An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh

摘要

A point interpolation method (PIM) with continuous strain field (PIM-CS) is developed for mechanics problems using triangular background mesh, in which PIM shape functions are used to construct both displacement and strain fields. The strain field constructed is continuous in the entire problem domain, which is achieved by simple linear interpolations using locally smoothed strains around the nodes and points required for the interpolation. A general parameterized functional with a real adjustable parameter a are then proposed for establishing PIM-CS models of special property. We prove theoretically that the PIM-CS has an excellent bound property: strain energy obtained using PIM-CS lies in between those of the compatible FEM and NS-PIM models of the same mesh. Techniques and procedures are then presented to compute the upper and lower bound solutions using the PIM-CS. It is discovered that an extended Galerkin (x-Galerkin) model, as special case resulted from the extended parameterized functional with alpha = 1, is outstanding in terms of both performance and efficiency. Intensive numerical studies show that upper and lower bound solutions can always be obtained, there exist a values at which the solutions of PIM-CS are of superconvergence, and the x-Galerkin model is capable of producing superconvergent solutions of ultra accuracy that is about 10 times that of the FEM using the same mesh.