An inexact interior point method for the large-scale simulation of granular material

Abstract

Non-smooth contact dynamics provides an increasingly popular simulation framework for granular material. In contrast to classical discrete element methods, this approach is stable for arbitrary time steps and produces visually acceptable results in very short computing time. Yet when it comes to the prediction of draft forces, non-smooth contact dynamics is typically not accurate enough. We therefore propose to combine the method class with an interior point algorithm for higher accuracy. Our specific algorithm is based on so-called Jordan algebras and exploits the relation to symmetric cones in order to tackle the conical constraints that are intrinsic to frictional contact problems. In every interior point iteration a linear system has to be solved. We analyze how the interior point method behaves when it is combined with Krylov subspace solvers and incomplete factorizations. We show that efficient preconditioners and efficient linear solvers are essential for the method to be applicable to large-scale problems. Using BiCGstab as a linear solver and incomplete Cholesky factorizations, we substantially improve the accuracy in comparison to the projected Gauß-Jacobi solver.

Dynamic behaviour of solids and granular materials: a force potential-based particle method

“…Multiscale analysis of solids has also been proposed in recent papers [7,8].On the other hand, such a discrete nature is well evident for some classes of materials, such as the granular one, that consist of several low deformable particles, usually interacting each other through elastic contact, cohesive and friction forces [9][10][11][12][13]. Among the different problems involving the simulation of mechanical materials response, geomechanical and powders, one can be naturally studied by exploiting their significant granular nature [14][15][16][17].At the macroscale, the discrete methods have relevant applications in mineral processing, rock blasting, crushing, phenomena involving sand mechanics, powders technology, failure of compact or granular bodies [15,17]. It is worth mentioning that materials like gases and liquids can be simulated with this class of methods [18,19], that is, by using the simple scheme of interacting particles.Mesh-free Lagrangian methods (such as the well-known smooth particle hydrodynamics [20]), in which the material is discretised with particles with variable nodal connectivity interacting each other with interpolation functions, have been developed and successfully applied to different physical problems.A generic solid may always be assumed to have a particle structure: the average particles size adopted and the particular nature of the particles' interaction forces allow the description of different materials; in such a way, the possible behaviours can be made to range from the very incoherent cases up to the compact material ones, typical of granular (or powder) and of polycrystalline materials, respectively.Particles can be viewed as objects carrying the physical characteristics of the mechanical system; therefore, the system can be simulated through the study of the motion evolution of the properties carried by the particles (position, velocity, forces, : : :).…”

A conical interior point method for nonsmooth rigid body dynamics

“…IPMs have a very good theoretical complexity bound, but they rely on effective preconditioners and linear solvers. A detailed description and analysis of our IPM is given in [9]. We implemented a prototype of the conical IPM in C++ to test its applicability.…”

Nonsmooth Contact Dynamics for the large-scale simulation of granular material

“…This contribution continues the work [12] and the thesis [13]. For brevitiy, basic concepts such as Jordan algebras and Lebesgue-Stieltjes measures are not included but can be found in [13].…”

“…From a physical point of view, γ j can is the reaction impulse of the constraint at time t j . The equations of motion (12) hold in the same way for a rigid body system with m rigid bodies in three dimensions, i.e., for trajectories q : [0, T ] → R 6m :…”

“…We propose the use of a matrix-free interior point method and provide here a short outline. For the details we refer to [12,13].…”