Two-dimensional Cellular Automata Research Articles

Modeling of morphological evolution of columnar dendritic grains in the molten pool of gas tungsten arc welding

A macro–micro coupled model for epitaxial nucleation and the subsequent competitive dendrite growth was developed to study the morphological evolution of both dendrite and grain structures in molten pool of the gas tungsten arc welding (GTAW) for Fe–C alloy. The simulation of heat and mass transfer in molten pool was conducted by the three-dimensional finite element (FE) model to obtain the transient solidification conditions. The process of epitaxial nucleation and the competitive dendrite growth was simulated by a two-dimensional cellular automata (CA) model. The size and random preferential orientations of substrate grains were considered in this model. The transient thermal conditions used in the CA model were obtained from the results of FE model through the interpolation method. In addition, the effects of the substrate grain size and the welding speed on the morphologies of both dendrite and grain structures were investigated. The simulated results indicate that dendrites with the preferential orientations parallel to the direction of the highest temperature gradient are more competitive during the competitive dendrite growth, and the morphology of resulting columnar grains is determined by the competition between different dendritic arrays. Under the same welding conditions, with the increase of substrate grain size, the average width of resulting columnar grains becomes larger, and the characteristics of dendrite structure within the columnar grains do not change obviously. Without considering the new nucleation in the melt, with the increase of welding speed, the dendrite structure in weld seam becomes much finer, and the average columnar grain width within the calculation domain of the CA model does not change obviously. The trend of the simulated results of dendrite arm spacing under various welding conditions are consistent with the analytical and experimental data.

A Modified Cellular Automaton Model for the Quantitative Prediction of Equiaxed and Columnar Dendritic Growth

Since the characteristic of dendrite is an important factor determining the performance of castings, a two-dimensional cellular automaton model with decentered square algorithm is developed for quantitatively predicting the dendritic growth during solidification process. The growth kinetics of solid/liquid interface are determined by the local equilibrium composition and local actual liquid composition, and the calculation of the solid fraction increment is based on these two compositions to avoid the solution of growth velocity. In order to validate the developed model, quantitative simulations of steady-state dendritic features over a range of undercooling was performed and the results exhibited good agreement with the predictions of LGK (Lipton–Glicksman–Kurz) model. Meanwhile, it is demonstrated that the proposed model can be applied to simulate multiple equiaxed dendritic growth, as well as columnar dendritic growth with or without equiaxed grain formation in directional solidification of Al–Cu alloys. It has been shown that the model is able to simulate the growth process of multi-dendrites with various preferential orientations and can reproduce a wide range of complex dendritic growth phenomena such as nucleation, coarsening of dendrite arms, side branching in dendritic morphologies, competitive growth as well as the interaction among surrounding dendrites.

A novel construction of chaotic dynamics base gliders in diffusion rule B2/S7

The dynamical behaviors of gliders (mobile localizations) in diffusion rule B2/S7 are quantitatively analyzed from the theory of symbolic dynamics in two-dimensional symbolic sequence space. Their intrinsic complexity is demonstrated by exploiting the relationship between one-dimensional and two-dimensional subshifts. Based on this rigorous approach and technique, the underlying chaos of the extant gliders and their combinations is characterized in a subtle way. It is demonstrated that they can be identified to distinct subsystems with very rich and complicated dynamics; that is, diffusion rule is topologically mixing and possesses positive topological entropy on each subsystem. This analytical assertion provides the fact that diffusion rule is covered with complex subsystems “almost everywhere”. Finally, it is worth mentioning that the procedure proposed in this paper is also applicable to all other gliders arising from the two-dimensional cellular automata therein. It is an extended discovery in both cellular automata and chaos theory.

Platform for Two-Dimensional Cellular Automata Models Implemented by Living Cells of Electrically Controlled Green Paramecia Designed for Transport of Micro-Particles

Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics. Here, we attempted to establish an experimental platform for mimicking microscopic traffic flow models at microscopic dimensions. We achieved this, by monitoring the flow of micro-sized particles transported by the motile cells of living microorganisms. Some researchers have described the cells of protozoan species as “swimming neurons” or “swimming sensory cells” applicable to biological micro-electro-mechanical systems or micro-biorobotics. Therefore these cells, in a controlled environment, may form a good model system for bio-implementable cellular automata for traffic simulation. The living cells of theParameciumspecies including those of green paramecia (Paramecium bursaria), actively migrate towards a negatively charged electrode when exposed to an electric field. This type of cellular movement is known as galvanotaxis.P. bursariawas chosen as amodel organismsince the ideal micro-vehicles required for micro-particle transport must have a particular particle packing capacity within the cells. The present study establishes that the movement of cells with or without the loading of microspheres (Φ, 9.75 µm) can be controlled on a two-dimensional plane under strict electrical controls. Lastly, implementation of microchips equipped with optimally sized micro-flow channels that allow the single-cell traffic of swimmingP. bursariawas proposed for further studies and mathematical modeling.

