Discrete element method: Difference between revisions

Revision as of 13:26, 4 December 2009

Latest revision as of 08:36, 2 August 2025

(305 intermediate revisions by more than 100 users not shown)

Line 1:

{{short description| Numerical methods for computing the motion and effect of a large number of small particles}}

The term '''discrete element method''' (aka, distinct element method or DEM) is a family of [[numerical analysis|numerical]] methods for computing the motion of a large number of particles of micron-scale size and above. Though DEM is very closely related to [[molecular dynamics]], the method is generally distinguished by its inclusion of rotational degrees-of-freedom 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.

{{distinguish|finite element method}}

{{More citations needed|date=November 2019}}

A '''discrete element method''' ('''DEM'''), also called a '''distinct element method''', is any of a family of [[numerical analysis|numerical]] methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to [[molecular dynamics]], the method is generally distinguished by its inclusion of rotational [[Degrees of freedom (statistics)|degrees-of-freedom]] as well as stateful contact, particle deformation 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, ice and [[rock mechanics]]. DEM has been extended into the [[Extended Discrete Element Method]] taking [[heat transfer]],<ref name="Peng">{{cite journal |last1=Peng |first1=Z. |last2=Doroodchi |first2=E. |last3=Moghtaderi |first3=B. |date=2020 |title=Heat transfer modelling in Discrete Element Method (DEM)-based simulations of thermal processes: Theory and model development |journal=Progress in Energy and Combustion Science |volume=79,100847 |page=100847 |doi=10.1016/j.pecs.2020.100847|s2cid=218967044 }}</ref> [[chemical reaction]]<ref name="Papadikis">{{cite journal |last1=Papadikis |first1=K. |last2=Gu |first2=S. |last3=Bridgwater |first3=A.V. |date=2009 |title=CFD modelling of the fast pyrolysis of biomass in fluidised bed reactors: Modelling the impact of biomass shrinkage |journal=Chemical Engineering Journal |volume=149 |issue=1–3 |pages=417–427|doi=10.1016/j.cej.2009.01.036 |url=https://eprints.soton.ac.uk/149223/1/Paper.pdf }}</ref> and coupling to [[Computational fluid dynamics|CFD]]<ref name="Kafui">{{cite journal |last1=Kafui |first1=K.D. |last2=Thornton |first2=C. |last3=Adams |first3=M.J. |date=2002 |title=Discrete particle-continuum fluid modelling of gas–solid fuidised beds |journal=Chemical Engineering Science |volume=57 |issue=13 |pages=2395–2410|doi=10.1016/S0009-2509(02)00140-9 |bibcode=2002ChEnS..57.2395K }}</ref> and [[Finite element method|FEM]]<ref name="Trivino">{{cite journal |last1=Trivino |first1=L.F. |last2=Mohanty |first2=B. |date=2015 |title=Assessment of crack initiation and propagation in rock from explosion-induced stress waves and gas expansion by cross-hole seismometry and FEM–DEM method |journal=International Journal of Rock Mechanics & Mining Sciences |volume=77 |pages=287–299|doi=10.1016/j.ijrmms.2015.03.036 |bibcode=2015IJRMM..77..287T }}</ref> into account.

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 mechanics|continuum]]. In the case of [[solid]]-like granular behavior as in [[soil mechanics]], the continuum approach usually treats the material as [[Elasticity (physics)|elastic]] or [[Plasticity (physics)|elasto-plastic]] and models it with the [[finite element method]] or a [[Meshfree methods|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]]. Drawbacks to [[Homogenization (chemistry)|homogenization]] of the granular scale physics, however, are well-documented{{Citation needed|date=August 2025}} and should be considered carefully before attempting to use a continuum approach.

