# On the contact treatment of non-convex particles in the granular element method

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## Abstract

We present a new contact algorithm that endows the granular element method [1] with the ability to model non-convex particles using non-uniform rational basis splines. This significant extension allows for the representation of particle morphological features, namely, sphericity and angularity, to their fullest extent, with local contact rolling resistance and interlocking emanating directly from grain geometry. Both particle elasticity and friction at the contact level are treated implicitly and simultaneously, and the contact algorithm is cast into a mathematical programming-based contact dynamics framework. The framework provides the advantages of implicit time integrators (for e.g., stability and larger time steps) and ability to handle both rigid and highly stiff particles. By allowing for particle non-convexity, modeling flexibility is significantly enhanced, to a level that is comparable with isogeometric methods. As such, the transition from image data to particle shapes is greatly streamlined. More importantly, increased macroscopic strength in granular packings comprising of non-convex particles is fully captured. All the above capabilities are achieved under a very modest implementation effort.

### Keywords

Non-convex particles Discrete element method Granular element method Contact dynamics NURBS## 1 Introduction

The objective of this paper is to develop a new contact algorithm that endows the granular element method (GEM) [1] with the ability to model non-convex particles using non-uniform rational basis splines (NURBS). This significant extension allows for the representation of particle morphological features, namely, sphericity and angularity [2], to their fullest extent, with local contact rolling resistance and interlocking emanating directly from grain geometry. Both particle elasticity and friction at the contact level are treated implicitly and simultaneously, and the contact algorithm is cast into a mathematical programming-based contact dynamics framework. The framework provides the advantages of implicit time integrators (for e.g., stability and larger time steps) and ability to handle both rigid and highly stiff particles. By allowing for particle non-convexity, modeling flexibility is significantly enhanced, to a level that is comparable with isogeometric methods [3]. As such, the transition from image data to particle shapes is greatly streamlined. More importantly, increased macroscopic strength in granular packings comprising of non-convex particles is fully captured. All the above capabilities are achieved under a very modest implementation effort.

Recent discrete simulation approaches include techniques to represent complex particle morphology or shape, beyond discs/spheres and ellipses/ellipsoids, which are based mostly on the clustering or clumping technique [4, 5] and polyhedra approach [6, 7, 8, 9, 10]. Alternatives, which are essentially combinations of the aforementioned techniques, include spheropolyhedra [11, 12] and potential particles [13, 14]. All development work to date, however, have focused almost exclusively on the treatment of convex particles. Polyhedra have the capability to handle non-convex particles, but are still limited in resolution and discrete models tend to be very blocky, a result of the primitive simplexes underlying the geometry basis, as well as the complexity of the associated contact detection algorithms [15]. As a result, the shape representation capabilities of polyhedra have not been fully realized. By and large, clustering appears to be the most popular approach taken to handle non-convexity due to its simplicity (see for e.g., [16, 17]), although in actual simulations, the number of disks or spheres clustered to form a particle is usually small to minimize computational cost. While simple, clustering is unappealing because of its lack of continuity in the curvatures and tangents on the particle boundaries, which results in clustered particles appearing ‘clumpy’ at locations where spheres overlap or clump, and the curvature at any point on the resulting boundary always positive. The geometrical anomalies associated with the clustering technique prevents it from enabling higher fidelity contact mechanics calculations without further numerical treatment.

While the influence of particle morphology on properties such as strength, permeability, etc. is well established [2, 18], it appears that grain-scale modeling and characterization efforts have remained compartmentalized, as exemplified by a relative lack of connection between real experiments and numerical studies. In cases where discrete modeling of real granular materials with non-trivial geometries were attempted, crude discrete models with large geometrical biases and significantly calibrated parameters have been widely employed. Interestingly, the effects of geometrical bias on grain-scale response from use of simplified geometries are largely not discussed or quantified in the literature. It appears that the current gap between grain-scale modeling and characterization technologies is quite large. While imaging techniques are becoming increasingly sophisticated [19, 20, 21, 22, 23], there continues to be a lack of effort to bring discrete granular simulation technology closer to the engineering application level. The development to be discussed in this paper may help in narrowing the gap between grain-scale modeling and characterization.

