The Discrete Element Method Lecture 1: Fundamentals Carlos Labra International Center for Numerical Methods in Engineering Barcelona, Spain DEM, Barcelona 2010

“The term discrete element method (DEM) is a family of numerical methods for computing the motion of a large number of particles like molecules or grains.” Wikipedia DEM, Barcelona 2010

Overview DEM, Barcelona 2010 grains of sand SAG mill Conveyor Belts Sand sorting tower at a gravel extraction pit. DEM, Barcelona 2010

Overview DEM, Barcelona 2010 Operation of a jaw crusher Blasting Cohesionless Soil Slope Uniaxial compression test DEM, Barcelona 2010

Fundamentals DEM, Barcelona 2010

Materials structure Solid materials are characterized with a certain internal structure – in some materials, like rocks or sand, it can be seen without any instruments, in some cases the microstructure can be seen under microscope. Observations can be performed at atomistic scale. Crystals of gold Sandstone Steel microstructure Sand DEM, Barcelona 2010

Continuum vs. discrete material models Continuous models: Molecular structure, microstructure or granularity of materials disregarded; mathematical functions are continuous except possibly at a finite number of interior surfaces separating regions of continuity Discrete models: Discontinuities and discrete internal structure taken into account, material treated as discrete medium Discrete models DEM, Barcelona 2010

Numerical methods based on discrete models Physical model of discrete medium Discrete mathematical model Discrete model is obtained directly applying equations describing discrete medium Numerical methods based on discrete material models: Molecular dynamics Particle-in-cell method Cellular automata Lattice models Discrete element method DEM, Barcelona 2010

Numerical modelling Remark: It must be remembered that some particle methods, like SPH (Smoothed Particle Methods) are derived from continuous material models. Solution obtained by Particle Finite Element Method DEM, Barcelona 2010

Development of the discrete element method P.A. Cundall and Strack, University of Minnesota A discrete numerical model for granular assemblies, Géotechnique, 1979. O.R. Walton, LLNL Explicit particle dynamics for granular materials, Num. Meth. in Geomechanics, 1982. Other authors: G. Mustoe, J.P. Bardet, J.R. Williams Powders and Grains, 1st International Conference on Micromechanics of Granular Media 1st US Conference on Discrete Element Methods, 1989 Initial period, 1980s, small capabilities of computers limit practical applications of the discrete element method Fast development since late 1980s Growing interest in the recent years because of possible new applications, for instance, in the design of new materials DEM, Barcelona 2010

Most important centres developing the DEM Itasca Consulting Group, Inc., University of Minnesota, P. Cundall (commercial programs PFC2D, PFC3D) - Colorado School of Mines, Geomechanics Research Center, G.W. Mustoe - MIT Intelligent Engineering Systems Lab, J.R. Williams - Lawrence Livermore National Laboratory, O.R. Walton, R.L. Braun - Sandia National Laboratories, Geomechanics Department, D.S. Preece - University of Southern California, J.P. Bardet - University of Florida, Particle Engineering Research Center - University of Wales, Swansea, Department of Civil Engineering, A. Munjiza, D.R.J. Owen - Universität Stuttgart, Institute of Structural Mechanics, E.Ramm University of Grenoble, Discrete Element Group for Hazard Mitigation, France, F. Donze, YADE code - DEM Solutions, Scotland, J. Favier, EDEM - CIMNE, Barcelona, DEMpack software (DEM since 1999) DEM, Barcelona 2010

Basic assumptions of the discrete element method Material is represented by a large collection of rigid or deformable discrete elements interacting among one another with contact forces Discrete elements can be of arbitrary shape: cylinders (discs), spheres (advantage: efficient contact detection) ellipses, ellipsoides polygons, polyhedra other shapes Discs Ellipses Rombes DEM, Barcelona 2010

Equations of rigid body motion Discrete element motion is described by the rigid body dynamics equations Rigid body kinematics DEM, Barcelona 2010

Equations of rigid body motion The principle of virtual work – mass centre DEM, Barcelona 2010

Equations of rigid body motion Newton - Euler equations where − inertia tensor − resultant force − resultant moment DEM, Barcelona 2010

Equations of motion of a discrete element The principle of virtual work for a cylindrical or spherical discrete element: Newton - Euler equations: DEM, Barcelona 2010

Contact constraints Kuhn-Tucker conditions where For two spheres: DEM, Barcelona 2010