The various branches of the DEM family are the [[distinct element method]] proposed by [[Peter A. Cundall|Cundall]] in 1971, the [[generalized discrete element method]] proposed by [[Grant Hocking|Hocking]], [[John R. Williams|Williams]] and [[Graham Mustoe|Mustoe]] in 1985, the [[Discontinuous Deformation Analysis|discontinuous deformation analysis]] (DDA) proposed by [[Gen-hua Shi|Shi]] in 1988 and the finite-discrete element method concurrently developed by several groups (e.g., [[Ante Munjiza|Munjiza]] and [[Roger Owen|Owen]]). The general method was originally developed by [[Peter A. Cundall|Cundall]] in 1971 to problems in rock mechanics. The theoretical basis of the method was established by Sir Isaac Newton in 1697<!-- 1967 is INCORRECT! --~~~~Aclariont Sept 25, 2009 (assumed a typo for now)-->. [[John R. Williams|Williams]], [[Grant Hocking|Hocking]], and [[Graham Mustoe|Mustoe]] in 1985 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.. 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 [[John R. Williams|Williams]], [[Nenad Bicanic|Bicanic]], and [[Bobet]] et. al. (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 [[Ante Munjiza|Munjiza]].

==The DEM family==

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 mechanics|continuum]]. In the case of [[solid]]-like granular behavior as in [[soil mechanics]], the continuum approach usually treats the material as [[Elasticity (physics)|elastic]] or [[Plasticity (physics)|elasto-plastic]] and models it with the [[finite element method]] or a [[Meshfree methods|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]]. Drawbacks to [[homogenization]] of the granular scale physics, however, are well-documented and should be considered carefully before attempting to use a continuum approach.

The various branches of the DEM family are the [[distinct element method]] proposed by [[Peter A. Cundall]] and Otto D. L. Strack in 1979,<ref>{{Cite journal|last1=Cundall|first1=Peter. A.|last2=Strack|first2=Otto D. L.|date=1979|title=Discrete numerical model for granular assemblies|url=http://websrv.cs.umt.edu/classes/cs477/images/0/0e/Cundall_Strack.pdf|journal=Géotechnique|volume=29|issue=1|pages=47–65|doi=10.1680/geot.1979.29.1.47}}</ref> the [[generalized discrete element method]],<ref name="WHM85">{{cite journal |last1=Williams |first1=J. R. |last2=Hocking |first2=G. |last3=Mustoe |first3=G. G. W. |title=The Theoretical Basis of the Discrete Element Method |journal=NUMETA 1985, Numerical Methods of Engineering, Theory and Applications |publisher=A.A. Balkema |___location=Rotterdam |date=January 1985|url=https://docs.google.com/document/d/1ljujwjib2h2NwYksdh9wONZhEpNljGQdAmehXANFJw4}}</ref> the [[Discontinuous Deformation Analysis|discontinuous deformation analysis]] (DDA) {{harv|Shi|1992}} and the finite-discrete element method concurrently developed by several groups (e.g., [[Ante Munjiza|Munjiza]] and [[Roger Owen (mathematician)|Owen]]). The general method was originally developed by Cundall in 1971 to problems in rock mechanics.

Williams<ref name="WHM85" /> showed that DEM could be viewed as a generalized finite element method, allowing deformation and fracturing of particles. Its application to [[geomechanics]] problems is described in the book ''Numerical Methods in Rock Mechanics''.{{sfn|Williams|Pande|Beer|1990}} 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 and O'Connnor,<ref>{{cite journal |last1=Williams |first1=J. R. |last2=O'Connor |first2=R. |title=Discrete element simulation and the contact problem |journal=Archives of Computational Methods in Engineering |date=December 1999 |volume=6 |issue=4 |pages=279–304 |doi=10.1007/BF02818917|citeseerx=10.1.1.49.9391 |s2cid=16642399 }}</ref> [[Nenad Bicanic|Bicanic]], and [[Antonio Bobet|Bobet]] et al. (see below). A comprehensive treatment of the combined Finite Element-Discrete Element Method is contained in the book ''The Combined Finite-Discrete Element Method''.<ref name="Munjiza 2004">{{cite book |last1=Munjiza |first1=Ante |title=The Combined Finite-Discrete Element Method |date=2004 |publisher=Wiley |___location=Chichester |isbn=978-0-470-84199-0}}</ref>

