Three-dimensional Cellular Automaton Model Research Articles

Cellular automaton model of phase separation during annealing of nonstoichiometric silicon oxide layers

A cellular automaton (CA) model is proposed that simulates the structural evolution of a nonstoichiometric SiOm (m < 2) layer with a thickness of 3–30 nm, involving the formation of silicon nanoclusters during thermal annealing at temperatures within 900–1200°C. The model does not take into account the crystalline or amorphous structure of the nanoclusters. The three-dimensional CA model implemented on a cubic 0.54-nm mesh grid by the SoftCAM software is synchronous, does not use the Margolus block neighborhood, and is open to the incorporation of ab initio calculations for SixOy clusters. The state of the CA cell is determined by three variables (x, y, z), which take discrete values 0, 1, 2, …, and 255 (corresponding to the numbers of silicon and oxygen atoms and conditional vacancies in the cell), and the fourth variable δ, which takes values 0, 1, and 2, indicating that the given cell belongs to a silicon nanocluster, SiOx matrix, or their interface. The local transition rules are defined based on the following considerations: (i) each cell is characterized by a scalar “free energy” dependent only on the cell state (analogous to thermodynamic potentials); (ii) this free energy is defined as the sum of three components: internal energy U(x, y), elastic energy G(z), and surface energy E(δ); and (iii) the exchange of matter between the cells is determined by probabilities dependent on the difference of the free energies according to the Fermi–Dirac relation. The model traces the evolution of the total number of clusters, their average size, and the average distance between them. The results of the numerical simulation are consistent with the published experimental data.

Simulation of dendritic growth of Al-4 wt.% Cu alloy from an undercooled melt

Abstract Three-dimensional dendritic growth of an Al-4 wt.% Cu alloy during solidification in the presence of forced flow was simulated by using a three-dimensional cellular automaton model. The three-dimensional Lattice–Boltzmann method was used in computing the flow dynamics. The effects of forced flow on the growth Péclet number, dendrite tip selection parameter, and secondary arm spacing were investigated. The simulated relationships of growth Péclet number and dendrite tip selection parameter with the forced flow are in good agreement with the Oseen–Ivantsov solution. The results indicate that with the increase in forced flow velocity, the secondary arm spacing first increases and then decreases.

Cellular automaton simulation of three-dimensional dendrite growth in Al–7Si–Mg ternary aluminum alloys

Due to the extensive applications in the automotive and aerospace industries of Al–7Si–Mg casting alloys, an understanding of their dendrite microstructural formation in three dimensions is of great importance in order to control the desirable microstructure, so as to modify the performance of castings. For this reason, a three-dimensional cellular automaton model (3-D CA) allowing for the prediction of dendrite growth of ternary alloys is presented. The growth kinetics of the solid/liquid (S/L) interface is calculated based on the solutal equilibrium approach. This proposed model introduces a modified decentered octahedron algorithm for neighborhood tracking, in order to eliminate the effects of mesh dependency on dendrite growth. The thermodynamic and kinetic data needed in the simulations are obtained through coupling with the Pandat software package in combination with thermodynamic/kinetic/equilibrium phase diagram calculation databases. The solute interactions between alloying elements are considered in the model. The model was first used to simulate the Al–7Si dendrite, followed by a validation using theoretical predictions. The influence of Mg content on the dendrite growth dynamics and dendrite morphologies was investigated. Also, the model was applied to Al–7Si–0.5Mg dendrite simulation both with and without a consideration of solute interactions between the Si and Mg alloying elements and the effects on dendrite growth process was analyzed using the simulation results. This model was finally used in order to simulate the dendrite growth in different crystallographic orientations in an Al–7Si–0.36Mg ternary alloy during polycrystalline solidification, resulting in a predicted secondary dendrite arm spacing (SDAS) and dendrite volume fraction data that show a reasonable agreement with experimental results. The single dendrite and polycrystalline growth simulations effectively demonstrate the capability of this model in predicting the three-dimensional dendrite microstructure of ternary alloys.

