Simulating granular materials by energy minimization

Abstract

Discrete element methods are extremely helpful in understanding the complex behaviors of granular media, as they give valuable insight into all internal variables of the system. In this paper, a novel discrete element method for performing simulations of granular media is presented, based on the minimization of the potential energy in the system. Contrary to most discrete element methods (i.e., soft-particle method, event-driven method, and non-smooth contact dynamics), the system does not evolve by (approximately) integrating Newtons equations of motion in time, but rather by searching for mechanical equilibrium solutions for the positions of all particles in the system, which is mathematically equivalent to locally minimizing the potential energy. The new method allows for the rapid creation of jammed initial conditions (to be used for further studies) and for the simulation of quasi-static deformation problems. The major advantage of the new method is that it allows for truly static deformations. The system does not evolve with time, but rather with the externally applied strain or load, so that there is no kinetic energy in the system, in contrast to other quasistatic methods. The performance of the algorithm for both types of applications of the method is tested. Therefore we look at the required number of iterations, for the system to converge to a stable solution. For each single iteration, the required computational effort scales linearly with the number of particles. During the process of creating initial conditions, the required number of iterations for two-dimensional systems scales with the square root of the number of particles in the system. The required number of iterations increases for systems closer to the jamming packing fraction. For a quasistatic pure shear deformation simulation, the results of the new method are validated by regular soft-particle dynamics simulations. The energy minimization algorithm is able to capture the evolution of the isotropic and deviatoric stress and fabric of the system. Both methods converge in the limit of quasi-static deformations, but show interestingly different results otherwise. For a shear amplitude of 4 %, as little as 100 sampling points seems to be a good compromise between accuracy and computational time needed.

Un-jamming due to energetic instability: statics to dynamics

“…Recent research also considers soft particles [74,118,121] for which jamming changes from a sharp to a rather smooth transition. One objective of this paper is to bring together fundamental theoretical concepts of continuum mechanics [21,32,35,48,107,122,123] with observations made from particle simulations for simple granular systems in the gas, fluid, and solid states, including also the transitions between those states [8,49,116,118,[124][125][126].…”

A static discrete element method with discontinuous deformation analysis

“…To better model the static problem, various static discrete element models have been proposed, in which different formulations are employed to remove inertial effects in DEM . As a consequence, the integration of the equation of motion is avoided, and the system evolves through a series of static equilibrium states under varied loads.…”

Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions

“…Discrete element method (soft-particle method) is one of the typical molecular dynamics 2 HAOLEI WANG AND LEI ZHANG methods in granular applications [11], where a set of Langevin equations of motion are formulated, with relaxation and random fluctuation terms to quantify the physical relaxation time scale and thermal effect, respectively. Related methods include event-driven method, non-smooth contact dynamics, etc, see references in [10]. On the other hand, energy minimization methods can be applied to the situation of quasi-static deformation, when the applied load changes slowly over time with respect to the inertial forces.…”

“…al. have applied trust region methods to study the relaxation and shear of granular system [10]. In this paper, we will study the performance of several typical minimization methods for the granular simulation, such as conjugate gradient methods and quasi-Newton type methods.…”

“…The molecular details of molecular dynamics methods such as softparticle methods and energy minimization methods are usually not the same. However, it is important to note that the equilibrium physical quantities such as fabric, stress and energy remain similar [10]. Also, there is no temperature in the granular model simulated by energy minimization methods, namely, we are only interested in athermal granular materials .…”