Efficient implementation of superquadric particles in Discrete Element Method within an opensource framework
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Abstract
Particle shape representation is a fundamental problem in the Discrete Element Method (DEM). Spherical particles with well known contact force models remain popular in DEM due to their relative simplicity in terms of ease of implementation and low computational cost. However, in real applications particles are mostly nonspherical, and more sophisticated particle shape models, like superquadric shape, must be introduced in DEM. The superquadric shape can be considered as an extension of spherical or ellipsoidal particles and can be used for modeling of spheres, ellipsoids, cylinderlike and box(dice)like particles just varying five shape parameters. In this study we present an efficient C++ implementation of superquadric particles within the opensource and parallel DEM package LIGGGHTS. To reduce computational time several ideas are employed. In the particle–particle contact detection routine we use the minimum bounding spheres and the oriented bounding boxes to reduce the number of potential contact pairs. For the particle–wall contact an accurate analytical solution was found. We present all necessary mathematics for the contact detection and contact force calculation. The superquadric DEM code implementation was verified on test cases such as angle of repose and hopper/silo discharge. The simulation results are in good agreement with experimental data and are presented in this paper. We show adequacy of the superquadric shape model and robustness of the implemented superquadric DEM code.
Keywords
Discrete Element Method Granular flow Contact detection Superquadric Nonspherical particles Angle of repose Hopper discharge1 Introduction
In many engineering applications different types of particles have to be stored, transported, mixed, or segregated. Despite that, knowledge of the static and dynamic behavior of particulate solids is still not fully understood. Such knowledge is of major importance for a proper design of processing units of silos, rotating drums, and others [23]. The Discrete Element Method (DEM), developed by Cundall and Strack [14], has proven to be an efficient tool for modeling granular materials. In DEM granular material is treated as a system of distinct interacting particles. Each particle has own mass, velocity, position, and contact properties; it obeys Newton’s second law and is tracked individually. Together with the rapidly increasing computational power available, DEM becomes more and more popular among engineers and researchers. A comprehensive overview of major DEM applications can be found in [56].
Many DEM codes still employ disks (in 2D) and spheres (in 3D) to represent particle shapes due to their implementation simplicity and efficiency in speed of contact detection, which results in faster code development and lower computational time. The rolling friction correction can be theoretically linked to various physical effects to model particle nonsphericity using spherical particles, as emphasized by Ai et al. [2]. Moreover, the contact force models that include normal and tangential forces for a pair of interacting spheres are already established. An overview of the most popular contact forces is given in [55]. However, particles in granular and powder materials in nature and industry are mostly nonspherical. Moreover, spherical particles behave differently than complexshaped particles, not only on the single grain level but also as an assembly. As summarized by Cleary [10], nonspherical particles differ from the spherical ones in the following ways: compacity of packed heap, resistance to shear and roll, and, as a result, ability to block the flow. Therefore, the physical validity of results obtained from simulations using spherical particles is usually questionable [31, 48].
Many approaches have been suggested in the literature to handle particle nonsphericity. Previously, Lu et al. [34] have summarized the main theoretical developments in nonspherical DEM and reviewed its applications. The most popular approach in the DEM community, according to Lu et al. [34], is the multisphere (MS) approach [1, 19, 30, 31, 37, 50]. In this method simple spheres are allowed to overlap and glued together to represent complex shapes. The method has the advantages that any shape can be represented by a set of glued (or prime) spheres and contact detection together with force calculation is based on that for spheres. One of the disadvantages is the fact that high accuracy of the shape approximation requires a significant number of prime spheres. Markauskas et al. [37] showed that approximation of ellipsoidal particles by 25 prime spheres increases CPU time by factor of 17. Marigo and Stitt [36] studied the influence of particle shape representation by the MS approach for a system of cylindrical pellets and found that about 160 primary spheres are required to be in agreement with experimental data. Another disadvantage of the MS method is the occurrence of multiple contact points [31] since an approximation of a convex particle (e.g., ellipsoid) by MSparticle is always nonconvex if the number of primary spheres is more than one. The number of interparticle contacts increases with the increase of the number of the prime spheres [37]. Thus, a reasonable number of prime spheres should be chosen.
