INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 85:693–722 Published online 13 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.2991 A generalized approximation for the meshfree analysis of solids C. T. Wu 1, ∗, † , C. K. Park 2 and J. S. Chen 3 1 Livermore Software Technology Corporation, Livermore, CA 94550, U.S.A. 2 Department of Civil and Environmental Engineering, The George Washington University, Washington, DC 20052, U.S.A. 3 Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, U.S.A. SUMMARY This paper presents a new approach in the construction of meshfree approximations as well as the weak Kronecker-delta property at the boundary, referred to as a generalized meshfree (GMF) approximation. The GMF approximation introduces an enriched basis function in the original Shepard’s method. This enriched basis function is introduced to meet the linear or higher order reproducing conditions and at the same time to offer great flexibility on the control of the smoothness and convexity of the approximation. The construction of the GMF approximation can be viewed as a special root-finding scheme of constraint equations that enforces that the basis functions are corrected and the reproducing conditions with certain orders are satisfied within a set of nodes. By choosing different basis functions, various convex and non- convex approximations including moving least-squares (MLS), reproducing kernel (RK), and maximum entropy (ME) approximations can be obtained. Furthermore, the basis function can also be translated or blended with other functions to generate a particular approximation for a special purpose. One application in this paper is to incorporate a blending function at the boundary based on the concept of local convexity for the non-convex approximation, such as MLS, to acquire the weak Kronecker-delta property. To achieve the higher order GMF approximation, two possible methods are also introduced. Several examples are presented to examine the effectiveness of various GMF approximations. Copyright  2010 John Wiley & Sons, Ltd. Received 9 March 2010; Revised 10 June 2010; Accepted 12 June 2010 KEY WORDS: meshfree method; convex approximation; non-convex approximation; basis function; Kronecker-delta property; root-finding scheme 1. INTRODUCTION The construction of shape functions based on scattered data is a fundamental task in meshfree methods. In general, the construction of the meshfree shape functions can be categorized into two groups, the interpolation methods and the approximation methods [1]. A well-known interpolation method is the radial basis functions (RBF), which are invariant under all Euclidean transformations. The classic form of RBF is the multi-quadrics (MQ) [2]. There are other forms of RBF, such as thin plate spline (TPS) [3] and Gaussian radial function [4]. The RBF offers an exponential convergence rate and works well with highly scattered data points [5, 6]. However, they are limited to small- scale problems because of their non-locality and have been enhanced by the weighted radial basis collocation method [7] and the combined reproducing kernel (RK) and RBF approximation [8]. ∗ Correspondence to: C. T. Wu, Livermore Software Technology Corporation (LSTC), 7374 Las Positas Road, Livermore, CA 94551, U.S.A. † E-mail: ctwu@lstc.com Copyright  2010 John Wiley & Sons, Ltd.