ss
Kishore Vaigyanik Protsahan Yojana (KV P Y ) (Junior Scientist Promotion Scheme)
(Department of Science and Technology, Government of India) 01 November, 2011 Summer Camp Report on
Studies and Implementation of Discrete Element Method (DEM )
by
Rahul Kumar Soni (
[email protected]) Department of Fuel and Mineral Engineering Indian School of Mines, Dhanbad
under the guidance of
Prof. (Dr.) B. K. Mishra (
[email protected]) (Professor, Indian Institute of Technology, Kanpur) Director, Institute of Minerals and Materials Technology, Bhubaneswar Council of Scienti c & Industrial Research, Govt. of India
1
ss
2
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Acknowledgment My sincere thanks to Prof. lucrative project.
B. K. Mishra for having given me this opportunity to work on such a
I would like to thank Dr.
C. Eswaraiah and Dr.
A. Tripathi for taking care of
me and co-guiding through out the period of camp. I wish to thank the provided a great work environment. and timely inputs from Prof.
KV P Y
program for having
Project would have not been complete without the guidance
B. K. Mishra.
My thanks to Prof.
R. Venugopal (Dean, Research &
Development, Indian School of Mines), Dr. N. Suresh (Head of Department, Department of Fuel and Mineral Engineering, Indian School o Mines), for helping at administrative levels. I am thankful to the entire work force at
IM M T
Bhubaneswar and for the part of contribution
they played in making the period of work a joyful experience. I thank to Indian Institute of Science Bangalore and Department of Science and Technology, Government of India for their support, nance and encouragement through the
3
KV P Y
fellowship program.
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Abbreviations and meaning of symbols DEM: discrete element method CSR: cubic spline regression DFR: discrete function representation DESS: double ended spatial sorting
CFT: curve tting toolbox in MATLAB IMMT: Institute of Minerals and Materials Technology CMTPD: Computer monitored twin pendulum device
Machine con gurations Simulation and regression done in the machine of following con guration: Operating System: Microsoft Windows 7 Ultimate, 32 bit OS Processor: Intel(R) Core(TM)2Duo CPU Frequency: 1.67GHz , 2GB RAM, 160GB HDD. MATLAB used for Computation operations: Version: 7.6.0.324 (R2008a)
Disclaimer 1. The part of programs reproduced in this report are the copyright of Dr. R. K. Rajamani, University of Utah. Therefore only the part of programs are produced in this report and readers hold their ethical responsibility to not reproduce or use it any manner possible. 2. Author/guide of report or Dr. Rajamani himself can be contacted for the details of the programs. 3. In some places in programs '...' this notation has used at last of the lines. Notation is for continuation purpose, in running program this may replace by continuing next line. c copyright provisions, important data set and plots acquired are not shown in 4. Due to this report.
4
Contents 1
2
Review: DEM Methodologies, Stages, Parameters, Implementation
9
1.1
Particle Shape in 2D or 3D
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2
Particle De nitions in 2D or 3D . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3
Contact Search
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.4
Contact Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5
Contact Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.6
Parameter(s) Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.7
Force Calculations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.8
Updating Position & Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.9
Step Size
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
1.10 Recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Source Code (Copyright: Dr. R.K. Rajamani) Analysis
35
5
List of Figures 1.1
Nomenclature for contact location between two ellipses (Ting, 1992 [25]) . . . . . . . . .
11
1.2
Scheme for contact detection
11
1.3
The
1.4
Mechanism of
1.5
De nition of distances and sign convention of a point to a plane (Nezami, 2004 [67])
. .
15
1.6
An example of contact graph (Erleben, 2004 [71]) . . . . . . . . . . . . . . . . . . . . . .
16
1.7
A typical example of contact group (Erleben, 2004 [71])
. . . . . . . . . . . . . . . . . .
16
1.8
Box list corresponding to an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.9
Contact list for example in gure 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.10 The
DF R
DF R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sampling grid and discrete sphere example (presented in Williams, 1995 [22])
Octree
data structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
contact resolution steps (Williams et al., 1995 [61]) . . . . . . . . . . . . . . .
1.11 Simple contact bounds test (William et al., 1995 [61])
13 14
18
. . . . . . . . . . . . . . . . . . .
19
1.12 Zone Index set from clip zone (Williams, 1995 [22]) . . . . . . . . . . . . . . . . . . . . .
21
1.13 A region quadtree with point data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.14 A quadtree based representation of disc in 2D . . . . . . . . . . . . . . . . . . . . . . . .
22
1.15 Graphical representation of equation (6), showing the pressure distribution over the contact area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Dependency of
Fn
and
kn
on
α
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.17 Double spring and damping contact model (Zhang et al., 1998 [43]
23 23
. . . . . . . . . . . .
24
1.18 Forces acting on two particles rolling on each other (Zhang et al., 1999 [48]) . . . . . . .
26
1.19 Forces and velocities for two particles sliding on each other
30
. . . . . . . . . . . . . . . .
1.20 Trajectory of the simulation with variable step size and xed step size (Zhang et al., 2001 [87])
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.21 The error in trajectory in gure 19 with two simulations
33
. . . . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . .
25
List of Tables 1
Parameters used in simulation by Mishra et al., 1994 [45]
2
Simulation parameters used in Ting et al., 1993 [49]
3
The parameters used in simulation by Kun et al., 1996 [89]
6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28 29
sdhj
Abstract In recent past, these has been rapid development in the understanding of Discrete Element Method (DEM) through computer simulation, and its implementation in Mineral Engineering. Larger portions of the report covers the review of development of DEM and its implementation in Mineral processing techniques.
Review evaluates the development in DEM, various methods adopted, their merits,
improvements necessary to be adopted and current status of DEM in Mineral Engineering.
Various
programs in C and fortran had been developed by scientists and engineers in channelizing the use of DEM technique in Mineral Engineering.
The critical research points in DEM are particle shape
determination, contact detection, geometric resolution and step size determination, among which the contact detection is the most critical technique to be carefully adopted in any DEM coding. DEM models granular materials numerically as a collection of particles rather than a continuum. Di culties can arise when using the more common circular and polygon shapes for modeling real granular materials. This report presents the detailed formulations and their comparison of using dierent shapes for representation of granular materials in DEM. Discrete Element Method (DEM) models simulate the behavior of uid and solids by assembly of discrete elements. In the discrete element models, the forces between discrete components are calculated and used to determine the motion of the discrete components.
During the simulation process, the
simulation time is discretized into small time intervals. The velocity of each discrete component in each time interval is calculated. The positions and velocities of these discrete components are updated and then the simulation progresses to the next small time interval. This report is an attempt to brief the developments done in the eld of DEM wher major developments concerned with implementation of DEM simulation in granular media ow. In support of the same, some articles from the streams of civil engineering and rock mechanics are quoted. Report highlights the common practices in simulation of granular ow. In this report along with the review of DEM articles the methodology and steps of the program developed for simulation of tumbling mills by Dr. R. K. Rajamani and Dr. B. K. Mishra at University of Utah. The source code is written in C language, and was studied by the author of this report.
Energy spectra inside mill
plays signi cant role in prediction of breakage and its distribution, the ultimate perfomance criteria for milling operation. Due to copyright provisions the respect towards the privacy of program is maintained while presenting the parts of them whenever seems required. The structure of developed source code is presented in the remaining part of the report.
7
Preamble The discrete (or distinct) element method has been used over decades for numerically modeling the mechanical behavior of granular materials. Instead of a continuum, the DEM treats granular material as an assemblage of distinct particles, each governed by fundamental laws of physics.
Each particle
interacts with its neighborhood particles through particle-to-particle contacts, which can be formed or broken at each time step.
As no restriction exist regarding mode of deformation or amount of
displacement of each particle, the DEM is uniquely suited for modeling large deformation processes and bifurcation-type behavior in granular materials. This numerical technique was rst adapted to geo-mechanics by Cundall, 1974 [1]; Cundall et al., 1978 and 1979 [2], who studied the dynamic behavior of rock masses. As this technique has evolved, it has been used in a wide variety of research applications in engineering mechanics and geotechnical engineering. These applications include statistical micromechanics by Cundall, 1988 [3]; Bathurst et al., 1988 [4]; Zhang et al., 1986 [5], behavior of granular soils by Ting et al., 1988 [6]; Dobry et al., 1989 [7]; Issa et al., 1989 [8]; Kuhn et al., 1989 [9], creep of soils by Walton et al., 1989 [10], ow of granular materials by Campbell et al., 1983 [11]; Ghaboussi et al., 1990 [12]; Hakuno et al., 1989 [13]; Lorig et al., 1984 [14]), analysis of rock-support interaction by Butkovich et al., 1988 [15], blast loading of grounds by Taylor et al., 1989 [16]; Barbosa et al., 1987 [17] and large deformation geotechnical modeling by Ting et al., 1989 [18]. The greatest problems tackled to date with the DEM involve up to a few ten thousands of twoor three-dimensional particles simulated with a few tens of thousands of time steps. The particles are usually of simple geometric shape either two dimensional discs, polygons, three-dimensional spheres or polygonal blocks.
While circular particles have the great advantage of computational simplicity,
they also possess an inherent tendency to roll.
