2D Cellular Automata Research Articles

Oscillations and surface domain formation in a 3D lattice gas simulation study of metal passivation

We have performed a simulation study of a passivating metal surface in contact with electrolyte solution, using recently published 3D cellular automaton model. In the simulations, we observe oscillations of current and potential under potentiostatic and galvanostatic control, respectively. We have also rendered snapshots of the metal surface during several periods of oscillation. In the presented case the surface consists of active (bare metal) domains surrounded by passivated area. In analogy to the breathing mode of 1D models this pattern vanishes and reappears during the cycles.

State-efficient realization of fault-tolerant FSSP algorithms

The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) cellular arrays. It is shown that, under some constraints on the length and distribution of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in $$2n-2+p$$ steps and the algorithm is realized on a 1D cellular automaton of length $$n, 2 \le n \le 50$$, having 164 states and 4792 transition rules. In addition, we give by far a smaller-state implementation of a 2D FSSP algorithm that can synchronize any 2D rectangular array of size $$m \times n$$, possibly including at most O(mn) isolated defective zones, exactly in $$2(m+n)-4$$ steps on a cellular automaton with only 6 states and 935 transition rules.

Evaluation of the kinetics of recrystallization based on scaling 1D cellular automaton by DSC measurements

There are several methods to study the kinetics of the recrystallization process. DSC measurement is one of the most effective measurement methods. Cellular automata simulations are frequently used in computational studies. Both methods give and use kinetic constants. The aim of this study to find the relation between the kinetic constants. Additionally an evaluation method is introduced based on the DSC evaluation and cellular automaton which can compute a more detailed kinetic description of recrystallization.

A P system model of swarming and aggregation in a Myxobacterial colony

Bacterial communities provide an interesting subject for the study of emergence and complexity as the consequence of many local interactions. In particular, the soil-dwelling social bacterium Myxobacteria demonstrates two distinct types of motility, social motility via the sensing of bacterial slime deposits and adventurous motility. Both modes of motility are governed by local interactions. Using P systems, a membrane computing methodology based on compartmental rewrite rules for modelling computational processes; this work demonstrates how minimal set of rules can model swarming and aggregating behaviour in Myxobacteria bacterial populations. Our model uses a multi-environment P system similar a 2D cellular automaton to represent the substrate environment whilst stochastic rule selection dictates Myxobacterial motion according to behaviour observed in vitro. The rules account for both mechanisms of motility, the deposit and detection of slime, a change in direction due to C-signal induction and the mixing of population numbers. Simulations demonstrate an extensible computational framework for the modelling of bacterial behaviour, with the potential for extension into additional emergent behaviours.

A cellular automaton finite volume method for microstructure evolution during additive manufacturing

Additive manufacturing (AM) processes produce unique microstructures compared with other manufacturing processes because of the large thermal gradient, high solidification rate and other local temperature variations caused by the repeated heating and melting. However, the effect of these thermal profiles on the microstructure is not thoroughly understood. In this work, a 3D cellular automaton method is coupled to a finite volume method to predict the grain structure of an alloy, e.g. Inconel 718, fabricated by AM. The heat convection due to thermocapillary flow inside the melt pool is resolved by the finite volume method for a real and accurate temperature field, while an enriched grain nucleation scheme is implemented to capture epitaxial grain growth following the mechanism identified from experiments. Simulated microstructure results are shown to be in qualitative agreement with experimental result and the effects of the process parameters on both thermal characteristics and the grain structure are identified. The 3D cellular automaton finite volume method results establish our approach as a powerful technique to model grain evolution for AM and to address the process-structure-property relationship.

