Discrete element method
Encyclopedia
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 micrometre-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.
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 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
. Drawbacks to homogenization of the granular scale physics, however, are well-documented and should be considered carefully before attempting to use a continuum approach.
and Mustoe in 1985, the discontinuous deformation analysis
(DDA) proposed by Shi in 1988 and the finite-discrete 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 Element-Discrete Element Method is contained in the book The Combined Finite-Discrete Element Method by Munjiza.
Typical industries using DEM are:
. 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:
On a molecular level, we may consider
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:
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:
has been further developed to various irregular and deformable particles in many applications
including pharmaceutical tableting, packaging and flow simulations, and concrete and impact analysis, and many other applications.
Disadvantages
Commercially available DEM software packages include PFC3D, EDEM and Passage/DEM:
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
methods for computing the motion of a large number of particles of micrometre-scale size and above. Though DEM is very closely related to molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
, 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.
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
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
. In the case of solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
-like granular behavior as in soil mechanics
Soil mechanics
Soil mechanics is a branch of engineering mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids and particles but soil may also contain organic solids, liquids, and gasses and other...
, the continuum approach usually treats the material as elastic
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
or elasto-plastic
Plasticity (physics)
In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...
and models it with the finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
or a mesh free method
Meshfree methods
Meshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for calculations. Meshfree methods exist...
. In the case of liquid-like or gas-like granular flow, the continuum approach may treat the material as a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
and use computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
. 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 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, WilliamsJohn R. Williams
John R Williams was an American soldier, merchant, and politician who is most well known for serving as the first mayor of Detroit, Michigan. In total, he served as Detroit's mayor for five other terms...
and Mustoe in 1985, the discontinuous deformation analysis
Discontinuous Deformation Analysis
Discontinuous Deformation Analysis is a type of discrete element method originally proposed by Shi in 1988. DDA is somewhat similar to the finite element method for solving stress-displacement problems, but accounts for the interaction of independent particles along discontinuities in fractured...
(DDA) proposed by Shi in 1988 and the finite-discrete element method concurrently developed by several groups (e.g., Munjiza and Owen
Roger Owen
Edward Roger Owen is a British historian who has authored several classic works on the history of the modern Middle East. His research interests include the economic history of the Middle East, especially Egypt, from 1800 to the present, as well as the political and socioeconomic history of the...
). 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
Bobet
Bobet may refer to:in people:*Jean Bobet , French cyclist*Louison Bobet , French cyclistin other:*A flag which monitors which way the wind is blowing, like a weather vane...
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 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:- liquidLiquidLiquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
s and solutionSolutionIn chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
s, for instance of sugarSugarSugar is a class of edible crystalline carbohydrates, mainly sucrose, lactose, and fructose, characterized by a sweet flavor.Sucrose in its refined form primarily comes from sugar cane and sugar beet...
or proteinProteinProteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...
s; - bulk materials in storage siloStorage siloA silo is a structure for storing bulk materials. Silos are used in agriculture to store grain or fermented feed known as silage. Silos are more commonly used for bulk storage of grain, coal, cement, carbon black, woodchips, food products and sawdust. Three types of silos are in widespread use...
s, like cerealCerealCereals are grasses cultivated for the edible components of their grain , composed of the endosperm, germ, and bran...
; - granular matter, like sandSandSand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...
; - powdersPowder (substance)A powder is a dry,thick bulk solid composed of a large number of very fine particles that may flow freely when shaken or tilted. Powders are a special sub-class of granular materials, although the terms powder and granular are sometimes used to distinguish separate classes of material...
, like tonerTonerToner is a powder used in laser printers and photocopiers to form the printed text and images on the paper. In its early form it was simply carbon powder. Then, to improve the quality of the printout, the carbon was melt-mixed with a polymer...
. - 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 metallurgyPowder metallurgyPowder metallurgy is the process of blending fine powdered materials, pressing them into a desired shape , and then heating the compressed material in a controlled atmosphere to bond the material . The powder metallurgy process generally consists of four basic steps: powder manufacture, powder...
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 velocityVelocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
. 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:
- frictionFrictionFriction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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 cohesionCohesionCohesion may refer to:* Cohesion , the intermolecular attraction between like-molecules* Cohesion , a measure of how well the lines of source code within a module work together...
, adhesionAdhesionAdhesion is any attraction process between dissimilar molecular species that can potentially bring them in close contact. By contrast, cohesion takes place between similar molecules....
, 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 chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
; - Pauli repulsionPauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
, when two atoms approach each other closely; - van der Waals forceVan der Waals forceIn physical chemistry, the van der Waals force , named after Dutch scientist Johannes Diderik van der Waals, is the sum of the attractive or repulsive forces between molecules other than those due to covalent bonds or to the electrostatic interaction of ions with one another or with neutral...
.
All these forces are added up to find the total force acting on each particle. An integration method
Numerical ordinary differential equations
Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
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
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
. 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 algorithmVerlet integrationVerlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics...
, - velocity Verlet,
- symplectic integratorSymplectic integratorIn mathematics, a symplectic integrator is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations...
s, - 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 quadraticallyQuadratic growth
In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position, in the limit as the argument or sequence position goes to infinity...
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
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
: 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
Quadtree
A quadtree is a tree data structure in which each internal node has exactly four children. Quadtrees are most often used to partition a two dimensional space by recursively subdividing it into four quadrants or regions. The regions may be square or rectangular, or may have arbitrary shapes. This...
in the two-dimensional case and an octree
Octree
An octree is a tree data structure in which each internal node has exactly eight children. Octrees are most often used to partition a three dimensional space by recursively subdividing it into eight octants. Octrees are the three-dimensional analog of quadtrees. The name is formed from oct + tree,...
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 methodFast Multipole MethodThe fast multipole method is a mathematical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem...
.
Combined finite-discrete element method
Following the work by Munjiza and Owen's earlier work, the combined-discrete element methodhas been further developed to various irregular and deformable particles in many applications
including pharmaceutical tableting, packaging and flow simulations, and concrete and impact analysis, 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 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.
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).
Software
Open source and non-commercial software:- Ascalaph Molecular dynamicsMolecular dynamicsMolecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
with fourth ordersymplectic integratorSymplectic integratorIn mathematics, a symplectic integrator is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations...
. - 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, ...).
- 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.
- LAMMPSLAMMPSLAMMPS is a molecular dynamics program from Sandia National Laboratories.. LAMMPS makes use of MPI for parallel communication and is a free open-source code, distributed under the terms of the GNU General Public License.LAMMPS was originally developed under a Cooperative Research and Development...
is a very fast parallel open-source 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 multi-physics 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 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.
- 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, real-time control and post-processing, and allows introspection of all internal data. Can run in parallel on shared-memory machines using OpenMPOpenMPOpenMP is an API that supports multi-platform shared memory multiprocessing programming in C, C++, and Fortran, on most processor architectures and operating systems, including Linux, Unix, AIX, Solaris, Mac OS X, and Microsoft Windows platforms...
. - 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) General-purpose 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.) 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.
- 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 three-dimensional 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 methodFinite element methodThe finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...