Particle-particle Particle-mesh Method Research Articles

Accelerating molecular dynamics codes by performance and accuracy modeling

Scientific software is often driven by multiple parameters that affect both accuracy and performance. Since finding the optimal configuration of these parameters is a highly complex task, it is extremely common that the software is used suboptimally. In a typical scenario, accuracy requirements are imposed, and attained through suboptimal performance. In this paper, we present a methodology for the automatic selection of parameters for simulation codes, and a corresponding prototype tool. To be amenable to our methodology, the target code must expose the parameters affecting accuracy and performance, and there must be formulas available for error bounds and computational complexity of the underlying methods. As a case study, we consider the particle–particle particle–mesh method (PPPM) from the LAMMPS suite for molecular dynamics, and use our tool to identify configurations of the input parameters that achieve a given accuracy in the shortest execution time. When compared with the configurations suggested by expert users, the parameters selected by our tool yield reductions in the time-to-solution ranging between 10% and 60%. In other words, for the typical scenario where a fixed number of core-hours are granted and simulations of a fixed number of timesteps are to be run, usage of our tool may allow up to twice as many simulations. While we develop our ideas using LAMMPS as computational framework and use the PPPM method for dispersion as case study, the methodology is general and valid for a range of software tools and methods.

Reconsidering Dispersion Potentials: Reduced Cutoffs in Mesh-Based Ewald Solvers Can Be Faster Than Truncation

Long-range dispersion interactions have a critical influence on physical quantities in simulations of inhomogeneous systems. However, the perceived computational overhead of long-range solvers has until recently discouraged their implementation in molecular dynamics packages. Here, we demonstrate that reducing the cutoff radius for local interactions in the recently introduced particle-particle particle-mesh (PPPM) method for dispersion [Isele-Holder et al., J. Chem. Phys., 2012, 137, 174107] can actually often be faster than truncating dispersion interactions. In addition, because all long-range dispersion interactions are incorporated, physical inaccuracies that arise from truncating the potential can be avoided. Simulations using PPPM or other mesh Ewald solvers for dispersion can provide results more accurately and more efficiently than simulations that truncate dispersion interactions. The use of mesh-based approaches for dispersion is now a viable alternative for all simulations containing dispersion interactions and not merely those where inhomogeneities were motivating factors for their use. We provide a set of parameters for the dispersion PPPM method using either ik or analytic differentiation that we recommend for future use and demonstrate increased simulation efficiency by using the long-range dispersion solver in a series of performance tests on massively parallel computers.

Characteristics of thermal conductivity in classical water models

The thermal conductivities of common water models are compared using equilibrium (EMD) and non-equilibrium molecular dynamics (NEMD) simulation. A complete accounting for electrostatic contributions to the heat flux was found to resolve the previously reported differing results of NEMD and EMD Green-Kubo measurements for the extended simple point-charge (SPC/E) model. Accordingly, we demonstrate the influence of long-range electrostatics on the thermal conductivity with a simple coulomb cutoff, Ewald summation, and by an extended particle-particle particle-mesh method. For each water model, the thermal conductivity is computed and decomposed in terms of frequency-dependent thermodynamic and topological contributions. The rigid, three-site SPC, SPC/E, and transferable intermolecular potential (TIP3P-Ew) water models are shown to have similar thermal conductivity values at standard conditions, whereas models that include bond stretching and angle bending have higher thermal conductivities.

Particle-particle particle-mesh method for dipolar interactions: On error estimates and efficiency of schemes with analytical differentiation and mesh interlacing

The interlaced and non-interlaced versions of the dipolar particle-particle particle-mesh (P(3)M) method implemented using the analytic differentiation scheme (AD-P(3)M) are presented together with their respective error estimates for the calculation of the forces, torques, and energies. Expressions for the optimized lattice Green functions, and for the Madelung self-forces, self-torques and self-energies are given. The applicability of the theoretical error estimates are thoroughly tested and confirmed in several numerical examples. Our results show that the accuracy of the calculations can be improved substantially when the approximate (mesh computed) Madelung self-interactions are subtracted. Furthermore, we show that the interlaced dipolar AD-P(3)M method delivers a significantly higher accuracy (which corresponds approximately to using a twice finer mesh) than the conventional method, allowing thereby to reduce the mesh size with respect to the non-interlaced version for a given accuracy. In addition, we present similar expressions for the dipolar ik-differentiation interlaced scheme, and we perform a comparison with the AD interlaced scheme. Rough tests for the relative speed of the dipolar P(3)M method using ik-differentiation and the interlaced/non-interlaced AD schemes show that when FFT computing time is the bottleneck, usually when working at high precisions, the interlaced AD-scheme can be several times faster than the other two schemes. For calculations with a low accuracy requirement, the interlaced version can perform worse than the ik and the non-interlaced AD schemes.

