CAFE模型机理研究及应用

 摘 要：研究和分析了CAFE法模拟凝固过程微观组织的物理本质、数值计算方法。在CAFE模型中,形核密度用高斯分布来描述;枝晶尖端生长动力学用KGT模型进行计算;枝晶生长的择优取向是<100>方向,并可实现枝晶生长的竞争机制;FE与CA的耦合是通过FE节点和CA元胞之间的插值实现的。应用CAFE法模拟了易切削钢9SMn28的三维微观凝固组织,模拟结果与实验吻合较好。对易切削钢9SMn28进行了成分优化,并对优化结果进行了模拟,有效的改善了9SMn28的凝固组织。     

Prediction of grain structures in various solidification processes

Grain structure formation during solidification can be simulatedvia the use of stochastic models providing the physical mechanisms of nucleation and dendrite growth are accounted for. With this goal in mind, a physically based cellular automaton (CA) model has been coupled with finite element (FE) heat flow computations and implemented into the code3- MOS. The CA enmeshment of the solidifying domain with small square cells is first generated automatically from the FE mesh. Within each time-step, the variation of enthalpy at each node of the FE mesh is calculated using an implicit scheme and a Newton-type linearization method. After interpolation of the explicit temperature and of the enthalpy variation at the cell location, the nucleation and growth of grains are simulated using the CA algorithm. This algorithm accounts for the heterogeneous nucleation in the bulk and at the surface of the ingot, for the growth and preferential growth directions of the dendrites, and for microsegregation. The variations of volume fraction of solid at the cell location are then summed up at the FE nodes in order to find the new temperatures. This CAFE model, which allows the prediction and the visualization of grain structures during and after solidification, is applied to various solidification processes: the investment casting of turbine blades, the continuous casting of rods, and the laser remelting or welding of plates. Because the CAFE model is yet two-dimensional (2-D), the simulation results are compared in a qualitative way with experimental findings.     

Interaction of porosity with a planar solid/liquid interface

In this article, an investigation of the interaction between gas porosity and a planar solid/liquid (SL) interface is reported. A two-dimensional numerical model able to accurately track sharp SL interfaces during solidification of pure metals and alloys is proposed. The finite-difference method and a rectangular undeformed grid are used for computation. The SL interface is described through the points of intersection with the grid lines. Its motion is determined by the thermal and solute gradients at each particular point. Changes of the interface temperature because of capillarity or solute redistribution as well as any perturbation of the thermal and solute field produced by the presence of non-metallic inclusions can be computed. To validate the model, the dynamics of the interaction between a gas pore and a solidification front in metal alloys was observed using a state of the art X-ray transmission microscope (XTM). The experiments included observation of the distortion of the SL interface near a pore, real-time measurements of the growth rate, and the change in shape of the porosity during interaction with the SL interface in pure Al and Al-0.25 wt pct Au alloy. In addition, porosity-induced solute segregation patterns surrounding a pore were also quantified.     

Three-dimensional probabilistic simulation of solidification grain structures: Application to superalloy precision castings

A two-dimensional (2-D) probabilistic model, previously developed for the prediction of microstructure formation in solidification processes, is applied to thin section superalloy precision castings. Based upon an assumption of uniform temperature across the section of the plate, the model takes into account the heterogeneous nucleation which might occur at the mold wall and in the bulk of the liquid. The location and crystallographic orientation of newly nucleated grains are chosen randomly among a large number of sites and equiprobable orientation classes, respectively. The growth of the dendritic grains is modeled by using a cellular automaton technique and by considering the growth kinetics of the dendrite tips. The computed 2-D grain structures are compared with micrographie cross sections of specimens of various thicknesses. It is shown that the 2-D approach is able to predict the transition from columnar to equiaxed grains. However, in a transverse section, the grain morphology within the columnar zone differs from that of the experimental micrographs. For this reason, a three-dimensional (3-D) extension of this model is proposed, in which the modeling of the grain growth is simplified. It assumes that each dendritic grain is an octaedron whose half-diagonals, corresponding to the <100> crystallographic orientations of the grain, are simply given by the integral, from the time of nucleation to that of observation, of the velocity of the dendrite tips. All the liquid cells falling within a given octaedron solidify with the same crystallographic orientation of the parent nucleus. It is shown that the grain structures computed with this 3-D model are much closer to the experimental micrographie cross sections.     

