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On the Factors Affecting the Accuracy and Robustness of Smoothed-Radial Point Interpolation Method

Part of: Partial differential equations, boundary value problems Mathematical modeling, applications of mathematics

Published online by Cambridge University Press:  11 October 2016

Abderrachid Hamrani ,
Idir Belaidi ,
Eric Monteiro  and
Philippe Lorong
Abderrachid Hamrani*
Affiliation:
Research team MISP, LEMI, University of M'Hamed Bougara de Boumerdes, 35000, Algérie PIMM, Arts et Métiers ParisTech, CNRS, 151 bd de l'Hôpital, 75013 Paris, France
Idir Belaidi*
Affiliation:
Research team MISP, LEMI, University of M'Hamed Bougara de Boumerdes, 35000, Algérie
Eric Monteiro*
Affiliation:
PIMM, Arts et Métiers ParisTech, CNRS, 151 bd de l'Hôpital, 75013 Paris, France
Philippe Lorong*
Affiliation:
PIMM, Arts et Métiers ParisTech, CNRS, 151 bd de l'Hôpital, 75013 Paris, France
*
*Corresponding author. Email:hamrani.abderrachid@gmail.com (A. Hamrani), idir.belaidi@gmail.com (I. Belaidi), eric.monteiro@ensam.eu (E. Monteiro), philippe.lorong@ensam.eu (P. Lorong)
*Corresponding author. Email:hamrani.abderrachid@gmail.com (A. Hamrani), idir.belaidi@gmail.com (I. Belaidi), eric.monteiro@ensam.eu (E. Monteiro), philippe.lorong@ensam.eu (P. Lorong)
*Corresponding author. Email:hamrani.abderrachid@gmail.com (A. Hamrani), idir.belaidi@gmail.com (I. Belaidi), eric.monteiro@ensam.eu (E. Monteiro), philippe.lorong@ensam.eu (P. Lorong)
*Corresponding author. Email:hamrani.abderrachid@gmail.com (A. Hamrani), idir.belaidi@gmail.com (I. Belaidi), eric.monteiro@ensam.eu (E. Monteiro), philippe.lorong@ensam.eu (P. Lorong)
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Abstract

In order to overcome the possible singularity associated with the Point Interpolation Method (PIM), the Radial Point Interpolation Method (RPIM) was proposed by G. R. Liu. Radial basis functions (RBF) was used in RPIM as basis functions for interpolation. All these radial basis functions include shape parameters. The choice of these shape parameters has been and stays a problematic theme in RBF approximation and interpolation theory. The object of this study is to contribute to the analysis of how these shape parameters affect the accuracy of the radial PIM. The RPIM is studied based on the global Galerkin weak form performed using two integration technics: classical Gaussian integration and the strain smoothing integration scheme. The numerical performance of this method is tested on their behavior on curve fitting, and on three elastic mechanical problems with regular or irregular nodes distributions. A range of recommended shape parameters is obtained from the analysis of different error indexes and also the condition number of the matrix system. All resulting RPIM methods perform very well in term of numerical computation. The Smoothed Radial Point Interpolation Method (SRPIM) shows a higher accuracy, especially in a situation of distorted node scheme.

Keywords

Radial Basis FunctionRadial Point Interpolation Methodsstrain smoothing nodal integrationGalerkin weak form

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 97M50: Physics, astronomy, technology, engineering
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 43 - 72
Copyright
Copyright © Global-Science Press 2017 

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