In particular, computed normal contact forces never
contribute to the moment acting on a particle, as each normal contact force always acts through the particle centroid. As a result, it is di cult to simulate 'real' material behavior when the basic particle is even slightly angular (Dobry et al., 1989 [7]). In the simulation of multiple discrete bodies, the design of contact detection algorithm is an important consideration as the process can be computationally expensive. Contact detection is usually divided into two separate phases, neighbor searching and geometric resolution. The goal of neighbor search is to identify and list objects within a certain zone around a target object. The resulting list is often called the neighbor list or contact list. The geometric resolution phase then compares the target object geometry against the geometry of the objects in the neighbor list in detail. The computational cost of geometric resolution depends on the complexity of the geometric representation. By separating the neighbor searching and geometric resolution algorithms, it is possible to consider the problem of neighbor searching without reference to the details of local geometry. This separation achieved through the use of bounding volumes, ensures that the neighbor search algorithm will remain valid for any geometric representation and resolution scheme chosen. This robustness is essential in the development of the generalized DEM simulation system, where any number of geometric representation may be used concurrently. The implications of allowing arbitrary geometries to coexist in the system, however, must not be ignored. If the neighbor search algorithm is designed with speci c geometry in mind, its performance may degrade when it is used outside those restricted settings. The step size of simulation determines the accuracy of discrete element simulation. The position and velocity updating calculations uses a pre-calculated table through leap-frog method, and the control of step size can not use the integration formulas for step size control. A step size control scheme for use with the table driven velocity and position calculation uses the dierence between the calculation result from one big step and what from two small steps. The size of time step is critical to the calculations since it determines the stability and accuracy of the calculation in the discrete element method. A time step size which is too small results in an unnecessarily long simulation time. However, a time step size which is too large produces incorrect simulation results, with unstability of simulation. The contact force calculation using in discrete element simulation is typically calculated by single spring and damping models. The commonly used linear model gives unrealistic behavior. Single spring and damping does not allow particles with dierent surface properties. The step size needed to give the required accuracy typically varies in dierent parts of discrete element simulation and at dierent times in the simulation. The extra computation needed to check the accuracy of a given step size can oer be justi ed by the reduction in the computation obtained by being able to use larger steps in parts of the calculation. In addition, the controlled accuracy avoids the need to choose a conservative step size and to make multiple runs to check accuracy. Both of these require additional calculation.
8
1
Review: DEM Methodologies, Stages, Parameters, Implementation
The DEM refers to a numerical scheme that allows nite rotation and displacements of discrete bodies which interact with their nearest neighbors through local contact laws, where loss of contacts and formation of new contacts between bodies take place as the calculation cycle progresses.
DEM on behavior of Flow of Granular Media The important steps in DEM simulation for granular ow are as follows: 1. Particle Shape in 2D or 3D 2. Particle De nitions in 2D or 3D 3. Contact Search 4. Contact Resolution 5. Contact Models 6. Parameters Determination 7. Force Calculation 8. Updating Position & Velocity 9. Step Size 10. Recursions It is very important to select the appropriate shape of the particle. In reality particles are very nonuniform in shape and while simulating them it is di cult to represent particles. Representation can be made by disc, ellipse, polygon in 2-D, and sphere, ellipsoid, polygonal in 3-D. Each one includes the merit and demerit of their own. For example it is easy to perform force calculations for spheres but spheres have common tendency to roll which is contradicting in the nature of real particles. Ellipsoid may fall with some di culty in calculations but they represent real particles in better way than spheres represent them. There are various ways to de ne particles in space represented by spheres or ellipsoid. De nition by analytical equation of particle shape results in accurate results but for large number of regressions, it was found extremely di cult to update the position and velocity of particle in space. On the other hand de nition of particle with the use of some parameters are better representation and faster way for large number of regressions. These parameters can be center, radius of sphere, in case of sphere or maxima, minima, center of ellipsoid, in case of ellipsoid. Ellipse or ellipsoid can also be de ne by their spherical bounding volumes of radius equal to maxima of ellipse or ellipsoid. A proper de nition is very important to fastly update the new positions and velocities of the particles. Contact search prepares a list that contains the de nition of particles in contact with a particular targeted particle.
Various techniques of boxing, plane de nition have been attempted among which
boxing in virtual environment resulted in similar results as of practical results. The de nition of all particles are checked against the position of particle for which contact list has to be prepared and particles in direct contact with the concerned particles are enlisted. The particles enlisted in contact list are checked for their exact positions, overlaps, bounding volumes and therefore their extent of contact with the concerned particle. Separate list for each contact containing the contact information is prepared. After having prepared the list of contacts and gathering the contact information, the next step is the calculation of all applicable forces.
Several models have been attempted to calculate forces,
such models are single spring-dashpot model, double spring-dashpot model, non-linear spring dashpot model, Hertzian force model, elastic and plastic model etc.
A model guides the mode of force
calculation and decides the factors which needs to be taken in consideration, and factors that needs to be eliminated. After having selected the parameters which deemed important for force calculations, next step would be xing the values for parameters. This step is very important to arrive to appropriate results
9
of simulation. Small variation may lead to signi cant errors in results, as large number of recursions have to be performed. These parameters may be coe cient of friction, coe cient of restitution, viscosity, speci c gravity, stiness, step size etc., depending upon the simulation environment. Simulation may now lead to calculation of forces.
Force calculation may involve calculation of
normal forces, tangential forces, moment due to forces. Force calculation step is followed by implementation of their aects on positions and velocities. After calculating the forces acting on each particle the linear and angular accelerations are calculated followed by new velocities of the particles which will follow the new positions of the particles. Above all steps complete the simulation for one time step size. Step size is the elemental time for which calculations have been made and accordingly changes for velocities, positions and other variables have to be recorded. Selection of step size is very critical step for simulation. Various attempts have been made by researchers to calculate the appropriate step size; a smaller step size may lead to high computation load while a larger step size lead to unstable simulation. Selection of parameter of spring constant and mass of particles are critical for calculation of step size. The new positions of the particles again recur to step number 2 of de ning particles in space and then follows the remaining steps. Process will go on until the time reaches the completion of simulation time.
1.1 Particle Shape in 2D or 3D The discrete element method (DEM) models granular materials numerically as a collection of particles rather than as continuum. Di culties can arise when using the more common circular and polygonal shapes for modeling real granular materials. In particular, use of discs and spheres leads to typically low aggregate internal friction angles compared with real materials. Robust algorithms for computing ellipse-ellipse and ellipse-wall have been reviewed in this report. Cundall et al. 1978 and 1979 [2], the pioneer of discrete element technology have rst introduced the use of spherical shape to represent particles in space. represented by balls in 3-dimention.
Heterogeneous, irregular particles were
Researchers have rst coded for simulation in 2-dimension and
named the program as BALL, additions have been continuously made to the program BALL, which led to simulation in 3-dimention as well. The later version of BALL was named as TRUBAL. Williams, 1994 [19] has simulated the particle ow in ink-jet printers through CAD models and spacial sorting algorithm. Eect of shape and packing on dynamic wave propagation was analyzed by Williams, 1995 [20]. Mithra et al., 2010 [21] studied simulation of spherical particles in spiral concentrators. Williams et al., 1995 [22] has described the
discrete function representation (DFR)
scheme in details for object
representation of a ball. Problems with spherical representation include unrealistic way of presentation and unnatural sliding or spinning tendency of balls. So, a dierent approach of representing particles with ellipse was adopted by several researchers. Ng, 1994 [23] has represented particles in 2-D with ellipse shape after developing the program named ELLIPSE2. Developed program actually analyzes the stress-strain behavior of mixture of circular and elliptical shaped particles. Spheres and discs have major de ciencies against use of ellipse as particle representative, this has been shown by Ting et al., 1993 [24].
Some researchers in past have also suggested to use general analytical equation of body
shapes for particle representation but complexity in updating particles position, velocity and de nition led to very complex calculations and high computational load. A general analytical equation used by several researchers to represent ellipse can be given as
f (x, y) = a1 x2 + a2 xy + a3 y 2 + a4 x + a5 y + a6 Ting, 1992 [25] has shown the robust algorithm for development of programs for simulation with elliptical particles. Figure 1 shows the elliptical representation of two particles, gure also indicates towards contact detection, apart from which some authors have also followed a slightly dierent manner of nding contact point as the mid-point of the line joining the extreme points of two ellipses in contact. Contact point determination is important for calculations of all applicable forces.
10
Figure 1.1: Nomenclature for contact location between two ellipses (Ting, 1992 [25])
Article compares between particle de nition with its center and other parameters to the particle de nition by its analytical equation.
A general quadratic equation with transformed coordinates to
represent one ellipse, given in Ting, 1992 [25] is:
A(X − dx)2 + B(Y − dy) + 2C(X − dx)(Y − dy) − 1 = 0 Nomenclature for parameters in equation can be referred to the paper. Article represents particle with quadratic equations which on their solution to intersection point of two ellipses lead to a quartic equation having four solution of it, and is declared to be an ill-conditioned equation. Since the particles being simulated are usually fairly sti, the amount of overlap is normally quite small.
Article also
suggested a new method for determination of contact point by the locus de nition of ellipses. Ellipses are presented by their dierential equations and contact point is determined by their common slope. Figure 2 shows the two ellipses with their locus representation. In gure f(x) and g(x) are the locus of two ellipses while h(x) is the locus of common slope point of two ellipses near intersection area. The mid point of intersection of h(x) with f(x), and h(x) with g(x) is projected to be the contact point between two ellipses.
Figure 1.2: Scheme for contact detection
In addition to above Trent et al., 1989 [26]; Ishibashi et al., 1989 [27]; Bathurst et al., 1989 [28]; Taylor et al., 1989 [29]; Hakuno et al. [30]; Kuhn et al. [31]; Hakuno et al. [32]; Thornton, 1989 [33]; Salami et al., 2002 [34]; Cheng et al., 2002 [35]; Sakaguchi et al., 1993 [36]; Songfack et al., 1993 [37]; Liao et al., 1993 [38]; Lin et al., 1993 [39]; Trent et al. [40]; Chang et al., 1993 [41]; Mishra, 2003 [42]; Zhang et al., 1998 [43]; Fortin et al., 2002 [44]; Mishra et al., 1994 [45]; Mishra et al., 1994 [46] and Mishra, 2003 [47] have also worked on the assemblies of spheres as the representation of particles in simulation. Zhang et al., 1999 [48] explains the type of forces and their derivations acting on spherical particles; details of tangential forces acting on spheres were presented in graphical manner. Ting et al., 1993 [49] have discussed the discrete element algorithm on ellipse based model, interaction of ellipses with each other, contact point detection algorithm based on common slopes between locus of ellipses, contact list and grid box list scheme. Ting, 1991 [50] has shown the ellipse based complete mathematical model to calculate common point coordinates, to update velocity, angular velocity, acceleration and forces.
