DEM models in 2D

Engineering Computations On the validation of DEM and FEM/DEM models in 2D and 3D Jiansheng Xiang Antonio Munjiza John-Paul Latham Romain Guises

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To cite this document: Jiansheng Xiang Antonio Munjiza John-Paul Latham Romain Guises, (2009),"On the validation of DEM and FEM/DEM models in 2D and 3D", Engineering Computations, Vol. 26 Iss 6 pp. 673 - 687 Permanent link to this document: http://dx.doi.org/10.1108/02644400910975469 Downloaded on: 02 March 2015, At: 06:10 (PT) References: this document contains references to 23 other documents. To copy this document: [email protected] The fulltext of this document has been downloaded 1283 times since 2009*

Users who downloaded this article also downloaded: A. Munjiza, D.R.J. Owen, N. Bicanic, (1995),"A combined finite-discrete element method in transient dynamics of fracturing solids", Engineering Computations, Vol. 12 Iss 2 pp. 145-174 http:// dx.doi.org/10.1108/02644409510799532 A. Munjiza, J.P. Latham, (2004),"Comparison of experimental and FEM/DEM results for gravitational deposition of identical cubes", Engineering Computations, Vol. 21 Iss 2/3/4 pp. 249-264 http:// dx.doi.org/10.1108/02644400410519776 Y. Sheng, C.J. Lawrence, B.J. Briscoe, C. Thornton, (2004),"Numerical studies of uniaxial powder compaction process by 3D DEM", Engineering Computations, Vol. 21 Iss 2/3/4 pp. 304-317 http:// dx.doi.org/10.1108/02644400410519802

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On the validation of DEM and FEM/DEM models in 2D and 3D Jiansheng Xiang

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Department of Earth Science and Engineering, Imperial College London, London, UK Downloaded by IMPERIAL COLLEGE LONDON At 06:10 02 March 2015 (PT)

Antonio Munjiza Department of Engineering, Queen Mary, University of London, London, UK, and

John-Paul Latham and Romain Guises Department of Earth Science and Engineering, Imperial College London, London, UK Abstract Purpose – As particulate systems evolve, sliding, rolling and collision contacts all produce forces that discrete element method (DEM) methods aim to predict. Verification of friction rarely takes high priority in validation studies even though friction plays a very important role in applications and in mathematical models for numerical simulation. The purpose of this paper is to address sliding friction in finite element method (FEM)/DEM and rolling friction in DEM. Design/methodology/approach – Analytical solutions for ‘‘block sliding’’ were used to verify the authors’ tangential contact force implementation of 2D FEM/DEM. Inspired by the kinetic art work Liquid Reflections by Liliane Lijn, which consists of free balls responding within a rotating shallow dish, DEM was used to simulate rolling, sliding and state-of-rest of spherical particles relative to horizontal and inclined, concave and flat spinning platforms. Various material properties, initial and boundary conditions are set which produce different trajectory regimes. Findings – Simulation output is found to be in excellent agreement when compared with experimental results and analytical solutions. Originality/value – The more widespread use of analytically solvable benchmark tests for DEM and FEM/DEM codes is recommended. Keywords Finite element analysis, Friction Paper type Research paper

1. Introduction Validation of our discrete element method (DEM) community’s numerical simulation tools has taken many forms. Reasonably accurate mathematical representations of the contact physics between particles and container walls including friction effects have been implemented with varying sophistication in DEM codes. Li et al. (2005) produced a classic validation study. They performed experiments to complete the determination of all material properties including friction coefficients applicable to glass and steel sphere simulations so that ‘‘sandpile’’ experiments with spherical particles could be numerically simulated with DEM. In their DEM implementation, energy dissipation occurs through mechanisms governed by accurately measurable material parameters, rather than invoking numerical damping through un-measurable coefficients. Their results for angle of repose, when compared with the equivalent experimentally produced angles are therefore all the more convincing. Theirs is one of many cases of an emergent property of the granular system being used to compare and validate the simulator’s accuracy. Other pseudo-static emergent properties include porosity,

Engineering Computations: International Journal for ComputerAided Engineering and Software Vol. 26 No. 6, 2009 pp. 673-687 # Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644400910975469

