In this work, a consistent and physically accurate implementation of the general framework of unified second-order time accurate integrators via the well-known GSSSS framework in the Discrete Element Method is presented. The improved tangential displacement evaluation in the present implementation of the discrete element method has been derived and implemented to preserve the consistency of the correct time level evaluation during the time integration process in calculating the algorithmic tangential displacement. Several numerical examples have been used to validate the proposed tangential displacement evaluation; this is in contrast to past practices which only seem to attain the first-order time accuracy due to inconsistent time level implementation with different algorithms for normal and tangential directions. The comparisons with the existing implementation and the superiority of the proposed implementation are given in terms of the convergence rate with improved numerical accuracy in time. Moreover, several schemes via the unified second-order time integrators within the framework of the GSSSS family have been carried out based on the proposed correct implementation. All the numerical results demonstrate that using the existing state-of-the-art implementation reduces the time accuracy to be first-order accurate in time, while the proposed implementation preserves the correct time accuracy to yield second-order.

In order to numerically describe the behavior of granular materials (not continuous body) represented as a collection of large number of distinct particles, the discrete/distinct element method (DEM), proposed in [

One of the common applications for the DEM is particle packing. Various packing methods using DEM have been studied previously such as, Formation of a pile, i.e., sand piling process [

Furthermore, the DEM not only simulates granular flow, but it can also be used for solid body analysis and assessing fracture as studied in [

In transient dynamic simulation of the DEM type problems, contact forces and particle motion need to be evaluated and updated at every time step. The widely used time integrations in the DEM numerical simulations include the explicit Euler method, fourth-order Runge-Kutta (RK-4) method, and central difference method/velocity verlet method, etc. It is straightforward to implement these methods in the DEM simulation; however, these explicit time stepping techniques possess several numerical disadvantages in addition to the conditionally stable features; for example, the Euler methods are first-order accurate in time, RK-4 method requires additional computations of high-order derivatives of differential variables [

In this work, the main objective is to show the consistent implementation of the GSSSS family of algorithms to the simulation of granular assemblies with DEM while retaining all the advantages of numerous methods within the present framework of algorithms. It is worthy to mention that one major drawback of the DEM technique, as well as the other particle based methods or meshless methods [

DEM is a numerical simulation technique for a system of small “distinct” particles/elements, such as powders and granular materials. For the sake of simplicity, we assume the shape of every particle/element in a system is a disk in the two-dimensional Euclidean space

Let

Note that

Hence,

Similarly, we can derive the relative acceleration vectors of point

The resultant contact force and torque acting on particle

The viscous damping coefficient

Similarly, the tangential contact force,

We usually take

However, as opposed to the linear model,

On the other hand, the tangential contact force,

Applying the Newton-Euler equation to particle

In a unified format for all particles (

Time Discretization

Consider the temporal discretization of the governing equations for a time interval

The algorithmic parameters, also known as the DNA [discrete numerically assigned] markers, are defined with the input scalar algorithmic parameters

Suppose particles

Note that

If the contact starts at time

When

When slip occurs at the algorithmic time level

The evaluations of

It is worth noting that the tangential acceleration is usually not included during the computations of the tangential contact forces, i.e., for the central difference method, for example,

For the evaluation of the tangential displacement

To verify and validate the concepts described in the previous section, two numerical experiments are given in this section. The main objective of

To valid our proposed approach and the DEM code, two cases are designed to test the performance of the proposed tangential displacement evolution. Case I is designed to extract the tangential displacement calculation in the general discrete element method. The influence of normal direction is avoided in Case I via fixing a two dimensional disk with only initial angular velocity and constant normal force. Case II gives a general ball falling problem with gravity and the comparisons of traditional and consistent tangential displacement evolution.

The comparison of the results with different tangential displacement evaluation methods is given in

Problem Description: The numerical example chosen in this section is that of a simple granular cube drop of particles in three-dimensional space, as shown in

In this section, the consistent time level implementation and preserving the second-order time accuracy for the presented results in this section are as follows for a wide variety of time integrators in the present unified framework. They yield no overshoot in displacement or no overshoot in velocity or no overshoot in both displacement and velocity for a given set of initial conditions:

CDM-T: Central Difference Method with traditional tangential displacement evaluation. Note that the central difference method has been proved to be equivalent to the GS4-II U0-based family with

CDM-C: Central Difference Method with consistent tangential displacement evaluation.

V0(1, 1, 0)-C: GS4-II V0-based family with

U0/V0(0, 1, 0)-C: GS4-II U0-based family or V0-based family with

In order to capture particle contacts, the time step size used in DEM type simulations is always required to be very small. This implies that the implicit time integration schemes are not necessary in the general DEM simulations. Therefore, in this work, all the schemes listed are explicit and the time step size is

Schemes | n | Time | ||
---|---|---|---|---|

CDM-C ( |
3375 | 1500 | 100 | 0.740 |

CDM-T ( |
3375 | 1500 | 100 | 0.732 |

V0 (1, 1, 0)-C ( |
3375 | 1500 | 100 | 0.725 |

U0/V0 (0, 1, 0)-C ( |
3375 | 1500 | 100 | 0.737 |

Numerical Results and Discussion

Three-dimensional snapshots with the granular velocity filed values at six different times (

The time accuracy for translational and rotational position, velocity and acceleration of the central difference method with the traditional tangential displacement evaluation shows only first-order time accuracy in

In this work, a consistent tangential displacement evaluation approach is proposed under the framework of the unified second-order time integrators GS4-II family for improving discrete element method (DEM) simulations. The newly proposed approach is derived and designed to maintain the simplicity and generalization of the GS4-II family. Comparing it with the traditional tangential displacement evaluation method, the proposed consistent time level implementation approach can be easily implemented and yields the correct algorithmic tangential displacement at the same time level as in the time integrations via using the same parameters (

We appreciate all the constructive comments and suggestions from editors and reviewers.