DEM Application

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WARNING: This page is still under construction.

The DEM Kratos Team


Contents

Theory

This application solves the equations.... Mathematical approach to the problems.

Nothing numerical

Integration Schemes

Forward Euler Scheme

Contact Laws

Concept of indentation HMD, LSD

Normal Force Laws
Linear Repulsive Force

The most simple representation of a repulsive contact force between a sphere and a wall is given by a linear law, where the force acting on the sphere when contacting a plane is a linear function of the indentation, which in turn would bring a quadratic dependence with the contact radius. The next figure shows this simple law:

Jkr cohesion linear force.png


Hertzian Repulsive Force

On the other hand, Hertz solved in 1882 the non-cohesive normal contact between a sphere and a plane. In 1971 Johnson, Kendall and Roberts presented the solution (JKR-Theory) for the same problem in this case adding cohesive behaviour. Not much later, Derjaguin, Müller and Toporov published similar results (DMT-Theory).

Both theories are very close and correct, the main difference being that the JKR theory is oriented to the study of flexible, large spheres, while the DMT theory is specially suited to represent the behaviour of rigid, small ones.

Jkr cohesion hertz.jpeg

The previous figure shows the standard representation of a Linear or Hertzian contact between a sphere and a wall. The distribution of contact pressures between both bodies follow a parabolic law.

JKR Cohesive Force

Jkr cohesion jkr.jpeg

The preceding capture shows the a representation of a JKR contact between a sphere and a wall. In this case, the distribution of pressures between both bodies is more complex due to the formation of a neck at the boundaries of the contact. A later figure shows a detailed view of the pressures involved.

Jkr cohesion forces.png

In the previous graphic, it is very interesting to note the existence of two singular values of contact radius: one for which the total forces acting on the contacting sphere is zero, and another for which the maximum value of adhesion is achieved.


Jkr cohesion pressures.png

In the previous figure, the blue area represents the distribution of pressures acting on the sphere when contacting a wall if a Hertzian Law is followed. On the other hand, the sum of both green and blue areas represents the JKR distribution. Note the larger values and the existence of adhesive behaviour at both sides of the pressures distribution.


An example of granular simulation without cohesive forces:

JKR no cohesion 1 33.png JKR no cohesion 2 33.png JKR no cohesion 3 33.png JKR no cohesion 4 33.png

The same simulation as before, this time with cohesive forces in both sphere-sphere and sphere-plane contacts.

JKR cohesion 1 33.png JKR cohesion 2 33.png JKR cohesion 3 33.png JKR cohesion 4 33.png


References:

V. L. Popov. Contact Mechanics and Friction (2010).

Tangential Force Laws
Damping Force Laws

(restit. coef)

Numerical approach

This section is dedicated to describe the numerical methods used to solve.


Dem manual main scheme 66 discontinuum.png


DEM elements

Spheric Particle
Spheric Continuum Particle
Spheric Swimming Particle

DEM walls (Kratos Conditions)

DEM Inlets

A DEM Inlet is a source of new DEM Elements. It is a cloud of Nodes, where each Node is the center of a Generator Sphere. In a random order, the Nodes are chosen to create a new DEM Element (both spherical elements or clusters) whose center coincides with the Node and the whole element is fully included in the Generator Sphere. The newly generated DEM Element has a non-zero imposed velocity which eventually makes the DEM Element get outside the Generator Sphere. Until this moment, the Generator Sphere is not allowed to generate another DEM-Element. From this moment on, the newly created DEM Element no longer has the velocity imposed, it moves freely and cannot penetrate its Generator Sphere or any other. Actually, the Generator Sphere is seen as any other Sphere of the domain by the DEM Element, and the contact between them is calculated as usual. In other words, the Generator Spheres reserve the necessary space to create a new DEM Element.

DEM strategies

Non-cohesive materials Strategy
Evaluation of Forces

Once contact between two spheres has been detected (see next figure), the forces occurring at the contact point are computed. The interaction between the two contacting spheres can be represented by forces Fij and Fji, which have the same module but opposite directions.


Dem application interaction forces.png


At the same time, this force Fij can be decomposed into its normal and shear components Fijn and Fijs, respectively. The next figure shows a detailed view of this decomposition.

Fij = Fijn + Fijs = Fnnij + Fijs


Dem application forces decomposition.png


The nij vector represents the unitary normal with respect to the surfaces of both particles at the contact point. This vector lies along the line connecting the centers of the two particles and is directed outwards from particle i.


The contact forces Fn, Fs1 and Fs2 are obtained using a constitutive model formulated for the contact between two rigid spheres (or discs in 2D). The contact interface for the simplest formulation is characterized by the normal and tangential stiffness Kn and Ks, respectively, a frictional device obeying the Couloumb law with a frictional coefficient mu, and a dashpot defined by the contact damping coefficient cn as shown in the next figure.


Dem application forces rheological.png

Continuum Strategy

DEM schemes

Integration of Motion

The standard translational and rotational equations for the motion of rigid bodies are used to compute the dynamics of the spheres. For the i-th particle we have:

miui = Fi Ii i = Ti


where u represents the element centroid displacement in a xed (inertial) coordinate frame X, ! the angular velocity, m the element mass, I the moment of inertia, Fi the resultant force, and Ti the total moment arount the central axes.


Vectors Fi and Ti are the sum of all forces and moments applied to the i-th particle due to external loads, Fexti and Texti , respectively, the contact interactions with neighbour spheres Fc i, j = 1;; nc, where nci is the total number of elements in contact with the i-th discrete element, and the forces and moments resulting from external damping, Fdampi and Tdampi , respectively, which can be written as:


Fi = Fexti +nciXj=1Fci + Fdampi

Ti = Texti +nciXj=1(rci Fci + qci ) + Tdampi


Dem application motion.png


where rci is the vector connecting the centre of mass of the i th particle with the contact point c (see next figure) and qci are the torques due to rolling and/or torsion (not related to the tangential forces). The next figure shows the motion of a rigid particle. Note that the form of the rotational equation is only valid for spheres and cylinders (in 2D). It is simplified with respect to a general form for an arbitrary rigid body with the rotational inertial properties represented by a second order tensor. In the general case it is more convenient to describe the rotational motion with respect to a co-rotational frame x which is embedded at each element since in this frame the tensor of inertia is constant.


The previous set of equations are integrated in time using a simple central difference scheme. The time integration operator for the translational motion at the n-th time step is as follows:


The first two steps in the integration scheme for the rotational motion are identical to those given by the previous equations:


On the other hand, the vector of incremental rotation is computed as


Knowledge of the incremental rotation is enough to update the tangential contact forces. It is also possible to track the rotational position of particles when necessary. If that is the case, the rotation matrices between the moving frames embedded in the particles and the fixed global frame must be updated incrementally by using an adequate multiplicative scheme. Explicit integration in time yields high computational efficiency and it enables the solution of large models. The main disadvantage of this explicit integration scheme is its conditional numerical stability which imposes a limitation on the time step delta_t.

Search Strategies

The contact search is a very important part of the method in terms of computational cost (range 60%-90% of simulation time in simulations with large number of particles) and it is possibly the most difficult part to treat when dealing with particles that have no spherical/circular shape.

The contact detection basically consists in determining, for every target object, what other objects are in contact with, and then, judge for the correspondent interaction. It is usually not needed to perform a search every time step, which is generally limited for the stability of the explicit integration of the equations of motion. Due to the fact that the search is an expensive step a lower search frequency can be selected without much loss of accuracy.

The most naïve method of search that can be set is the brute search; for every element, the method does a loop for any other element checking for the contact. The order of the number of operations needed is quadratic: n2. Normally, the strategy to avoid such an expensive scheme is to divide the contact search in two basic stages, a global search and a subsequent local resolution of the contact; in this case the computation time of the contact search is proportional to n log n. In the DEMApplication a Grid/Cell based algorithm is used in this purpose.

Global Search

In a generic way, there are two types of elements: searcher elements (particles or finite elements) and target elements (particles or finite elements). Hereafter searcher elements will be called S.E. and target elements T.E.

Global search1.png

The steps needed to perform contact search are:

a) Build bounding box of S.E. (Figure 2(a)).

b) Build bins cells based on size and position of S.E. (Figure 2(b)).

c) Collocate S.E. in bins and construct hash table with relates coordinates with cells which point to the contacting S.E. (Figure 2(c)).

d) Build bounding box of T.E. (Figure 2(d)).

e) Loop over T.E., detect the intersecting cells to each T.E., check the intersection with the possible found cells and add the entire S.E. contained in the cells intersected by each T.E. (Figure 2(e)).

f) Solve the contact with local resolution (Figure 2(f)).

Note: In the case of FE and DE the FE are selected as the S.E. to construct the Bins and the Spheres are T.E. to be found in that bins.

Global search2 bigger.png

Local Search

Once the possible neighbours are detected, the local resolution check takes place. For the case of two spherical particles, the check is easy; only the sum of the radius has to be compared against the distance between centres. Other geometries may demand a much complicated check. The followed strategy is to mesh all the geometries with a discretization of triangles. In 3D, surface meshes are used for contact detection. Now, the contact detection should be performed between particles and triangles; if no contact is found, particle contact against lines is searched for; and if contact is still not found, contact against points is performed. Figure 3 shows how the local search is performed. Particle i searches contact against element j, then against lines k, l and m and finally against points n, o and p.

This is known as a hierarchical algorithm. For further explanation please refer to the paper: "3D contact algorithms for particle DEM with rigid and deformable solids" - M.Santasusana, J.Irazábal, E.Oñate, J.M.Carbonell. Where the advantges and the drawback of this method and other proposed algorithms are detailed and analysed in complicated situations like multicontact.

Local search1.png

Fig. 3 Particle-Face contact detection.


Programming Implementation

Structure of the code (Strategy, Scheme, Element, Node, Utilities, functions frequently used like FastGet,...)

The source code is accessible through this site.

Dem manual code scheme 66.png


Benchmarks

The DEM Benchmarks consist of a set of 9 simple tests which are run every night and whose object is to make sure both that the application performs correctly and that the code did not break after the daily changes. They are the following:

Test1: Elastic normal impact of two identical spheres

Check the evolution of the elastic normal contact force between two spheres with time.


Benchmark1 1.png Benchmark1 graph 66.png

If the coefficient of restitution is 1, the module of the initial and final velocities should be the same. Also, by symmetry, velocities should be equal for both spheres.

_

Test2: Elastic normal impact of a sphere against a rigid plane

Check the evolution of the elastic normal contact force between a sphere and a plane.

Benchmark2 66.png Benchmark2 graph 66.png

If the coefficient of restitution is equal to 1, the module of the initial and final velocity should remain unchanged.

_

Test3: Impact of a sphere against a rigid plane with different coefficients of restitution

Check the effect of different restitution coefficients on the damping ratio.

Benchmark3 66.png Benchmark3 graph 66.png

If total energy is conserved, the restitution coefficient and the damping ratio values should be identical.

_

Test4: Oblique impact of a sphere with a rigid plane with constant velocity module and variable incident angles

Check the tangential restitution coefficient, final angular velocity and rebound angle of the sphere.

Benchmark4 66.png Benchmark4 graph1 66.png Benchmark4 graph2 66.png

_

Test5: Oblique impact of a sphere with a rigid plane with constant normal velocity and different angular velocities

Check the final linear and angular velocities of the sphere.

Benchmark5 66.png Benchmark5 graph1 66.png

_

Test6: Impact of a sphere with a rigid plane with a constant normal velocity and variable angular velocities

Check the final linear and angular velocities of the sphere.

Benchmark6 66.png Benchmark6 graph1 66.png

_

Test7: Impact of two identical spheres with a constant normal velocity and different angular velocities

Check the final linear and angular velocities of both spheres.

Benchmark7 66.png Benchmark7 graph1 66.png

By symmetry, the tangential final velocity of both spheres should be zero. Additionally, for a coefficient of restitution of 1, there should be no changes in the modules of both linear and angular velocities and their values should conserve symmetry.

_

Test8: Impact of two differently sized spheres with a constant normal velocity and variable angular velocities

Check the final linear and angular velocities of both spheres.

Benchmark8 66.png Benchmark8 graph1 66.png

In this case, it is interesting to note that, the bigger and/or denser sphere 2 is, the more this test resembles the sphere versus plane simulation.

_

Test9: Impact of two identical spheres with a constant normal velocity and different coefficients of restitution

Check the effect of different restitution coefficients on the damping ratio.

Benchmark9 66.png Benchmark9 graph1 66.png

If total energy is conserved, the restitution coefficient and the damping ratio values should be identical.


References:

Y.C.Chung, J.Y.Ooi. Benchmark tests for verifying discrete element modelling codes at particle impact level (2011).

How to analyse using the current application

Pre-Process

GUI's & GiD

D-DEMPack

D-DEMPack is the package that allows a user to create, run and analyze results of a DEM simulation for discontinuum / granular / little-cohesive materials. It is written for GiD. So in order to use this package, you should install GiD first.

You can read the D-DEMPack manual or follow the D-DEMPack Tutorials for fast learning on how to use the GUI.

C-DEMPack

Continuum / Cohesive

F-DEMPack

Fluid coupling

Post-Process

Application Dependencies

The Swimming DEM Application depends on the DEM application

Other Kratos Applications used in current Application

FEM-DEM


Problems!

What to do if the Discrete Elements behave strangely

In the case you notice that some discrete elements cross walls, penetrate in them or simply fly away of the domain at high velocity, check the following points:


In the case of excessive penetration:

  • Check that the Young Modulus is big enough. A small Young Modulus makes the Elements and the walls behave in a very smooth way. Sometimes they are so soft that total penetration and trespass is possible.
  • Check the Density of the material. An excessive density means a big weight and inertia that cannot be stopped by the walls.
  • Check the Time Step. If the time step is too big, the Elements can go from one side of the wall to the other with no appearence of a reaction.
  • Check the frequency of neighbour search. If the search is not done frequently enough, the new contacts with the walls may not be detected soon enough.


In the case of excessive bounce:

  • Check that the Young Modulus is not extremely big. An exaggerated Young Modulus yields extremely large reactions that can make the Elements bounce too fast in just one time step. Also take into account that the stability of explicit methods depends on the Young Modulus (the higher the modulus, the closer to instability).
  • Check the Density of the material. A very low density means a very small weight and inertia, so any force exerted by other elements or the walls can induce big accelerations on the element.
  • Check the Time Step. If the time step is too big, the method gains more energy, and gets closer to instability.
  • Check the restitution coefficient of the material. Explicit integration schemes gain energy noticeably, unless you use a really small time step. In case the time step is chosen to be big (but still stable), use the restitution coefficient to compensate the gain of energy and get more realistic results.

Contact

Contact us for any question regarding this application:


-Miguel Angel Celigueta: maceli@cimne.upc.edu

-Guillermo Casas: gcasas@cimne.upc.edu

-Salva Latorre: latorre@cimne.upc.edu

-Miquel Santasusana: msantasusana@cimne.upc.edu

-Ferran Arrufat: farrufat@cimne.upc.edu

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