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co-ordination number statistics, fabric tensor statistics, force chain characteristics. The dynamics of granular systems in flow, flow rates through apertures, velocity field statistics, and the special emergent properties associated with tumbling mills, including power draw, all have been subject to experimental vs DEM comparisons e.g. (Cleary et al., 2003). Unlike the work of Li et al. (2005), Asmar et al. (2002) produced a numerical validation study. In their study, they set up eight ‘‘mathematical tests’’ based on artificial situations. Although they did not compare their results with that of experiments, their study shows that some simple simulations can help to verify code and reveal bugs. These tests constitute a standard benchmark. However, the effect of rolling friction was neglected and not implemented in their model. Zhou et al. (1999) investigated the mathematical importance of rolling friction on the formation of a heap of spheres. They demonstrated that the rolling friction plays a critical role in achieving physically or numerically stable results, and the angle of repose increases with rolling friction coefficient and decreases with particle size. It is therefore perhaps surprising that to date, validation of rolling friction rarely takes high priority in validation studies. Using finite element method (FEM)/DEM, Latham and Munjiza (2004) presented a comparison of cube-packing experiments and their equivalent numerical simulation. They found sensitivity to initial conditions to be even more prevalent for these nonspherical bodies, but emergent bulk behaviour in close agreement. One underused tool of validation for accuracy of contact physics might be to subject the motion of simple systems involving few particles to deeper scrutiny. This is usually considered impractical because there is so much sensitivity to initial conditions (chaos) in inter-particle mechanics, especially for non-spherical shapes and increased particle numbers. The advantage of a rotating container simulation is that there is a constant supply of energy to the system, the particle(s) can be spatially constrained and there may be some interesting boundary conditions for which analytical solutions exist or experiments can be devised with a reasonably high degree of parameter control. Friction plays a very important role in most technical applications and from a numerical view point, friction is an important part of the mathematical model. Verification of friction effects during simulation should take high priority in the work of validation. This paper begins by addressing sliding friction in FEM/DEM and rolling friction in DEM. 2. Computer implementation In this paper, there are two groups of validation tests carried out based on two programs, respectively. One is named Y code developed by Munjiza (2004) using a coupled FEM/ DEM. Details of this method are presented in the references (Komodromos and Williams, 2004; Munjiza et al., 1995) and are further explained with reference to topics in computational mechanics of discontinue in Munjiza’s book (Munjiza, 2004). Another one is a DEM code called 3DSYdeveloped by Xiang using a linear-spring and dashpot model. The details of this model are presented in cited references (Mustoe and Miyata, 2001; Xiang and McGlinchey, 2004; Xiang, 2004; Tsuji et al., 1993). 2.1 DEM with rolling friction The DEM, sometimes called the distinct element method, is becoming widely used in simulating granular flows. Pioneering work in the application of the method to granular systems was carried out by Cundall and Strack (1979). Many researchers further developed and modified this method in searching algorithm (Williams and O’Connor, 1995; Perkins and Williams, 2001; Williams et al., 2004; Feng et al., 2006),

algorithm for calculating contact forces (Tsuji et al., 1993) and time integration scheme (Munjiza, 2004; Feng, 2005) etc. Individual particles have two types of motion, translational and rotational. The motions of particles are influenced by the total forces. Forces acting on particles i include: the gravitational force, mg, normal elastic contact force between particles i and j, fn,ij, and the tangential elastic contact force between particles i and j, ft,ij. According to Newton’s second law of motion the translational and rotational motions of particle at any time t can be written as i X   dv i ¼ mi g þ f t;ij þ f n;ij dt j¼1

k

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mi

i   d vi X ¼ T t;ij þ T r;ij dt j¼1

ð1Þ

k

Ii

ð2Þ

where mi and Ii are the mass and moment of inertia, respectively. vi and vi are translational and rotational velocities of particle i, respectively. Tt,ij is the tangential contact force torque. Tr,ij is the rolling friction torque. For multiple interactions, the inter-particle forces and torques are summed for ki particles interacting with particle i. In calculating contact forces, the contact of particles is modelled by a pair of linear spring–dashpot–slider contact model (Cundall and Strack, 1979) in both the normal and tangential directions. The contact forces are given by, f n;ij ¼ ðkn dn;ij  n ðv ij  n ij ÞÞn ij

ð3aÞ

f t;ij ¼ minfkt dt;ij  t v t;ij ; f jf n;ij jt ij g

ð3bÞ

v ij ¼ v i  v j þ ðvi  R i  vj  R j Þ

ð3cÞ

v n;ij ¼ ðv ij  n ij Þn ij

ð3dÞ

Vt;ij ¼ v ij  v n;ij

ð3eÞ

where dn,ij and dt,ij are normal and tangential displacement vectors between particles i and j, and vij is the relative velocity vector of particle i to j at the contact point, nij is the unit vector from the centre of particle i to particle j, tij is the unit vector perpendicular to nij, vn,ij and vt,ij are the relative velocities of contact point in normal and tangential directions, respectively, f is coefficient of sliding friction n and t are the normal and tangential spring stiffness constants of particle i; respectively. n and t are the normal and tangential viscous contact damping coefficients of particle i; respectively. Ri is a vector from the centre of particle i to the contact point with its magnitude equal to the radius of particle i, Ri.