Modeling on Dendrite Growth of Medium Carbon Steel during Continuous Casting

Dendrite growth is an important phenomenon during steel solidification. In the current paper, a numerical method was used to analyse and calculate the dendrite tip radius, dendrite growth velocity, liquid concentration, temperature gradient, cooling rate, secondary dendrite arm spacing, and the dendrite tip temperature in front of the solid/liquid (S/L) interface for the solidification process of medium carbon steels during continuous casting. The current model was well validated by published models and measurement data. The results show that in the continuous casting process, the dendrite growth rate is dominated by the casting speed. Dendrite growth rate, liquid concentration at the S/L interface, temperature gradient and cooling rate decrease with proceeding solidification and solid shell thickness growth, while other parameters such as dendrite tip radius, secondary dendrite arm spacing, and dendrite tip temperature in front of the S/L interface become larger with solidification progress and solid shell thickness growth. Parametric investigations were carried out. The effects of the stability coefficient, temperature gradient and casting speed on the micro‐structural parameters were discussed. Under the same conditions, higher casting speed promotes coarser secondary dendrite arm spacing and enlarges the dendrite tip radius, while decreasing temperature gradient, reducing the dendrite growth rate and making the solute distribute more uniform.     

A three-dimensional cellular automation-finite element model for the prediction of solidification grain structures

A three-dimensional (3-D) model for the prediction of dendritic grain structures formed during solidification is presented. This model is built on the basis of a 3-D cellular automaton (CA) algorithm. The simulation domain is subdivided into a regular lattice of cubic cells. Using physically based rules for the simulation of nucleation and growth phenomena, a state index associated with each cell is switched from zero (liquid state) to a positive value (mushy and solid state) as solidification proceeds. Because these physical phenomena are related to the temperature field, the cell grid is superimposed to a coarser finite element (FE) mesh used for the solution of the heat flow equation. Two coupling modes between the microscopic CA and macroscopic FE calculations have been designed. In a so-called “weak” coupling mode, the temperature of each cell is simply interpolated from the temperature of the FE nodes using a unique solidification path at the macroscopic scale. In a “full” coupling mode, the enthalpy field is also interpolated from the FE nodes to the CA cells and a fraction of solid increment is computed for each mushy cell using a truncated Scheil microsegregation model. These fractions of solid increments are then fed back to the FE nodes in order to update the new temperature field, thus accounting for a more realistic release of the latent heat (i.e., the solidification path is no longer unique). Special dynamic allocation techniques have been designed in order to minimize the computation costs and memory size associated with a very large number of cells (typically 10⁷ to 10⁸). The potentiality of the CAFE model is demonstrated through the predictions of typical grain structures formed during the investment casting and continuous casting processes.     

Simulation of Microsegregation in Multicomponent Alloys During Solidification

A phase‐field model is applied to the simulation of microsegregation and microstructure formation during the solidification of multicomponent alloys. The results of the one‐dimensional numerical simulations show good agreement with those from the Clyne–Kurz equation. Phase‐field simulations of non‐isothermal dendrite growth are examined. Two‐dimensional computation results exhibit different dendrites in multicomponent alloys for different solute concentrations. Changes in carbon concentration appear to affect dendrite morphology. This is due to a larger concentration and a lower equilibrium partition coefficient for carbon. On the other hand, changes in phosphorus concentration affect the dendrites and interface velocity in multicomponent alloys during solidification when phosphorus content is increased from 10^?3 mol% P. With additional manganese, the solidification kinetics slow down; dendrite morphology, however, is not affected. The potential of the phase‐field model for applications pertaining to solidification has been demonstrated through the simulations herein.     

