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Selected papers from the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Finland, July 2004
Vaughan Voller
1 Introduction
A moving boundary problem is a problem in which one of the domain boundaries is an unknown. The classic example is the Stefan melting problem ([4] Crank, 1984) a heat transfer problem requiring the tracking of the a priori unknown melting front. Since, the melt front position needs to be determined as part of the solution the problem formulation requires an additional boundary condition - the Stefan condition, obtained by balancing the net heat flux arriving at the melting front with the rate of evolution of latent heat.
Typically moving boundary problems only have a limited number of analytical solutions ([2] Carslaw and Jaeger, 1986) and as a result, from the advent of digital computers, see [5] Eyres et al. (1946) and [13] Price and Slack (1954), there has been extensive development of numerical methods. The key feature in these methods is the mechanisms used to track the continuously moving boundary over the discrete grid of nodes that define the numerical method. Very broadly speaking, these mechanisms fall into one of three classes.
Fixed grid methods. These methods employ a grid of nodes that remain fixed in space and track the boundary by use of an auxiliary variable. An example is the so-called enthalpy methods, used in the analysis solid-liquid phase change ([5] Eyres et al. , 1946; [13] Price and Slack, 1954; [17] Voller and Cross, 1981; [4] Crank, 1984; [18] Voller et al. , 1990; [16] Voller, 1996). In these methods, the melting front is tracked by the evaluation of a nodal liquid farction field, the elements of which take values 0≤f ≤1.
Deforming grid methods. In these methods, a line of nodes is located on the moving boundary and as the solution evolves the space grid deforms to ensure that these nodes remain on the boundary ([8] Lynch and O'Neill, 1981; [7] Lynch, 1982; [1] Beckett et al. , 2001).
Hybrid methods. Hybrid methods employ elements of both fixed and deforming grids, e.g. local front tracking ([15] Udaykumar et al. , 1999; [3] Crank, 1957) which uses a fixed background grid and employs local front tracking schemes to...