Numerical solution of moving boundary problems using a new hybrid grid and Meshless method

Abstract In the present paper, a new hybrid numerical technique is developed to solve phase change problems with moving boundaries. A hybrid collocation and grid method based radial basis function is derived herein with simultaneous numerical iterative algorithm to solve moving boundary problems. The new proposed method has the advantages of both grid and mesh-less methods. Two test practical problems are solved, the first is the dissolution in binary alloys and the second is the solidification in a two dimensions prism. For the first test problem, the computed results are compared with available analytical solution, while the second one, no analytical solution available, but the computed results are compared with available numerical results. The results due to the present method were accepted compared with available results.

[1]  S. G. Ahmed,et al.  A collocation method using new combined radial basis functions of thin plate and multiquadraic types , 2006 .

[2]  S. G. Ahmed,et al.  A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations , 2010, Int. J. Comput. Math..

[3]  Zahari Zlatev,et al.  Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants , 1984 .

[4]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[5]  G. W. Evans.  A note on the existence of a solution to a problem of Stefan , 1951 .

[6]  F. Vermolen,et al.  A comparison of numerical models for one-dimensional Stefan problems , 2006 .

[7]  Richard H. Gallagher,et al.  Dispersion analysis in homogeneous lakes , 1980 .

[8]  J. Parlange,et al.  Water Transport in Soils , 1980 .

[9]  D. Juric,et al.  A Front-Tracking Method for Dendritic Solidification , 1996 .

[10]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[11]  S. Atluri,et al.  A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems , 1999 .

[12]  J. Hoopes,et al.  Dispersion in Anisotropic, Homogeneous, Porous Media , 1985 .

[13]  Masataka Tanaka,et al.  On exponential variable transformation based boundary element formulation for advection-diffusion problems , 2000 .

[14]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[15]  Jim Douglas,et al.  A uniqueness theorem for the solution of a Stefan problem , 1957 .

[16]  S. Ahmed.  A new semi-analytical method for phase transformations in binary alloys , 2007 .

[17]  C Vuik,et al.  Numerical solution of an etching problem , 1985 .

[18]  F. Vermolen,et al.  A mathematical model for the dissolution of stoichiometric particles in multi-component alloys , 2002 .

[19]  J. Isenberg,et al.  Heat transfer to a draining film , 1973 .

[20]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[21]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[22]  K. M. Liew,et al.  Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform , 2005 .

[23]  S. Osher,et al.  A Simple Level Set Method for Solving Stefan Problems , 1997, Journal of Computational Physics.