INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2005; 64:1111–1131 Published online 11 July 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1403 Parametric meshless Galerkin method Hossein Hosseini-Toudeshky ∗, † and Mehdi Musivand-Arzanfudi ‡ Aerospace Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), P.O. Box 15875-4413, Hafez Avenue, Tehran, Iran SUMMARY In meshless methods, generation of meshless shape functions is usually a complicated and time- consuming task. In this paper, a new meshless method called parametric meshless Galerkin method (PMGM) is presented. In this method, meshless shape functions are constructed on meshless parametric domains (MPD), before running to solve the problem. For modelling the new problems, MPDs are mapped to the physical space. Therefore the shape functions constructing time can be saved. Mapping is simply performed by defining a linear function. Also, the integration grids are defined in the MPD and it is not necessary to create background integration grids separately for each problem. The method is described for two-dimensional problems, but it can be applied to three-dimensional problems in the same way. It is shown that using the PMGM, a time saving as much as 21% is achieved with respect to the element-free Galerkin method for the numerical examples and the obtained results show efficiency and convergence of the method. Copyright  2005 John Wiley & Sons, Ltd. KEY WORDS: parametric; mapping; meshless; computational time; time saving 1. INTRODUCTION Since 1976, a new set of computational methods called meshless or meshfree methods have emerged in the field of computational methods [1, 2], but major improvements in these methods occurred after 1994 with publishing of the so-called diffuse element method [3] and element- free Galerkin methods (EFGM) [4, 5]. The EFGM has been highly developed and applied to different problems by other authors [6–12]. In recent years, many authors have worked on different aspects of meshless methods. Meshless methods use shape functions that can be smoother than FEM shape functions and this property serves some advantages such as increasing accuracy and rate of convergence, avoiding volumetric locking in near compressible problems, decreasing the need to postprocess- ing and giving more accurate results near the boundaries. Meshless methods do not use elements ∗ Correspondence to: Hossein Hosseini-Toudeshky, Aerospace Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), P.O. Box 15875-4413, Hafez Avenue, Tehran, Iran. † E-mail: hosseini@aut.ac.ir ‡ Ph.D. Student. Received 14 February 2005 Revised 30 March 2005 Copyright  2005 John Wiley & Sons, Ltd. Accepted 11 May 2005