Polygonal (in 2D) and Polyhedral (in 3D) particles as introduced by Cundall [13] have been widely used in DEM to model granular materials. In this approach the particle surface is approximated by line segments (in 2D) or by triangles (in 3D), thereby providing a high level of versatility in particle shape representation. Different algorithms for collision detection were developed by Cundall [13], Chang and Chen [8], Boon et al. [5], and Nezami and Hashash [41]. The major drawback of this method is that the question of how contact forces between two colliding polyhedral bodies are calculated is still not completely answered.
Unfortunately, there is a lack of literature that could provide detailed descriptions of algorithms necessary for implementation of superquadric particles in DEM. In this study we will present the nonspherical DEM approach providing all necessary mathematical tools for an efficient implementation of superquadric particles in the DEM based on opensource DEM package LIGGGHTS [28], such that the reader can understand the underlying algorithms and analytical expressions for particle–wall contact and minimum bounding sphere. We show good versatility of the approach for the practical range of blockiness parameters. Validation work along with several application examples is presented.
2 Numerical model
2.1 Motion of an arbitrarily shaped particle
Equation (7) in conjunction with Eq. (5) can be solved by various methods, such as described by Miller et al. [38], Walton and Braun [51] and Omelyan [42, 43, 44, 45]. Corresponding analytical expressions for volume and principal moments of inertia can be found in works by Jaklič et al. [25, 26].
2.2 Neighbor search
While the bounding sphere is an orientation invariant approximation of a particle, the oriented bounding box (OBB) can capture orientation and aspect ratio of the particle. The minimum oriented bounding box is a rectangular block with semiaxes a, b, c with the center located at the center of the particle in question and oriented as the particle. The intersection check methods between two OBBs are usually based on the concept of the separation axis and can be found in [17, 18, 47].
2.3 A contact detection algorithm
Equation (1) defines a superquadric surface in its local (canonical) coordinate system. We will refer to the function \(f(\varvec{x})\equiv \left( \left \frac{x}{a} \right ^{n_2}+\left \frac{y}{b} \right ^{n_2}\right) ^{n_1/n_2}+\left \frac{z}{c} \right ^{n_1}1\) as the shape function. If for a certain point \((x,y,z)^T\) the value \(f<0\) then the point is located inside the particle, if \(f>0\), then \((x,y,z)^T\) is outside the particle. If \(f=0\), then \((x,y,z)^T\) lies on the particle surface.
 (1)Find the contact point for a pair of volume equivalent spheres with radii \(r_1\) and \(r_2\) and centers located at the same points as for given particles, \(\varvec{X}_C^1\) and \(\varvec{X}_C^2\). These spheres defined as superquadrics have the following shape and blockiness parameters: \(a^1=b^1=c^1=r^1\), \(a^2=b^2=c^2=r^2\), \(n_1^1=n_2^1=n_1^2=n_2^2=2\). The analytical solution for the sphere–sphere contact point isUse this point as a starting point.$$\begin{aligned} \varvec{X}=(r_2 \varvec{X}_C^1+r_1 \varvec{X}_C^2)/(r_1+r_2). \end{aligned}$$(24)
 (2)Choose number of steps N and calculate$$\begin{aligned} \begin{aligned}&\delta a^i = (a_0^i  r^i)/N \\&\delta b^i = (b_0^i  r^i)/N \\&\delta c^i = (c_0^i  r^i)/N \\&\delta n_1^i = (n_{10}^i  2)/N \\&\delta n_2^i = (n_{20}^i  2)/N \\&i=1,2 \\&k = 1. \end{aligned} \end{aligned}$$(25)
 3.Modify shape and blockiness parameters.$$\begin{aligned} \begin{aligned}&a^i:=r^i+k\cdot \delta a^i \\&b^i:=r^i+k\cdot \delta b^i \\&c^i:=r^i+k\cdot \delta c^i \\&n_1^i:=2+k\cdot \delta n_1^i \\&n_2^i:=2+k\cdot \delta n_2^i \\&i:=1,2 \\&k:=k+1. \end{aligned} \end{aligned}$$(26)
 (4)
Calculate the contact point for particles with shape parameters (\(a^1\), \(b^1\), \(c^1\), \(n_{1}^1\), \(n_{2}^1\)) and (\(a^2\), \(b^2\), \(c^2\), \(n_{1}^2\), \(n_{2}^2\)) using the iterative algorithm described above and the last computed starting point. Use the found contact point as a new starting point for the next step.
 (5)
Repeat steps 3 and 4 for \(N1\) times
2.4 Particle–wall contact
2.5 Contact force calculation
In the spherical Discrete Element Method the two following approaches are common: the hardsphere and the softsphere approach. In the hardsphere approach (eventdriven), particles are assumed as rigid bodies, a sequence of collisions is processed, one collision at a time without the contact forces being explicitly considered. In the softsphere approach (timedriven) particles are allowed to deform slightly(overlap), and the contact forces are calculated as functions of the overlap [55]. This overlap is not real but intends to model the deformation of the interacting particles at a contact point in an indirect way.

