The natural neighbour Petrov-Galerkin method for elasto-statics

In this paper, an efficient and accurate meshless natural neighbour Petrov–Galerkin method (NNPG) is proposed to solve elasto-static problems in two-dimensional space. This method is derived from the generalized meshless local Petrov–Galerkin method (MLPG) as a special case. In the NNPG, the local supported trial functions are constructed based on the non-Sibsonian interpolation and test functions are taken as the three-node triangular FEM shape functions. The local weak forms of the equilibrium equation and the boundary conditions are satisfied in local polygonal sub-domains. These sub-domains are constructed with Delaunay tessellations and domain integrals are evaluated over included Delaunay triangles by using Gaussian quadrature scheme. As this method combines the advantages of natural neighbour interpolation with Petrov–Galerkin method together, no stiffness matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Several numerical examples are presented and the results show the presented method is easy to implement and very accurate for these problems.