Excitable Delaunay triangulations

In an excitable Delaunay triangulation every node takes three states (resting, excited and refractory) and updates its state in discrete time depending on a ratio of excited neighbours. All nodes update their states in parallel. By varying excitability of nodes we produce a range of phenomena, including reflection of excitation wave from edge of triangulation, backfire of excitation, branching clusters of excitation and localized excitation domains. Our findings contribute to studies of propagating perturbations and waves in non-crystalline substrates

Theoretical Study of the Conditions and the Mechanism of Shear Crack Acceleration towards the Longitudinal Wave Velocity

AbstractThe question about physically admissible velocity of dynamic crack growth is of significance to safety engineering as well as to earthquake dynamics. Recent researches including numerical simulations, experimental observations and the analysis of strong earthquakes have shown a possibility of propagation of shear cracks in supershear regime, namely at velocities comparable with dilatational wave speed. The present paper is devoted to the theoretical (numerical) study of some fundamental aspects of this problem. It is shown that development of a sub Raleigh shear crack is connected with a vortex traveling ahead of the crack tip at a shear wave velocity. The stress concentration area ahead of the crack tip revealed by different authors (Burridge, Andrews, Geubelle, Rosakis and others) is connected with this vortex. Acceleration of a shear crack towards the longitudinal wave velocity is concerned with formation of a daughter crack by the mechanism of shearing (the daughter crack is formed in the center of vortex). Analysis of sub Raleigh to intersonic transition has shown that development of shear cracks is self-similar and depends on dimensionless parameters

A study of fragmentation processes using a discrete element method

We present a model of solids made from polygonal cells connected via beams. We calculate the macroscopic elastic moduli from the beam and cell parameters. This modellisation is particularly suited for the simulation of fragmentation processes. We study the effects of an explosion inside a circular disk and the impact of a projectile and obtain the fragment size distribution. We find that if breaking only happens under tensile forces a layer on the free wall opposed to impact is first ejected. In that case the distribution follows a power-law with an exponent that in most cases is around two.Comment: 16 pages in LaTex format, 17 PostScript figures. Figures are available upon request from the authors. Submitted to Int. J. of Mod. Phys.

Cellular Automaton for Realistic Modelling of Landslides

A numerical model is developed for the simulation of debris flow in landslides over a complex three dimensional topography. The model is based on a lattice, in which debris can be transferred among nearest neighbors according to established empirical relationships for granular flows. The model is then validated by comparing a simulation with reported field data. Our model is in fact a realistic elaboration of simpler ``sandpile automata'', which have in recent years been studied as supposedly paradigmatic of ``self-organized criticality''. Statistics and scaling properties of the simulation are examined, and show that the model has an intermittent behavior.Comment: Revised version (gramatical and writing style cleanup mainly). Accepted for publication by Nonlinear Processes in Geophysics. 16 pages, 98Kb uuencoded compressed dvi file (that's the way life is easiest). Big (6Mb) postscript figures available upon request from [email protected] / [email protected]

Numerical analysis of the geometrical and material criteria of acceleration of shear crack to supershear velocity in brittle nanoporous solids

The paper is devoted to the study of dynamic propagation of mode II cracks in porous brittle materials with nanoscale pore size. We compared static (shear strength) and dynamic parameters of crack growth in dry and fluid saturated nanoporous brittle materials at different degrees of confinement. We have shown that pore fluid in nanoporous brittle materials influences mainly the dynamics of crack propagation. This leads in particular to pronounced peculiarities of the dependence of the critical value of dimensionless geometrical parameter of the initial crack (it majorizes the interval of the ratios of length to thickness for the cracks that are capable to accelerate to intersonic velocity) on applied crack normal stress. The results of the study are relevant for understanding the conditions of supershear regime of propagation of mode II cracks as well as for assessment of the ability of mode II cracks in brittle materials (including nanoporous fluid-saturated solids) to develop in supershear regime

Role of vortex-like motion in fracture of coating-substrate system under contact loading

Deformation of a heterogeneous material containing internal interfaces or/and free surfaces is accompanied by collective vortex motion near these boundaries. One should expect that rotational motion in nanomaterials takes place at different scales, from the atomic scale to the macroscopic one. Nevertheless such a fundamental factor as elastic vortex motion in material formed during dynamic loading still remains out of discussion. The aim of this paper is revealing the role of vortex displacements in contact interaction of the strengthening coating with a hard counter-body by means of 3D modeling using movable cellular automata (MCA). MCA method is an efficient numerical method in particle mechanics, which assumes that the material is composed of a certain amount of elementary objects interacting among each other according to many-particle forces. In this paper MCA method is applied to 3D modeling deformation of the coating-substrate system under its contact loading by the rigid indenter. Main attention of the research is focused on the role of vortex structures in the velocity fields in elastic and non-elastic deformation of the strengthening coating and substrate. The mechanical properties of the model coating correspond to multifunctional nanostructured film and the properties of the substrate, to nanostructured titanium. The loading is performed by a hard conical indenter with various ratios of normal and tangential components. The peculiarities of the velocity vortex formation and propagation, as well as interaction with the structural elements are studied

Validation and Calibration of Models for Reaction-Diffusion Systems

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps ($\Delta x$ and $\Delta t$) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite $\Delta x$ and $\Delta t$, if the parameter $\gamma_N=D \Delta t/(\Delta x)^2$ assumes a fixed constant value, where $N$ is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as $(\Delta x)^{2(N+2)}$ and the values of $\gamma_N$ are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of $10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of $10^{-3}$.Comment: 33 pages, 8 figures, to appear in Int. J. Bifurcation and Chao

Simulating Three-Dimensional Hydrodynamics on a Cellular-Automata Machine

We demonstrate how three-dimensional fluid flow simulations can be carried out on the Cellular Automata Machine 8 (CAM-8), a special-purpose computer for cellular-automata computations. The principal algorithmic innovation is the use of a lattice-gas model with a 16-bit collision operator that is specially adapted to the machine architecture. It is shown how the collision rules can be optimized to obtain a low viscosity of the fluid. Predictions of the viscosity based on a Boltzmann approximation agree well with measurements of the viscosity made on CAM-8. Several test simulations of flows in simple geometries -- channels, pipes, and a cubic array of spheres -- are carried out. Measurements of average flux in these geometries compare well with theoretical predictions.Comment: 19 pages, REVTeX and epsf macros require