\(\varvec{F}_{n,ij}=k_{n,ij}\varvec{\delta }_{n,ij}+\gamma _{n,ij}\varvec{v}_{n,ij}\) is the normal force component,

\(\varvec{F}_{t,ij}=k_{t,ij}\varvec{\delta }_{t,ij}+\gamma _{t,ij}\varvec{v}_{t,ij}\) is the tangential force component,

\(\varvec{\delta }_{n,ij}=\delta _{n,ij}\varvec{n}_{ij}\) is the normal overlap vector,

\(\varvec{n}_{ij}=(\varvec{X}_{Cj}\varvec{X}_{Ci})/\varvec{X}_{Cj}\varvec{X}_{Ci}\) is the normal overlap direction,

\(\delta _{n,ij}=R_i + R_j  d_{ij}\) is the normal overlap distance,

\(d_{ij}=\varvec{X}_{Cj}\varvec{X}_{Ci}\) is the distance between particles’ centers,

\(\varvec{v}_{n,ij}=((\varvec{v}_j\varvec{v}_i)\cdot \varvec{n}_{ij})\varvec{n}_{ij}\) is the normal component of the relative velocity,

\(\varvec{v}_{t,ij} = \varvec{v}_j\varvec{v}_i  \varvec{v}_{n,ij}\) is the tangential component of the relative velocity,

\(\varvec{\delta }_{t,ij}=\int _{T_{0}}^T \varvec{v}_{t,ij} d\tau \) is the tangential overlap [28].
However, the models above are only applicable for spherical particles. Feng and Owen [20] proposed theoretical framework for developing energyconserving normal contact models for arbitrarily shaped particles. It has been established that the normal force must be a potential field vector associated with a potential function \(\phi \) that is a function of the overlap volume [20]. However, the accurate calculation of the overlap volume may be computationally expensive and thus become not applicable for a case with millions of particles.
These equations with respect to \(\alpha _i\) and \(\alpha _j\) are easier to solve if moved to their own local reference frames:
\(f_i(\varvec{x}_{i0}+\alpha _i \varvec{\hat{n}}^i_{ij})=0\),
\(f_j(\varvec{x}_{j0}+\alpha _j \varvec{\hat{n}}^j_{ij})=0\),
Particle parameters for ellipsoids
Parameter  Value 

Three semiaxes, a, b, c (mm)  5, 2.5, 2.5 
Young’s modulus (GPa)  10 
Poisson ratio  0.3 
Density (kg/m\(^3\))  2500 
One of the disadvantages of the proposed methodology (and hence of the possible extension of the method by Zheng et al. [54] to superquadrics) is that the curvature coefficients may become zero leading to infinite curvature radii if superquadric exponents \(n_1\) and \(n_2\) are more than 2, especially in facetoface contact. The mean curvature radius becomes infinite if both principal curvature coefficients are zero. This occurs at 6 points on the particle surface: \(x=y=0,z=\pm c\), \(y=z=0,x=\pm a\) and \(x=z=0,y=\pm b\) (in the local reference frame). The Gaussian curvature radius becomes infinite if any of the principal curvature coefficients is zero. This occurs if \(x=0\), \(y=0\) or \(z=0\) (in the local reference frame). For this reason, we limit the curvature radius: \(R_{curvature}=\min (R_{curvature},qR_{vol})\), where \(R_{vol}\) is radius of the volume equivalent sphere, q is the limiting coefficient that must be chosen in advance. In the current implementation of LIGGGHTS \(q=10\) is used. The influence of the choice of the curvature radius and the limiting coefficient q on the simulation results is to be studied in the future publications.
3 Validation
3.1 Contact force between two ellipsoidal particles
Simulation parameters used in settling simulation
Parameter  Value 