Molecular dynamics simulation of the contact angle of liquids on solid surfaces

In this work, Lennard-Jones liquid and water droplets are simulated adjacent to a solid surface using molecular dynamics. The contact angle is obtained by using direct simulation. The particle-particle particle-mesh method [B. Shi, S. Sinha, and V. K. Dhir, J. Phys. Chem. 124, 204715 (2006)] for long range force correction is used in the simulation. The contact angle is studied as a function of system temperature and the solid/fluid interaction potential. It is shown that the contact angle decreases with increasing system temperature and increases when the potential decreases. At high system temperature (pressure), the contact angle drops to zero. The predictions are compared with data from experiments and a reasonable agreement is found.

Molecular dynamics simulation of the density and surface tension of water by particle-particle particle-mesh method

In this work, molecular dynamics simulation is performed to study the density and surface tension of water for a range of temperatures from 300 to 600 K. The extended simple point charge interaction potential for water is used. The particle-particle particle-mesh method, which automatically includes untruncated long-range terms, is used for the Lennard-Jones and the Coulombic terms. The results show that the long-range correction for the Lennard-Jones term is very important for the calculation of surface tension. It is found that the calculated density and surface tension of water fit well with experimental data for temperatures less than 500 K. Near the critical temperature, the simulation results are off from the experimental data.

Calculating Electrostatic Interactions Using the Particle−Particle Particle−Mesh Method with Nonperiodic Long-Range Interactions

The particle−particle particle−mesh (PPPM) method is an accurate and computationally efficient method for calculating interactions in molecular simulations. The PPPM method is based on separating the total interaction between particles into the sum of short-range interactions, which are computed by direct particle−particle summation, and long-range interactions, which are calculated by solving Poisson's equation using periodic boundary conditions (PBCs). For nonperiodic systems there is a concern that use of PBCs may introduce an artificial periodicity into the system. In this article we propose, and test, the use of boundary conditions for the long-range interactions which mimic surrounding the explicit system by high dielectric material such as water.

Lattice-sum methods for calculating electrostatic interactions in molecular simulations

When simulating a periodic molecular system, lattice sum methods may be used to evaluate the electrostatic interactions without resorting to an unphysical truncation of the potential. In a previous publication, we compared the computational efficiency of the Ewald and Particle-Particle Particle-Mesh (PPPM) lattice-sum methods. Because the PPPM method discretizes the field equations and utilizes the highly efficient fast Fourier transform algorithm, it requires significantly less computational effort than the Ewald method and scales almost linearly with system size. In this paper, we take a more detailed look into the theory behind the lattice-sum methods to clarify the underlying similarities between the Ewald method and the PPPM method. We also investigate the errors introduced by different approximation used in the discretization of the field equations in the PPPM method.

A Fast Adaptive Vortex Method in Three Dimensions

The method of local corrections (MLC) developed by Anderson for two spatial dimensions is a particle-particle particle-mesh method, in which the calculation of the velocity field induced by a collection of vortices is split into two parts: (i) a finite difference velocity field calculation using a fast Poisson solver, the results of which are used to represent the velocity field induced by vortices far from the evaluation point; and (ii) an N-body calculation to compute the velocity field at a vortex induced by nearby vortices. We present a fast vortex method for incompressible flow in three dimensions, based on the extension of the MLC algorithm from two to three spatial dimensions and the use of adaptive mesh refinement in the finite difference calculation of the MLC. Calculations with a vortex ring in three dimensions show that the break-even point between the MLC with AMR and the direct method is at N ≈ 3000 on a Cray Y-MP; for N ≈ 64,000 MLC with AMR can be 12 times faster than the direct method. Results from calculations of two colliding inviscid vortex rings demonstrate the increased resolution which can be obtained using fast methods.