 Discrete element method

A discrete element method (DEM), also called a distinct element method is any of family of numerical methods for computing the motion of a large number of particles of micrometrescale size and above. Though DEM is very closely related to molecular dynamics, the method is generally distinguished by its inclusion of rotational degreesoffreedom as well as stateful contact and often complicated geometries (including polyhedra). With advances in computing power and numerical algorithms for nearest neighbor sorting, it has become possible to numerically simulate millions of particles on a single processor. Today DEM is becoming widely accepted as an effective method of addressing engineering problems in granular and discontinuous materials, especially in granular flows, powder mechanics, and rock mechanics.
Discrete element methods are relatively computationally intensive, which limits either the length of a simulation or the number of particles. Several DEM codes, as do molecular dynamics codes, 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 solidlike granular behavior as in soil mechanics, the continuum approach usually treats the material as elastic or elastoplastic and models it with the finite element method or a mesh free method. In the case of liquidlike or gaslike granular flow, the continuum approach may treat the material as a fluid and use computational fluid dynamics. Drawbacks to homogenization of the granular scale physics, however, are welldocumented and should be considered carefully before attempting to use a continuum approach.
Contents
The DEM family
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 finitediscrete element method concurrently developed by several groups (e.g., Munjiza and Owen). The general method was originally developed by Cundall in 1971 to problems in rock mechanics. The theoretical basis of the method was established by Sir Isaac Newton in 1697. Williams, Hocking, and Mustoe in 1985 showed that DEM could be viewed as a generalized finite element method. Its application to geomechanics problems is described in the book Numerical Modeling in Rock Mechanics, by Pande, G., Beer, G. and Williams, J.R.. The 1st, 2nd and 3rd International Conferences on Discrete Element Methods have been a common point for researchers to publish advances in the method and its applications. Journal articles reviewing the state of the art have been published by Williams, Bicanic, and Bobet et al. (see below). A comprehensive treatment of the combined Finite ElementDiscrete Element Method is contained in the book The Combined FiniteDiscrete Element Method by Munjiza.
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.
 Blocky or jointed rock masses
Typical industries using DEM are:
 Agriculture and food handling
 Chemical
 Civil Engineering
 Oil and gas
 Mining
 Mineral processing
 Pharmaceutical
 Powder metallurgy
Outline of the method
A DEMsimulation is started by first generating a model, which results in spatially orienting all particles and assigning an initial velocity. The forces which act on each particle are computed from the initial data and the relevant physical laws and contact models. Generally, a simulation consists of three parts: the initialization, explicit timestepping, and postprocessing. The timestepping usually requires a nearest neighbor sorting step to reduce the number of possible contact pairs and decrease the computational requirements; this is often only performed periodically.
The following forces may have to be considered in macroscopic simulations:
 friction, when two particles touch each other;
 contact plasticity, or recoil, when two particles collide;
 gravity, the force of attraction between particles due to their mass, which is only relevant in astronomical simulations.
 attractive potentials, such as cohesion, adhesion, liquid bridging, electrostatic attraction. Note that, because of the overhead from determining nearest neighbor pairs, exact resolution of longrange, compared with particle size, forces can increase computational cost or require specialized algorithms to resolve these interactions.
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.
Longrange forces
When longrange 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. Socalled 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 twodimensional case and an octree in the threedimensional 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 socalled cutoff 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 longrange force include:
 Barnes–Hut simulation,
 the fast multipole method.
Combined finitediscrete element method
Following the work by Munjiza and Owen's earlier work, the combineddiscrete element method has been further developed to various irregular and deformable particles in many applications including pharmaceutical tableting,^{[1]} packaging and flow simulations,^{[2]} and concrete and impact analysis,^{[3]} and many other applications.
Advantages and limitations
Advantages
 DEM can be used to simulate a wide variety of granular flow and rock mechanics situations. Several research groups have independently developed simulation software that agrees well with experimental findings in a wide range of engineering applications, including adhesive powders, granular flow, and jointed rock masses.
 DEM allows a more detailed study of the microdynamics of powder flows than is often possible using physical experiments. 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.
Disadvantages
 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 on large cluster computing resources have only recently been able to approach this scale for sufficiently long time (simulated time, not actual program execution time).
References
 ^ R W Lewis, D T Gethin, XS Yang, R. Rowe, A Combined FiniteDiscrete Element Method for Simulating Pharmaceutical Powder Tableting, Int. J. Num. Method in Engineering, 62, 853–869 (2005)
 ^ D T Gethin, XS Yang and R W Lewis; A Two Dimensional Combined Discrete and Finite Element Scheme for Simulating the Flow and Compaction of Systems Comprising Irregular Particulates, Computer Methods in Applied Mechanics and Engineering, 195, 2006, 5552–5565 (2006)
 ^ I. M. May, Y. Chen, D. R. J. Owen, Y.T. Feng and P. J. Thiele: Reinforced concrete beams under dropweight impact loads, Computers and Concrete, 3 (2–3): 79–90 (2006).
Bibliography
Book
 Ante Munjiza, The Combined FiniteDiscrete Element Method Wiley, 2004, ISBN 0470841990
 Bicanic, Ninad, Discrete Element Methods in Stein, de Borst, Hughes Encyclopedia of Computational Mechanics, Vol. 1. Wiley, 2004. ISBN 0470846992.
 Griebel, Knapek, Zumbusch, Caglar: Numerische Simulation in der Molekulardynamik. Springer, 2004. ISBN 3540418563.