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The torque due to tangential contact force is given by T t;ij ¼ f t;ij  R i

ð4Þ

The rolling friction torque in Equation (2) is calculated using Beer and Johnson’s model (1976) which is described below. ^ T r;ij ¼ r jf n;ij j!

ð5Þ

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^ is the direction of angular velocity. where r is coefficient of rolling friction, and ! 2.2 FEM/DEM with sliding friction One of the most powerful developments in DEM that began in the late 1980s is the combination of FEM and DEM. Decades of FEM modelling technology designed for modelling stressed and deformed solids has now been combined with the motion tracking capability of DEM. Combined FEM/DEM was first proposed by A. Munjiza and the first working FEM/DEM code (the RG program written in Cþþ) was developed by Munjiza in 1990. Originally, combined FEM/DEM incorporated finite strain elasticity–plasticity coupled with a smeared crack model for fracture and fragmentation. Important advantages over DEM models based on spheres, ellipsoids or even superquadrics are that complex particle shapes can be introduced. Furthermore, a vast range of alternative e.g. nonlinear constitutive or internally fracturing properties can be introduced for the individual particles. Thus, if stresses are sufficient to propagate cracks and initiate failure in the particles, they will fragment and the DEM formulations will continue to track the fragment motions. Such FEM/DEM approaches have been successfully applied to modelling the key processes of stress wave propagation and expanding gas-driven fragmentation in rock blasting. In FEM/DEM, a penalty function method is employed to calculate the normal contact force when two particles are in contact. The penalty function method in its classical form assumes that two particles penetrate each other. The elemental contact force is directly related to the overlapping area of finite element in contact. The distributed contact force approach takes into account the shape and the size of the overlap area in order to be distributed among the surrounding nodes. Munjiza (2004) showed that integration over finite elements was equivalent to integration over finite element boundaries, the contact force is given by, fc ¼

n X m ð X i¼1 j¼1

c

n c \t ð’ci  ’tj Þd i

i

\t j

j

ð6Þ

Where c and  t are the contactor and target discrete elements, respectively, n is the outward unit normal to the boundary of the overlapping area, the integration over finite element boundaries can be written as summation of integration over the edges of finite elements. Munjiza (2004) implemented discretised contact force in using the simplest possible finite element in 2D, i.e. a linear three-noded triangle. The total contact force exerted by

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the target triangle onto an edge of the element is given by the area of the diagram of potential over the edge, ðL 1 ð7Þ f c;edge ¼ 2 u p’ðvÞdv u 0 Where p is the penalty term, the term u2 comes from the fact that vector u is not unit vector. Then the calculated elemental contact force is distributed around the nodes surrounding the contact in order to preserve the system from artificial stress concentration. In this paper, we develop further the FEM/DEM method by taking account of the sliding friction force. The well-known classic Coulomb-type friction is implemented and described as follows, f t ¼ kt dt  v t

ð8Þ

where  is the coefficient of viscous dissipation, ft is the tangential elastic contact force and vt is the tangential relative velocity. If ft is bigger than the friction force obeying the Coulomb-type friction law, jf t j > jf n j the particles slide over each other and the tangential force is calculated using the total normal elastic contact force fn: f t ¼ f n

ð9Þ

where  is the coefficient of sliding friction. 3. Verification for FEM/DEM The case of a rectangle which is paced on a horizontal plane with initial horizontal velocities is carried out as a benchmark test for 2D FEM/DEM with sliding friction. The stop distance L can be theoretically derived and is given by the following equation, L¼