基于元胞自动机有限元法对方坯凝固组织的数值模拟

 基于元胞自动机有限单元法(CAFE)对国内某钢厂220mm×220mm方坯的三维显微凝固组织进行模拟,分析了CAFE法模拟凝固过程显微组织的物理本质,对形核密度、枝晶尖端生长动力学、枝晶生长的择优取向以及FE与CA耦合的实现分别进行了探讨。用该方法对方坯的三维显微组织进行模拟,并对结晶器出口处方坯的角部温度、中心表面温度及坯壳厚度进行了计算。模拟结果表明：当拉速为0. 85m/min,浇铸温度为1535℃,浇钢过热度为30℃时,结晶器出口处方坯角部温度在850~950℃之间,中心表面温度在1050~1170℃之间,坯壳厚度在15mm左右,铸坯柱状晶发达,等轴晶比率较小。模拟的铸坯组织的等轴晶比例与低倍试验结果吻合较好,可以很好地预测方坯实际凝固组织。     

Growth of solutal dendrites: A cellular automaton model and its quantitative capabilities

A model based on the cellular automaton (CA) technique for the simulation of dendritic growth controlled by solutal effects in the low Péclet number regime was developed. The model does not use an analytical solution to determine the velocity of the solid-liquid (SL) interface as is common in other models, but solves the solute conservation equation subjected to the boundary conditions at the interface. Using this approach, the model does not need to use the concept of marginal stability and stability parameter to uniquely define the steady-state velocity and radius of the dendrite tip. The model indeed contains an expression for the stability parameter, but the process determines its value. The model proposes a solution for the artificial anisotropy in growth kinetics valid at zero and 45° introduced in calculations by the square cells and trapping rules used in previous CA formulations. It also introduces a solution for the calculation of local curvature, which eliminates mesh dependency of calculations. The model is able to reproduce qualitatively most of the dendritic features observed experimentally, such as secondary and tertiary branching, parabolic tip, arms generation, selection and coarsening, etc. Computation results are validated in two ways. First, the simulated secondary dendrite arm spacing (SDAS) is compared with literature values. Then, the predictions of the classic Lipton-Glicksman-Kurz (LGK) theory for steady-state tip velocity are compared with simulated values as a function of melt undercooling. Both comparisons are found to be in very good agreement.     

Numerical simulation of solidification structure of high carbon SWRH77B billet based on the CAFE method

Abstract

Based on the coupled method of cellular automaton (CA) and finite element (FE), the solidification structure of 160×160 mm cast billet of high carbon SWRH77B steel was simulated. The nucleation density, kinetics of the dendrite tip growth, crystallographic orientation and coupling of CA and FE methods are discussed. In the current study, the influence of superheat on the solidification structure of the billet is researched in detail. The results show that for an increase in superheat extent from 20 to 30°C, the density of grain in billet decreased from 4·662×10⁶ to 3·087×10⁶ m^?2, and the grain mean radius increased from 295·1 to 346·3 μm. The three-dimensional microstructure of high carbon SWRH77B steel was simulated by the CAFE method, and there was good agreement with the results from industrial billet.     

Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

A comprehensive model is developed for solving the heat and solute diffusion equations during solidification that avoids tracking the liquid—solid interface. The bulk liquid and solid phases are treated as regular solutions and an order parameter (the phase field) is introduced to describe the interfacial region between them. Two-dimensional computations are performed for ideal solutions and for dendritic growth into an isothermal and highly supersaturated liquid phase. The dependence upon various material and computational parameters, including the approach to conventional sharp interface theories, is investigated. Realistic growth patterns are obtained that include the development, coarsening, and coalescence of secondary and tertiary dendrite arms. Microsegregation patterns are examined and compared for different values of the solid diffusion coefficient.     

Cellular Automaton Study of Hydrogen Porosity Evolution Coupled with Dendrite Growth During Solidification in the Molten Pool of Al-Cu Alloys

Welding porosity defects significantly reduce the mechanical properties of welded joints. In this paper, the hydrogen porosity evolution coupled with dendrite growth during solidification in the molten pool of Al-4.0 wt pct Cu alloy was modeled and simulated. Three phases, including a liquid phase, a solid phase, and a gas phase, were considered in this model. The growth of dendrites and hydrogen gas pores was reproduced using a cellular automaton (CA) approach. The diffusion of solute and hydrogen was calculated using the finite difference method (FDM). Columnar and equiaxed dendrite growth with porosity evolution were simulated. Competitive growth between different dendrites and porosities was observed. Dendrite morphology was influenced by porosity formation near dendrites. After solidification, when the porosities were surrounded by dendrites, they could not escape from the liquid, and they made pores that existed in the welded joints. With the increase in the cooling rate, the average diameter of porosities decreased, and the average number of porosities increased. The average diameter of porosities and the number of porosities in the simulation results had the same trend as the experimental results.