[[File:Cundall DEM.gif|thumb|upright=1|Discrete-element simulation with particles arranged after a photo of [[Peter A. Cundall]]. As proposed in Cundall and Strack (1979), grains interact with linear-elastic forces and Coulomb friction. Grain kinematics evolve through time by temporal integration of their force and torque balance. The collective behavior is self-organizing with discrete shear zones and angles of repose, as characteristic to cohesionless granular materials.]]

==Applications==

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

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

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:

* bulk materials in storage silos, like cereal;

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

* granular matter, like sand;

* [[bulk material]]s in [[storage soil]]s, like [[cereal]];

* [[granular matter]]powders, like [[sand]];toner.

* [[Powder (substance)|powders]], like [[toner]].

* Blocky or jointed rock masses

Typical industries using DEM are:

* Mining

* Mineral processing

* Pharmaceutical

* Civil Engineering

* Oil and gas

* Agriculture and food handling

* Chemical

*Detergents<ref>{{Cite journal|last1=Alizadeh|first1=Mohammadreza|last2=Hassanpour|first2=Ali|last3=Pasha|first3=Mehrdad|last4=Ghadiri|first4=Mojtaba|last5=Bayly|first5=Andrew|date=2017-09-01|title=The effect of particle shape on predicted segregation in binary powder mixtures|journal=Powder Technology|volume=319|pages=313–322|doi=10.1016/j.powtec.2017.06.059|issn=0032-5910|url=http://eprints.whiterose.ac.uk/118401/3/Manuscript-Mohammadreza%20Alizadeh%20et%20al.pdf}}</ref>

* Oil and gas

* Mining

* Mineral processing

* Pharmaceutical industry<ref>{{Cite journal|last1=Behjani|first1=Mohammadreza Alizadeh|last2=Motlagh|first2=Yousef Ghaffari|last3=Bayly|first3=Andrew|last4=Hassanpour|first4=Ali|date=2019-11-07|title=Assessment of blending performance of pharmaceutical powder mixtures in a continuous mixer using Discrete Element Method (DEM)|url=http://www.sciencedirect.com/science/article/pii/S0032591019309313|journal=Powder Technology|volume=366|pages=73–81|doi=10.1016/j.powtec.2019.10.102|s2cid=209718900 |issn=0032-5910|archive-url=http://eprints.whiterose.ac.uk/157493/|archive-date=21 Feb 2020|url-access=subscription}}</ref>

* [[Powder metallurgy]]

==Outline of the method==

A DEM-simulation is started by first generating a model, which results in spatially orienting all particles and assigning an initial [[velocity]]. The 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 time-stepping, and post-processing. The time-stepping 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 (chemistry)|cohesion]], [[adhesion]], [[liquid bridging]], [[electrostatic attraction]]. Note that, because of the overhead from determining nearest neighbor pairs, exact resolution of long-range, 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 exclusion principle|Pauli repulsion]], when two atoms approach each other closely;

Line 45 ⟶ 52:

* [[symplectic integrator]]s,

* the [[leapfrog method]].