A Three Dimensional Cellular Automata Model for Dendrite Growth in Non-Equilibrium Solidification of Binary Alloy

A three-dimensional (3D) cellular automata (CA) model has been developed for simulation of dendrite growth during non-equilibrium solidification. The heat and mass transports are calculated using 3D finite element (FE) method. The migration of solid/liquid (SL) interface is associated with effective free energy difference incorporating solute trapping and relaxation effects. The non-equilibrium solute partition coefficient involving solute trapping and relaxation effects is used to calculate solute redistribution at SL interface. The interface curvature and anisotropy of surface energy are introduced to represent the characteristics of dendrite tip. The CA model has been applied to simulate the microstructure evolution of Fe-1.5 wt.% C alloy. The grain morphology, solute and temperature distributions, and velocity of dendrite tip are characterized during both isothermal and non-isothermal conditions. The velocity of dendrite tip and secondary dendrite arm spacing during directional solidification are validated and are in good agreements with previous experimental and mathematical results.

The pedestrian evacuation model with collision probability in three-dimensional space

In order to effectively depict the evacuation process in a three-dimensional space, a novel three-dimensional cellular automata model was proposed with the stair factors. Based on the position attractiveness, collision probability, and vacancy degree, the calculation formula for transition probability was provided by the model, and the evacuation strategy was explained through defining the cellular evolutionary process. Meanwhile, the experiments were conducted on the simulation platform to study the relationships of evacuation time, exit width, pedestrian density, as well as average system velocity. The results showed that evacuation time had a positive correlation with pedestrian density, and a negative correlation with exit width; in addition, concerning the optimum evacuation time, there exists a desirable threshold value for vacancy parameter and collision parameter.

The effects of group and position vacancy on pedestrian evacuation flow model

Abstract In order to reduce the simulation error of pedestrian evacuation model, a novel three-dimensional cellular automata model was proposed with ladder factor. The calculation formula for transition probability was given in the model based on floor field, position vacancy degree and group attraction, and the evacuation strategy was presented. Meanwhile, the experiment studied the relationships of evacuation time, mean system velocity as well as pedestrian density. The result showed that the evacuation time could be reduced effectively when the mean system velocity and position vacancy had proper degree, and the bigger group had negative effect with the evacuation time, so the bigger group should be avoided in actual evacuation process.

Physical Improvements to a Mesoscopic Cellular Automaton Model for Three-Dimensional Snow Crystal Growth

We motivate and derive the dynamical rules for a computationally feasible three-dimensional cellular automaton model of snow crystal growth. The model improves upon points of weak physical connections identified in other similar models which have produced morphological features observed in many snow crystal photographs. A systematic survey of the morphologies resulting from our model illustrates the degree to which these features persist in our results, and the trends that appear as model parameters are varied.

A cellular automata model of pedestrian evacuation in three-dimensional space

In order to effectively depict the evacuation process in three-dimensional space, combining with the ladder factors, a novel three-dimensional cellular automata model is proposed. Based on the position attraction and collision possibility, the model presents a formula for calculating the transition probability, and expounds the evacuation strategy through defining the cellular evolutionary process. Meanwhile, experiments are conducted using the simulation platform to study the relationships between evacuation time, exit flow rate, exit width, original pedestrian density and system average velocity. The results show that evacuation time and exit flow rate have a positive correlation with original pedestrian density, and a negative correlation with exit width; in addition, concerning the optimum evacuation time, there exists a desirable threshold value for system average velocity and exit width.

