(388o) Development of a Boundary Model for Complex Systems in the Discrete Element Method

Introduction

In the industrial
powder handling processes, large amount of powder is treated in complicated
shaped devices. Application of computational simulations is desired to reduce
the cost in designing or planning operational conditions of these devices. The
discrete element method(DEM)[1] is widely used in the powder simulations. The
location of each particle of the powder is calculated with its movement in the
DEM, and these movements are calculated in results of particle-particle
interactions and particle-boundary interactions. When the wall boundary
conditions of the complicated shape is modeled, there are some methods, namely,
meshes and signed distance functions(SDF). The meshes are widely used in the computer
aided design (CAD) systems, and they are also well used in the DEM, such as our
previous study[2] or an open source code[3]. On the other hand, the SDF is
volume based method using a scalar field based on the level set function. The
SDF was originally used in the level set method[4], and recently it was applied
in two dimensional DEM simulation[5]. The SDF is reported as a robust method,
and it might be useful in powder-fluid coupling simulations. However, the SDF
was applied only in two dimensions, and it is not applied in the
three-dimensional simulations.

In the current study,
we show that the SDF is able to be applied to the three-dimensional
simulations. Besides it, we compare the accuracy and the computation time
between the mesh and the SDF in the usage of the wall boundary. Consequently we
illustrate the SDF is effective for DEM simulations with complicated shapes.

Conventional wall boundary

In designing of these
complicated shaped devices, the CAD systems are used with meshes which are widely
used to represent shapes of objects; these meshes are composed of multiple
triangles or squares in some cases. These meshes generated by CAD systems
usually represent only the surfaces of objects, unlike the meshes used in the
finite element method. The mesh can express almost all shapes; even smooth
curves can be expressed by the mesh if the resolution is high enough. In our
previous study we performed DEM simulations using CAD generated meshes as the
complicated boundary walls[4]. In the calculation, particle-boundary
interaction is performed by detecting collisions between each triangle element
in the boundary mesh and a particle.
The collision detection cost may grow with the number
of particles and with the number of triangular elements of the mesh. Therefore
DEM simulations in a complicated shaped system might require high calculation
cost.

Signed distance function

The signed distance function (SDF) is the mathematical expression of
surfaces by using the scalar field based on distance. In the previous studies,
SDF is used in the level set method[2] and distance scalar field was used in
DEM simulation in two dimensions[3]. The SDF is a kind of implicit function and
it is defined as ö(x) = d(x)s(x), where d(x) is the minimal distance from the point x to the surface of the shape, and s(x) is the sign positive if x is outside of
the shape, and negative if x is inside. The function gives the distance from the point x, and the sign of the function gives the information of if the point
is inside or outside of the shape. The normalized normal vector of a SDF
expressed shape is given as a normalized gradient of ö(x). The SDF can be defined on general form of a CAD generated mesh.
In the simulations, the values of SDF are discretized, i.e., they are
calculated in advance on each sampling point before the simulation starts. This
pre-calculation is equivalent to the pre-calculation of the distance between
the sampling point and the mesh.

Accuracy comparison between shapes by the mesh and
the SDF

In the present study,
first we compared the results of DEM simulation between using shapes expressed
by the mesh(Case 1-M) and the SDF(Case 1-S). In the comparison, DEM simulations
were performed with only one particle in a very simple sloped box (slope angle
is 45 degrees) as shown in Fig. 1. At the initial condition, the particle was placed at the
same location among two cases. The calculation conditions are shown in Table 1. In the
simulation, the movements of the particles were measured as shown in Fig. 2. It was found that the movements
of two cases matched very well and they were almost the same.

Comparison of computational time between two
boundary expressions

In the current study,
we compare calculation times between the mesh and the SDF with a screw
conveyer, which is an example of complicated systems. Some numerical analyses
were performed with the shape varying the number of particles, and measured the
particle-boundary interaction time of each case.

The screw conveyer used
in this study has a pair of helical fins fixed inside a cylinder as shown in Fig. 3. It can contain some powder
inside the cylinder, and it can rotate along the height axis of the cylinder to
convey powder. It is an example of a powder conveyer system generated by a CAD
system. The mesh consists of 448 triangles. In this study, the cylinder is
placed with the rotation axis parallel to the ground. In the simulation, the
particles were placed on one bottom of the cylinder with the gravity along the
axis of the cylinder, and when the calculation begins, the gravity is set to
cross the axis of the cylinder and the screw conveyer rotates at 30 rpm. In the
simulation, the number of particles and the particle radius was changed for
three cases as shown in Table 2. The total
mass of the particles were kept to be the same among all cases.

In the calculation
results, the particles were transported with the time in all cases. Typical
snapshots are shown in Fig. 4. The
calculation times consumed for particle-boundary interactions in one step was
measured as the average of first 200 steps. The results are shown in Fig. 5. The interaction time grows
longer with the number of the particles becomes much in all cases, meanwhile
times spent in cases using the SDF was always shorter than the times spent in
cases using the mesh.

Conclusion

We performed
three-dimensional DEM simulations using the mesh and the SDF for boundary
expressions. It was shown that the SDF can be applied in three-dimensional
simulations. We shows that the calculation time can be reduced drastically by
using the SDF, though the accuracy were regarded to be equivalent between them.
Eventually, the SDF is effective for DEM simulation with complicated shapes.

 References

[1]:
P.A.Cundall and O.D.L. Strack: "A Discrete Numerical Model for Granular
Assembles", Geotechnique, 29, 47-65 (1979)

[2]: Y. Shigeto, M. Sakai, "Parallel Computing of
the Discrete Element Method on Multi-Core Processors", Particuology, 9(4),
pp.398-405 (2011)

[3]: CFDEM Project, "LIGGGHTS Open Source Discrete
Element Method Particle Simulation Code", http://www.liggghts.com/

[4]: S.Osher and R.Fedkiw: "Level Set Methods and
Dynamic Implicit Surfaces", Applied Mathematical Sciences, Vol.153, Springer,
Berlin, 2003

[5]: K.Yokoi: "Numerical method for interaction
between multiparticle and complex structures", Physical Review E, 72, 046713
(2005)

Fig. 1 Schematic diagram of a sloped box

Fig. 2 Trajectory of the particle movement in Case 1-M
and 1-S

Table 1 Calculation
Conditions of Case 1

Fig. 3 Schematic diagram of the screw conveyer

Table 2 Calculation
conditions of the screw conveyer, cases 2, 3 and 4

Fig. 4 Typical snapshots of Case 3-S

Fig. 5 Time consumed for
particle-wall collision in cases 2, 3 and 4