==Thermal DEM==

The discrete element method is widely applied for the consideration of mechanical interactions in many-body problems, particularly granular materials. Among the various extensions to DEM, the consideration of heat flow is particularly useful. Generally speaking in Thermal DEM methods, the thermo-mechanical coupling is considered, whereby the thermal properties of an individual element are considered in order to model heat flow through a macroscopic granular or multi-element medium subject to a mechanical loading.<ref>{{cite journal | arxiv=1406.4199 | doi=10.13182/FST13-727 | title=Thermal Discrete Element Analysis of EU Solid Breeder Blanket Subjected to Neutron Irradiation | year=2014 | last1=Gan | first1=Yixiang | last2=Hernandez | first2=Francisco | last3=Hanaor | first3=Dorian | last4=Annabattula | first4=Ratna | last5=Kamlah | first5=Marc | last6=Pereslavtsev | first6=Pavel | journal=Fusion Science and Technology | volume=66 | issue=1 | pages=83–90 | bibcode=2014FuST...66...83G | s2cid=51903434 }}</ref> Interparticle forces, computed as a part of classical DEM, are used to determined areas of true interparticle contact and thus model the conductive transfer of heat from one solid element to another. A further aspect that is considered in DEM is the gas phase conduction, radiation and convection of heat in the interparticle spaces. To facilitate this, properties of the inter-element gaseous phase need to be considered in terms of pressure, gas conductivity and the mean-free path of gas molecules.<ref>{{cite journal | url=https://www.sciencedirect.com/science/article/pii/S0032591013002702 | doi=10.1016/j.powtec.2013.04.013 | title=Thermal DEM–CFD modeling and simulation of heat transfer through packed bed | year=2013 | last1=Tsory | first1=Tal | last2=Ben-Jacob | first2=Nir | last3=Brosh | first3=Tamir | last4=Levy | first4=Avi | journal=Powder Technology | volume=244 | pages=52–60 | url-access=subscription }}</ref>

==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. TheBoth the number of interactions, and with it the cost of the computation, [[quadratic growth|increasesincrease 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 space|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-HutBarnes–Hut simulation]],

* the [[fast multipole method]].

==Combined Finitefinite-Discretediscrete Element Methodelement method==

Following the work by Munjiza and Owen, the combined finite-discrete element method has been further developed to various irregular and deformable particles in many applications including pharmaceutical tableting,<ref>{{Cite journal |last1=Lewis |first1=R. W. |last2=Gethin |first2=D. T. |last3=Yang |first3=X. S. |last4=Rowe |first4=R. C. |title=A combined finite-discrete element method for simulating pharmaceutical powder tableting |doi=10.1002/nme.1287 |journal=International Journal for Numerical Methods in Engineering |volume=62 |issue=7 |pages=853 |year=2005|bibcode = 2005IJNME..62..853L |arxiv=0706.4406 |s2cid=122962022 }}</ref> packaging and flow simulations,<ref>{{Cite journal |last1=Gethin |first1=D. T. |last2=Yang |first2=X. S. |last3=Lewis |first3=R. W. |doi=10.1016/j.cma.2005.10.025 |title=A two dimensional combined discrete and finite element scheme for simulating the flow and compaction of systems comprising irregular particulates |journal=Computer Methods in Applied Mechanics and Engineering |volume=195 |issue=41–43 |pages=5552 |year=2006 |bibcode = 2006CMAME.195.5552G }}</ref> and impact analysis.<ref>{{Cite journal |last1=Chen |first1=Y. |last2=May |first2=I. M. |doi=10.1680/stbu.2009.162.1.45 |title=Reinforced concrete members under drop-weight impacts |journal=Proceedings of the ICE - Structures and Buildings |volume=162 |pages=45–56 |year=2009 }}</ref>

Following the work by Munjiza and Owen's earlier work, the combined-discrete element method

has been further developed to various irregular and deformable particles in many applications

including pharmaceutical tableting <ref>R W Lewis, D T Gethin, X-S Yang, R. Rowe, A Combined Finite-Discrete Element Method for Simulating Pharmaceutical Powder Tableting, Int. J. Num. Method in Engineering, 62, 853-869 (2005)</ref>, packaging and flow simulations <ref>D T Gethin, X-S 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)</ref>, and concrete and impact analysis <ref>

I. M. May, Y. Chen, D. R. J. Owen, Y.T. Feng and P. J. Thiele: Reinforced concrete beams under drop-weight impact loads, Computers and Concrete, 3 (2-3): 79-90 (2006). </ref>, and many other applications.

