Coarse-graining

In most cases where the Discrete Element Method (DEM) is used, it is necessary to carry out some averaging of the data to a scale more representative of what is required. As such it is quite common for spatial averaging, temporal averaging, or both, to be applied to the DEM data. To aid the process, a module for the processing of DEM data both temporally and spatially has been implemented in Iota. The module provides both a coarse graining method to calculate bulk quantities and project those onto a continuum field. Some of the results are shown in Figure 50.

The coarse-grained density ρ is provided by equation:

Coarse-grained density

where

• r is a point in space where the values are to be evaluated,

• ri(t) is a vector to the centre of mass of the ith particle at a given timestep t,

• mi is that particle's mass, and

• ϕ is the coarse-graining or spatial averaging function which is subject to the condition of its integral over space being unity.

The coarse-grained velocity V is provided from equation

Coarse-grained velocity

where p is the coarse-grained momentum density:

where vi(t) is the particle's velocity vector.

The stress tensor is given by equation:

Stress tensor

where fijα is the interaction force between two particles, rijβ is the branch vector, s is the integral of the branch vector and v' is the fluctuating velocity of the particle.

Figure 50 - Iota-post toolbox in use - Post-processing on a fluidized bed. a) Particles visualization. b) Spatial averaged density. c) Spatial and temporal averaged density over 20 sec.