Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions J. Amani n , M.H. Afshar, M. Naisipour Enviro-Hydroinformatics COE, School of Civil Engineering, Iran University of Science and Technology (IUST), P.O. Box 16765-163, Narmak, Tehran, Iran article info Article history: Received 22 February 2010 Accepted 9 September 2011 Available online 15 December 2011 Keywords: Discrete least squares meshless Mixed formulation Planar elasticity Meshless methods MLS approximation abstract A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The meshless methods have shown to enjoy some distinct advantages over the mesh-based approaches. The ability to solve the problems only using a set of arbitrarily distributed nodes not requiring any certain connectivity is the most important of these. Other capabilities such as successful application to elastic fracture problems with excellent accuracy can also be attributed to the meshless methods [1,2]. The meshless methods have, therefore, become an important tool in computational solid and fluid mechanics, and especially for solving problems with severe distortion, discontinuities and moving boundaries. The main idea of these methods is to approximate the unknown field by a linear combination of shape functions built without having resource to mesh the domain. Instead, nodes are scattered in the domain and a certain weight function with a local support is associated with each of these nodes. The shape functions associated with a given node is then built considering the weight functions whose support overlaps the weight function of this node. In the last decade, several well-known meshless methods have been proposed and used to solve a variety of problems in the literature. The meshless method has become a promising alter- native to mesh-based methods and in particular the widely used in finite element method. Smoothed Particle Hydrodynamics (SPH) was the first meshless method proposed by Gingold and Monaghan [3] in which formulated using strong form conception. Then Nayroles et al. [4] proposed Diffuse Element Method (DEM) as a first weak form meshless method. Belytschko et al. [5] extended the DEM to more solid foundation within the framework of Galerkin weak form and developed an Element-Free Galerkin (EFG) method. Liu et al. [6] proposed Reproducing Kernel Particle Meshless (RKPM) method. Also, Moving Particle Semi-implicit (MPS) [7], hp-Meshless clouds method [8], Local Boundary Integral Equation (LBIE) method [9], Meshless Local Petrov–Galerkin (MLPG) method [10] and Finite Point Method (FPM) [11] are then proposed. Zhang et al. [12] developed Least Squares Collocation Method (LSCM) for the solution of elliptic problems. Moving Particle Finite Element Method (MPFEM) [13], Particle Finite Element Method (PFEM) [14] and more recently Arzani and Afshar [15] proposed discrete least squares meshless (DLSM) method. The DLSM method is based on minimizing a lest squares func- tional is formed as the weighted squared residual of the governing differential equations and its boundary conditions at nodal points. Stability and accuracy of the method for irregular distribution of nodes were improved by employing some auxiliary points, named sampling points [16]. The method has been successfully used to solve a variety of fluid mechanics problems such as solution of transient Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2011.09.012 n Corresponding author. E-mail addresses: jamani@iust.ac.ir (J. Amani), mhafshar@iust.ac.ir (M.H. Afshar), m.naisipour@hotmail.com (M. Naisipour). Engineering Analysis with Boundary Elements 36 (2012) 894–902