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Original Articles

Radial basis function meshless method for the steady incompressible Navier–Stokes equations

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Pages 1509-1521 | Received 24 Jul 2006, Accepted 30 Jan 2007, Published online: 24 Sep 2007

References

  • Belytschko , T. , Lu , Y. and Gu , L. 1994 . Element free Galerkin methods . International Journal for Numerical Methods in Engineering , 37 : 229 – 256 .
  • Perrone , N. and Kao , R. 1975 . A general finite difference method for arbitrary meshes . Computers and Structures , 5 : 45 – 47 .
  • Lucy , L. B. 1977 . A numerical approach to the testing of fission hypothesis . Astronomical Journal , 8 : 1013 – 1024 .
  • Nayroles , B. , Touzot , G. and Villon , P. 1992 . Generalizing the finite element method: diffuse approximation and diffuse elements . Computational Mechanics , 10 : 307 – 318 .
  • Liu , W. K. , Jun , S. and Belytschko , T. 1995 . Reproducing kernel particle methods . International Journal for Numerical Methods in Fluids , 20 : 1081 – 1106 .
  • Melenk , J. M. and Babuska , I. 1996 . “ The partition of unity finite element method: basic theory and applications ” . Texas Institute for Computational and Applied Mathematics . Technical Report
  • Onate , E. , Idelsohn , S. , Zienkiewicz , O. C. , Taylor , R. L. and Sacco , C. 1996 . A stabilized finite point method for analysis of fluid mechanics problems . Computer Methods in Applied Mechanics and Engineering , 139 : 315 – 346 .
  • Atluri , S. N. and Zhu , T. 1998 . New meshless local Petrov–Galerkin approach in computational mechanics . Computational Mechanics , 22 : 117 – 127 .
  • Kansa , E. J. 1990 . Multiquadrics—A scattered data approximation scheme with applications to computation fluid dynamics—I. Surface approximations and partial derivatives estimates . Computers and Mathematics with Applications , 19 : 127 – 145 .
  • Kansa , E. J. 1990 . Multiquadrics—A scattered data approximation scheme with applications to computation fluid dynamics—II. Solution to parabolic, hyperbolic and elliptic partial differential equations . Computers and Mathematics with Applications , 19 : 147 – 161 .
  • Fornberg , B. , Wright , G. and Larsson , E. 2004 . Some observations regarding interpolants in the limit of flat radial basis functions . Computers and Mathematics with Applications , 1 : 37 – 55 .
  • Fries , T. P. and Matthies , H. G. 2003 . “ Classification and overview of meshfree methods ” . Brunswick, , Germany : Institute of Scientific Computing, Technical University Braunschweig . Technical Report
  • Hon , Y. C. and Mao , X. Z. 1997 . A multiquadric interpolation method for solving initial value problems . Scientific Computing , 12 : 51 – 55 .
  • Hon , Y. C. , Lu , M. W. , Xue , W. M. and Zhu , Y. M. 1997 . Multiquadric method for the numerical solution of a biphasic mixture model . Applied Mathematics and Computation , 88 : 153 – 175 .
  • Chinchapatnam , P. P. , Djidjeli , K. and Nair , P. B. 2006 . Unsymmetric and symmetric meshless schemes for the unsteady convection–diffusion equation . Computer Methods in Applied Mechanics and Engineering , 195 : 2432 – 2453 .
  • Chinchapatnam , P. P. , Djidjeli , K. and Nair , P. B. 2006 . Domain decomposition for time-dependent problems using radial based meshless methods . Numerical Methods for Partial Differential Equations , published online: 25 May 2006
  • Power , H. and Barraco , V. 2002 . A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations . Computers and Mathematics with Applications , 43 : 551 – 583 .
  • Hon , Y. C. and Mao , X. Z. 1998 . An efficient numerical scheme for Burger's equation . Applied Mathematics and Computation , 95 : 37 – 50 .
  • Wong , A. S.M. , Hon , Y. C. , Li , T. S. , Cheng , S. L. and Kansa , E. J. 1999 . Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme . Computers and Mathematics with Applications , 37 : 23 – 43 .
  • Hon , Y. C. and Mao , X. Z. 1999 . A radial basis function method for solving options pricing model . Financial Engineering , 8 : 31 – 50 .
  • Cheng , A. H.-D. , Golberg , M. A. , Kansa , E. J. and Zammito , G. 2003 . Exponential convergence and H-c multiquadric collocation method for partial differential equations . Numerical Methods for Partial Differential Equations , 19 : 571 – 594 .
  • Thom , A. 1928 . “ An investigation of fluid flow in two dimensions ” . UK : Aerospace Research Centre . Technical Report
  • Jensen , V. G. Viscous flow round a sphere at low Reynolds number (≤40) . Proceedings of the Royal Society of London: Series A , Vol. 249 , pp. 346 – 366 .
  • Woods , L. C. 1954 . A note on the numerical solution of fourth order differential equations . Aeronautical Quarterly , 5 : 176
  • Spotz , W. F. 1995 . “ High-order compact finite difference schemes for computational mechanics ” . University of Texas at Austin . PhD thesis
  • Mai-Duy , N. and Tanner , R. I. 2005 . Solving high-order partial differential equations with indirect radial basis function networks . International Journal for Numerical Methods in Engineering , 63 : 1636 – 1654 .
  • Larsson , E. and Fornberg , B. 2003 . A numerical study of some radial basis function based solution methods for elliptic PDEs . Computers and Mathematics with Applications , 46 : 891 – 902 .
  • Schaback , R. 1995 . Error estimates and condition numbers for radial basis interpolation . Advances in Computational Mathematics , 3 : 251 – 264 .
  • Hardy , R. L. 1990 . Theory and application of the multiquadric-biharmonic method . Computers and Mathematics with Applications , 19 : 163 – 208 .
  • Coleman , T. F. and Li , Y. 1994 . On the convergence of reflective Newton methods for large scale nonlinear minimization subject to bounds . Mathematical Programming , 67 : 189 – 224 .
  • Coleman , T. F. and Li , Y. 1996 . An interior trust region approach for nonlinear minimization subject to bounds . SIAM Journal on Optimization , 6 : 418 – 445 .
  • Ghia , U. , Ghia , K. N. and Shin , C. T. 1982 . High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method . Journal of Computational Physics , 48 : 387 – 411 .
  • Bruneau , C.-H. and Jouron , C. 1990 . An efficient scheme for solving steady incompressible Navier–Stokes equations . Journal of Computational Physics , 89 : 389 – 413 .
  • Gartling , D. K. 1990 . A test problem for outflow boundary conditions—flow over a backward-facing step . International Journal for Numerical Methods in Fluids , 11 : 953 – 967 .
  • Barragy , E. 1993 . “ Parallel finite element methods and iterative solution techniques for viscous incompressible flows ” . University of Texas at Austin . PhD thesis

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