# Discrete element method

The term discrete element method (DEM) is a family of numerical methods for computing the motion of a large number of particles like molecules or grains of sand. The method was originally applied by Cundall in 1971 to problems in rock mechanics. The theoretical basis of the method was detailed by Williams, Hocking, and Mustoe in 1985 who showed that DEM could be viewed as a generalized finite element method. Its applications to geomechanics problems is described in the book Numerical Modeling in Rock Mechanics, by Pande, G., Beer, G. and Williams, J.R.. Good sources detailing research in the area are to be found in the 1st, 2nd and 3rd International Conferences on Discrete Element Methods. Journal articles reviewing the state of the art have been published by Williams, and Bicanic (see below). A comprehensive treatment of the combined Finite Element-Discrete Element Method is contained in the book The Combined Finite-Discrete Element Method by Munjiza. The method is sometimes called molecular dynamics (MD), even when the particles are not molecules. However, in contrast to molecular dynamics the method can be used to model particles with non-spherical shape. The various branches of the DEM family are the distinct element method proposed by Cundall in 1971, the generalized discrete element method proposed by Hocking, Williams and Mustoe in 1985, the discontinuous deformation analysis (DDA) proposed by Shi in 1988 and the finite-discrete element method proposed by Munjiza and Owen in 2004.

Discrete element methods are processor intensive and this limits either the length of a simulation or the number of particles. Advances in the software are beginning to take advantage of parallel processing capabilities (shared or distributed systems) to scale up the number of particles or length of the simulation. An alternative to treating all particles separately is to average the physics across many particles and thereby treat the material as a continuum. In the case of solid-like granular behavior as in soil mechanics, the continuum approach usually treats the material as elastic or elasto-plastic and models it with the finite element method or a mesh free method. In the case of liquid-like or gas-like granular flow, the continuum approach may treat the material as a fluid and use computational fluid dynamics.

Applications

The fundamental assumption of the method is that the material consists of separate, discrete particles. These particles may have different shapes and properties. Some examples are:

* liquids and solutions, for instance of sugar or proteins;

* bulk materials in storage silos, like cereal;

* granular matter, like sand;

* powders, like toner.

Typical industries using DEM are:

* Mining

* Pharmaceutical

* Oil and gas

* Agriculture and food handling

* Chemical

Outline of the method

A DEM-simulation is started by putting all particles in a certain position and giving them an initial velocity. Then the forces which act on each particle are computed from the initial data and the relevant physical laws.

The following forces may have to be considered in macroscopic simulations:

* friction, when two particles touch each other;

* recoil, when two particles collide;

* damping, when energy is lost during the compression and recoil of grains in a collision;

* gravity (the force of attraction between particles due to their mass), which is only relevant in astronomical simulations.

* cohesion and/or adhesion (when two particles collide and stick to each other)

* liquid bridging (wet particles in contact may have a thin liquid film exerting a force on both particles)

Any other type of inter-particular forces may be considered, example electrostatic attraction or repulsion, etc. However, the computational cost increases as the particle-particle force model is made more complex.

On a molecular level, we may consider

* the Coulomb force, the electrostatic attraction or repulsion of particles carrying electric charge;

* Pauli repulsion, when two atoms approach each other closely;

* van der Waals force.

All these forces are added up to find the total force acting on each particle. An integration method is employed to compute the change in the position and the velocity of each particle during a certain time step from Newton's laws of motion. Then, the new positions are used to compute the forces during the next step, and this loop is repeated until the simulation ends.

Typical integration methods used in a discrete element method are:

* the Verlet algorithm,

* velocity Verlet,

* symplectic integrators,

* the leapfrog method.

Long-range forces

When long-range forces (typically gravity or the Coulomb force) are taken into account, then the interaction between each pair of particles needs to be computed. The number of interactions, and with it the cost of the computation, increases quadratically with the number of particles. This is not acceptable for simulations with large number of particles. A possible way to avoid this problem is to combine some particles, which are far away from the particle under consideration, into one pseudoparticle. Consider as an example the interaction between a star and a distant galaxy: The error arising from combining all the stars in the distant galaxy into one point mass is negligible. So-called tree algorithms are used to decide which particles can be combined into one pseudoparticle. These algorithms arrange all particles in a tree, a quadtree in the two-dimensional case and an octree in the three-dimensional case.