11
In addition to above, Mustoe et al., 2000 [51] attempted to present particles in general shapes. Shapes are presented by general equation of body. Also, article presented the contact stiness forces matrices for spherical bodies. It has been observed from review that mostly authors have preferred simulations with spherical shaped bodies due to easiness of calculation against other shapes. The common problem of unnecessary rolling obtained by spherical bodies during simulation have directed authors to model for elliptical shaped particles. The main drawback of using circular particles in modeling of granular soil is their monotonous geometric shape and tendency to roll. Thus, it is di cult to simulate certain important behaviors of a soil assembly in which particles are at least slightly elongated or tabular. Replacing circular shapes with elliptical shapes seems natural step towards improving the simulations. Ellipsoid have a degree of freedom to simulate dierent particle shapes in reality while avoiding geometric singularities like those with a polygon shape, and it is still relatively easy to handle mathematically. Elliptical particles lead calculations di cult, complex simulation but they better represents a realistic particle when compared with disc shape representation. In general elliptical algorithm have simulated the assembly in 2-dimensional environment, arriving to the algorithm for ellipsoid in 3-dimension was found di cult. Spherical representation may compensate this complexity at the cost of some ideal move.
However,
attempts have shown that it is nearly impossible to represent particles in their natural irregular form of shape til we are sticking to conventional algorithms and way of presentation.
A new and robust
approach or algorithm may design in future to computerize the particles in their realistic form. Particle modeling in their possible realistic form is the core feature of DEM and any such development in this area will be highly appreciable.
1.2 Particle De nitions in 2D or 3D There are two major classes of particle representation called consider, the
boundary representation
and the
Object Representation
implicit representation.
that we should
The boundary representation
breaks the surface of each body into facets and patches and interpolates within the patch. Intersection checks between two bodies then requires the projection of each patch of one body onto each patch of the other body. The order of the algorithm for contact resolution between two bodies is then
O(N )2 ,
where N is the number of patches per body. Mustoe et al. [52]; Williams et al. [53]; Williams et al. [54] shown the representation used in FEM and CAD, and has been applied to both 2D and 3D discrete element. In computational complexity theory, the element distinctness problem or element uniqueness problem is the problem of determining whether all the elements of a list are distinct or not. It is a well studied problem in many dierent models of computation. Computational complexity algorithm is deterministic factor for speed of computation. that, on average, makes
O(N logN )
Quicksort is a sorting algorithm developed by Tony Hoare N items. In the worst case, it makes O(N 2 ) Quicksort is often faster in practice than other O(N logN )
comparisons to sort
comparisons, though this behavior is rare.
algorithms. Additionally, quicksort's sequential and localized memory references work well with a cache. Quicksort can be implemented as an in-place sort, requiring only
O(logN )
additional space. Quicksort
(also known as "partition-exchange sort") is a comparison sort and, in space e cient implementations, is not a stable sort. The implicit representation, which represents the whole geometry in single function
x2 + y 2 − R2 = 0),
f (x, y) = 0 (e.g.
is popular in DEM for analysis of discs and spheres because of the speed with which
intersections can be checked and the compactness of representation, is given in articles Cundall, 1979 [55]; Taylor et al., 1992 [56], Trent et al., 1992 [57]. It has been extended to two dimension ellipse by Ting [49]. This implicit representation have been generalized to a wider range of two and three dimensional shapes by these authors and the others using superquadrics [58, 59, 60]. In the case of superquadrics the intersection check relies on the sampling of the surface at discrete points.
Each point on the
superquadric of one body is checked against the implicit function representing the other body. order of the contact algorithm is then
O(N )
where
N
The
is the number of points per body. Although the
order of complexity gives information about how the algorithm scales, it does not take into account the number of calculations required for each check. In the case of patches the calculation of surface normals etc. is relatively expensive. Similarly the root nding for implicit equations is also expensive. In order to support both boundary and implicit representations, Williams et al., 1995 [61] developed a low level boundary representation called
discrete f unction representation (DF R)
which is both
general, in its ability to handle a wide variety of shapes, and demonstrably faster than many existing
12
techniques. They used the superquadric representation for ease of input and then use the
DF R
for
contact checking.
DF R scheme is presented in Williams et al., 1995 [22]. Article describes DF R scheme and how it is used to store the surface geometry of a 3D object in form of the look-up
A detailed discussion of the
tables.
The rst phase of the scheme given is to tessellate the surface geometry of a 3D object on
a regular grid of fashion.
The tessellation process starts by discretizing the space occupied by the
object into an ordered set of cubes or voxels. Cubes that intersect the object's surface are then used to approximate this surface by calculating the location of the intersection points of the surface with the cube edges. These points then form of the facets of the tessellate surface. The input surface geometry can be derived from implicit functions, from raw scalar eld data or inferred from existing unstructured polygonal surface. A family of implicit functions called superquadrics presented in Barr, 1981 [62] were used. Graphical representation of the procedure given in Williams et al., 1995 [22] is very similar to given in Williams et al., [63] and is reproduced here in gure 3.
Figure 1.3: The
DF R
sampling grid and discrete sphere example (presented in Williams, 1995 [22])
Figure 3 shows the computational representation of a sphere described by John R. Williams.
DF R
follows a data structure set corresponds to a cage of cells that completely enclose the 3D surface. Refer to article for detailed three data sets called slice, prism and cell. In the mentioned way any spherical or dierent shape particles can be described geometrical. Although the set of cells describe the body but it is not accurate way of representation as it is only a set of cubes in 3D. Cubes de ning the surface of particles are having their own shape and corners which only approximates the surface of particle. Object representation in 3D in this way leads to more accuracy as the size of cubes reduces but the same also results in high computational cost. Such computational cost increases rapidly as the size of cells reduces so an optimized selection of size of smallest cubes is important for accuracy of results, computational cost and time requirement for simulation. Though there are several models of data structures known as Spatial Sorting Algorithm and is an important step for contact search. Such a method is used to sub-divide shapes in cells to represent them and then performs the contact detection as well.
William et al., 1995 [61] discussed various Spatial
Sorting Algorithm such as grid subdivision, adaptive grid methods, octree method, body based cells, spacial heapsort etc. Among them the
Octree
method is highly adaptable among researchers. Figure 3
that sub-divides the body into cells and grids is an example of
Octree
algorithm. The
Octree
method
is conceptually perhaps the most elegant technique to tackle the problems of spatial reasoning and resolution. Method adopts rectilinear cells to formulate the body shapes. How is shown graphically in gure 4.
13
Octree
algorithm work
Figure 1.4: Mechanism of
Octree
data structure
The 3D space is repetitively sub-divided into 8 cubes/cuboids of equivalent volume, however only those cubes at every stage is recorded which consist of full/partial part of body. Signs such as +, - or 0 can be assign to cube to denote partial or full part of body contained by cubes. For example, if a cube completely consist a body then it may have -ve number, if a cube contains no part of the body then it will assign the number 0 and if contains partial part of the body or if cube is completely inside the body then it may have a +ve number assigned to it.
In this way it can be elaborated that the
-ve number assigned, is a sign of possibility of further sub-division, zero assigned to cube illustrates no requirement of further sub-division of that cube, while the set of cubes having positive number leads to de nition of body in 3D. Bodies for which simulation needs to be performed may assigned unique numbers to them and then, accordingly during particle de nition +ve and -ve numbers to cubes might have their magnitudes equivalent to body number. Consider an elemental cube having a positive number assigned, now during body de nition of another particle forcing the same elemental cube to assign a positive number again would mean that it is common cube between particles. So, there can be two meaning of same, rst, during initial (before starting simulation) particles de nition it would mean an impractical condition i.e. overlap of bodies, and might force to change the position of particle while in second case, during simulation it would indicate the particle overlap on another and would be considered for force calculations. Williams et al. [61] mentioned computation complexity for de nition of particle and contact search in 3D as
O(M logM ).
In addition to above Preece et al., 1993 [64]
developed aect of packing angle among packing of spherical bodies on blasting induced rock motion at Sandia National Laboratory.
1.3 Contact Search The most important part of discrete element simulation is contact search and then preparing the list of contacts because of its consumption of major part of computational load. In the simulation of multiple discrete bodies, the design of contact detection algorithms is having important consideration, as the process can be computationally expensive. Contact search is usually divided into two phases,
searching
and
geometric resolution.
neighbor
The goal of the neighbor search is to identify and list objects
within a certain neighborhood or zone around a target object. The resulting list is often called neighbor list. The geometric resolution phase then compares the target object geometry against the geometry of the objects in the neighbor list in detail. The computational cost of geometric resolution depends on the complexity of geometric representation. For example, if there are M facets then a naive scheme may take order
O(N 2 )
computations.
operations.
More sophisticated schemes, such as
DF R
scheme requires
O(M 1/2 )
Similarly, the cost of neighbor searching depends on the number of objects and has
the potential for poor scaling; in a naive implementation, where every object is checked against every other object the cost is
O(N 2 )
operations. Because the number of objects treated in discrete element
simulation is often large, neighbor searching can become a computational bottleneck problem. For this reason, the development of e cient neighbor search algorithm is crucial to the overall performance of the simulation. The implications of allowing arbitrary geometries to coexist in the system, however, must not be ignored. If the neighbor search algorithm is designed with a speci c geometry in mind, its performance may degrade when it is used outside that restricted settings.
