Three-dimensional Cellular Automaton Model Research Articles

The impact of inter-dune space and volume ratio on barchan dune collision patterns: A numerical simulation study

Collision among barchan dunes is a main cause for the dune swarm to achieve size and speed selection, and it can be divided into two patterns: coalescence and ejection. Here, a three-dimensional (3D) cellular automaton model is applied to collectively study the effects of inter-dune space and relative dune size on these collision patterns. A parameter space is established that includes a transition surface distinguishing coalescence and ejection. Confirmed by field observations and subaquatic experiments, a comprehensive depiction of transition between coalescence and ejection is provided, which may bring new constraints on the morphodynamics of barchan dune fields.

Predicting hydrogen microporosity in long solidification range ternary Al-Cu-Li alloys by coupling CALPHAD and cellular automata model

Aluminum-lithium alloys have wide applications in aerospace industries in the 21st century but their manufacturing is extremely difficult due to the 10 times more equilibrium hydrogen concentration in the liquid than the solubility limit in traditional Al alloys (0.036 mL/100gSTP). The reduction of solubility from liquid to solid by 95% leads to hydrogen porosity being hard to control. In this work, a three-dimensional multicomponent cellular automaton (CA) model is coupled with CALPHAD calculations to simulate the nucleation and growth of hydrogen porosity and its interaction with surrounding dendritic structures during the solidification of Al-Cu-Li alloys. By quantifying the effects of hydrogen concentration, cooling rate, and Li content, it was found the solidification conditions can effectively reduce porosity size. To validate the model, X-ray computed tomography (XCT) has been used to obtain not only the size but also the morphology of porosity as a function of cooling conditions. It was found that porosity grows elongated and tortuous shape at slow cooling rates between columnar dendrites, filling up the empty spaces of secondary arms, while it tends to be dispersed spherical shape when its surrounding grains are refined to equiaxed structures at high cooling rates.

Cellular automaton simulation and experimental validation of eutectic transformation during solidification of Al-Si alloys

The morphology of eutectic silicon in solidification microstructure is critical to the performance of Al-Si-based alloys. Simulating eutectic Si phase formation has been a challenge in ICME (integrated computational materials engineering) based design and manufacturing of solidification products of Al-Si-based alloys. In this study, our previous three-dimensional (3-D) cellular automaton (CA) model for α-Al dendritic growth was extended to include eutectic (α-Al + Si) transformation in multi-dendrite domains, providing a complete solidification simulation of critically important Al-Si based alloys. The quantitative results of the Si phase in the eutectic microstructure were experimentally validated using scanning electron microscopy and deep etching techniques. The simulation results show a good agreement with the experimental observations and calculations by the Scheil model and lever rule. This 3-D CA model is useful for predicting and optimizing the solidification microstructure including eutectic transformation during solidification processing such as casting, potentially welding, and additive manufacturing.

Simulation Approach for Random Diffusion of Chloride in Concrete under Sustained Load with Cellular Automata

Steel bar corrosion caused by chloride is the major reason for concrete structure durability failures in a corrosive environment. An accurate simulation of chloride ion diffusion in concrete is hence critical to durability design, maintenance, and reinforcement of concretes in erosive environments. To accurately simulate actual chloride ion diffusion in concretes, an improved three-dimensional neighborhood type is proposed according to the mechanism of chloride ion diffusion in concrete, and a three-dimensional cellular automaton model (3D CA model) for describing the diffusion process of chloride in concrete is established based on this neighborhood type. The accuracy and correctness of simulation results obtained from the 3D CA model were verified by comparison with Fick’s second law analytical solutions. Based on the 3D CA model, an improved modified 3D CA model is developed (3D RTCA model) which takes into account random chloride ion distribution in concrete, the time dependence of the coefficient of chloride ion diffusion, and the structure stress level effect on chloride ion diffusion. Numerical simulation results reveal that the 3D RTCA model has higher calculation accuracy in predicting long-term concentration of chloride in concretes, and the simulation results are closer to experimental findings than analytical results obtained based on Fick’s second law. Compared with Fick’s second law analytical solutions, the 3D RTCA model can reflect more truly the cross-sectional stress level effect on chloride ion diffusion through simple local evolution rules. Besides, the 3D RTCA model can genuinely describe the randomness and uncertainty of the chloride diffusion process. The 3D RTCA model developed in the current study provides a novel perspective and method to investigate chloride ion diffusion in concrete from structural level.

