Mixed finite-element methods for elliptic convection-dominated problems arising in semiconductor physics

In this work we develop and analyze an adaptive numerical scheme for simulating a class of macroscopic semiconductor models. At first the numerical modelling of semiconductors is reviewed in order to classify the Energy-Transport models for semiconductors that are later simulated in 2D. In this class of models the flow of charged particles, that are negatively charged electrons and so-called holes, which are quasi-particles of positive charge, as well as their energy distributions are described by a coupled system of nonlinear partial differential equations. A considerable difficulty in simulating these convection-dominated equations is posed by the nonlinear coupling as well as due to the fact that the local phenomena such as "hot electron effects" are only partially assessable through the given data. The primary variables that are used in the simulations are the particle density and the particle energy density. The user of these simulations is mostly interested in the current flow through parts of the domain boundary - the contacts. The numerical method considered here utilizes mixed finite-elements as trial functions for the discrete solution. The continuous discretization of the normal fluxes is the most important property of this discretization from the users perspective. It will be proven that under certain assumptions on the triangulation the particle density remains positive in the iterative solution algorithm. Connected to this result an a priori error estimate for the discrete solution of linear convection-diffusion equations is derived. The local charge transport phenomena will be resolved by an adaptive algorithm, which is based on a posteriori error estimators. At that stage a comparison of different estimations is performed. Additionally a method to effectively estimate the error in local quantities derived from the solution, so-called "functional outputs", is developed by transferring the dual weighted residual method to mixed finite elements. For a model problem we present how this method can deliver promising results even when standard error estimator fail completely to reduce the error in an iterative mesh refinement process.

Numerical coupling of thermal-electric network models and energy-transport equations including optoelectronic semiconductor devices

In this work the numerical coupling of thermal and electric network models with model equations for optoelectronic semiconductor devices is presented. Modified nodal analysis (MNA) is applied to model electric networks. Thermal effects are modeled by an accompanying thermal network. Semiconductor devices are modeled by the energy-transport model, that allows for thermal effects. The energy-transport model is expandend to a model for optoelectronic semiconductor devices. The temperature of the crystal lattice of the semiconductor devices is modeled by the heat flow eqaution. The corresponding heat source term is derived under thermodynamical and phenomenological considerations of energy fluxes. The energy-transport model is coupled directly into the network equations and the heat flow equation for the lattice temperature is coupled directly into the accompanying thermal network. The coupled thermal-electric network-device model results in a system of partial differential-algebraic equations (PDAE). Numerical examples are presented for the coupling of network- and one-dimensional semiconductor equations. Hybridized mixed finite elements are applied for the space discretization of the semiconductor equations. Backward difference formluas are applied for time discretization. Thus, positivity of charge carrier densities and continuity of the current density is guaranteed even for the coupled model.

MikroSISAK für die Chemie der schwersten Elemente

ZusammenfassungrnDie vorliegende Arbeit beschreibt Experimente mit einer Apparatur namens Mikro-rnSISAK, die in der Lage ist, eine Flüssig-Flüssig-Extraktion im Mikroliter-Maßstab durchzuführen. Dabei werden zwei nicht mischbare Flüssigkeiten in einer Mikrostruktur emulgiert und anschließend über eine Teflonmembran wieder entmischt.rnIn ersten Experimenten wurden verschiedene Extraktionssysteme für Elemente derrnGruppen 4 und 7 des Periodensystems der Elemente untersucht und die Ergebnisse mit denen aus Schüttelversuchen verglichen. Da die zunächst erreichten Extraktionsausbeuten nicht ausreichend waren, wurden verschiedene Maßnahmen zu deren Verbesserung herangezogen.rnZunächst hat man mit Hilfe eines an die MikroSISAK-Apparatur angelegten Heizelements die dort für die Extraktion herrschende Temperatur erhöht. Dies führte wie erhofft zu einer höheren Extraktionsausbeute.rnDes Weiteren wurde MikroSISAK vom Institut für Mikrotechnik Mainz, welches derrnEntwickler und Konstrukteur der Apparatur ist, durch eine Erweiterung modifiziert, um den Kontakt der beiden Phasen zwischen Mischer und Separationseinheit zu verlängern. Auch dies verbesserte der Extraktionsausbeute.rnNun erschienen die erzielten Ergebnisse als ausreichend, um die Apparatur für online-Experimente an den TRIGA-Reaktor Mainz zu koppeln. Hierfür wurden durch Kernreaktion erzeugte Spaltprodukte des Technetiums MikroSISAK zugeführt, um sie dort abzutrennen und anschließend über ihren Zerfall an einem Detektor nachzuweisen. Neben erfolgreichen Ergebnissen lieferten diese Experimente auch die Belege für die Funktionsfähigkeit eines neuen Entgasers und für die Möglichkeit sowohl diesen als auch ein adäquates Detektorsystem an die MikroSISAK-Apparatur anzuschließen.rnDies schafft die Voraussetzung für die eigentliche Anwendungsidee, die hinter der Entwicklung von MikroSISAK steckt: Die Untersuchung der chemischen Eigenschaften von kurzlebigen superschweren Elementen (SHE) an einem Schwerionenbeschleuniger. Es liegt nahe, solche Experimente für das schwere Homologe des Technetiums, Element 107, Bohrium, ins Auge zu fassen.

Analysis and numerical solution of the Peterlin viscoelastic model

Liquids and gasses form a vital part of nature. Many of these are complex fluids with non-Newtonian behaviour. We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. In two space dimensions we prove global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme. We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the pressure-stabilized characteristic finite element scheme is confirmed by a series of numerical experiments.