2D Triangular von Neumann Cellular Automata with Periodic Boundary

Cellular automata (CA) theory is a very rich and useful model of a discrete dynamical system that focuses on their local information relying on the neighboring cells to produce CA global behaviors. Although the main structure of CA is a discrete special model, the global behaviors at many iterative times and on big scales can be close to nearly a continuous system. The mathematical points of the basic model imply the computable values of the mathematical structure of CA. After modeling the CA structure, an important problem is to be able to move forwards and backwards on CA to understand their behaviors in more elegant ways. This happens in the possible case if CA is a reversible one. In this paper, we investigate the structure and the reversibility cases of two-dimensional (2D) finite, linear, and triangular von Neumann CA with periodic boundary case. It is considered on ternary field [Formula: see text] (i.e. 3-state). We obtain the transition rule matrices for each special case. It is known that the reversibility cases of 2D CA is generally a very challenging problem. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CA is reversible or not. In other words, the reversibility problem of 2D triangular, linear von Neumann CA with periodic boundary is resolved completely over ternary field. However, the general transition rule matrices are also presented to establish the reversibility cases of these special 3-states CA. Since the main CA structures are sufficiently simple to investigate in mathematical ways and also very complex for obtaining chaotic models, we believe that these new types of CA can be found in many different real life applications in special cases e.g. mathematical modeling, theoretical biology and chemistry, DNA research, image science, textile design, etc. in the near future.

Direct Macroscopic Modeling of Grain Structure and Macrosegregation with a Cellular Automaton–Finite Element Model

Grain structure and macrosegregation are two main factors determining mechanical properties of components and are strongly coupled during alloy solidification. A two-dimensional (2D) cellular automaton (CA)–finite element (FE) model is developed to achieve a direct macroscopic modeling of grain structure and macrosegregation during the solidification of binary alloys. With the conservation equations of mass, momentum, energy, and solute solved by a macroscopic FE model and the grain structure described by a microscopic CA model, a two-way coupling between the CA and FE models is applied. Furthermore, the effect of the fluid flow on the dendrite tip growth velocity is considered by modified dendrite tip growth kinetics. The CAFE model is applied to a quasi-2D benchmark solidification experiment of a Sn–3.0wt.%Pb alloy, and the grain structure and macrosegregation are predicted simultaneously. It is demonstrated that the model has a capacity to describe the undercooling ahead of the growth front. The growth directions of columnar grains, grain sizes, and columnar-to-equiaxed transition (CET) position are obviously modified by the fluid flow, and obvious segregated channels almost aligned with the orientations of the columnar grains are found. Qualitatively good agreement is obtained between the predicted segregation profiles and experimental measurements.

Numerical investigations of freckles in directionally solidified nickel-based superalloy casting with abrupt contraction in cross section

Abstract Evolution of grain structure and formation of freckle defects were simulated in directionally solidified nickel-based superalloy casting with abrupt contraction in cross-section area. A two-dimensional (2D) Cellular Automaton (CA) – Finite Element (FE) model was presented. The transport of liquid momentum, heat and solute mass at macroscopic scale, the development of grain envelopes at mesoscopic scale and microsegregation were fully coupled. Influences of alloying elements, cooling rate and geometrical shape of casting on freckle formation were studied. CAFE simulations show that through reducing the initial concentration of destabilizing element (W), the formation of freckle along casting edges was suppressed. Elements such as Co, Ti and Ta are not critical elements that affect density inversion in mushy region. Increasing cooling rate apparently decreased the freckling tendency. Without geometric contraction, there was no obvious formation of freckles along casting edges. It was numerically showed that the contraction in cross section introduced the high-permeability path for the upward fluid flow during directional solidification, which enhanced the freckling tendency.

Cellular automaton simulation of molten pool migration due to temperature gradient zone melting