Our earlier work on GEM has focused on particles geometries that are angular but strictly convex. While GEM has been shown to improve particle morphology representation beyond disks and spheres, it was still limited in two ways. First, the increase in rolling resistance of angular but convex geometries relative to disks is limited. For instance, rolling resistance provided by distributed contact reaction over flat boundaries cannot be represented using strictly convex shapes. Moreover, interlocking behavior between non-convex particles, which contributes significantly to mobilized strength and stability [16, 17], is not accounted for. Second, the generation of strictly convex NURBS shapes is very difficult and restrictive from a modeling perspective. This is even more so when dealing image data of real particle shapes and obtaining strictly convex boundaries through a fitting procedure is not possible in most cases. We emphasize that this difficulty is not due to the limitation of NURBS itself but rather the need to deal with contact algorithms for NURBS, which were undeveloped in GEM at the time. A contact algorithm capable of dealing with general non-convex NURBS particles, to be described in this paper, would eliminate the above two limitations. As a result, a more faithful representation on the contact force distributions over particle boundaries is obtained and the image data-to-analysis pipeline (see for e.g., [24]) is significantly streamlined.

We extend a variational formulation of granular contact dynamics (CD) recently developed for convex particles [25, 26, 27, 28]. This particular formulation, which is employed in this paper, is appealing because it provides a way for CD to be easily implemented and solved using off-the-shelf mathematical programming solvers. The most prominent advantage of this formulation is its automatic inclusion of the quasi-static limit, enabling quasi-static modeling without the need for adjusting damping parameters or time step. The advantages of CD over DEM in the context of our current applications, as well as their differences, have been discussed in [28].

We generalize the node-to-surface approach typically used in the contact treatment of finite element models [29] to a ‘knot-to-surface’ approach to enable the contact treatment of non-convex particles described using NURBS. The latter is similar to that of a recent approach used in the contact treatment of frictionless bodies in isogeometric analysis [30]. The key difference and novelty here is on the simultaneous treatment of contact elasticity and frictional contact within the aforementioned CD formulation, as well as the ability to perform contact calculations for granular systems, which contain a large number of particles.

*Remark 1*

Isogeometric analysis is a computational mechanics technology that uses basis functions emanating from computer aided geometric design (CAGD), such as B-splines, NURBS, and T-splines. It has been shown that isogeometric analysis provides more precise and efficient geometric representations [3]. The advantages of isogeometric basis functions for the representation of particle morphology in discrete simulations have been discussed in [1, 31].

## 2 Governing equations for frictional particles

### 2.1 General contact problem definition

### 2.2 Notation for general multi-particle system

A quantity, which at the initial time is denoted by \(\Box _0\), would then be denoted at time \(t_0+\Delta t\) by \(\Box \). For example, \(\varvec{x}_0\) and \(\varvec{v}_0\) are the known positions and velocities at time \(t_0\) while \(\varvec{x}\) and \(\varvec{v}\) are the corresponding quantities at time \(t_0+\Delta t\). With this notation, an increment of a quantity will be denoted by \(\Delta \Box = \Box -\Box _0\).

### 2.3 Discrete update equations

### 2.4 Variational formulation of contact problem

In Eq. (7), the matrix \(\varvec{R}^q_0\) contains the contribution of the total angular momentum balance from the tangential forces and contains entries \(\varvec{R}^T_{iI}\varvec{n}_0\) where \(\varvec{R}_{iI}\) is the moment arm vector extending from the centroid of particle \(i\) to the contact point \(\varvec{\bar{y}}\). The matrix \(\varvec{R}^p_0\) contains the contribution of the total angular momentum balance from the normal contact forces and contains entries \(-\varvec{R}^T_{iI}\varvec{t}_0\). Both \(\varvec{R}^T_{iI}\varvec{n}_0\) and \(-\varvec{R}^T_{iI}\varvec{t}_0\) are signed moment arms and their signs depend on whether the associated contact force induces a positive (clockwise) or negative moment on the particle. A similar description applies to the slave particle using its contact normal \(-\varvec{n}_0\) and tangent \(-\varvec{t}_0\). The matrices \(\varvec{C}_N\) and \(\varvec{C}_T\) contain the compliances \(1/k_N\) and \(1/k_T\) on the diagonal, where \(k_N\) and \(k_T\) are the normal and tangential contact stiffnesses, respectively. Finally, the Coulomb criterion is imposed with \(\mu = \tan \phi \) being the effective inter-particle friction coefficient and \(\phi \) is the effective friction angle at the scale below the particle angularity level.

