Examining the smooth and nonsmooth discrete element
approaches to granular matter


M. Servin, D. Wang, C. Lacoursière, K. Bodin, Examining the smooth and nonsmooth discrete element approaches to granular matter,
accepted for publication in International Journal for Numerical Methods in Engineering (2013). 
Drum rotating with 0.13 rad/s and 7500 particles  
This simulation shows a drum rotating with 0.13 rad/s, drum diameter 0.8 m and 7500 spherical particles of 0.01 m diameter size.
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video channel which has more simulations of many configuration and played with different speeds.


Drum rotating with 0.63 rad/s and 7500 particles  
This simulation shows a drum rotating with 0.63 rad/s, drum diameter 0.8 m and 7500 spherical particles of 0.01 m diameter size.
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video channel which has more simulations of many configuration and played with different speeds.


Drum rotating with 2.51 rad/s and 7500 particles  
This simulation shows a drum rotating with 2.5 rad/s, drum diameter 0.8 m and 7500 spherical particles of 0.01 m diameter size.
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video channel which has more simulations of many configuration and played with different speeds.


Formula: 
\begin{equation*} \begin{array}[l] \\ \frac{\tau_{{\tiny NDEM}}}{\tau_{{\tiny SDEM}}} = \sqrt{\frac{ \max\left(\epsilon^2 m v{n}^2, 2\epsilon mgd\right)}{kd^2}} \frac{K^{{\tiny GS}}_{{\tiny NDEM}}}{K{{\tiny SDEM}}} \frac{N_{c}}{N_\text{p}}\cdot\nonumber\\ \quad\quad\quad\cdot\frac{c_0 (1 + c_1 I )}{\epsilon} \left\{ \begin{array} [l]{lr}% w( 1  \exp[\tfrac{c_2 l} {w}]) & \text{ , if } w \gtrsim 5\\ c_2 l & \text{ , if } w < 5 \end{array} \right.\\ \frac{}{}\\ \frac{}{}\\ \tau_{\tiny SDEM} = \sqrt{\frac{k}{m}} K_{\tiny SDEM}N_{\text{p}}\tau_{\text{real}}\\ \frac{}{}\\ \frac{}{}\\ \tau_{\tiny NDEM} = \frac{K^{\tiny GS}_{\tiny NDEM}N_{\text{it}} N_{c}} {\min\left(\epsilon\frac{ d}{v_{n}}, \sqrt{\frac{ 2\epsilon d}{g}}\right)}\tau_{real}\\ N_{\text{it}}^{\epsilon} (l,w,I) = \tfrac{c_0 (1 + c_1 I )}{ \epsilon } \left\{ \begin{array} [l]{lr}% w( 1  \exp[\tfrac{c_2 l} {w}]) & \text{ , if }w \gtrsim 5\\ c_2 l & \text{ , if }w < 5 \end{array} \right. \end{array} \end{equation*} 
[${\tiny m/s^{2}}$]  [${\tiny kg/m^3}$]  
0.3  2.0  0.44 


Force network  
The contact forces form force networks. These are weighted
graphs with the particles as nodes and
contact forces as edges. We use the normal force
magnitude for the edge weight. In granular materials
with too few iterations the strong force chains that are
a characteristic feature of granular materials do not appear.
Instead, the force distributes as the hydrostatic pressure
in a fluid, i.e., increases linearly with depth from the top
surface. When increasing the number of iterations,
strong force chain structures emerge and with this the
pressure force saturate and become independent of depth in the
column. This is the wellknown Janssen effect of
granular materials,
which is due to an arching effect of the force chains whereby the
container walls carry part of the weight of the material
Click here for more force network figures.


Velocity fields  
On sufficiently large scales of length and time a granular
material may be considered as a continuous solid represented
by continuous fields of mass distribution, flow velocity and stresses and
strains. The averaging process of sampling microscopic particle properties
and interparticle forces to macroscopic quantities is referred to as
coarse graining (or homogenization). A cubic grid is introduced and
the fields are evaluated at the grid points. The smoothing length is chosen
to 1.5 times particle diameter and we use grid size L = 0.5 d.
Click here for more velocity fields


Scalar fields  
On sufficiently large scales of length and time a granular
material may be considered as a continuous solid represented
by continuous fields of mass distribution, flow velocity and stresses and
strains. The averaging process of sampling microscopic particle properties
and interparticle forces to macroscopic quantities is referred to as
coarse graining (or homogenization). A cubic grid is introduced and
the fields are evaluated at the grid points. The smoothing length is chosen
to 1.5 times particle diameter and we use grid size L = 0.5 d.
Click here for more scalar field plots of mass density, pressure, strain rate and inertial number.
