Cellular Automaton-finite Difference Research Articles

Modeling and Simulation Investigations on Microstructure Evolution during Additive Manufacturing of AlSi10Mg Alloy

Microstructure has significant effects on the mechanical properties of AlSi10Mg alloy. Therefore, an in-depth understanding of microstructure evolution, such as dendrite and Al-Si eutectic, is of great significance to obtain the desirable microstructure and manage the performance of AlSi10Mg components. In the current work, an integrated dendrite and eutectic evolution model based on the cellular automaton–finite difference (CA-FD) method, taking account of solute distribution, growth kinetics, and nucleation mechanism, was established. Microstructures of the as-built selective laser melted (SLMed) samples were characterized by optical microscopy (OM), scanning electron microscopy (SEM) and electron backscatter diffraction (EBSD) techniques, and the experimental results showed that the microstructure consisted of Al grains and Al-Si eutectic networks in the individual melt pool. Dendrite growth, solute redistribution in ternary alloy and dendritic morphologies with different cooling rates were numerically investigated. In addition, the proposed model was also applied to predict the Al-Si eutectic evolution, and eutectic morphologies under eutectic undercooling in a range of 5 K to 20 K were also simulated. The simulated results indicated that dendrites were refined with the increasing of the cooling rates, and Al-Si eutectic morphology was sensitive to eutectic undercooling such that higher eutectic undercooling refined the eutectic microstructures. Model validations were performed, and the experimental results agreed well with the simulation results, indicating that the proposed model can successfully reproduce both dendrite and eutectic microstructures.

Grain Growth of Ti–6Al–4V Alloy Predicted with Cellular Automaton-Finite Difference (CAFD) Model

Grain Growth of Ti–6Al–4V Alloy Predicted with Cellular Automaton-Finite Difference (CAFD) Model

Competitive dendrite growth during directional solidification of a transparent alloy: Modeling and experiment.

A two-dimensional (2-D) cellular automaton-finite difference method (CA-FDM) model and in situ observation experiments of directional solidification using a transparent alloy of SCN-2wt.% ACE are employed to investigate various microstructural evolution of columnar dendrites during directional solidification. In the present model, the growth of columnar dendrites is simulated using a CA technique. The solute diffusion is solved using the FDM. The model is capable of visualizing the interaction between the formation of dendrite arrays with identical or different growth orientations, and the evolving solute concentration field. Several dendritic competitive growth modes between two converging and diverging dendrite arrays are reproduced. The simulation results agree well with the experimental observations. The simulations are also performed to study the effects of temperature gradient and cooling rate on the growth morphology of diverging dendrites. It is found that with the increase of temperature gradient and cooling rate, the tertiary branches produced from the well-developed side branches of the unfavorably oriented grain at the divergent grain boundaries are more likely to become the new primary dendrite arms.

Cross-scale simulation for MnS precipitation of Fe-C alloy with cooling rate variation

The cooling rate is a key factor controlling casting solidification, which determines the microstructure and performance of a metal and inevitably affects the formation of inclusion in steel. MnS is a common inclusion in Fe-C alloy, and it is indispensable to determine the variations in MnS precipitation with respect to the solidification conditions. This work develops a cross-scale cellular automaton-finite difference (CA-FD) model to simulate MnS precipitation in two Fe-C-Mn-S systems, Fe-0.6%C-0.8%Mn-0.005%S (High-Mn) and Fe-0.6%C-0.2%Mn-0.02%S (High-S) with the constant factor of concentration product of elements Mn and S. The model divides the simulated area into three different grid sizes to calculate the macroscopic heat transfer, mesoscopic matrix growth and microscopic MnS precipitation. The results show that most of the MnS precipitates occur within regular belt-shaped regions in the solidified matrix. The cooling rate has a significant effect on MnS precipitation in the belt-shaped regions by affecting the matrix solidification and solute concentration in the liquid phase. High cooling rates delay MnS precipitation and reduce the number of precipitates, which narrows the belt-shaped regions and increases the number of precipitate per unit of belt area (N ave). Reprecipitation was observed in the High-Mn system, and the reprecipitation decreases with the increase of cooling rate.

