1138 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
and rotational motions of a particle i obey Newton’s law of
motion and the Euler equation, given by:
m
F
m Fi Fiext) 1
i
i
i
ij
p p/w f p /(F =++--(1)
)I
M
I
1
i
i
i
ij /(M =(2)
The above equations describe the translation and rotational
motions of a particle i with mass mi and rotational inertia
Ii. For DEM to accurately predict the relevant forces act-
ing on a particle, the following are considered: the particle-
particle/wall interaction forces (F
ij
p p/w -)and torque (M
ij )
acting on particle i exerted by particle j or wall boundaries,
the particle-fluid interaction force (F
i
f p -),and the exter-
nal forces (F
i
ext ).
The spherical particle shape model can be described
using the radius and center-of-mass coordinates of each
particle. The multi-sphere model represents the particle
shape as a cluster of over-lapping spheres with fixed posi-
tions relative to each other, representing un-even smooth
surfaces with no sharp edge or vertex. This model is
described by using the radius and center coordinates of all
the individual elemental spheres that make up the non-
spherical shape with respect to the center of mass (Favier
et al., 1999). Each sphere is tested for the contact with all
spheres in other particles and with the wall. Polyhedron is
defined by a set of vertices, edges, and faces, with vertices
being the points where the edges meet and edges being the
line segments that connect the vertices. The faces are the
planar areas bounded by the edges. The polyhedral DEM
approach involves approximating the particle shape using a
set of convex polyhedra which are defined using their ver-
tices and edges (Cundall 1988). The contact force between
the particles is calculated using the surface normal vectors
and the penetration depth. The non-spherical particles
with sharp edges and vertices are well represented using the
polyhedral DEM approach. Non-spherical contact detec-
tion is the most challenging and tricky step. A two-stage
approach was used to simplify this: a broad-phase method
generates a list of potential contact pairs, and a narrow-
phase algorithm determines which potential contact pairs
are actually in contact. The broad-phase method allows for
faster overlap detection by first simplifying the actual non-
spherical particle shape to bounding volume shape such as
bounding spheres, axis-aligned bounding boxes (AABBs),
and oriented bounding boxes (OBBs). Using the bounding
volume shapes rather than the actual shape of the particle
in the broad search allows us to create scalable and general
codes that can handle particles of any arbitrary shape. The
contact detection for spherical particles is very straightfor-
ward. If the distance between the center of the spherical
particles is less than the sum of their radii, then particles are
in contact. The multi-sphere model is an extension of the
spherical model in which each elemental sphere is checked
for overlap using the same approach being used in detecting
contact between spheres. In the present polyhedral DEM,
the GJK algorithm (Gilbert, Johnson, and Keerthi 1988)
is used to estimate the contact between the particles. The
GJK algorithm represents each convex polyhedra as the
Minkowski difference of its vertices, which is another con-
vex polyhedra. The Minkowski difference is essentially the
difference between the two shapes, and it contains informa-
tion about the relative position and orientation of the two
shapes. Once the Minkowski difference is calculated, the
GJK algorithm iteratively constructs a simplex (a geometric
object that generalizes a triangle to higher dimensions) that
contains the origin of the coordinate system. The simplex
is constructed by adding one vertex at a time, and each
new vertex is chosen to be the point in the Minkowski sum
that is farthest in the direction of the origin. If the origin is
contained within the simplex, then the two convex polyhe-
dra intersect. The GJK algorithm is efficient and accurate
and can handle arbitrary convex objects in any number of
dimensions. For the study of the flow behavior of the cubes,
the shape representation in DEM is carried out using the
multi-sphere and polyhedral DEM. In the present study,
the cube shape in multi-sphere DEM is represented using 9
Figure 2. Particle shapes representation using DEM in this study (a) Spheres (b) Cube
using polyhedral approach (c) Cube using multi-sphere DEM
and rotational motions of a particle i obey Newton’s law of
motion and the Euler equation, given by:
m
F
m Fi Fiext) 1
i
i
i
ij
p p/w f p /(F =++--(1)
)I
M
I
1
i
i
i
ij /(M =(2)
The above equations describe the translation and rotational
motions of a particle i with mass mi and rotational inertia
Ii. For DEM to accurately predict the relevant forces act-
ing on a particle, the following are considered: the particle-
particle/wall interaction forces (F
ij
p p/w -)and torque (M
ij )
acting on particle i exerted by particle j or wall boundaries,
the particle-fluid interaction force (F
i
f p -),and the exter-
nal forces (F
i
ext ).
The spherical particle shape model can be described
using the radius and center-of-mass coordinates of each
particle. The multi-sphere model represents the particle
shape as a cluster of over-lapping spheres with fixed posi-
tions relative to each other, representing un-even smooth
surfaces with no sharp edge or vertex. This model is
described by using the radius and center coordinates of all
the individual elemental spheres that make up the non-
spherical shape with respect to the center of mass (Favier
et al., 1999). Each sphere is tested for the contact with all
spheres in other particles and with the wall. Polyhedron is
defined by a set of vertices, edges, and faces, with vertices
being the points where the edges meet and edges being the
line segments that connect the vertices. The faces are the
planar areas bounded by the edges. The polyhedral DEM
approach involves approximating the particle shape using a
set of convex polyhedra which are defined using their ver-
tices and edges (Cundall 1988). The contact force between
the particles is calculated using the surface normal vectors
and the penetration depth. The non-spherical particles
with sharp edges and vertices are well represented using the
polyhedral DEM approach. Non-spherical contact detec-
tion is the most challenging and tricky step. A two-stage
approach was used to simplify this: a broad-phase method
generates a list of potential contact pairs, and a narrow-
phase algorithm determines which potential contact pairs
are actually in contact. The broad-phase method allows for
faster overlap detection by first simplifying the actual non-
spherical particle shape to bounding volume shape such as
bounding spheres, axis-aligned bounding boxes (AABBs),
and oriented bounding boxes (OBBs). Using the bounding
volume shapes rather than the actual shape of the particle
in the broad search allows us to create scalable and general
codes that can handle particles of any arbitrary shape. The
contact detection for spherical particles is very straightfor-
ward. If the distance between the center of the spherical
particles is less than the sum of their radii, then particles are
in contact. The multi-sphere model is an extension of the
spherical model in which each elemental sphere is checked
for overlap using the same approach being used in detecting
contact between spheres. In the present polyhedral DEM,
the GJK algorithm (Gilbert, Johnson, and Keerthi 1988)
is used to estimate the contact between the particles. The
GJK algorithm represents each convex polyhedra as the
Minkowski difference of its vertices, which is another con-
vex polyhedra. The Minkowski difference is essentially the
difference between the two shapes, and it contains informa-
tion about the relative position and orientation of the two
shapes. Once the Minkowski difference is calculated, the
GJK algorithm iteratively constructs a simplex (a geometric
object that generalizes a triangle to higher dimensions) that
contains the origin of the coordinate system. The simplex
is constructed by adding one vertex at a time, and each
new vertex is chosen to be the point in the Minkowski sum
that is farthest in the direction of the origin. If the origin is
contained within the simplex, then the two convex polyhe-
dra intersect. The GJK algorithm is efficient and accurate
and can handle arbitrary convex objects in any number of
dimensions. For the study of the flow behavior of the cubes,
the shape representation in DEM is carried out using the
multi-sphere and polyhedral DEM. In the present study,
the cube shape in multi-sphere DEM is represented using 9
Figure 2. Particle shapes representation using DEM in this study (a) Spheres (b) Cube
using polyhedral approach (c) Cube using multi-sphere DEM