14
Perkins et al., 2001 [65] reported contact detection, contact resolution and
sorting (DESS)
double − ended spatial
in details. DESS according to author designed for high performance and robustness
in arbitrary simulations. To better highlight the features of this algorithm, authors compared it with the spatial hashing based
N BS
algorithm by Munjiza et al., 1998 [66]. Authors also de ned the use
of bounding volumes as representative of local body geometry. This bounding volumes are generally sphere or ellipsoid. The goal of neighbor search algorithm design is to minimize computational costs in order to achieve less than
O(N 2 )
cost. Perkins et al., 2001 [65] states about the NBS algorithm, the
NBS algorithm overcomes the problem of sparse simulations through careful bookkeeping, and object based traversal of the bucket lists; instead of visiting each bucket in order, the NBS algorithm checks the bucket for each object, and visits the 'occupied' bucket once. If the current object's bucket has not yet been visited, that bucket, and the lower adjacent one are both hashed on the next axis. Author have compared, and shown superiority of NBS algorithm with spatial hashing, Spatial hashing which they summarized as, in one hashing pass the object can be divided into buckets according to their discrete coordinates along a given axis. Each of those buckets can be divided on another axis. By keeping the most recent as well as the current bucket in each dimension, half of the contact neighborhood is always accessible, and all of the contacts can be evaluated using only two bucket lists for each dimension. Because each object is only visited a constant number of times, the computational cost of the whole algorithm is proportional to
O(N ).
An exhaustive description of methodologies available for contact detection such as spatial sort, exhaustive spatial sorting schemes, spatial sorting using heapsort and the prior knowledge required for spatial sorting algorithms is given in Williams et al., 1995 [61]. Article also compared the heapsort and octree algorithm. Williams et al., 1995 [61] and Williams et al., 1995 [22] reported detection algorithm of computational complexity of
O(N ). DF R
DF R
based contact
based contact detection determines
whether pairs of objects intersect and calculates a precise geometric description of the contact region. Each object maintains a reference frame describing the position and orientation of the object's local coordinate system. The local frame is in turn de ned with respect to a xed global reference frame. The geometric description of each object is expressed using the
DF R scheme, in addition to a bounding
box of the surface, both aligned with the local frame. Nezami et al., 2004 [67] have declared that the applicability of the a DEM code is directly related to the e ciency of the employed contact detection scheme. Article emphasis
N o Binary Search [N BS]
and DESS algorithm for contact detection. Also, briefs about the Barbosa, 1990 [68] simple algorithm for contact detection between polyhedron that requires comparing all the vertices of one particle to all faces of other one and vice versa. This algorithm is having the computational complexity of
O(N 2 ).
Nezami worked out the details of various method for contact detection under the methodology of common plane (CP), rst proposed by Cundall, 1988 [69]. A common plane is a plane that, in some sense, bisects the space between the two contacting particles.
If the two particles are in contact,
then both will intersect the CP. As a result of using CP, the expensive particle-to-particle contact detection problem reduces to a much faster plane-to-particle contact detection problem.
Once, the
CP is developed the normal to the CP de nes the direction of contact normal, which in turn de nes the direction of the contact force acting between two particles.
This is especially an advantage to
edge-to-vertex or vertex-to-vertex contacts, where the de nition of contact normal is a non-trivial problem.
Method is having the computational complexity of order
O(N ).
Paper vastly details a
method for obtaining CP for polygonal (2-D) and polyhedral (3-D) particles. Figure 5 shows here the sign convention used by Nezami in his work.
Figure 1.5: De nition of distances and sign convention of a point to a plane (Nezami, 2004 [67])
15
Contact graphs are used to form contact groups in rigid body simulations. Contact graphs provide an e cient underlying data structure for keeping information about the entire con guration. Contact groups are used for splitting objects into disjoints group that can be simulated independently. Mirtich, 2000 [70] has rstly used the contact graphs in advance manner for other than contact force computation. Erleben, 2004 [71] have presented detailed research on contact graphs as an alternative method for contact detection and contact listing, through a unique data structure used in computation.
Paper
de nes a contact graph as a set consisting of nodes, where a node is an entity in the con guration, such as a rigid body or a xed body. Typical example of contact graph and contact group are presented in gure 6 and 7 respectively, symbolic notation for which can be found in Erleben, 2004 [71].
Figure 1.6: An example of contact graph (Erleben, 2004 [71])
Figure 1.7: A typical example of contact group (Erleben, 2004 [71])
Some of the authors have given a lot emphasis in contact search algorithm in the light of consumption of computational load. Mishra, 2003 [42] have considered the earlier method of keeping track of contact of one particle with all others at every time step, as a
N (N − 1)/2 computational complexity algorithm,
regardless of the shape of particle. Reviewer enlightens the implementation of cell and box in working space to divide the space into cells or boxes in 2D and 3D respectively. A box in this technique is having box length equivalent to the maximum particle diameter. A particle is having entries to all those boxes where the corners of circumscribing box have an entry. Once the box list (the list of all bboxes having entries of a particle) is generated, only those particles that have entries into boxes associated with a
16
given particle are assumed to be in potential contact with it.
Actual particle-to-particle contact is
calculated by knowing the particle coordinates. Mishra, 1991 [72] described contact listing and type of data structure used, in details. According to his PhD thesis, the box list is a linked-list data structure. For a given box, a module is made that consists of two array calls containing the numeric identity of discrete body present in that box and an address pointing to next module of two array cells. The contact-list is also a linked-list data structure of having array cells in each module. For a given ball, a module contains the numeric identity of the discrete body in contact, the contact number associated with the contact, and an address pointing to the next module of three array cells. Figure 8 and 9 below shows typical examples of box-list and contact-list, details of which can be found in Mishra, 1991 [72].
Figure 1.8: Box list corresponding to an example
Figure 1.9: Contact list for example in gure 8
Also, Peng et al., 2005 [73] concentrates their study on e cient mesh construction in 3D with proposed 3D triangulation mesh encoder.
Ferrez et al., 2005 [74] discusses the use of Delaunay tri-
angulation for contact detection in 3D. Williams et al., 1995 [61] briefs available methods for object representation and contact search as discussed in previous sub-section as well. Researchers worldwide follow any of the technique for contact search described in this sub-section, and so it also brings the overall background of contact search.
17
1.4 Contact Resolution Contact resolution is the stage in DEM simulation which follows the stage of contact search/detection. Immediately after the contact-list is prepared after any of the applied contact search method, the next stage is to analyze each one of them for the sack of force calculations. Though, the researchers do not discuss this part in details or else they discuss the contact resolution as an integral part of contact search. Williams et al., 1995 [61] have describes contact resolution using the
DF R
scheme. In two dimension
discrete function representation (DF R) scheme represents the boundary as a single valued function of one local parameter, for example
y = f (x).
The function is then discretized at xed intervals
gives an ordered set of points which are fast to search. The simplicity of
DF R
dx.
This
makes it suitable as a
low level representation in the sense that it is easily derived from higher level abstractions such as spline patches, implicit functions, or nite elements.
DF R
based contact detection can also be incorporated
into existing software systems which use other object representation schemes. In two dimension, closed regions are rst separated into two sub-regions each of which is described by a single valued function. The separating line in this case is taken to be the local X axis of the object. Each half region is then quantized with a uniformly spaced sampling,
dx.
Simple linear or spline regressions can be considered
between boundary points. Once a search point is determined to lie between two of the quantized X coordinates, all other boundary points can be eliminated from search. The
DF R
contact algorithm can
now be described. Two polygonal objects, A and B are considered, each represented with
N
vertices
and a local coordinate frame. A bounding box is also maintained for each object, with respect to the local frame. The contact algorithm in this method follows number of steps, William et al., 1995 [61] can be referred for the same. A similar method for contact resolution is described in Connor et al., 1993 [76]. Paper also compares the computational requirements of the
Figure 1.10: The
DF R
DF R
and
Cyrus − Beck algorithm.
contact resolution steps (Williams et al., 1995 [61])
In gure, four stages of contact resolution through
DF R
scheme are shown.
First, the body is
split into two along X-axis and formed with their respective implicit equations. Figure 10a and 10b graphically represent the same.
Transformation of bounding box B onto A, and project extents of
transformed box in the local X-axis of frame of bounding box A. Test is performed if, the extents of projection lie outside the bounding box A then, no contact is possible otherwise determine the minimum and maximum X-coordinates of the projection inside bounding box A. Figure 10c describes the step followed.
Then, the projected X-coordinates inside bounding box A are used to create one reduced
bounding box and then transformed onto the X-axis in the frame of B, as shown in gure 10d.
A A Xmin = M AX(xA lb , xB,min ) A A Xmax = M AX(xA rb , xB,max ) ......(1)
18
where,
A xA lb , xrb are left and right bounds of A A xB,min , xA B,max are min/max transformed bounds A xA , x max are min/max sub-region bounds min A A If, xmin the > xmax then no overlap is develop has
occurred and so examination is nished. Oth-
erwise, clipping bounds are used to calculate the corresponding and
half − space
start
and
end
addresses of the
upper
of the object A:
A ius = SxA (xA rb − xmax ) A iuf = SxA (xA rb − xmin ) + 1 ......(2) where ius and iuf are the start, (s), and nish, (f ), indices of the sub-region lying in the upper, (u), half space of object A.
SxA
is the scaling factor between the physical space to address space for object
A. This is simply the inverse of the discretization step size,
dx.
