The natural neighbour Petrov–Galerkin method for thick plates

In this paper, a meshless natural neighbour Petrov–Galerkin method (NNPG) is presented for a plate described by the Mindlin theory. The discrete model of the domain Ω consists of a set of distinct nodes N, and a polygonal description of the boundary. In the NNPG, the trial functions on a local domain are constructed using natural neighbour interpolation and the three-node triangular FEM shape functions are taken as test functions. The natural neighbour interpolants are strictly linear between adjacent nodes on the boundary of the convex hull, which facilitate imposition of essential boundary conditions. The local weak forms of the equilibrium equations and the boundary conditions are satisfied in local polygonal sub-domains in the mean surface of the plate. These sub-domains are constructed with Delaunay tessellations and domain integrals are evaluated over included Delaunay triangles by using Gaussian quadrature scheme. Both elasto-static and dynamic problems are considered. The numerical results show the presented method is easy to implement and very accurate for these problems.