Modeling of Ferrite-Austenite Phase Transformation Using a Cellular Automaton Model

A two-dimensional (2D) cellular automaton (CA) model is proposed to simulate the ferrite-austenite transformation in binary low-carbon steels. In the model, the preferential nucleation sites of austenite, the driving force of phase transformation coupled with thermodynamic parameters, solute partition at the ferrite/austenite interface, and carbon diffusion in both the ferrite and austenite phases are taken into consideration. The proposed model is applied to simulate the ferrite-to-austenite transformation during isothermal heating at 760°C that is in the ferrite and austenite two-phase range, the austenite-to-ferrite transformation during continuous cooling, and carbon diffusion during tempering at different temperatures for an Fe-0.2969 mol.% C alloy. The results show that during the isothermal heating, austenite nucleates and grows. The austenite grains are mostly located at the boundaries of ferrite grains. The carbon concentration in austenite is higher than that in ferrite. The simulated microstructure agrees reasonably well with the experimental observation. During the continuous cooling process, the austenite-to-ferrite transformation occurs accompanied with carbon diffusion. After cooling from the heating temperature of 760°C to room temperature with a cooling rate of 2°C/s, the carbon concentration field is nearly uniform, while a higher cooling rate of 5°C/s results in a non-uniform carbon concentration field. After tempering at different temperatures for 20 min, the uniformity of carbon distribution increases with increasing tempering temperature. The simulation results are used to understand the mechanisms of the observed experimental phenomena that a cold-rolled low-carbon enameling steel presents different yield strengths after different heat treatment processes.

Coupled Cellular Automaton (CA) – Finite Element (FE) Modeling of Directional Solidification of Al-3.5 wt% Ni Alloy: A Comparison with X-ray Synchrotron Observations

Development of dendritic grain structure and mesa-segregation leading to localized intergranular eutectic is simulated during directional solidification of an Al - 3.5 wt% Ni alloy with two pulling velocities (4 mu m s(-1) and 8 mu m s(-1)) using a two-dimensional (2D) Cellular Automaton Finite Element (CAFE) model. 2D CAFE simulations are compared with in situ and real-time experiments characterized by means of X-ray radiography at the European Synchrotron Radiation Facility (Grenoble, France). Two nucleation models are used. The first model is established by listing all measured positions and orientations of experimentally-observed nucleation events. The undetermined value of the nucleation undercooling is then set to 0 degrees C for all nucleation sites. The other model considers a stochastic model with a Gaussian distribution of the nucleation site as a function of the undercooling. It is derived from series of experiments. The influences of the nucleation models and liquid convection on the grain structure characteristics are numerically investigated. A good agreement with experimental observations is achieved concerning the evolutions of both the dendritic and the eutectic growth fronts (i.e., the size of the mushy zone). The predicted grain size and elongation are compared with measurements for the two nucleation laws. Simulations using the list of nucleation events reach better agreement with the longitudinal profiles of the grain equivalent diameter and elongation factor compared to the stochastic nucleation model. Direct tracking of the eutectic growth front as well as three-dimensional analyses are found to be required for improvement of the predictions.

Cellular Automaton Modeling of Microporosity Formation during Solidification of Aluminum Alloys

A two-dimensional (2D) cellular automaton (CA)-finite difference method (FDM) model is proposed to simulate the dendrite growth and microporosity formation during solidification of aluminum alloys. The model involves a three-phase system of liquid, gas, and solid. The growth of both dendrite and gas pore is simulated using a CA approach. The diffusion of solute and hydrogen is calculated using the FDM. The model is applied to simulate the formation and interactions of dendrites and micropores in an Al-7wt.%Si alloy. The effects of initial hydrogen concentration and cooling rate on microporosity formation are investigated. It is found that the porosity nuclei with larger size grow preferentially, while the growth of the small porosity nuclei is restrained. The competitive growth between porosities and dendrites is also observed. With the increase of initial hydrogen concentration, the incubation time of porosity nucleation and growth decreases, and the percentage of porosity increases, while porosity density does not increase apparently. With the decrease of cooling rate, porosity nucleates and starts to grow at higher temperatures, and the percentage of porosity increases, but the porosity density displays a decreasing trend. In addition, at a slower cooling rate, the competitive growth between porosities and dendrites becomes more evident, leading to a more non-uniform distribution of porosity size, and an increased maximum porosity size. The simulation results agree reasonably with the experimental data in the literature.

