Download multi-sphere models of particles in discreete element simulations

World Journal of Engineering
STOCHASTIC APPROACH TO SIMULATION OF INTERACTION
OF MULTI-SPHERICAL PARTICLES
Liudas Tumonis1, Rimantas Kačianauskas2 and Algis Džiugys1
1
Laboratory of Combustion Processes, Lithuania Institute of Energy,
Breslaujos str. 3, LT-44403 Kaunas
2
Laboratory of Numerical Modeling, Vilnius Gediminas Technical University,
Saulėtekio al. 11, LT-10223 Vilnius, Lithuania.
Introduction
Particle-based numerical simulation methods are widely
applied in many areas. The particle shape is important
factor playing decisive role in evaluation of particle’s
interaction behavior on various scales. A general so
called multi-sphere (MS) approach for representation of
non-spherical particles by rigidly connected multispheres is recently explored in the Discrete Element
(DEM) and applied to various shapes [1-3]. Contact
detection efficiency and simplicity of implementation
using sphere-to-sphere contact, is the main advantage of
the multi-sphere model.
Approximation the non-smooth spherical particle by
applying MS models was considered in [4]. The studies
performed shows that the multi-sphere method has
certain limitations when used for the approximation of a
non-spherical body. Moreover, increase of the
approximation degree may introduce new errors itself at
least on the single grain level and can be
computationally inefficient. In the light of the above
findings and the scattering of the initial data,
application of the MS models to complex shapes in a
frame of deterministic approach remains questionable.
The stochastic approach is suggested in this paper to
evaluate the performing of the MS models. Although
our investigation is limited to normal sphere-plane
contact, the concept may be easy extended to other
shapes and hopefully to multi-particle problems.
Modeling approach
Stochastic approach elaborated is aimed to evaluate the
mechanical behavior of contacting particles. A threedimensional particle is considered to be solid body with
the prescribed mechanical properties. Particles surface is
assumed to be continuous random function uniquely
defined by a discrete set of random as well as determined
parameters and/or variables. Consequently, particle
surface may be characterized by a set of random
functions while particle as a whole is characterized by a
set of random parameters. Degree of uncertainty will
depend on available data which may be obtained on the
measurements or artificially generated by random
generators. Interacting of particle with target (particle,
wall, etc.) will yield random response.
MS model of spherical particle
Concept of the stochastic approach will be illustrated by
the randomly generated MS model of the perfectly
smooth sphere. The MS model presents non-convex
quasi-spheroid usually known as clump being
composed by rigidly connected overlapping spheres
termed hereafter as sub-spheres. Composition of them is
done on basis of following concept. Firstly, axial
symmetry and isotropy is assumed to be imposed. Axial
isotropy means that the centers of two identical subspheres with random radius are located symmetrical on
each randomly oriented axis of symmetry.
Continuous surface of the MS model of the sphere with
the radius Rmax is described in spherical coordinates and
defined by the random radius R which is function of
spherical angles θ and φ. Finally, the MS particle is
enveloped by a perfect sphere, thus R(θ, φ)  Rmax.
a
b
Fig. 1 Geometry of generated MS particles: a) sample 1
(N = 48); b) sample 2 (N = 164).
Two randomly generated samples of MS particles with
N sub-spheres are shown in Fig. 1.
Geometry of the MS model may be represented by a
discrete set of N points. Histogram, which shows
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World Journal of Engineering
0.25
0.2
0.15
0.1
0.05
0
N=48
0.4
probability
probability
distribution of the relative surface radius R / Rmax is
presented in Fig. 2.
N=164
N=164
N=1
0.2
0.1
0
0
0.9
0.95
relative radius
0.5
0.4
0.3
0.2
0.1
0
N=48
1
2
3
4
5 6
contact points
1
2
3
relative contact time
4
1
Fig. 4 Distribution of the relative spheroid-wall contact
duration.
7
probability
Fig. 2 Distribution of the relative radiuses of the MS
particles.
Numerical tests and results
Stochastic properties of the generated spheroids were
studied by conducting numerical tests. A particle is
considered to be homogeneous isotropic elastic body
with the prescribed deterministic elasticity constants.
Discrete set of the random orientations of the particle
was examined during the experiments. The normal
impact of particle with the rigid wall is considered by
applying the elastic Hertz contact model.
Because of surface deviation, two considerable
differences compared to the perfect normal contact
occurred. Firstly, normal contact may be transformed to
oblique contact; secondly, multiple contacts are
observed because of deformation of particle surface.
Selected results of numeric simulation characterizing
normal contact behavior are presented below. The
distribution of contacting points is shown in Fig. 3
illustrates occurrence of the multiple contacts.
Distributions of contact duration and relative maximal
contact depth are shown in Fig. 4 and Fig. 5,
respectively. Here, the single dashed column indicates
solution of the perfect sphere.
probability
N=48
0.3
N=48
0
N=164
N=1
relative contact depth
1
Fig. 5 Distribution of the relative maximal contact depth.
Simulation results show that MS reflect merely
influence of discrete contacts rather than continuous
surface. As a consequence, substantial reduction of
contact depth is observed which should be properly
evaluated on statistical analysis.
Conclusions
Stochastic approach gives the trend of a random nature
of contacting particles which may be different for
various parameters and will depend on discrete
character of the shape of contacting subjects.
References
1. Ferellec, J.-F. and McDowell, G.R. A method to
model realistic particle shape and inertia. Granular
Matter, 12 (2010) 459–467.
2. Garcia, X., Latham, J.P., Xiang, J. and Harrison, J. A
clustered overlapping sphere algorithm to represent
real particles in distinct element modeling.
Geotechnique, 59 (2009) 779–784.
3. Markauskas, D., Kačianauskas, R., Džiugys, A. and
Navakas R. Investigation of adequacy of multisphere approximation of elliptical particles for DEM
simulations, Granular Matter, 12 (2010) 107-123.
4. Kruggel-Emden, H., Rickelt, S., Wirtz, S. and
Scherer, V. A study on the validity of the multisphere Discrete Element Method. Powd. Techn., 188
(2008) 153-165.
N=164
8
0.25
0.2
0.15
0.1
0.05
0
9
Fig. 3 Distribution of the maximal count of contacting
sub-spheres of the particles.
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