Treatment of collisions in the DEM Hard contact, also called hard particles, event driven approach contact time short, it is neglected instantaneous momentum change is assumed velocity after collision determined using the restitution coefficient in the normal direction r, the Coulomb friction coefficient μ , and the maximum tangential restitution β0 Soft contact, also called soft particles, time driven, molecular dynamics approach finite time of contact change of momentum determined by a potential or interaction law Remark: Hard contact algorithms are inefficient if the time between events (collisions) is very small, which can occur in densely packed particle models. Soft particle model is considered in the formulation presented. DEM, Barcelona 2010

Equations of motion of a discrete element Newton - Euler equations DEM, Barcelona 2010

Enforcement of contact constraints Lagrange multipliers method Penalty function – this method is adopted in the soft particle formulation of the discrete element method, a certain particle penetration is allowed − penalty parameter DEM, Barcelona 2010

Contact model Decomposition of the contact force – normal contact force – tangential contact force Pair of interacting particles DEM, Barcelona 2010

Frictional contact model Normal contact force Elastic part of the contact force - Elastic part - Contact damping Rheologic model - contact stiffness in the normal direction DEM, Barcelona 2010

Frictional contact model Hertz model for the elastic normal contact force Contact stiffness in the normal direction DEM, Barcelona 2010

Frictional contact model Normal contact force Contact damping Rheologic model – relative normal velocity at contact point , – damping coefficient , – critical damping for the system of two masses with a spring DEM, Barcelona 2010

Frictional contact model Tangential contact force – Coulomb friction model Relative tangential velocity Slip function Slip law Stick/slip conditions: Coulomb friction model DEM, Barcelona 2010

Frictional contact model Tangential contact force – Regularized Coulomb friction model Decomposition of the relative tangential velocity Slip law Slip function Stick/slip conditions: if if Regularized Coulomb friction model DEM, Barcelona 2010

Frictional contact model Tangential contact force – Numerical algorithm Given: - old tangential force: Fsold - normal force: Fn - relative velocity: Vrs Find: - new tangential velocity: Fsnew Trial friction force Stick/slip condition - stick - slip Regularized Coulomb friction model DEM, Barcelona 2010

Contact models with cohesion Discrete element models of rocks, cohesive soils and other cohesive materials require contact models with cohesion (tensile resistance) Cohesive bonds are applied to pairs of elements being in contact Cohesive bonds can be broken – initiation and propagation of cracks can be modelled After debonding frictional contact model is assumed DEM, Barcelona 2010

Elastic perfectly brittle contact model Normal contact force (elastic part) Failure (decohesion) criterion Fn kn – contact stiffness in the normal direction g – gap or penetration compression tension Elastic perfectly brittle model In the absence of cohesion (or after decohesion) no tensile interaction is allowed – bond strength in the normal direction DEM, Barcelona 2010

Elastic perfectly brittle contact model Tangential contact force (cohesive): Failure (decohesion) criterion: In the absence of cohesion: g < 0 us Fs Rs ks ks - contact stiffness in the tangential direction us - relative tangential displacement g ≥ 0 FS Rs - bond strength in the normal direction Elastic perfectly brittle model In the absence of cohesion it is either zero (for separattion) or friction force RS us ks g = 0 Coulomb friction g ≥ 0 DEM, Barcelona 2010

Other contact models with cohesion Elasto-plastic contact models with softening Elastic damage models Normal contact force Tangential contact force Elastic perfectly brittle model In the absence of cohesion it is either zero (for separattion) or friction force DEM, Barcelona 2010

Rotational interaction Interaction moment Relative Slip law Slip function Stick/slip conditions: if if Regularized Coulomb friction model DEM, Barcelona 2010

Rotational interaction Interaction moment Relative angular velocity Decomposition of the relative angular velocity – normal (torsional) component – tangential component DEM, Barcelona 2010

Rotational interaction without cohesion Torsion Regularized slip law Slip function Stick/slip conditions: t DEM, Barcelona 2010

Rotational interaction without cohesion Rotation about the tangential axis (rolling friction) Regularized slip law Slip function Stick/slip conditions: t DEM, Barcelona 2010

