Two-dimensional Cellular Automata Research Articles

Modeling and simulation of grain growth for FGH96 superalloy using a developed cellular automaton model

Through heat treatment experiments and numerical simulations, the effects of the heating temperature (1313–1423 K) and holding time (10–240 min) on the grain growth behavior of the extruded FGH96 alloy were investigated. A two-dimensional cellular automata (CA) model that considered the dissolution of the γ′ phase over time and the distribution characteristics with different sizes was developed to explore the grain growth behavior above the γ′ phase over-solution temperature (1423 K) and below the γ′ sub-solution temperature (1383 K), respectively. The results showed that the rate of grain growth of FGH96 alloy was obviously enhanced when the heating temperature exceeded 1363 K, which was mainly related to the dissolution of the γ′ phase, and the grain growth of FGH96 alloy mainly occurred during the initial stage of insulation. The grain growth model of the extruded FHG96 alloy could accurately predict the grain growth behavior, and the simulation results were in good agreement with the experimental results at over-solution temperature or sub-solution temperature. The effects of volume fraction and radius of γ′ phase on the grain growth behavior of FGH96 alloy were studied by simulating the grain growth behavior of FGH96 alloy under different sizes and volume fractions of γ′ phase. The results follow the Zener relation, and the coefficient n in the Zener relation was determined by fitting the simulation results.

A novel zero-watermarking scheme based on two-dimensional cellular automata

A novel zero-watermarking scheme based on two-dimensional cellular automata

The game of life as a species model

Conway's classic game of life is a two-dimensional cellular automaton in which each cell is alive or dead and evolves according to simple rules that depend solely on the number of live cells in its immediate neighborhood. The emergence of complex multi-cellular objects provides a fascinating vehicle for exploration. A variant of the classic game of life is presented, the generalized semi-classical game of life, in which each cell contains a qubit that evolves by repeated application of birth, death, and survival operators. Species are characterized by just two parameters: a preferred neighborhood liveness representing the tendency to herd and a resilience parameter representing species' vulnerability to environmental changes. This generalized model provides the opportunity to model the fortune of species and to compare to available data. The model is shown to mimic environmental catastrophes and is illustrated by the model's prediction of a return to the pre-hunting level of the global whale population by 2140. A student-designed predator–prey model is shown to qualitatively describe the fate of strongly- and weakly coupled predator–prey systems (snowshoe hare/lynx and rabbit/fox, respectively) and sudden and slow predatory impact (dodo and diprotodon, respectively).

2D Linear CA with Mixing Boundary Conditions and Reversibility

In this paper, we consider two-dimensional cellular automata (CA) with the von Neumann neighborhood. We study the characterization of 2D linear cellular automata defined by the von Neumann neighborhood with new type of boundary conditions over the field [Formula: see text]. Furthermore, we investigate the rule matrices of 2D von Neumann CA by applying the group of permutations [Formula: see text]. Moreover, the algorithm for computing the rank of rule matrices is given. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for considered two-dimensional cellular automata are obtained.

Chaos and Cellular Automata-Based Substitution Box and Its Application in Cryptography

Substitution boxes are the key factor in symmetric-key cryptosystems that determines their ability to resist various cryptanalytic attacks. Creating strong substitution boxes that have multiple strong cryptographic properties at the same time is a challenging task for cryptographers. A significant amount of research has been conducted on S-boxes in the past few decades, but the resulting S-boxes have been found to be vulnerable to various cyberattacks. This paper proposes a new method for creating robust S-boxes that exhibit superior performance and possess high scores in multiple cryptographic properties. The hybrid S-box method presented in this paper is based on Chua’s circuit chaotic map, two-dimensional cellular automata, and an algebraic permutation group structure. The proposed 16×16 S-box has an excellent performance in terms of security parameters, including a minimum nonlinearity of 102, the absence of fixed points, the satisfaction of bit independence and strict avalanche criteria, a low differential uniformity of 5, a low linear approximation probability of 0.0603, and an auto-correlation function of 28. The analysis of the performance comparison indicates that the proposed S-box outperforms other state-of-the-art S-box techniques in several aspects. It possesses better attributes, such as a higher degree of inherent security and resilience, which make it more secure and less vulnerable to potential attacks.

Irreversibility of 2D Linear CA and Garden of Eden

In this paper, we consider a pentagonal lattice and we investigate the rule matrix with null boundary condition for two-dimensional cellular automata with the field [Formula: see text] (the set of integers modulo [Formula: see text]) and analyze their characteristics. Moreover, an algorithm of computing the rank of rule matrix with null boundary condition for von Neumann neighborhood is developed. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for two-dimensional cellular automata are obtained.

Seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one.

We consider two-dimensional cellular automata with the von Neumann neighborhood that satisfy two properties of interest from a modeling viewpoint: rotation symmetry (i.e., the local rule is invariant under rotation of the neighborhood by 90^{∘}) and number conservation (i.e., the sum of all the cell states is conserved upon every update). It is known that if the number of states k is smaller than or equal to six, then each rotation-symmetric number-conserving cellular automaton is isomorphic to some k-ary one, i.e., one with state set {0,1,...,k-1}. In this paper, we exhibit an example of a seven-state rotation-symmetric number-conserving cellular automaton that is not isomorphic to any septenary one. This example strongly supports our plea that research into multistate cellular automata should not only focus on those that have {0,1,...,k-1} as a state set.

Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain

The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Markov chain. The master equation for the probability distribution is derived and solved exactly using the probability-generating function method. The probability distribution is expressed analytically in terms of Jacobi polynomials. The moments of the obtained solution allowed us to derive the exact analytical formula for the parametric dependence of the diffusion coefficient in the two-dimensional cellular automaton with the Margolus neighbourhood. Our analytic results agree with earlier empirical results of other authors and refine them. The results are of interest for the modelling two-dimensional diffusion using cellular automata especially for the multicomponent problem.

The Effect of Nanoflare Flows on EUV Spectral Lines

The nanoflare model of coronal heating is one of the most successful scenarios to explain, within a single framework, the diverse set of coronal observations available with the present instrument resolutions. The model is based on the idea that the coronal structure is formed by elementary magnetic strands which are tangled and twisted by the displacement of their photospheric footpoints by convective motions. These displacements inject magnetic stress between neighboring strands that promotes current sheet formation, reconnection, plasma heating, and possibly also particle acceleration. Among other features, the model predicts the ubiquitous presence of plasma flows at different temperatures. These flows should, in principle, produce measurable effects on observed spectral lines in the form of Doppler shifts, line asymmetries and nonthermal broadenings. In this work we use the two-dimensional cellular automaton model (2DCAM) developed in previous works, in combination with the enthalpy-based thermal evolution of loops (EBTEL) model, to analyze the effect of nanoflare heating on a set of known EUV spectral lines. We find that the complex combination of the emission from plasmas at different temperatures, densities and velocities, in simultaneously evolving unresolved strands, produces characteristic properties in the constructed synthetic lines, such as Doppler shifts and nonthermal velocities up to tens of km s−1 for the higher analyzed temperatures. Our results might prove useful to guide future modeling and observations, in particular, regarding the new generation of proposed instruments designed to diagnose plasmas in the 5–10 MK temperature range.

Cellular automata that generate symmetrical patterns give singular functions

In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we normalize the number of nonzero states of the patterns, take the limits, and give one-variable functions for the limit sets. We can obtain a one-variable function for each limit set and show that the resulting functions are singular functions, which are non-constant, are continuous everywhere, and have a zero derivative almost everywhere. We show that for Rule 90, a one-dimensional elementary cellular automaton (CA), and a two-dimensional elementary CA, the resulting functions are Salem’s singular functions. We also discuss two nonlinear elementary CAs, Rule 22 and Rule 126. Although their spatio-temporal patterns are different from that of Rule 90, their resulting functions from the number of nonzero states equal the function of Rule 90.

Cryptographic Services Based on Elementary and Chaotic Cellular Automata

In this paper, a security framework based on cellular automata is proposed, which is composed of three parts: entity authentication, data encryption, and decryption. Authentication is based on a zero-knowledge protocol, and its robustness is supported by the fact that the shared secret is not only an NP-complete problem but also dynamically transformed by a two-dimensional cellular automaton into a more complex secret. Thus, the authentication’s output is the key to data encryption/decryption in the following steps. One more strength of this system is the sensibility of the cellular automata to initial conditions—which is why we must know the initial configuration in order to obtain the key and the times that cellular automata evolved. All those characteristics give the system both robustness and security.

RAIN SURFACE RUNOFF MODELING USING CELLULAR AUTOMATON

The numerical solution of a system of two-dimensional shallow water (Saint-Venant) differential equations in partial derivatives for modeling the dynamics of surface runoff using digital elevation models obtained by laser altimetry leads to a significant increase in the computational load on the computer. The article offers a view on reasonable simplifications for the shallow water model, the two-dimensional cellular automaton for the modeling of rain surface runoff is proposed along with adequacy analysis of the calculations performed in their practical application.

