146
D. Markauskas and H. Kruggel-Emden
degree with the multi-sphere method, it is well suited for modeling of screening
of non-spherical particles. Due to the relatively low computational effort the multisphere approach is the most popular method for representing non-spherical particles.
However, the number of spheres required to properly approximate the real particle
must be estimated on a case by case basis [27] which is further addressed in Sect. 2.5.
2.2 Equations of Motion Governing the DEM
The DEM is routinely used to model systems with spherical particles [13, 14]. To
obtain the translational and rotational motion in such a system, the Newton’s and
Euler’s equations are integrated
m i
d
2
x i
dt 2 =
F
c
i + m i
g +
F
l
i ,
(1)
I i
d
ω i
dt
=
M i ,
(2)
with particle mass m i , particle acceleration d
2
x i /dt
2 , contact force
F
c
i , forces arising
from a possible liquid amount
F
l
i , gravitational force m i
g, moment of inertia I i ,
angular acceleration d
ω i /dt, angular velocity
ω i and external moments resulting out
of contact and other sources
M i .
When non-spherical particles are used in the DEM, the integration of the Euler’s
equation reads
ˆ
I i
d
W i
dt
+
W i ×
ˆ
I i
W i
= Λ
−1
i
M i ,
(3)
where d
W i /dt is the angular acceleration,
W i is the angular velocity in the body
fixed frame,
M i is the external moment resulting out of contact and other sources, ˆ
I i
is the inertia tensor along the principal axis and Λ
−1
i is the rotation matrix converting
a vector from the inertial into the body fixed frame. Explicit integration schemes
are used to solve the equations for translational and rotational motion (see [28]).
The relevant forces required in Eqs. (1)–(3) are described in Sects. 2.3 and 2.4.
Contributions due to rolling friction are neglected.
2.3 Contact Forces
After determining a pair of colliding particles or a contact between a particle and
a boundary object, the resulting forces are evaluated by applying a suitable contact
force model. In this sense Fig. 4a shows a collision of two spherical particles i and
D. Markauskas and H. Kruggel-Emden
degree with the multi-sphere method, it is well suited for modeling of screening
of non-spherical particles. Due to the relatively low computational effort the multisphere approach is the most popular method for representing non-spherical particles.
However, the number of spheres required to properly approximate the real particle
must be estimated on a case by case basis [27] which is further addressed in Sect. 2.5.
2.2 Equations of Motion Governing the DEM
The DEM is routinely used to model systems with spherical particles [13, 14]. To
obtain the translational and rotational motion in such a system, the Newton’s and
Euler’s equations are integrated
m i
d
2
x i
dt 2 =
F
c
i + m i
g +
F
l
i ,
(1)
I i
d
ω i
dt
=
M i ,
(2)
with particle mass m i , particle acceleration d
2
x i /dt
2 , contact force
F
c
i , forces arising
from a possible liquid amount
F
l
i , gravitational force m i
g, moment of inertia I i ,
angular acceleration d
ω i /dt, angular velocity
ω i and external moments resulting out
of contact and other sources
M i .
When non-spherical particles are used in the DEM, the integration of the Euler’s
equation reads
ˆ
I i
d
W i
dt
+
W i ×
ˆ
I i
W i
= Λ
−1
i
M i ,
(3)
where d
W i /dt is the angular acceleration,
W i is the angular velocity in the body
fixed frame,
M i is the external moment resulting out of contact and other sources, ˆ
I i
is the inertia tensor along the principal axis and Λ
−1
i is the rotation matrix converting
a vector from the inertial into the body fixed frame. Explicit integration schemes
are used to solve the equations for translational and rotational motion (see [28]).
The relevant forces required in Eqs. (1)–(3) are described in Sects. 2.3 and 2.4.
Contributions due to rolling friction are neglected.
2.3 Contact Forces
After determining a pair of colliding particles or a contact between a particle and
a boundary object, the resulting forces are evaluated by applying a suitable contact
force model. In this sense Fig. 4a shows a collision of two spherical particles i and