A point interpolation method with least square strain field (PIM-LSS) for solution bounds and ultra-accurate solutions using triangular mesh

Abstract

A point interpolation method with least square strain field (PIM-LSS) is developed for solid mechanics problems using triangular background mesh. In the PIM-LSS, PIM shape functions are used for displacement field construction that may or may not be compatible, and a least square fitting technique is adopted to construct the strain field. A strain constructed Galerkin (SC-Galerkin) weak formulation is then proposed for establishing discretized PIM-LSS models that have a number of special properties. We proved theoretically (1) the PIM-LSS provides a “softening” effect to the FEM model, and a “stiffing” effect to the node-based smoothed point interpolation method (NS-PIM) model; (2) the exact solution is bounded from both by PIM-LSS solutions with strain field of zero-order fitting and that with higher order fitting; (3) There exists a preferred order of fitting for the strain field such that ultra-accurate (one order higher accuracy) solution can be obtained using the PIM-LSS. These theorems and properties have been confirmed in numerical examples.