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 model for simulation of dendritic growth of magnesium alloy

A three-dimensional cellular automaton model for simulation of dendritic growth of magnesium alloy

Influence of Grain Boundary Mobility on Microstructure Evolution during Recrystallisation

The paper introduces first investigations on how low angle grain boundaries can influence the recrystallisation behaviour of crystalline metallic materials. For this purpose a three-dimensional cellular automaton model was used. The approach in this study is to allow even low angle grain boundaries to move during recrystallisation. The effect of this non-zero mobility of low angle grain boundaries will be analysed for the recrystallisation of deformed Al single crystals with Cube orientation. It will be shown that low angle grain boundaries indeed influence the kinetics as well as the texture evolution of metallic materials during recrystallisation.

Orientation selection of equiaxed dendritic growth by three-dimensional cellular automaton model

A three-dimensional (3-D) adaptive mesh refinement (AMR) cellular automata (CA) model is developed to simulate the equiaxed dendritic growth of pure substance. In order to reduce the mesh induced anisotropy by CA capture rules, a limited neighbor solid fraction (LNSF) method is presented. It is shown that the LNSF method reduced the mesh induced anisotropy based on the simulated morphologies for isotropic interface free energy. An expansion description using two interface free energy anisotropy parameters (ε1, ε2) is used in the present 3-D CA model. It is illustrated by present 3-D CA model that the positive ε1 favors the dendritic growth with the 〈100〉 preferred directions, and negative ε2 favors dendritic growth with the 〈110〉 preferred directions, which has a good agreement with the prediction of the spherical plot of the inverse of the interfacial stiffness. The dendritic growths with the orientation selection between 〈100〉 and 〈110〉 are also discussed using the different ε1 with ε2=−0.02. It is found that the simulated morphologies by present CA model are as expected from the minimum stiffness criterion.

A three-dimensional cellular automaton model for dendritic growth in multi-component alloys

A three-dimensional (3-D) cellular automaton model for dendritic growth in multi-component alloys is developed. The velocity of advance of the solid/liquid (S/L) interface is calculated using the solute conservation relationship at the S/L interface. The effect of interactions between the alloying elements on the diffusion coefficient of solutes in the solid and liquid phases are considered. The model is first validated by comparing with the theoretical predictions for binary and ternary alloys, and then applied to simulate the solidification process of Al–Cu–Mg alloys by a coupling of thermodynamic and kinetic calculations. The numerical results obtained show both the free dendrite growth process as well as the directional solidification process. The calculated secondary dendrite arm spacing in the directionally solidified Al–Cu–Mg alloy is in good agreement with the experimental results. The effect of interactions between the various alloying elements on dendritic growth is discussed.

Microstructure Modelling of Solid-State Transformations in Low-Alloy Steel Production &lt;sup&gt;&lt;/sup&gt;

In this work the formation of microstructural banding in a dual-phase steel is investigated by using a three-dimensional cellular automata model for phase transformations. Originally developed for describing the metallurgical processes occurring during the annealing stage of cold-rolled strips, this model is presently applied to investigate microstructural-band formation during the austenite-to-ferrite transformation kinetics during cooling after hot rolling. The recent incorporation in the model of an inhomogeneous concentration of Mn, the alloying element most responsible for the development of microstructural banding, and the local nucleation behaviour dependent on the Mn concentration allows the study of the effect of material and process parameters on the banding formation.

Three-Dimensional Modeling and Simulation of Dendrite Morphology of Cast Mg Alloys

A three-dimensional (3-D) cellular automaton (CA) model for simulating the dendrite morphology of cast Mg alloys has been developed. In the model a technique based on two sets of mesh is utilized to perform the simulation to reproduce the texture of Mg dendrites. The CA calculation is performed using a set of mesh that is defined by the hexagonal close-packed (HCP) crystal lattice, and other computations are carried out by using a cubic mesh. The two sets of mesh are coupled by using interpolation method. The kinetics of the solid-liquid interface is obtained directly by the difference between local equilibrium composition and local actual composition given by the solute transport equation. The model was used to simulate 3-D columnar growth of sixteen grains and 3-D equiaxed growth of a single dendrite of AZ91D alloy. Permanent mold castings of AZ91D alloy were produced and sampled for optical metallographic examinations, and the simulated results were compared with the metallographic results.