Rolling friction Similarly to interaction between particles, respective models for the interaction particle-plane can be defined Rolling resistance moment can be interpreted as an effect of the off-set of the normal reaction Rolling friction compensates the deficiency of the model using particles of ideal shape (spheres, cylinders) − rolling friction coefficient Friction , Fricción, frotamiento , rozamiento ; friction brake, freno de Prony; friction clutch, embrague de fricción; friction coefficient, coeficiente de rozamiento; friction disc, disco de fricción; friction head, pérdida de carga (hidráulica); frictionless, sin rozamiento; frictionless bearings, rodamientos sin rozamiento; friction of rolling or rolling, rozamiento de rodadura; friction of sliding, rozamiento de deslizamiento; friction (screw) press, prensa a fricción; friction shoe, patínde rozamiento, zapata; friction socket, cono de friccíón; air friction , frotamiento de¡ aire; angle of friction , ángulo de rozamiento; box of a friction coupling, manguito de fricción; magnetic friction, fricción magnética; skidding friction , rozarniento de derrape; skin friction , rozamiento superficial (aviación); sliding friction, rozamiento de deslizamiento. Friction bearing, cojinete de fricción , ( Ingeniería mecánica ) Cojinete enterizo que soporta y está en contacto con el extremo de un eje.Friction brake, freno de fricción , ( Ingeniería Mecánica ) Freno en el que la resistencia o fuerza de frenado se obtiene por rozamiento o fricción. Friction clutch, embrague de fricciónFriction coefficient, coeficiente de fricción , ( Mecánica ) See friction coefficientFriction damping, amortiguamiento por rozamiento, ( Mecánica ) La conversión de la energía vibracional mecánica de sólidos en energía calorífica producida por el desplazamiento de un elemento seco sobre otro elemento.Friction drive, transmisión por fricciónFriction factor, factor de fricción , ( Mecánica de los fluidos  ) Cualquiera de los números adimensionales utilizados para el estudio de la fricción de los fluidos en tuberías, que es igual al factor de fricción de Fanning multiplicado por cierta constante adimensional. Friction flow, flujo de fricción , ( Mecánica de los fluidos ) Flujo de un fluido en el cual una cantidad significativa de energía mecánica se disipa en forma de calor por la acción de la viscosidad.  Friction gear, rueda de fricción Friction head, carga de fricción, ( Mecánica de los fluidos ) Pérdida de carga hidrostática de un flujo, en una corriente o conducto, debido a perturbaciones friccionales establecidas por el fluido en movimiento y por la fricción intermoiecular.Friction horsepower, potencia absorbida por rozamientoFriction loss, pérdida por rozamientoFriction saw, sierra de fricciónFriction sawing ,serrado por fricciónFriction torque, par de rozamientoFrictional , A fricción, de rozamiento; - losses, pérdidas por rozamiento.Frictional grip, adherencia friccional, ( Mecánica ) Adherencia por rozamiento de una locomotora o adhesión que se produce entre las ruedas y las vías en una línea férrea.Frictional secondary flow, flujo friccional secundario , ( Mecánica de los fluidos ) See Secondary flow. Frictionless flow, flujo sin fricción , ( Mecánica de los fluidos  ) See Inviscid flow. FRICCION de rodamiento FRICCION ADHERENTE, DESLIZANTE Y MIXTA. TENSION TRASERA Y DELANTERA. DEM, Barcelona 2010

Equations of motion of a discrete element Newton - Euler equations background (global) damping DEM, Barcelona 2010

Integration methods for ordinary differential equations Implicit integration methods – dependent variables are computed in terms of unknown quantities, for equations of motion – new configuration is obtained from the equations taken at the new (unknown) configuration; implicit solution is almost always iterative (possible problems with convergence), better control of accuracy of solution, unconditional numerical stability, large integration steps can be used Explicit integration methods – direct computation of the dependent variables is made in terms of known quantities, for equations of motion – new configuration is obtained from the equation taken at the known (last) configuration Non-iterative solution (no problems with convergence), conditional numerical stability, large number of small steps, more efficient solution.

Integration methods – example Problem to be solved (initial value problem): Explicit integration scheme: forward Euler method Implicit integration scheme: backward Euler method

Integration methods for first order Explicit Implicit Hybrid Forward Euler Backward Euler Milnov Runge-Kutta Trapezoidal Numerov Merson Simpson Gears Adams-Bashfort Adams-Moulton Others Others Linninger Willoughby Others

Integration methods for second order Numerical methods have been developed to deal with second order ordinary differential equations directly. Explicit: Implicit: Explicit central difference method Implicit central difference method Verlet method Newmark method Velocity Verlet method Numerov (Cowell) method (leapfrog method) Others Runge-Kutta method for 2nd order others Integration algorithm typically used in the DEM