### 2.5 Optimality conditions

### 2.6 Force-based problem

### 2.7 Static limit

### 2.8 Solution procedure and computational complexity

We observe that Eqs. (14) and (15) are essentially standard quadratic programming problems. In this work, the primal-dual interior-point solver in MOSEK [35] is used for the solution of both problems. The solution and storage costs associated with these problems are usually justified by the larger analysis steps that can be taken when using implicit algorithms. This is more so for systems comprised of rigid or highly stiff particles in which explicit solution procedures perform poorly or simply fails. Moreover, large-scale mathematical programming solvers with sparse storage (for e.g., [35, 36]) are becoming widely available and increasingly efficient and robust. More recent solvers such as MOSEK [35] also include multi-core or multi-threaded capabilities.

## 3 Contact algorithm

### 3.1 Non-uniform rational basis-splines (NURBS)

- 1.
A set of \(n+1\) control points \(\mathbf {P}_i\)\((0 \le i \le n)\),

- 2.
A knot vector \(U\) of \(m+1\) knots \((0 = u_0 \le u_1 \le \cdots \le u_{m-1} \le u_m = 1)\), and

- 3.
A degree \(p\) satisfying \(m=n+o\) where \(o=p+1\) is the order of the curve.

Finally, we close the curve using the following approach: the first and last control points are made coincident, i.e., \(\varvec{P}_0=\varvec{P}_n\) and the two coincident and adjacent control points are made collinear to maintain curvature continuity.

*Remark 2*

Polygonal bodies can be generated using NURBS curves having degree \(p=1\) and full multiplicity \(k=p\) at each internal knot. In this case, existing polyhedra-based contact algorithms are fully applicable. In this paper, we focus on the case of smooth non-convex bodies described using NURBS, with \(p > 1\) where the normals and tangents are well-defined everywhere on the boundary.

### 3.2 Boundary knot positioning

We generalize the node-to-surface approach typically used in the contact treatment of finite element models [29] to a ‘knot-to-surface’ approach to enable the contact treatment of non-convex particles described using NURBS. Following a master-slave approach, the contact points associated with the slave particle are represented by knots (i.e., parameter values) while the master boundary serves as the constraint boundary for non-penetration of contact points. This is necessary for computational tractability, as well as for tracking the incremental slip and contact gain or loss around non-convex boundaries of potentially contacting particles.

While similar to nodal discretization in FEM, we emphasize that the key difference here is that the positioning of contact points by knots does not change the particle geometry; the knot or parameter value simply runs continuously along the boundary. For simplicity, we have selected the knot positions a priori and in sufficient number (between 40 and 60 in our examples here) to represent the particle geometry and contact problem appropriately. More complicated schemes such as the adaptive positioning of contact points to reduce the number of knots, as well as more advanced interface contact discretization strategies may be possible; further research, however, is required to determine the feasibility of these ideas in the context of granular simulations.

### 3.3 Contact elasticity

In this paper, we have only considered the case of linear contact elasticity as shown in Fig. 3 but extension to nonlinear elasticity is entirely possible as is the consideration of more complex contact models incorporating hardening, viscous effects, etc.

The static problem described by Eq. (15) reveals a number of interesting properties related to the indeterminacy of force networks in granular media. It is well known that rigid particles lead to a situation where the force network solution is non-unique [43, 44, 45, 46]. Setting \(\varvec{C}_N = \varvec{C}_T = \varvec{0}\) in Eq. (15) leads to a linear program where global optimality may be achieved by more than one set of forces. Conversely, for finite values of \(\varvec{C}_N\) and \(\varvec{C}_T\), the solution is unique, i.e. there is a unique set of contact forces leading to the optimal value of the objective function.