A microstructure model for recrystallisation and phase transformation during the dual-phase steel annealing cycle

A three-dimensional cellular automata model is developed for the description of the relevant metallurgical mechanisms occurring in the annealing stage of dual-phase steels: ferrite recrystallisation, pearlite-to-austenite and ferrite-to-austenite transformation on heating and austenite-to-ferrite transformation on cooling. Based on the local grain-boundary and interface velocity, the latter controlled by both interface mobility and carbon diffusivity, the model is able to simulate the microstructure development throughout the annealing stage. The model also provides information on the carbon gradient in austenite at the end of the cycle, which is relevant for the prediction of martensite formation during the subsequent quenching. The simulated structure thus provides a realistic representation of many microstructural aspects.

Investigating CTL Mediated Killing with a 3D Cellular Automaton

Cytotoxic T lymphocytes (CTLs) are important immune effectors against intra-cellular pathogens. These cells search for infected cells and kill them. Recently developed experimental methods in combination with mathematical models allow for the quantification of the efficacy of CTL killing in vivo and, hence, for the estimation of parameters that characterize the effect of CTL killing on the target cell populations. It is not known how these population-level parameters relate to single-cell properties. To address this question, we developed a three-dimensional cellular automaton model of the region of the spleen where CTL killing takes place. The cellular automaton model describes the movement of different cell populations and their interactions. Cell movement patterns in our cellular automaton model agree with observations from two-photon microscopy. We find that, despite the strong spatial nature of the kinetics in our cellular automaton model, the killing of target cells by CTLs can be described by a term which is linear in the target cell frequency and saturates with respect to the CTL levels. Further, we find that the parameters describing CTL killing on the population level are most strongly impacted by the time a CTL needs to kill a target cell. This suggests that the killing of target cells, rather than their localization, is the limiting step in CTL killing dynamics given reasonable frequencies of CTL. Our analysis identifies additional experimental directions which are of particular importance to interpret estimates of killing rates and could advance our quantitative understanding of CTL killing.

Three-dimensional cellular automaton simulation of tumour growth in inhomogeneous oxygen environment

Cellular automaton theory has previously been used to study cell growth. In this study, we present a three-dimensional cellular automaton model performing the growth simulation of normal and cancerous cells. The necessary nutrient supply is provided by an artificial arterial tree which is generated by constrained constructive optimization. Spatial oxygen diffusion is approximated again by a cellular automaton model. All results could be illustrated dynamically by three-dimensional volume visualization. Because of the chosen modelling approach, an extension of the model to simulate angiogenic processes is possible.

Prediction of isothermal crystallization parameters in monomer cast nylon 6

A three-dimensional cellular automaton (CA) model is applied to simulate isothermal crystallization of monomer casting (MC) nylon 6. The kinetics is formulated based on the Kim and Kim's rate theory for spherulite expansion. The probability of distribution of crystalline nuclei and impinging were determined by Monte Carlo method. The spherulite impinging was considered in the simulation through the deduction of overlay of spherulite volume. The number of spherulite nuclei and bulk density of MC nylon 6 after completing crystallization, easily obtained from experiment, were used as input data, allowing one to make predictions on a real time and space scale. Finally, the model was verified by quantitative comparison between the simulation results in this work and experimental data in literatures.

Microstructure Simulation of Die Casting AZ91D Alloy

Physical and mathematical models of microstructure evolution during the solidification process of die casting AZ91D alloy were investigated in this paper. Coupled with solute concentration, a modified three-dimensional cellular automaton (CA) model was proposed. Considering the solute enrichment and the formation of eutectic microstructure, these models can reproduce the whole microstructure evolution process of Mg alloy, from the formation of primary phase to the eutectic transformation. The microstructure of the AZ91D alloy cylinder head cover die casting was simulated with the proposed models. The simulated results are in agreement with the experimental.