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The Navier–Stokes equations are the basis for describing the flow of a viscus material and are used to model fluid motion from weather to air flow over a wing. These equations, however, tend to be notoriously difficult to solve, whether analytically or computationally, even for small systems due to their highly nonlinear nature. Thus, less complicated methods that are more computationally tractable are desirable in domains as varied as hydroelectric power to race car construction. At a fundamental level, fluids are composed of interacting molecules. Lattice Gas Cellular Automata (LGCA) represents an efficient way to simulate these interacting fluid particles on a lattice. LGCA captures the microscopic behavior of the fluid by applying simple collision and propagation rules at each lattice site. This leads to realistic macroscopic behavior that can be used to build insight about real fluid flow. Here, we show the Hardy, Pomeau, and de Pazzis model for simulating a lattice gas. The computational framework presented in this case study can also be expanded to more complicated LGCA.

Penelitian ini bertujuan untuk menghitung porositas efektif ( ф eff) dan permeabilitas ( k ) menggunakan model segitiga dengan variasi tinggi yaitu 3, 4, 5, 6 dan 7 cm. Perhitungan porositas dan permeabilitas yang efektif dilakukan dengan menggunakan model Lattice Gas Automata (LGA), yang diimplementasikan dengan bahasa pemrograman Delphi 7.0. Untuk model segitiga penghalang dengan tinggi 3, 4, 5, 6 dan 7 cm, nilai porositas efektif dan permeabilitas, masing-masing: ф eff (T1) = 0,1690, k (T1) = 0 , 001339 pixel2; ф eff (T2) = 0,1841, k (T2) = 0,001904 pixel2; ф eff (T3) = 0,1885, k (T3) = 0,001904 pixel2; ф eff (T4) = 0,1938, k (T4) = 0001925 pixel2; dan ф eff (T5) = 0,2053, k (T5) = 0,002400 pixel2. Dari hasil simulasi, diperoleh tinggi segitiga akan berpengaruh signifikan terhadap nilai porositas efektif dan permeabilitas. Pada segitiga lebih tinggi, menyebabkan tabrakan model aliran fluida LGA mengalami lebih banyak hambatan untuk penghalang, sehingga porositas efektif dan permeabilitas menurun. Sebaliknya, jika segitiga lebih rendah, menyebabkan tabrakan model aliran fluida LGA mengalami lebih sedikit hambatan untuk penghalang, sehingga porositas efektif dan permeabilitas meningkat. This research purposed to calculate the effective porosity ( f e ff ) and permeability (k) using the barrier model of the triangle with a high varying are 3, 4, 5, 6 and 7 cm. Effective porosity and permeability calculations performed using the model Lattice Gas Automata (LGA), which is implemented with Delphi 7.0 programming language. For model the barrier triangle with a high of 3, 4, 5, 6 and 7 cm, the value of effective porosity and permeability, respectively: f eff (T 1 )=0,1690, k(T 1 )=0,001339 pixel 2 ; f eff (T 2 )=0,1841, k(T 2 )=0,001904 pixel 2 ; f eff (T 3 )=0,1885, k(T 3 )=0,001904 pixel 2 ; f eff (T 4 )=0,1938, k(T 4 )= 0001925 pixel 2 ; and f eff (T 5 )=0,2053, k(T 5 )=0,002400 pixel 2 . From the simulation results, obtained by the high of the triangle will be a significant effect on the value of effective porosity and permeability. If the triangle highest, causing the collision of fluid flow models LGA experience more obstacles to the barrier, so that the effective porosity and permeability decrease. Conversely, if the triangle lower, causing the collision of fluid flow models LGA experience less obstacles to the barrier, so that the effective porosity and permeability increases. Keywords : Effective p orosity, permeability, model triangle , model LGA

