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Aspects of the use of orthogonal basis functions in the element‐free Galerkin method
Author(s) -
Zhuang Xiaoying,
Augarde Charles
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2696
Subject(s) - basis function , inversion (geology) , galerkin method , polynomial basis , orthogonal basis , basis (linear algebra) , mathematics , moving least squares , polynomial , finite element method , matrix (chemical analysis) , algorithm , mathematical analysis , mathematical optimization , computer science , geometry , physics , paleontology , materials science , structural basin , quantum mechanics , composite material , biology , thermodynamics
The element‐free Galerkin (EFG) method is probably the most widely used meshless method at present. In the EFG method, shape functions are derived from a moving least‐squares approximation using a polynomial basis, a calculation involving the inversion of a small matrix. A new implementation of the EFG method was published soon after the original where an alternative approach using an orthogonal basis was proposed to avoid matrix inversion in the formulation of the shape functions. In this paper we revisit this topic and show that the difficulties associated with the use of a polynomial basis remain present in the orthogonal case. We also show that certain terms in the derivative expressions are omitted in the new implementation of the EFG, which can lead to errors. Finally, we propose a new approach that avoids inversion while maintaining accuracy. Copyright © 2009 John Wiley & Sons, Ltd.