Pedestrian Evacuation with a Improved Dynamic Parameters Model

A improved dynamic parameters model is presented based on cellular automata in this paper.The dynamic parameters: Direction-parameter, Empty-parameter and Cognition-parameter are formulated to simplify tactically the decision-making process of pedestrians, which can reflect the pedestrian judgment on the surrounding conditions and decide the pedestrians choice of action. Simulation of pedestrian evacuation and pedestrian moving rules were established, according to two-dimensional cellular automaton Moore neighborhood. In the improved model, the impact of the pedestrian density around exits is considered, the simulation and experimental results prove that this improvement makes sense, because besides the spatial distance to exits, people may also choose the exit according to the pedestrian density around exits. And the simulation results of improved model is compared with experiment, comparison shows that the improved model can reproduce the experiment well. The improved model is useful for further study, it is instructional significant for pedestrian evacuation, avoiding or reducing the number injuries.

Cellular-local Algorithm for Localizing and Estimating Changes in Binary Images

In this paper we consider the problem of detection of changes and estimation of the degree of these changes in a dynamically changing binary image. The authors introduce the numerical characteristic degree of change areas in dynamically changing binary images, based on the Jaccard similarity coefficient. To calculate this characteristic the authors developed an original architecture of a two-dimensional cellular automaton with the diffusion dynamics. We establish that cellular automaton configurations converge to a stationary configuration. The stationary configuration of a cellular automaton defines the desired characteristics for each area in dynamically changing binary images. The result can be presented as a grayscale image, that greatly facilitates the visual analysis of the dynamics of changes in dynamically changing binary images. The suggested approach can be used to detect and numerically estimate changes in the case when a number of brightness gradation comprises more than two values.

Self-Replicating Patterns in 2D Linear Cellular Automata

This paper studies the theoretical aspects of two-dimensional cellular automata (CAs), it classifies this family into subfamilies with respect to their visual behavior and presents an application to pseudo random number generation by hybridization of these subfamilies. Even though the basic construction of a cellular automaton is a discrete model, its macroscopic behavior at large evolution times and on large spatial scales can be a close approximation to a continuous system. Beyond some statistical properties, we consider geometrical and visual aspects of patterns generated by CA evolution. The present work focuses on the theory of two-dimensional CA with respect to uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) conditions. In total, there are 512 linear rules over the binary field ℤ2for each boundary condition and the effects of these CA are studied on applications of image processing for self-replicating patterns. After establishing the representation matrices of 2D CA, these linear CA rules are classified into groups of nine and eight types according to their boundary conditions and the number of neighboring cells influencing the cells under consideration. All linear rules have been found to be rendering multiple self-replicating copies of a given image depending on these types. Multiple copies of any arbitrary image corresponding to CA find innumerable applications in real life situation, e.g. textile design, DNA genetics research, statistical physics, molecular self-assembly and artificial life, etc. We conclude by presenting a successful application for generating pseudo numbers to be used in cryptography by hybridization of these 2D CA subfamilies.

COLOR IMAGE ENCRYPTION BASED ON TWO-DIMENSIONAL CELLULAR AUTOMATA

Cellular automaton (CA) has a lot of inherent features, such as simple regular structure, local interaction, random-like behavior and massive parallelism, which make it a good candidate to design cryptosystems. Therefore, a number of CA-based image encryption systems have been proposed, though the drawbacks of small key space and weak security in one-dimensional (1D) CA cryptosystems are obvious. In this paper, a novel image encryption scheme is presented using a two-dimensional (2D) CA with nonlinear balanced rules. During the whole process of encryption, the confusion operation is performed by the nonlinear rule of CA, while the diffusion operation is achieved by the local interactions among cells. So confusion and diffusion are well integrated in our proposed scheme. The corresponding simulations and analyses illustrate that the scheme has quite prominent cryptographic properties as well as high security.

