Applications of the Discrete Element Method in Mechanical Engineering

Multibody Syst Dyn (2007) 18: 81–94DOI 10.1007/s11044-007-9066-2

Applications of the discrete element methodin mechanical engineering

Florian Fleissner · Timo Gaugele · Peter Eberhard

Received: 28 February 2007 / Accepted: 9 May 2007 /Published online: 1 June 2007© Springer Science+Business Media B.V. 2007

Abstract Compared to other fields of engineering, in mechanical engineering, the DiscreteElement Method (DEM) is not yet a well known method. Nevertheless, there is a varietyof simulation problems where the method has obvious advantages due to its meshless na-ture. For problems where several free bodies can collide and break after having been largelydeformed, the DEM is the method of choice. Neighborhood search and collision detectionbetween bodies as well as the separation of large solids into smaller particles are naturallyincorporated in the method. The main DEM algorithm consists of a relatively simple loopthat basically contains the three substeps contact detection, force computation and integra-tion. However, there exists a large variety of different algorithms to choose the substeps tocompose the optimal method for a given problem. In this contribution, we describe the dy-namics of particle systems together with appropriate numerical integration schemes and givean overview over different types of particle interactions that can be composed to adapt themethod to fit to a given simulation problem. Surface triangulations are used to model com-plicated, non-convex bodies in contact with particle systems. The capabilities of the methodare finally demonstrated by means of application examples.

Keywords Discrete element method · Numerical integration · Quaternions · Surfacetriangulation · Multibody system coupling

Commemorative Contribution.

F. Fleissner (�) · T. Gaugele · P. EberhardInstitute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9,70569 Stuttgart, Germanye-mail: [email protected]

T. Gaugelee-mail: [email protected]

P. Eberharde-mail: [email protected]

82 F. Fleissner et al.

1 Introduction

The Discrete Element Method (DEM) is a well established tool for physicists and engineersin geophysics and in mining, civil and chemical engineering, where it is used to simulateparticle flows of granules and powders and to investigate shear effects and the nature ofgranular packings. In contrast to most other methods from the growing group of meshlessmethods, which are mainly designed to simulate continuum effects described by partial dif-ferential equations, the DEM accounts for the simulation of inter-particle contacts. However,a hybrid application for the simulation of continua in contact seems to be quite promising.In mechanical engineering the method is hardly known as many typical engineering prob-lems can be solved with the Finite Element Method (FEM). Even though the FEM may besuperior for most problems where small elastic deformations are in the focus or for the in-vestigation of mode shapes of structural oscillations, it is worth to have a look at the DEM’spower and flexibility when it comes to breakage, rupture and large deformations, togetherwith contacts of multiple bodies. In engines and gears, abrasion yields small particles, some-times mixed with cooling- or lubrication-liquid that can cause clamping of mechanisms orplugging in pipes and cavities and can thus have a significant impact on the performanceof mechanical systems in terms of reliability and durability. In production engineering, e.g.,in the simulation of cutting processes, the DEM is used to observe the effects of processparameters on the plastic deformation and fragmentation of material to optimize the processoutcome. Rapidly deformed membrane structures which are permanently subject to self-contacts can also be efficiently simulated with the DEM, especially when ruptures are partof the systems functionality. All of these examples yield the same dynamic equations ofmotion (see Sect. 2), and can thus be simulated using the same numerical integration ap-proaches which we present in Sect. 3. For modeling the different large scale behavior that isnecessary to adapt the method to a variety of simulation problems, several particle interac-tions, transient or persistent, can be applied. Some important aspects are described in Sect. 4.Finally, in Sect. 5, we present some recent DEM simulation examples. This paper presents adigest of our work in progress and it is meant to give an overview about the potential of theDEM in mechanical engineering. The determination of physical parameters, e.g., by meansof standard material tests is still an ongoing work.

2 Dynamics of particle systems

Particle systems consist of a set of free, not necessarily rigid bodies which interact mostly interms of unilateral forces due to contacts. The Discrete Element Method [1, 2] usually em-ploys a description of the dynamics on a force-acceleration-level, which enables, in contrastto an impulse-velocity based modeling, a mixture of various different contact—or, moregeneral, interaction-force laws.

