Cellular Automaton Approach to Arching in Two-Dimensional Granular Media

2014 ◽  
pp. 310-319 ◽  
Author(s):  
Takumi Masuda ◽  
Katsuhiro Nishinari ◽  
Andreas Schadschneider
Keyword(s):  
Cellular Automaton ◽  
Granular Media ◽  
Two Dimensional

THE CRITICAL BEHAVIOR OF THE 2D SPIN-1 ISING MODEL WITH THE BILINEAR AND POSITIVE BIQUADRATIC NEAREST NEIGHBOR INTERACTIONS ON A CELLULAR AUTOMATON

2004 ◽  
Vol 15 (10) ◽  
pp. 1425-1438 ◽  
Author(s):  
A. SOLAK ◽  
B. KUTLU
Keyword(s):  
Cellular Automaton ◽  
Phase Boundary ◽  
Critical Behavior ◽  
Nearest Neighbor ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Beg Model

The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D/K>0. The values of static critical exponents (α, β, γ and ν) are estimated within the framework of the finite size scaling theory along D/K=-1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.

A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

1993 ◽  
Vol 03 (02) ◽  
pp. 293-321 ◽  
Author(s):  
JÜRGEN WEITKÄMPER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Parallel Computers ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
Logistic Mapping ◽  
Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

THE SIMULATION OF 2D SPIN-1 ISING MODEL WITH POSITIVE BIQUADRATIC INTERACTION ON A CELLULAR AUTOMATON

2003 ◽  
Vol 14 (10) ◽  
pp. 1305-1320 ◽  
Author(s):  
BÜLENT KUTLU
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Cellular Automaton ◽  
Intermediate Phase ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Antiferromagnetic Spin

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

A Two-Dimensional Cellular Automaton Crystal with Irrational Density

2003 ◽  
Author(s):  
David Griffeath ◽  
Dean Hickerson
Keyword(s):  
Cellular Automaton ◽  
Asymptotic Density ◽  
Coarse Grain ◽  
Constant Density ◽  
Mathematical Framework ◽  
Two Dimensional ◽  
Empty Cell ◽  
Asymptotic Shape ◽  
Conway’S Game Of Life ◽  
Asymptotic Densities

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

CHARACTERIZING THE ABILITY OF PARALLEL ARRAY GENERATORS ON REVERSIBLE PARTITIONED CELLULAR AUTOMATA

1999 ◽  
Vol 13 (04) ◽  
pp. 523-538 ◽  
Author(s):  
KENICHI MORITA ◽  
SATOSHI UENO ◽  
KATSUNOBU IMAI
Keyword(s):  
Pattern Recognition ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Inverse System ◽  
Two Dimensional ◽  
Turing Machines ◽  
Recognition Mechanism ◽  
Parallel Array ◽  
Parallel Pattern ◽  
Monotonic Constraint

A PCAAG introduced by Morita and Ueno is a parallel array generator on a partitioned cellular automaton (PCA) that generates an array language (i.e. a set of symbol arrays). A "reversible" PCAAG (RPCAAG) is a backward deterministic PCAAG, and thus parsing of two-dimensional patterns can be performed without backtracking by an "inverse" system of the RPCAAG. Hence, a parallel pattern recognition mechanism on a deterministic cellular automaton can be directly obtained from a RPCAAG that generates the pattern set. In this paper, we investigate the generating ability of RPCAAGs and their subclass. It is shown that the ability of RPCAAGs is characterized by two-dimensional deterministic Turing machines, i.e. they are universal in their generating ability. We then investigate a monotonic RPCAAG (MRPCAAG), which is a special type of an RPCAAG that satisfies monotonic constraint. We show that the generating ability of MRPCAAGs is exactly characterized by two-dimensional deterministic linear-bounded automata.

Kinetics of segregation of granular media in a two-dimensional rotating drum

1997 ◽  
Vol 93 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Florence Cantelaube ◽  
Daniel Bideau ◽  
Stéphane Roux
Keyword(s):  
Granular Media ◽  
Two Dimensional ◽  
Rotating Drum ◽  
Kinetics Of

Euler–Lagrange Two-Phase Model for Simulating Live-Bed Scour Beneath Marine Pipelines

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
A. Yeganeh-Bakhtiary ◽  
M. Zanganeh ◽  
E. Kazemi ◽  
L. Cheng ◽  
A. K. Abd Wahab
Keyword(s):  
Numerical Model ◽  
Granular Media ◽  
Fluid Shear Stress ◽  
Distinct Element ◽  
Fluid Phase ◽  
Point Of View ◽  
Two Dimensional ◽  
Two Phase ◽  
Drag Forces ◽  
Bed Scour

In this study, an Euler–Lagrange coupling two-phase flow model, namely movable bed simulator (MBS)-two-dimensional (2D) model was employed to explore the current-induced live-bed scour beneath marine pipelines. The fluid phase characteristics, such as velocity and pressure, were obtained by the Reynolds-averaged Navier–Stokes (RANS) equations with a k-ε turbulence closure model in a two-dimensional Eulerian grid, whereas the seabed beneath pipelines was traced as an assembly of discrete sand grains from the Lagrangian point of view. The live-bed scour was evolved as the motion of a granular media based on distinct element method (DEM) formulation, in which the frequent interparticle collision was described with a spring and dashpot system. The fluid flow was coupled to the sediment phase, considering the acting drag forces between. Comparison between the numerical result and experimental measurement confirms that the numerical model successfully estimates the bed profile and flow velocity field. It is evident that the fluid shear stress decreases with the increasing of gap ratio e/D. The numerical model provides a useful approach to improve mechanistic understanding of hydrodynamic and sediment transport in live-bed scour beneath a marine pipeline.

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