Unsteady seepage analysis using local radial basis function-based differential quadrature method

Abstract Numerical simulation of two-dimensional transient seepage is developed using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to seepage analysis. For the general case of irregular geometry and unstructured node distribution, the local form of RBF-DQ was used. The multiquadric type of radial basis functions was selected for the computations, and the results were compared with analytical, finite element method, and existing numerical solutions from the literature. Results of this study show that localized RBF-DQ can produce accurate results for the analysis of seepage. The method is meshfree and easy to program, but as with previous applications of RBFs, requires careful selection of suitable shape parameters. A practical method for estimating suitable shape parameters is discussed. For time integration, Crank–Nicolson, Galerkin and finite difference methods were applied, leading to stable results.

[1]  C.-S. Huang,et al.  On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs , 2010 .

[2]  Jean-Pierre Bardet,et al.  A practical method for solving free-surface seepage problems , 2002 .

[3]  Guangyao Li,et al.  Shape variable radial basis function and its application in dual reciprocity boundary face method , 2011 .

[4]  A. Krowiak METHODS BASED ON THE DIFFERENTIAL QUADRATURE IN VIBRATION ANALYSIS OF PLATES , 2008 .

[5]  H. Ding,et al.  Error estimates of local multiquadric‐based differential quadrature (LMQDQ) method through numerical experiments , 2005 .

[6]  Masoud Darbandi,et al.  A moving‐mesh finite‐volume method to solve free‐surface seepage problem in arbitrary geometries , 2007 .

[7]  R. L. Hardy Multiquadric equations of topography and other irregular surfaces , 1971 .

[8]  Chang Shu,et al.  Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method , 2007 .

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

[10]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[11]  Kwok Fai Cheung,et al.  Multiquadric Solution for Shallow Water Equations , 1999 .

[12]  C. Shu Differential Quadrature and Its Application in Engineering , 2000 .

[13]  M. J. Abedini,et al.  Tidal and surge modelling using differential quadrature: a case study in the Bristol Channel. , 2008 .

[14]  Ahad Ouria,et al.  An Investigation on the Effect of the Coupled and Uncoupled Formulation on Transient Seepage by the Finite Element Method , 2007 .

[15]  C. Shu,et al.  Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations , 2003 .

[16]  H. Ding,et al.  Solution of partial differential equations by a global radial basis function-based differential quadrature method , 2004 .

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

[18]  M. J. Abedini,et al.  A differential quadrature analysis of unsteady open channel flow , 2007 .

[19]  Y. Hon,et al.  Geometrically Nonlinear Analysis of Reissner-Mindlin Plate by Meshless Computation , 2007 .

[20]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[21]  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 .

[22]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[23]  Shiang-Woei Chyuan,et al.  Boundary element analysis and design in seepage problems using dual integral formulation , 1994 .

[24]  G. Barani,et al.  Modeling of water surface profile in subterranean channel by differential quadrature method (DQM) , 2009 .

[25]  Gregory E. Fasshauer,et al.  On choosing “optimal” shape parameters for RBF approximation , 2007, Numerical Algorithms.

[26]  C. Shu,et al.  An upwind local RBF-DQ method for simulation of inviscid compressible flows , 2005 .

[27]  C. Micchelli Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[28]  Guirong Liu,et al.  Smoothed Particle Hydrodynamics: A Meshfree Particle Method , 2003 .

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

[30]  Chang Shu,et al.  Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method , 2006 .