With regards to the finite element modeling of phase transformations, the foremost concern is ``capturing'' the solution in the vicinity of the moving interface, where the solution exhibits sharp gradients. Some pioneering work was developed by [Lynch and O'Neill, 1981], who wrote the shape functions as functions of time as well as the spatial coordinates. The nodes on the freezing front thus moved with the interface, such that the discontinuity in temperature gradient could always be captured. Twenty years later, this basic technique continues to be adopted for a wide range of problems. For directional solidification problems, the technique has been considerably refined by Zabaras and his coworkers (see [Sampath and Zabaras, 1999]).
A number of problems exist with the moving-mesh approach. Firstly, it is best applied to directional solidification problems, wherein the front essentially moves unidirectionally. The main reason is mesh distortion. It is easy to visualize that for freezing fronts which are geometrically complex, the moving mesh approach eventually leads to mesh entanglement. Secondly, for thermo-mechanical problems, we are concerned with the evolution of stress and strain in the object. A moving mesh approach requires projection techniques, and crucially in the vicinity of sharp gradients, leading to a significant loss in accuracy. In part, these concerns motivated the ALE formulation developed by [Ghosh and Moorthy, 1993]. The work proposed herein shares several features with this approach, though it is suggested to be much simpler and more flexible.
The main philosophy of the eXtended Finite Element Method (X-FEM) developed by [Dolbow, 1999] and [Moës et al., 1999], is to model moving features (cracks, voids, interfaces), with moving enrichment strategies. The basic idea is to augment a subset of the nodal basis functions with products of those basis functions and prescribed enrichment functions. It therefore belongs to the general class of partition-of-unity methods as described by [Melenk and Babuška, 1996]. When the prescribed functions contain discontinuities, the result is a method capable of representing cracks whose morphology is independent of the finite element boundaries. As a result, no remeshing is required to simulate crack growth; one merely changes the subset of enriched nodes and updates (if necessary) the enrichment functions. Likewise, its application to phase transformations offers the possibility of using a fixed mesh.
For phase transformation problems, we seek to enrich with functions whose derivatives are discontinuous across the interface. While the precise form of the functions which provide the greatest accuracy has yet to be determined, the basic concept is appealing. With the X-FEM the mesh and interface descriptions are independent, and yet we can still maintain accuracy at the interface, and allow it to move arbitrarily. Thus the method is applicable to freezing front motions beyond those of directional solidification. Moreover, with regard to the thermo-mechanical problem, the projections are postulated to be much more accurate.
This report is organized as follows. In the next section, we describe the governing equations which describe the heat transfer problem. We then discuss the weak form, and the discretization with the X-FEM. We after provide some one-dimensional experiments of the general method, and the last section describes the phase transformation problem and shows some results in two dimension.