The state of a particle i can be described by its position r i , its velocity vi , its rotationunit quaternion q i = [qi

0 qi1 qi

2 qi3]T, ‖q i‖ = 1 (Euler–Rodriguez parameters) and its angular

velocity ωi . Changes of the state variables are induced by the forces f i and the torques li

acting on the particle. The particle’s dynamics is described by the Newton and the Eulerequations [3]

mi v̇i = f i , (1)

I i · ω̇i + ωi × I i · ωi = li . (2)

Applications of the discrete element method in mechanical engineering 83

In a body fixed principal frame, the inertia tensor I i simplifies to the constant diagonal tensor

I i =⎡⎢⎣

I ix 0 0

0 I iy 0

0 0 I iz

⎤⎥⎦ . (3)

Written in the principal frame, (2) are called the Dynamic Euler Equations which can besolved with little effort for the angular acceleration

ω̇i =

⎡⎢⎢⎢⎢⎣

lix

I ix

+ ωiyω

iz

I iy−I i

z

I ix

liy

I iy

+ ωixω

iz

I iz−I i

x

I iy

liz

I iz

+ ωixω

iy

I ix−I i

y

I ix

⎤⎥⎥⎥⎥⎦

, (4)

as the constant diagonal inertia tensor is simple to invert. The time derivative of the orienta-tion quaternion can be expressed using the angular velocity, expanded to a pure quaternion�i = [0 ωi], and the orthogonal quaternion matrix [4]

Q(q) =

⎡⎢⎢⎣

q0 −q1 −q2 −q3

q1 q0 −q3 q2

q2 q3 q0 −q1

q3 −q2 q1 q0

⎤⎥⎥⎦ , (5)

which is used to express the quaternion product in matrix-vector notation, as

q̇ i = 1

2Q

(q i

) · �i . (6)

In the same manner, the second derivative of the orientation quaternion can be written as

q̈ i = 1

2

(Q

(q̇ i

) · �i + Q(q i

) · �̇i). (7)

Using the quaternion approach, rotations can be expressed without the need to care for sin-gularities of Jacobian matrices as they occur if Euler or Cardan angles are used [5]. Based ona set of initial conditions, this set of equations can be solved by different kinds of integrationschemes.

3 Numerical integration for particle dynamics

In contrast to celestial mechanics or potential based molecular dynamics, particle systemsare usually characterized as dissipative and non-smooth, mainly due to viscous dampingforces which may induce jumps in the forces acting between colliding particles. Therefore,particle systems require specialized numerical algorithms.

The non-smoothness of the contact forces prohibits the use of higher order integrationschemes, such as Radau formulae [6, 7] or Backward Differencing Formulae (BDF) [8],as they are too sensitive to jumps of the right-hand sides. Moreover, there is another reasonwhy large time step sizes are undesired anyway. As it is important to ensure that all collisionsbetween particles are resolved, particles must not move further than a certain distance withina timestep, which is limited by the particles’ extension and velocity. This imposes a statedependent upper limit to the time step size.


3.1 Verlet integrator and its derivates

The well established Verlet integrator [9] and its derivates are widely used in the contextof DEM simulations. However, one of its characteristic features, the reversibility, vanishesif it is applied to dissipative dynamical systems. If reversibility is not required anyway,the Verlet scheme is less useful for particle dynamics than other integration schemes withbetter stability properties. That it is used anyway for the simulation of particle systems maybe due to its simplicity and the fact that DEM evolved from potential driven moleculardynamics, where the method is well suited. For a quick start or the simulation of smallsystems, however, the method proves to be a good choice. The basic Verlet scheme is directlybased on the second order equations of motion. It is mainly applied for the integration oftranslational motion. A particle position at the end of a time step is calculated from theposition at the end of the previous two time steps and the force induced acceleration at theend of the last time step as

r(t + �t) = 2r(t) − r(t − �t) + �t2a(t). (8)

The velocity is then evaluated from the central finite difference

v(t) = r(t + �t) − r(t − �t)

2�t. (9)

There are different derivates of the Verlet scheme, namely the velocity Verlet and the leap-frog algorithm [10] which differ in the points in time where positions and velocities areevaluated. Unfortunately, all Verlet derivates are only conditionally stable which makes thechoice of appropriate time step sizes a difficult and tedious task.