Three semiaxes, a, b, c (m)  0.0025, 0.0025, 0.005 
Blockiness \(N=n_1=n_2\)  Varied: 2, 4, 6, 8, and 10 
Density (kg/m\(^3\))  2500 
Young’s modulus (GPa)  1 
Poisson’s ratio  0.3 
Coefficient of friction  0.5 
Coefficient of restitution  0.5 
Sizes of the simulation domain  0.1 \(\times \) 0.1 \(\times \) 0.25 m 
Time step \(\Delta t\) (s)  \(10^{5}\) 
Number of time steps  \(10^5\) 
3.2 Settling of particles under gravity and simulation speed
3.3 Particle–wall impact
Simulation parameters in particle–wall impact simulation
Parameter  Value 

Cylinder diameter, D (m)  \(8\times 10^{3}\) 
Cylinder length, L (m)  \(5.3\times 10^{3}\) 
Volume (m\(^3\))  \(2.49\times 10^{7}\) 
Density (kg/m\(^3\))  1245 
Mass (kg)  \(3.1\times 10^{4}\) 
Moment of inertia \(I_{xx}\) (kg/m\(^2\))  \(1.834\times 10^{9}\) 
Moment of inertia \(I_{yy}\) (kg/m\(^2\))  \(1.834\times 10^{9}\) 
Moment of inertia \(I_{zz}\) (kg/m\(^2\))  \(2.362\times 10^{9}\) 
Shear modulus (GPa)  1.15 
Poisson’s ratio  0.35 
Coefficient of friction  0.0 
Coefficient of restitution  0.85 
Time step \(\Delta t\) (s)  \(5\times 10^{7}\) 
where m is the mass of the cylinder, \(\varepsilon \) is the coefficient of restitution at the point of contact, \(V^{}_{z}\) is the preimpact translational velocity of the cylinder, \(\alpha \) is the angle between the cylinder’s face and the line joining the contact point and the center of the particle, \(\theta \) is the angle between the cylinder’s face and the wall, and \(I_{yy}\) is the moment of inertia around the yaxis. The parameter r is the distance between the cylinder’s center and the corner point C (Fig. 9), which is assumed to be fixed. Particle parameters are listed in Table 3.
The postimpact angular and translational velocities were calculated for various orientation angles \(\theta \) for the DEM simulations and compared with analytical expressions in Fig. 10. The wall was removed immediately after collision to prevent the secondary contact that occurs in reality at low and high impact angles and is not taken into account in Eqs. (41) and (42).
Simulation parameters used in pilling test case
Parameter  Value 

Density (kg/m\(^3\))  957 
Young’s modulus (Pa)  \(2.5\times 10^7\) 
Poisson’s ratio  0.25 
Coefficient of friction  0.5 
Coefficient of rolling friction  0 
Coefficient of restitution  0.5 
Sizes of the simulation domain  0.31 \(\times \) 0.03 \(\times \)0.24 m 
Time step \(\Delta t\) (s)  \(5\times 10^{6}\) 
Number of time steps  600, 000 
3.4 Piling of particles
For the second validation test case superquadric particles with the following shape parameters were used: \(a=2.0\) mm, \(b=2.0\) mm, \(c=1\) mm, \(n_1=n_2=4\). Particle parameters along with simulation setup data are listed in Table 4. They were compared with volume equivalent spherical particles of radius \(R=1.836\) mm. Domain boundaries are represented by rigid walls of the same material as the particles. In both cases the heap was formed by continuously dropping particles from a small area located above the center of the heap. As can be seen from Fig. 11 the heap shape for spherical particles (angle of repose \(31^{\circ }\)) differs from the heap shape for superquadric particles (angle of repose \(40^{\circ }\)). The heap becomes stable 1s after the dropping is stopped with almost zero maximum angular and translational velocity, which testifies the stability of the algorithms.
These simulations show importance of using nonspherical particle shapes in the Discrete Element Method. Having the same material properties the nonspherical particles can demonstrate different behavior in comparison to spherical ones just by changing the shape of the particles.
4 Numerical experiments
4.1 Angle of repose
Superquadric shape parameters for particles in the “Angle of repose” test case
Particle  a (mm)  b (mm)  c (mm)  \(n_1\)  \(n_2\) 

Sugar cube  8.5  7.5  6  10.0  10.0 
“M&M’s” dragee  6.5  6.5  3  2.0  2.0 
Chewing gum  9.5  3.25  6.4  3.0  2.0 
Material properties of particles in the “Angle of repose” test case
Parameter  Value 

