A mixed selective edge-based smoothed PFEM with second-order cone programming for geotechnical large deformation analysis

Traditional particle finite element method (PFEM) with a six-node triangular (T6) element using an iteration algorithm often suffers from high computational costs and non-convergence problems. This paper proposes a novel dynamic implicit optimisation based smoothed PFEM for solving geotechnical large deformation problems. The governing equations are treated as an equivalent min–max optimisation that is recast as a second-order cone programming (SOCP) problem. Owing to the particularity of the min–max optimisation problem, which needs to simultaneously interpolate displacement and stress field, a novel mixed constant-stress selective edge-based smoothed strain element is proposed in which strain and stress are constant in the smoothing domain. The two-level mesh repartitioning scheme is implemented into the proposed method to overcome volumetric locking. Next, a mass lumping scheme using nodal integration is developed to improve computational efficiency. An efficient and simple variable mapping scheme is finally proposed and implemented into PFEM to tackle large deformation problems. Several classical statics and dynamics examples are used to validate the proposed method. All results demonstrate that the proposed method only using the lower-order element offers high accuracy, high efficiency, mesh insensitivity and is free of volumetric locking for geotechnical problems under both small and large deformation.