3.2 Newmark-β method

A group of integration schemes we found to be much better suited than the Verlet integratorfor application in dissipative DEM simulations is the group of schemes of the Newmark-βmethod [11]. The bandwidth of the integrator equations

r(t + �t) = r(t) + �tv(t) +(

1

2− β

)�t2a(t) + �t2βa(t + �t), (10)

v(t + �t) = v(t) + (1 − γ )�ta(t) + γ�ta(t + �t), (11)

reaches from fully explicit (β = γ = 0) to fully implicit schemes (β = 1/2, γ = 1), depend-ing on the choice of the two parameters β and γ . The choice of parameters also influencesthe amount of numerical damping as well as the approximation error which can be of secondor third order. The parameters β = 1/4 and γ = 1/2 yield an unconditionally stable implicitscheme with third order accuracy and negligible numerical damping.

3.3 Integration of rotational dynamics

The two integration schemes we presented in (8), (9) or (10), (11) can both be modified tointegrate rotational dynamics with quaternions. Omelyan [4] proposed an explicit velocityVerlet-like scheme that requires, even though it is fully explicit, an iteration to calculatea consistent set of angular velocities and angular accelerations from the non-linear Euler


equations. Moreover, to fulfill the unit-quaternion constraint, in every integration step a pro-jection on the constraint manifold is necessary. We modified Omelyan’s integration schemeto be used with the Newmark-β scheme, yielding the integrator equations

q(t + �t) = q(t) + �t q̇(t) +(

1

2− β

)�t2q̈(t) + �t2βq̈(t + �t) + ∂c

∂qλ, (12)

c(q) = q · q − 1 (13)

where c(q) is the unity constraint that is imposed via the Lagrangian parameter λ whichis calculated explicitly. For the preferable choice of β = 1/4 and γ = 1/2, these equationsbecome implicit, thus requiring an iterative solution which we obtain via a simple fixedpoint iteration in an outer loop together with (4), (6) and (7), which are solved also viafixed point iteration in an inner loop, to yield consistent values of ω and ω̇. After conver-gence of both iterations is reached, the newly calculated quaternion satisfies the constraintc(q(t + �t)) = 0. Different evaluation points of the constraint gradient in terms of energyconservation are compared in [4], namely ∂c(q(t + �t))/∂q and ∂c(q(t))/∂q . As we aredealing with dissipative systems, we found the former, a simple scaling of the quaternion, tobe sufficient as the general existence of a solution for the latter is not guaranteed.

3.4 Step size control

Up to now, we treated the integration time step size as constant. However, to speed upsimulations it is desirable to have a robust step size control. In [12] a step size controlalgorithm is proposed for use with an implicit second order scheme. Unfortunately, the inte-gration scheme features strong numerical damping. As it is very similar to the Newmark-βscheme, we successfully applied the step size control part with our Newmark-integrators fortranslation and rotation. This simple and robust step size control algorithm uses the rate ofconvergence of the fixed point iteration as controller input. If for a given tolerance a largenumber of iteration cycles is needed to reach convergence, the time step size is reduced andotherwise increased if convergence is reached quickly.

4 Particle interactions

4.1 Normal contact

The interactions between particles in a DEM simulation are treated with a unilateral penaltyforce approach. Contacting particles are temporarily connected with spring-damper systemsto exert forces that prevent the particles from further penetrations. For spherical particles,such as granules or pebbles, the Hertz contact law

Kij = 4

3π(hi + hj )

[RiRj

Ri + Rj

] 12

, (14)

hl = 1 − ν2l

πEl

, l = i, j, (15)

Fij = Kij δ32ij + Dδ̇ij (16)


Fig. 1 Schematic representation of contact situations in sphere-surface contact

with the radii Rl , Young’s moduli El and Poisson ratios νl of the spheres i and j and thedamping parameter D is used as a physically motivated non-linear penalty force law thatyields the contact force Fij based on the penetration depth δij [13]. For more complexgeometries, such as polyhedra, there exist no analytical force laws. For still being able toprevent particle penetrations, stiff linear springs are used to model quasi-rigid behavior [14,15].