Directional solidification is a common and important process in both scientific research and industrial practice. Owing to the presence of temperature gradients during directional solidification, local remelting and solidification in the mushy zone occurs, resulting in some typical phenomena such as temperature gradient zone melting (TGZM). The TGZM influences the solidifying microstructure and microsegregation significantly. In the present work, a two-dimensional (2D) cellular automaton (CA) model involving the mechanism of both solidification and melting is adopted to investigate the migration phenomena of molten liquid pools in the mushy zone due to the TGZM. The effect of pulling velocity, initial liquid pool position, temperature gradient, and alloy composition on the TGZM kinetics are studied. The simulation results are compared with the analytical predictions, and good agreement between two models is obtained. It is found that under a temperature gradient, the liquid pool always migrates towards the high temperature direction. When the pulling velocity is lower than the critical velocity, the liquid pool migrates through the liquidus into the bulk liquid and the time required for a liquid pool to reach the liquidus increases with pulling velocity increasing. On the other hand, when a pulling velocity higher than the critical value is adopted, the liquid pool moves towards the solidus and the time required for migrating liquid pool to reach the solidus decreases with pulling velocity increasing. For a given pulling velocity, the liquid pools located above the critical position move towards the liquidus, while the others gradually approach to the solidus. When a molten liquid pool migrates towards the liquidus, the migration velocity and liquid pool thickness are found to gradually increase, while the liquid pool composition decreases with time. Inversely, for the molten liquid pool that moves towards the solidus, the migration velocity and liquid pool thickness gradually decrease, while the liquid pool composition increases with time going by. The average migration velocity of liquid pool caused by the TGZM effect increases with temperature gradient increasing and alloy composition decreasing. The CA simulations provide an insight into the complicated interactions among the local temperature, solute distribution and diffusion, and the kinetics of local remelting and solidification in the TGZM process.

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).

Effect of thermal convection on columnar-to-equiaxed transition during solidification of Al-Cu alloy

To investigate the effect of three-dimension (3D) thermal convection on columnar-to-equiaxed transition (CET), the CET transition during the solidification of an Al-Cu alloy was simulated by 3D cellular automaton model coupled with the finite element method (CAFE). The thermal convection in the liquid phase was considered. The results show that the thermal convection in the liquid phase promotes the CET. When the convection is present, the temperature gradient at the start position of CET increases and the growth velocity of columnar dendrite decreases. The convection influences the formation of elongated equiaxed grain through changing the local temperature gradient and dendritic growth velocity.

3D Cellular Automata as a Computational Tool to Generate Artificial Porous Media

3D Cellular Automata as a Computational Tool to Generate Artificial Porous Media

ERT monitoring of gas injection into water saturated sands: Modelling and inversion of cross-hole laboratory data

A set of laboratory experiments were performed in order to investigate the capacity of the electrical resistivity tomography method to detect and monitor the circulation of gaseous CO2 into geological media. The experiments consisted of injecting reactive (CO2) or inert (N2) gases into reactive (carbonate) or inert (silica) sands saturated with water. The laboratory setup was a metric-scale cylindrical tank instrumented with electrodes placed on vertical rods, hence simulating a borehole to borehole monitoring network. The results demonstrated that the ERT method can efficiently detect, locate and image gas circulation plumes. Two main behaviors were identified: either an increase of the medium resistivity due to the invasion of the porous space by an electrically insulating gas phase, or a resistivity decrease induced by dissolution processes that enhance ionic concentrations in the saturating water. The observed anomalies showed varying geometric shapes depending on which type of sand was used. A 3D cellular automaton model was built and managed to reproduce different gas plume circulation patterns and the electrical resistivity response associated. This modeling work confirmed the link existing between the shapes of the resistivity anomalies, the gas plume geometry and the particle size distribution of the sand. Through a numerical sensitivity analysis, several measuring protocols were tested and developed in order to improve the imaging capacity of the electrical resistivity method. Results demonstrated that the most efficient protocol was different depending on the shape of the gas plume, thus stressing the importance of having a priori knowledge on the geological, lithological and hydraulic characteristics of the subsurface, in order to correctly assess the most likely gas plume geometry and hence design the most adapted monitoring protocol.

Domino pattern formation by cellular automata agents

Considered is a 2d cellular automaton with moving agents in a multi-agent system. The objective is to find agents controlled by a finite-state program (FSP) that can form domino patterns. The quality of a formed pattern is measured by the degree of order computed by counting matching $$3 \times 3$$ templates. An agent reacts on its own color, the color in front and whether it is blocked or not. It can change the color, move or not and turn into any direction. Four FSPs were evolved with k agents ($$1 \le k \le 4$$) initially placed in the corners of the field. For a $$12 \times 12$$ training field, the aimed pattern could be formed with a 100% degree of order. The performance was also high with other field sizes. Livelocks are avoided by using three different variants of the evolved FSP. The degree of order usually fluctuates after reaching a certain threshold, but it can also be stable, and the agents may show the termination by running in a cycle or by stopping their activity.

