The extended finite element method (XFEM) for solidification problems

An enriched finite element method for the multi‐dimensional Stefan problems is presented. In this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface. The approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element. The phase interface can be evolved without re‐meshing or the use of artificial heat capacity techniques. The interface is described by a level set function that is updated by a stabilized finite element scheme. Several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method. Copyright © 2001 John Wiley & Sons, Ltd.

[1]  Ted Belytschko,et al.  The extended finite element method for rigid particles in Stokes flow , 2001 .

[2]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[3]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[4]  Thomas J. R. Hughes,et al.  On modelling thermal oxidation of Silicon II: numerical aspects , 2000 .

[5]  I. Babuska,et al.  The design and analysis of the Generalized Finite Element Method , 2000 .

[6]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[7]  T. Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[8]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[9]  Jaroslav Mackerle,et al.  Finite elements and boundary elements applied in phase change, solidification and melting problems. A bibliography (1996–1998) , 1999 .

[10]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..

[11]  J. Sethian,et al.  Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains , 1998 .

[12]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[13]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[14]  Peter Hansbo,et al.  Explicit streamline diffusion finite element methods for the compressible Euler equations in conservation variables , 1993 .

[15]  Brian G. Thomas,et al.  Fixed grid techniques for phase change problems: A review , 1990 .

[16]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[17]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[18]  Kevin O'Neill,et al.  Moving boundary‐moving mesh analysis of phase change using finite elements with transfinite mappings , 1986 .

[19]  Karl-Hermann Tacke,et al.  Discretization of the expliclt enthalpy method for planar phase change , 1985 .

[20]  Graham E. Bell,et al.  A refinement of the heat balance integral method applied to a melting problem , 1978 .

[21]  K. A. Rathjen,et al.  Heat Conduction With Melting or Freezing in a Corner , 1971 .

[22]  Anastas Lazaridis,et al.  A numerical solution of the multidimensional solidification (or melting) problem , 1970 .

[23]  J. Dolbow,et al.  Modeling dendritic solidification with the extended finite element method , 2001 .

[24]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[25]  V. Voller.  FAST IMPLICIT FINITE-DIFFERENCE METHOD FOR THE ANALYSIS OF PHASE CHANGE PROBLEMS , 1990 .

[26]  Klaus-Jürgen Bathe,et al.  AN EFFICIENT ALGORITHM FOR ANALYSIS OF NONLINEAR HEAT TRANSFER WITH PHASE CHANGES , 1982 .

[27]  Roland W. Lewis,et al.  Finite element solution of non‐linear heat conduction problems with special reference to phase change , 1974 .