Integration over curves and surfaces defined by the closest point mapping

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wish to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation - initially derived to integrate over manifolds of codimension one - to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.Comment: Revised the pape

An Implicit Interface Boundary Integral Method for Poisson’s Equation on Arbitrary Domains

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions

Numerical study of laser micro- and nano-processing of nanocomposite porous materials

This thesis is focused on numerical simulations of the laser interaction with porous materials. A possibility of well-controlled processing is particularly important for the laser based micro-structuring of porous glass and nano-machining of semiconducting porous materials in the presence of metallic nanoparticles. To understand the periodic micro-void structures produced inside porous glass by femtosecond laser pulses, a detailed numerical thermodynamic analysis was performed. The calculation results show the possibility to control laser micro-machining in volume of SiO2. The obtained characteristic dimensions of the structures correlate with the experimental results. Comparing to the porous glass, the mesoporous TiO2 films loaded by Ag ions and nanoparticles support localized plasmon resonances. To identify the optimum parameters of the continuous-wave laser, a multi-physical model considering Ag nanoparticle growth, photo-oxidation, reduction was developed. The performed simulations show that the laser writing speed controls the Ag nanoparticles size. The thermally activated fast growth followed by the photo-oxidation was found to be the main reason for the writing speed-dependent size-change and temperature rises. Writing of mesoporous TiO2 films loaded with Ag nanoparticles by a pulsed laser is, furthermore, promising to provide additional possibilities in the generation of two kinds of nanostructures: laser-induced periodic surface grooves and Ag nanogratingsinside the TiO2 film. To better understand the effects of a pulsed laser, two multi-pulses models are developed to simulate the Ag nanoparticle growth. The obtained results provided new insights into laser micro-processing of porous material and better laser controlling over nanostructuring in porous semiconducting films loaded with metallic nanoparticles

An Energy Formulation of Surface Tension or Willmore Force For Two-Phase Flow

The motion of a biological cell in liquid is a rich subject for modeling. In the early 1970’s, it was realized by Canham that biological vesicles with lipid bilayer membranes reach a steady state shape that minimizes bending. Helfrich soon after mathematically quantified the related bending energy and showed that the shapes from minimizing this bending energy match the types of shapes observed in nature. The resulting Canham-Helfrich energy, consisting of bending energy and a constant surface area and volume constraint, is a major component of any model of cellular motility. To this end, we consider the cellular vesicle to be a closed interface between two fluids and we present a finite element model for a two-phase flow coupling the minimization of some given energy defined on the interface to the incompressible flow of the two fluids, which is then advected according to the resulting velocity field. We provide a general framework for incorporating the energies on the interface and then focus on three applications of energy on the interface: the first is surface tension minimizing the surface area energy, the second minimizes the bending energy without explicit surface area or volume constraints, the third minimizes the Canham-Helfrich energy including the constraints. We present a semi-implicit model for bending energy which uses an implicit levelset formulation for the interface and couples the forces from the interface to the two phase incompressible Navier-Stokes system through the use of an approximate Dirac delta function defined on a band around the interface. By using energies to describe the motion, our model is immediately provided with a sense of energy stability. We provide various numerical simulations and validations of flow under these three energies in two and three dimensions. Our simulations confirm that enforcing the volume constraint in the incompressible flow is vital to achieve the desired steady state shapes

Methods for higher order numerical simulations of complex inviscid fluids with immersed boundaries

Within this thesis, we study inviscid compressible flows of fluids modelled by several equations of state. Namely, these are the ideal gas law, the stiffened gas law, Tait's law and the covolume gas law. In their entirety, these equations of state can be used as models for the behaviour of many gases and liquids. After deriving new exact solutions for the corresponding variants of the Euler equations, we use the results as a tool for the verification of a higher-order accurate numerical scheme that has been implemented during the course of this thesis. The scheme is based on a Runge-Kutta Discontinuous Galerkin Method and the presented verification results show that we are able to obtain the expected rates of convergence in both, space and time. In the main part of this thesis, we consider an important building block for the extension of this conventional discretization by means of a treatment for generic immersed boundaries, namely the numerical integration of general functions over domains that are at least partly defined by the zero iso-contour of a level set function defining the domain boundary. Here, we study two new, generally applicable approaches in terms of their robustness and convergence behaviour. The first approach is based on a classical adaptive strategy, while the second approach is based on a hierarchical moment-fitting strategy with variable Ansatz order P. Both methods have been designed such that they are applicable on general element types. Most notably, the results of our numerical experiments suggest that the moment-fitting procedure leads to integration errors that decrease with a rate of O(h^(P+1)), thus allowing for a severe increase of integration accuracy at constant computational effort

Efficient Solvers for the Phase-Field Crystal Equation: Development and Analysis of a Block-Preconditioner

A preconditioner to improve the convergence properties of Krylov subspace solvers is derived and analyzed in this work. This method is adapted to linear systems arising from a finite-element discretization of a phase-field crystal equation