## 76M10 Finite element methods

This thesis is concerned with the design and analysis of an interface fitted finite element strategy
for the simulation of free- and moving boundary problems with sharp interfaces. The presented
strategy falls into the class of interface tracking methods based on the Arbitrary Lagrangian-Eulerian (ALE) formulation. The main goal is to resolve the fundamental drawback of mesh
degeneration typical of moving mesh strategies while preserving all other benefits, in particular a
well-defined, accurate, and explicit representation of the moving geometry in space and time. This
representation allows for a straight-forward definition and implementation of problem tailored finite element spaces.
Parametric finite element spaces based on curved elements serve as the baseline of the presented
approach and are introduced and discussed in the context of a prototypical elliptic interface problem. A solution to the associated problem of generating admissible, interface fitted parametrizations is presented in terms of a variational mesh optimization approach. In this approach, a mesh
quality functional subject to an alignment constraint is minimized which leads to interface aligned,
optimal, and non-degenerate parametrizations.
For time-dependent problems, interface aligned parametrizations are constructed such that they
behave smoothly in time within each time slab, while a re-parametrization taking place in between
two successive time slabs is permitted. This strategy yields parametric finite element spaces that
are discontinuous in time. Consequently, a discontinuous Galerkin (dG(k)) approach in time is
used to construct fully discrete schemes. For the lowest order variant dG(0), an a priori error
analysis is conducted to show that the proposed strategy leads to convergence rates that are sub-
optimal by one order with respect to space in the worst case. A strategy to solve arising space-time
systems for higher-order dG(k) methods is proposed. This strategy is based on a transformation of
the space-time system into real block diagonal form, such that an efficient, preconditioned Schur
complement formulation for the arising 2 × 2 blocks can be used. It is shown that the preconditioned system possesses a condition number that is uniformly bounded by 2.
The last part of this thesis addresses the application of the strategy to free boundary problems. In
the context of a general flow problem in the presence of an interface force, governing equations
are introduced, a fully discrete space-time formulation is derived and an efficient first order in
time method is proposed. Two different application scenarios, a two-phase flow example with
surface tension, and a fluid-structure interaction problem are considered to validate and evaluate
the proposed approach, and to show its benefits and limitations.

In this thesis, a numerical method for the simulation of two–phase flow problems with a deformable interface and coupled species transport is investigated. The coupling of species concentration and fluid dynamics originates from a concentration–dependent interfacial tension coefficient – an effect known as concentration–induced Marangoni convection.
For this purpose, a mathematical model of a single drop was formulated using subspaces of the conventional function spaces, a problem–specific nondimensionalization, an accelerated frame of reference, and a divergence formulation of interfacial stresses.
The numerical method is based on a finite element method within an arbitrary Lagrangian– Eulerian framework. This class of methods results in an explicit interface representation, and forms the basis of the subspace projection method, a novel method for the implementation of various interface conditions in two–phase flow problems via problem–specific projections. Interface conditions that are investigated in this thesis comprise the flow around a rigid body, the simulation of spherically–shaped fluid particles, interface conditions of deformable drops with and without Marangoni convection, and the simulation of spherical stagnant caps – a limiting case observed in single drop flow under the presence of surfactants.
An essential feature of the subspace projection method is its ability to represent discontinuous functions, enabled by using a computational grid with doubled interfacial nodes. Spurious oscillations –often observed in two–phase flow problems where the pressure is approximated by globally continuous functions– are considerably reduced.
Concentration–induced Marangoni convection (resulting from a non–vanishing tangential gradient of interfacial stresses) is a challenging application in the numerical simulation of two– phase flows with a deforming interface. The implementation of interfacial stresses is based on a divergence formulation. This formulation accounts for normal and tangential stresses without the necessity of approximating surface derivatives of the interfacial tension.
A validation of the numerical method is performed for different axisymmetric single drop flow applications.
Different binary systems (single, buoyant, deformable drops with constant interfacial tension) are numerically investigated in terms of terminal rise velocities, drag coefficients, and deformation, comprising the range from low to high interfacial tension coefficients and drop diameters below the onset of shape oscillations. An excellent agreement of simulation results and experimental data is obtained. In case of terminal rise velocities, the deviation is below 4%.
Simulations of ternary systems (binary systems with species transport) are performed with and without Marangoni convection. In case of high Peclet numbers, finite element simulations often suffer from instabilities due to convection dominance. Stabilizing methods are investigated with special regard to two–phase flow applications.
Simulations of fluid dynamics and species transport in single drop flow applications like rigid particles, spherically shaped drops, spherical stagnant caps, and thermocapillary migration, show good to excellent agreement to correlations from literature in systems with moderate Peclet numbers. The deviation of the thermocapillary rise velocity is below 0.2% to the prediction by Young, Goldstein and Block (1959).
The ternary system toluene/acetone/water with concentration–induced Marangoni convection is numerically investigated and the results are compared to experimental data. In spite of the convection dominance of the corresponding species transport problem and the assumption of axisymmetry despite the inherent three–dimensional nature of the Marangoni effect, a reasonable qualitative agreement was achieved.

A novel finite element method for the 3d simulation of (many) particles in a Newtonian carrier liquid is presented. The method features the celebrated one domain approach to simplify the spatial discretization, a newly developed subspace projection method to account for the rigid body motion within the particles and an operator splitting to decouple the nonlinearities. Combined with local mesh refinement the method results in a fast and accurate algorithm which is, in addition conceptually simple to implement. Validation is achieved using the sedimentation of a single particle and comparing the resulting drag coefficient with theoretical and experimental results. Furthermore, a viscometer is considered where the effective viscosity of a particle laden fluid is compared with analytic results. Furthermore, a method for the solution of the Nernst–Planck–Poisson equations is presented. These equations describe the distribution of the concentration of charged substances in a fluid. Validation is carried out by stationary solutions of the equations. Finally, both methods are combined for the simulation of particulate electrodynamic flows.