The X-FEM Approximation

The standard finite element approximation to the temperature field takes the form:

where are nodal shape functions and are the corresponding coefficients. The sum is taken over all nodes in the mesh. When the shape functions are piecewise linear, the above construction has the gradient

which can only be discontinuous across element boundaries. This is, essentially, one of the motivations for using a moving mesh technique, where the element and interface boundaries always coincide.

In the X-FEM, we extend the above using the construction

where is the enrichment function, are enriched degrees of freedom and is a subset of the nodes in the mesh. In the case of phase transformations, we take to consist of all nodes whose shape function supports contain the liquid-solid interface:

where is the location of the freezing front. This construction has a gradient

which can be discontinuous wherever is discontinuous. This represents the basic idea for this work: the motion of the interface is modeled through a change in the enrichment function .