacing selection in constrained dendrite growth[J]. Acta Mater.,2008,56:4965-4971
[113]Amoorezaei M, Gurevich S, Provatas N. Spacing characterization in Al-Cu alloys directionally solidified under transient growth conditions[J]. Acta Mater.,2010,58:6115-6124
[114]Amoorezaei M, Gurevich S, Provatas N. Orientation selection in solidification patterning[J]. Acta Mater.,2012,60:657-663
[115]于艳梅.过冷熔体枝晶生长的相场法模拟[D].西北工业大学博士学位论文.西安,2002.6
[116]Yu Yan-Mei, Yang Gen-Cang, Zhao Da-Wen et al. Influence of isothermal approximation on the phase-field simulation of directional growth in undercooled melt[J]. Chinese Physics,2003,12(02):211-217
[117]Guo Jingjie, Li Xinzhong, Su Yanqing et al. Phase-field simulation of structure evolution at high growth velocities during directional solidification of Ti55Al45 alloy[J]. Intermetallics,2005,13:275-279
[118]李新中,苏彦庆,郭景杰等Ti-45%Al合金界面形态及微观结构演化的相场模拟[J].中南大学学报,2006,37(5)：856-861
[119]李梅娥,杨根仓,周尧和.二元合金高速定向凝固过程的相场法数值模拟[J].物理学报,2005,54(1)：454-458
[120]王锦程,李俊杰,杨玉娟等.定向凝固界面形态演化及其稳定性的相场法研究[J].中国科学E,2008,38(1)：16-23
[121]Li Junjie, Wang Zhijun, Wang Yaqin, Wang Jincheng. Phase-field study of competitive dendritic growth of converging grains during directional solidification[J]. Acta Mater.,2011,60(4):1478-1493
[122]王雅琴,王锦程,李俊杰.定向倾斜枝晶生长规律及竞争行为的相场法研究[J].物理学报,2012,61(11)：118103
[123]庄建平,龙文元,方立高等.铝合金定向凝固过程枝晶生长的相场法模拟[J].南昌航空工业学院学报,2005,19(3)：4-9
[124]庄建平,龙文元,方立高等.过冷熔体定向凝固过程枝晶生长的相场法模拟[J].热加工工艺,2005,(12)：5-7
[125]Ode M, LEE J S, Kim S G et al. Phase-field Model for Solidification of Ternary Alloys[J]. ISIJ International,2000,40(9):870-874
[126]Kobayashi H, Ode M, Kim S G et al. Phase-field model for solidification of ternary alloys coupled with thermodynamic database[J]. Acta Mater.,2003,48:689-694
[127]Qin R S, Wallach E R, Thomoson R C. A Phase-field Model for the Solidification of Multicomponent and Multiphase alloys[J]. J. of Cryst. Growth,2005,279(8):163-169
[128]Alexandre F F, Leonardo O F. Microsegregation in Fe-C-P Ternary Alloys Using a Phase-Field Model[J]. J. of the Braz. Soc. of Mech. Sci. & Eng.,2009,16(3):173-180
[129]Garcke H, Stinner B. Multicomponent alloy solidification: Phase-field modeling and simulations[J]. Phys. Rev. E,2005,71: 041609(06)
[130]Cha P R, Yeon D H, Yoon J K. Phase-field model for multicomponent alloy solidification[J]. J. of Cryst. Growth,2005,274:281-293
[131]Zhang R J, He Z, Wang X Y et al. A Simple Model for Dendrite Arm Coarsening During Solidification in Multicomponent Alloys. J. of Mater. Sci.,2004,39(20):6379-6381
[132]Zhang R J, Jing T, Jie W Q. Phase-field Simulation of Solidification in Multicomponent Alloys Coupled with thermodynamic and diffusion mobility databases[J]. Acta Mater.,2006,54(8):8055-8061
[133]Zhang R J, Li M, Allison J. Phase field study for the influence of solute interactions on solidification process in multicomponent alloys[J]. Comput. Mater. Sci.,2010,47:832-837.
