Hierarchical X-FEM for n-phase flow (n>2)

Author (s): Zlotnik, S., Díez, P.
Journal: Computer Methods in Applied Mechanics and Engineering

Volume: 198 Issues 30-32
Pages: 2329 – 2338
Date: 2009

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 quasistatic 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.



author = "Zlotnik, S. and Díez, P.“,
title = "Hierarchical X-FEM for n-phase flow ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "198",
number = "30–32",
pages = "2329 - 2338",
year = "2009",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2009.02.025",
url = "http://www.sciencedirect.com/science/article/pii/S0045782509000978",