4.2 Triangulated surfaces

In our simulations, we focus on spherical particles. More complex bodies can be composedof elastically bonded spheres if rigidity is not required. As particles usually interact withsome sort of bounding geometry or other more complex large scale bodies, a force law isrequired for the contact between these spherical particles and planar surfaces. According tothe Hertz law, a contact between a sphere and a rigid wall is equivalent to the constellationwhere two identical spheres collide. As both spheres are identical, (14) reduces to

Kij = E√

2R

3(1 − ν2). (17)

A simple, yet flexible way to resolve complex bounding geometries is the approximationwith triangular surface meshes. For implementation reasons, it is desirable to treat surfacetriangles as individual particles in terms of collision detection and data storage. Especially, inparallel implementations where the particles are distributed to processors, it is advantageousto divide the triangle mesh up into independent triangle-particles. The emerging contact sit-uations become, therefore, slightly more complicated as the contact region may be anythingfrom non-convex over planar to convex where multiple triangles can be in contact with thesame sphere, see Fig. 1.

For the calculation of the individual contact forces, we take again Hertz’s law to modelthe contact between the sphere and the triangle plane. However, in order to obtain a consis-tent model, it is important, at least for the planar contact case, that the sum of the contactforces of the individual contacts matches the contact force that would emerge from a con-tact between a sphere and a single triangle. To ensure this, the contact force of the individualcontacts is multiplied with the ratio between the surface of the intersection circle Ac betweensphere and plane and the surface of the overlap between the intersection circle and the trian-gle Ao, see Fig. 2. Compared to the evaluation of a simple sphere-sphere Hertz-contact, the


Fig. 2 Overlap regions of the triangle-sphere contact

calculation of the overlap regions between the triangles and the intersection circle of sphereand plane is not trivial, as there are in total eight possible overlap cases.

We calculate the contact force for an overlap between a sphere and the triangles of aoverlapping set of triangles T from the individual contact forces as

F i =∑j∈T

Aoij

Acij

(Kδ

32ij + Dδ̇ij

)nj , (18)

with the normals ni of the triangles. If all triangles in T are coplanar, i.e.

Aci=

∑j∈T

Aoij, Acij

= Acik= Aci

, nj = nk, ∀j, k ∈ T (19)

it is obvious that the total contact force is equal to the force from a contact with a singletriangle that is fully overlapped, i.e., with a circular intersection region between sphere andtriangle.

4.3 Friction

In most particle systems, friction plays an important role as it can, e.g., lead to plugging inpipe flows even for almost ideally spherical particles. The appropriate friction model mustbe chosen with respect to the application. In highly dynamic systems, slipping friction isdominant and it is often legitimate to neglect sticking friction. In quasi-static or static cases,e.g., the simulation of particle piles, sticking friction is usually not negligible.

Pure slipping friction like

F slip = −μ‖F n‖ v

‖v‖ , (20)

see [16], with normal force F n, relative velocity v and friction coefficient μ is, however,hard to simulate. This is due to the jump at zero velocity which can cause numerical insta-bility as an infinitesimal relative velocity v leads to large forces. Therefore, the modifiedslipping friction force

F slip = −μ‖F n‖ v

‖v‖ tanh(k‖v‖) (21)


is used that smoothes the jump at zero velocity. The parameter k is a shape parameter thatserves to tune the slope of the slip friction force close to zero velocity. It is important tomention, that for explicit integrators such as the Verlet integrator there exists a stability limitthat depends on k.

Modeling sticking friction is more complicated. For a force-acceleration based modeling,Cundall and Strack [2] proposed a penalty sticking friction model that inserts a tangentialspring-damper system between the contacting bodies at the initial contact point. Drillingfriction can be modeled accordingly.

For modeling friction between spheres and triangles, we expand our approach for thenormal contact in (18) by an additional friction term F fij

to

F i =∑j∈T

Aoij

Acij

[(Kδ

32 + Dδ̇

)nj + F fij

]. (22)

4.4 Macro particles

For particles being coupled with a multibody simulation it is of great interest to allow thesimulation boundaries to move and have their own dynamics. Therefore, we define rigidmacro particles that are represented by a triangulated surface geometry. All contact forcesand torques between spherical particles and the macro particle’s triangles are accumulatedwith respect to the macro particle’s center of gravity. Within the particle simulator, the macroparticle’s motion is then integrated by means of (2) to (7) but it is also possible to couple theparticle simulator with any other multibody simulation software. This corresponds to usingthe particle simulation as a very special force element in the MBS program.