The corresponding address in the lower region are calculated from the upper address with the simple relationship:
ils = N − iuf − 1 ilf = N − ius ......(3) where
ils
and
ilf
are the lower (l), start (s) and nish (f ) indices, and
N
is the total number of
elements in the array storing the boundary points of object A. The preceding steps are then applied in projecting the
reduced
bounding box in A onto the local X-axis of object B, as shown in gure
10d. The bounds of the sub-region found in B are calculated using equation (1) and (2) with the B as primary frame of reference. Williams et al., 1995 [61] further discusses the test of candidate contact points. Due to the implicit ordering of the objects boundary representation, the contact detection step is simply a one to one to bounds test as follows: Transform each point from its reference frame into the frame of the object to be tested against. Project the point onto local X-axis and transform to the address space using the scaling factor as in equation (1) and (2). in which the point potentially lies.
Calculate the array index of the bounding edges of the partition Since, the boundary representation is limited to a single valued
function, each partition corresponds uniquely to a single trapezoidal strip. Transformed point is then tested for presence in a trapezoidal panel, as in gure 11.
Figure 1.11: Simple contact bounds test (William et al., 1995 [61])
19
Geometric resolution algorithms strongly depend on complexity of the geometric representation of particles. For example, if the boundaries of the particles are implicitly represented by a single function
f (x, y, z) = 0,
then a closed form solution is likely to be available (for example see Cundall et al., 1979
[55] for contacts between discs and spheres, see Ting et al., 1993 [49] for two-dimensional ellipses, and Lin et al., 1997 [75] for three-dimensional ellipsoids).
E ciency of these contact detection schemes
are mostly controlled by the simplicity of, the resulting equations. Nezami et al., 2004 [67] somehow considers contact resolution as a part of common plane algorithm and thus, described in brief. Ting et al., 1993 [49] presented robust algorithm for ellipse shaped particle simulation. Such a derivation is of very important to consider contact resolution and nding contact point. DEM considers particles to be quasi-rigid with deformable contacts, one can arbitrarily de ne the point of contact
(xc , yc )
between
two ellipses as the midpoint of the line connecting the two ellipse-ellipse intersection points. Equation for ellipse in local coordinate can be de ned as follows:
x x ( )2 + ( )2 = 1 a b where a and b are the length of major and minor axis of ellipse. Transformation of this equation in world coordinate system can be performed as:
cos(θ) x y = sin(θ) 1 world 0 where
θ
is the orientation of the local system with respect to the world, measured counterclockwise
positive. For an ellipse with center at to the
x
−sin(θ) dx x cos(θ) dy y 0 1 1 world
(dx, dy)
in world coordinates and major axis rotated at angle
θ
axis, measured counterclockwise, the general equation in world coordinates would be:
f (x, y) = A(x − dx)2 + B(y − dy)2 + 2C(x − dx)(y − dy) − 1 = 0 where
sin(θ) 2 2 A = ( cos(θ) a ) +( b ) sin(θ) 2 2 B = ( a ) + ( cos(θ) b ) C = cos(θ)sin(θ)( a12 − b12 ) For two ellipses i and j with axes ai , bi and aj , bj , centers (xci , yci ) and (xcj , ycj ) with rotations θi and θj , the equation for ellipse j expressed in the local coordinate system of ellipse i is simply the previous equation, except that aj , bj replace a and b in the values of factor A, B and C , so the new θ , dx and dy would be:
θ = θj − θi dx = (xcj − xci )cos(θi ) + (ycj − yci )sin(θi ) dy = −(xcj − xci )sin(θi ) + (ycj − yci )cos(θi ) Further, derivation that can be referred from Ting et al., 1993 [49], leads to following quartic equation for
X
coordinate, which can be solved for its root to nd the smallest one lies closes to intersection
point. Therefore, coordinate
Y
can also be derived as,
X
accounts to a known value. Further, reverse
transformation results in the value of point of contact in world coordinate.
¯ 2 )X 4 + (4P¯ R ¯ + 2S¯Q)X ¯ 3 + (R ¯ 2 − 4P¯ 2 − Q ¯ 2 + S¯2 )X 2 + (−2P¯ R ¯ − 2S¯Q)X ¯ (4P¯ 2 + Q + (P¯ 2 − S¯2 ) = 0 where
¯ − C¯ dx ¯ ; S¯ = A¯dx ¯ + C¯ dy ¯ ¯=B ¯ − A¯; R ¯ = −B ¯ dy P¯ = C¯ ; Q ¯ = dx , dy ¯ = dy ¯ = b2 B , C¯ = ai bi C , dx A¯ = a2i A, B i ai bi X = axi , Y = byi and
X2 + Y 2 = 1
is the equation of ellipse
i.
Williams et al., 1995 [22] de ned the merit of
DF R
scheme that, it determines whether the pair
of objects intersect and calculates a precise geometric description of the contact region. Each object maintains a local reference frame describing the object's local coordinate system, and the object's
20
geometry is presented by
DF R scheme, in addition to a bounding box of the surface, both aligned with
the local frame. Algorithm rst checks for the bounding box overlap in the overlap region, parallel to local coordinate axes, the upper and lower bound of the overlap are calculated and are saved as 3D vector i.e.
(x, y, z)min
and
(x, y, z)max .
This vector pair is considered as zones of object and shows a
possible zone pair from a bounding box overlap test as shown in gure 12. The zones now bound the region of each object that needs to be further considered.
Figure 1.12: Zone Index set from clip zone (Williams, 1995 [22])
Wensel et al., 1993 [77] have dealt with a quadtree algorithm as a substitute of octree algorithm for simulation of polygon shape particles in 2D. It is notable that an octree algorithm repetitively sub-divides the space in 3D in order to de ne particles, where sub-division is subjected to any of the condition i.e. particle in a cube, cube intersecting the particle or cube inside the particle. In a similar manner, a quadtree algorithm sub-divides 2D space to de ne polygons, discs or ellipse. example of same is given in gure 13.
Figure 1.13: A region quadtree with point data
21
A typical
Figure 1.14: A quadtree based representation of disc in 2D
1.5 Contact Models Contact models are the guiding principles, based on which the forces in the simulation are computed. It is important to remember that DEM simulations are performed mainly to calculate the particles behavior in motion or under the action of complex set of forces, to predict the energy consumption in a similar experiment/operation performed in reality, to predict the breakage pattern of particle under those forces etc.
The results of simulation i.e.
energy consumption, particle motion and breakage
behavior; can be directly or indirectly computed from the forces acting on particles. Also, in granular system or in reality particles are highly heterogeneous in their composition and mechanical properties, and they possess a wide range of non-uniform shape in them. Under such conditions it is di cult to represent particles in virtual environment, still several attempts have been made to represent particles with uniform shape viz.
disc, ellipse, polygon in 2D and sphere, ellipsoid, polyhedral in 3D. Such
assumption/idealistic representation leads to corresponding errors, where such errors have non-linear growth. In a system of granular assembly mechanical properties of individuals are important. Once again contradicting to nature, simulation procedures assigns uniform mechanical properties to one particle, leading to the cause of one more error. On the other hand, for clear understanding and less computational load it is important to keep simulation simple, but not compensating with the representation of natural environment wherever possible. Vast details of contact model as good examples is described in Mishra, 2003 [42].
Paper details
in, various inter-particle contact models viz. Linear-spring dashpot model, Non-linear dashpot model, The elastic perfectly plastic contact model.
Also, it briefs the Tangential interaction between two
particles. Simplest model among all is the linear contact model in which the spring stiness is constant. Improvement over this law can be made by considering
Hertz theory
to obtain the force deformation
relation, where bodies are subjected to deform under the action of forces. Interaction between particles and their deformation leads to dissipation of energy, which may re ect with the
f orce − def ormation equation.
damping term
in the
Interaction between two particles assumed to be linearly elastic, can
be modeled with Linear-spring dashpot model, for which the force in terms of incremental step is given as
Fn (t + ∆t) = Fn (t) − vn kn ∆t + Cn vn ......(4) where
Fn (t) is the normal force at the end of the previous time step. Here, kn is the normal stiness vn is the normal component of the relative velocity of the particles.
which assumes a constant value and
Cn
is the normal damping coe cient, which is modeled by Ting et al., 1992 [78] as
22
r Cn = −2ln(e){
kn m } 2 ln e + π 2 ......(5)
where
kn , m
and
e
is the normal contact stiness, normalized mass and coe cient of restitution
respectively.
Hertz
theory is employed in implemented of Nonlinear-spring dashpot model. Johnson 1985 [79]
had proposed the Hertzian pressure distribution over the contact area of radius contact of radii
(Ri ),
(Ei )
elastic properties
and Poisson's ratio
(vi )
(a),
for two spheres in
as
r p(r) = po [1 − ( )2 ]1/2 a ......(6)
Figure 1.15: Graphical representation of equation (6), showing the pressure distribution over the contact area
Further, derivations in this paper derives the force normalized elastic stiness
∗
(E )
(Fn )
as a function of normalized radii
and relative approach of the centroids
Fn =
(α)
(Ri∗ ),
as
4 ∗ ∗ 1/2 3/2 E (R ) α 3 ......(7)
Most important part of the Non-linear dashpot model is the variation in value of contact stiness
(kn ).
Paper derivates the equation for
kn
kn =
dF = 2E ∗ a = 2E ∗ (R∗ α)1/2 dα ......(8)
Figure 1.16: Dependency of
23
Fn
and
kn
on
α
Equation (4) incorporates the value of damping constant from coe cient of restitution
(e).
(Cn ) while equation (5) shows its derivation
In reality, coe cient of restitution is a function of colliding velocity,
as coe cient of restitution is lower for higher impacting velocities, due to higher dissipation of energy. So it is di cult to x the value of damping constant, as an alternative Mishra et al., 2002 [80] had proposed to replace the viscous dissipation term (damping constant) with the plastic dissipation term, which is easily derivable from material's stress-strain curve.
