Link-cell Method Research Articles

A hierarchical linked cell method based on scale-proportional multi-level DEM grids

The linked cell method is widely used in the discrete element method (DEM) to detect contact between particles because of its high efficiency for uniform particles. However, the efficiency is reduced for polydisperse particles which are common in nature and engineering applications and have a wide size range. In this paper, a hierarchical linked cell method based on scale-proportional multi-level DEM grids is developed to improve the efficiency for polydisperse particles. The performance of the hierarchical linked cell method is verified by simulating different polydisperse particles. The results show that the hierarchical linked cell method can significantly reduce the elapsed CPU time of contact detection in all simulated cases. The scale proportion and the number of levels of the DEM grids are found to be two important parameters influencing the efficiency improvement. The elapsed CPU time decreases as the scale proportion decreases or as the number of levels increases. For a given scale proportion, the optimal number of levels to achieve maximum efficiency should satisfy two conditions: (1) the grid cell granularity is less than 10; (2) the exponential decrease of the grid cell granularity with the number of levels is broken.

Optimal parameters for a hierarchical grid data structure for contact detection in arbitrarily polydisperse particle systems

The objective of this paper is to find the optimum number of hierarchy levels and their cell sizes for contact detection algorithms based on a versatile hierarchical grid data structure, for polydisperse particle systems with arbitrary distribution of particle radii. These algorithms perform as fast as $$O(N)$$ for $$N$$ particles, but the prefactor can be as large as $$N$$ for a given system, depending on the algorithm parameters chosen, making a recipe for choosing these parameters necessary. We estimate theoretically the calculation time of two distinct algorithms for particle systems with various packing fractions, where the sizes of the particles are modelled by an arbitrary probability density function. We suggest several methods for choosing the number of hierarchy levels and the respective cell sizes, based on truncated power-law radii distributions with different exponents and widths. The theoretical estimations are then compared with simulation results for particle systems with up to one million particles. The proposed recipe for selecting the optimal hierarchical grid parameters allows to find contacts in arbitrarily polydisperse particle systems as fast as the commonly-used linked-cell method in purely monodisperse particle systems, i.e., extra work is avoided in presence of polydispersity. Furthermore, the contact detection time per particle even decreases slightly with increasing polydispersity or decreasing particle packing fraction.

A fast multilevel algorithm for contact detection of arbitrarily polydisperse objects

An efficient algorithm for contact detection among many arbitrarily sized objects is developed. Objects are allocated to cells based on their location and size within a nested hierarchical cell space. The choice of optimal cell sizes and the number of hierarchies for best performance is not trivial in most cases. To overcome this challenge, a novel analytical method to determine the optimal hierarchical cell space for a given object size distribution is presented. With this, a decision can be made between using the classical Linked-Cell method and the contact detection algorithm presented. For polydisperse systems with size ratios up to 50, we achieved 220 times speed-up compared to the classical Linked-Cell method. For larger size ratios, even better speed-up is expected. The complexity of the algorithm is linear with the number of objects when the optimal hierarchical cell space is chosen. So that the problem of contact detection in polydisperse systems essentially is solved.

Faster neighbour list generation using a novel lattice vector representation

In any many-body simulation where particles are coupled using short-range potentials, a key part of the simulation is to find which particles { j } interact with particle i. The set of such particles is known as the neighbour list of particle i. A novel algorithm is developed here which efficiently returns a neighbour list. A partially occupied reference lattice may be constructed for any simulation, with the position of particles defined as being a short vector separation from a node. A lattice vector which preserves translational symmetry in a periodic supercell under addition and subtraction operations can then be constructed from a single 32-bit integer number. A novel neighbour list algorithm is then developed which uses a single set of lattice vectors to return all nodes, and therefore all particles associated with the nodes, within a fixed radius sphere of particle i. This new algorithm preserves translational symmetry in a periodic supercell, requires a small memory overhead, and is shown to be faster than the well-known Linked-Cell method in all cases considered here.

Monte Carlo data of dilute solutions of large spheres in binary hard sphere mixtures

Monte Carlo (MC) simulation data for additive binary hard sphere mixtures are reported for dilute concentrations of the large sphere. Using a single occupancy linked cell method, binary hard sphere solutions with a size ratio of 5 are simulated at high reduced density and low concentration of the large sphere. Data for the solute-solvent pair distribution function show that at the lowest concentrations of the large sphere simulated, the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation underestimates the contact value; whereas the recently proposed Henderson-Chan (HC) equation gives a good prediction. For the solute-solute contact value at the colloidal limit, the MC data lie between the two predictions. The BMCSL equation underestimates, while the HC equation overestimates, the correct solute-solute contact value.