Parallel implementation of cellular automata model of electron-hole transport in a semiconductor

A parallel implementation of a three-dimensional cellular automaton (CA) model of electron — hole transport in a semiconductor is presented. Carriers transport is described by a nonlinear system of drift-diffusion-Poisson equations. This system includes the drift-diffusion equations in divergence form for electrons and holes and the Poisson equation for the potential, the gradient of which enters the drift-diffusion equations as the drift velocity. We solve the drift-diffusion-Poisson system for the three-dimensional case using the CA approach. A regular mesh is introduced in the three-dimensional domain, and the solution is calculated in all lattice cells. The drift-diffusion-Poisson system is solved by an iterative algorithm consisting of two alternating steps. In the first step, the electron and hole concentrations are calculated. In the second step, the drift velocity is calculated as the gradient of the solution to the Poisson equation with the right-hand side depending on the electron and hole concentrations. The correctness of both CA models is tested against the exact solutions of the drift-diffusion and Poisson equations for some special cases. A parallel implementation of the iterative CA algorithm using the domain decomposition method is presented. The efficiency of the parallel code is analyzed. The simulation results are obtained for the model parameters specific to GaN semiconductors.

Multi-GPU implementation of a cellular automaton model for dendritic growth of binary alloy

A parallel three-dimensional (3D) cellular automaton (CA) model is developed for dendritic growth simulation during solidification of binary alloy. The CA model is implemented using the graphic processing unit (GPU). Compute Unified Device Architecture (CUDA) is combined with Message Passing Interface (MPI) to perform large-scale simulation on the multi-GPU architecture. A CUDA-Aware OpenMPI is used to reduce the communication cost between different GPU cards. The accuracy and the speedup ratio of the CA model are investigated. First, the simulated steady-state tip velocity under different undercooling is compared with the LGK analytical solution. Then, dendritic growth during directional solidification and equiaxed multi-dendrite growth are simulated. Dendritic growth simulation with a domain size up to 453 million grids is performed. Finally, the speedup ratio of the CA model using four GPUs is compared with that using four CPU processes via MPI, which leads to a maximal speedup ratio of 152.19. The developed CA model achieves high parallel performance compared with the MPI-based parallel model. The developed CA model is capable of predicting the columnar to equiaxed transition (CET) event during directional solidification. The results indicate that increasing cooling rate promotes the occurrence of the equiaxed dendrites in the melt ahead of the solidification front, and equiaxed dendrites are prone to nucleate in the columnar dendrite groove zone when a middle cooling rate is applied.

Microstructure evolution during dendrite coarsening in an isothermal environment: 3-D cellular automaton modeling and experiments

Microstructure evolution of columnar dendrites during isothermal coarsening is simulated by a three-dimensional (3-D) cellular automaton (CA) model. The 3-D CA model incorporates the thermodynamics, and kinetics aspects influenced by the factors of temperature, composition, and interfacial weighted mean curvature. After quantitative validation through grid independence test and the liquid pool migration problem, the 3-D CA model is applied to simulate the dendrite coarsening phenomena caused by the simultaneous melting and solidification for a SCN-2.0 wt.% ACE alloy in an isothermal environment. Several typical features of dendrite coarsening occurring in the 3-D space are reproduced by the CA model, including melting of tertiary and small secondary branches, interdendritic groove base advancement, and coalescence of neighboring dendrite arms. The simulated dendritic microstructure compares well with the in situ experimental observation. Besides, the simulated specific surface area of the solid/liquid interface decreases with time following a power-law relation with an exponent of 1/3. The simulated distribution curve of the interfacial weighted mean curvatures becomes narrower with time, and the peak gradually shifts to zero, showing that the S/L interface is flattened during the isothermal dendrite coarsening. The average thickness of the primary dendrite arm and the secondary dendrite arm spacing (λ2) are found to increase with time, and the third power of λ2 shows a nearly linear relationship with time. The CA model not only visualizes morphological evolution of the dendrite arms, but also provides insight into the complex interaction among local melting/solidification, solute diffusion, and variation of weighted mean curvature in the 3-D space.

Three Dimensional Cellular Automata Sand Castle construction Model Based on Step Factor

It seems that everyone has the experience of making sandcastles on the leisure beach. In order to know how to build a beautiful and sturdy beach castle, this article simulates and analyzes the process of sand castles being eroded by water, and obtains some factors that extend the life of sand castles. First, we model the waves. Simplify the waves into a sine wave. Using calculus and fluid mechanics, the water wave impact force on a particle on the sand castle was calculated. Second, we model the sandcastle. Discretizing time, space, and state, we innovatively introduce the concepts of friction and impact into the model, and propose a new Three-dimensional Cellular Automaton model combined with step factors. We consider each gravel or water molecule as a cell, and simultaneously introduce the moving probability and collision probability to describe the moving trend and collision between the gravel cells, respectively. In order to effectively describe the loss of sand castles in the waves, the gravel cells that are in direct contact with the water cells will receive an impact force calculated by the first model. According to the law of large numbers, we use enough cells to simulate the erosion of sandcastles by the waves. It was found that within a certain range, the farther from the ocean and the more the total gravel, the stronger the anti-erosion ability of the sand castle.

