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@PHDTHESIS{Lehrenfeld:462743,
author = {Lehrenfeld, Christoph},
othercontributors = {Reusken, Arnold and Behr, Marek and Schöberl, Joachim},
title = {{O}n a {S}pace-{T}ime {E}xtended {F}inite {E}lement
{M}ethod for the {S}olution of a {C}lass of {T}wo-{P}hase
{M}ass {T}ransport {P}roblems},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {Publikationsserver der RWTH Aachen University},
reportid = {RWTH-2015-00673},
pages = {IX, 219 S. : Ill., graph. Darst.},
year = {2015},
note = {Aachen, Techn. Hochsch., Diss., 2015},
abstract = {In the present thesis a new numerical method for the
simulation of mass transport in an incompressible immiscible
two-phase flow system is presented. The mathematical model
consists of convection diffusion equations on moving domains
which are coupled through interface conditions. One of those
conditions, the Henry interface condition, prescribes a jump
discontinuity of the solution across the moving
interface.For the description of the interface position and
its evolution we consider interface capturing methods, for
instance the level set method. In those methods the mesh is
not aligned to the evolving interface such that the
interface intersects mesh elements. Hence, the moving
discontinuity is located within individual elements which
makes the numerical treatment challenging. The
discretization presented in this thesis is based on
essentially three core components. The first component is an
enrichment with an extended finite element (XFEM) space
which provides the possibility to approximate discontinuous
quantities accurately without the need for aligned meshes.
This enrichment, however, does not respect the Henry
interface condition.The second component cures this issue by
imposing the interface condition in a weak sense into the
discrete variational formulation of the finite element
method. To this end a variant of the Nitsche technique is
applied. For a stationary interface the combination of both
techniques offers a good way to provide a reliable method
for the simulation of mass transport in two-phase flows.
However, the most difficult aspect of the problem is the
fact that the interface is typically not stationary, but
moving in time. The numerical treatment of the moving
discontinuity requires special care. For this purpose a
space-time variational formulation, the third core component
of this thesis, is introduced and combined with the first
two components: the XFEM enrichment and the Nitsche
technique. In this thesis we present the components and the
resulting methods one after another, for stationary and
non-stationary interfaces. We analyze the methods with
respect to accuracy and stability and discuss important
properties. For the case of a stationary interface the
combination of an XFEM enrichment and the Nitsche technique,
the Nitsche-XFEM method, has been introduced by other
authors. Their method, however, lacks stability in case of
dominating convection. We combine the Nitsche-XFEM method
with the Streamline Diffusion technique to provide a stable
method also for convection dominated problems. We further
discuss the conditioning of the linear systems arising from
Nitsche-XFEM discretizations which can be extremely
ill-conditioned.For the case of a moving interface we
propose a space-time Galerkin formulation with trial and
test functions which are discontinuous in time and combine
this approach with an XFEM enrichment and a Nitsche
technique resulting in the Space-Time Nitsche-XFEM method.
This method is new. We present an error analysis and discuss
implementation aspects like the numerical integration on
arising space-time geometries.The aforementioned methods
have been implemented in the software packages DROPS for the
spatially three-dimensional case. The correctness of the
implementation and the accuracy of the method is analyzed
for test cases. Finally, we consider the coupled simulation
of mass transport and fluid dynamics for realistic
scenarios.},
keywords = {Dissertation (GND)},
cin = {111710 / 110000},
ddc = {510},
cid = {$I:(DE-82)111710_20140620$ / $I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
urn = {urn:nbn:de:hbz:82-rwth-2015-006739},
url = {https://publications.rwth-aachen.de/record/462743},
}
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