Donald Greenspan

Conservative numerical methods for ẍ = f(x)

Abstract Two distinctly different numerical methods are developed for solving the conservative initial value problem x = f(x), x(O)= α, dot x (O)= β . Both methods conserve exactly the same total energy as does the given differential equation. Computer examples are described and

Completely Conservative, Covariant Numerical Methodology

For a distance dependent potential o, Newtonian dynamics is characterized by conservation of energy, linear momentum, and angular momentum. In addition, the basic dynamical equations are covariant, that is, invariant under fundamental coordinate transformations. In this paper, it is shown that related N-body problems can be solved numerically in such a fashion that, independently of the time step, the identical conservation laws continue to be valid for the approximating difference equations. In addition, the numerical method is covariant.

Supercomputer Simulation of Cracks and Fractures by Quasimolecular Dynamics

Abstract A modern method for predicting the time and place of crack and fracture development is through molecular dynamics, which studies reactions to external forces in the small, that is, the molecular level. In quasimolecular dynamics, which is an attempt to bridge the gap between atomistic and continuum simulations, molecules are aggregated into larger units, called quasimolecules, to simulate the large scale behavior. In this first paper, quasimolecular dynamics is applied to the study of crack and fracture generation in a stressed, slotted plate made of pure copper. Conservation of mass and energy is basic to the methodology.

An analysis of stress wave propagation in slender bars using a discrete particle approach

Abstract In this paper, a discrete particle approach is developed for the quantitative analysis of stress wave propagation in metal bars. Though linear forces are emphasized, nonlinear forces are also considered. Cylindrical, tapered, homogeneous, and nonhomogeneous bars are studied. Computer results show most favourable agreement with available theoretical and experimental results.

Quasi-molecular, particle modeling of crack generation and fracture

Abstract In this paper a particle type approach is developed for the simulation of crack generation and fracture. The major objective is to develop a new modeling approach which will allow experiments in fracture phenomena directly on a digital computer. Rectangular plates are modeled as particle aggregates and then stressed by using molecular type formulas. Computer examples are described and discussed which relate to customary wedge and punched hole phenomena.

Computer studies in particle modelling of fluid phenomena

Abstract A new, numerical approach is developed for modelling fluid phenomena. Unlike the continuum and statistical mechanics approaches, it uses relatively small sets of quasi-molecular particles which interact in accordance with classical, molecular-type formulas. Computer examples are described and discussed, as is the potential for modelling turbulent behavior.

MOLECULAR CAVITY FLOW

Using molecular mechanics, cavity flow is studied in a basin of 4235 water molecules at 15°C. Primary vortices are generated with wall speeds V = −20,−40,−100,−250 A/ps. The vortex motions agree with experimental results in the large. Fully turbulent flow is generated with wall speed V = −3000 A/ps. The mechanisms for primary vortex generation are clearly delineated, as are those for turbulent flow.

A classical molecular approach to computer simulation of biological sorting

Steinbergs theory of sorting is modified by replacing the free energy minimization principle with dynamical equations of a molecular nature. Correct cellular sorting then follows in all cases where the mixture is in a liquid or a near-liquid state. Computer examples are described and discussed, primarily for two dimensional, but also for three dimensional, interactions.

Supercomputer simulation of colliding microdrops of water

Abstract Microdrop collisions are important in fast kinetic, microwave, nucleation and rainstorm studies. In this paper a molecular model of colliding microdrops of water is simulated on a CRAY X-MP/24. A least squares fit of the force field allows relatively large time steps in the numerical procedure. Rotating “raindrop” modes, “dumbell” modes and oscillatory oblateness modes are demonstrated, as are nonadhesive soft and hard collisions.

Deterministic Computer Physics

Completely arithmetic formulations, which possess exactly the same conservation laws and symmetry as their continuum counterparts, are given for both Newtonian and special relativistic mechanics. Applications are made to new models of fluid flow, vibration, diffusion, planetary evolution, biological self-reorganization, and relativistic oscillation. Computer examples are described and discussed.