Porous microstructure is a common surface morphology that is widely used in antifouling, drag reduction, adsorption, and other applications. In this paper, the lattice gas automata (LGA) method was used to simulate the non-uniform electrochemical machining of porous structure at the mesoscopic level. In a cellular space, the metal and the electrolyte were separated into orderly grids, the migration of corrosive particles was determined by an electric field, and the influences of the concentration gradient and corrosion products were considered. It was found that different pore morphologies were formed due to the competition between dissolution and diffusion. When the voltage was low, diffusion was sufficient, and no deposit was formed at the bottom of the pore. The pore grew faster along the depth and attained a cylindrical shape with a large depth-to-diameter ratio. As the voltage increased, the dissolution rates in all directions were the same, therefore, the pore became approximately spherical. When the voltage continued to increase, corrosion products were not discharged in time due to the rapid dissolution rate. Consequently, a sedimentary layer was formed at the bottom of the pore and hindered further dissolution. In turn, a disc-shaped pore with secondary pores was formed. The obtained simulation results were verified by experimental findings. This study revealed the causes of different morphologies of pores, which has certain guiding significance for non-uniform electrochemical machining.

Interactive simulations of water and cloth have important applications in many fields. However, because of the high time and space complexity of water–cloth interaction simulations, current simulation methods cannot meet the requirements of being both real time and realistic. In this paper, lattice gas automata (LGA)-based simulation method for water diffusion in cloth is proposed. In the proposed method, a three-dimensional mass–spring cloth model is constructed to simulate cloth with an arbitrary thickness and the geometric structure of the cloth is unified via the LGA lattice structure. The diffusion process of water in cloth is simplified as the movement process of water particles in the lattice structure, and the diffusion direction of the water is determined by the differences in the vertex saturation. The advantage of this method is that the physical parameters of the cloth are integrated into the diffusion equation via the three-dimensional model of the cloth and the saturation of the vertices. The D3Q27 diffusion model is used to simplify the complexity of the water particle diffusion algorithm, and the applicability of the algorithm to different types of cloths is improved. By experimenting with this method and comparing our results with those of other methods, it is found that the proposed method can effectively improve the simulation speed and realism and can realistically simulate the effects of wetting diffusion, water deformation, seepage dripping and cloth folding when the water and cloth interact.

From the perspective of cellular-automaton and by exploiting the interactions between travelling and localized waves the Lorentz Excitable Lattice Gas Automata (LELGA) is proposed and applied to the optimization of Lennard-Jones (LJ) atomic clusters. Our approach exploits the fact that a reaction-diffusion phenomenon with self-localized excitations cells which behave like quasi-particles, could be potentially used to implement dynamical computation. As an example of collision-based computation in reaction-diffusion systems based on cellular automata we provided our experimental results for unbiased optimization of Lennard-Jones (LJ) atomic clusters. We also demonstrated that the proposed method has successfully located the known global minima of LJ clusters for configuration space size of N ≤ 383.

In some gas–solid reactions, a new solid substance is produced. The product acts as a shield and prevents the collision between gas and solid reactants which further causes an incomplete reaction. If the molar volume of the new product differs from the solid reactant, the inner structure of porous media is changed as well. In this paper, we discuss such gas–solid reactions in porous media using the two-dimensional lattice gas cellular automata FHP-III model. We simulate the fluid flow and chemical reaction in different porous media. We also show the effects of porosity and morphology of the solid, and reaction probability on the reaction process. Results obtained from the simulations agree closely with the theory of gas–solid reactions and diffusion theories. Hence, the proposed model is a good choice to simulate gas–solid chemical reactions in porous media at the mesoscopic level.

A cellular automaton model is used for 3D simulations of metal corrosion in electrolyte solutions in potentiostatic conditions. The focus of this research is the formation of the passive layer on the metal surface. A general, mesoscopic-scale model is used. It contains only the elements necessary to reproduce the most important features of passivation. It is a 3D version of the 2D model presented in our previous papers. The simulations are run on graphics processing units which allow an effective operation on large 3D lattices. We consider the influence of three factors: the rate of corrosion (related to the electric potential), solubility of the corrosion products and their adhesion or absence of adhesion to the metal surface. For each set of the parameters we wait in the simulation run for the system to attain the steady state. Then we collect data characterizing steady-state properties such as electric current density, passive layer thickness, its roughness and overall morphology. The qualitative properties of passivation are reproduced in our 3D simulations confirming the conclusions based on earlier, 2D simulations.

