Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Shaw, Amit and Kaushik, KN and Roy, D (2009) Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks. In: International Journal of Non-Linear Mechanics, 44 (4). pp. 417-431.

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Abstract

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.

Item Type:	Journal Article
Publication:	International Journal of Non-Linear Mechanics
Publisher:	Elsevier Science
Additional Information:	Copyright of this is article belongs to Elsevier Science.
Keywords:	NURBS;Convex hulls;Error reproducing kernels;Mesh-free methods;Approximations of non-differentiable functions;Error estimates;Burgers equation;Shocks.
Department/Centre:	Division of Mechanical Sciences > Civil Engineering
Date Deposited:	15 Dec 2009 05:27
Last Modified:	19 Sep 2010 05:32
URI:	http://eprints.iisc.ac.in/id/eprint/20425

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