[134]龙文元,蔡启舟,魏伯康等.相场法模拟多元合金过冷熔体中的枝晶生长[J].物理学报,2006,55(3)：1341-1345
[135]Long Y Q, Liu P, Liu Y, Zhang W M, Pan J S. Simulation of Recrystallization Grain Growth During Re-aging Process in the Cu-Ni-Si Alloy Based on Phase Field Model[J]. Materials Letters,2008, 62:3039-3042
[136]袁栋世,王锦程,杨根仓.非对角扩散项对三元合金枝晶生长影响的相场法研究[J].铸造技术,2008,29(4)：461-465
[137]崔海霞,张玉妥,孙强等Fe-Mn-C三元合金等温凝固过程枝晶生长的相场法模拟[J].铸造设备研究,2008,(6)：39-41
[138]Wang Gang, Zeng De-chang, Liu Zhong-wu. Phase field calculation of interface mobility in a ternary alloy[J]. Trans. Nonferrous Met. Soc. China,2012,22:1711-1716
[139]Diepers H J et al. Solidification Processing[M]. Sheffield, Ranmoon House,1997
[140]Diepers H J, Beckermann C, Steinbach D M. Simulation of Convection and Ripening in a Binary Alloy Mush Using the Phase-field Method[J]. Acta Mater.,1999,47(13):3663-3678
[141]Diepers H J, Ma D, Steinbach D M. History effects during the selection of primary dendrite spacing. Comparison of phase-field simulations with experimental observations[J]. J. Crystal Growth,2002, 237:149-153
[142]Tonhardt R, Amberg G. Phase-field simulation of dendritic growth in a shear flow[J]. J. Cryst. Growth,1998,194(7):406-425
[143]Tonhardt R. Convective Effects on Dendritic Solidification[D]. Royal Institute of Technology, Sweden,1999
[144]Tonhardt R, Amberg G. Dendritic Growth of Randomly Oriented Nuclei in a Shear Flow[J]. J Cryst. Growth.2000,213:161-187
[145]Beckermann C, Diepers H J, Steinbach I et al. Modeling Melt Convection in Phase-Field Simulations of Solidification[J]. J. of Comput. Phys.,1999,154:468-496
[146]Tong X, Beckermann C, Karma A. Velocity and Shape Selection of Dentritic Crystals in a Forced Flow. Phys. Rev. E,2000,61:49-53
[147]Tong X, Beckermann C, Karma A et al. Phase-filed Simulations of Dendritic Crystal Growth in Forced Flow. Phys. Rev. E,2001,63: 061601(16)
[148]Anderson D M, Mcfadden G B, Wheeler A A. A phase-field model of solidification with convection[J]. Physica D,2000,135:175-194
[149]Anderson D M, Mcfadden G B, Wheeler A A. A phase-field model with convection:Sharp-Interface Asymptotics[J]. Physica D,2001,151: 305-331
[150]Li Q, Beckermann C. Modeling of free dendritic growth of succinonitrile-acetone alloys with thermosolutal melt convection[J]. J Cryst. Growth.2002,236:482-498
[151]Jeong J H, Goldenfeld N, Dantzig J A. Phase Field Model for Three Dimensional Dendritic Growth with Fluid Flow[J], Phys. Rev. E,2001, 64:041602(14)
[152]Jeong J H, Goldenfeld N, Dantzig J A. Phase Field Model for Three-dimension Dendritic Growth with Fluid Flow[J]. Metall. Mater. Transit. A 2003,34:459-470
[153]Lan C W, Hsu C M, Liu C C. Efficient Adaptive Phase Field Simulation of Dendritic Growth in a Forced Flow[J]. J. Cryst. Growth, 2002,241:379-386
[154]Lan C W, Hsu C M, Liu C C et al. Adaptive Phase Field Simulation of Dendritic Growth in a Forced Flow at Various Supercoolings[J]. Phys. Rev. E,2002,65:061601(12)
[155]Lan C W, Liu C C, Hsu C M. An Adaptive Finite Volume Method for Incompressible Heat Flow Problems in Solidification[J]. J. Comput. Phys.,2002,178:464-491
[156]Lan C W, Lee M H, Chuang M H et al. Phase field modeling of convective and morphological instability during directional solidification of an alloy[J]. J. Cryst. Growth,2006,295:202-208
[157]陈玉娟,陈长乐.相场法模拟对流速度对上游枝晶生长的影响[J].物理学报,2008,57(7)：4585-4589
[158]陈玉娟,陈长乐,邱文旭等.强迫对流下各相场参数对枝晶生长的影响[J].特种铸造及有色合金,2008,28(7)：513-515
[158]袁训锋,丁雨田.强制对流作用下纯Ni枝晶生长的相场法研究[J].铸造技术,2012,33(8)：913-917
[159]袁训锋,丁雨田.强制对流作用下多枝晶生长的相场法研究[J].材料工程,2011,(10)：5-10
[160]袁训锋,丁雨田,郭廷彪.强制对流作用下镁合金枝晶生长的相场法数值模拟[J].中国有色金属学报,2010,20(8)：1474-1480
[161]陈志,陈长乐,郝丽梅.有/无强制流动下定向凝固界面形貌的数值模拟研究[J].稀有金属材料与工程,2010,39(12)：2117-2121
[162]徐祖耀.相变原理[M].科学出版社,1988
[163]田卫星.纯金属凝固过程枝晶生长的相场法研究[D].山东大学博士毕业论文.济南,2007.6
[164]Mullins W W. Morphological Stability of a Particle Growing by Diffusion of Heat Flow[J]. App. Phys. Lett.,1963,34:323-329
[165]常国威,王建中.金属凝固过程中的晶体生长与控制[M].冶金工业出版社,2002
[166]王献孚,熊鳌魁.高等流体力学[M].武汉：华中科技大学出版社,2003
[166]刘全坤.材料成型基本原理[M].北京：机械工业出版社,2004
[167]栾贻国.材料加工中的计算机应用基础[M].哈尔滨：哈尔滨工业大学出版社,2005
[168]陈铁.科学数据图显分析软件Tecplot[J].软件世界,1995,(06)： 28-28
[169]田薇,孙秦,贺海鸥.基于Tecplot的飞行器有限元图形处理技术[J].机械设计与制造,2006,(2)：125-126
[170]Aoyama T, Kuribayashi K. Influence of undercooling on solid/liquid interface morphology in semiconductors[J]. Acta Mater.,2000,48(14): 3739-3744
[171]Henry S, Jarry P, Jouneau P H, Rappaz M. Electron backscattered diffraction investigation of the texture of feathery crystals in aluminum alloys[J]. Metall. Mater. Trans. A,1997,28:207-213
[172]Henry S, Jarry P, Rappaz M.<110> dendrite growth morphologies in aluminum alloys[J]. Metall. Mater. Trans. A,1998,29:2807-2817
[173]Henry S, Minghetti T, Rappaz M. Dendrite growth morphologies in aluminum alloys[J]. Acta Mater.,1998,46:6431-6443
[174]Ahmad N A, Wheeler A A, Boettinger W J et al. Solute trapping and solute drag in a phase-field model of rapid solidification[J]. Phys. Rev. E,1998,58(3):3436-3450