 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
 Pande, G., Beer, G. and Williams, J.R., Numerical Modeling in Rock Mechanics, John Wiley and Sons, 1990.
 Farhang Radjaï and Frédéric Dubois, "Discreteelement Modeling of Granular Materials", Wiley, 2011, ISBN 9781848212602
 Thorsten Pöschel and Thomas Schwager, Computational Granular Dynamics, models and algorithms. Springer, 2005. ISBN 3540214852.
Periodical
 A. Bobet, A. Fakhimi, S. Johnson, J. Morris, F. Tonon, and M. Ronald Yeung (2009) "Numerical Models in Discontinuous Media: Review of Advances for Rock Mechanics Applications", J. Geotech. and Geoenvir. Engrg., 135 (11) pp. 1547–1561
 P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies. Geotechnique, 29:47–65, 1979.
 Kawaguchi, T., Tanaka, T. and Tsuji, Y., Numerical simulation of twodimensional fluidized beds using the discrete element method (comparison between the two and threedimensional models) Powder Technology, 96(2):129–138, 1998.
 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
 Zhu HP, Zhou ZY, Yang RY, Yu AB. Discrete particle simulation of particulate systems: Theoretical developments. Chemical Engineering Science. 2007;62:33783396
 Zhu HP, Zhou ZY, Yang RY, Yu AB. Discrete particle simulation of particulate systems: A review of mayor applications and findings. Chemical Engineering Science. 2008;63:57285770.
Proceedings
 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.
 2nd International Conference on Discrete Element Methods, Editors Williams, J.R. and Mustoe, G.G.W., IESL Press, 1992 ISBN 0918062888
Software
Open source and noncommercial software:
 Ascalaph Molecular dynamics with fourth ordersymplectic integrator.
 BALL & TRUBAL (1979–1980) distinct element method (FORTRAN code), originally written by P.Cundall and currently maintained by Colin Thornton.
 dp3D (discrete powder 3D), DEM code oriented toward materials science applications (powder compaction, powder sintering, ...).
 ESySParticle ESySParticle is a highperformance computing implementation of the Discrete Element Method released under the Open Software License v3.0. To date, development focus is on geoscientific applications including granular flow, rock breakage and earthquake nucleation. ESySParticle includes a Python scripting interface providing flexibility for simulation setup and realtime data analysis. The DEM computing engine is written in C++ and parallelised using MPI, permitting simulations of more than 1 million particles on clusters or highend workstations.
 LAMMPS is a very fast parallel opensource molecular dynamics package with GPU support also allowing DEM simulations. LAMMPS Website, Examples .
 LIGGGHTS is a code based on LAMMPS with more DEM features such as wall import from CAD, a moving mesh feature and granular heat transfer. LIGGGHTS Website
 SDEC Spherical Discrete Element Code.
 LMGC90 Open platform for modelling interaction problems between elements including multiphysics aspects based on an hybrid or extended FEM – DEM discretization, using various numerical strategies as MD or NSCD.
 Pasimodo PASIMODO is a program package for particlebased simulation methods. The main field of application is the simulation of granular media, such as sand, gravel, granulates in chemical engineering and others. Moreover it can be used for the simulation of many other Lagrangian methods, e.g. fluid simulation with SmoothedParticleHydrodynamics.
 Yade Yet Another Dynamic Engine (historically related to SDEC), modular and extensible toolkit of DEM algorithms written in c++. Tight integration with Python gives flexibility to simulation description, realtime control and postprocessing, and allows introspection of all internal data. Can run in parallel on sharedmemory machines using OpenMP.
 MechSys Although it is initially a package dedicated to the FEM method, it also contains a DEM module. It uses both spherical elements and spheropolyhedra to model collision of particles with general shapes. Both elastic and cohesive forces are included to model damage and fracture processes. Parallelization is achieved mostly by the new std::thread library of the new C++ standard. There is also a module dealing with the coupling between DEM and LBM still under development.
Commercially available DEM software packages include PFC3D, EDEM and Passage/DEM:
 Bulk Flow Analyst (Applied DEM) Generalpurpose 3D DEM tool for mechanical engineering applications. Imports DXF files and integrates with AutoCAD and SolidWorks.
 Chute Analyst (Overland Conveyor Company) 3D DEM tool for transfer chute engineering applications. Imports DXF files and integrates with AutoCAD.
 Chute Maven (Hustrulid Technologies Inc.) Spherical Discrete Element Modeling in 3 Dimensions. Directly reads in AutoCad dxf files and interfaces with SolidWorks.
 EDEM (DEM Solutions Ltd.) Generalpurpose DEM simulation with CAD import of particle and machine geometry, GUIbased model setup, extensive postprocessing tools, progammable API, couples with CFD, FEA and MBD software.
 ELFEN
 GROMOS 96
 MIMES a variety of particle shapes can be used in 2D
 PASSAGE/DEM (PASSAGE/DEM Software is for predicting the flow particles under a wide variety of forces.)
 PFC (2D & 3D) Particle Flow Code.
 SimPARTIX DEM and SPH simulation package from Fraunhofer IWM
 StarCCM+ Engineering analysis suite for solving problems involving flow (of fluids or solids), heat transfer and stress.
 UDEC and 3DEC Two and threedimensional simulation of the response of discontinuous media (such as jointed rock) that is subject to either static or dynamic loading.
 DEMpack Discrete / finite element simulation software in 2D and 3D, user interface based on GiD.
See also
 Movable Cellular Automata
 Finite element method
Categories: Numerical differential equations
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