Reduction of discrete element models by Karhunen–Loève transform: a hybrid model approach

The term “discrete element method” (DEM) in engineering science comprises various approaches to model physical systems by agglomerates of free particles. While shapes, sizes and properties of particles may vary, in most DEM models, particles are not confined by constraints, but subject to applied forces derived from potential fields and/or contact laws. This general approach allows for widespread use of DEM models for physical phenomena including gas dynamics, granular flow, fracture and impact analysis. However, its characteristic feature, combining particle restraints and forces into applied forces, does not only provide for flexible adaption of DEM to different physics, but also creates the most limiting restriction: Evaluation of the applied forces for each particle is computational expensive restraining the time sequence and sample size for numerical analyses. As an ansatz to circumvent this obstacle for a class of DEM models, we propose a model order reduction method based on coherency in the dynamics of particles. While initial flexibility of DEM is conserved, computational effort can be reduced significantly.     

Three-dimensional simulation of the particle distribution in a downer using CFD–DEM and comparison with the results of ECT experiments

A three-dimensional numerical model of the down-flow fluidized bed (Downer) with a newly designed distributor was applied to investigate the particle distribution profiles using combined computational fluid dynamics (CFD) and the discrete element method (DEM). A realistic model of DEM, which calculates the contact force acting on the individual particles, is used to monitor the movement of individual particles in the bed. The contact force is calculated using the concepts of the spring, dash-pot, and friction slider. The flow field of gas is predicted by the Navier–Stokes equation. This CFD–DEM model provides information regarding the particle movement and distribution, the particle velocity, and the gas velocity in the bed under different air-particle mixture conditions. The results demonstrate that the air supply conditions directly influence the particle distribution uniformity. Furthermore, the numerical predictions for the axial and radial profiles of the particle distribution were found to agree well with the experimental results obtained by electrical capacitance tomography (ECT).     

Numerical simulation of particle motion in dense phase pneumatic conveying

A gas-solids two-dimensional mathematical model was developed for plug flow of cohesionless particles in a horizontal pipeline in dense phase pneumatic conveying. The model was developed based on the discrete element method (DEM). For the gas phase, the Navier-Stokes equations were integrated by the semi-implicit method for pressure-linked equations (SIMPLE) scheme of Patankar employing the staggered grid system. For the particle motion the Newtonian equations of motion of individual particles were integrated, where repulsive and damping forces for particle collision, the gravity force, and the drag force were taken into account. For particle contact, a nonlinear spring and dash pot model for both normal and tangential components was used. In order to get more realistic results, the model uses realistic pneumatic system and material values.     

Critical time step for DEM simulations of dynamic systems using a Hertzian contact model

The discrete element method (DEM) typically uses an explicit numerical integration scheme to solve the equations of motion. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time step. Currently, there are no comprehensive techniques for estimating appropriate DEM time steps when a nonlinear contact interaction is used. It is common practice to apply a large factor of safety to these estimates to ensure stability, which unnecessarily increases the computational cost of these simulations. This work introduces an alternative framework for selecting a stable time step for nonlinear contact laws, specifically for the Hertz-Mindlin contact law. This approach uses the fact that the discretised equations of motion take the form of a nonlinear map and can be analysed as such. Using this framework, we analyse the effects of both system damping and the initial relative velocity of collision on the critical time step for a Hertz-Mindlin contact event between spherical particles.     

Linear-frictional contact model for 3D discrete element simulations of granular systems

The linear-frictional contact model is the most commonly used contact mechanism for discrete element (DEM) simulations of granular materials. Linear springs with a frictional slider are used for modeling interactions in directions normal and tangential to the contact surface. Although the model is simple in two dimensions, its implementation in 3D faces certain subtle challenges, and the particle interactions that occur within a single time step require careful modeling with a robust algorithm. The paper details a three-dimensional algorithm that accounts for the changing direction of the tangential force within a time step, the transition from elastic to slip behavior within a time step, possible contact sliding during only part of a time step, and twirling and rotation of the tangential force during a time step. Without three of these adjustments, errors are introduced in the incremental stiffness of an assembly. Without the fourth adjustment, the resulting stress tensor is not only incorrect but it is also no longer a tensor. The algorithm also computes the work increments during a time step, both elastic and dissipative.     

