Comparisons of two meshfree local point interpolation methods for structural analyses G. R. Liu, Y. T. Gu Abstract As truly meshless methods, the local point in- terpolation method (LPIM) and the local radial point in- terpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are ex- amined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical imple- mentations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formula- tions are developed for structural analyses of 2-D elasto- dynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics. Keywords Structural Analysis, Meshless Method, Local Meshless Method, Interpolation Function, Weak Form, Strong Form 1 Introduction Numerical methods have been developed and used to solve partial differential equations for long time. In recent years, a new class of numerical methods, namely, the element-free method or the meshless method, has been developed in order to overcome drawbacks in the conversional numerical methods, e.g. finite element method (FEM), finite difference method (FDM) and boundary element method (BEM). According to their computational modelings used, meshfree methods may be largely categorized into two different categories (Li and Liu, 2001): meshfree methods based on strong forms of partial differential equations (PDEs) and meshfree methods based on Galerkin weak forms of PDEs. There are exceptions to this classification because some meshfree methods can be used in both strong form as well as weak form, such as and the repro- ducing kernel particle method (RKPM) (Aluru, 2000; Liu et al., 1995). One of the most famous meshfree methods based on the strong forms is the smooth particle hydro- dynamics (SPH) (Gingold and Monaghan, 1977). The Galerkin meshless methods based on the global weak forms proposed include diffuse element method (DEM) (Nayroles et al., 1992), element-free Galerkin (EFG) method (Belytschko et al., 1994), point interpolation method (PIM) (Liu and Gu, 2001a), and so on. The Galerkin meshless methods, which are proposed much later than SPH, have been successful widely used in computational mechanics. In addition, techniques for coupling meshless methods with other established nu- merical methods have also been proposed, such as the coupled EFG/FEM (Belytschko and Organ, 1995), EFG/ BEM (Gu and Liu, 2001a; Liu and Gu, 2000a). The difference in above-mentioned global Galerkin meshless methods comes mainly from the interpolation techniques used. In particular, the above-mentioned meshless methods are ‘‘meshless’’ only in terms of the interpolation of the field variables, as compared to the usual FEM. Most of them have to use background cells to integrate a weak form over the global problem domain. The requirement of integration background cells makes these methods not ‘‘truly’’ meshless. In order to alleviate the global integration background mesh, a group of so called ‘‘truly’’ meshless methods are proposed and devel- oped, such as, the meshless local Petrov-Galerkin (MLPG) method, the local boundary integral equation (LBIE) method (Zhu et al., 1998a, b), the finite spheres method (De and Bathe, 2000), and so on. Te MLPG method is proposed and developed by Atluri and his co-workers (Atluri and Zhu, 1998, 2000; Atluri et al., 1999). In the MLPG method, the moving least squares (MLS) approximation is used to construct shape functions. The most important contribution of the MLPG method is developing a local weak form integrated in a regular-shaped local domain, which can be as simple as possible, such as circles, ellipses, rectangles, or triangles in 2-D; spheres, rectangular parallelepipeds, or ellipsoids in 3-D. The MLPG method has been successfully used in analyses of solids (Atluri and Zhu, 1998, 2000; Atluri et al., 1999; Gu and Liu, 2001b, c). In addition, the MLPG method has also been combined with FEM and BEM (Liu and Gu, 2000b). Liu and his co-workers applied the local concept and developed two ‘‘truly’’ meshless methods, the local point interpolation method (LPIM) (Liu and Gu, 2001b) and the local radial point interpolation method (LR-PIM) (Liu Computational Mechanics 29 (2002) 107–121 Ó Springer-Verlag 2002 DOI 10.1007/s00466-002-0320-4 107 Received 31 August 2001 / Accepted 04 March 2002 G. R. Liu (&), Y. T. Gu Singapore-MIT alliance (SMA), Centre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 E-mail: mpeliugr@nus.edu.sg