A Cartesian-grid collocation technique with integrated radial basis functions for mixed boundary value problems

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...

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Bibliographic Details
Published in:	International journal for numerical methods in engineering Vol. 82; no. 4; pp. 435 - 463
Main Authors:	Le, Phong B. H., Mai-Duy, Nam, Tran-Cong, Thanh, Baker, Graham
Format:	Journal Article
Language:	English
Published:	Chichester, UK John Wiley & Sons, Ltd 23-04-2010 Wiley
Subjects:	Approximation Boundary conditions Cartesian grid collocation method Collocation methods Convergence Dirichlet problem elasticity Exact sciences and technology first-order system Fundamental areas of phenomenology (including applications) incompressibility Mathematical analysis Mathematical models mixed formulation Physics Radial basis function RBF Solid mechanics Static elasticity (thermoelasticity...) Structural and continuum mechanics volumetric locking Solid mechanics elasticity Cartesian coordinate Incompressible material Experimental study Modeling Temperature gradient first-order system Radial basis function volumetric locking incompressibility mixed formulation Boundary value problem Strain gradient RBF Convergence rate Dirichlet problem Mixed problem Cartesian grid Collocation method Differential method Structural analysis
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Summary:	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.
Bibliography:	The Australian Research Council - No. 98-1846389 ark:/67375/WNG-X3F59WTW-3 istex:EB9425D9CA9E811C7C6629C6C384ACBC2A74192B ArticleID:NME2771 ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0029-5981 1097-0207
DOI:	10.1002/nme.2771