Numerical simulations of solidification structures and macrosegregation by a cellular automaton model coupled with flow calculations

In this study, a numerical model was developed, and direct simulations were performed to predict solidification grain structures and macrosegregation based on a three-dimensional cellular automaton finite difference (CAFD) method coupled with flow calculation of natural convection and shrinkage flow. First, to evaluate the model coupled with natural convention, simulations of unidirectional solidification for Al-10wt% Mg alloy were performed. Mg-rich plumes rising in the melt were seen due to subsequent upward flow, and Mg-rich channels forming in the mushy zone were observed. Columnar grains were then formed, and they became coarse afterwards. Next, to evaluate the model coupled with shrinkage flow, simulations of casting Al-10wt% Cu alloy in a unique mold, which can form macrosegregation in the central region of the small ingot, were performed. The bridging of columnar grains formed during solidification, and the positive segregation was generated in the region below the bridging. Thus, the main factor for this macrosegregation is the shrinkage flow with bridging. From the comparison of simulation results with and without the chill for the unique mold, it was established that the shrinkage flow and the bridging of solidification structures play an important role in macrosegregation.

Modeling of hydrogen porosity formation during solidification of dendrites and irregular eutectics in Al–Si alloys

A two-dimensional multi-phase cellular automaton-finite difference method (CA-FDM) model is proposed for the simulation of hydrogen porosity formation during solidification of dendrites and irregular eutectics in aluminum alloys. The present model encompasses a multi-phase system of liquid, gas porosity, dendrites and eutectic phases. The growth of gas pores and solid phases is simulated using a CA technique. The diffusion of hydrogen and silicon is calculated using the FDM. The proposed model is applied to simulate the temporal evolution of casting gas porosity in an Al-7 wt% Si alloy. It allows visualizing the interactions between the formation of microporosity and the growth of solid phases, as well as the evolving concentration fields of hydrogen and silicon. The competitive growth mode between the gas pores with different sizes is analyzed in detail by comparing the local hydrogen concentration and saturation at the gas/liquid interface. The pores with relative larger sizes are found to grow preferentially. The simulations are also performed to investigate the influences of cooling rate and initial hydrogen concentration on microporosity formation. The simulated time-dependent variations of porosity percentage and solid fraction indicate that the main portion of porosity is formed in the stage of eutectic solidification. When applying a higher cooling rate, the final pore size is smaller and more uniform, and the porosity percentage is lower, while the porosity density is higher. A decreased initial hydrogen concentration evidently reduces the porosity percentage. The simulation results compare reasonably well with the experimental data reported in the literature.

A cellular automaton-finite difference simulation of the ausferritic transformation in ductile iron

ABSTRACTThe development of the ausferritic transformation of a ductile iron was analysed using a novel cellular automaton-finite difference model, which considers geometrical details of the microstructure, nucleation of the new phase at graphite nodule surface, contact between growing phases, and carbon diffusion in austenite. The role of nucleation, austenite carbon enrichment, and contact between phases in the different stages of the growth kinetics was studied. Moreover, a parametric study was performed to investigate the influences of graphite nodule size, and austenitizing and austempering temperatures on the required time to end the transformation and final phase fractions. The obtained results are in agreement with experimental data reported in the literature.

Multiscale Modeling and Simulation of Directional Solidification Process of Ni-Based Superalloy Turbine Blade Casting

Ni-based superalloy turbine blades have become indispensable structural parts in modern gas engines. An understanding of the solidification behavior and microstructure formation in directional solidified turbine blades is necessary for improving their high-temperature performance. The multiscale simulation model was developed to simulate the directional solidification process of superalloy turbine blades. The 3D cellular automaton-finite difference (CA-FD) method was used to calculate heat transfer and grain growth on the macroscopic scale, while the phase-field method was developed to simulate dendrite growth on the microscopic scale. Firstly, the evolution of temperature field of an aero-engine blade and a large industrial gas turbine blade was studied under high-rate solidification (HRS) and liquid-metal cooling (LMC) solidification processes. The varying withdrawal velocity was applied to change the curved mushy zone to a flat shape. Secondly, the grain growth in the aero-engine blade was simulated, and the grain structures in the starter block part and the spiral selector part in the HRS process were compared with those in the LMC process. The simulated grain structures were generally in agreement with experimental results. Finally, the dendrite growth in the typical HRS and LMC solidification process was investigated and the simulation results were compared with the experimental results in terms of dendrite morphology and primary dendritic spacing.