Mishra, 2003 [42] further derived for
elastic-perfectly plastic contact model. Equation (9) given in Zhang et al., 1998 [43] is similar to equation (4) presented by Mishra, 2003 [42], and can be further compared with equation (10) proposed by Tsuji et al., 1992 [81], for the eect of damping constant.
m
dx d2 x + Cn + kx = 0 dt2 dt ......(9)
m
dx d2 x + Cn k 0.5 x0.75 + kx1.5 = 0 dt2 dt ......(10)
Equation (10) according to Zhang et al., 1998 [43] is more realistic towards behavior of contact loss, as contact should be terminated when the force returns to zero rather than when the distance between two ball centers becomes greater than the sum of the ball radiis. Which is necessary as the balls do not regain the original shape as fast as they are separating and if the constant is not terminated the force between balls become attractive (negative value force). Contact force starts from zero, while following the equation (10) whereas equation (9) starts the force from a non-zero and highest ever value, which is not true in nature. Force patterns by these two equations in graphical format can be referred from the article. Paper also claims the ine ciency of single spring-dashpot model on the modeling forces at the same time, including the surface properties viz. distinction between ball-ball contact and ball-wall/liner contact. Authors have derivated the force model equation for double spring-dashpot model following the basis of Newton's third law.
Figure 17 below shows graphical model of double spring-dashpot
model. Authors have also analyzed and compared the single and double spring-dashpot models in the conclusion part of their work.
Figure 1.17: Double spring and damping contact model (Zhang et al., 1998 [43]
Mishra et al., 1994 [45] have also reviewed the force model equations for single spring-dashpot model. Authors have shown the importance of keeping simulation time step lower than the value of Where
m
and
k
p 2 m/k .
are the normalized mass and spring stiness values. Authors have also re-explained
24
the model for normal spring stiness
(kn )
given in equation (11), and quoted by Chang et al., 1989
[82]. Table 1 shows an example for parameters used in paper to simulate the mill conditions.
kn =
2πG (1 − ν)[22ln(2r/A) − 1] ......(11)
where
1 r
1 r2 ) 2r(1−ν)fn 1/2 A = ( πG ) where G and ν are the shear modulus and Poisson's ratio respectively for the discs,
= 12 ( r11 +
radius and
fn
r is the combined
is the contact force. Here, stiness values are obtained experimentally for metal-metal
and metal-particle collisions. Authors have suggested that coe cient of restitution changes with the velocity of colliding bodies, so they have chosen single value to minimize the computational load. Paper predicts various results for ball mill simulation including the path of motion, product size distribution inside the mill and power draw.
Table 1: Parameters used in simulation by Mishra et al., 1994 [45]
400, 000N/m 300, 000N/m 0.4
Normal stiness Shear stiness Coe cient of friction Coe cient of restitution ball-ball impact ball-wall impact Critical time step
0.9 0.3 1.0 × 0−4 sec
Ting et al., 1993 [49] claimed that, in particular, use of discs and spheres leads to typically low aggregate internal friction angles compared with real materials. Robust algorithm for computing ellipseellipse and ellipse wall interactions were presented by authors.
However, the similar algorithm is
presented by most of the authors who have claimed ellipse shape particle to be better for simulation. In addition, Yang et al., 2003 [83] describe the derivation for rotation of bodies, in principal. Kremmer et al., 2000 [84] have shown the importance of inertia tensor in force calculations. Authors have shown inertia tensor
(I)
for an arbitrary shape particle as:
´ I=
−Ixy −Ixz Iyy −Iyz −Izy Izz ´ ´ − ´ xzdm ´ 2− xydm ´ 2 x dm ´ 2 ´ + z dm ´ 2− yzdm − zydm x dm + y dm
Ixx I = −Iyx −Izx
´ 2 y 2 dm ´ + z dm − ´ yxdm − zxdm
where the diagonal elements are the moments of inertia coordinate system with axes
XX , Y Y
and
ZZ ,
Ixx , Iyy
Izz with respect to an arbitrary −Ixy , −Ixz and −Iyz are the mass-dierential and x, y and z are the and
the o-diagonal elements
products of inertia with respect to the same frame,
dm
is the
distances of the mass dierential to the respective axes of the coordinate system. The inertia tensor is symmetric and has three non-complex eigenvalues which are the principal moments of inertia
Iyy∗
and
Izz∗ .
Ixx∗ ∗ 0 I = 0
0 Iyy∗ 0
0 0 I
zz ∗
Zhang et al., 1998 [85] shown the force as a component of is the contact force
(fe )
Ixx∗ ,
The principal inertia tensor is then:
and gravitational force
(fg ).
25
(fn ),
external force
m
d2 X(t) = fc + fe + fg dt2
Authors have also recommended the spring-dashpot contact model as similar to Zhang et al., 1998 [43], Mishra, 2003 [42] and Tsuji et al., 1992 [81]. Similar recommendation is also found in Zhang et al., 1999 [48] who have shown force, as a summation of normal and tangential components separately. Also, shown the equations for rolling friction as the coe cient times the normal force.
µr N = −M µr N r = I w
dv(t) dt
dw(t) dt
in the model represents the angular velocity. Authors have also proposed an additional imbalance
force on the contact area caused by damping which leads to additional component of force in equation. The contact between two spheres is not a single point contact due to deformation of both components is a nite area as shown in gure 18. The forces acting on this area can be resolved into a tangential force and normal forces acting at dierent locations. The force on the side where the surface is expanding is less than that on the other side due to damping friction. This imbalance creates a moment at the contact which is the additional component needed in the equation of motion. The fractional dierence from the mean normal force on each side of the contact is assumed to be proportional to the normal force and to increase slowly, from an initial value, with increasing velocity. The forces on each side of the contact can be written as:
where
q
F1 =
1 + qv(t) + qo N 2
F2 =
1 − qv(t) − qo N 2
is assigned as the coe cient of rolling.
The contact also exerts a tangential force which adjusts to be consistent with the equations of motion and the rolling condition. This force can take values up to a maximum, that is typically proportional to the normal force, and if the condition are such that the maximum would be exceeded rolling is replaced by sliding.
Figure 1.18: Forces acting on two particles rolling on each other (Zhang et al., 1999 [48])
The rolling friction moment cause the angular velocity of one particle to decrease, and the angular velocity of the other particle to increase. The modi ed rolling equations for two particles are:
dv T (t) 1 1 = −fr ( + ) dt M1 M2 I1
dw1 (t) = fr r1 − (qv T (t) + qo )N (t)s(t) dt
26
I2
dw2 (t) = fr r2 + (qv T (t) + qo )N (t)s(t) dt
where
v T (t) = V1T (t) − V2T (t) Authors have also derivated equations for the time when the sliding switches to rolling, the time when rolling stops. When the relative velocity at the contact point changes from positive to negative or vise versa, it indicates that sliding ends and rolling starts. Linear approximation between time and velocity gives:
ts2r =
(tn − tn+1 )vnc + t1 c − vnc vn+1
The relative velocity at contact point becomes zero when sliding switches into rolling. The relative velocity at the contact point during the sliding stage is:
v c (t) = V1T (t) − w1 (t)r1 − (V2T (t) + w2 (t)r2 ) = −µD(
1 r2 1 r2 + 1 + + 2 ) + voc M1 I1 M2 I2
where D is the switching point and can be calculated as:
Dvc=0 =
voc µ( M11 +
r12 I1
+
1 M2
+
r22 I2 )
Similarly, rolling stops when particles rotate together as a solid, i.e.
w1 = w2 .
Applying this
condition to the rolling, it becomes:
v T = V1T − V2T = w1 (r1 + r2 ) A linear approximation can be used to estimate the rolling stop time.
trs =
t2 (w2n − w1n ) − t1 (w2n+1 − w1n+1 ) w1n+1 − w2n+1 − w1n + w2n
and the common angular velocity at this is
w=
w1n+1 w2n − w1n w2n+1 w1n+1 − w2n+1 − w1n + w2n
Sakaguchi et al., 1993 [36] also focused on single-spring dashpot model. Cundall, in his pioneering work used global damping in the calculations of rotational motion as the rolling resistance:
I θ¨ = where
I
is the moment of inertia and
C∗
X
M − C ∗ θ˙
is the coe cient of global damping.
usually formulated by the term of resisting moment, now may be expressed as
Rolling friction is
MRf
MRf = b ∗ N where
b
is the coe cient of rolling friction and
N
is the normal contact force at each point.
1.6 Parameter(s) Determination Parameters incorporated in suitable force models leads to require, force and moment calculations. Though, these parameters are very sensitive towards such calculations, and are tested by several researchers for their eects. It is also true that mostly authors do not disclose the parameters that they have chosen to perform simulation. However, available data and their comparison is presented in this report. For example, a typical set of parameters for simulation of a system of ellipse of ratio 2:1, is presented in table 2.
Before proceeding to examples, it is fruitful to brief the eects of parameters
explained by Mishra, 2003 [42].
Discussing the eect of coe cient of restitution, authors suggested
that coe cient of restitution is not a constant value but in most of the simulation cases it is assumed
27
constant, accordingly the energy dissipation due to viscous damping has been made proportional to the coe cient of restitution. Coe cient of restitution is the function of material of the bodies, their surface geometry, and the impact velocity.
It was found that, the ratio of energy dissipated during
loading to that during unloading depends on the magnitude of the impact velocity and therefore the coe cient of restitution. On the other hand, contact stiness is a key parameter that determines the
(k)
overall dynamic behavior of the particles. The value of stiness
is chosen in such a way that the
fraction of overlap in most severe collision expected is a small fraction of the diameter of the colliding element. It was analyzed that parameters (damping and contact) values were very large that limit the critical time step, which when used as parameters in the DEM model make the simulation too slow. Authors have recommended to use linearized parameters because which not even allow a larger time step, but also the use of a linear-dashpot-model to signi cantly reduce the computational eort. Coef cient of friction is the next important parameter to be considered, which actually controls the power draw to the mill, and which is particularly sensitive at higher mill speeds. However, the coe cient of friction does not match with the predicted power draw. Conclusions also found that torque drawn to mill decreases with increase in coe cient of restitution.
