1
Calibration of DEM Parameters

Corné Coetzee1 and André Katterfeld2

1Stellenbosch University, Department of Mechanical and Mechatronic Engineering, Granular Materials Research Group, Joubert Street, Stellenbosch, 7600, South Africa

2Otto von Guericke University Magdeburg, Chair of Material Handling, Institute of Logistics and Material Handling Systems, Universitätsplatz 2, 39106 Magdeburg, Germany

Abbreviations and Acronyms

AoR

Angle of Repose

BCA

Bulk Calibration Approach

CFD

Computational Fluid Dynamics

CoR

Coefficient of Restitution

CPU

Central Processing Unit

DEM

Discrete Element Method/Discrete Element Modelling

DMA

Direct Measurement Approach

FEM

Finite Element Method

GPR

Gaussian Process Regression

GPU

Graphical Processing Unit

GSMC

Generalised Surrogate Modelling-based Calibration

MPM

Material Point Method

PSD

Particle Size Distribution

UCT

Uniaxial Compression Test

1.1 Introduction

Mesh-based simulation techniques in continuum mechanics, such as the Finite Element Method (FEM) or Computational Fluid Dynamics (CFD), require the body or volume of interest to be discretised by a mesh. The mesh closely represents the real object with small differences or idealisations, mostly directly proportional to the size of the mesh. These approaches show the convergence of the results with mesh refinement. However, Lagrangian approaches, such as FEM, suffer from severe mesh distortion when the body experiences large deformation. In these cases, the solution can become unstable, and the results inaccurate. The so-called meshless continuum-based methods such as the Material Point Method (MPM) are capable of modelling larger deformation [1]. However, these methods still assume a continuum body and might still rely on a non-deforming mesh. As a result, these methods cannot model the discrete nature of granular materials such as mixing and segregation or single particles separated from the bulk of the material in a screening process, for example.

The Discrete Element Method (DEM) was developed by Cundall and Strack [2] in the 1970s in order to solve problems associated with rock mechanics. The potential of DEM was quickly recognised for research purposes in a number of areas such as physics, nanotechnology, chemical engineering, and materials handling. DEM is completely meshfree (or meshless) and can easily model the large deformation typically associated with the handling and flow of bulk granular materials (particulate matter). DEM can also model the discrete nature of the individual particles, during screening for example.

To use DEM to analyse the behaviour of bulk materials, for example, in conveyor systems, during transportation and storage and flow through processing equipment, an accurate simulation model should be generated. A DEM model should define the geometric properties of the particles, such as the size and shape distributions, as well as the geometry of any structure or equipment. The interaction or contact properties (particle-particle and particle-wall) also need to be defined, which is a major component of the modelling process. As in all the numerical simulation methods, the experience (know-how) and sometimes the art applied by the user are critical to define and create a model capable of producing the most accurate simulation results in the shortest possible time frame.

In a DEM model, the discretisation of the bulk material is directly related to the size of the considered particles, which have a significant influence on the behaviour of the modelled material. DEM is also computationally intensive, and for this reason, most practical models are simplified in terms of particle size, shape, and contact properties. This idealisation is the reason why established bulk material properties (e.g. the angle of internal friction and the angle of repose [AoR]) cannot be directly used as input parameters. Hence, it is necessary to reverse engineer the parameters by comparing the modelled bulk behaviour to that observed in the experimental tests. This procedure is called the 'calibration of DEM parameters' and is the key to produce realistic simulation results.

1.2 Basic DEM Theory

A typical DEM model consists of particles and walls. The particles can make contact with one another and with walls, but wall-wall contact is usually undefined. The particles represent the granular material and can in theory take on any shape and size. However, in practice, spherical and multi-sphere particles are the most commonly used. Walls are used to define all the structures with which the particles can interact, such as the walls of equipment and machines. Contact models are used to calculate the contact forces and moments based on the contact kinematics.

Figure 1.1 DEM computation cycle.

1.2.1 The Calculation Cycle

A DEM calculation cycle consists of four steps as illustrated in Figure 1.1, namely (1) contact detection, (2) contact resolution, (3) solving the equation of motion, and (4) updating of the particle velocity and position.

The contacting pieces (particles and walls) are allowed to overlap, and in the first step of the computation cycle, all particle-particle and particle-wall contacts are identified. The overlap is assumed to be relatively small compared to the particle size. Although contact detection happens automatically, without any user intervention, the particle shape selected by the user has a significant effect on the efficiency of this step. Spherical particles are computationally the best, followed by multi-sphere particles and more complex shape representations such as super-quadrics and polyhedra.

In the second step, the force and moment vectors are calculated at each contact, based on one of a number of available contact models selected by the user. The contact force and moment are dependent on the relative contact displacement or overlap (elastic force), velocity (viscous force) and the contact parameter values specified by the user.

In the third step, the resultant force and moment acting on each particle are calculated. This includes the forces, Fc, and moments, Mc, due to the contacts and the body force due to gravity Fg. Based on the particle's mass, m, and moment of inertia, Ig, the translational acceleration, , and the rotational acceleration, , can be calculated using the equations of motion,

In the fourth and last step, the particle velocity (translational and rotational ) is first updated using an explicit time integration scheme,

where ?t is the timestep. This is followed by the particle's position and orientation update,

The explicit time integration scheme is conditionally stable and requires a timestep smaller than the critical timestep. Using the analogy of a single degree-of-freedom mass-spring system, it can be shown that the stable timestep is proportional to the particle mass and inversely proportional to the effective contact stiffness. For slight variations in the explicit time integration scheme and the calculation of the timestep, see O'Sullivan [3] for example.

This step concludes the basic time cycling sequence, after which the time is incremented, followed by a new contact detection step.

1.2.2 Contact Models

A contact model defined at each contact describes the force-displacement relation. There are a number of contact models from which the user can select. Figure 1.2 shows the basic elements of a contact model, namely springs, dashpots, frictional sliders, and tension elements. A combination of these elements act in each of the normal and shear (tangential) directions.

The spring elements define the elastic force component and can have linear behaviour (as in the linear model) or non-linear behaviour (as in the Hertz-Mindlin model). The viscous dashpots dissipate energy, and the frictional slider allows for Coulomb-like frictional behaviour in the shear direction. For the modelling of spheres, rolling resistance models are very important; however, they are not visualised in Figure 1.2. Cohesive behaviour can be modelled by allowing tensile forces in the normal direction. The details of the different contact models are not presented here, and the interested reader should consult other sources such as O'Sullivan [3] and Thornton [4].

Figure 1.2 Typical elements of a contact model.

At each timestep, the relative motion between two contacting pieces is used, in combination with the elements defined above, to update the contact force components. The user should specify the parameter values for the spring stiffness, for example the damping (dashpot) constants and the coefficient of friction. Obtaining a set...