Keywords: DEM, discrete element, computational geometry, ellipsoid, contact, collision detection.

Discrete Element Modeling (DEM) has become an increasingly important tool for evaluating behavior in highly discontinuous materials such as fractured rock, erodible sedimentary material, pharmaceutical powders, and agricultural grains. The practical limits of DEM in both time for simulation and analyzable number of particles, however, have limited its use in analyzing systems of interest in many industrial applications. These limits arise due to several key computational components of the simulation pipeline, including parallelization strategy, particle representation, contact law, and neighbor-sorting algorithm. The focus of this paper is on representing a particle's geometry as a general ellipsoid in a computationally efficient manner.

Because of the limits of currently available algorithms for determining the exact contact between non-spherical primitives, spheres form the basis for the most popular codes in DEM, including TRUBAL (Cundall [1]), PFC3D (Cundall [2]), and DMC (Taylor and Preece [3]) as well as the application developed in Cleary [4]. However, the sphere has several attributes that, while attractive for computational purposes, limit its ability to capture key behaviors of real particle systems, which typically contain non-spherical particles. Spheres tend to exhibit excessively rolling when subjected to small moments. Because spheres cannot exert a resistive couple (i.e., friction is the only channel by which a resistive couple can be applied), they are unable to represent the particle interlocking observed in many real systems. Some numerical implementations address this by introducing couples artificially by either perturbing the normal force direction so that it does not pass through the centroid or by adding an adjustable "rolling friction" factor (e.g., Vu-Quoc [5]). Spheres also tend to organize into dense, ordered packings as the coefficient of friction decreases (Grest [6]), which is not representative of the structure found in many naturally occurring materials. Despite these drawbacks, sphere-based DEM implementations remain the most common for 3-D modeling, since they are simple and compact (only 4 floating point numbers are required to describe the geometry, and contact resolution between spheres is trivial) and hence are fast computationally.

A new contact resolution algorithm based on the Method of Equivalent Spheres is presented, which takes advantage of the convexity and smoothly varying curvature of the ellipsoidal geometry in combination with an efficient iterative, vector-based search algorithm. This algorithm seeks to exploit the smoothly varying convexity of the ellipsoidal geometry through a vector operation based search algorithm, which displays favorable computational resource requirements based on initial estimates of floating point operations per contacting pair.

1

Cundall, P.A. "Computer Simulations of Dense Sphere Assemblies", Proceedings of the U.S./Japan Seminar on the Micromechanics of Granular Materials, Sendai-Zao, Japan, October 26-30, 1987.

2

Cundall, P.A., Konietzky, H., and Potyondy, D.O. "PFC Ein Neues Werkzeug f'ur Numerische Modellierungen", Bautechnik, 73(8), 492-498, 1996.

3

Taylor L.M., Preece D.S. "Simulation of blasting induced rock motion using spherical element models", Engineering Computations, 9(2), 1992. doi:10.1108/eb023863

4

Cleary, P.W. "Discrete Element Modeling of Industrial Granular Flow Applications", TASK. Quarterly Scientific Bulletin, 2, 385-416, 1998.

5

Vu-Quoc, L., and Zhang, X. "An accurate and efficient tangential force-displacement model for elastic frictional contact in particle-flow simulations", Mechanics of Materials, 31, 235-269, 1999. doi:10.1016/S0167-6636(98)00064-7

6

Grest, G., Landry, J., Silbert, L., and Plimpton, S. "Rheology of Granular Flow", Lecture at the Kavli Institute for Theoretical Physics, Santa Barbera, CA, 4.June, 2001.

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