Table 2: Simulation parameters used in Ting et al., 1993 [49]
Ellipse 1
Ellipse 2
Ellipse 3
Simulation parameters
Gradation information a(mm)
9.76
6.52
4.35
Time step (s) Gravity (m/s2 )
2 × 10−5
b(mm)
4.88
3.26
2.17
number
228
228
228
ρ(g/cm2 )
1.5
1.5
1.5
kn (kN/m2 )
3000
3000
3000
30000
ks (kN/m2 )
2000
2000
2000
20000
Cn (kN/m2 /s)
0.48
0.33
0.216
1.08
0
Cs (kN/m2 /s)
0
0
0
0
0
φµ
26
26
26
cohesion
0
0
0
e
0.5
0.5
0.5
Contact properties
Wall
Cn
Global
26 0 0.5
where a and b are the length of major and minor axes of ellipses, is the coe cient of restitution used to calculate
0
φµ
is the angle of friction,
e
and number is the number of particles of ellipses
in simulation. In the simulation, samples were initially dense with void ratios of about 0.19. Zhang et al., 2001 [87] used stiness rolling coe cient
(q):
0.4,
(qo ):
(k):
2000, damping
(C):
0.4, gravity: 9.8, sliding coe cient
(µ):
0.7,
0.3, static friction coe cient: 10, time step size: 0.01 seconds. Though,
authors have also tested for time step size as 0.1, 0.001 seconds, out of which they found 0.01 seconds most suitable. Buchholtz et al., 2000 [88] followed elastic constant
(Y ) = 8∗106 g/s2 , and γN = 800s−1 ,
−1
γT = 3000s as damping in normal and tangential direction. They assumed Coulomb friction constant to be µ = 0.5. Mishra et al., 1994 [46] claimed 1.5 more power draw due to increase of coe cient of friction from 0.2 to 0.7, in particular conditions of mill speed, mill lling percent and certain number of balls of particular sizes. Ting et al., 1991 [50] implemented the coe cient of restitution 0.5, disc-disc and disc-wall normal spring stiness
105 kN/m2
and
106 kN/m2 ,
respectively. Time step of integration
0.00025 seconds. Table 3 is another example for the set of parameters used to perform simulation of explosion of a disc-shaped solid.
28
Table 3: The parameters used in simulation by Kun et al., 1996 [89]
Parameter Density
Unit
3
(ρ) (Y ) (E)
Grain bulk Young modulus Beam Young modulus Time step
(dt)
(d) (n) Energy of the explosion (Eo ) Ave. initial speed (vo ) Estimated sound speed (c) Diameter of the solid
g/cm dyn/cm2 dyn/cm2 s cm
Ave. no. of polygons
erg m/s m/s
Value
5 1010 5 × 109 10−6 40 1100 5 × 109 200 900
Zhang et al., 1998 [85] compared bouncing of single and two ball system, on a rigid surface. For which, ball-ball stiness, ball-surface stiness, ball-ball damping and ball-surface damping were 50000, 50000, 0.9 and 0.4 respectively.
Another example of set of parameters is already shown in table 1.
Similarly, number of examples can be presented in this report.
1.7 Force Calculations Force calculation step in the simulation does not need much attention as forces and moments can easily be calculated if force models are known and parameters of simulations are determined. In sub-section 1.5 possible and frequently used force models are presented in this report, similarly parameters required in force calculation are discussed in sub-section 1.6, with their examples. Though, the combined form of sub-section 1.5 and 1.6 is enough to calculate forces for further updating the positions, velocities and accelerations of the particles, but care must be taken and values must be repeatedly checked regularly. Repetitive unexpected or large values may lead to instability of simulation.
1.8 Updating Position & Velocity Numerous numbers of algorithm for updating position and velocity are recommended by researchers. These schemes are also known as Numerical integration scheme. Mishra, 2003 [42] has implemented successfully the Leapfrog type integration scheme where the velocity of a particle at time step, say,
n + 1/2
can be calculated, by knowing acceleration at the
and velocity of
i
th
nth
time step. As an example, here position
particle in an assembly are updated as follows
(xi )N +1 = (xi )N + (x˙ i )N +1/2 × ∆t (x˙ i )N +1/2 = (x˙ i )N −1/2 + (¨ xi )N × ∆t ......(12) In the equations,
(x˙ i )N
and
(x˙ i )N −1/2
are known values from previous update. Therefore, algorithm
takes three major steps, calculation of (i) lastly the updated position (iii)
(x˙ i )N +1 .
(¨ xi )N × ∆t
followed by updating velocity (ii)
The instantaneous velocity at the
(x˙ i )N =
N
th
(x˙ i )N −1/2 and
time step is:
(x˙ i )N +1/2 + (x˙ i )N −1/2 2
The above explicit integration scheme is the
2nd order accurate, and it is also regarded as the best for
overall accuracy, stability, and e ciency. Leapfrog integration is equivalent to calculating positions and velocities at interleaved time points, interleaved in such a way that they 'leapfrog' over each other. For example, the position is known at integer time steps and the velocity is known at integer plus half time steps. In DEM, Leapfrog scheme was rst implemented by Cundall and Strack, the pioneers of DEM. Leapfrog integration is a second order method and hence usually works better than Euler integration, which is only of rst order. There are several researchers who implemented Leapfrog scheme in updating
29
position and velocity in their work, some of them can be quoted here are, Ting et al., 1993 [49], Ting et al., 1991 [50]. In Zhang et al., 1998 [85] a dierent method, using velocity and position summation to calculate the particle motion in normal direction was introduced. For an accurate calculation, the particle motion calculation was divided into normal motion calculation and tangential motion calculation. The velocity and displacement summation method takes the force equation:
m where,
fc , fe
and
fg
d2 x = fc + fe + fg dt2
are the contact force, external force and gravitational force respectively.
1 m
VN +1 = VN +
XN +1 = XN + VN ∆t +
1 m
ˆ
t+∆t
ˆ
ˆ
t+∆t
f dt + t
t+∆t
fc dtdt + t
t
ˆ
1 m
1 m
t+∆t
+ t
ˆ
t+∆t
ˆ
1 m
ˆ
t+∆t
fg dt t
t+∆t
fe dtdt + t
t
1 m
ˆ
t+∆t
ˆ
t+∆t
fg dtdt t
t ......(13)
where
´ t+∆t 1 ∆Vc = m fc dt ´t t+∆t ´ t+∆t 1 ∆Xc = m fc dtdt t t ∆Vg = g∆t ∆Xg = 12 g∆t2 considering the eect of external forces
∆Ve = fme ∆t fe ∆Xe = 2m ∆t2 combining the acceleration due to gravity and acceleration force
an = g + fe /m Substituting all above newly de ned terms in equation (13), we get nal model for position and velocity update as per Zhang et al., 1998 [85]
VN +1 = VN + ∆Vc + an ∆t 1 XN +1 = XN + VN ∆t + ∆Xc + an ∆t2 2 Models above, describe the position and updating method suggested by Zhang et al., 1998 [85], for eect of normal forces on the particle under consideration. Zhang et al., 1999 [48] voids this limit by extending models to cause the aects of tangential forces, such as sliding and rolling. Figure 19 below shows graphical representation of force and velocity directions acting on the pair of spherical particles.
Figure 1.19: Forces and velocities for two particles sliding on each other
30
According to Newton's third law, the friction forces on two particles are equal and with opposite direction. The integral form of velocities of the particles are given as follows:
v1TN +1 (t) = −
µ D + v1TN M1
v2TN +1 (t) =
µ D + v2TN M2
w1N +1 (t) =
µr1 D + w1N I1
w2N +1 (t) =
µr2 D + w2N I2
where
D=
´ t+∆t t
where
v
n
n n N (t)dt = m∆v n = m(vN +1 − vN )
is the velocity in normal direction,
angular velocity,
M
is the mass of particle,
I
m
is the mass,
µ
is the coe cient of friction,
is the moment of inertia and
r
w
is the
is the radius of particle.
Though, most of the researchers implement the mentioned methods, their are other models proposed by few researchers in their work. In Zhang et al., 1993 [91], a system of two dimensional circular discs was used to simulate the dynamic motion of the body water. The dynamic equilibrium equations for a circular disc in the DEM model are de ned:
where
y
and
θ
m
is the mass of the particle,
Ic
m¨ x=
X
Fx
m¨ y=
X
Fy
Ic θ¨ =
X
Mc
is the moment of inertia with respect to center of mass,
are the global centroidal coordinates of the particle respectively, and
Fx , Fy
and
Mc
x,
are the
resultant forces and moments acting on the particle. A numerical solution to above equation of motion may be obtained by using an explicit
Euler time stepping scheme.
For two dimensional planar discs,
the explicit time velocity update equations are:
x˙ N +1/2 = x˙ N −1/2 +
∆t X Fx m
y˙ N +1/2 = y˙ N −1/2 +
∆t X Fy m
∆t X θ˙N +1/2 = θ˙N −1/2 + Mc Ic Thus, the corresponding position update equations are de ned by:
xN +1 = xN + x˙ N +1/2 ∆t yN +1 = yN + y˙ N +1/2 ∆t θN +1 = θN + θ˙N +1/2 ∆t
31
1.9 Step Size Step size is one of the most important parameter to determine the stability of simulation. Importance of step size in simulation is dedicately discussed in this sub-section.