Computational modeling of degradation process on the mechanical performance of Poly-lactic acid /Magnesium composite

In this study, the corrosion behavior of biodegradable composites is modeled. These composites are made of Poly-Lactic acid and Magnesium with different volume fractions. The scanning electron microscopy images of these composites were taken, and statistical reconstruction of the composite based on scanning electron microscopy images was done by the phase recovery algorithm, the three-dimensional structure of this composite was extracted with this reconstruction, then a three-dimensional cellular automata model was developed to predict the corrosion of this composite. Results of experiments for the composite with 10% of Magnesium volume fraction were used to calibrate the parameters of the cellular automata model, and with these parameters, the results for the composite with 5% of Magnesium volume fraction was obtained, and with comparing these data with the results of the experiment, our model was validated. In the end, we estimated the mechanical properties of these composites and analyzed the results.

Three-Dimensional Evacuation Model with Applicability Based on the Theory of Cellular Automata

With the increasing number of terror attacks in France, evacuation plans at the Louvre, which receives tens of thousands of visitors every year, has become the key to the safety of visitors. Our goal is to make a reasonable emergency evacuation plan. First, we establish a two-dimensional evacuation model, based on the model, we build a three-dimensional cellular automaton model, which takes the speed of moving up and down the stairs of the occupants, positional attraction and probability of collisions into consideration. Second, we use Pathfinder for simulation. Finally, we find that the exit 1(secondary entrance) should be opened when the visitor density is high, while the exit 3 should be opened when the visitor density is low; and the Passage Richelieu entrance has the greatest influence on evacuation time, and emergency personnel should choose the Pyramid entrance as their emergency entrance.

Predicting gas and shrinkage porosity in solidification microstructure: A coupled three-dimensional cellular automaton model

Porosity formation during solidification of aluminum-based alloys, due to hydrogen gas and alloy shrinkage, has been a major issue adversely affecting the performance of solidification products such as castings, welds or additively manufactured components. A three-dimensional cellular automaton (CA) model has been developed, for the first time, to couple the predictions of hydrogen-induced gas porosity and shrinkage porosity during solidification microstructure evolution of a binary Al-Si alloy. The CA simulation results are validated under various cooling rates by porosity measurements in an experimental wedge die casting using X-ray micro computed tomography (XMCT) technique. This validated porosity moel provides a critical link in integrated computation materials engineering (ICME) design and manufacturing of solidification products.

Three-dimensional cellular automaton simulation of austenite grain growth of Fe-1C-1.5Cr alloy steel

A three-dimensional (3D) cellular automaton model describing the austenite grain growth process of Fe-1C-1.5Cr alloy steel is established. This model proposes a discrete representation of the three-dimensional grain boundary curvature, and calculates the 26 neighbor lattice points to the center lattice point in three-dimensional space. The strength of neighbor lattice point’s influence is determined by the distance from the center lattice point. The paper presents the calculation steps of the three-dimensional cellular automaton model and the calculation methods of key parameters such as grain boundary curvature and grain growth speed, and validates the model calculation results. The results show that the model established in this paper can well describe the austenite grain growth process of Fe-1C-1.5Cr alloy steel at different temperatures. The model can be used to study the grain growth process of Fe-1C-1.5Cr alloy steel under three-dimensional conditions.

Three-Dimensional Cellular Automata Simulation of the Austenitizing Process in GCr15 Bearing Steel.

On the basis of the two-dimensional cellular automaton model, a three-dimensional cellular automaton model of austenitizing process was established. By considering the orientation of pearlite layer and the direction of austenite grain growth, the velocity of the interface was calculated during the austenitizing process. The austenitizing process of GCr15 steel was simulated, and the anisotropy of grain growth rate during austenitization was demonstrated by simulation results. By comparing the simulation results with the experimental data, it was found that the calculated results of the three-dimensional cellular automaton model established in this paper were in good agreement with the experimental results. By using this model, the three-dimensional austenitizing process of GCr15 steel at different temperatures and under different processing times can be analyzed, and the degree of austenitization can be predicted.

Three-dimensional cellular automaton simulation of coupled hydrogen porosity and microstructure during solidification of ternary aluminum alloys

Hydrogen-induced porosity formed during solidification of aluminum-based alloys has been a major issue adversely affecting the performance of solidification products such as castings, welds or additively manufactured components. A three-dimensional cellular automaton model was developed, for the first time, to predict the formation and evolution of hydrogen porosity coupled with grain growth during solidification of a ternary Al-7wt.%Si-0.3wt.%Mg alloy. The simulation results fully describe the concurrent nucleation and evolution of both alloy grains and hydrogen porosity, yielding the morphology of multiple grains as well as the porosity size and distribution. This model, successfully validated by X-ray micro-tomographic measurements and optical microscopy of a wedge die casting, provides a critical tool for minimizing/controlling porosity formation in solidification products.