A highly vectorised “link-cell” FORTRAN code for the DL_POLY molecular dynamics simulation package

Highly vectorised FORTRAN subroutines, based on the link-cell algorithm for DL_POLY molecular dynamics simulation package, are developed. For several specific benchmark systems the efficiency of the proposed codes on a Fujitsu VPP700/128E vector computer has been tested. It is shown that in constructing the neighbor list and in calculating atomic forces our link-cell method is significantly faster than the original code.

FAST PARTICLE PAIR DETECTION ALGORITHMS FOR PARTICLE SIMULATIONS

New algorithms with O(N) complexity have been developed for fast particle-pair detections in particle simulations like the discrete element method (DEM) and molecular dynamic (MD). They exhibit robustness against broad particle size distributions when compared with conventional boxing methods. Almost similar calculation speeds are achieved at particle size distributions from is mono-size to 1:10 while the linked-cell method results in calculations more than 20 times. The basic algorithm, level-boxing, uses the variable search range according to each particle. The advanced method, multi-level boxing, employs multiple cell layers to reduce the particle size discrepancy. Another method, indexed-level boxing, reduces the size of cell arrays by introducing the hash procedure to access the cell array, and is effective for sparse particle systems with a large number of particles.

Determination of vapour-liquid equilibrium using cavity-biased grand canonical Monte Carlo method

In a previous paper a new simulation method was introduced for the determination of vapour–liquid equilibrium (VLE) of pure fluids in the grand canonical ensemble (Boda, D., Liszi, J., and Szalai, I., 1996, Chem. Phys. Lett., 256, 474). Its basic idea is the extrapolation of the pressure in the directions of reciprocal temperature and configurational chemical potential via third-order Taylor series expansion: p(β, μ). The coefficients of the series can be obtained from fluctuation formulae by performing grand canonical Monte Carlo (GC MC) simulations on both vapour and liquid sides. It was found that the main shortcoming of the method originates from the inaccurate calculation of the pressure on liquid side because of the slow convergence of the original GC MC simulation used. It is now shown that the application of the cavitybiased GC MC method of Mezei on the liquid side can overcome this difficulty. The linked-cell method to fasten the cavity-searching algorithm is proposed. The results of test calcula...

Determination of vapour-liquid equilibrium using cavity-biased grand canonical Monte Carlo method

In a previous paper a new simulation method was introduced for the determination of vapour–liquid equilibrium (VLE) of pure fluids in the grand canonical ensemble (Boda, D., Liszi, J., and Szalai, I., 1996, Chem. Phys. Lett., 256, 474). Its basic idea is the extrapolation of the pressure in the directions of reciprocal temperature and configurational chemical potential via third-order Taylor series expansion: p(β, μ). The coefficients of the series can be obtained from fluctuation formulae by performing grand canonical Monte Carlo (GC MC) simulations on both vapour and liquid sides. It was found that the main shortcoming of the method originates from the inaccurate calculation of the pressure on liquid side because of the slow convergence of the original GC MC simulation used. It is now shown that the application of the cavitybiased GC MC method of Mezei on the liquid side can overcome this difficulty. The linked-cell method to fasten the cavity-searching algorithm is proposed. The results of test calculations for the Lennard-Jones system show excellent agreement with those obtained by the NpT plus test particle method. The method can be implemented for the determination of phase separation in mixtures in a straightforward manner.

Massively parallel molecular dynamics simulations with EAM potentials

Molecular dynamics of cascades in pure iron and iron–copper alloys using embedded atom method type of interatomic potentials are presented. Reliable simulations of radiation damage at the atomic scale with high energy Primary Knocked Atoms (PKA) need systems with large numbers of particles and very long computational time. To perform the simulation in a reasonable amount of time high-performance computer systems such as massively parallel machines need to be used. This paper presents the parallelisation strategy applied to a serial classical Molecular Dynamics code: DYMOKA. The original sequential Fortran code CDCMD from the University of Connecticut was first improved algorithmically by applying a link cell method for the neighbour list construction of the Verlet list, resulting in a fully linear algorithm. The parallelisation strategy adopted is a multidimensional domain decomposition of the simulation box using a link cell method and a Verlet list method for each subdomain independently. The program paradigm is based on explicit message passing, and the standard Message-Passing Interface (MPI) was chosen in order to achieve portability. First measurements have demonstrated that the simulation of a system of 2.000.000 (750.000) atoms on 128 (32) processors costs 2.5 (10.) μs per atom per step. The current implementation has proven good scalability up to 32 processors on a NEC Cenju-3 machine. To study the effects of irradiation on copper segregation, simulations with up to 1.000.000 atoms in iron and iron–copper were performed with PKA energies up to 20 keV.