Integration of DEM equations of motion Explicit leapfrog (velocity Verlet, variant of CD) method Equations of motion for the known configuration at tn Determination of the new configuration at tn+1 Translational motion Remark: Velocities are shifted by with respect to displacements Δt/2

Integration of DEM equations of motion Explicit leapfrog (velocity Verlet, variant of CD) method Determination of the new configuration at tn+1 Rotational motion Remarks: Angular velocity in 3D is not a derivative of any vector – cannot be integrated, the vector of small incremental rotations is evaluated only. For spherical discrete elements the rotational configuration need not be evaluated If necessary, the rotational configuration can be determined

Stability of the time integration scheme Explicit time integration schemes are conditionally stable Time step length Δt must be smaller than a certain critical time step Δtcr For the leapfrog algorithm: − highest natural frequency of the whole discrete system N - number of discrete elements DEM, Barcelona 2010

Stability of the time integration scheme Natural frequencies for the i-th discrete element can be determined by solution of the following eigenproblem 2D Example DEM, Barcelona 2010

Stability of the time integration scheme Single element contacting with a rigid plane Usually in DEM: DEM, Barcelona 2010

Background (global) damping Viscous damping Non-Viscous damping (dissipation does not depend on velocity) DEM, Barcelona 2010

Equations of motion of a discrete element DEM equations – system of ordinary differential equations Solution – integration with respect to time Integration methods: Explicit, Implicit, Hybrid (combine explicit and implicit steps) DEM, Barcelona 2010

Stability of the time integration scheme Single element contacting with a rigid plane Influence of the damping damping given by a fraction of the critical damping for the frequency DEM, Barcelona 2010

Influence of damping – numerical test Model details: Gravitational load Normal interaction only (friction not considered) Two cases: without damping with damping (viscous) Conclusions Without damping – no energy dissipation in the simulated process Damping allows us to simulate a quasistatic process No damping Damped

Rotational motion Rotational degrees of freedom test t = 0.0054 s

Search of contacts DEM, Barcelona 2010

Search of contacts Many searches during the simulation High computational cost involved Different techniques/algorithms can be selected DEM, Barcelona 2010

Search of contacts Algorithms: Brute force DEM, Barcelona 2010

Search of contacts Algorithms: Brute force Quadtree / Octtree DEM, Barcelona 2010

Search of contacts Algorithms: Brute force Quadtree / Octtree KdTree DEM, Barcelona 2010

Search of contacts Algorithms: Brute force Quadtree / Octtree KdTree Bins (Hash) DEM, Barcelona 2010

Particles generation Katalin Bagi Hungarian Academy of Sciences DEM, Barcelona 2010

Particles generation Assembly preparation Methods to generate random dense arrangements of particles: Dynamic techniques Follow the particles motions through several time steps Constructive techniques Assembly preparation with purely geometrical calculations Application experiences DEM, Barcelona 2010

Particles generation Dynamic techniques Isotropic compression Particle expansion Gravitational deposition Multi-layer Undercompaction Method Collective rearrangement techniques Optimization scheme … DEM, Barcelona 2010

Particles generation Much faster!!! Constructive techniques Regular arrangements (WHY NOT USE THEM?) Sequential inhibition model Sedimentation techniques Closed front technique Inwards packing algorithm … Much faster!!! DEM, Barcelona 2010

Dynamic techniques Isotropic compression Particle expansion DEM, Barcelona 2010

Dynamic techniques Gravitational deposition Multi-layer undercompaction method DEM, Barcelona 2010

Dynamic techniques Collective rearrangement techniques Optimization scheme (Labra & Oñate 2009) DEM, Barcelona 2010

Optimization scheme Dense random particle configurations obtained by minimization of porosity of a loose random collection of spheres (Labra & Oñate 2009). Unstructured irregular mesh Initial particle configuration (overlapping is eliminated during the optimization) Final particle configuration DEM, Barcelona 2010

Optimization scheme Objective function where: Constraints DEM, Barcelona 2010

Optimization scheme Generation of a particle assembly using the optimization scheme DEM, Barcelona 2010

Constructive techniques Regular arrangements Very easy to create!! Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure DEM, Barcelona 2010

Constructive techniques Sequential inhibition model (PFC2D) Very easy to create!! Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure DEM, Barcelona 2010

Constructive techniques Sedimentation techniques (Jodrey & Tory 1985, Bagi 1993, Feng & Owen 2003) Stable position of a grain: Very easy to create!! Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure DEM, Barcelona 2010

Constructive techniques Closed front techniques (Feng & Owen, 2003) Very easy to create!! Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure DEM, Barcelona 2010

