Natural neighbour Galerkin methods

N. Sukumar*, Brian Moran, A. Yu Semenov, V. V. Belikov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

301 Scopus citations

Abstract

Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariate data fitting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkin method is known as the natural element method (NEM). In the natural element method, natural neighbour co-ordinates are used to construct the trial and test functions. Recent studies on NEM have shown that natural neighbour co-ordinates, which are based on the Voronoi tessellation of a set of nodes, are an appealing choice to construct meshless interpolants for the solution of partial differential equations. In Belikov et al. (Computational Mathematics and Mathematical Physics 1997; 37(1):9-15), a new interpolation scheme (non-Sibsonian interpolation) based on natural neighbours was proposed. In the present paper, the non-Sibsonian interpolation scheme is reviewed and its performance in a Galerkin method for the solution of elliptic partial differential equations that arise in linear elasticity is studied. A methodology to couple finite elements to NEM is also described. Two significant advantages of the non-Sibson interpolant over the Sibson interpolant are revealed and numerically verified: the computational efficiency of the non-Sibson algorithm in 2-dimensions, which is expected to carry over to 3-dimensions, and the ability to exactly impose essential boundary conditions on the boundaries of convex and non-convex domains.

Original languageEnglish (US)
Pages (from-to)1-27
Number of pages27
JournalInternational Journal for Numerical Methods in Engineering
Volume50
Issue number1
DOIs
StatePublished - Jun 20 2000

Keywords

  • Essential boundary conditions
  • Meshless Galerkin methods
  • Natural element method
  • Natural neighbour co-ordinates
  • Non-Sibsonian interpolation

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics

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