Keywords: moving least squares approximation, gaussian and polynomial weight functions, basis functions, radius of support, meshless methods.

The moving least square (MLS) was introduced by Shepard [1] as approximation method in the lowest order case and generalized to a higher degree by Lancaster and Salkauskas [2]. The object of those works was to provide an alternative to classic interpolation useful to approximate a function from its values given at irregularly spaced points by using a weighted least squares approximation. This method is nowadays widely used for constructing meshless shape functions. It was first used by Nayroles et al. [3] to construct shape functions for their proposed diffuse element method (DEM). Belytschko et al. [4] modified the DEM and named it the element free galerkin method (EFG) in which the MLS approximation is also employed. MLS approximation is also used to construct shape functions for the "finite point method" [5], the "local Petrov-Galerkin method" [6] and the "point interpolation method" [7]. Thus, regarding the application of the MLS approximation in many of the mesh free methods, it is necessary to consider the influence of the effective parameters on these methods.

The MLS method is employed to approximate a function and its derivatives in the near neighbourhood of a node. In this work the capability of the MLS technique, regarding the approximation of some different functions such as polynomial, exponential, harmonic and their combinations is discussed. To do this, the effect of the density of sample points, the radius of support, weight function, and the order of basis functions for a regular arrangement of sample points are studied. Moreover, the suitable ranges for some parameters which include the Gaussian coefficient of the weight function and also the density term of sample points are determined. Then, Gaussian and polynomial weight functions are separately applied and the results are compared.

1

D. Shepard, "A Two-Dimensional Interpolation Function for Irregularly Spaced Points", Proc. A.C.M. Natl. Conf., 517-524, 1968.

2

P. Lancaster and K. Salkauskas, "Surfaces Generated By Moving Least Squares Methods", Mathematics of Computation, 37, 141-158, 1981. doi:10.2307/2007507

3

B. Nayroles, G. Touzot and P. Villon, "Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements", Computational Mechanics, 10, 307-318, 1992. doi:10.1007/BF00364252

4

T. Belytschko, Y.Y. Lu and L. Gu, "Element Free Galerkin Methods", International Journal for Numerical Methods in Engineering, 37, 229-256, 1994. doi:10.1002/nme.1620370205

5

E. Onate et al., "A Finite Point Method in Computational Mechanics Applications to Convective Transport and Fluid Flow", International Journal for Numerical Methods in Engineering, 39, 3839-3866, 1996. doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R6

S.N. Atluri and T. Zhu, "A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics", Computational Mechanics, 22, 117-127, 1998. doi:10.1007/s004660050346

7

J.G. Wang and G.R. Liu, "A Point Interpolation Method for Simulating Dissipation Process of Consolidation", Computer Methods in Applied Mechanics and Engineering, 190, 5907-5922, 2001. doi:10.1016/S0045-7825(01)00204-3

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £120 +P&P)