Simulation and Experimental Studies on Grain Selection and Structure Design of the Spiral Selector for Casting Single Crystal Ni-Based Superalloy

Grain selection is an important process in single crystal turbine blades manufacturing. Selector structure is a control factor of grain selection, as well as directional solidification (DS). In this study, the grain selection and structure design of the spiral selector were investigated through experimentation and simulation. A heat transfer model and a 3D microstructure growth model were established based on the Cellular automaton-Finite difference (CA-FD) method for the grain selector. Consequently, the temperature field, the microstructure and the grain orientation distribution were simulated and further verified. The average error of the temperature result was less than 1.5%. The grain selection mechanisms were further analyzed and validated through simulations. The structural design specifications of the selector were suggested based on the two grain selection effects. The structural parameters of the spiral selector, namely, the spiral tunnel diameter (dw), the spiral pitch (hb) and the spiral diameter (hs), were studied and the design criteria of these parameters were proposed. The experimental and simulation results demonstrated that the improved selector could accurately and efficiently produce a single crystal structure.

Numerical simulation of dendrite growth in nickel-based superalloy and validated by in-situ observation using high temperature confocal laser scanning microscopy

Dendritic structures are the predominant microstructural constituents of nickel-based superalloys, an understanding of the dendrite growth is required in order to obtain the desirable microstructure and improve the performance of castings. For this reason, numerical simulation method and an in-situ observation technology by employing high temperature confocal laser scanning microscopy (HT-CLSM) were used to investigate dendrite growth during solidification process. A combined cellular automaton-finite difference (CA-FD) model allowing for the prediction of dendrite growth of binary alloys was developed. The algorithm of cells capture was modified, and a deterministic cellular automaton (DCA) model was proposed to describe neighborhood tracking. The dendrite and detail morphology, especially hundreds of dendrites distribution at a large scale and three-dimensional (3-D) polycrystalline growth, were successfully simulated based on this model. The dendritic morphologies of samples before and after HT-CLSM were both observed by optical microscope (OM) and scanning electron microscope (SEM). The experimental observations presented a reasonable agreement with the simulation results. It was also found that primary or secondary dendrite arm spacing, and segregation pattern were significantly influenced by dendrite growth. Furthermore, the directional solidification (DS) dendritic evolution behavior and detail morphology were also simulated based on the proposed model, and the simulation results also agree well with experimental results.

Multi-scale coupling simulation in directional solidification of superalloy based on cellular automaton-finite difference method

Casting microstructure evolution is difficult to describe quantitatively by only a separate simulation of dendrite scale or grain scale, and the numerical simulation of these two scales is difficult to render compatible. A three-dimensional cellular automaton model couplling both dendritic scale and grain scale is developed to simulate the microstructure evolution of the nickel-based single crystal superalloy DD406. Besides, a macro–mesoscopic/microscopic coupling solution algorithm is proposed to improve computational efficiency. The simulation results of dendrite growth and grain growth of the alloy are obtained and compared with the results given in previous reports. The results show that the primary dendritic arm spacing and secondary dendritic arm spacing of the dendritic growth are consistent with the theoretical and experimental results. The mesoscopic grain simulation can be used to obtain results similar to those of microscopic dendrites simulation. It is indicated that the developed model is feasible and effective.

Large-scale simulation of dendrite growth in directional solidification castings

In this study, the growth and distribution of directional solidification (DS) dendrites in entire cross-sections of DS superalloy castings are simulated based on a modified shape function dendritic (MSFD) model coupled with the cellular automaton-finite difference (CA-FD) method. The macro-thermal transfer and columnar grain growth are calculated using CA-FD with a macro time-step, and then, the original and boundary conditions for the dendrite-growth simulation are determined via the interpolation of the macro temperature and the projection of the 3D preferential grain orientation using a matrix, M. The dendrite growth calculation cyclically proceeds based on the MSFD model. It is used to predict the growth of hundreds of dendrites throughout the cross-section of a DS casting by considering the macro-solidification parameters. The MSFD model is further applied to predict the competitive growth of DS dendrites and their distribution in the cross-sections of a spiral selector and a fake single crystal turbine blade. The results of the simulation correlate with the results of the DS experiment. Hence, the modified model coupled with the macro-simulation models can be used to predict the morphology and distribution of DS dendrites with respect to large-scale castings produced by the DS process.

Multi-scale simulation of directional dendrites growth in superalloys

The directional dendrite growth behavior and morphology were simulated based on the cellular automaton-finite difference (CA-FD) considering macro directional solidification (DS) parameters such as withdrawal rate and pouring temperature. Two types of directional dendrite growth models were proposed to realize the multi-scale simulation. A 3D dendrite growth model based on heat transfer and solute diffusion was built to simulate the dendrite growth during a real DS process, which is proved useful for the detail dendrite growth and morphology prediction at a micro 3D scale. A modified shaped function model was proposed to predict the evolution behavior of a large number of dendrites in the entire section of a DS cast sample at a macro scale, based on which the evolution behavior of thousands of dendrites was predicted in the 2D section during the DS process. In these models, the temperature interpolation algorithm and the dendrite orientation mathematical projection were used to transfer information between macro DS process calculation and micro dendrite growth simulation. The corresponding DS experiment was performed to verify the numerical models, and they agreed well. The DS dendritic evolution behavior and detail morphology, especially thousands of dendrites distribution at a large scale, were successfully simulated based on the two models proposed on the condition of a real DS process.