This technical report discusses the application of the Lattice Boltzmann Method (LBM) and Cellular Automata (CA) simulation in fluid flow and particle deposition. The current work focuses on incompressible flow simulation passing cylinders, in which we incorporate the LBM D2Q9 and CA techniques to simulate the fluid flow and particle loading respectively. For the LBM part, the theories of boundary conditions are studied and verified using the Poiseuille flow test. For the CA part, several models regarding simulation of particles are explained. And a new Digital Differential Analyzer (DDA) algorithm is introduced to simulate particle motion in the Boolean model. The numerical results are compared with a previous probability velocity model by Masselot [Masselot 2000], which shows a satisfactory result.

In this work a mathematical model and simulation for the gluing of wood particles designated for particleboards is presented. The aim is to obtain a better understanding of the gluing process. Thus, the behaviour of wood particles during gluing is investigated and the resulting adhesive distribution across the surface of the wood particles is analysed. For developing a mathematical model, the modelling methods &ldquo, lattice gas cellular automata&rdquo, and &ldquo, random walk&rdquo, were used. The model was implemented in MATLAB and different scenarios were simulated for answering the main questions of the behaviour during gluing. The influences of different parameters on the adhesive distribution were investigated and quantitatively determined by several key figures. Based on these key figures, the effects of the mixing arm, realistic size distributions of wood particles and adhesive droplets, the transfer of adhesive, and the total mass of adhesive are discussed. Furthermore the results are compared with experimental measurements. The simulation results show that the model can feasibly be used for studying the gluing of wood particles. For a possible industrial application, additional research for developing a three-dimensional model is needed.

Lattice Gas Cellular Automata (LGCA) simulations are typical High-Performance Computing (HPC) applications commonly used to simulate fluid flows. Due to the computational locality and discretization of LGCA, these simulations can achieve high performance by using parallel computing devices like GPUs or multi-core CPUs. Nevertheless, many studies also have shown that state-of-the-art Field Programmable Gate Arrays (FPGAs) have enormous parallel computing potential and power-efficient for high-performance computations. In this paper, we present an FPGA-based fluid simulation architecture design for the LGCA method. Our design exploits both temporal and spatial parallelism inside the LGCA algorithm to scale up the performance on FPGA. We also propose an application-specific cache structure to overcome the memory bandwidth bottleneck. Furthermore, our development process is based on the High-Level Synthesis (HLS) approach that increases productivity. Experimental results on a Xilinx Vcu 1525 FPGA show that our design is able to achieve 17130.2 Million Lattice Updates Per Second (MLUPS).

The dynamics of wealth distribution plays a critical role in the economic market, hence an understanding of its nonequilibrium statistical mechanics is of great importance to human society. For this aim, a simple and efficient one-dimensional (1D) lattice gas automaton (LGA) is presented for wealth distribution of agents with or without saving propensity. The LGA comprises two stages, i.e., random propagation and economic transaction. During the former phase, an agent either remains motionless or travels to one of its neighboring empty sites with a certain probability. In the subsequent procedure, an economic transaction takes place between a pair of neighboring agents randomly. It requires at least 4 neighbors to present correct simulation results. The LGA reduces to the simplest model with only random economic transaction if all agents are neighbors and no empty sites exist. The 1D-LGA has a higher computational efficiency than the 2D-LGA and the famous Chakraborti–Chakrabarti economic model. Finally, the LGA is validated with two benchmarks, i.e., the wealth distributions of individual agents and dual-earner families. With the increasing saving fraction, both the Gini coefficient and Kolkata index (for individual agents or two-earner families) reduce, while the deviation degree (defined to measure the difference between the probability distributions with and without saving propensities) increases. It is demonstrated that the wealth distribution is changed significantly by the saving propensity which alleviates wealth inequality.