     

Phase Field Simulation of Binary Alloy Dendrite Growth Under Thermal- and Forced-Flow Fields: An Implementation of the Parallel–Multigrid Approach

Dendrite growth and morphology evolution during solidification have been studied using a phase field model incorporating melt convection effects, which was solved using a robust and efficient parallel, multigrid computing approach. Single dendrite growth against the flow of the melt was studied under a wide range of growth parameters, including the Lewis number (Le) and the Prandtl number (Pr) that express the relative strengths of thermal diffusivity to solute diffusivity and kinematic viscosity to thermal diffusivity. Multidendrite growths for both columnar and equiaxed cases were investigated, and important physical aspects including solute recirculation, tip splitting, and dendrite tilting against convection have been captured and discussed. The robustness of the parallel–multigrid approach enabled the simulation of dendrite growth for metallic alloys with Le ~ 10⁴ and Pr ~ 10^?2, and the interplay between crystallographic anisotropy and local solid/liquid interfacial conditions due to convection on the tendency for tip splitting was revealed.     

强制对流影响下Fe?C合金定向凝固微观组织的相场法研究

定向凝固技术能够获得特定柱状晶结构,对于优化合金轴向力学性能具有非常显著的效果。本文采用耦合流场的相场模型模拟了定向凝固过程中枝晶的生长过程,研究了各向异性系数、界面能对定向凝固枝晶生长的影响以及强制对流作用下枝晶的生长行为。数值求解过程中,选用基于均匀网格的有限差分方法对控制方程进行离散,实现了格子中标记点算法（MAC）和相场离散计算方法的联合求解。处理微观速度场和压力场耦合时,采用MAC算法求解Navier-Stokes方程和压力Poisson方程,采用交错网格法处理复杂的自由界面。结果表明:随着各向异性系数的增大,枝晶尖端生长速度增大,曲率半径减小,枝晶根部溶质浓度逐渐降低;随着界面能的增大,枝晶尖端曲率半径增大,当界面能为最大（0.6 J·m?2）时,凝固呈现平界面的凝固方式向前推进;强迫对流对定向凝固枝晶生长方向影响较大,上游方向定向凝固枝晶粗大且生长速度更快,其现象随流速的增大而愈加明显。      

Application of the phase-field method to the solidification of hot-dipped galvanized coatings

The growth of dendrites during the solidification of thin metallic films has been modeled using the phase-field method, with appropriate boundary conditions to take into account wetting effects. The model was applied to the growth of zinc dendrites during the solidification of hot-dipped coatings of steel, and the simulation results were compared to recent experimental observations of Strutzenberger and Faderl. It has been found that the presence of a boundary modifies the usual crystallographic growth directions of the dendrite arms as well as their growth velocity. In the case of hcp zinc dendrites in galvanized coatings, wetting effects at the boundary decrease the growth velocity as the inclination angle of the basal plane increases. This model also shows that shiny regions of the coating, characterized by a low density of lead particles and a smooth surface, result from the growth of the dendrite along the outer surface, while dimpled regions, characterized by a high density of lead particles and a rough surface, are due to the growth of the dendrite along the steel substrate.     

Evolution of Specific Surface Area with Solid Volume Fraction During Coalescence with Parabolic Shape Assumption for Dendrite

Coalescence of secondary dendrite arms in terms of the evolution of specific surface area for a dendritic envelope with respect to the fraction solid during directional solidification is studied analytically in 2-D and 3-D frame of reference. For 2D case study, the dendritic growth has been captured by gradual penetration of a parabola with fixed tip radius (??) inside a square domain, while for 3D case study, we replace the parabola and square domain for 2-D case with a paraboloid of revolution and cubic domain, respectively. The dimensions of the enclosing volumes are characterized by the secondary dendrite arm-spacing. Coalescence obtained by the analytical solution is compared with existing dendrite growth simulation results obtained by phase-field method.     

