Explicit form and efficient computation of MLS shape functions and their derivatives.

*(English)*Zbl 0965.65015This 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 \([10^{-6},10^{-3}]\).

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 \([10^{-6},10^{-3}]\).

Reviewer: Jesus Illán González (Vigo)

##### MSC:

65D05 | Numerical interpolation |

65D25 | Numerical differentiation |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

##### Keywords:

diffuse derivatives; moving least-squares approximation; meshless methods; interpolation; consistency; shape functions generation; stability; algorithm
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\textit{P. Breitkopf} et al., Int. J. Numer. Methods Eng. 48, No. 3, 451--466 (2000; Zbl 0965.65015)

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##### References:

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