Discrete element method

The term Discrete Element Method (Engl. Discrete element method) or ( DEM ) is nowadays used for two numerical methods.

The most common use is the numerical calculation method developed by Cundall in 1971 , with which the motion of a large number of particles can be calculated. The method is sometimes referred to as the Distinct Element Method . Originally it was used for various calculations of molecular dynamics (MD). Since its inception, its field of application has expanded, such as B. on simulations from particle process engineering, geotechnical engineering and mechanical engineering. The extended discrete element method is an extension of the method .

On the other hand, the term DEM is also used for a bar lattice model. This approach - depiction of a body through rods - goes back to work by EG Kirsch from 1868 and was further developed by Felix Klein and Karl Wieghardt at the beginning of the 20th century. Today the bar lattice method is used, among other things, to simulate the material behavior of composite materials, in particular fabric structures. In addition, it was shown that the DEM is suitable for estimating the service life of metallic and ceramic materials. DEM is also used to describe material behavior in geomechanics.

The following text is limited to the DEM according to Cundall, as it is currently more relevant in research.

Procedure

Areas of application

The basic assumption of the method is based on the fact that the matter to be calculated (physics) is composed of individual, closed elements. These elements can have different shapes and properties.

The method is used in the following areas:

• Simulation of the behavior of atoms and molecules (see also chemical molecular dynamics )
• Development of processing methods for bulk materials in silos , e.g. B. Cereals
• Development of processing methods for bulk raw materials , e.g. B. sand
• Simulation of dust development , e.g. B. Toner
• Simulation of processes in geodynamics , e.g. B. Dynamics of Accretion Wedges

procedure

In a DEM simulation , all particles are positioned in a certain starting geometry and given an initial speed . From these initial data and the physical laws that are relevant for the particles, the forces that act on each particle are calculated.

Forces that come into question here are, for example, in the macroscopic case:

• Frictional forces when two particles brush against each other
• Repulsive forces when two particles meet and are slightly reversibly deformed
• Gravitational forces , i.e. the attraction of the particles due to their masses (only relevant for astronomical simulations)

or at the molecular level

• Coulomb forces , i.e. the electrostatic attraction or repulsion of the particles if they carry an electrical charge .
• Pauli Repulsion when two atoms come close to each other
• Van der Waals forces

All these forces are summed up and then with the help of a numerical integration process from Newton's equation of motion, the change in particle speed and position is calculated, which results in a certain time step. Then the forces are calculated again with the changed positions and speeds and this loop is repeated until the simulation period has ended.

Long-range powers

If long-range forces (typically gravitational forces or electrostatic forces) are taken into account, the interaction of each particle with all other particles must be calculated. The number of interactions and thus also the computational effort increases quadratically with the number of particles. With high numbers of particles, the computing time increases unacceptably. One way to avoid this is to combine several particles that are far away from the current particle to form a pseudoparticle and only calculate an interaction between the current particle and the pseudoparticle. The interaction between a star and a distant galaxy can serve as an example: The error that arises when all the stars of the distant galaxy are combined into a single mass point is negligible under normal requirements. So-called tree methods are used to decide which particles can be grouped into pseudoparticles. The particles are arranged in a hierarchical tree, in the two-dimensional case a quadtree, in the three-dimensional case an octree . In molecular dynamics simulations, on the other hand, the room in which the simulation is to take place is divided into simulation cells. Both the forces and the particles, if they go beyond the edge of the cell, are simply reinserted on the other side of the cell ( periodic boundary condition ). To prevent a particle from being captured by both the actual force and its mirror image on the other side, this force is no longer taken into account from the so-called cutoff distance (usually half the length of the cell). In order to increase the number of particles involved, the simulation cell is simply multiplied as required.

Algorithms

Integration algorithms

• Verlet algorithm
• Velocity-Verlet
• Leap Frog
• Predictor-corrector method

Long-range powers

• Barnes-Hut Algorithm
• Fast Multipole Method
• Ewald sum
• Particle Mesh Ewald

See also

• Molecular dynamics

literature

• MP Allen, DJ Tildesly: Computer Simulation of Liquids . Oxford University Press, 1989, ISBN 0-19-855645-4 .
• Griebel, Knapek, Zumbusch, Caglar: Numerical simulation in molecular dynamics . Springer, 2004. ISBN 3-540-41856-3 .
• Nenad Bicanic: Discrete Element Methods. In: Stein, de Borst, Hughes Encyclopedia of Computational Mechanics, Vol. 1 . Wiley, 2004. ISBN 0-470-84699-2 .

software

• Aspherix (DCS Computing, further development of LIGGGHTS)
• Chute Maven (Hustrulid Technologies Inc.)
• EDEM from DEM Solutions Ltd.
• GROMACS
• GROMOS 96
• ELVES
• LAMMPS (open source)
• LIGGGHTS (Open Source, DCS Computing GmbH)
• YADE (open source)
• PASIMODO particle simulation package
• ROCKY DEM
• SimPARTIX from Fraunhofer IWM
• PFC2D and 3D from ITASCA
• ThreeParticle (BECKER 3D GmbH)

Web links

• RWTH Aachen
• ISD University of Stuttgart

Individual evidence

1. ↑ PA Cundall: A computer model for simulating progressive large scale movements in blocky rock systems. In: Proceedings Symposium Int. Soc. Rock Mech. , Nancy Metz, vol. 1, S. Paper II-8, 1971.
2. ↑ EG Kirsch: The fundamental equations of the theory of elasticity of solid bodies, derived from the consideration of a system of points which are connected by elastic struts. Volume 7 (1868), Issue 8, In: Journal of the Association of German Engineers , vol. 7, no. 8, pp. 481-487, 553-570, 631-638, 1868.
3. ↑ FK Wittel: Discrete elements - models for determining the strength evolution in composite materials. University of Stuttgart, Faculty of Aerospace Engineering and Geodesy, Institute for Statics and Dynamics of Aerospace Structures, 2006.
4. ↑ D. Ballhauser: Discrete modeling of the deformation and failure behavior of tissue membranes. University of Stuttgart, Faculty of Aerospace Engineering and Geodesy, Institute for Statics and Dynamics of Aerospace Structures, 2006.
5. ↑ M. Hahn, M. Bouriga, B.-H. Kröplin, T. Wallmersperger: Life time prediction of metallic materials with the Discrete-Element-Method. In: Computational Materials Science , vol. 71, no.0, pp. 146–156, 2013.
6. ↑ M. Hahn: Estimation of the service life of metallic structures using the discrete element method in a coupled thermo-mechanical field. University of Stuttgart, Faculty of Aerospace Engineering and Geodesy, Institute for Statics and Dynamics of Aerospace Structures, 2012.
7. ↑ Discrete Element Method, University of Stuttgart ( Memento from September 17, 2014 in the Internet Archive ).