A fixed grid finite element technique for modelling phase change in steady-state conduction–advection problems

A new fixed grid finite element technique for modelling phase-change effects in steady-state conduction–advection problems is described. The method is appropriate for materials that transform at a fixed critical temperature and is based on the assumption that the enthalpy experiences a jump discontinuity at this temperature, yielding a singular specific heat. This singularity is described using a Dirac delta function and the appropriate finite element matrix equations are developed for isoparametric elements. A simple first order upwinding scheme is used to avoid numerical oscillations in advection dominated problems. The utility of the method is illustrated through the solution of a one-dimensional test problem using eight-node hexahedral isoparametric finite elements. The technique may be applied to a wide range of finite element types and offers a number of advantages over other schemes for modelling phase-change effects.

[1]  Juan C. Heinrich,et al.  Petrov-Galerkin method for multidimensional time-dependent convective-diffusion equations , 1987 .

[2]  Daniel R. Lynch,et al.  Unified approach to simulation on deforming elements with application to phase change problems , 1982 .

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

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

[5]  Nicholas Zabaras,et al.  An analysis of solidification problems by the boundary element method , 1987 .

[6]  Kevin O'Neill,et al.  Boundary integral equation solution of moving boundary phase change problems , 1983 .

[7]  Z. Abdullah,et al.  On the numerical modelling of heat transfer during solidification processes , 1988 .

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

[9]  O. C. Zienkiewicz,et al.  An ‘upwind’ finite element scheme for two‐dimensional convective transport equation , 1977 .

[10]  M. N. Özişik,et al.  Exact Solution for Freezing in Cylindrical Symmetry with Extended Freezing Temperature Range , 1979 .

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

[12]  J. Heinrich.  Finite element techniques for fluid flow, J. J. Connor and C. A. Brebbia, Newnes–Butterworth, London/Boston, 1976. No. of pages: 310. Price: $10·00 , 1977 .

[13]  O. C. Zienkiewicz,et al.  Finite element methods for second order differential equations with significant first derivatives , 1976 .

[14]  A. Patera.  A FINITE-ELEMENT/GREEN'S FUNCTION EMBEDDING TECHNIQUE APPLIED TO ONE-DIMENSIONAL CHANGE-OF-PHASE HEAT TRANSFER , 1984 .

[15]  Anthony T. Patera,et al.  A Legendre spectral element method for the Stefan problem , 1987 .

[16]  M. Cross,et al.  An enthalpy method for convection/diffusion phase change , 1987 .

[17]  Sergio Idelsohn,et al.  A temperature‐based finite element solution for phase‐change problems , 1986 .

[18]  M. Salcudean,et al.  On numerical methods used in mathematical modeling of phase change in liquid metals , 1988 .

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

[20]  T. Saitoh,et al.  High Rayleigh Number Solutions to Problems of Latent Heat Thermal Energy Storage in a Horizontal Cylinder Capsule , 1982 .

[21]  François Thomasset,et al.  Implementation of Finite Element Methods for Navier-Stokes Equations , 1981 .

[22]  A. Patera.  a Finite-Element Function Embedding Technique Applied to One-Dimensional Change-Of Heat Transfer , 1984 .

[23]  Mario A. Storti,et al.  Making curved interfaces straight in phase‐change problems , 1987 .