Simulation of corrosion behavior of Fe–Cr–Ni alloy in binary NaCl–CaCl2 molten salt using a cellular automata method

In this paper, a two-dimensional cellular automata (CA) method is used to simulate the corrosion behavior of Fe–Cr–Ni alloy in NaCl–CaCl2 molten salt. Simulation results and experimental measurements show good agreement, revealing the detailed change and migration of components in the corrosion process of Fe–Cr–Ni alloy in molten chloride salt over time. The simulation revealed that the grain size, elemental composition of the alloy, and O2 and H2O contents in the molten salt are important factors that significantly affect the corrosivity of the NaCl–CaCl2 molten salt with respect to the Fe–Cr–Ni alloy. Additionally, the CA method shortens the period of experimental research and the progression of corrosion is predicted.

Simulation and experimental study of dynamic recrystallization process during friction stir vibration welding of magnesium alloys

A two-dimensional cellular automaton (CA) model was utilized to analyze the effect of mechanical vibration on microstructure evolution of AZ91 alloy during friction stir welding (FSW). The simulated results, namely grain topology, grain size distribution, average grain size, and also the dynamic recrystallization (DRX) fraction were compared with measured data. The adequate comparability between FEM and experimental data shows that the CA method can be applied to the analysis of the microstructure progression during the friction stir welding of AZ91 alloy. It is concluded that the dislocation density during the friction stir vibration welding (FSVW) is higher than that in the FSW process and the process of nucleation and grain growth is faster for samples during FSVW compared to FSW. The grain size modification and DRX phenomenon with various vibration frequencies were also simulated in detail during FSVW. It is found that vibration makes nucleation start earlier and decreases the proportion of the incubation period and the percentage of recrystallization as vibration frequency improves.

Ink Art Three-Dimensional Big Data Three-Dimensional Display Index Prediction Model

This paper starts with the study of realistic three-dimensional models, from the two aspects of ink art style simulation model and three-dimensional display technology, explores the three-dimensional display model of three-dimensional model ink style, and conducts experiments through the software development platform and auxiliary software. The feasibility of the model is verified. Aiming at the problem of real-time rendering of large-scale 3D scenes in the model, efficient visibility rejection method and a multiresolution fast rendering method were designed to realize the rapid construction and rendering of ink art 3D virtual reality scenes in a big data environment. A two-dimensional cellular automaton is used to simulate a brushstroke model with ink and wash style, and outlines are drawn along the path of the brushstroke to obtain an effect close to the artistic style of ink and wash painting. Set the surface of the model with ink style brushstroke texture patterns, refer to the depth map, normal map, and curvature map information of the model, and simulate the drawing effect of the method by procedural texture mapping. Example verification shows that the rapid visualization analysis model of ink art big data designed in this paper is in line with the prediction requirements of ink art big data three-dimensional display indicators. The fast visibility removal method is used to deal with large-scale three-dimensional ink art in a big data environment. High efficiency is achieved in virtual reality scenes, and the multiresolution fast rendering method better maintains the appearance of the prediction model without major deformation.

Numerical Simulation of Microstructure Evolution in Solidification Process of Ferritic Stainless Steel with Cellular Automaton

In order to study the basic principles of vibration-excited liquid metal nucleation technology, a coupled model to connect the temperature field calculated by ANSYS Fluent and the dendritic growth simulated by cellular automaton (CA) algorithm was proposed. A two-dimensional CA model for dendrite growth controlled by solute diffusion and local curvature effects with random zigzag capture rule was developed. The proposed model was applied to simulate the temporal evolution of solidification microstructures under different degrees of surface undercooling and vibration frequency of the crystal nucleus generator conditions. The simulation results showed that the predicted columnar dendrites regions were more developed, the ratio of interior equiaxed dendrite reduced and the size of dendrites increased with the increase of the surface undercooling degrees on the crystal nucleus generator. It was caused by a large temperature gradient formed in the melt. The columnar-to-equiaxed transition (CET) was promoted, and the refined grains and homogenized microstructure were also achieved at the high vibration frequency of the crystal nucleus generator. The influences of the different process parameters on the temperature gradient and cooling rates in the mushy zone were investigated in detail. A lower cooling intensity and a uniform temperature gradient distribution could promote nucleation and refine grains. The present research has guiding significance for the process parameter selection in the actual experimental.

