A residual based error estimator using radial basis functions

In this paper, a novel residual based error estimator using radial basis functions (RBFs) is proposed. The error estimator evaluates the residual in the strong-form governing equation in the local domain through direct integration. Due to the higher order continuous feature of the RBFs, the higher derivatives of the field function in the strong-form governing equation can be obtained using RBFs. The numerical examples show that the new residual based error estimator is simple, versatile robust and yet effective in the adaptive analyses. It is not only suitable for adaptive analysis that uses numerical method formulated based on mesh, e.g. finite element method, but also meshfree methods where the conventional residual based and recovery based error estimator cannot be used. Furthermore the present error estimator is also feasible for numerical method that is formulated based on both strong and weak formulation in the adaptive analyses.

[1]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[2]  Ivo Babuška,et al.  Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements , 1992 .

[3]  J. Oden,et al.  A unified approach to a posteriori error estimation using element residual methods , 1993 .

[4]  Wolfram Volk,et al.  From discrete element simulations to a continuum model , 2000 .

[5]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[6]  O. C. Zienkiewicz,et al.  The superconvergent patch recovery (SPR) and adaptive finite element refinement , 1992 .

[7]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[8]  Kang Tai,et al.  Radial point interpolation collocation method (RPICM) for partial differential equations , 2005 .

[9]  Guirong Liu,et al.  A meshfree radial point interpolation method (RPIM) for three-dimensional solids , 2005 .

[10]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[11]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[12]  R. L. Hardy.  Theory and applications of the multiquadric-biharmonic method : 20 years of discovery 1968-1988 , 1990 .

[13]  K. Y. Dai,et al.  A LINEARLY CONFORMING POINT INTERPOLATION METHOD (LC-PIM) FOR 2D SOLID MECHANICS PROBLEMS , 2005 .

[14]  E. J. Kansa,et al.  Multi-quadrics-a scattered data approximation scheme with applications to computational fluid dynamics-II , 1990 .

[15]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[16]  YuanTong Gu,et al.  A matrix triangularization algorithm for the polynomial point interpolation method , 2003 .

[17]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[18]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[19]  Guirong Liu,et al.  On the optimal shape parameters of radial basis functions used for 2-D meshless methods , 2002 .

[20]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[21]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[22]  Ted Belytschko,et al.  Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements , 1994 .

[23]  Guirong Liu.  Meshfree Methods: Moving Beyond the Finite Element Method, Second Edition , 2009 .

[24]  Gui-Rong Liu,et al.  A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis , 2007 .

[25]  Alvaro L. G. A. Coutinho,et al.  A posteriori error estimate for stress analysis of homogeneous and heterogeneous materials: an engineering approach , 2005 .

[26]  Nils-Erik Wiberg,et al.  Enhanced Superconvergent Patch Recovery incorporating equilibrium and boundary conditions , 1994 .

[27]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[28]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[29]  Nils-Erik Wiberg,et al.  Patch recovery based on superconvergent derivatives and equilibrium , 1993 .

[30]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .