A Consistent Time Level Implementation Preserving Second-Order Time Accuracy via a Framework of Unified Time Integrators in the Discrete Element Approach

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.