==Advantages and Limitationslimitations==

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 micro-dynamics 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.

* The general characteristics of force-transmitting contacts in granular assemblies under external loading environments agree with experimental studies using Photo-stress analysis (PSA).<ref>{{cite journal |doi=10.1098/rsta.2007.0004 |title=Link between single-particle properties and macroscopic properties in particulate assemblies: Role of structures within structures |date=2007 |last1=Antony |first1=S.J |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=365 |issue=1861 |pages=2879–2891 |pmid=17875541 |bibcode=2007RSPTA.365.2879A }}</ref><ref>{{cite journal |last1=S. J. Antony, D. Chapman, J. Sujatha and T. Barakat |title=Interplay between the inclusions of different sizes and their proximity to the wall boundaries on the nature of their stress distribution within the inclusions inside particulate packing |journal=Powder Technology |date=2015 |volume=286 |pages=286, 98–106|doi=10.1016/j.powtec.2015.08.007 |url=https://eprints.whiterose.ac.uk/89428/1/Manuscript-FinalV.pdf }}</ref>

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).

* DEM is computationally demanding, which is the reason why it has not been so readily and widely adopted as continuum approaches in [[computational engineering]] sciences and industry. However, the actual program execution times can be reduced significantly when graphical processing units (GPUs) are utilized to conduct DEM simulations, due to the large number of computing cores on typical GPUs. In addition GPUs tend to be significantly more energy efficient than conventional computing clusters when conducting DEM simulations i.e. a DEM simulation solved on GPUs requires less energy than when it is solved on a conventional computing cluster.<ref>{{Cite journal|last1=He|first1=Yi|last2=Bayly|first2=Andrew E.|last3=Hassanpour|first3=Ali|last4=Muller|first4=Frans|last5=Wu|first5=Ke|last6=Yang|first6=Dongmin|date=2018-10-01|title=A GPU-based coupled SPH-DEM method for particle-fluid flow with free surfaces|journal=Powder Technology|volume=338|pages=548–562|doi=10.1016/j.powtec.2018.07.043|issn=0032-5910|doi-access=free}}</ref>

== See also ==

*[[Compaction simulation]]

*[[Movable Cellular Automata]]

==References==

Line 75 ⟶ 88:

==Bibliography==

* P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies. ''Geotechnique,'' '''29''':47&ndash;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&ndash;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

* 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

'''Book'''

==Software==

* {{cite encyclopedia |last=Bicanic |first=Ninad |title=Discrete Element Methods |editor1-last=Stein |editor1-first=Erwin |editor2-last=De Borst |editor3-last=Hughes |editor3-first=Thomas J.R. |encyclopedia=Encyclopedia of Computational Mechanics |volume=1 |publisher=Wiley |date=2004 |isbn=978-0-470-84699-5}}

Open source and non-commercial software:

* {{cite book|last1=Griebel|first1=Michael|title=Numerische Simulation in der Moleküldynamik |date=2003 |publisher=Springer |___location=Berlin |isbn=978-3-540-41856-6|display-authors=etal}}

* {{cite journal |last1=Williams |first1=J. R. |last2=Hocking |first2=G. |last3=Mustoe |first3=G. G. W. |title=The Theoretical Basis of the Discrete Element Method |journal=NUMETA 1985, Numerical Methods of Engineering, Theory and Applications |publisher=A.A. Balkema |___location=Rotterdam |date=January 1985 }}

* {{cite book |last1=Williams|first1=J. R.|last2=Pande |first2=G. |last3=Beer |first3=J.R. |title=Numerical Methods in Rock Mechanics |date=1990 |publisher=Wiley |___location=Chichester |isbn=978-0471920212 }}

