An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems

This paper presents a novel element-free smoothed radial point interpolation method (EFS-RPIM) for solving 2D and 3D solid mechanics problems. The idea of the present technique is that field nodes and smoothing cells (SCs) used for smoothing operations are created independently and without using a background grid, which saves tedious mesh generation efforts and makes the pre-process more flexible. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form that requires only discrete values of shape functions that can be created using the RPIM. By varying the amount of nodes and SCs as well as their ratio, the accuracy can be improved and upper bound or lower bound solutions can be obtained by design. The SCs can be of regular or irregular polygons. In this work we tested triangular, quadrangle, n-sided polygon and tetrahedron as examples. Stability condition is examined and some criteria are found to avoid the presence of spurious zero-energy modes. This paper is the first time to create GS-Galerkin weak-form models without using a background mesh that tied with nodes, and hence the EFS-RPIM is a true meshfree approach. The proposed EFS-RPIM is so far the only technique that can offer both upper and lower bound solutions. Numerical results show that the EFS-RPIM gives accurate results and desirable convergence rate when comparing with the standard finite element method (FEM) and the cell-based smoothed FEM (CS-FEM).