The natural neighbour Petrov–Galerkin method for thick plates S.L. Li n , K.Y. Liu, S.Y. Long, G.Y. Li State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha 410082, China article info Article history: Received 8 August 2010 Accepted 5 November 2010 Available online 24 December 2010 Keywords: Meshless Natural neighbour interpolation Petrov–Galerkin method Thick plates Delaunay triangulation abstract 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 O 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. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction The finite element method (FEM) is a well established and most successful numerical method which has been widely applied in different fields of engineering and the applied sciences. In spite of its well-known specific features and numerous advantages, there is an on-going thrust in the development. In recent years, consider- able efforts have been devoted to develop various meshless methods to avoid the mesh-related difficulties for certain problems such as large deformation or crack propagation. Many meshless methods have been proposed, such as the smooth particle hydro- dynamics (SPH) [1], the element-free Galerkin method (EFG) [2], the reproducing kernel particle method (RKPM) [3], the partition of unity finite element method [4], the meshless local Petrov– Galerkin method (MLPG) [5], and several others. The advantages of these meshless methods are apparent, the approximations are constructed entirely on a set of scattered nodes, and any predefined nodal connectivity is not needed, and therefore avoid the numerical difficulties of mesh entanglement in FEM. However they pay for the high cost in the computational time, the imposition of essential boundary conditions, and the treatment of material discontinuities. Special techniques, such as Lagrange multiplier [2], the penalty method [5], the modified variational principle [6], and the efficient computation of shape functions [7], have been proposed to over- come the problems. But the progress is miniscule. The natural element method (NEM) [8,9] and natural neighbour Galerkin method (NNGM) [10] are another class of meshless methods. The interpolation scheme used in this class of methods is known as natural neighbour interpolation. Unlike the moving least square (MLS) technique which plays a crucial role in meshless methods, the natural neighbour interpolation shape functions have Kronecker Delta function property, which enables the meshless methods have the ability of easy imposition of essential boundary condition. The meshless local Petrov–Galerkin method (MLPG) which is based on local weak formulation proposed a new integration method in a local domain and permits trial and test functions from different spaces. The integrals are evaluated over sub-domain and no extra quadrature background cells are needed. Remarkable successes of MLPG and their variation have been reported in elastic dynamic problem [11,12], contact analysis of elastomers [13], analysis of cracks in the isotropic functionally graded material [14], elastic transient analysis [15], etc. Recently, the natural neighbour Petrov–Galerkin method (NNPG) was pro- posed by Wang et al. [16], which intends to combine the advantage of easy imposition of essential boundary conditions of NEM with some prominent features of the MLPG. Plate structures are widely used in many engineering structures. It is well known that the classical thin plate theory of Kirchhoff gives rise to certain non-physical simplifications mainly related to the omission of the shear deformations and rotary inertia, which are growing significantly for increasing thickness of the plate. The effects of shear deformation and rotary inertia are taken into account in the Reissner–Mindlin plate bending theory [17,18]. Recently, geometrically nonlinear analysis [19] and thermal bend- ing [20] of Reissner–Mindlin plate by meshless method have been Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.11.003 n Corresponding author. Tel.: + 86 0731 8822114. E-mail address: shunlili66@163.com (S.L. Li). Engineering Analysis with Boundary Elements 35 (2011) 616–622