REVERSIBILITY ALGORITHMS FOR 3-STATE HEXAGONAL CELLULAR AUTOMATA WITH PERIODIC BOUNDARIES

This paper presents a study of two-dimensional hexagonal cellular automata (CA) with periodic boundary. Although the basic construction of a cellular automaton is a discrete model, its global level behavior at large times and on large spatial scales can be a close approximation to a continuous system. Meanwhile CA is a model of dynamical phenomena that focuses on the local behavior which depends on the neighboring cells in order to express their global behavior. The mathematical structure of the model suggests the importance of the algebraic structure of cellular automata. After modeling the dynamical behaviors, it is sometimes an important problem to be able to move backwards on CAs in order to understand the behaviors better. This is only possible if cellular automaton is a reversible one. In the present paper, we study two-dimensional finite CA defined by hexagonal local rule with periodic boundary over the field ℤ3 (i.e. 3-state). We construct the rule matrix corresponding to the hexagonal periodic cellular automata. For some given coefficients and the number of columns of hexagonal information matrix, we prove that the hexagonal periodic cellular automata are reversible. Moreover, we present general algorithms to determine the reversibility of 2D 3-state cellular automata with periodic boundary. A well known fact is that the determination of the reversibility of a two-dimensional CA is a very difficult problem, in general. In this study, the reversibility problem of two-dimensional hexagonal periodic CA is resolved completely. Since CA are sufficiently simple to allow detailed mathematical analysis, also sufficiently complex to produce chaos in dynamical systems, we believe that our construction will be applied many areas related to these CA using any other transition rules.

Influence of natural convection during upward directional solidification: A comparison between in situ X-ray radiography and direct simulation of the grain structure

This paper presents a comparison between the in situ and real time observation of a directional solidification experiment carried out at the European Synchrotron Radiation Facility and a direct simulation of grain structure formation in the sample using a two-dimensional cellular automaton–finite element (CA-FE) model. In situ characterization of the columnar-to-equiaxed transition in a refined Al–3.5wt.% Ni alloy is achieved by synchrotron X-ray radiography. Two main characteristics are derived from the radiographs for each nucleated grain, namely the nucleation position in the sample and the orientation of the main trunk relative to the vertical temperature gradient. These data are then used as input for the CA-FE simulations. The CA-FE model takes into account the effects of macroscopic transport of heat, liquid momentum and solute mass on the development of the dendritic grain structure and vice versa. The influence of convection on macroscopic shape of the growth front, the grain structure, the microstructure distribution and macro-segregation is determined by a comparison between the experimental observations and results from the numerical simulations with and without fluid flow. Good qualitative agreement is obtained and limitations that are linked to the two-dimensional approximation and the need for direct tracking of the eutectic grain structure are pointed out.

Modeling and Simulation of Dynamic Recrystallization of GCr15 Steel Using Cellular Automaton Method

A modified two-dimensional (2-D) cellular automaton (CA) model was constructed to simulate dynamic recrystallization (DRX) process of GCr15 steel. Particle stimulated nucleation (PSN) was incorporated into the CA model to determine the influence of dispersed particles on the nucleation of DRX. In addition, the model included the effects of particles on increasing the dislocation density and pinning the grain boundaries for accurate determination of micro-structural evolution during DRX. The model was applied to simulate the DRX process of GCr15 bearing steel. DRX grain size and volume fraction were simulated using the CA model. The simulated results indicated that the simulated stable grain size of particle-containing model is closer to measured value than particle-free model. It was observed that DRX kinetics depends on both thermo-mechanical parameters and initial grain sizes. The calculated results were compared with the experimental findings in GCr15 bearing steel, the predictions show very good agreement with the experimental results.

Simulation of Grain Coarsening Using One-Dimensional Cellular Automaton

Numerous literature [1,4,5] has reported on the effective use of cellular automaton method for the simulation of short-range diffusion. Using this model for the simulation of short-range diffusional phase transformations therefore is a resolved issue. It is proven that two- or three-dimensional automata can reflect the course of the abovementioned processes realistically. What our study demonstrates more than in the past [1] is that two-dimensional stochastic cellular automaton simulation already presented before has been simplified. This time our automaton operates in one dimension [2], which has consequently reduced running time, thus, made it possible to enhance the efficiency of the scaling of simulation. In our previous work the results of scaling of one-dimensional simulation of the recrystallization process [3] were demonstrated, in our current study fitting is performed for measurement results of grain coarsening using one-dimensional cellular automaton.

