# Stochastic simulations of two-dimensional composite packings

## Abstract

In recent years, dense packings of two- and three-dimensional objects have been studied intensely in the context of computational physics and material sciences. For example, computer simulations of disordered solids usually employ a two-dimensional model which is based on hexagonal networks of elastic and rigid bonds or arrangements of mixed soft and hard disks, respectively. Both types of bonds/disks are distributed randomly. Large systems of equations have to be solved at any simulation step for the calculation of local displacements or particle velocities. The simulations start from equidistant nodes of the hexagonal network or centers of disks, respectively, which, in general, may not be in an equilibrium state. We suggest an extension of the model where first a near-equilibrium packing of randomly distributed bonds/disks is calculated. Then, we can compute the displacement caused by external forces from this near-equilibrium initial packing of the elementary units. To this end, we propose a stochastic simulation of the external impact by incorporating the computation of near-equilibrium states as well as specific boundary conditions. Our methodology is based on a two-step approach consisting of a preprocessing stage, where physical properties of different types of particles are analyzed by numerical methods, and a second stage of stochastic (annealing-based) simulations which exploits approximate formulas for local interactions. We have implemented two types of cooling schedules with an expected serial run-timen· ln2nandn3/2· ln5/2n, respectively, to reach near-equilibrium states forndisks. The algorithms were parallelized on a 20-processor machine, and for a sufficiently large number of objects the speedup is close to the number of processors. For example, the parallel run-time for computing near-equilibrium states is about 212h for 449 disks, using the first cooling schedule, and about 37 h for 1068 disks, using the second cooling schedule. We performed a number of computer simulations calculating the average displacement in near-equilibrium states from regular, equidistant initial packings. The underlying physical model for our implementations is very similar to the model used for the analysis of granular composites which involves arrangements of mixed soft and hard disks. However, our emphasis is on the computational aspects rather than on particular systems of physical interactions, because substituting a system of physical interactions by another one does not affect significantly the run-time or the overall approach. © 1997 Academic Press.