4.5 Granular solid

Various different phenomena like, e.g., stress wave propagation, elastic-plastic deforma-tion or breaking can be observed when loading mechanical structures. Traditionally usedmethods to analyze these scenarios like FEM or BEM solve partial differential equations ofcontinuum mechanics. If fracture and fragmentation are to be incorporated in the scenario itis difficult to deal with the resulting discontinuities while using these classical methods. Adifferent approach to model damage and failure is based on the DEM. In recent years, thisapproach was adopted in geomechanics and civil engineering to model fracture in elastic-brittle materials, e.g., [17, 18]. Using this approach, the material is considered as beingfully discontinuous and made up by assembling and bonding adjacent discrete elements.The particle bonds are represented by force laws and can sustain only a specified stress untilfailure. Different types of force laws can be used depending on the considered problem to bemodeled as well as the type of particles used [17, 19]. In this paper, we focus on sphericalparticles bonded by rods.

As a rod can only bear tensile loads and the rods are fixed in the center of mass of eachparticle there is no need to introduce rotational degrees of freedom for the spheres. Theforce laws representing the rods contain an elastic-plastic part as well as a viscous part. Asthe DEM model is also used to model solids of ductile metal, plastic deformation has tobe considered. The basic stress-strain relationship of a ductile specimen which is uniaxiallyloaded behaves qualitatively as schematically depicted in Fig. 3(a). If the specimen is loadedbeyond the yield point σ0.2 plastic flow occurs. A subsequent loading in opposite directioncauses plastic flow for σ < σy2 where |σ0.2| > |σy2|. This is called Bauschinger effect.


Fig. 3 Stress strain curves for plasticity laws: (a) hysteresis, (b) bilinear hardening [20], (c) bilinear harden-ing with shifted yield function

The elastic-plastic part of the force law which is used in our model is based on a piecewiselinear hardening model, [20], schematically depicted in Fig. 3(b). Characteristic parametersof this model are the initial yield limit ε0.2, Young’s modulus E and the slope 0 < k < E ofthe yield function beyond the initial yield point.

If a specimen is loaded within the elastic limit σ0.2 the resulting stress is governed byYoung’s modulus, i.e., σ = Eε. Further loading to σ1 beyond the yield point σ0.2 causesplastic flow and stress is governed by

σ1 = Eε0.2 + k(ε1 − ε0.2). (23)

A subsequent loading in opposite direction causes plastic flow if σ ≤ −σ1 and so on.Using this model, it is not possible to reproduce the Bauschinger effect which one can

observe in experiments. Starting from this model we introduce some changes to includethe Bauschinger effect, see Fig. 3(c). For a material without previous plastic deformationit is assumed that the stress-strain relationship is governed by a piecewise linear functionσ = F(ε) which is symmetrical about the origin, i.e.

σ(ε) = −σ(−ε). (24)

If the considered specimen is loaded to σy beyond the elastic limit σ0.2 the materialflows plastically and the stress-strain relationship is given according to (23). A subsequentunloading follows the straight line g(ε) which is given by

g(ε) = (ε − εy)E + σy. (25)

Furthermore the plastic deformation shifts the yield function in the σ − ε-plane alongthe line through the origin given by h(ε) = kε. The shift parameters σ0 and ε0 can then becalculated by setting g(ε) = h(ε). This yields

ε0 = Eεy − σy

E − k(26)


and

σ0 = kε0. (27)

As a result the yield function (24) is transformed to

F̂ = σ(ε) = σ0 + F(ε − ε0). (28)

The stress then can be readily calculated as

σ = σ0 + E(ε − ε0). (29)

The forces f i acting on the connected particles are calculated by

|f i | = σA (30)

where A is the cross sectional area of the rod.As known from experience a loaded specimen does not flow infinitely but fails once

stress reaches tensile strength. In order to reproduce realistic behavior, the tensile strengthσmax is introduced in the model. Consequently, if

σ > σmax (31)

the rod connecting two considered particles is removed. Currently an interspherical bondcan be removed only if the overall strain is positive, i.e., in case of tensile loading. A failureof a bond in case of compression demands a modification of the used algorithm to preventan overlap of particles when removing the bond. Otherwise, a jump of the system energymight occur by a jump of the forces when switching from rod forces to contact forces.