NUMERICAL SIMULATION OF DENDRITE GRAIN GROWTH OF DD6 SUPERALLOY DURING DIRECTIONAL SOLIDIFICATION PROCESS

Modern aero and power industry needs high- performance gas turbine. Directional solidification(DS) columnar grain and single crystal(SX) blade as key parts of gas turbine serve in heavy stress and high temperature conditions. The DS and SX blade are mainly produced by high rapid solidification(HRS) method, and HRS is one of useful DS technology, which has a property that the heat dissipating ways are changing during the process and the temperature gradients will vary correspondingly. The dendrite grain arrays were the substructure of a DS or SX blade. The structure of the dendrite grain arrays influences the mechanical property of the final casting very much, but is seriously affected by the solidification parameters, such as temperature gradient. In this work, the dendrite grain growth of DD6 superalloy was studied based on cellular automaton-finite difference(CA- FD) model concerning the HRS method's macro solidification parameters. Mathematic models for dendrite grain growth controlled by temperature field and solute field were built to describe the competitive growth and morphology evolution of dendrite grains. Then the dendrite calculation model was coupled with the models of DS process calculation, and some HRS solidification parameters were included, such as withdrawal rate, pouring temperature, etc. The coupled models were used to predict the dendrite grain competitive growth of DD6 superalloy during the DS process. The variation of solute distribution and the dynamic adjustment of dendritic spacing during the process could be predicted by simulating calculation. The DS experiment was carried out with a cylinder sample, and dendrite grains' distribution in the transverse and longitude section was observed by OM and SEM. Then the simulated dendritic morphology was compared with that by experiment. The primary and secondary dendritic spacing by experiment and simulation were measured, and the compared results revealed that as the DS process going on the temperature gradient decreased gradually and the primary dendritic spacing was increasing. So simulation results of the DS dendritic competitive growth were validated by the experiment results, and the proposed models could predict the dendrite grain morphology and the adjustments of DS dendritic spacing of DD6 superalloy very well.

Numerical Simulation and Optimization of Directional Solidification Process of Single Crystal Superalloy Casting.

The rapid development of numerical modeling techniques has led to more accurate results in modeling metal solidification processes. In this study, the cellular automaton-finite difference (CA-FD) method was used to simulate the directional solidification (DS) process of single crystal (SX) superalloy blade samples. Experiments were carried out to validate the simulation results. Meanwhile, an intelligent model based on fuzzy control theory was built to optimize the complicate DS process. Several key parameters, such as mushy zone width and temperature difference at the cast-mold interface, were recognized as the input variables. The input variables were functioned with the multivariable fuzzy rule to get the output adjustment of withdrawal rate (v) (a key technological parameter). The multivariable fuzzy rule was built, based on the structure feature of casting, such as the relationship between section area, and the delay time of the temperature change response by changing v, and the professional experience of the operator as well. Then, the fuzzy controlling model coupled with CA-FD method could be used to optimize v in real-time during the manufacturing process. The optimized process was proven to be more flexible and adaptive for a steady and stray-grain free DS process.

Three-dimensional Cellular Automaton Model for the Prediction of Microsegregation in Solidification Grain Structures

A 3-D cellular automaton finite difference (3-D CAFD) model to predict the solidification grain structure and the microsegregation was developed. In the present model, a new approach to calculate the solute concentration distribution was developed. Moreover, the CA cell including the grain boundary was newly defined. The probabilistic approach was adopted as the nucleation model, and the decentered octahedron growth algorithm was adopted as the grain growth model. The growth kinetics of the dendrite tip to calculate the growth of the dendrite envelope was calculated by a 3-D Kurz-Giovanola-Trivedi (KGT) model. In the 3-D KGT model, the local undercooling estimated by the local temperature and local solute concentration was used. To evaluate the validity of the present model, we carried out simulations of the solidification grain structure under directional solidification. The different solidification grain structures were obtained by the different solute concentration distributions. In addition, we compared our model with the microsegregation predicted by Scheil’s equation. The simulated results of microsegregation by the present model were in fairly good agreement with those by Scheil’s equation. From these results, we confirmed that the calculation of solute concentrations has to be considered and that the present model can simultaneously simulate microstructure and microsegregation.

