# Finite element simulation of solidification problems

DOI: 10.1007/BF00412007

- Cite this article as:
- Lewis, R.W. & Roberts, P.M. Applied Scientific Research (1987) 44: 61. doi:10.1007/BF00412007

## Abstract

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.

### Nomenclature

*c*heat capacity

**C**capacitance matrix

*f*time function

**F**loading term

*h*heat convection coefficient

*H*specific enthalpy

- |
*J*| Jacobian determinant

- |
*Ĵ*| patch approximation to |

*J*|*k*thermal conductivity

**K**conductance matrix

*L*latent heat

- \(\hat n\)
unit outward normal

*N*_{i}nodal shape function

*q*known heat flux

*R*_{i}nodal heat capacity

*S*phase change interface

*t*time

*T*temperature

- \(\hat T\)
known boundary temperature

**T**vector of nodal temperatures

*T*_{a}ambient temperature

*T*_{c}solidification temperature

*T*_{L}liquidus temperature

*T*_{0}initial temperature

*T*_{s}solidus temperature

**x**space coordinates

**α**interface heat transfer coefficient

*γ*iteration parameter

- Γ
boundary of domain

*δT*solidification range

*Δt*timestep magnitude

- ▽
vector gradient operator

*ε*convergence tolerance

*θ*timestepping parameter

- tΘ
known vector in alternating-direction formulation

*λ*Laplace modifying parameter

- (
*ξ, η*) local space coordinates

*ρ*density

*τ*time limit

*φ*(*ξ*)shape function factor

*ψ*(*η*)shape function factor

- Ω
domain of interest