Applied Scientific Research

, Volume 44, Issue 1–2, pp 61–92 | Cite as

Finite element simulation of solidification problems

  • R. W. Lewis
  • P. M. Roberts


The modelling of liquid-solid phase change phenomena is extremely important in many areas of science and engineering. In particular, the solidification of molten metals during various casting methods in the foundry, provides a source of important practical problems which may be resolved economically with the aid of computational models of the heat transfer processes involved. Experimental design analysis is often prohibitively expensive, and the geometries and complex boundary conditions encountered preclude any analytical solutions to the problems posed. Thus the motivation for numerical simulation and computer aided design (CAD) systems is clear, and several mathematical/computational modelling techniques have been brought to bear in this area during recent years.

This paper reports on the application of the finite element method to solidification problems, principally concerning industrial casting processes. Although convective heat transfer has been modelled, the work herein considers only heat conduction, for clarity. The heat transfer model has also been coupled with thermal stress analysis packages to predict mechanical behaviour including cracking and eventual failure, but this is reported elsewhere.

Following the introduction, the mathematical and computational modelling tools are described in detail, for completeness. A discussion on the handling of the phase change interface and latent heat effects is then presented. Some aspects of the solution procedures are examined next, together with special techniques for dealing with the mold-metal interface. Finally, some numerical examples are presented which substantiate the capabilities of the finite element model, in both two and three dimensions.



heat capacity


capacitance matrix


time function


loading term


heat convection coefficient


specific enthalpy


Jacobian determinant


patch approximation to |J|


thermal conductivity


conductance matrix


latent heat

\(\hat n\)

unit outward normal


nodal shape function


known heat flux


nodal heat capacity


phase change interface





\(\hat T\)

known boundary temperature


vector of nodal temperatures


ambient temperature


solidification temperature


liquidus temperature


initial temperature


solidus temperature


space coordinates


interface heat transfer coefficient


iteration parameter


boundary of domain


solidification range


timestep magnitude

vector gradient operator


convergence tolerance


timestepping parameter

known vector in alternating-direction formulation


Laplace modifying parameter

(ξ, η)

local space coordinates




time limit


shape function factor


shape function factor


domain of interest


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Copyright information

© Martinus Nijhoff Publishers 1987

Authors and Affiliations

  • R. W. Lewis
    • 1
  • P. M. Roberts
    • 1
  1. 1.Institute for Numerical Methods in Engineering, University College of SwanseaUniversity of WalesSwanseaUnited Kingdom

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