Development of micromechanical models for granular media

Micromechanical analysis has the potential to resolve many of the deficiencies of constitutive equations of granular continua by incorporating information obtained from particle-scale measurements. The outstanding problem in applying micromechanics to granular media is the projection scheme to relate continuum variables to particle-scale variables. Within the confines of a projection scheme that assumes affine motion, contact laws based on binary interactions do not fully capture important instabilities. Specifically, these contact laws do not consider mesoscale mechanics related to particle group behaviour such as force chains commonly seen in granular media. The implications of this are discussed in this paper by comparison of two micromechanical constitutive models to particle data observed in computer simulations using the discrete element method (DEM). The first model, in which relative deformations between isolated particle pairs are projected from continuum strain, fails to deliver the observed behaviour. The second model accounts for the contact mechanics at the mesoscale (i.e. particle group behaviour) and, accordingly, involves a nonaffine projection scheme. In contrast with the first, the second model is shown to display strain softening behaviour related to dilatancy and produce realistic shear bands in finite element simulations of a biaxial test. Importantly, the evolution of microscale variables is correctly replicated. This paper is dedicated to Professor Ching S. Chang on the occasion of his 60th birthday.     

The discrete element method with deformable particles

This work presents a new original formulation of the discrete element method (DEM) with deformable cylindrical particles. Uniform stress and strain fields are assumed to be induced in the particles under the action of contact forces. Particle deformation obtained by strain integration is taken into account in the evaluation of interparticle contact forces. The deformability of a particle yields a nonlocal contact model, it leads to the formation of new contacts, it changes the distribution of contact forces in the particle assembly, and it affects the macroscopic response of the particulate material. A numerical algorithm for the deformable DEM (DDEM) has been developed and implemented in the DEM program DEMPack. The new formulation implies only small modifications of the standard DEM algorithm. The DDEM algorithm has been verified on simple examples of an unconfined uniaxial compression of a rectangular specimen discretized with regularly spaced equal bonded particles and a square specimen represented with an irregular configuration of nonuniform‐sized bonded particles. The numerical results have been verified by a comparison with equivalent finite element method results and available analytical solutions. The micro‐macro relationships for elastic parameters have been obtained. The results have proved to have enhanced the modeling capabilities of the DDEM with respect to the standard DEM.     

Study on a large-scale discrete element model for fine particles in a fluidized bed

The discrete element method (DEM) is widely used to comprehend complicated phenomena such as gas–solid flows. This is because the DEM enables us to investigate the characteristics of the granular flow at the particle level. The DEM is a Lagrangian approach where each individual particle is calculated based on Newton’s second law of motion. However, it is difficult to use the DEM to model industrial powder processes, where over a billion particles are dealt with, because the calculation cost becomes too expensive when the number of particles is huge. To solve this issue, we have developed a coarse grain model to simulate the non-cohesive particle behavior in large-scale powder systems. The coarse grain particle represents a group of original particles. Accordingly, the coarse grain model makes it possible to perform the simulations by using a smaller number of calculated particles than are physically present. As might be expected, handling of fine particles involving cohesive force is often required in industry. In the present study, we evolved the coarse grain model to simulate these fine particles. Numerical simulations were performed to show the adequacy of this model in a fluidized bed, which is a typical gas–solid flow situation. The results obtained from our model and for the original particle systems were compared in terms of the transient change of the bed height and pressure drop. The new model can simulate the original particle behavior accurately.     

Study of sedimentation of non-cohesive particles via CFD–DEM simulations

The sedimentation process of granular materials exists ubiquitously in nature and many fields which involve the solid–liquid separation. This paper employs the coupled computational fluid dynamics and discrete element method (CFD–DEM) to investigate the sedimentation process of non-cohesive particles, including the hindered settling stage and the deposition stage. Firstly, the coupled CFD–DEM model for sedimentation is validated by the hindered settling velocity at different solid volume concentrations of suspension \(\phi _{0} \), i.e., \(\phi _0 =\) 0.05–0.6. Two typical modes of sedimentation are also presented by the concentration profiles and the equal-concentration lines. Then, the comparisons between mono- and poly-dispersed particle system are detailed. In the sedimentation of the poly-dispersed particle system, the segregation phenomenon is simulated. Furthermore, this segregation effect reduces with the increase of the initial solid concentration of suspension. From the simulations, the contact force between every pair of particles can be obtained, hence we demonstrate the “effective stress principle” from the view of the particle contact force by giving the correspondence between the particle contact force and the “effective stress”, which is a critical concept of soil mechanics. Moreover, the deposition stage can be simulated by CFD–DEM method, therefore the solid concentrations of sediment bed \(\phi _{\mathrm{max}} \) on different conditions are studied. Based on the simulation results of \(\phi _{\mathrm{max}} \) and the theory of sedimentation, this paper also discusses a method to calculate the critical time when sedimentation ends of two typical modes of sedimentation.     

Particle screening phenomena in an oblique multi-level tumbling reservoir: a numerical study using discrete element simulation

Due to their wide usage in industrial and technological processes, granular materials have captured great interest in recent research. The related studies are often based on numerical simulations and it is challenging to investigate computational phenomena of granular systems. Particle screening is an essential technology of particle separation in many industrial fields. This paper presents a numerical model for studying the particle screening process using the discrete element method that considers the motion of each particle individually. Dynamical quantities like particle positions, velocities and orientations are tracked at each time step of the simulation. The particular problem of interest is the separation of round shape particles of different sizes using a rotating tumbling vertical cylinder while the particulate material is continuously fed into its interior. This rotating cylinder can be designed as a uniform or stepped multi level obliqued vertical vessel and is considered as a big reservoir for the mixture of particulate material. The finer particles usually fall through the sieve openings while the oversized particles are rebounded and ejected through outlets located around the machine body. Particle–particle and particle–boundary collisions will appear under the tumbling motion of the rotating structure. A penalty method, which employs spring-damper models, will be applied to calculate the normal and frictional forces. As a result of collisions, the particles will dissipate kinetic energy due to the normal and frictional contact losses. The particle distribution, sifting rate of the separated particles and the efficiency of the segregation process have been studied. It is recognized that the screening phenomenon is very sensitive to the machines geometrical parameters, i.e. plate inclinations, shaft eccentricities and aperture sizes in the sieving plates at different levels of the structure. The rotational speed of the machine and the feeding rate of the particles flow have also a great influence on the transportation and segregation rates of the particles. In an attempt to better understand the mechanism of the particle transport between the different layers of the sifting system, different computational studies for achieving optimal operation have been performed.     

A combined physical and DEM modelling approach to investigate particle shape effects on load movement in tumbling mills

The discrete element method (DEM) which is used to simulate granular flows often assumes spherical shape for particles. This assumption is legitimized by the added complexity of non-spherical shape representation, contact detection and computational cost. In this work, the difference between the dynamics of non-spherical and spherical particles was studied in detail by a combined physical and DEM modeling approach. An in-house developed DEM software called KMPC_DEM©, which was coded to handle non-spherical particles, was used to simulate the behavior of particles. To calibrate the model parameters, a model tumbling mill (100 cm diameter and 10.8 cm length) with one transparent end was used which made accurate photography possible. The tests were performed at filling of 20% and mill speed of 85% of critical speed with steel balls and wood cubes. In the simulation, each cubical particle was represented with clusters of spheres (with identical size) by particle packing algorithm for contact detection and contact-force calculation. Comparison of the simulation and experimental results showed that the difference between the measured and predicted impact toe, shoulder angle and bulk toe angle were 3, 4 and 5°, respectively. The significant change in the charge movement and structure on account of non-spherical particles was reflected in the amount of in-flight charge, and positions of shoulder, impact toe and bulk toe. It found that there was a 17% difference in the amount of in-flight of charge between cubical and spherical particles. The marked difference was attributed to higher interlocking of non-spherical particles in comparison to spherical balls. The results showed that cubical particles participated 5% more in the high energy impact action compared to that of the spherical particles. The simulation computation time increased by 35 times when the shape of particles changed from spherical to cubical.     