v2i 2g

ð10Þ

where  is the coefficient of sliding friction, g is the gravity acceleration and vi is the initial horizontal velocity. The FEM/DEM simulation of a sliding rectangle is performed for a square rectangle with length, l ¼ 0.05 m, density  ¼ 2650 kg/m3, frictional coefficient  ¼ 0.5 and Young’s modulus E ¼ 1.0  109 Pa. The rectangle is given an initial velocity vi then slows down due to the frictional effect with the base. Finally it stops at a distance L. Two sets of simulation tests were repeated with different initial velocity with two different time steps, t ¼ 1.0  107 s and t ¼ 1.0  108 s. Results are compared with the analytical solution (see Figure 1). They show that with a small time step, the numerical results are in excellent agreement with the theoretical values. It is worth noting that with the larger of the two time steps, the errors become significant. However, using the larger time step, the calculation of FEM/DEM with zero friction is fairy stable. This somewhat alarming conclusion suggests that in order to reduce the numerical error for calculation of tangential forces, the smaller time step is required. 4. Validation for DEM Ehrlich and Tuszynski (1995) carried out several experimental tests to validate their theoretical model. In their study, they analysed the motion of a ball on both a flat and a

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Figure 1. Stop distance plotted against initial velocity for a rectangle with frictional coefficient,  ¼ 0.5

parabolically curved rotating turntable. Compared with DEM, their model is quite simple. In their model, it is assumed that sliding friction force and rolling friction torque are constants. In our DEM simulation, the contact forces are calculated using a spring, dashpot and slider (Xiang, 2004). Soodak and Tiersten (1996) further explained E&T’s experimental results and provided an explanation of the effects of rolling friction and other ‘‘perturbations’’, e.g. a tilt of the turntable. In this paper, we choose E&T’s experimental results of a flat turntable to validate our DEM code. The works of art Liquid Reflections by Liliane Lijn (Mellor, 2005), were made in 1966-68. They incorporate a rotating circular platform and large acrylic spheres combined with spot lighting to produce an intriguing example of 1960s kinetic art that can be seen in many national modern art collections (see Figure 2 and the video Force Fields at: www.lilianelijn.com/vid02.html). Drawing inspiration from Lijn’s experiment setup, we did several tests and found that if a ball was released from rest, the ball tended to stay at a certain height and finally ran on a closed circular orbit. This phenomenon is explained and verified by Equation (12) which explains that with a certain angular velocity of the turntable, at a certain height, the contact force between the ball and turntable can balance centrifugal force and gravity to maintain circular motion. Using this equation, we validate our numerical results. The critical speed for a banked curve with zero friction is v2 ¼ rg tan .ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As for spherically curved turntable, v ¼ !r; r ¼ R2  ðR2  h2 Þ and tan  ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2  ðR2  h2 Þ=R  h, then we can obtain h¼R

g !2

ð11Þ

According to Equation (11), height increases with the increasingpofffiffiffiffiffiffiffiffi angular velocity. When the angular velocity is unlimited, the height equals R. If !  g=R; h  0. This is not physical correct, so we make a criteria for the calculation and modify Equation (11),

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Figure 2. Liquid Reflections by Liliane Lijn

8 g > > < R  2 ; if ! h¼ > > : 0; if

rffiffiffiffi g ! Rffi rffiffiffi g !< R

ð12Þ

In our DEM simulation, we examine spherical particle motions in flat and spherically curved rotating platform geometries, beginning with behaviour for which analytical solutions can be derived for the motion, progressing to more complex regimes. In comparing with E&T’s experimental results and Lijn’s visual results, we set up the same set of conditions. The physical properties of ball and geometry of the turntable are shown in Table I. 4.1 Flat turntable Test 1. Several studies (Weltner, 1979; Ehrlich and Tuszynski, 1995) show that if the ball is given an initial push, the ball runs on a stable circular orbit. The angular velocity is 2/7 Turntable Shape Radius R (m) Ball Shape Density  (kg m3) Radius rs (mm) Frictional coefficient  Co-efficient of restitution e Co-efficient of rolling friction r Spring stiffness k

Flat 0.2

Spherically curved 20.0

Spherical 7,800 6.35 0.15 0.8 1.5  106 1.0  105

Spherical 7,500 3.00 0.5 0.8 5.0  105 1.0  103

Table I. Turntable and ball properties used in the DEM models

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Figure 3. A numerically computed trajectory of moving ball on flat turntable with tilt angle 0.00

