Surfaces Generated by Moving Least Squares Methods

Abstract

Abstract. An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finiteelement schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.1. Introduction. While the theory and practice of interpolation and approximation of functions of a single variable on the basis of a finite amount of information is well developed, the same is not true for functions of several variables. When the information consists of function values at the points of a rectangular grid, tensor product and blended interpolants based on univariate schemes can be employed. For irregularly distributed function-value data, the situation is much worse. Finiteelement techniques are of value, although, in order to produce C1 interpolants, more than just function-value information is required. As well, if the distribution of data is irregular, only triangular elements seem feasible. Their use has become more attractive in view of the recent development of fast and efficient triangulation algorithms. Nevertheless, the additional nodal information consisting of derivative data must somehow be concocted.Least squares approximation by polynomials is in widespread use and is, of course, not expected to produce an interpolant. However, as has been shown by McLain [7]

Analysis of Cohesive Crack Growth by the Element-Free Galerkin Method

“…The construction of the shape functions in the EFG method is based on an approximation technique using the moving least square (MLS) method [11]. The MLS method is an effective technique for approximating a function using a set of scattered data.…”

“…Defined by Lancaster and Salkauskas [11], the moving least-square (MLS) approximation which originated in scattered data fitting is chosen to construct EFG shape functions and their derivatives. The core concept of the EFG method is that there is no longer a finite element mesh.…”

Reliability analysis based on gradient and heuristic optimization techniques of composite laminates using element-free Galerkin method

“…In this section a brief for moving least-square (MLS) [14] approximation is given. More details are referred to in references [2] and [13].…”

“…(2)). This approximation is based upon MLS in curve and surface fitting [14]. The shape function in equation (19) is constructed as:…”

Free mesh method: A new meshless finite element method

“…The element free Galerkin method (EFGM) (Belytschko et al 1994) is one of them with the use of integration by background-cell instead of by element, based on the moving least square method (Lancaster and Salkauskas, 1981) and the diffuse element method (Nayroles et al, 1992). The reproducing kernel particle methods (RKPM) (Liu et al 1995) is another meshless scheme, which is based on particle method (Monaghan 1983) and wavelets.…”