A quantitative dendrite growth model and analysis of stability concepts

While a number of cellular automaton (CA) based models for dendrite growth have been proposed, none so far have been validated, casting doubt on their quantitative capabilities. All these models are mesh dependent and cannot correctly describe the influence of crystallographic orientation on growth morphology. In this work, we present an improved version of our previously developed CA based model for dendrite growth controlled by solutal effects in the low Péclet number regime. The model solves the solute and heat conservation equations subject to the boundary conditions at the interface, which is tracked with a new virtual front tracking method. It contains an expression equivalent to the stability constant required in analytical models, termed stability parameter, which is not a constant. The process determines its value, changing with time and angular position during dendrite formation. The article proposes solutions for the evaluation of local curvature, solid fraction, trapping rules, and anisotropy of the mesh, which eliminates the mesh dependency of calculations. Several tests were performed to demonstrate the mesh independence of the calculations using Fe-0.6 wt pct C and Al-4 wt pct Cu alloys. Computation results were validated in three ways. First, the simulated secondary dendrite arm spacing (SDAS) was compared with literature values for an Al-4.5 wt pct Cu alloy. Second, the predictions of the classic Lipton-Glicksman-Kurz (LGK) analytical model for steady-state tip variables, such as velocity, radius, and composition, were compared with simulated values as a function of melt undercooling for Al-4 wt pct Cu alloy. In this validation, it was found that the stability parameter approaches the experimentally and theoretically determined value of 0.02 of the stability constant. Finally, simulated results for succinonitrile-0.29 wt pct acetone (SCN-0.29 wt pct Ac) alloy are compared with experimental data. Model calculations were found to be in very good agreement with both the analytical model and the experimental data. The model is used to simulate equiaxed and columnar growth of Fe-0.6 wt pct C and Al-4 wt pct Cu alloys offering insight into microstructure formation under these conditions.     

Modeling of casting solidification stochastic or deterministic?

Solidification modeling has had a phenomenal impact on metalcasting in the last decade. Following its initial success in predicting the occurrence of porosity defects, it has grown to be an essential casting engineering tool. As more complex models have been developed (in response to the realization that simple heat flow models were not adequate to solve the shrinkage problem) , they are being used to design more efficient gating and risering systems, which minimize the amount of metal poured to produce a good casting. Models today include predictions of fluid flow during mold filling, casting distortion, mold–metal interface reactions and cast structure.Until a few years ago solidification simulation was based only on deterministic models. As prediction of microstructural evolution became a contemporary problem, the limitation of deterministic models in predicting such features as dendrite coherency, columnar-to-equiaxed transition, dendrite fragmentation and movement of dendrites by the liquid, became evident. Recently developed stochastic models for solidification are capable of simulating and displaying the growth of columnar and equiaxed grains. However, the physics of dendritic growth is rather approximate. The growth of dendrite arms and their branching are ignored, and only a bulk representation of the grain growth is provided.A micro-scale approach for more accurate dendritic growth simulation in casting processes is presented in this paper. The model couples stochastic modeling at a length scale of 10⁻⁶ m, with deterministic modeling at a length scale of 10⁻⁴ m. A deterministic tip-velocity model is used to calculate the advance of the dendrite tip. Arm thickening is also calculated with a deterministic law derived from the dendrite tip velocity law and crystallographic considerations in combination with a deterministic coarsening model. However, the overall growth of dendrite arms is derived from probabilistic calculations. Branching of dendrites arm is allowed to occur based on morphologic instability. Thus the dendrite morphology, rather than the gain structure can be simulated.A discussion on the advantages and limitations of contemporary deterministic and stochastic models is also included.     

An integrated model for dendritic and planar interface growth and morphological transition in rapid solidification

Rapid solidification can be achieved by quenching a thin layer of molten metal on a cold substrate, such as in melt spinning and thermal spray deposition. An integrated model is developed to predict microstructure formation in rapidly solidified materials through melt substrate quenching. The model solves heat and mass diffusion equations together with a moving interface that may either be a real solid/liquid interface or an artificial dendrite tip/melt interface. For the latter case, a dendrite growth theory is introduced at the interface. The model can also predict the transition of solidification morphology, e.g., from dendritic to planar growth. Microstructure development of Al-Cu alloy splats quenched on a copper substrate is investigated using the model. Oscillatory planar solidification is predicted under a critical range of interfacial heat-transfer coefficient between the splat and the substrate. Such oscillatory planar solidification leads to a banded solute structure, which agrees with the linear stability analysis. Finally, a microstructure selection map is proposed for the melt quenching process based on the melt undercooling and thermal contact conditions between the splat and the substrate.