A temperature‐based finite element solution for phase‐change problems

A finite element procedure for solving multidimensional phase change problems is described. The algorithm combines a temperature formulation with a finite element treatment of the differential equation and discontinuous integration within the two-phase elements to avoid the necessity of regularization. A new criterion for the computation of the iteration matrix is proposed. It is based on a quasi-Newton correction of the Jacobian matrix for conduction problems without change of phase. A set of test problems with exact solution is analysed and demonstrates that the procedure can accurately evaluate the front position and temperature history with a reasonable computational effort.

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

[2]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[3]  Sergio Idelsohn,et al.  Nonlinear structural dynamics via Newton and quasi-Newton methods , 1980 .

[4]  J. Z. Zhu,et al.  The finite element method , 1977 .

[5]  Ephraim M Sparrow,et al.  ANALYSIS OF MULTIDIMENSIONAL CONDUCTION PHASE CHANGE VIA THE ENTHALPY MODEL. , 1975 .

[6]  Graham E. Bell,et al.  On the performance of the enthalpy method , 1982 .

[7]  C. M. Elliott,et al.  Weak and variational methods for moving boundary problems , 1982 .

[8]  R. E. White.  A modified finite difference scheme for the Stefan problem , 1983 .

[9]  Paul T. Boggs,et al.  Moving boundary problems , 1978 .

[10]  Sergio Idelsohn,et al.  Computational strategies for the solution of large nonlinear problems via quasi-newton methods , 1981 .

[11]  Joseph W. Jerome,et al.  Error estimates for the multidimensional two-phase Stefan problem , 1982 .

[12]  J. T. Oden,et al.  Discontinuous finite-element approximations for the analysis of shock waves in nonlinearly elastic materials , 1975 .

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

[14]  J. Ciavaldini.  Analyse Numerique d’un Probleme de Stefan a Deux Phases Par une Methode d’Elements Finis , 1975 .

[15]  M. Cross,et al.  Accurate solutions of moving boundary problems using the enthalpy method , 1981 .

[16]  Jean S. Roose,et al.  Modelization of phase changes by fictitious‐heat flow , 1984 .

[17]  L. Carter Wellford,et al.  A finite element free boundary formulation for the problem of multiphase heat conduction , 1977 .

[18]  Frank Kreith,et al.  Heat transfer with melting or freezing in a wedge , 1973 .

[19]  Antonio Fasano,et al.  Numerical solution of phase-change problems , 1973 .

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

[21]  G. Meyer.  Multidimensional Stefan Problems , 1973 .

[22]  R. White,et al.  An Enthalpty Formulation of the Stefan Problem , 1982 .

[23]  P Jamet,et al.  Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements , 1977 .

[24]  Grant P. Steven.  Internally discontinuous finite elements for moving interface problems , 1982 .

[25]  Gunter H. Meyer.  The numerical solution of Stefan problems with front-tracking and smoothing methods , 1978 .

[26]  Roland W. Lewis,et al.  An improved algrorithm for heat conduction problems with phase change , 1978 .

[27]  L. E. Goodrich,et al.  Efficient numerical technique for one-dimensional thermal problems with phase change , 1978 .

[28]  D. R. Atthey.  A Finite Difference Scheme for Melting Problems , 1974 .