| A model is proposed to describe the influence of solidification shrinkage on the final shape of cast ingots. We regard it as a starting point towards the modelling of the phenomena producing thermal stresses. The model is applied to 'nearly' purely thermal solidification processes occurring in 'long' cylindrical molds. In this instance evolution equations are greatly simplified: their solution in a closed form is found in some technologically interesting instances by means of a semi-inverse method. This solution is used to validate the numerical procedure conceived for tracking the free moving boundary and for determining the final shape of the ingot in a wider range of temperature and heat flux boundary conditions. The above mentioned procedure is based on a nonconventional space-time finite difference discretization of the governing field and boundary equations. A locally uniaxial discretization in space is worked out along either direction of an orthogonal two-dimensional grid. It is refined by iteratively solving the sequence of two uniaxial linear problems, each of which exhibits a narrowly banded coefficient matrix: thus their solutions are found via a factorization algorithm. |
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