Purpose The holistic numerical model based on cellular automata (CA) and lattice Boltzmann method (LBM) are being developed as part of an integrated modelling approach applied to study the interaction of different physical mechanisms in laser-assisted additive layer manufacturing (ALM) of orthopaedic implants. Several physical events occurring in sequence or simultaneously are considered in the holistic model. They include a powder bed deposition, laser energy absorption and heating of the powder bed by the moving laser beam, leading to powder melting or sintering, fluid flow in the melted pool and flow through partly or not melted material, and solidification. The purpose of this study is to develop a structure of the holistic numerical model based on CA and LBM applicable for studying the interaction of the different physical mechanisms in ALM of orthopaedic implants. The model supposed to be compatible with the earlier developed CA-based model for the generation of the powder bed. Design/methodology/approach The mentioned physical events are accompanied by heat transfer in solid and liquid phases including interface heat transfer at the boundaries. The sintering/melting model is being developed using LBM as an independent numerical method for hydrodynamic simulations originated from lattice gas cellular automata. It is going to be coupled with the CA-based model of powder bed generation. Findings The entire laser-assisted ALM process has been analysed and divided on several stages considering the relevant physical phenomena. The entire holistic model consisting of four interrelated submodels has currently been developed to a different extent. The submodels include the CA-based model of powder bed generation, the LBM-CA-based model of heat exchange and transfer, the thermal solid-liquid interface model and the mechanical solid-liquid interface model for continuous liquid flow. Practical implications The results obtained can be used to explain the interaction of the different physical mechanisms in ALM, which is an intensively developing field of advanced manufacturing of metal, non-metal and composite structural parts, for instance, in bio-engineering. The proposed holistic model is considered to be a part of the integrated modelling approach being developed as a numerical tool for investigation of the co-operative relationships between multiphysical phenomena occurring in sequence or simultaneously during heating of the powder bed by the moving high energy heat source, leading to selective powder sintering or melting, fluid flow in the melted pool and through partly (or not) melted material, as well as solidification. The model is compatible with the earlier developed CA-based model for the generation of the powder bed, allowing for decrease in the numerical noise. Originality/value The present results are original and new for the study of the complex relationships between multiphysical phenomena occurring during ALM process based on selective laser sintering or melting, including fluid flow and heat transfer, identified as crucial for obtaining the desirable properties.

In this paper a model for the movement of wood chips and resin droplets within a resinating mixer is presented. The mixing of wood chips and resin droplets is a sub-process, which is called gluing or resin metering, of the manufacturing process of particleboards. The aim of this sub-process is to equally distribute the resin on the wood chips. As the process of gluing is not completely understood, two arising research questions are “How is the resin distributed across the surface of the wood chips?” and “How much resin penetrated into the wood particles?”. These aspects can hardly be determined using real experiments. A mathematical model in two dimensions is developed. A modified lattice gas cellular automaton (LGCA) is used to model the moving resin droplets and wood chips. Due to the difference in size of the wood and resin particles, some modifications regarding the definition of states, collisions and streaming for the movement of the wood chips are necessary. In the common LGCA a particle is located at a cell of a specific node, whereas in the modified LGCA presented, a wood particle occupies several nodes. Due to the different states (e.g. wood or resin), several collision rules had to be defined. For the streaming, each node of the same wood particle has the same underlying velocity. The simulation is carried out in MATLAB. Finally, first simulation results are presented. Based on the simulation the percentage of glued wood chips is calculated.

Originating from lattice gas automata theory, the lattice Boltzmann method (LBM) is an interesting alternative to the solving of Navier-Stokes equations. In contrast to isothermal simulations, for a while thermal flow simulations were challenging for LBM. Thermal flow simulations are important task in various fields of research. Despite a large amount of work and research the dynamics of thermally induced flows are still highly demanded. Motivation of this work is development of computational tool for simulation of the dynamics of thermal flows. To this purpose, we developed LES-LBM solver accelerated by the Graphics Processing Unit (GPU) on the CUDA computational platform, integrating LBM with Large Eddy Simulation (LES). Simplicity of coding is usually an appealing feature of the LBM. Conventional implementations of LBM suffer from high memory consumption and poor computational performance. The main advantage of the solvers based on GPU is their ability to perform significantly more floating point operations per unit time (FLOPS) than a Central Processing Unit (CPU) and a good scalability of explicit parallel algorithms. LES-LBM code was tested on the NVIDIA GeForce GTX 1050 ti and NVIDIA TESLA K80 GPUs.

In this paper, we develop an efficient lattice Boltzmann model for the two-phase moving contact line problem with variable density. The NavierStokes and CahnHilliard equations are recovered from the lattice Boltzmann model proposed by Fakhari and Rahimian [5]. To improve numerical stability, we present a semi-implicit lattice Boltzmann method together with a mixed finite difference scheme. In order to describe the behavior of the contact line motion on the boundary, we incorporate the generalized Navier boundary condition [25] by the nonequilibrium extrapolation method [8]. The proposed method is easy to implement and retains the advantage of the standard lattice Boltzmann method. Numerical tests are carried out to verify the proposed method. Our numerical results show that the present approach is able to model two-phase flows with variable density and moving contact line.

