Simulating granular materials by energy minimization

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 quasi-static 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 quasi-static 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 $$4\,\%$$4%, as little as 100 sampling points seems to be a good compromise between accuracy and computational time needed.

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