Three-dimensional cellular automaton model for the destruction of brittle heterogeneous materials

The article investigates the evolution modes of cluster damage structure in brittle heterogeneous materials by using a three-dimensional probabilistic cellular automaton. By comparing the data of computer and physical experiments, there was established the essential role of the model parameter, which describes the intensity of the material destruction process under the influence of local overstress near the existing damage clusters - the probability of perimeter germination. The comparison of kinetic curves of damage accumulation and correlation functions showed that, depending on the probability value for damage cluster perimeter germination, two qualitatively different modes of evolution of damage accumulation process are observed. In this case, the best correspondence of correlation functions in model and physical experiment on pulsed electromagnetic emission is observed for perimeter germination probability values smaller than 0.2.

Examination of Dendritic Growth During Solidification of Ternary Alloys via a Novel Quantitative 3D Cellular Automaton Model

A three-dimensional (3D) quantitative cellular automaton (CA) model was developed to simulate dendritic growth during solidification processing of ternary alloys. A detailed method was proposed to solve solute diffusion and calculate the solid fraction for ternary alloys during solidification. The present model has been shown to accurately predict dendrite morphologies and solute distributions of both single equiaxed dendrite and columnar dendrites during solidification. The model was also used to study the influence of the concentration of a third component (Mg in the Al-Si-Mg system) on dendritic growth. With increasing Mg concentration, the steady-state dendrite tip growth velocity was shown to decrease, resulting in a shorter primary dendrite length. The 3D CA simulation results agree well with the prediction of the LGK theoretical model. Multi-columnar dendrite growth was simulated with different cooling rates. Primary and secondary dendrite morphology was shown to have little variation at low cooling rates (1.0 to 2.0 K/s) and a constant undercooling. The average secondary dendrite arm spacing was shown to decrease with the increase of cooling rate within this range. The 3D CA simulation results are in good agreement with the experimental directional solidification experiments of a commercial A356 (Al-7 wt pct Si-0.3 wt pct Mg).

Parallel implementation of a three-dimensional cellular automaton model of the electrochemical oxidation of carbon “Ketjenblack EC-600JD”

The paper presents a three-dimensional cellular automaton model of electrochemical oxidation of the carbon. The sample of the electro-conductive carbon black “Ketjenblack EC-600JD” consisting of granules of carbon is simulated. The electrochemical oxidation of the carbon granules occurs through a few successive stages. Parallel implementation of the three-dimensional cellular automaton model of carbon corrosion is developed. The efficiency and speedup of the parallel code are analyzed. The portions of surface carbon atoms and atoms with different degree of oxidation are computed by the parallel code. Based on the obtained values of atom portions the electrochemical capacity is calculated. The results of computer simulation are compared with the experimental data.

A parallelized three-dimensional cellular automaton model for grain growth during additive manufacturing

In this paper, a parallelized 3D cellular automaton computational model is developed to predict grain morphology for solidification of metal during the additive manufacturing process. Solidification phenomena are characterized by highly localized events, such as the nucleation and growth of multiple grains. As a result, parallelization requires careful treatment of load balancing between processors as well as interprocess communication in order to maintain a high parallel efficiency. We give a detailed summary of the formulation of the model, as well as a description of the communication strategies implemented to ensure parallel efficiency. Scaling tests on a representative problem with about half a billion cells demonstrate parallel efficiency of more than 80% on 8 processors and around 50% on 64; loss of efficiency is attributable to load imbalance due to near-surface grain nucleation in this test problem. The model is further demonstrated through an additive manufacturing simulation with resulting grain structures showing reasonable agreement with those observed in experiments.

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.

A three-dimensional cellular automaton model of dendrite growth with stochastic orientation during the solidification in the molten pool of binary alloy

A three-dimensional (3D) cellular automaton model is developed for the prediction of dendrite growth with stochastic orientation during solidification process in the molten pool of binary alloy. An angle-information transfer method is proposed for improving cellular automaton technique to simulate the growth of the dendrites whose preferred growth direction owns stochastic misorientation with respect to the direction of the coordinate system. Dendrite morphologies and solute distributions of single dendrite growth and multi-dendrite growth are able to be obtained by the simulation using present model. The model is also employed to study the difference between two-dimensional simulation and 3D simulation on solute segregation and dendritic growth. Using the established model, 3D multi-columnar dendrites with stochastic crystallographic orientations can be obtained efficiently, and the competitive growth and impinging of dendrites can be reproduced in practice. The simulation results agree well with the experimental results.