Generalized Farey trees, transfer Operators and phase transitions

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter map is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral analysis of generalized transfer operators. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.Comment: 39 pages, 10 figure

A parameter-free total Lagrangian smooth particle hydrodynamics algorithm applied to problems with free surfaces

This paper presents a new Smooth Particle Hydrodynamics computational framework for the solution of inviscid free surface flow problems. The formulation is based on the Total Lagrangian description of a system of first-order conservation laws written in terms of the linear momentum and the Jacobian of the deformation. One of the aims of this paper is to explore the use of Total Lagrangian description in the case of large deformations but without topological changes. In this case, the evaluation of spatial integrals is carried out with respect to the initial undeformed configuration, yielding an extremely efficient formulation where the need for continuous particle neighbouring search is completely circumvented. To guarantee stability from the SPH discretisation point of view, consistently derived Riemann-based numerical dissipation is suitably introduced where global numerical entropy production is demonstrated via a novel technique in terms of the time rate of the Hamiltonian of the system. Since the kernel derivatives presented in this work are fixed in the reference configuration, the non-physical clumping mechanism is completely removed. To fulfil conservation of the global angular momentum, a posteriori (least-squares) projection procedure is introduced. Finally, a wide spectrum of dedicated prototype problems is thoroughly examined. Through these tests, the SPH methodology overcomes by construction a number of persistent numerical drawbacks (e.g. hour-glassing, pressure instability, global conservation and/or completeness issues) commonly found in SPH literature, without resorting to the use of any ad-hoc user-defined artificial stabilisation parameters. Crucially, the overall SPH algorithm yields equal second order of convergence for both velocities and pressure

SPH and Eulerian underwater bubble collapse simulations

SPH (Smoothed Particle Hydrodynamics) is a gridless Lagrangian technique which is appealing as a possible alternative to numerical techniques currently used to analyze high deformation impulsive loading events. Previously, the SPH algorithm has been subjected to detailed testing and analysis to determine the feasibility of using the coupled finite-element/SPH code PRONTO/SPH for the analysis of various types of underwater explosion problems involving fluid-structure and shock-structure interactions. Here, SPH and Eulerian simulations are used to study the details of underwater bubble collapse, particularly the formation of re-entrant jets during collapse, and the loads generated on nearby structures by the jet and the complete collapse of the bubble. Jet formation is shown to be due simply to the asymmetry caused by nearby structures which disrupt the symmetry of the collapse. However, the load generated by the jet is a minor precursor to the major loads which occur at the time of complete collapse of the bubble

Experimental and numerical studies of high-velocity impact fragmentation

Developments are reported in both experimental and numerical capabilities for characterizing the debris spray produced in penetration events. We have performed a series of high-velocity experiments specifically designed to examine the fragmentation of the projectile during impact. High-strength, well-characterized steel spheres (6.35 mm diameter) were launched with a two-stage light-gas gun to velocities in the range of 3 to 5 km/s. Normal impact with PMMA plates, thicknesses of 0.6 to 11 mm, applied impulsive loads of various amplitudes and durations to the steel sphere. Multiple flash radiography diagnostics and recovery techniques were used to assess size, velocity, trajectory and statistics of the impact-induced fragment debris. Damage modes to the primary target plate (plastic) and to a secondary target plate (aluminum) were also evaluated. Dynamic fragmentation theories, based on energy-balance principles, were used to evaluate local material deformation and fracture state information from CTH, a three-dimensional Eulerian solid dynamics shock wave propagation code. The local fragment characterization of the material defines a weighted fragment size distribution, and the sum of these distributions provides a composite particle size distribution for the steel sphere. The calculated axial and radial velocity changes agree well with experimental data, and the calculated fragment sizes are in qualitative agreement with the radiographic data. A secondary effort involved the experimental and computational analyses of normal and oblique copper ball impacts on steel target plates. High-resolution radiography and witness plate diagnostics provided impact motion and statistical fragment size data. CTH simulations were performed to test computational models and numerical methods

Optimizing the Point-In-Box Search Algorithm for the Cray Y-MP(TM) Supercomputer

Determining the subset of points (particles) in a problem domain that are contained within certain spatial regions of interest can be one of the most time-consuming parts of some computer simulations. Examples where this 'point-in-box' search can dominate the computation time include (1) finite element contact problems; (2) molecular dynamics simulations; and (3) interactions between particles in numerical methods, such as discrete particle methods or smooth particle hydrodynamics. This paper describes methods to optimize a point-in-box search algorithm developed by Swegle that make optimal use of the architectural features of the Cray Y-MP Supercomputer