### 3.4 Implementation of contact algorithm

The key procedure required for the implementation of the proposed contact algorithm is the closest point projection (CPP) as described in Algorithm 1. Here, the contact points on the slave boundary are projected onto the master curve, from which the signed gap values can then be calculated. The minimization problem in Algorithm 1 can be solved using either gradient-based or derivative-free constrained optimization procedures. Here, we choose the derivative-free procedure since it is simpler and does not require the evaluation of the derivatives of the objective function. As a simple additional optimization, Algorithm 2 first isolates the spans that are nearest to a particular contact point by checking that the dot product of normals at a contact point and its corresponding closest-projected point is negative. This means that only the spans on the master curve that are within the direct sight of a contact point on the slave boundary are used in the CPP operation. Combining the above two algorithms, Algorithm 3 describes the global procedure to determine all the signed gaps over all particles, which are then stored in the vector \(\varvec{g}_0\) in either Eq. (14) or (15). The broad-phase contact detection aspect of Algorithm 3 is standard (see for e.g., [15]).

By considering the knot spans as segments, our proposed contact algorithm has essentially the same computational complexity as polyhedra-based approaches employed in either CD [9] or finite element contact algorithms [29]. We note that for a given prescribed resolution, however, NURBS has the advantage of carrying significantly less geometric information compared to polyhedra, i.e., less number of NURBS control points compared to number of polyhedra nodes [1]. For quasistatic applications, further efficiency enhancement to the proposed contact algorithm can be achieved by taking advantage of temporal coherence in which the configuration of particles from one time step to the next changes little. In this case, an incremental approach similar to that of [47] may be adapted to minimize the number of nearest span computations (Algorithm 2).

*Remark 3*

If the curves are locally convex, the separating hyperplane algorithm described in [31] can be employed as an additional refinement. For general smooth non-convex bodies, however, there is no local convexity everywhere in general. The proposed contact algorithm would handle this situation effectively and robustly.

## 4 Numerical examples

In this section, we present two examples that highlight the effects of particle morphology on the macroscopic response of granular assemblies, as well as the robustness of our proposed method. In particular, we compare responses of assemblies with three levels of particle angularity: discs, angular but (strictly) convex, and non-convex particles.

### 4.1 Biaxial compression

Biaxial compression simulations using the static limit formulation in Eq. (15) are carried out on a rectangular assembly of initial width \(W_0\) and initial height \(H_0\) containing 1,520 particles. First, an assembly with GEM non-convex particles is prepared using 16 different shapes. Then, two additional assemblies—one with angular convex GEM particles and another with discs—are also prepared. The particles in these assemblies are obtained by matching average particle diameters of the 16 non-convex shapes. Effectively, sphericity is kept constant and a comparison of effects of angularity is made. The three assemblies with their corresponding particle shapes are shown in Fig. 4.

The non-convex and angular GEM assemblies have an initial porosity of approximately 0.152, while the disc assembly has an initial porosity of approximately 0.176. The higher porosity of the disc assembly points to the inability of discs to match porosity by just simply matching average particle diameters. Indeed, a wider distribution of disc sizes would be required in this case to match the initial porosity of 0.152, which in turn would substantially increase the number of particles. In this regard, the use of discs to represent particle geometry introduces an unavoidable geometrical bias, which leads to packings with higher porosities [17]. Nevertheless, in this example, we retain the disc assembly for comparison with the other two GEM assemblies. Two inter-particle friction coefficients \(\mu =0.3\) and 0.5 are used to gauge the effect of inter-particle friction. The upper wall is moved downwards while the applied stress on the right wall \(\sigma _3\) is maintained at 125 units. A total of about 150 steps are used to impose a total axial strain of approximately 0.21. The left and bottom walls are stationary. All walls are frictionless.

To show the effects of particle elasticity, we perform the tests with several values of particle elasticity: \(k_N=\infty ,\; 10^8,\; 10^6,\; 10^5\). The tangential stiffness is set at \(k_T=2 k_N/3\), which is within the range for a physically consistent volumetric response in granular materials [26, 48]. At every time step, the current width \(W\) and current height \(H\) of the assembly box are tracked, and the stresses \(\sigma _1\) and \(\sigma _3\) computed using the contact forces of the particles impinging on the top and right walls, respectively. The results in terms of deviatoric stress \(\sigma _1 - \sigma _3\) versus axial strain, \(\epsilon _a = 1 - H/H_0\), and volumetric strain, \(\epsilon _v = 1 - WH/(W_0H_0)\), versus axial strain are shown in Figs. 5, 6, 7 and 8. We see that the macroscopic response at \(k_N=10^8\) is close to rigid. As \(k_N\) is lowered, a more elastic initial response is observed in which the sharp initial peak is progressively suppressed and the peak response lowers slightly. At \(k_N=10^5\), an initial slope in the deviatoric stress becomes visible and the corresponding volumetric strain response shows an initial compaction followed by volume expansion. The deformed configurations of the three assemblies for the case of \(k_N=10^8\) are shown in Fig. 4.

