Discrete element method: Difference between revisions

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The '''discrete element method''' (DEM) is a family of [[numerical analysis|numerical]] methods for computing the motion of a large number of particles like molecules or grains of sand. The method is sometimes ''molecular dynamics'' (MD), even when the particles are not molecules. Two prominent members of the DEM family are the [[distinct element method]] proposed by [[Peter A. Cundall | Cundall]] in [[1971]], and the [[discontinuous deformation analysis]] (DDA) proposed by [[Gen-hua Shi | Shi]] in [[1988]].

'''Discrete element methods''' (DEM) comprise a family of related mathematical tools for determining the [[kinematic]] behavior of finite-sized [[particle]]s.

As defined by Cundall, ''discrete element methods'' allow:

DEMDiscrete iselement methods are processor intensive and this limits either the length of a simulation (given the small timestep) 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 treat the material as a continuum and use [[computational fluid dynamics]].

#finite [[displacement (distance)|displacement]], [[rotation]] and separation of elements, and<br>

#new particle contacts are recognized [[algorithm]]ically as the [[computer simulation|simulation]] proceeds.

==Applications==

Two prominent members of the DEM family are the [[distinct element method]] proposed by [[Peter A. Cundall | Cundall]] in [[1971]], and the [[discontinuous deformation analysis]] (DDA) proposed by [[Gen-hua Shi | Shi]] in [[1988]].

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:

'''Discrete element modeling''' (also abbreviated to DEM) is a computational method for simulating the behaviour of a systems of discrete objects (particles). It tracks every individual particle (and any bounding geometry) subjecting them to contact and body forces, and updating the system incrementally in time. As opposed to [[CFD]] a much smaller time step is necessary to accurately simulate the contacts between objects, typically of the order ot 10^-5 - 10^-6. DEM software allows the user to understand existing behaviour and make predictions on new systems. Typical industries using DEM are:

* [[liquid]]s and [[solution]]s, for instance of [[sugar]] or [[protein]]s;

* [[bulk material]]s in [[silo]]s, like [[cereal]];

* [[granular matter]], like [[sand]];

* [[powder]]s, like [[toner]].

Typical industries using DEM are:

#* Mining

#* Pharmaceutical

#* Oil and gas

#* Agriculture and food handling

#* Chemical

==Outline of the method==

DEM is processor intensive and this limits either the length of a simulation (given the small timestep) 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.

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;

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

On a molecular level, we may consider

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

* [[Pauli exclusion principle|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 [[numerical ordinary differential equations|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 [[program loop|loop]] is repeated until the simulation ends.

Typical integration methods used in a discrete element method are:

* the [[Verlet integration|Verlet algorithm]],

* [[velocity Verlet]],

* 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 an 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 condition]]s); 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]],

* the [[fast multipole algorithm|fast multipole method]].

==Bibliography==

* P.A. Cundall, O.D.L. Strack:, ''A distinct element model for granular assemblies''. ''Geotechnique,'' '''29''':47,&ndash;65, 1979.

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

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

* Shi, G (1989)., 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, 16pp1989.

* Kawaguchi, T., Tanaka, T. and Tsuji, Y., [http://www-mupf.mech.eng.osaka-u.ac.jp/paper_pdf/PT98,v96,129 Kawaguchi, T., Tanaka, T. and Tsuji, Y., "Numerical simulation of two-dimensional fluidized beds using the discrete element method (comparison between the two- and three-dimensional models)]," ''Powder Technology,'' Vol. '''96, No. '''(2 (1998):129&ndash;138, pp1998. 129-138.]

* [http://www.itascacg.com/pfc_pubs.html itasca DEM pubs.html]

==Software==

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Open source and non-commercial software:

* BALL & TRUBAL (1979-1980) distinct element method (fortranFORTRAN code), authored by [http://www.iem.bham.ac.uk/computation/granular/colthor.htm Colin Thornton].

* [http://geo.hmg.inpg.fr/frederic/Research_project_Discrete_Element_Software.html SDEC] Spherical Discrete Element Code.

* [http://yade.berlios.de/ YADE] Yet Another Dynamic Engine, second incarnation of SDEC written from ground-up, GPL license.

Commercially available DEM software packages include PFC3D and EDEM:

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*[http://www.itascacg.com/pfc.html PFC2D and PFC3D (Particle Flow Code in 2 Dimensions; Particle Flow Code in 3 Dimensions)]

*[http://www.dem-solutions.com/introduction.html EDEM (DEM Solutions Ltd.)]

*[http://www.igc.ethz.ch/gromos/ GROMOS 96]

*[http://www.rockfield.co.uk/elfen.htm ELFEN]

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[[Category:Numerical analysis]]