The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more than two phases, the X-FEM has to be further eXtended. The proposed approach is presented in the case of a quasi-static Stokes n-phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases. A level set hierarchy allows describing triple junctions avoiding overlapping or “voids” between materials. Moreover, an enriched solution accounting for several simultaneous phases inside one element is proposed. The interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets.
The computational implementation of this scheme involves calculating integrals in elements having several materials inside. An adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals.
Examples of the hierarchical X-FEM approach are given for a n-phase Stokes problem in 2 and 3 dimensions.