Premium
A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems
Author(s) -
Le Phong B. H.,
MaiDuy Nam,
TranCong Thanh,
Baker Graham
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.2771
Subject(s) - mathematics , discretization , boundary value problem , collocation method , rate of convergence , collocation (remote sensing) , mathematical analysis , dirichlet boundary condition , radial basis function , boundary (topology) , interpolation (computer graphics) , orthogonal collocation , cartesian coordinate system , partial differential equation , singular boundary method , convergence (economics) , ordinary differential equation , geometry , differential equation , finite element method , boundary element method , computer science , physics , computer graphics (images) , economic growth , machine learning , channel (broadcasting) , computer network , artificial neural network , animation , economics , thermodynamics
In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) ( Numer. Meth. Partial Differential Equations 2007; 23 :1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.