Energy Minimization Methods For Granular Materials Simulation

Granular materials are collections of discrete,macroscopic particles and they are ubiquitous in nature,daily life and industries.Granular mate-rials can exhibit very rich and complex phenomena,as a new matter form different from solid,liquid and gas.The research on granular materials can help understand many natural process such as the migration of sand dunes and the onset of earthquakes,and improve the efficiency to handle granular raw materials in industry.Although many insightful results have been ob-tained by experiments and simulations in recent decades,there are still no comprehensive theories to govern the behavior of granular materials.In this thesis,we adopt the energy minimization method to simulate the quasi-static process of granular materials,which is enlightened by the dissipative nature of the interactions between particles.During quasi-static process,the energy minimization methods search the minimum of potential energy to find the equilibrium state,which is more effective than the discrete element methods demanding strict time-step.There is no kinetic energy during the simulation and the system is always in complete mechanical equi-librium,which helps the judgement of critical states and the searching of critical points in high accuracy.Furthermore,we develop two precondition-ing techniques to enhance the performance of energy minimization methods,which are Exp(exponential)preconditioner based on the configuration's ge-ometry information and FF(Force Field)preconditioner based on the pair potentials between grains.Those methods are validated with two benchmark problems:quasi-static compression and shear deformation.In 2D numerical simulations,we show that the Exp preconditioner can have more than 50%reduction in terms of iteration number in quasi-static compression and shear simulation,while 30%in random perturbation simulation.There exits a scaling law between the iteration numbers and volume fractions close to the jamming point.In 3D numerical simulations,we focus on the performance of the preconditioners to typical minimization methods,such as conjugate gradient methods and quasi-Newton methods.During the quasi-static shear simulation,both 2D and 3D jamming con-figuration reveal the plastic deformation.We apply the shear transformation zone(STZ)theory to investigate the plastic behavior of granular materials.We find an inverse correlation between the simplified STZ quadrilaterals'number and the stress-strain curve,which is similiar as the experiment re-sults^[4]and simulations for metal glasses^[5],this demonstrate the universality for disorder materials.In the future work,we can investigate the application of the energy minimization method for granular systems with more realistic force field,the construction of more efficient preconditioners,and the development of machine learned coarse graining models with simulation and experimental data.