of the turntable’s angular velocity. This phenomenon is not related to initial position and radius of ball. In our simulation test, before the ball is released on the rotating turntable it is given the initial spin velocity such that there is no slipping between ball and rotating turntable. If the rolling friction is turned off, we get a stable circular orbit (see Figure 3). The angular velocity is 2.855 rad/s, 2/7 of turntable’s angular velocity. Test 2. Figure 4(b) shows the case where the turntable is slightly tilted (0.18 ) towards North with non-zero rolling friction, the orbit radius grows in size, and its centre drifts at fairly constant speed in the direction of 225.00 which is the direction of ^k  g^. (note: we define 0.00 represents East !, 90.00 represents North ", 180.00 represents West and 270.00 represents South #, see Figure 4(b)) ^k is the unit vector perpendicular to the surface of the turntable, directed from the surface to the ball centre. g^ is the unit vector in the direct of inclination. The numerical results agree with the experimental results in Figure 4(a) (Ehrlich and Tuszynski, 1995) and theoretical explanations (Soodak and Tiersten, 1996). Test 3.When the turntable is given a slightly greater tilt degree, its orbit centre drifts at fairly constant speed in the direction of 180.00 which is the direction of ^k  ^g. The simulation reproduces the trajectory in Figure 5(b) in E&T’s paper (see Figure 5(a)) (Ehrlich and Tuszynski, 1995). When the rolling friction is increased to 2.5  105 (see Figure 6), the orbit centre drifts away from the centre of the turntable and orbit radius grows in size. The output simulation is in good agreement with the numerical results in the paper (Soodak and Tiersten, 1996). Test 4. Figure 7(b) shows a special trajectory when the orbit has a nearly fixed centre which grows slowly in size. The drift is very small but in the direction of ^k  ^g. The simulation successfully reproduces the trajectory of experimental results (see

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Figure 4.

Figure 5.

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Figure 6. A numerically computed trajectory of moving ball on flat turntable with tilt angle 0.52 in the direction of 90.00

Figure 7.

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Figure 7(a)) (Ehrlich and Tuszynski, 1995). Figure 8 shows the trajectory when the turntable tilt is changed from 0.17 to 0.00 . The numerical results in Figure 7 of E&T’s paper, compare very well with our result in Figure 8. Test 5. Figure 9(b) shows an interesting trajectory which is called, ‘‘Approach-to-thecentre trajectory’’ (Ehrlich and Tuszynski, 1995). The translational velocity of the ball was set to 0.00 m/s and the ball was released under no slipping condition. Numerical results fit experimental results (see Figure 9(a)) (Ehrlich and Tuszynski, 1995) which shows that the ball approaches a certain point along a curved path (note that this certain point is not exactly the centre of the turntable). The ball first reaches centreline of the turntable, x-axis at a certain angle. Then it approaches to the point very slowly after the inflection point. After it reaches the point, the orbit radius grows in size and its centre drifts in the direction of ^k  g^.

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4.2 Spherically curved turntable A ball was released onto a rotating spherical (i.e. constant curvature) turntable with zero initial velocity. When the turntable rotates slowly, the ball spirals to the centre of the turntable quickly. Finally, it comes to rest at the centre (see Figure 10(a)). When the turntable rotates faster, the ball spirals outwards and finally stays at a certain height and on a stable circular orbit (see Figure 10(d)). Several tests were carried out with a wide range of angular velocity of the turntable. Figure 11 shows the comparison of numerical results of stable heights with the analytical solution according to Equation (6). The figure shows that when the angular velocity is below 0.7 rad/s, which is the critical value calculated by Equation (6), the ball tends to stay at the centre of turntable and remains stationary. When the angular velocity is above 0.7 rad/s, the stable height increases quickly with the increase of angular velocity. When angular velocity

Figure 8. A numerically computed trajectory of moving ball on flat turntable with tilt angle 0.00 in the direction of 90.00

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Figure 9.

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Figure 10. A numerically computed trajectory of moving ball after the ball is released from static on spherically curved turntable with tilt angle 0.00

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Figure 11. Stable height to which ball will climb against angular velocity of turntable

