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Positivity conditions in meshless collocation methods Xiaozhong Jin, Gang Li, N.R. Aluru *
 

Summary: Positivity conditions in meshless collocation methods
Xiaozhong Jin, Gang Li, N.R. Aluru *
Department of Mechanical and Industrial Engineering, Beckman Institute for Advanced Science and Technology,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 19 March 2003; received in revised form 3 October 2003; accepted 4 December 2003
Abstract
Collocation meshless methods are conceptually simple, easy-to-implement and fast numerical methods. The
robustness of collocation methods has, however, been an issue especially for scattered set of points. In this paper we
show that the robustness of collocation meshless methods can be improved by ensuring that certain conditions, defined
as the positivity conditions, are satisfied when constructing approximation functions and their derivatives. The sig-
nificance of positivity conditions is pointed out by an error analysis of the finite cloud method, which is a collocation
based meshless method. We propose techniques, based on modification of weighting functions, to ensure satisfaction of
positivity conditions on the approximation function and its derivatives when using a scattered set of points. Several
types of weighting functions are tested for 1D and 2D problems on scattered points. Numerical results demonstrate the
effectiveness of collocation methods when positivity conditions are satisfied.
2004 Published by Elsevier B.V.
Keywords: Positivity conditions; Collocation methods; Finite cloud method; Error analysis; Weighting function
1. Introduction
Meshless methods have emerged as a class of effective numerical methods which are capable of avoiding
the difficulties encountered in conventional computational mesh based methods, such as, meshing complex

  

Source: Aluru, Narayana R. - Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign

 

Collections: Engineering; Materials Science