Constructive techniques The inwards packing method (Bagi 2005) Initial front Interior Very easy to create!! Unreliable for the simulation of the mechanical behavior of materials with random internal microstructure DEM, Barcelona 2010

Characterization of mesh/assembly Contact graph Average radius Coordination number Void ratio Fabric tensor DEM, Barcelona 2010

Characterization of mesh/assembly Contact bond graphs Contact bond graph represents connections in the particle assembly - visual confirmation of irregularity and compactness DEM assembly Contact bond graph DEM, Barcelona 2010

Characterization of mesh/assembly Contact bond graphs Detail of the DEM assembly and the contact bond graph DEM assembly DEM, Barcelona 2010

Characterization of mesh/assembly Radius: Average value & distribution DEM, Barcelona 2010

Characterization of mesh/assembly Coordination number Average number of contacts per particle Void ratio / porosity The volume of the voids divided by the total volume of the domain Constructive algorithms ~ 3.5 – 4.5 (2D) High density packing ~ 5.0 – 5.5 (2D) Constructive algorithms ~ 0.5 – 0.7 (2D) High density packing ~ 0.8 – 0.9 (2D) DEM, Barcelona 2010

Characterization of mesh/assembly Fabric tensor The distribution of contact orientations in the assembly. Based on the unit vectors of the contact directions Isotropic anisotropic assembly assembly The sum of its eigenvalues always equals to 1, and the difference between them reflects the strength of anisotropy (if the two eigenvalues are equal, the contact distribution is perfectly isotropic) DEM, Barcelona 2010

Discrete – continuum relationship DEM, Barcelona 2010

Discrete-continuum relationships Definition of DEM parameters Estimation of DEM parameters based on macro behaviour Continuum equivalence Stress-strain relationships DEM, Barcelona 2010

Relationship micro – macro parameters ?   Micromechanical constitutive laws Macroscopic stress-strain relationships DEM, Barcelona 2010

Relationship micro – macro parameters Simulation of standard laboratory tests (unconfined compression, Brazilian test, direct shear test) using a discrete (micromechanical) model and determination of macroscopic properties corresponding to a micro parameters used in a discrete model Homogenization and averaging procedures ? DEM, Barcelona 2010

Relationship micro – macro parameters Microscopic level Constitutive parameters kn – contact stiffness in normal direction ks – tangential contact stiffness Rn – normal contact strength Rs – tangential contact strength μ – Coulomb friction coefficient αt – translational damping coefficient αr – rotational damping coefficient State variables g – penetration/gap us – relative tangential displacement Fn – normal contact force Fs – tangential contact force Geometrical and physical parameters r – average element radius ρ – density others … Macroscopic level Constitutive parameters E – Young’s modulus ν – Poisson ratio σc – compressive strength σt – tensile strength c – viscosity State variables ε – strain σ – stress Geometrical and physical parameters L – sample size n – porosity ρ – effective density others … DEM, Barcelona 2010

Relationship micro – macro parameters Dimensionless numbers Mesh influence should be considered!! f(nc,ϕ,…) : function of the mesh characterization parameters DEM, Barcelona 2010

Relationship micro – macro parameters Mesh influence Average radius Coordination number Fabric tensor Poisson ratio (Best fit approach) DEM, Barcelona 2010

Relationship micro – macro parameters Unconfined compression test DEM, Barcelona 2010

Relationship micro – macro parameters Brazilian test DEM, Barcelona 2010

Discrete - continuum relationships Stress and strain Stress Forces Relative displacement Strain Averaging Contact constitutive relation Kinematic localisation assumption DEM, Barcelona 2010

Discrete - continuum relationships Stress Macroscopic stress is generated by contact forces and configuration Average stress i j l f DEM, Barcelona 2010

Discrete - continuum relationships Transition from the microscopic discrete to macroscopic continuum description by averaging over RVE Around each point we define a representative volume element (RVE) Averaging of the quantity Q over RVE The average is assigned to the point , if Q constant for the particle: DEM, Barcelona 2010

Discrete-continuum relationships Strain (equivalent-continuum strain) Can’t be obtained in a direct way Different aproachs Kruyt & Rothemburg (1996) Bagi (1993) (1996) Kuhn (1999) Cambou et al (2000) Kruyt (2003) Differences between the versions Differences in the applied equivalent continuum Differences in how to define a translational field in them DEM, Barcelona 2010

Lecture 2: Application and advanced topics DEM, Barcelona 2010