Analysis of thin beams, using the meshless local Petrov±Galerkin method, with generalized moving least squares interpolations

In this paper, the conventional moving least squares interpolation scheme is generalized, to incorporate the information concerning the derivative of the ®eld variable into the interpolation scheme. By using this generalized moving least squares interpolation, along with the MLPG (Meshless Local Petrov±Galerkin) paradigm, a new numerical approach is proposed to deal with 4th order problems of thin beams. Through numerical examples, convergence tests are performed; and problems of thin beams under various loading and boundary conditions are analyzed by the proposed method, and the numerical results are compared with analytical solutions. Introduction In the past decade, a considerable attention has been given to meshless computational methods, due to their ̄exibility in solving boundary value problems, especially in problems with discontinuities, or with moving boundaries, or with severe deformations. The driving force for research on the meshless method is the desire to minimize, or alleviate, the human labor and error involved in meshing the entire structure. As a result, several so-called meshless methods have been proposed, such as smooth particle hydrodynamics (SPH) (Lucy, 1977), diffuse element method (DEM) (Nayroles et al., 1992), element free Galerkin method (EFG) (Belytschko et al., 1994; Organ et al., 1996), reproducing kernel particle method (RKPM) (Liu et al., 1995 and 1996), hp-clouds method (Duarte and Oden, 1996), partition of unity method (PUM) (Babu ska and Melenk, 1997), local boundary integral equation method (LBIE) (Zhu, Zhang, and Atluri, 1998a, b), meshless local Petrov±Galerkin method (MLPG) (Atluri and Zhu, 1998a, b). Of these, as discussed in Atluri and Zhu (1998a, b), and in Zhu, Zhang and Atluri (1998a, b), only the MLPG and LBIE methods are truly meshless. To be a truly meshless method, the two characteristics should be guaranteed: One is a non-element interpolation technique, and the other is a non-element approach for integrating the weak form. Most of the meshless methods are based on the non-element interpolation techniques, such as the Shepard interpolation technique (Shepard, 1968), moving least square interpolation (MLS) (Lancaster and Salkauskas, 1981), reproducing kernel particle method (RKPM), and the partition of unity method (PUM), which do not need any elements for constructing the interpolation functions for the unknown variables. However, because most of the so-called meshless methods, such as the EFG, RKPM, and hp-clouds method, still require a global background mesh for numerical integration of the global weak-form, they cannot be classi®ed as being truly meshless. From this point of view, only the recently proposed LBIE and MLPG (meshless local Petrov±Galerkin) methods can be labeled as being truly meshless. As distinct from the other meshless methods based on the global weak form, the elegant paradigm of MLPG method is based on the local symmetric weak form (LSWF). Through the LSWF, one is naturally lead to a local non-element integration in local sub-domains such as spheres, cubes, and ellipsoids, in 3-D, without any dif®culty. Because of this pioneering truly meshless nature of the MLPG method, the present work is aimed at extending the MLPG method for 4th order boundary value problems governing thin beams or thin plates. Furthermore, in dealing with 4th order boundary value problems by a meshless computational method such as the EFG method, only a few works (Krysl and Belytschko, 1995 and 1996) were reported. Although only a thin beam will be addressed in this work, it is noted that the present approach is quite general and can easily deal with 2D plate problems. In the 4th order boundary value problems, displacement and slope boundary conditions can be imposed at the same point, while such is impossible in 2nd order boundary value problems. Therefore, it is natural to introduce the slope as another independent variable in the interpolation schemes, in the 4th order problem. Due to this necessity, the conventional moving least square interpolation scheme is generalized, to incorporate the independent slope information, and it is used as a meshless interpolation technique in the MLPG method for 4th order boundary value problems. (It should be noted that the MLPG concept is independent of a meshless interpolation technique, and it can be combined with any meshless interpolation technique, such as PUM, or RKPM). To study the accuracy of the present method, convergence tests are carried out, and several problems of thin beams under various loading and boundary conditions are analyzed. From these tests, it is con®rmed that the proposed Computational Mechanics 24 (1999) 334±347 Ó Springer-Verlag 1999