An updated Lagrangian framework with quadratic element formulations for FDEM

Abstract

In this study, quadratic element formulations (i.e., ten-node tetrahedral elements and twelve-node cohesive elements) are proposed to improve the computational accuracy of the three-dimensional combined finite-discrete element method (3D FDEM). The ten-node tetrahedral element formulation is developed based on the isoparametric transformation. As for the twelve-node cohesive element formulation, the opening displacement is described by quadratic interpolation within the element, which is fully compatible and connected with the adjacent ten-node tetrahedral elements. To integrate quadratic element formulations into FDEM, a novel and more generalized updated Lagrangian framework (UL-FDEM) is developed to replace the non-Cartesian framework in the original FDEM. The equilibrium equation of UL-FDEM is established based on the principle of virtual work. Subsequently, the governing equation of UL-FDEM is derived by employing the isoparametric discretization techniques. Several numerical examples are provided to illustrate the effectiveness of the proposed approach. The results demonstrate that this approach is well-suited for the geometrically nonlinear analysis and achieves higher accuracy in stress and fracturing predictions than the original FDEM under the same mesh size. The proposed approach can use fewer elements to perform fracturing analysis, thus improving computational efficiency.