5 Application examples

We have applied the DEM to simulate a variety of different problems, from highly dynamicalsystems, such as granular flows, to systems with slow dynamics, such as loaded continua.As an example for a granular flow with dynamically moving boundaries, we present thesimulation of a particle driven water wheel. By introducing rod elements as lasting bilateralinteractions between particles, that break for a certain load, it is possible to model a con-tinuum. This approach is demonstrated by means of a simulation of an orthogonal cuttingprocess where the DEM has obvious advantages compared to the FEM as particle separa-tion that enables chips is naturally included in the model. The same elastic rods are also usedto model elastic membranes which we present as a final example. Video sequences show-ing these and other applications can be found at www.itm.uni-stuttgart.de. Thesimulation examples we present in this paper were mostly carried out to demonstrate the be-havior and applicability of the method. Therefore, we used just roughly estimated materialparameters.

5.1 Particle driven dynamics

To demonstrate the capability to couple a particle simulation with a multibody system(MBS), we chose an ancient water wheel as a simulation example. In the particle simulation


Fig. 4 Snapshots of a simulation of a particle driven water wheel with the wheel body modeled as a rigidmacro particle. The particle velocities are color coded

the MBS bodies are represented as rigid macro particles that consist of a set of triangularsurface elements which are in turn handled as individual particles in the particle simulation,as described in Sect. 4.2. The water wheel body with center of mass on the rotation axisis fixed in a static frame. The driving torque that is applied is negative proportional to theangular velocity of the wheel. Thus, after a short transient phase, the angular velocity of thewheel reaches a steady state. To be able to feed the wheel with a constant stream of particles,we use a particle inflow and an outflow. Thus, particles are created and removed during thesimulation to save memory and CPU expense. In Fig. 4 several snapshots from simulationsof the water wheel are depicted. The motion of the water wheel is computed by integratingthe equations of motion of the MBS with the particles considered as force elements.

5.2 Orthogonal cutting processes

To show the applicability of the DEM model to simulate cutting processes we considered thecase of orthogonal cutting. This is a relatively simple case of cutting with a two-dimensionalstate of stress. As a pre-processing step, the solid representing the workpiece is generatedas a bulk of identical spheres arranged in a regular face-centered cubic lattice. All adja-cent spheres are then connected with visco-elastic-plastic rods represented by force lawsas described in Sect. 4.5. Thereby one generates a three-dimensional model of a break-able and granular solid which is made up by bonding rigid, unbreakable spherical particles.The workpiece is machined using a tool represented by triangles and moved according toa function of time. If the tensile strength of the material is reached, the rods representingthe cohesive forces of the material are removed and are never again restored. As an implicitintegration scheme is used, rods are only plastically deformed or removed if convergencehas been reached. Some snapshots of intermediate states are shown in Fig. 5.


Fig. 5 Snapshots of a simulation of orthogonal cutting of elastic-plastic material. The fraction of brokenrods is color coded

5.3 Elastic membranes

By bonding particles with viscoelastic elements, membranes can be simulated whose dis-tributed mass is represented by spherical particles. Between bonded particles only the springdamper forces act, whereas between unbonded particles any other contact law such asHertz’s law or a friction model can be applied. In our simulation example, we used arbitrarybut complicated and non-convex 3D-models as input for the surface geometry of obstaclesthat interact with the membrane. An example of a membrane consisting of 8,300 particlesthat interacts with a rigid bust is shown in Fig. 6. The membrane falls under the influence ofgravity on the bust and deforms elastically contouring the shape of the bust’s face.

If membranes are clamped one can observe wave phenomena, see Fig. 7. In this kind ofsimulations it is important to account for the Courant–Friedrichs–Lewy condition [21] byrestricting the time step size to enforce the numerical sound speed to be greater than thephysical sound speed to be able to resolve the wave phenomena correctly.

6 Conclusion

In this work, we presented an overview about some applications of the Discrete ElementMethod in mechanical engineering. The equations of motion for particles with six degreesof freedom were presented, where the rotation coordinates where represented by quater-nions. By means of this, the Newton and Euler equations form a system of seven differentialequations with the single scalar unit-quaternion invariant. Several numerical methods werepresented that can be used to solve the equations of motion, with focus on the solution of therotational part. A simple, yet robust step size controller was developed for the Newmark-βscheme that is based on the rate of convergence of the integrator’s fixed-point iteration. We


Fig. 6 Snapshots of a simulation of an elastic membrane falling on a rigid bust with color coded particlevelocity

Fig. 7 Snapshots of a simulation of a pig doll falling on a clamped elastic membrane with color codedparticle velocity

presented a penalty contact model for contacts between spherical particles and triangulatedsurface meshes where the triangles are treated as individual particles. Rigid dynamic macroparticles accumulate contact forces and torques acting on their surface triangles. Thus, theparticle simulations can be coupled to a MBS program as very powerful force elements.


To demonstrate the bandwidth of different fields of applications, we presented three sim-ulation examples. The first one, a simulation of a water wheel, showed how a particle sim-ulation can be coupled with a multibody simulation. The great advantages of the method,the absence of a discretization mesh that naturally allows for a separation of a continuumwas shown by means of an orthogonal cutting process. Finally, we presented simulations ofelastic membranes that feature large deformations and traveling waves.

Acknowledgements Some of the 3D-surface models for the simulations presented in this paper were takenfrom the wonderful web page of the gamma project, http://www-c.inria.fr/gamma/gamma.php. The work oncutting processes modeled by DEM is funded by the DFG in the framework of the SPP 1180. This support ishighly appreciated.

References

1. Cundall, P.A.: A computer model for simulation of progressive, large-scale movements in blocky rocksystems. In: Proceedings of the Symposium of the International Society of Rock Mechanics, Nancy(1971)

2. Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Geotechnique 29,47–56 (1979)

3. Shabana, A.A.: Computational Dynamics. Wiley, New York (2001)4. Omelyan, I.P.: On the numerical integration of motion for rigid polyatomics: the modified quaternion

approach. Comput. Phys. 12(1), 97–103 (1998)5. Roberson, R.E., Schwertassek, R.: Dynamics of Multibody Systems. Springer, Berlin (1988)6. Hairer, E., Wanner, G.: Stiff differential equations solved by Radau methods. J. Comput. Appl. Math.

111, 93–111 (1999)7. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic

Problems. Springer, Berlin (1991)8. Petzold, L.R.: A description of DASSL: a differential/algebraic system solver. In: Stepleman, R. (ed.)

Scientific Computing: Application of Mathematics and Computing to the Physical Sciences, vol. 2,pp. 65–68 (1983)

9. Verlet, L.: Computer experiments on classical fluids. Phys. Rev. 159, 98–103 (1967)10. Allen, M.P., Tildesley, D.J.: Computer Simulations of Liquids. Clarendon, Oxford (1989)11. Geradin, M., Rixon, D.: Mechanical Vibrations. Theory and Application to Structural Dynamics. Wiley,

Chichester (1997)12. Zohdi, T.I.: Charge-induced clustering in multifield particulate flows. Int. J. Numer. Methods Eng. 62,

870–898 (2004)13. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of

multibody systems. J. Mech. Des. 112, 369–376 (1990)14. Muth, B., Eberhard, P.: Collisions between particles of complex shape. In: Garcia-Rojo, R., Herr-

mann, H.J., McNamara, S. (eds.) Powders and Grains, pp. 1379–1383 (2005)15. Muth, B., Eberhard, P.: Investigation of large systems consisting of many spatial polyhedral bodies. In:

van Campen, D.H., Lzurko, M.D., van den Oever, W.P.J.M. (eds.) Proceedings of the ENOC-2005, FifthEUROMECH Nonlinear Dynamics Conference, pp. 1644–1650 (2005)

16. Brendel, L., Dippel, S.: Lasting Contacts in Molecular Dynamics Simulations, p. 313ff. Kluwer Acad-emic, Dordrecht (1998)

17. Herrmann, H., D’Adetta, G.A., Kun, F., Ramm, E.: Two-dimensional dynamic simulation of fracture andfragmentation of solids. Comput. Assis. Mech. Eng. Sci. 6, 385–402 (1999)

18. Potapov, A.V., Hopkins, M.A., Campbell, C.S.: A two-dimensional dynamic simulation of solid fracture.Int. J. Mod. Phys. C 6(3), 371–398 (1995)

19. D’Addetta, G.A., Ramm, E.: A microstructure-based simulation environment on the basis of an interfaceenhanced particle model. Granul. Matter 8, 159–174 (2006)

20. Liu, K., Gao, L., Tanimura, S.: Application of discrete element method in impact problems. JSME Int. J.Ser. A 47(2), 138–145 (2004)

21. Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. IBMJ. 11, 215–234 (1967)

Applications of the Discrete Element Method in Mechanical Engineering

Documents

Transcript of Applications of the Discrete Element Method in Mechanical Engineering