Direct numerical simulations of the transition process from steady laminar to chaotic flow are considered in this study with the relatively new incompressible lattice Boltzmann equation. Numerically, a multiple relaxation time fully incompressible lattice Boltzmann equation is implemented in a 2D driven cavity. Spatial discretization is 2nd-order accurate, and the Kolmogorov length scale estimation based on Reynolds number (Re) dictates grid resolution. Initial simulations show the method to be accurate for steady laminar flows, while higher Re simulations reveal periodic flow behavior consistent with an initial Hopf bifurcation at Re 7,988. Non-repeating flow behavior is observed in the phase space trajectories above Re 13,063, and is evidence of the transition to a chaotic flow regime. Finally, flows at Reynolds numbers above the chaotic transition point are simulated and found with statistical properties in good agreement with literature.

The accuracy of the Moment Method for imposing no-slip boundary conditions in the lattice Boltzmann algorithm is investigated numerically using lid-driven cavity flow. Boundary conditions are imposed directly upon the hydrodynamic moments of the lattice Boltzmann equations, rather than the distribution functions, to ensure the constraints are satisfied precisely at grid points. Both single and multiple relaxation time models are applied. The results are in excellent agreement with data obtained from state-of-the-art numerical methods and are shown to converge with second order accuracy in grid spacing.

Cellular automata (CA) are discrete time, space, and state models which are extensively used for modeling biological phenomena. CA are "on-lattice" models with low computational demands. In particular, lattice-gas cellular automata (LGCA) have been introduced as models of single and collective cell migration. The interaction rule dictates the behavior of a cellular automaton model and is critical to the model's biological relevance. The LGCA model's interaction rule has been typically chosen phenomenologically. In this paper, we introduce a method to obtain lattice-gas cellular automaton interaction rules from physically-motivated "off-lattice" Langevin equation models for migrating cells. In particular, we consider Langevin equations related to single cell movement (movement of cells independent of each other) and collective cell migration (movement influenced by cell-cell interactions). As examples of collective cell migration, two different alignment mechanisms are studied: polar and nematic alignment. Both kinds of alignment have been observed in biological systems such as swarms of amoebae and myxobacteria. Polar alignment causes cells to align their velocities parallel to each other, whereas nematic alignment drives cells to align either parallel or antiparallel to each other. Under appropriate assumptions, we have derived the LGCA transition probability rule from the steady-state distribution of the off-lattice Fokker-Planck equation. Comparing alignment order parameters between the original Langevin model and the derived LGCA for both mechanisms, we found different areas of agreement in the parameter space. Finally, we discuss potential reasons for model disagreement and propose extensions to the CA rule derivation methodology.

Recently, a new formulation of quantum mechanics, based on the concept of signed particles, has been suggested. In this paper, we introduce a cellular automaton which mimics the dynamics of quantum objects in the phase-space in a time-dependent fashion. This is twofold: it provides a simplified and accessible language to non-physicists who wants to simulate quantum mechanical systems, at the same time it enables a different way to explore the laws of Physics. Moreover, it opens the way towards hybrid simulations of quantum systems by combining full quantum models with cellular automata when the former fail. In order to show the validity of the suggested cellular automaton and its combination with the signed particle formalism, several numerical experiments are performed, showing very promising results. Being this article a preliminary study on quantum simulations in phase-space by means of cellular automata, some conclusions are drawn about the encouraging results obtained so far and the possible future developments.

