Estimate Simulation Time

It is challenging to estimate the exact real time required for a DEM simulation, as each simulation and each computer is different.

Time Step

One of the key numbers in DEM simulation is the Rayleigh Time Step which is the time taken for a shear wave to propagate through a solid particle. It is, therefore, a theoretical maximum Time Step for a DEM simulation of a quasi-static particulate collection in which the coordination number (total number of contacts per particle) for each particle remains above 1 and is defined as:

$T_{R} = \frac{π R \sqrt{\frac{ρ}{G}}}{0.1631 υ + 0.8766}$

where R is the particle’s radius, ρ its density, G the Shear modulus, and v the Poisson’s ratio. This formula assumes that the relative velocity between contacting particles is very small. Other than for quasi-static systems, in practice some fraction of this maximum value is used, and for high coordination numbers (4 and above) a typical Time Step of 0.2T_R (20%) is recommended. For lower coordination numbers, 0.4T_R (40%) is more appropriate.

Hertzian Contact

While the Rayleigh Time Step is a suitable starting point for quasi-static simulations, a shorter Time Step is required for systems undergoing flow. Consider two elements approaching each other at a speed v. In one Time Step, t, the maximum possible overlap is defined as:

$d_{\max} = υ t$

In DEM, particles undergoing elastic (Hertzian) contact are treated as overlapping and this overlap is equated to a surface compression. t in the above equation must be such that the maximum overlap is lower than the theoretical maximum overlap for Hertzian contact. In practice, to get a good numerical integral to the contact graph, at least six time points should occur - three during approach and three during separation (though ten is more desirable) .

Elastic Impact

From the Hertz theory of elastic collision, the total time of contact is defined as:

$T_{h} = 2.87 {(\frac{m^{* 2}}{R^{*} E^{* 2} V_{z}})}^{\frac{1}{5}}$

where

$\frac{1}{m^{*}} = \frac{1}{m_{i}} + \frac{1}{m_{j}}$

where m_i and m_j are the masses of the two elements.

$\frac{1}{R^{*}} = \frac{1}{R_{i}} + \frac{1}{R_{j}}$

where R_i and R_j are the radii of the two elements.

$\frac{1}{E^{*}} = \frac{1 - υ^{2}_{i}}{E_{i}} + \frac{1 - υ^{2}_{j}}{E_{j}}$

where ν_i and ν_j are the Poisson’s ratios of the two materials.

$V_{z} = V_{i} - V_{j}$

where V_z is the relative velocity and V_i and V_j are the velocities of the two elements.