In all cases, we note the following observations. For a particular assembly, the macroscopic deviatoric stress reaches a constant value that is independent of the elastic properties while the rate of volumetric strain tends to zero, in agreement with standard continuum plasticity theories. Comparing across the three assemblies with different particle angularity levels, however, we observe that the both the peak strength and dilatancy increase with increasing angularity i.e. from discs to non-convex. This latter observation is consistent with experimental evidence of increased strength with increasing angularity of the particles [2].

### 4.2 Column drop test

We compare the response evolutions of the angular GEM and disc columns against the non-convex GEM column as shown in Figs. 9 and 10. The final configurations of the three columns for \(\mu = 0.5\) are shown in Fig. 11. The slopes of the final spreads of the angular GEM and disc columns are approximately \(12^{\circ }\) and \(9^{\circ }\), respectively. More prominently, the slope in the non-convex GEM column is \(17^{\circ }\). This is a \(5^{\circ }\) and \(8^{\circ }\) increase from the angular GEM and disc columns, respectively, which are quite significant. Relative to the non-convex GEM column, the final spreads of the angular GEM and disc columns are approximately 14 and 45 percent wider. These observations are consistent with the increase of rolling resistance with increasing angularity.

For the case of \(\mu = 0\), the response evolutions of the angular GEM and disc columns as compared with the non-convex GEM column are shown in Figs. 12 and 13, with the final configurations of the GEM columns shown in Fig. 14. At Step 3,200, the two GEM columns have stopped flowing while the disc column continues to flow and as the simulation is progressed, a final layer thickness of 1 particle is reached. Essentially, without rolling resistance, the disc column simply ‘melts’ away. On the other hand, the GEM columns maintain a well-defined spread, even at zero inter-particle friction, due to the rolling resistance provided by angular and non-convex particles. As expected, the non-convex GEM column has a smaller spread due to increased angularity in the non-convex particles.

## 5 Conclusion

We have presented a new contact algorithm that enables the GEM to model non-convex particles. Its implementation is very simple, with the main operation being the closest point projection. The overall simplicity of GEM is retained by adopting a variational framework of contact dynamics, which allows the use of realistic contact stiffnesses and the system can be solved using reasonably large time steps. With this new contact algorithm, GEM is able to represent, through Non-Uniform Rational Basis Splines (NURBS), particle morphological features of sphericity and angularity to their fullest extent. Modeling flexibility is also greatly enhanced, streamlining the image data-to-analysis pipeline.

Through numerical simulations of the biaxial compression and column drop tests, we demonstrated and highlighted the ability of GEM to capture the effects of increased rolling resistance, associated with increased angularity in and interlocking between non-convex particles, on the macroscopic response. These effects are manifested macroscopically through an increase in the mobilized shear strength and dilatancy under biaxial compression, and a smaller spread and higher angle of response under a column drop test. These observations are consistent with reported experimental observations. The effect of geometrical bias from the use of discs to match average particle diameter on packing porosity, which in turn affects mobilized strength, is also noted.

In the present work, we have focused on the numerical verification aspect of GEM. Future research on GEM would include further algorithmic improvements on contact algorithms and solution procedures, as well as on the extension of non-convexity into the three-dimensional setting. At the present state, however, GEM has sufficient capabilities to allow for its application in lab-scale validation with experiments similar to that described in [49], and this endeavor is currently under preparation. While the comparison of numerical and experimental micromechanical quantities remains challenging due to system sensitivities and uncertainties, it is expected that GEM would be able to provide access to contact topologies, forces and kinematics that are unique to natural granular media with realistic particle morphologies. This would bring us a step closer toward the goal of complete characterization of natural granular media for engineering applications.

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