A variable property-based lattice Boltzmann flux solver (VPLBFS) is proposed in this paper for thermal flows with partial or total variation in fluid properties in the low Mach number limit. In the solver, one particle distribution function is introduced for pressure and momentum, and another for fluid temperature. The fluid properties are assumed to be functions of only the local fluid temperature. The macroscopic variables at cell centers are determined from the solution of macroscopic governing equations by the finite volume method. The fluxes at cell interfaces are evaluated by the local construction of the solution for the standard lattice Boltzmann equation. The additional terms due to fluid property variations and body forces are regarded as source terms and treated by the finite volume discretization. These features make its application on non-uniform grids with a fixed time interval more flexible in comparison with conventional lattice Boltzmann models. The VPLBFS is validated by several numerical examples, including fluid flows between two parallel plates, with one plate to be isothermally heated from a certain moment, the natural convection in a square cavity with a large temperature difference and the natural convection of supercritical carbon dioxide. Numerical results show the reliability of the VPLBFS for thermal flows with partial or total variation in fluid properties.

FPGA shows its potential in heterogeneous reconfigurable computing systems in recent years. However, the traditional design method of FPGA has often been described as Hardware Description Languages (HDL), which blocked the way that adopting it to many applications. Nevertheless, both academia and industry attempt to improve the programmability of FPGA by using high-level languages as a development tool. In this paper, we use the Xilinx OpenCL platform SDAccel to accelerate a classic hydrodynamic simulation called Lattice Gas Cellular Automata (LGCA). We introduce a vectorization method of LGCA that binding the degree of execution and data bandwidth together by exploiting the flexibility of FPGA. This work also employs spatial and temporal parallelism to avoid simulation size restrictions and increase the simulation performance. We evaluate our technique on Xilinx VCU1525 board under FHP-III collision rule sets. The speed of LGCA simulation can reach to maximum 11073 million hits per second. Comparison reports to an Intel Core i7-6700 processor and Nvidia Quadro P5000 GPU are also given in this paper.

We consider quantum extensions of classical hydrodynamic lattice gas models. We find that the existence of local conserved quantities strongly constrains such extensions. We find the only extensions that retain local conserved quantities correspond to changing the local encoding of a subset of the bits. These models maintain separability of the state throughout the evolution and are thus efficiently classically simulable. We then consider evolution of these models in the case where any of the bits can be encoded and measured in one of two local bases. In the case that either encoding is allowed, the models are efficiently classically simulable. In the case that both encoding and measurement is allowed in either basis, we argue that efficient classical simulation is unlikely. In particular, for classical models that are computationally universal such quantum extensions can encode Simon&rsquo, s algorithm, thus presenting an obstacle to efficient classical simulation.

A problem of development of user-friendly interfaces for high performance computing (HPC) applications is addressed. The HPC Community Cloud (HPC2C) service that provides a RESTful application programming interface for unified control of HPC jobs was used to develop a prototype of a web-based UI for cellular automata simulation package. The UI allows a user to easily run multiple simulations on remote HPC resources and, this way, study a parameter space of a cellular automaton. The interface was used to organize a series of numerical experiments resulting in reproduction of the Karman vortex street.

It was shown previously (Shan, 2010) that the lattice weights of the lattice Boltzmann method are the solutions to an optimization problem subjecting to a set of linear equality and inequality constraints. Here, a solution strategy is developed and complete solutions for such optimization problem are obtained in one-, two-, and three-dimensions up to 9th order. In addition to recovering the well-known lattices, a number of high-order lattices are also discovered for the first time. These lattices can be used to construct high-order lattice Boltzmann schemes for non-equilibrium flow simulation.

In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

In this work a lattice Boltzmann flux scheme for the convection-diffusion equation (CDE) is proposed. In this scheme, the fluxes across the cell interface are calculated by using the local solution of the multiple-relaxation-time lattice Boltzmann equation for CDE. The present method is suitable for simulating both isotopic and anisotropic diffusion processes. Meanwhile, through applying the midpoint time integration technique, the present method relaxes the time step constraint in the original lattice Boltzmann flux scheme. Four convection-diffusion problems are simulated to validate the present scheme. The obtained results agree well with the analytical or previous published solutions.

Describing complex phenomena by means of cellular automata (CAs) has shown to be a very effective approach in pure and applied sciences. Most of the applications, however, rely on multidimensional CAs. For example, lattice gas CAs and lattice Boltzmann methods are widely used to simulate fluid flow and both share features with two-dimensional CAs. One-dimensional CAs, on the other hand, seem to have been neglected for modeling physical phenomena. In the present paper, we demonstrate that some one-dimensional CAs are equivalent to a stable linear finite difference scheme used to solve advection---diffusion partial differential equations (PDEs) by relying on the so-called iota-delta representation. Consequently, this work shows an important link between continuous and discrete models in general, and PDEs and CAs more in particular.

