# Computing elastic moduli of two-dimensional random networks of rigid and nonrigid bonds by simulated annealing

## Abstract

In recent years, there has been a growing interest in the elastic properties of percolating random networks because mixing a normal elastic material and a rigid material with large stiffness moduli could lead to a new class of elastic synthetic materials. Computer simulations of such disordered solids usually employing a two-dimensional model which is based on hexagonal networks of randomly distributed elastic and rigid bonds, where the ratio of the two types of bonds is close to the percolation threshold. The networks are considered under the impact of external forces, and the corresponding elastic moduli of the mixed material are determined by the displacements of network nodes. The force/displacement relationship is calculated, in general, by an iterative method, where each step requires to solve large systems of linear equations which are derived from minimum energy conditions. The procedure has to be repeated for a large number of randomly generated assignments of rigid and nonrigid bonds. We propose a stochastic approach, where near-equilibrium configurations of percolation networks are calculated by simulated annealing algorithms. The impact of external forces is simulated by initial deformations of the boundary. We have implemented two types of cooling schedules with an expected serial run-time nln2n and n3/2In5/2n, respectively, to reach a configuration of minimum force for a network with n nodes. The algorithms were parallelized on a 20-processor machine, with the speed-up being close to half the number of processors for sufficiently large networks. For example, the parallel run-time for computing near-equilibrium states is ca. 1 h for a network of ≈ 500 nodes, using the first cooling schedule, and ca. 8 h for ≈ 1000 nodes, using the second cooling schedule. We performed a number of computer simulations calculating the average displacement in near-equilibrium states from regular, equidistant initial configurations under the impact of external forces. Since our emphasis is on computational aspects rather than on particular systems of physical interactions, we have considered the case of bond-stretching forces because our analysis reveals that substituting a system of physical interactions by another one (e.g. including angle-bending forces) only affects the run-time by a constant factor. From the force/displacement relationship we calculated Youngs' modulus and Poisson's ratio, and our results are close to published estimations obtained by deterministic methods for similar models and parameter settings.