However, simulations in molecular dynamics divide the space in which the simulation take place into cells. Particles leaving through one side of a cell are simply inserted at the other side (periodic boundary conditions); the same goes for the forces. The force is no longer taken into account after the so-called cut-off distance (usually half the length of a cell), so that a particle is not influenced by the mirror image of the same particle in the other side of the cell. One can now increase the number of particles by simply copying the cells.

Algorithms to deal with long-range force include:

* Barnes-Hut simulation,

* the fast multipole method.

* DEM can be used to simulate a wide variety of granular flow situations. The results obtained by competent researchers agree well with experimental findings.

* DEM allows a more detailed study of the micro-dynamics of powder flows. For example, the force networks formed in a granular media can be visualized using DEM. Such measurements are nearly impossible in experiments with small and many particles.

* The maximum number of particles, and duration of a virtual simulation is limited by computational power. Typical flows contain billions of particles, but contemporary DEM simulations have been able to simulate of the order of a million particles for sufficiently long time (virtual time, not actual program execution time).

* Even though any random particle geometry can be simulated, simulations are generally limited to spherical particles due to the increase in cost of computation with increasing complexity of geometry.

Bibliography

* P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies. Geotechnique, 29:47–65, 1979.

* Williams, J.R., Hocking, G., and Mustoe, G.G.W., “The Theoretical Basis of the Discrete Element Method,” NUMETA 1985, Numerical Methods of Engineering, Theory and Applications, A.A. Balkema, Rotterdam, January 1985

* Shi, G, Discontinuous deformation analysis - A new numerical model for the statics and dynamics of deformable block structures, 16pp. In 1st U.S. Conf. on Discrete Element Methods, Golden. CSM Press: Golden, CO, 1989.

* Williams, J.R. and Pentland, A.P., "Superquadric and Modal Dynamics for Discrete Elements in Concurrent Design," National Science Foundation Sponsored 1st U.S. Conference of Discrete Element Methods, Golden, CO, October 19-20, 1989.

* Pande, G., Beer, G. and Williams, J.R., Numerical Modeling in Rock Mechanics, John Wiley and Sons, 1990.

* Kawaguchi, T., Tanaka, T. and Tsuji, Y., [http://www-mupf.mech.eng.osaka-u.ac.jp/paper_pdf/PT98,v96,129 Numerical simulation of two-dimensional fluidized beds using the discrete element method (comparison between the two- and three-dimensional models) Powder Technology, 96(2):129–138, 1998.

* Griebel, Knapek, Zumbusch, Caglar: Numerische Simulation in der Molekulardynamik. Springer, 2004. ISBN 3-540-41856-3.

* Bicanic, Ninad, Discrete Element Methods in Stein, de Borst, Hughes Encyclopedia of Computational Mechanics, Vol. 1. Wiley, 2004. ISBN 0-470-84699-2.

* 2nd International Conference on Discrete Element Methods, Editors Williams, J.R. and Mustoe, G.G.W., IESL Press, 1992 ISBN 0-918062-88-8

* Williams, J.R. and O’Connor, R., Discrete Element Simulation and the Contact Problem, Archives of Computational Methods in Engineering, Vol. 6, 4, 279-304, 1999

* Ante Munjiza, The Combined Finite-Discrete Element Method Wiley, 2004, ISBN 0-470-84199-0

Software

Open source and non-commercial software :

• BALL & TRUBAL (1979-1980) distinct element method (FORTRAN code), originally written by P.Cundall and currently maintained by C.Thornton.
• SDEC Spherical Discrete Element Code.
• YADE Yet Another Dynamic Engine, second incarnation of SDEC written from ground-up, GPL license.
• Pasimodo Multi-purpose particle simulation software.
• LMGC90 Open platform for modelling interaction problems between elements including multi-physics aspects based on an hybrid or extended FEM - DEM discretization, using various numerical strategies as MD or NSCD.

Commercially available DEM software packages include PFC3D, EDEM and Passage/DEM: