Particle-Particle Hard Sphere Modeling

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Chapter 2 Methodology

Modeling three dimensional particulate systems is heavily dependent on the choice of technique to generate the geometry. The description of particle shape and size influences the choice of collision detection and resolution algorithms. Furthermore, the particle shape descriptors govern the applicability, accuracy and flexibility of the model. For instance, a DEM model described only by spheres or circles (in 2D) will have a higher computational efficiency which may not work well for non-spherical particles. On the other hand, non-spherical particle shapes offer a wider range of flexibility in terms of the shapes that can be simulated, with computational efficiency turning out to be a bottleneck in most cases. The choice of the surface definition consequently affects the collision algorithms and the post-collision kinematics. In this work, superellipsoids are generated using their surface function as this class of particles allows for a wide range of shapes to be simulated as illustrated in Figure 2-1.

Geometry Definition

In general, particles in a DEM simulation can be generated by the following techniques:

• Polyhedrons / Spheres Computational Mesh
• Composite particle approach
• Continuous Function Representation Discrete Function Representation

The easiest method to describe particles’ geometry is to use polyhedrons and spheres. In the case of polyhedrons, the surface can be defined by straight edges and vertices joining the edges. For spheres, the only information that needs to be tracked, as the particle evolves in space and time, is its centroid. As a result, the simulation becomes highly efficient compared to other complex shapes. The collision detection and resolution methods are also well defined for polyhedrons and spheres. Collision between polyhedrons can be detected using one of the edge-edge, vertex-edge and vertex-vertex collision techniques depending on the type of collision encountered [23], [24]. An advantage of these techniques is that the supplemental information required to process the post collision kinematics are obtained with relative ease. Similarly, collisions between spheres are detected based on an inequality relation between the distance between the centroids and their respective radii. As is the case with polyhedrons, the collision detection algorithms for spheres yield sufficient information to resolve the collisions.
However, the use of polyhedrons and spheres severely limits the applicability of the DEM model. In the majority of granular assembly studies, the particles used are irregularly shaped and carrying out computational studies to recreate the tests, using polyhedrons or spheres, would ensue in significant loss of accuracy. In CFD-DEM studies, accuracy is of utmost importance as in many cases the necessary equipment to carry out an experimental study might not be available. Since the results would solely be based on numerical simulations, efforts have to be taken to ensure the fidelity of the results. In that regard, complex shape descriptors aid in reproducing the exact kinematics of particulate assemblies in practical applications.
Alternatively, an independent mesh could be generated around the particle’s surface which would take significantly longer owing to the absence of any functional form defining the surface. Storing the surface descriptors using vertices, triangular or quadrilateral elements and the associated connectivity lists consume a significant amount of memory as well. However, a meshed representation could be widely applied to any particle geometry irrespective of its shape and texture.
Additionally, if storage and simulation time take precedence over accuracy, the surface definition of the particles could also be done with a composite particle approach, wherein the non-spherical particle geometry is approximated by a cluster (or) collection of spheres, as shown in Figure 2-2. The multi-sphere approach – introduced by Favier et al (1999) – is easier to implement in terms of the collision detection and resolution algorithms as they employ spheres in their framework[26]. Also, since the composite structure of spheres do not change with time, the initially quantified physical properties could be used without much modification. However, an unfavorable characteristic of this model is its lack of applicability to a wide range of shapes. Since spheres can only be used to approximate other non-spherical shapes like ellipsoids and cylinders, the model is of little relevance to shapes with sharp corners. Furthermore, deciding upon the number of spheres that would be required to approximate a non-spherical particle is a laborious task. With increasing number of spherical particles in a composite structure, the non-spherical particle could be approximated better, although, due to the varying shapes and sizes, calculation of the moment of inertia turns out to be less accurate, in addition to the loss of computational efficiency[27].
When dealing with non-spherical particles, the most commonly used classes of particles are superellipsoids. The surface of a superellipsoid can be defined by both continuous and discrete function representations. A Continuous Function Representation (CFR) defines the surface by a mathematical relation, often in the particle’s local coordinate system. For instance, a superellipsoid’s surface can be described by the following explicit formulation:
This surface function is sometimes referred to as the ‘inside-outside’ function as any point evaluating to a value less than one lies inside the surface of the superellipsoid, whereas any point outside the surface of the superellipsoids evaluates to a value greater than one.
The biggest disadvantage of the continuous function representation is that the collision detection and resolution algorithms adjunct to the geometry definition are computationally intensive as they involve solving for non-linear equations, using Newton Raphson method, in global coordinate systems. Another limitation of the technique is that the solution can diverge when the initial estimates are incongruous with the final solution[19].
As a consequence, it is easier to model a superellipsoid with the implicit formulation given. The implicit function is convenient for discretely approximating the surface. A Discrete Function Representation (DFR) refers to the method of discretizing a surface with nodes or vertices in such a way that the approximation replicates the continuous surface to a reasonable extent. Making use of this formulation and with a linearly spaced interval for ₁ and ₂, an equal distribution of vertices on the surface of the superellipsoid can be obtained for the majority of cases. However, for certain geometries such as a cylinder, it results in a paucity of vertices between the enclosing end-surfaces. In other words, there is an uneven distribution of vertices between the surfaces, which could pose an issue in contact detection and in the accuracy of the resolution algorithm. In order to circumvent this possible problem, the interval for ₂ can be modified to be exponentially spaced as opposed to the conventional linear spacing as shown in Figure 2-3.
In practice, DFR method is implemented frequently when working with superellipsoids. The technique offers advantages in terms of accuracy and reliability over spherical surface description methods. The computational cost associated with DFR is higher than the spherical descriptors, however with the ability to parallelize the DEM model, the increased cost could be overcome to a manageable extent. Also, describing a surface using DFR would enable the use of simpler methods – in comparison with CFR – to detect and resolve the collisions. Specifically, the detection and collision resolution methods for superellipsoids have been well defined in the literature for geometries defined using DFR [2], [25]. In most cases, DFR is preferred to CFR as the solutions when detecting collisions are guaranteed and unique. Moreover, DFR is preferred to meshed techniques as the simulation time and the storage required are less in comparison. A brief summary of the methods and their advantages and disadvantages are shown in Table 2-1.

Physical Attributes Definition

The physical quantities such as mass, volume, density and inertia are essential parameters that govern the resolution of collisions. The mass and density of a superellipsoid have a scalar magnitude associated with them. The volume and inertia – in local coordinates – can be expressed in terms of the dimensional and squareness parameters of a superellipsoid. It is also possible to mathematically calculate the inertia and volume based on integrals but for the sake of simplicity, the ensuing expressions of the integration operations are used. For a detailed derivation of these expressions a perusal of Jaklič (2000) is recommended [28].

Kinematics

The particles once generated are displaced by an offset magnitude in the global coordinate system. The offset magnitudes are chosen such that the particles do not overlap initially, irrespective of the number of particles. Spatial and temporal advancement of the particles are done by displacing the particles by the designated linear and rotational velocities at each time step.
The rotation matrix can be computed using various methods. In this work, quaternions are used to compute the rotational matrix. This eliminates the problem of sequential (or) preferential rotation where the order of rotation matters. Using quaternions for computing the rotation matrix ensures that one single ‘resultant’ rotation is applied to the particle about a unit axis of rotation. At a given time step, an instantaneous unit quaternion is defined as:
at the initial time step is the quaternion resulting from the particle’s initial orientation. With the updated quaternion, the particle’s local coordinates are transformed into its global space equivalent by applying the transformation.

Particle-Particle Collision Detection

Overview of Collision Detection

Historically, collision detection has been a significant part of computer graphics and computational geometry. It refers to techniques involved in determining if two objects are in contact (merely touching) or have interpenetrated each other while undergoing specified actions. Popular applications wherein collision detection plays a vital role are robotics, computational biology, and in the development of gaming engines. In CFD-DEM, granular particles are under the influence of fluid forces in a confined computational domain. This leads to the possibility of encountering particles colliding with one another at multiple points. Detecting and resolving the collisions are essential to accurately represent the post-collision kinematics of the particles, subsequently effecting realistic changes in the flow-field through coupled CFD solvers. The representation of the geometry, in many ways, outlines the techniques that would be applicable to model collisions between the particles in CFD-DEM. Common practices have modeled particles as spheres as detecting collisions between them are fairly straightforward. Intuitively, since the particles are governed by the equations of motions and their interaction with fluid flow can also be modeled, collisions between particles can be accounted for by numerically solving the global spatial surface equations of the concerned particles. However, this method results in a lot of computational effort and the efficiency associated with this technique reduces as the number of particles increases.
Collision detection between non spherical particles have been, traditionally, not as conventional as that for spherical or polyhedral particles. For spherical and polyhedral particles, bounding volume hierarchies that spatially approximate the particle with spheres or polyhedral are effective in detecting collision. The underlying assumption is that if two objects are possibly colliding, the projection of their bounding volumes, on all the coordinate axes, have to be overlapping[29]. The bounding volumes by themselves, can take one of the usually preferred geometrical forms, namely, Axis Aligned Bounding Boxes (AABBs) and Objected Oriented Bounding Boxes (OOBBs). An AABB is a computationally inexpensive method of monitoring possible contacts between particles. Each particle has a bounding volume aligned with the global coordinate axes, generated around its surface. For non-rotating particles the performance of AABB is similar to that of OOBB, however for rotating particles, the bounding volumes tightly aligned to the coordinate axes offer poor performance in detecting collisions, in addition to an increase in its volume, as illustrated in Figure 2-6 . An improved alternative to AABBs are the OOBBs which have a bounding volume generated along the body-fitted (local) axes. While the particle is in rotation, the OOBBs also align in close approximation with the rotated surface, as depicted in Figure 2-7. Comparatively, OOBBs offer a better approximation of the particle surface and an enhanced performance in collision detection, however, they are slightly more expensive to generate.
A lesser known method of creating bounding volumes is the Discrete Oriented Polytopes (K-DOPs), with ‘K’ denoting the number of edges constituting the bounding volume. These bounding volumes are more compact than OOBBs and offer a better approximation by minimizing the area between the bounding volume and the actual particle’s surface. Nevertheless, they are complex to generate and would require increased computation effort to detect overlaps[10], [30].

Abstract 
General Audience Abstract 
Acknowledgments 
1. Introduction 
1.1. Motivation
1.2. Overview of the Research Problem
1.3. Objectives and Contributions of Research
2. Methodology
2.1. Geometry Definition
2.2. Physical Attributes Definition
2.3. Kinematics
2.4. Particle-Particle Collision Detection
2.5. Particle-Wall Collision Detection
3. Collision Resolution
3.1. Particle-Particle Hard Sphere Modeling
3.2. Particle-Particle Soft Sphere Modeling
3.3. Particle-Wall Collision Resolution
4. Results and Discussion 
4.1. Verification and Validation of the Collision model
4.2. Computational Performance Metrics
5. Conclusion and Future Work
5.1. Conclusion
5.2. Future Work
References
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CFD – DEM Modeling and Parallel Implementation of Three Dimensional Non- Spherical Particulate Systems