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 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.

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 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.

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:

Typical industries using DEM are:

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 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:

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 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:

Combined finite-discrete element method

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,[1] packaging and flow simulations,[2] and concrete and impact analysis,[3] and many other applications.

Advantages and limitations

Advantages

Disadvantages

References

  1. ^ 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)
  2. ^ 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)
  3. ^ 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).

Bibliography

Book

Periodical

Proceedings

Software

Open source and non-commercial software:

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

See also