This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for solid mechanics using three-node triangular meshes. In the ES-PIM, displacement fields are construed using the point interpolation method (polynomial PIM or radial PIM), and hence the shape functions possess the Kronecker delta property, facilitates the enforcement of Dirichlet boundary conditions. Strains are obtained through smoothing operation over each smoothing domain associated with edges of the triangular background cells. The generalized smoothed Galerkin weak form is then used to create the discretized system equations and the formation is weakened weak formulation. Four schemes of selecting nodes for interpolation using the PIM have been introduced in detail and ES-PIM models using these four schemes have been developed. Numerical studies have demonstrated that the ES-PIM possesses the following good properties: (1) the ES-PIM models have a close-to-exact stiffness, which is much softer than for the overly-stiff FEM model and much stiffer than for the overly-soft node-based smoothed point interpolation method (NS-PIM) model; (2) results of ES-PIMs are generally of superconvergence and "ultra-accurate"; (3) no additional degrees of freedom are introduced, the implementation of the method is straightforward, and the method can achieve much better efficiency than the FEM using the same set of triangular meshes.
Goal-oriented error estimation and mesh adaptation for shallow water modelling
