# zbMATH — the first resource for mathematics

Explicit form and efficient computation of MLS shape functions and their derivatives. (English) Zbl 0965.65015
This paper deals with the construction of efficient meshless methods based on moving least-squares (MLS) approximation and interpolation. Let $$u(x)= N^T(x)u(x_i)$$ be the MLS approximant, where $$N$$ is the shape function. The main contribution of this work is a new procedure for computing both diffuse and full derivatives of $$N$$, which have been obtained in explicit form in one, two, and three dimensions.
This work is based on an extension of the consistency approach introduced by the authors [Comput. Assist. Mech. Eng. Sci. 5, No. 4, 479-501 (1998)], and it includes a fast numerical procedure for shape functions generation and their full derivation without neither matrix inversion nor linear system solving. Besides, this approach is independent of the choice of the base functions and the constraints imposed to the shape functions.
Some results of a stability analysis of the algorithm as compared to the standard formula are given. Mainly they consist in estimating the quadratic norm of the error as a perturbation of a singular pattern varies within $$[10^{-6},10^{-3}]$$.

##### MSC:
 65D05 Numerical interpolation 65D25 Numerical differentiation 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Breitkopf, Computer Assisted Mechanics and Engineering Sciences 5 pp 479– (1998) [2] Nayroles, Computer Mechanics 10 pp 307– (1992) [3] Liszka, International Journal for Numerical Methods in Engineering 20 pp 1599– (1984) · Zbl 0544.65006 [4] Belytschko, Computer Methods in Applied Mechanics and Engineering 139 (1996) [5] Mukherjee Yu, International Journal for Numerical Methods in Engineering 40 pp 797– (1997) [6] Lancaster, Mathematics of Computations 37 (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.