A Microstructure-based Simulation Environment on the Basis of an Interface Enhanced Particle Model

The present contribution introduces enhanced discrete element simulation methodologies (DEM) with a special focus on a microstructure-based model environment. Therewith, it is possible to represent the failure of cohesive granular materials like concrete, ceramics or marl in a qualitative as well as quantitative manner. Starting from a basic polygonal two-dimensional particle model for non-cohesive granular materials, more complex models for cohesive materials are obtained by inclusion of beam or interface elements between corresponding particles. In particular, we will introduce an interface enhanced DEM methodology where a standard ingredient of computational mechanics, namely interface elements, are combined with the particle methodology contained in the DEM. The last step in the series of increasing complexity is the realization of a microstructure-based simulation environment which utilizes the interface enhanced DEM methodology. With growing model complexity a wide variety of failure features of cohesive as well as non-cohesive geomaterials can be represented. Finally, the plan of the paper is enriched by the validation of the newly introduced and re-developed discrete models with regard to qualitative and quantitative aspects.     

Influence of process parameters on average particle speeds in a vibratory finisher

Vibratory finishing (VF) employs vibrationally-fluidized granular media to finish the surfaces of workpieces that are entrained in the flowing media. Its application has been based mostly on experience and trial-and-error due to the complexity of the granular material behavior. The present study used discrete element modeling (DEM) to investigate how the movement of a commercial two-dimensional tub finisher influenced the average particle speed of the media in a bed of smooth, steel, spherical particles, and thus the work that would be done on an entrained workpiece. The parameters governing the tub wall motion (frequency, in-plane amplitudes, and phases of vibration) and the coefficient of friction between the media and the wall were systematically varied in 71 three-dimensional DEM simulations. The average particle speed was affected mostly by the vertical amplitude of tub motion rather than by the frequency, and was mostly independent of other parameters of motion and of the wall friction. A strong relationship was found between the average particle speed and the work done by the wall per cycle of vibration. The normal force on the wall was also found to correlate strongly with the normal component of the wall velocity. Together, these relationships offer the potential to enable the analytical prediction of the average particle speed based on the motion parameters of the tub alone. The paper provides a set of practical guidelines for the control of the average particle speed in VF that are explained by the forces between the media and walls of the tub finisher.     

A saturated discrete particle model and characteristic-based SPH method in granular materials

Based on the discrete particle model for solid-phase deformation of granular materials consisting of dry particulate assemblages, a discrete particle–continuum model for modelling the coupled hydro-mechanical behaviour in saturated granular materials is developed. The motion of the interstitial fluid is described by two parallel continuum schemes governed by the averaged incompressible N–S equations and Darcy's law, respectively, where the latter one can be regarded as a degraded case of the former. Owing to the merits in both Lagrangian and mesh-free characters, the characteristic-based smoothed particle hydrodynamics (SPH) method is proposed in this paper for modelling pore fluid flows relative to the deformed solid phase that is modelled as packed assemblages of interacting discrete particles. It is assumed that the formulation is Lagrangian with the co-ordinate system transferring with the movement of the solid particles. The assumed continuous fluid field is discretized into a finite set of Lagrangian (material) points with their number equal to that of solid particles situated in the computational domain. An explicit meshless scheme for granular materials with interstitial water is formulated. Numerical results illustrate the capability and performance of the present model in modelling the fluid–solid interaction and deformation in granular materials saturated with water. Copyright © 2007 John Wiley & Sons, Ltd.     

Contact forces of polyhedral particles in discrete element method

A general contact force law for arbitrarily shaped bodies is presented. At first an advanced contact force law is derived from the well know Hertz contact law. The obtained formulation of the Hertz contact law can be applied to the contact of arbitrarily shaped bodies. In a second step this contact model is applied to the contacts among polyhedral particles. The results are compared to finite element simulations. The model is extended by terms for damping and friction. The behaviour of the damping and friction model are demonstrated with simple examples. The force law is then implemented in the discrete element method (DEM). The application of this DEM is demonstrated by a simulation of the particle movement in a mixer.     

Force propagation speed in a bed of particles due to an incident particle impact

The force propagation speed in granular matter is a very difficult property to be measured. A new technique has been developed to calculate the force propagation speed in granular matter based on measuring experimentally the contact time. The contact time for a particle hitting a bed of particles is estimated as the time taken for a particle to strike a bed of particles till the time of its ejection, and it is calculated using the discrete element method. The speed of force propagation in a bed of particles is estimated by plotting the dependence of the path length of the contact force on the contact time and finding the gradient of such dependence. Such approach leads to accurate results if the impact speed is below the yield velocity, i.e. no plastic deformations. It is found that the force propagation speed in spherical granular matter is proportional to the impact speed of the incident particle, which is different from force propagation in continuum matter. It is also found that the propagation speed is dependent on the material and diameters ratio of the interacting particles, but it is not dependent on the number of bed layers. The propagation speed in granular matter is normalized by dividing it by a reference propagation speed, i.e. the propagation speed at an impact speed of 1 m/s. It is found that the normalized propagation speed is independent of the material and diameter of the interacting particles, but it is logarithmically proportional to the impact speed. The proportionality constant is equal to 0.16, which can be taken as a universal constant for force propagation in spherical granular matter.     

A fast first order model of a rough annular shear cell using cellular automata

The motivation of the current work is to simulate granular phenomena in a rough annular shear cell without load using a physics-based cellular automata (CA) modeling approach. A simple yet powerful two-dimensional (2D) cellular automata model was developed to model granular flows inside a 2D annular shear cell from a tribological perspective. Physics-based equations developed from first-principles to model collisions have been adopted to form the cellular automata local rules of interaction. This combines the computational efficiency of CA modeling with the accuracy and generalization of first-principle physics modeling. The local flow properties—solid fraction, velocity and granular temperature profiles- were predicted from the CA simulation and compared to the granular shear cell experiments. The resulting steady-state CA model profiles show good agreement with the experimental results. The ability of the CA model to probe the effect of each granular flow property is demonstrated by performing parametric studies of the particle–particle coefficient of restitution and roughness factor.     

On the particle–particle contact effects on the hole cleaning process via a CFD–DEM model

ABSTRACT

The accurate and precise computational models in order to predict the hole cleaning process is one of the helpful assets in drilling industries. Besides the bulk properties such as the flow velocity, particles average size, cleaning fluid properties, etc., that will affect the cleaning process, there is an unanswered question about the microscopic properties of the particles, particularly those which determines the contact characteristics: Do those play a major role or not? The rudimentary answer is not. The first purpose of the present work is to answer this question via a developed computational fluid dynamics coupled with discrete element method (CFD–DEM) in which the six unknown rolling and sliding friction coefficients of particle–particle contact, particle–wall contact, and particle–drill contact are considered as the main microscopic properties of the contacts. The second purpose is to search for optimum values of these coefficients in order to calibrate the CFD–DEM model with the experimental data for a near horizontal well cleaning available in the literature. The verification of the calibrated CFD–DEM model is checked by simulation of the hole cleaning process for different inclination angles of the deviated well. The results indicate the pivotal role of the microscopic properties of the particles on the characteristics of the particle transport mechanism.     

DEM analysis of acoustic wake agglomeration for mono-sized microparticles in the presence of gravitational effects

The application of the discrete element method (DEM) to numerical simulation of the acoustic agglomeration of micron-sized mono-dispersed aerosol particles is demonstrated. The conventional DEM technique used in granular dynamics is modified for the simulation of the acoustically induced attractive motion of particles in incompressible fluid. The relationship between the acoustic wake and the gravity is investigated by simulating the binary interaction of two particles for a wide range of particles’ orientations to the sound direction. The main finding obtained by simulation results shows that agglomeration time is solely predefined by the acoustic wake effect independently of gravity. On the other hand, the gravitational motion is contributed by the sound, and an increase of settling velocity due to the sound of individual particles was found. Additionally, two particles surrounded by a system of other particles are illustrated by numerical results. It was also shown that isolated particles demonstrate an overestimated increase of the agglomeration time, which was essentially reduced by the contribution of other particles in the multi-particle system.