Density (kg/\(m^3\))  957 
Young’s modulus, particle and wall (Pa)  \(1.0\times 10^6\) 
Poisson’s ratio, particle and wall  0.3 
Coefficient of friction, particle–particle  0.6 
Coefficient of friction, particle–wall  0.4 
Coefficient of restitution, particle and wall  0.2 
Static packing of cylinders, experimental and simulation data
Parameter  Experiment  Simulation 

Cylinder diameter, D (m)  \(8\times 10^{3}\)  \(8\times 10^{3}\) 
Cylinder length, L (m)  \(5.3\times 10^{3}\)  \(5.3\times 10^{3}\) 
Volume (m\(^3\))  \(2.664\times 10^{7}\)  \(2.49\times 10^{7}\) 
Density (kg/m\(^3\))  1160  1245 
Mass (kg)  \(3.1\times 10^{4}\)  \(3.1\times 10^{4}\) 
Shear modulus (GPa)  1.15  1.15 
Poisson’s ratio  0.35  0.35 
Coefficient of friction  0.5  0.5 
Coefficient of restitution  0.85  0.85 
Container diameter (mm)  50.6  50.6 
Container height (mm)  130  130 
Time step \(\Delta t\)(s)  −  \(10^{5}\) 
4.2 Static packing of cylinders
In this validation test we simulate a static packing of cylinders, defined as superquadric particles. The particles were dropped into a cylindrical container, and compared to the experimental data provided by Kodam et al. [29]. The DEM material properties along with particle properties are listed in Table 7. Particle size parameters, a, b, and c, were chosen such that particles in the simulation have the same diameter and length as in the experiment: \(a=b=D/2\), \(c=L/2\) (Table 7). Volume and principal moments of inertia of the cylinders defined as superquadrics have values smaller than those for true cylinders due to the rounded edges. The difference between a true cylinder and its superquadric approximation decreases with the increase of the blockiness/roundness superquadric shape parameter \(n_1\), however, leading to less stability of the method. Hence, a compromise value must be chosen. The superquadric cylinder in this simulation was set to have the same mass as the true cylinder by increasing density by 7 % and setting blockiness parameters to \(n_1=6.0, n_2=2.0\). DEM time step was set \(\Delta t=10^{5}\) s.
The image of the final state from the simulation is shown in Fig. 16. The final experimental fill height according to Kodam et al. [29] is \(53.3\pm 2.0\) mm, while superquadric DEM simulation gives the fill height of roughly \(52.0\pm 3.0\) mm which is in good agreement with the experiment. Kodam et al. [29] simulated packing of the cylinders with the multisphere approach and found that 9sphere particles underpredict the bed height by 21 %, while 54sphere particles underpredict the fill height by 8 %.
4.3 Hopper discharge
4.4 Hopper dischargeinfluence of the aspect ratio
5 Conclusion
The superquadric shape model was implemented as a separate surface model in the opensource DEM package LIGGGHTS®[28] which is an extension of the opensource package LAMMPS (Largescale Atomic/Molecular Massively Parallel Simulator) [46]. Both are massively parallel and written in C++. The program codes are available for public download.
The superquadric DEM has shown promising results along with qualitative and quantitative agreement with experimental data reported in the literature. This paper shows versatility and applicability of the superquadric DEM. The methods for contact detection, which can easily take up to 80 % of computational time [41], and contact force calculation between superquadric particles have been described in detail by using their implicit equations. The formulation employs the “contact point” which is midway and closest to both particles. The corresponding algorithms for particle–mesh interaction have also been developed but are not presented in this paper. The superquadric DEM code has been applied to various DEM problems which prove robustness and efficiency of the implemented algorithms. The methods have shown to be fast, and can be further optimized.
The superquadric particles are expected to give more accurate results than multisphere approximations for more reasonable computational time. Detailed comparison of superquadrics and multispheres in DEM is to be done in the future. The proposed methodology has the potential to be further extended for any other type of particles defined by a potential/shape function. The code is expected to be available for public download in 2017.
Notes
Acknowledgments
Open access funding provided by University of Innsbruck and Medical University of Innsbruck. This work has been carried out as a part of the TMAPPP project, an EUfunded Framework 7 Marie Curie Initial Training Network. The financial support provided by the European Commission is gratefully acknowledged.
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