A lattice Boltzmann model for solving the (2+1) dimensional cubic-quintic complex Ginzburg–Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a two-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg–Landau equations. Numerical results reproduce the phenomena of the fusion of necklace-ring pattern and the effect of non-linearity on the soliton in the CQCGLE.

A lattice Boltzmann model for spatiotemporal solitons is proposed.The model is in the cylindrical coordinate system.The vortex tori soliton in CQCGLE has been simulated. A lattice Boltzmann model for solving the three-dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Differently from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Examples show that the model accurately reproduces the vortex tori pattern in the CQCGLE.

A lattice Boltzmann model is proposed for solving the complex Ginzburg-Landau equation (CGLE) with curvilinear coordinates. The method maintains the algorithmic simplicity of the original lattice Boltzmann scheme, and does not require an interpolation or coarse-graining procedure. This lattice Boltzmann scheme is based on uniformly distributed lattice points in these curvilinear coordinate systems. The algorithm provides advantages similar to the previous lattice Boltzmann method in that it is easily adapted to CGLE. The numerical simulations show spiral wave on a disc, the surface of a sphere, and the inside of a sphere. Examples show that the model accurately reproduces the phenomena in the CGLE.

A spatiotemporal lattice Boltzmann model for solving the three-dimensional cubic–quintic complex Ginzburg–Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional spatiotemporal space, and the evolution of the model is about a spatial axis rather than time. The algorithm possesses advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg–Landau equations. Examples show that the model reproduces the phenomena in the CQCGLE accurately.

Most models used in reservoir simulation studies are on the scale of meters to hundreds of meters. However, increasing resolution in geological measurements results in finer geological models. Simulations study of particle movements provide an alternative to conventional reservoir simulation by allowing the study of microscopic and/or macroscopic fluid flow, which is close to the scale of geological models. In this paper, the FHP-II (Frisch, Hasslacher and Pomeau - FHP) model of lattice gas automata were developed to study fluid flow in order to estimate the properties of heterogeneous porous media. Heterogeneity simulated by placing solid obstacles randomly in a two-dimensional test volume. Properties of the heterogeneous porous media were estimated by the shape, size, number of the obstacles and by the distribution of the obstacles within the volume. Results of the effects of grain sizes and shapes, and its distribution in the porous media on the tortuosity, effective porosity, permeability and displacement efficiency were obtained. An investigation of fluid flow and comparison with laboratory experiment were also presented. Reasonably good agreement between the lattice gas automata simulation and laboratory experiment results were achieved.

We present a review of the evolutionary advancement of the lattice Boltzmann methods (LBM) and current status of their applications in dealing with diverse flow problems such as advection-diffusion, forced convection, natural convection, conjugate heat transfer, radiative heat transfer, thermal flow through porous media, phase transition and phase separation, aerodynamics and aeroacoustics, and thermal jet flows. We analyze and discuss various thermal LBM frameworks and methodologies since the time of Lattice Gas Automata (LGA) to recently developed methods to model simple and complex aerodynamic, aeroacoustic and thermal flow problems. Multi-speed (M-S) models, hybrid LBM models, and double distribution function LBM (DDF-LBM) models, along with their applications and improvements are discussed in details. Moreover, the recent developments in addressing various applications such as energy storage devices, systems with electro-thermo convection, aerodynamic flows, simulations of noise profiles in nozzles with and without chevrons, and thermal simulations of jet flow with LBM are presented in great detail.

This chapter discusses the ancestor of the Lattice Boltzmann, the Boolean formulation of hydrodynamics known as lattice Gas Cellular Automata. In 1986, Uriel Frisch, Brosl Hasslacher and Yves Pomeau sent big waves across the fluid dynamics community: a simple cellular automaton obeying nothing but conservation laws at a microscopic level was able to reproduce the complexity of real fluid flows. This discovery spurred great excitement in the fluid dynamics community. The prospects were tantalizing: around free, intrinsically parallel computational paradigm for fluid flows. However, a few serious problems were quickly recognized and addressed with great intensity in the following years.