The step size determines the
accuracy of a discrete element simulation. Zhang et al., 2001 [87] compared the simulations using xed time steps and variable time step method. According to Cundall and Strack, the pioneers of DEM, the following relation must be satis ed to give stable calculation:
∆t < 2 where
k
is the stiness,
m
p m/k
is the particle mass and
∆t
is the size of time step. Tsuji et al., 1993 [90]
found that the calculation become unstable when the time step is near to the limit value given by this equation. Therefore, Tsuji proposed that time step should be proportional to the oscillation period of the spring-mass system. The oscillation period of the system is given by:
∆t < 2π
p m/k
A suitable time step can be obtained by dividing one of the natural oscillation period by a small integer
n,
e.g.
5.
However, particle separation time is less than one half of the oscillation period,
because the impacting particles will separate when impact forces equals to zero, and a second oscillation or changes in other forces could happen within this time step to aect the particles impact. A similar time step may be needed to obtain accurate calculation of the particle motion. Zhang et al., 1998 [85] introduced a method of using accurately integrated tables of updates to the velocity and position of each discrete element. A step doubling scheme to control the accuracy of this velocity and position updating by varying the step size has been developed.
This method allows the time step size to be adjusted
to a suitable size automatically according to the calculation error in each part of the simulation. The step doubling scheme calculates each simulation step twice: rstly, it calculates the simulation results for whole step, and secondly, it calculates the results by two half steps. The calculation error for each step is de ned as the dierence of simulation results from one full step and the results from two half steps. If the calculation error is within the prede ned error tolerance, the simulation moves on to the next step. The step size used for the next step is related to the calculation error of the previous step. If the calculation error is larger than the error tolerance, the step size is reduced and the calculations repeated until the error is within the error tolerance. The step size needs to give the required accuracy typically varies in dierent parts of a discrete element simulation and at dierent times in the simulation. The extra computation needed to check the accuracy of a given step size can often be justi ed by the reduction in computation obtained by being able to use larger steps in parts of the calculation. In addition, the controlled accuracy avoids the need to choose a conservative step size and to make multiple runs to check the accuracy. In the calculation of total error, the error term for control of step size needs to involve both position and velocity errors and the optimum combination of these can vary with the type of simulation. Various values of time steps chosen, is already being presented in sub-section 1.6.
Zhang et al., 2001 [87]
performed the simulations for xed time step sizes viz. 0.1, 0.01, 0.001 and 0.0001 sec. Among which they found
∆t = 0.01
suitable and given similar accuracy to the variable step size calculation. Authors
also performed the simulations for variable time step (based on step-doubling scheme). Figure 20 and 21 shows the path of bouncing object and error associated to them respectively, for dierent step sizes
(∆t).
They found variable time step gives reliable solution while the xed time step method with
large step size gives inaccurate simulation results. Simulation with xed time step may require number of trials to nd its appropriate value while variable step size method (step doubling scheme) nds a suitable value for all parts of simulation.
32
Figure 1.20: Trajectory of the simulation with variable step size and xed step size (Zhang et al., 2001 [87])
Figure 1.21: The error in trajectory in gure 19 with two simulations
Though, in sub-section 1.6 i.e. parameter determination, already some examples are presented to show implemented values of step size. Those can be reconsidered here with other presentable examples. In example of Kun et al., 1996 [89] time step chosen was
10−6 seconds
for explosion of a disc-shaped
solid. Since an explosion itself completes in fraction of seconds, so such a small value of step size is fairly justi ed to simulate the system with signi cant number of simulation data. Comparing this with Zhang et al., 2001 [87], it concludes that a wide range of step size is possible, depending upon the conditions of simulation, accuracy and computational load of operation. Nezami et al., 2004 [67] claimed that 80% of the step time is consumed in contact detection subroutines. In the paper for two stage simulation i.e. rstly fall of assembly of particles in a closed box and then secondly, ow of fallen particles while one side of the box is removed, time step of
1.5 × 10−5
sec was chosen. In Fortin et al., 2002 [44] for
two examples of simulation, rstly bouncing ball experiment and secondly, rolling of a cylinder on a rigid surface, researchers implemented a step time value of
10−4 sec.
1.10 Recursions Performing recursions is an integral part of simulation. Statement is generally applicable to all type of simulation, and not only for DEM. It is simply the repetition of preceding cycle. This cycle includes all those steps which are due under to analyze the system performance under speci ed conditions. So, recursion is the process of repeating items in a self-similar way, and so a large number of repetitions may take place while simulating a problem. Number of recursions depends upon length of one cycle, where both number of recursion and length of one cycle, which are analogous to each other are the function of several factors. These factors can be number of steps in a cycle, computational cost of one cycle, accuracy required to perform the simulation, cost of updating position, velocity and acceleration, cost of contact searching and listing etc. However, the bene ts of suitable number of recursions is very important to obtain satisfactory results, to assure the stability of simulation, to provide enough time to complete each step in one cycle, to assure the accuracy of simulation etc.
33
In DEM, recursion repeats dierent stages of simulation which includes updating particle de nition in 2D or 3D, contact search followed by contact listing for each particle, contact resolution for each contact in each contact list, force calculation, updating position, velocity and acceleration, and updating step size (optional e.g. in case of step doubling scheme). Among, according to some authors 80% of the computational cost is consumed in contact search step, recalling our previous statement that the accuracy and e ciency of DEM simulation is signi cantly a function of contact search algorithm. Author of this report has already elaborated some examples of recursion i.e. involved in simulation, in previous sub-sections.
34
number of recursions
2
Source Code (Copyright: Dr. R.K. Rajamani) Analysis
Current section analysis the source code written by Dr. R.K. Rajamani and his team at University of Utah, and later improved and developed by number of researchers involved. One of the key developer of the program, Prof.
(Dr.)
B.K. Mishra who is the professor at IIT Kanpur, and on deputation
currently holding the position of Director of Institute of Minerals and materials Technology (IMMT), a Council of Scienti c and Industrial Research (CSIR) lab at Bhubaneswar, India. version was successfully tested in 1997, as an advancement of code
c 2D DEM
Program rst
written in 1995 by
Dr. B.K. Mishra, for simulation of planetary mill. Dr. Mishra with others also developed codes like
c M illSof t (1999)
to predict power draw, monitor collisions on lifters.
Considering a SAG or in operation, industrial SAG mills process material that enters and exits the mill on continuous basis. experienced a grinding
f ield
During the short time period the material spends inside the mill, it or eld of breakage that is a result of the grinding media in constant
motion inside the mill. When the material exits the mill, its size is reduced. To maintain an optimal level of grinding, keeping the capacity at its peak, it is imperative that the so-called grinding eld must be maintained at its best. This necessitates a good understanding of charge behavior. Deciding the charge pro le of the mill, a priori knowledge to various operating and design constraints is not a trivial task. One can always look for sensors, but the grinding environment inside the mill is so severe that none of the online sensors would withstand the impact of the large steel balls falling from a 10-cm height inside the mill.
Since direct observations by means of online sensors is impractical, the next
best option is numerical simulation. It is apparent that, DEM provides the best solution as a tool for charge motion analysis in tumbling mills. It is quite reliable because the underlying principles originates from the fundamental laws of physics. Consider, another problem of milling e ciency, it is believed that only 20% of the energy is utilized in comminution (Flavel et al., 1981 [92]). Therefore, there is huge potential of saving energy in milling operations. DEM analysis of the tumbling mills, not only provide an insight into the charge motion, it simultaneously gives a host of other information, such as distribution of impact energy, force transmission paths inside the ball load, stresses on the wall, etc. This opens various avenues of research encompassing but not limited to energy utilization, material ow, lifter design, scale-up, etc. For example, the distributions and time histories of forces on the liner and lifters allows calculation of wear and it is also true for the ball charge. The major disadvantage of the DEM is the requirement of an enormous amount computational time.
This is due to the explicit nature of the algorithm that requires a very small time step of
simulation to assure numerical stability and accuracy.
However, DEM naturally renders itself for
parallelization. One can develop a exible parallel computer code that is capable of generating external shell, internal surfaces and multibody assemblies for carrying out simulations to track particle trajectory and even fragmentation. In the current scenario, while comparing the other available options with DEM, DEM has advantage to use as a tool for predictions of Mineral Processing operations, as rapid growth in computational capacity is observed.
Nowadays,
Core i7 − 975 Extreme Edition type very fast F ujitsu k computer
computers are available in market, also there is availability of supercomputers like with maximum speed of
10.51 P F LOP S ,
for high level, accurate simulation with very slow pace (step
size) advancement. Successive work in this report will describe the execution steps of Dr. Rajamani's code. Code was studied by the author of this report, and the development stages in the program is described here. This will also supplement as a good example to the stages already explained in the review part of the report. A ow chart is given here which shows the executionns of the program to simulate a charged ball mill. Note: 1. Flow chart presented in this section is possible the very simpli ed form of representing execution sequence. 2. Chart is prepared to accommodate the major steps for simulation without compensating the presentable form of execution to understand. 3. Data structures and pointers in executions are not shown in this chart.
35
36
37
38
39
40
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Hakuno, M., Iwashita, K., Uchida, Y., DEM simulations of cli collapse and debris ow. Proc.
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Butkovich, T.R., Walton, O.R., Heuze, F.E., Insights in cratering phenomenology provided by discrete element modeling. Proc.
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U.S. Symp. Rock Mechanics, Minneapolis, MN,
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Ting, J.M., Corkum, B.T., Kauman, C.R., Greco, C., Discrete numerical model for soil mechanics. J. Geotech Eng., 115, pp. 379-398, 1989.
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[19]
Williams, J.R., Particulate mechanics of manufacturing processes. Intelligent Engg. Systems Laboratory, Messachusetts Institute of Technology, 1994.
[20]
Williams, J.R., Dynamic wave propogation in particulate materials with dierent particle shapes using a discrete element method. Intelligent Engg. Systems Laboratory, Messachusetts Institute of Technology, Engineering mechanics, pp. 493-496, 1995.
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