SPATIAL SYNCHRONIZATION OF CELLULAR AUTOMATA IN EVOLUTIONARY PROCESSES SIMULATION TASKS

Manyappliedtasksaresimulatedbydifferenceequationsthatdescribethevectorofsystemstatesevolutionintime. Howeveritisrequiredtotakeintoaccount the spatial structure of simulated processes or systems in some tasks. Inpaperthepossibilityofaspatio-temporalprocessessimulation by cellular automata is considered. The brief review of two-dimensional cellular automataproperties is provided.Theprincipleofthemostfamoustwo-dimensionalcellularautomata“Game of Life”is described.Alsothegeneralwaytosettheseautomatainananalyticalformby Reaction-Diffusion equation is considered. Concrete forms of the Reaction equation and Diffusion equation are constructed and invariant sets for this system are defined. Thegeneralizationofanalyticalcellularautomata representation in total is provided.Asanexample,themodelofpopulationdevelopmentisconsidered. ItutilizestheclassicFerhulstequation, inwhichthespatialstructureistakenintoaccounthavingformofthecumulativeneighbors’impactonpopulationchangesrate.As per using of analytical form of cellular automata, differentschemasofsystemspatio-temporalcharacteristicscontrol aresuggested. Theseschemasarebasedonfeedback: delayedfeedback(thatis one that uses previous system states)and predictive feedback (that is one that uses predicted system states). As aresultthereismanaged to synchronize spatial configuration of cellular automataand it can be interpreted as stable population development. Particularly, cellular automata could work in cycle with cycle length set earlier. For cellular automata evolution visualization the algorithms and their computer implementation aredeveloped.Discrepancy functionis suggested, due to which it is possible to evaluate the synchronization accuracy. Research results and examples of received configurations are presented.

EVOLUTION OF TWO-DIMENSIONAL CELLULAR AUTOMATA. NEW FORMS OF PRESENTATION

The paper considers cellular automata and forms of reflection of their evolution. Forms of evolution of elementary cellular automata are known and widely used, which allowed specialists to model different dynamic processes and behavior of systems in different directions. In the context of the easy construction of the form of evolution of elementary cellular automata, difficulties arise in representing the form of evolution of two-dimensional cellular automata, both synchronous and asynchronous. The evolution of two-dimensional cellular automata is represented by a set of states of two-dimensional forms of cellular automata, which complicates the perception and determination of the dynamics of state change. The aim of this work is to solve the problem of a fixed mapping of the evolution of a two-dimensional cellular automaton in the form of a three-dimensional representation, which is displayed in different colors on a two-dimensional image The paper proposes the evolution of two-dimensional cellular automata in the form of arrays of binary codes for each cell of the field. Each time step of the state change is determined by the state of the logical "1" or "0". Moreover, each subsequent state is determined by increasing the binary digit by one. The resulting binary code identifies the color code that is assigned to the corresponding cell at each step of the evolution iteration. As a result of such coding, a two-dimensional color matrix (color image) is formed, which in its color structure indicates the evolution of a two-dimensional cellular automaton. To represent evolution, Wolfram coding was used, which increases the number of rules for a two-dimensional cellular automaton. The rules were used for the von Neumann neighborhood without taking into account the own state of the analyzed cell. In accordance with the obtained two-dimensional array of codes, a discrete color image is formed. The color of each pixel of such an image is encoded by the obtained evolution code of the corresponding cell of the two-dimensional cellular automaton with the same coordinates. The bitness of the code depends on the number of time steps of evolution. The proposed approach allows us to trace the behavior of the cellular automaton in time depending on its initial states. Experimental analysis of various rules for the von Neumann neighborhood made it possible to determine various rules that allow the shift of an image in different directions, as well as various affine transformations over images. Using this approach, it is possible to describe various dynamic processes and natural phenomena.

A cellular automaton model for a pedestrian flow problem

The evacuation phenomena in the two dimensional pedestrian flow model are simulated. The intended direction of the escape of pedestrians in panic situations is governed by the Eikonal equation of the pedestrian flow model. A new two-dimensional Cellular Automaton (CA) model is proposed for the simulation of the pedestrian flow. The solution of the Eikonal equation is used to define the probability matrix whose elements express the probability of a pedestrian moving in finite set of directions. The novelty of this paper lies in the construction of the density dependent probability matrix. The relevant evacuation scenarios are numerically solved. Predictions of the evacuation behavior of pedestrians, for various room geometries with multiple exits, are demonstrated. The mathematical model is numerically justified by comparison of CA approach with the Finite Volume Method for the space discretization and Discontinuous Galerkin Method for the implicit time discretization of pedestrian flow model.

Cellular Automata in Covid-19 prediction

At the end of 2019 a new coronavirus emerged, turning into a world pandemic. The new coronavirus is called COVID-19. Different countries handled the pandemic differently and our main focus in this article is on Poland. For better counteracting and managing the situation a model for predicting the dynamics of the pandemic is needed. In this article we present a model for simulating future infections taking into account various preventive measures and locations in Poland. We based the model on a two-dimensional cellular automata, with spatial dependencies between regions, different population and size of simulated regions.