A THREE-DIMENSIONAL CELLULAR AUTOMATON SIMULATION FOR DENDRITIC GROWTH

Perhaps dendrite is the most observed solidification microstructure of many metal- lic materials. The dendritic morphologies show a dominating effect on the performance of casting products. A lot of work has been carried out to investigate the formation mechanism of dendritic microstructure. It is found that the development of dendritic microstructures is a complicated process controlled by the interplay of many factors such as thermal and solute transfer, capillary etc. Cellular automaton (CA) can simulate the solidification process with a high computational efficiency, thus, attracts great attentions. In recent years, progress has been made on the two dimensional CA models for the solidification microstructure formation. But up to date researches on three dimensional CA model are very limited. A combined cellular automaton-finite difference (CA-FD) model for the three dimensional simulation of dendritic growth was developed in this paper. Simulations were performed to investigate the dendritic growth in an undercooled Al-Cu alloy as well as in a directionally solidified Al-Cu alloy. The numerical results showed clearly the development of the free dendrite in the under- cooled melt and the microstructure evolution in the directionally solidified alloy and agreed well with the theoretical predictions and the experimental results.

Numerical modeling of crystal growth of a nickel-based superalloy with applied direct current

A cellular automaton-finite difference (CA-FD) model is developed to simulate crystal growth with applied direct current (DC) in this study. The simulated dendrite growth is controlled by the CA rules that involved the effects of the supercooling, interfacial curvature and crystal anisotropy. The electric current distribution, the temperature field and the electromagnetic flow caused by the Lorentz force are solved by the FD calculation. The directional solidification of a model nickel-based superalloy with DC field is simulated. The results show that Joule heat brought by electric current delays the dendrite growth and changes the primary dendrite arm spacing (PDAS). Excessive Joule heat will completely stop the crystal growth. The maximum allowable current intensity increases with the cooling efficiency of directional solidification. It is found that the dendrites growing with the DC field have a self-adjustment ability to achieve flat S/L interface in macroscale during directional solidification. The electromagnetic flow induced by the DC field may theoretically accelerate the dendrite growth, but usually the influence is negligible for the directional solidification of the superalloy.

Restrictions of Physical Properties on Solidification Microstructures of Al-based Binary Alloys by Cellular Automaton

The solidification microstructure evolution and the Columnar to Equiaxed Transition (CET) during Al–Si and Al–Cu binary alloy solidification processes are analyzed by the help of Cellular Automaton-Finite Difference (CA-FD) model. The effects of the physical properties, except the effects of the nucleation parameters and the operation parameters, on the cooling curves, the dendrite growth, the solidification morphologies and the CET of the Al–Si and Al–Cu binary alloys are emphatically discussed. Results show that the solidification morphologies are internally influenced by the physical properties related to the dendrite tip growth kinetics. Besides the solute diffusivity in liquid DL and the growth restriction parameter Q=mC0(kS/L−1), the Gibbs–Thomson coefficient Γ shows a great effect on dendrite tip growth rate. Their effects on the dendrite tip kinetics are ordered as m(kS/L−1)>Γ>DL. The growth rate can be predicted efficiently by the present simplified expression based on GGAN model combining several physical parameters and the local undercooling, which clearly shows the physical meaning of the constant coefficients in the simplified expression based on KGT model. The equiaxed ratio related to the solidification morphologies can also be evaluated as a function of those physical properties.

Simulation of the Columnar-to-Equiaxed Transition in Alloy Solidification - The Effect of Nucleation Undercooling, Density of Nuclei in Bulk Liquid and Alloy Solidification Range on the Transition

A coupled cellular automaton-finite difference (CA-FD) model is used to simulate the detailed dendritic structure evolution of the columnar-to-equiaxed transition (CET) for Al-Cu alloys during solidification. The effects of material properties (nucleation undercooling, density of nuclei in bulk liquid and alloy solidification range) on the CET are investigated. Simulated results reveal that: (1) equiaxed grains form at an earlier stage with a smaller critical nucleation undercooling; (2) CET is promoted if the density of nuclei in bulk liquid is increased; (3) extending the alloy solidification range promotes the CET. Finally, CET maps corresponding to different alloy concentrations are constructed, illustrating the relationship between processing conditions and the resulting grain structures for alloys with different solidification ranges.