* {{cite book |editor1-last=Radjai |editor1-first=Farang |editor2-last=Dubois |editor2-first=Frédéric |title=Discrete-element modeling of granular materials |date=2011 |publisher=Wiley-ISTE |___location=London |isbn=978-1-84821-260-2 |url=http://www.iste.co.uk/index.php?f=x&ACTION=View&id=384,}}

* {{cite book |last1=Pöschel |first1=Thorsten |last2=Schwager |first2=Thoms |title=Computational Granular Dynamics: Models and Algorithms |date=2005 |publisher=Springer |___location=Berlin |isbn=978-3-540-21485-4}}

'''Periodical'''

* BALL & TRUBAL (1979-1980) distinct element method (FORTRAN code), originally written by P.Cundall and currently maintained by Colin Thornton.

* [[LAMMPS]] is a very fast parallel open-source molecular dynamics package with GPU support also allowing DEM simulations. [http://lammps.sandia.gov/ Homepage ], [http://lammps.sandia.gov/movies.html#dem Examples ].

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

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

* [http://www.lmgc.univ-montp2.fr/~dubois/LMGC90/index.html 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.

*[https://launchpad.net/esys-particle/ ESyS-Particle] ESyS-Particle is a high-performance 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. ESyS-Particle includes a Python scripting interface providing flexibility for simulation setup and real-time 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 high-end workstations.

*[http://www.itm.uni-stuttgart.de/research/pasimodo/pasimodo_en.php Pasimodo] PASIMODO is a program package for particle-based 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 Smoothed-Particle-Hydrodynamics.

* {{cite journal |last1=Bobet |first1=A. |last2=Fakhimi |first2=A. |last3=Johnson |first3=S. |last4=Morris |first4=J. |last5=Tonon |first5=F. |last6=Yeung |first6=M. Ronald |title=Numerical Models in Discontinuous Media: Review of Advances for Rock Mechanics Applications |journal=Journal of Geotechnical and Geoenvironmental Engineering |date=November 2009 |volume=135 |issue=11 |pages=1547–1561 |doi=10.1061/(ASCE)GT.1943-5606.0000133}}

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

* {{cite journal |last1=Cundall |first1=P. A. |last2=Strack |first2=O. D. L. |title=A discrete numerical model for granular assemblies |journal=Géotechnique |date=March 1979 |volume=29 |issue=1 |pages=47–65 |doi=10.1680/geot.1979.29.1.47}}

*[http://www.chutemaven.com Chute Maven (Hustrulid Technologies Inc.)] Spherical Discrete Element Modeling in 3 Dimensions. Directly reads in AutoCad dxf files and interfaces with SolidWorks.

*{{cite journal | last1 = Kafashan | first1 = J. | last2 = Wiącek | first2 = J. | last3 = Abd Rahman | first3 = N. | last4 = Gan | first4 = J. | year = 2019 | title = Two-dimensional particle shapes modelling for DEM simulations in engineering: a review | journal = Granular Matter | volume = 21 | issue = 3| page = 80 | doi = 10.1007/s10035-019-0935-1 | s2cid = 199383188 }}

*[http://www.itascacg.com/pfc2d PFC2D] and [http://www.itascacg.com/pfc3d PFC3D] (Particle Flow Code in 2 Dimensions; Particle Flow Code in 3 Dimensions), PFC2D uses BALL codebase, PFC3D uses TRUBAL codebase.

* {{cite journal |last1=Kawaguchi |first1=T. |last2=Tanaka |first2=T. |last3=Tsuji |first3=Y. |title=Numerical simulation of two-dimensional fluidized beds using the discrete element method (comparison between the two- and three-dimensional models) |journal=Powder Technology |date=May 1998 |volume=96 |issue=2 |pages=129–138 |doi=10.1016/S0032-5910(97)03366-4 |url=http://www-mupf.mech.eng.osaka-u.ac.jp/paper_pdf/PT98,v96,129 |access-date=2005-08-23 |archive-url=https://web.archive.org/web/20070930212612/http://www-mupf.mech.eng.osaka-u.ac.jp/paper_pdf/PT98,v96,129 |archive-date=2007-09-30 |url-status=dead }}

*[http://www.dem-solutions.com EDEM (DEM Solutions Ltd.)] General-purpose DEM simulation with CAD import of particle and machine geometry, GUI-based model set-up, extensive post-processing tools, progammable API, couples with CFD, FEA and MBD software.