Cellular automata-based efficient method for the removal of high-density impulsive noise from digital images

Cellular automata (CA) are simple dynamical systems used for solving simple as well as complex problems. It uses a discrete space, like the space used in spatial domain for image processing, to store the values of the input data. Then, the input data can be processed according to the transition rules, functions, or, programmes, for the required output. This research work presents an efficient algorithm based on two dimensional cellular automata (2D CA), with hybrid rules under null and periodic boundary conditions, for filtering high-density impulsive noise from corrupted digital images. The most important advantage of the proposed method is that, it can be applied on all types of digital images (binary, greyscale, or, true color). The paper is organized as follows: Sect. 1 gives a brief introduction towards the problem of noise in digital images with emphasis on its solution. Further, this section presents a brief review of existing standard noise filtering methods based on general image processing and CA techniques. Section 2 discusses the basic concept of CA with special emphasis on 2D CA and related concepts. Section 3 presents the proposed algorithm for the removal of high-density impulsive noise from corrupted digital images. Section 4 discusses, in detail, the experimental results using both mathematical and subjective analysis; and, the extensive experimentation reveals that the proposed 2D CA based algorithm yields better results than the standard noise filtering algorithms. And, then finally Sect. 5 presents the conclusions and future scope.

A 3D Cellular Automaton for Cell Differentiation in a Solid Tumor with Plasticity

A model with spherical symmetry is proposed. We analyze the appropriate parameters of cell differentiation for different kinds of cells (Cancer Stem Cells (CSC) and Differentiated Cells (DC)). The plasticity (capacity to return from a DC to its previous state of CSC) is taken into account. Following this hypothesis, the dissemination of CSCs to another organ is analyzed. The location of the cells in the tumor and the plasticity range for possible metastasis is discussed.

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.

Work hardening behaviour in banded dual phase steel structures with improved formability

In this work, we show how the presence of microstructural banding and segregation affects the work-hardening behaviour of a dual phase steel with improved formability. This steel contains chemical segregation inherited from the casting process. Our previously developed 3D cellular automaton model allowed us to design thermo-mechanical processes to either promote or suppress banding. The bands are properly described as in-plane sheets of martensite grains. Mechanical testing data revealed a significant reduction in tensile strength in banded structures for a similar level of ductility. The work-hardening behaviour in the pre-yield regime, including the yield strength itself, is not correlated to the incidence of segregation and/or microstructural banding. The reduction in ultimate tensile strength in banded structures stems from a reduced work-hardening capacity in the post-yield regime. This is due to increased austenite stability in the banded steels, coupled to the anisotropic strain localisation in the ferritic matrix between martensite bands.

TrueLife – Evolving the Game of Life

John Conway’s Game of Life, published in Scientific American in 1970 is an attempt to model the behavior of life using a 2D cellular automaton. Although a breakthrough discovery for cellular automata and emergence theory, the game is restricted and incomplete due to its static, simplified rules. We will show that the game does not model life accurately and propose an alternative: TrueLife. TrueLife is a non­ deterministic, non­local, evolving Game of Life variant that we believe is more complete than Life for several key reasons. TrueLife is unique since at each generation a rule is chosen randomly from a list and applied to the current state. This allows the game to be inherently non­deterministic since it is impossible to know which rule is being applied at a given iteration. TrueLife will also be a learning simulation where rules that produce better results will be applied more frequently. Another unique aspect of TrueLife is the motivation behind the rules. The original Life rules are Darwinian and selfish acting only on local inputs that lead to local outputs. TrueLife’s rules will be non­local and act globally across the entire grid. TrueLife’s rules were formalized by drawing on much broader areas of science such as ecology, psychology and quantum theory. We are currently in the process of finding a model system to which TrueLife would be best suited.

3D AUTOCUT: a 3D segmentation algorithm based on cellular automata

An improved 3D cellular automaton segmentation algorithm is proposed. The purpose is to create a tri-directional automaton in order to segment 3D volumes. The improvement is that a force is created in the RGB space in order to improve the result of the 3D algorithm. Experiments on several synthetic volumes have shown that the proposed method achieves better segmentation results and efficiency than the other methods.