John W. Perram

Cutoff Errors in the Ewald Summation Formulae for Point Charge Systems

Abstract Closed formulae for both real and reciprocal space parts of cutoff errors in the Ewald summation method in cubic periodic boundary conditions are derived. Such estimates are useful in tuning parameters in molecular simulations. Errors in both the electrostatic energy and forces are considered. The estimates apply to a disordered configuration of point charges and, with some limitations, also to point-charge molecular models. The accuracy of our estimates is tested and confirmed using simulated configurations of two systems (molten salt and diethylether) under a variety of conditions.

Electrostatic lattice sums for semi-infinite lattices

The techniques for the rapid computation of energies of three-dimensional neutral periodic assemblies of charged particles are extended to semi-infinite arrays and assemblies of ions in infinite filsm. The results will be useful for simulation of ionic movements in fast-ion conductors and dense colloidal dispersions.

The very fast multipole method

The fast multipole method (FMM) has become an important alternative to traditional methods such as the Ewald method for computing the long‐range interactions necessary to simulate charged or dipolar systems. In this paper, we present an improvement of this method, which we shall call the very fast multipole method (VFMM). The VFMM is shown to be a factor of about 1.2 faster than the FMM for two‐dimensional systems and a factor about 2–3 times faster for three‐dimensional systems without losing any accuracy for the worst case error.

Task curve planning for painting robots. I. Process modeling and calibration

This paper reports the first phase of a project whose aim is the automatic generation of tool center trajectories for robots engaged in spray painting of arbitrary surfaces. The first phase consists of proposing a mathematical model for the paint flux field within the spray cone. We have called this quantity the paint flux field partly to emphasize that it is a vector field and partly to distinguish it from a paint flux distribution function, which describes the angular variation of the flux field within the spray cone. It is shown that this flux field can be derived from experimental measurements performed by robots, of coverage profiles of paint strips on flat plates by solving a singular integral equation. This flux field is derived both for published experimental data as well as two sets of data from experiments performed by the authors. The correctness of the model is demonstrated by using the underlying distribution function to predict coverage profiles for other experiments in which the spray gun is no longer vertical to the plane surface.

Hamilton's Equations for Constrained Dynamical Systems

We derive expressions for the conjugate momenta and the Hamiltonian for classical dynamical systems subject to holonomic constraints. We give an algorithm for correcting deviations of the constraints arising in numerical solution of the equations of motion. We obtain an explicit expression for the momentum integral for constrained systems.

Computer simulation of ionic systems. Influence of boundary conditions

The results of molecular dynamics calculations for systems of charged particles under periodic boundary conditions are reported for the case in which the periodic array of particles makes a macroscopically large sphere surrounded by a continuum of dielectric constant ϵ′. It is shown that thermodynamic and most dynamic properties are independent of the nature of the surrounding medium. The conductivity σ(ω) of the system depends strongly on the dielectric properties of the surrounding medium.

Statistical mechanics of two-dimensional coulomb systems

Abstract The statistical mechanics of systems acting via two-dimensional charge-charge and dipole-dipole interactions is studied in periodic boundary conditions. The two-dimensional lattice theta-function transformation is obtained and the sum shown to contain a polarization correction to the Ewald sum, analogous to that found in three dimensions. Using a different approach, these sums are evaluated in closed form, for bodies of arbitrary shape. The methodology for the computer simulation of two dimensional dielectrics is discussed and the two-dimensional analogue of a generalized Kirkwood-Clausius-Mosotti-Debye formula derived.

ERROR ESTIMATES FOR THE FAST MULTIPOLE METHOD. II: THE THREE-DIMENSIONAL CASE

The Greengard-Rokhlin algorithm is a new and interesting method for computing long-range interactions in particle systems. Although the method already has been implemented and claimed to be superior to traditional and other methods, no reliable estimates of the size of the error of the method have been given. We illustrate what the error actually is for the two-dimensional case, and derive an estimate for it. The estimate has a simple analytic form which will allow its use in tuning the algorithm for best efficiency.

Molecular dynamics on transputer arrays. I: Algorithm design, programming issues, timing experiments and scaling projections

We extend earlier work on implementing boxing algorithms for simulating very large systems of particles on massively parallel S.I.M.D. arrays to more modest ones. We report details and preliminary results obtained using an OCCAM 2 program running on a linear array of transputers.

Microscopic derivation of fluctuation formulas for calculating dielectric constants by simulation

A microscopic derivation using the average Maxwell electric field is given for fluctuation formulas for the dielectric constant of a simulation sample for both periodic and reaction field boundary conditions. The reaction field case is for a spherical cavity reaction field. The derivations put both boundary conditions on an equal footing of microscopic theory and the only nonrigorous part of the derivation is the assumption that the region used to average the electric field is large enough. The fluctuation formula for reaction field boundary conditions is rather different from that used heretofore. The method is applied to a subregion of an isolated spherical system.