International audience; Agent-based models (ABMs) of multicellular systems are models in which each cell is represented individually. These models allow taking the variability between individual cells and the spatial heterogeneity of tissues on histological scales into account. In this chapter we present an overview and methodology of ABMs that are used to simulate mechanical and physiological phenomena in cells and tissues. ABMs are categorized as lattice based (cellular automata, lattice gas cellular automata, cellular Potts models), lattice free (center-based models, deformable cell models, vertex models), or hybrid, which are combinations of each of the latter. This model variety has been used to simulate a broad scale of phenomena, ranging from monolayer growth, tumor growth, wound healing, to organ modeling. Each approach has his strengths and weaknesses that we discuss here and in the next chapter together with the numerical solution methods and implementations. This chapter focuses on lattice and the next chapter on off lattice-free ABMs. The aim is to assist researchers in choosing the optimal model for the phenomenon they might want to model and understand.

Various lattice gas automata have been proposed in the past decades to simulate physics and address a host of problems on collective dynamics arising in diverse fields. In this work, we employ the lattice gas model defined on the sphere to investigate the curvature driven dynamic structures and analyze the statistical behaviors in equilibrium. Under the simple propagation and collision rules, we show that the uniform collective movement of the particles on the sphere is geometrically frustrated, leading to several non-equilibrium dynamic structures not found in the planar lattice, such as the emergent bubble and vortex structures. With the accumulation of the collision effect, the system ultimately reaches equilibrium in the sense that the distribution of the coarse-grained speed approaches the two-dimensional Maxwell-Boltzmann distribution despite the population fluctuations in the coarse-grained cells. The emergent regularity in the statistical behavior of the system is rationalized by mapping our system to a generalized random walk model. This work demonstrates the capability of the spherical lattice gas automaton in revealing the lattice-guided dynamic structures and simulating the equilibrium physics. It suggests the promising possibility of using lattice gas automata defined on various curved surfaces to explore geometrically driven non-equilibrium physics., Comment: 8 pages, 7 figures

WOS: 000365031800018 The four-dimensional Ising model is simulated on the Creutz cellular automaton (CCA) near the infinite-lattice critical temperature for the lattice with the linear dimension 4 a (c) 1/2 L a (c) 1/2 22. The temperature dependence of Binder parameter (g (L)) are analyzed for the lattice with the linear dimension 4 a (c) 1/2 L a (c) 1/2 22. In this study conducted highly detailed, two different types of behavior were determined as a result of varying linear lattice dimension. The infinite lattice critical temperatures are obtained to be T (c) = 6.6845 +/- 0.0005 in interval 4 a (c) 1/2 L a (c) 1/2 12 and T (c) = 6.6807 +/- 0.0024 in interval 14 a (c) 1/2 L a (c) 1/2 22. The finite and infinite lattice critical exponents for the order parameter, the magnetic susceptibility and the specific heat are computed from the results of simulations by using finite-size scaling relations. Critical linear lattice size have been identified as L = 14. Ahi Evran UniversityAhi Evran University [PYO-Fen.4010.14.009] This study was supported by the Ahi Evran University under Scientific Support, project no. PYO-Fen.4010.14.009.

We obtained the exact solution of a probabilistic cellular automaton related to the diagonal-to-diagonal transfer matrix of the six-vertex model on a square lattice. The model describes the flow of ants (or particles), traveling on a one-dimensional lattice whose sites are small craters containing sleeping or awake ants (two kinds of particles). We found the Bethe ansatz equations and the spectral gap for the time-evolution operator of the cellular automaton. From the spectral gap we show that in the asymmetric case it belongs to the Kardar-Parisi-Zhang (KPZ) universality class, exhibiting a dynamical critical exponent value $z=\frac{3}{2}$. This result is also obtained from a direct Monte Carlo simulation, by evaluating the lattice-size dependence of the decay time to the stationary state., Comment: 19 pages, 8 figures

We propose, analyze, and validate a lattice Boltzmann model with a cumulant collision operator. The new model is analytically and numerically shown to poses smaller errors than a moment based Multiple Relaxation Time lattice Boltzmann model. We demonstrate the usability of the cumulant lattice Boltzmann model by simulations of flow around a sphere for Reynolds numbers from 200 to 105.