increases further to more than 2 rad/s, the stable height increases slowly and is close to its maximum value of 20 m. 5. Conclusions The paper illustrates the extent to which DEM and FEM/DEM codes agree with theoretical solutions. It also explains the need for further analytically solved benchmark tests for DEM and FEM/DEM codes as an extension to those proposed by Asmar et al. (2002). The potential exists to extend this approach to tackle code validation using physical experiments conducted with system collisions using more than one particle. Study of orbiting system geometries may also throw light on validation for non-spherical shapes. Future work will need to consider progress to more complex regimes with two spheres and a tilted rotation axis. References Asmar, B.N., Langston, P.A., Matchett, A.J. and Walters, J.K. (2002), ‘‘Validation tests on a distinct element model of vibrating cohesive particle systems’’, Computers & Chemical Engineering, Vol. 26, pp. 785-802. Beer, F.P. and Johnson, E.R. (1976), Mechanics for Engineers – Statics and Dynamics, McGrawHill, New York, NY. Cleary, P.W., Morrisson, R. and Morrell, S. (2003), ‘‘Comparison of DEM and experiment for a scale model SAG mill’’, International Journal of Mineral Processing, Vol. 68, pp. 129-65. Cundall, P.A. and Strack, O.D.L. (1979), ‘‘A discrete numerical model for granular assemblies’’, Geotechnique, Vol. 29, pp. 47-65. Ehrlich, R. and Tuszynski, J. (1995), ‘‘Ball on a rotating turntable: comparison of theory and experiment’’, American Journal of Physics, Vol. 63, pp. 351-9. Feng, Y.T. (2005), ‘‘On the central difference algorithm in discrete element modelling of impact’’, International Journal for Numerical Methods in Engineering, Vol. 64, pp. 1959-80.

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Feng, Y.T., Li, C.F. and Owen, D.R.J. (2006), ‘‘SMB: collision detection based on temporal coherence’’, Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 2252-69. Komodromos, P.I. and Williams, J.R. (2004), ‘‘Dynamic simulation of multiple deformable bodies using combined discrete and finite element methods’’, Engineering Computations, Vol. 21, pp. 431-48. Latham, J.P. and Munjiza, A. (2004), ‘‘The modelling of particle systems with real shapes’’, Philosophical Transactions of the Royal Society London, Series A (Mathematical, Physical and Engineering Sciences), Vol. 362, pp. 1953-72. Li, Y., Xu, Y. and Thornton, C. (2005), ‘‘A comparison of discrete element simulations and experiments for ‘sandpiles’ composed of spherical particles’’, Powder Technology, Vol. 160, pp. 219-28. Mellor, D. (2005), Liliane Lijn: Works 1959-80, Nicholson & Bass, Belfast. Munjiza, A. (2004), The Combined Finite-Discrete Element Method, John Wiley & Sons, London. Munjiza, A., Owen, D.R.J. and Bicanic, N. (1995), ‘‘Combined finite-discrete element method in transient dynamics of fracturing solids’’, Engineering Computations (Swansea, Wales), Vol. 12, pp. 145-74. Mustoe, G.G.W. and Miyata, M. (2001), ‘‘Material flow analyses of noncircular-shaped granular media using discrete element methods’’, Journal of Engineering Mechanics, Vol. 127, pp. 1017-26. Perkins, E. and Williams, J.R. (2001), ‘‘A fast contact detection algorithm insensitive to object sizes’’, Engineering Computations (Swansea, Wales), Vol. 18, pp. 48-61. Soodak, H. and Tiersten, M.S. (1996), ‘‘Perturbation analysis of rolling friction on a turntable’’, American Journal of Physics, Vol. 64, pp. 1130-9. Tsuji, Y., Kawaguchi, T. and Tanaka, T. (1993), ‘‘Discrete particle simulation of two-dimensional fluidized bed’’, Powder Technology, Vol. 77, pp. 79-87. Weltner, K. (1979), ‘‘Stable circular orbits of freely moving balls on rotating discs’’, American Journal of Physics, Vol. 47, pp. 984-6. Williams, J.R. and O’Connor, R. (1995), ‘‘Linear complexity intersection algorithm for discrete element simulation of arbitrary geometries’’, Engineering Computations (Swansea, Wales), Vol. 12, pp. 185-201. Williams, J.R., Perkins, E. and Cook, B. (2004), ‘‘A contact algorithm for partitioning N arbitrary sized objects’’, Engineering Computations, Vol. 21, pp. 235-48. Xiang, J. (2004), ‘‘Investigation of particle motion in dense phase pneumatic conveying’’, Glasgow Caledonian University, Glasgow. Xiang, J. and Mcglinchey, D. (2004), ‘‘Numerical simulation of particle motion in dense phase pneumatic conveying’’, Granular Matter, Vol. 6, pp. 167-72. Zhou, Y.C., Wright, B.D., Yang, R.Y., Xu, B.H. and Yu, A.B. (1999), ‘‘Rolling friction in the dynamic simulation of sandpile formation’’, Physica A, Vol. 269, pp. 536-53. Corresponding author Jiansheng Xiang can be contacted at: [email protected]

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