A two-dimensional model for the quantitative simulation of the dendritic growth with cellular automaton method

A two-dimensional cellular automaton model was developed for simulating the diffusion-controlled dendritic growth in the low Péclet regime. The growth velocity of the solid/liquid interface was determined by the local solute diffusion, and the modified decentered square growth was improved to capture the first eight neighboring cells of the interface cell. In order to quantitatively validate the developed model, the free dendritic growth from an undercooled melt was simulated and compared with the classic Lipton–Glicksman–Kurz (LGK) analytical model. The results show that the predicted steady state tip velocity is in a reasonable agreement with the analytical value, and the dendritic growth can be quantitatively determined by the present model. Meanwhile, the simulation of the dendritic growth with random crystallographic orientation shows that the effects of the mesh dependency of the model on the growth kinetics and crystallographic orientation can be reduced to an accepted level. Moreover, the single dendritic growth and the multi-dendritic growth from an undercooled melt and the competitive dendritic growth in a unidirectional solidification were simulated and the dendritic growth features, such as crystallographic orientation, dendrite arm growing and coarsening, side branching, and arms fusion were graphically revealed.

Cellular Automata Model for Elastic Solid Material

The Cellular Automaton (CA) modeling and simulation of solid dynamics is a long-standing difficult problem. In this paper we present a new two-dimensional CA model for solid dynamics. In this model the solid body is represented by a set of white and black particles alternatively positioned in the x- and y-directions. The force acting on each particle is represented by the linear summation of relative displacements of the nearest-neighboring particles. The key technique in this new model is the construction of eight coefficient matrices. Theoretical and numerical analyses show that the present model can be mathematically described by a conservative system. So, it works for elastic material. In the continuum limit the CA model recovers the well-known Navier equation. The coefficient matrices are related to the shear module and Poisson ratio of the material body. Compared with previous CA model for solid body, this model realizes the natural coupling of deformations in the x- and y-directions. Consequently, the wave phenomena related to the Poisson ratio effects are successfully recovered. This work advances significantly the CA modeling and simulation in the field of computational solid dynamics.

Simulation of Evacuation Process in a Supermarket with Cellular Automata

Abstract The evacuation process in a supermarket is simulated by a two-dimensional Cellular Automata (CA) model in this paper. In this model, the interactions among people and those between people and obstacles in forward direction are considered, and the ‘Danger-degree’ concept is introduced to solve the path select of people evacuation process. This paper studies the effect of different shelves’ orientation and personal familiarity degree of exits to evacuation time. The result shows if occupants are fewer in supermarket, shelves orientations have little influence on evacuation time; but when occupants are larger, the longitudinal shelves orientation is more appropriate for evacuation in supermarket than transverse shelves orientation. The study also found that the evacuation time reduces, the exits’ available time improve with the increase of personal familiar with exits by induction. This can provide theoretical foundation for arranging the shelves orientation in supermarket as well as drafting contingency plans of pedestrians’ induction evacuation.

Impact of Neighborhood Structure on Epidemic Spreading by Means of Cellular Automata Approach

The main goal of this work is to propose a mathematical model, based on two-dimensional cellular automata (CA), to simulate an infectious disease outbreak. Specifically, we revisit the general deterministic SEIR (Susceptible Exposed Infectious Recovered) model by adding terms describing the spatial spreading of disease. The state of each cell is a 4-tuple where each component represents the portion of each compartment in the cell at a given time step. We emphasis the role of the basic reproduction number and its relation with some spatial characteristics of CA models. We study in particular, the impact of the neighborhood structure in relation to the spread of such infectious diseases. The presented model will serve as a basis to simulate real epidemics and will also be concerned with the control problem where efficient strategies for disease spread have to be investigated.

Effect of interface energy anisotropy on the dendritic growth in directional solidification

The dendritic growth patterns in directional solidification with different amplitudes of solid-liquid interface energy anisotropy were investigated using the two-dimensional cellular automata (CA) model. It is shown that when the preferred growth direction of the crystal was the same as the direction of thermal gradient, the solidification pattern would transform from seaweed to dendrite with the increase of amplitude of interface energy anisotropy. The amplitude of interface energy anisotropy could also influence the morphology of dendritic tips. As the amplitude of interface energy anisotropy increased, the dendritic tip radius, the liquid concentration ahead of the tip and the tip undercooling decreased. A power law relationship evisted between the stability parameter of dendritic tip and the amplitude of interface energy anisotropy in directional solidification. The primary arm spacing changed little with the increase of interface energy anisotropy. When the angle between the preferred growth direction of the crystals and the direction of thermal gradient was-40, and the amplitude of interface energy anisotropy increased, the solidification pattern would transform from seaweed to degenerated dendrite and finally to tilted dendrite.