* {{cite journal |last1=Williams |first1=J. R. |last2=O'Connor |first2=R. |title=Discrete element simulation and the contact problem |journal=Archives of Computational Methods in Engineering |date=December 1999 |volume=6 |issue=4 |pages=279–304 |doi=10.1007/BF02818917|citeseerx=10.1.1.49.9391 |s2cid=16642399 }}

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

* {{cite journal |last1=Zhu |first1=H.P. |last2=Zhou |first2=Z.Y. |last3=Yang |first3=R.Y. |last4=Yu |first4=A.B. |title=Discrete particle simulation of particulate systems: Theoretical developments |journal=Chemical Engineering Science |date=July 2007 |volume=62 |issue=13 |pages=3378–3396 |doi=10.1016/j.ces.2006.12.089|bibcode=2007ChEnS..62.3378Z }}

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

* {{cite journal |last1=Zhu |first1=HP |last2=Zhou |first2=ZY |last3=Yang |first3=RY |last4=Yu |first4=AB |title=Discrete particle simulation of particulate systems: A review of major applications and findings |journal=Chemical Engineering Science |date=2008 |volume=63 |issue=23 |pages=5728–5770 |url=http://www.dem-solutions.com/papers/discrete-particle-simulation-of-particulate-systems-a-review-of-major-applications-and-findings/ |doi=10.1016/j.ces.2008.08.006|bibcode=2008ChEnS..63.5728Z }}

*[http://cee.mit.edu/index.pl?id=2393&isa=Category&op=show MIMES]a variety of particle shapes can be used in 2D

*[http://www.technalysis.us/dem_software.aspx PASSAGE/DEM] (PASSAGE/DEM Software is for predicting the flow particles under a wide variety of forces.)

'''Proceedings'''

*[http://www.itascacg.com/udec UDEC] and [http://www.itascacg.com/3dec 3DEC] Two- and three-dimensional simulation of the response of discontinuous media (such as jointed rock) that is subject to either static or dynamic loading.

* {{cite journal |last1=Shi |first1=Gen-Hua |title=Discontinuous Deformation Analysis: A New Numerical Model For The Statics And Dynamics of Deformable Block Structures |journal=Engineering Computations |date=February 1992 |volume=9 |issue=2 |pages=157–168 |doi=10.1108/eb023855 }}

*[http://www.agilemolecule.com/Products.html Ascalaph] [[Molecular dynamics]] with fourth order [[symplectic integrator]].

* {{cite journal|last1=Williams|first1=John R.|last2=Pentland|first2=Alex P.|title=Superquadrics and Modal Dynamics For Discrete Elements in Interactive Design|journal=Engineering Computations|date=February 1992|volume=9|issue=2|pages=115–127|doi=10.1108/eb023852}}

* {{cite book |editor1-last=Williams |editor1-first=John R. |editor2-last=Mustoe |editor2-first=Graham G. W. |title=Proceedings of the 2nd International Conference on Discrete Element Methods (DEM) |date=1993 |publisher=IESL Publications |___location=Cambridge, MA |isbn=978-0-918062-88-8 |edition=2nd}}

{{DEFAULTSORT:Discrete Element Method}}

[[Category:Numerical differential equations]]

[[de:Discrete element method]]

[[ja:個別要素法]]

[[pl:Metoda elementów dyskretnych]]

[[ru:Метод дискретного элемента]]