Parallel short-range molecular dynamics using the Ādhāra runtime system

We describe a parallel program for simulating molecular dynamics subject to short-ranged molecular interactions. The program uses a runtime system, Ādhāra, for atom definition, partitioning and distribution over the processors as well as dynamical load balancing between processors. A significant speedup is achieved by eliminating the usual test for the minimum image criterion, using conditional statements, that is applied to pairs of atoms during the force calculation. Instead, coordinates of boundary atoms are adjusted for the periodic boundary condition just before communication. The algorithm uses domain decomposition and the link-cell method as well as a neighbor list within cells. Benchmark simulations of systems with 864 and up to 5 000 000 atoms run on an Intel Paragon computer are described.

Optimisation of the Ewald Sum for Large Systems

Abstract Techniques are described for optimising the Ewald sum in the simulation of systems containing large numbers of charged particles. Expressions are given for the best choice of parameters to maximise performance for a specified accuracy. The algorithm for the real-space sum uses a “small-cell” version of the link-cell method. Various other computational details are discussed. In the simulation of real systems there is considerable scope for speeding the calculation by redcing cutoffs, without loss of accuracy.

A two-dimensional monotonical logical grid algorithm

This paper describes a two-dimensional implementation of a monotonical logical grid (MLG) algorithm for a Monte Carlo study of a two-dimensional Lennard-Jones fluid, and compares its performance with a conventional link-cell algorithm with the same purpose. It is obtained that the MLG scales approximately linear with the number of particles N, instead of the conventional N 2 cost for the link-cell method.

Comparison of link-cell and neighbourhood tables on a range of computers

The recent improvement to the link-cell method (Comput. Phys. Commun. 60 (1990) 195), making it faster on a scalar computer than the traditional neighbour table (NT) method, was tested on different vectorial computers. The efficiency of both methods depends greatly on the particular processor architecture. In contrast to scalar computer results, the NT method was found to be better on the vectorial computers.

“Checker Board” Periodic Boundary Conditions in Molecular Dynamics Codes

Abstract Checker board periodic boundary conditions for molecular simulation are proposed. They represent a simple geometric transformation of the well known periodic boundary conditions formulae for variety of periodic and regular computational boxes (e.g. the truncated octahedron, rhombic dodecahedron, hexagonal prism, etc.) into a shifted parallelepiped periodic grid. As a result, the choice of an optional computational box shape is equivalent to the proper choice of the basic parallelepiped sides lengths. The checker board periodic boundary conditions have been applied in a molecular dynamics (MD) code based on the link-cell method (using Lennard-Jones interactions). Simulation efficiency is approximately the same as for the classical cubic periodic boundary transformations.

C-language molecular dynamics program for the simulation of Lennard-Jones particles

The paper presents a molecular dynamics C-language program suitable for mixtures of mono-atomic molecules of different types include in a cuboid box with periodic boundary conditions. The molecules mutually interact with the short-range Lennard-Jones potential. To solve the Newtonian equations of motion the leapfrog scheme is applied. Neighbours of a particular molecule are searched using the link-cell method. The program has been developed for microcomputers/workstations so that it incorporates the sequential algorithm optimized in respect with CPU-time.

C-language program for the irregular close packing of hard spheres

The paper presents a C-language program for the irregular close packing of equal haed spheres in a cubic box with periodic boundary conditions using Jodrey and Tory's algorithm. For the nearest-neighbour searching of a given sphere the link-cell method is applied. The program has been developed for microcomputers/workstations so that it incorporates the sequential algorithm optimized for computational time demands.

A technique for improving the link-cell method

A method for speeding up the link-cell (LC) method for very large systems is described. The size of the subcells into which the systems are divided is less than the cutoff distance of the potential. One case among all those studied can be chosen, called the favourable case, which most efficiently improves the conventional LC and the traditional neighbour table methods.

Efficiency test of the traditonal MD and the link-cell methods

For very large systems, the earlier method of computer simulation by molecular dynamics (MD) has been compared with the newer link-cell (LC) method. The results show clearly that the LC method does not improve the efficiency of traditional MD simulations in spite of the great memory saving achieved.

Vectorized link cell Fortran code for molecular dynamics simulations for a large number of particles

A highly efficient procedure for calculating forces in a molecular dynamics simulation on a vector computer is described. The algorithm is feasible for several thousand particles on currently available machines. The efficiency comes about by a combination of a Verlet table with a link cell algorithm. By a double list data structure, that treats the pairs of interacting particles in a symmetrical manner, the vectorization can be improved significantly. Moreover, the “layering algorithm” recently described by Rapaport can be incorporated. A standard Fortran formulation of the basic procedure is given. We test the algorithm on a Cray XMP/416 for two different Lennard-Jones fluids with interactions truncated at 2 1 6 α and 2.5 α. The CPU time is shown to increase linearly with system size. Our link cell method proves to be more efficient than a straightforward search over all pairs as soon as the particle number exceeds about 500.