The Euler number of OGradys 10-dimensional symplectic manifold

Wir berechnen die Eulerzahl der 10-dimensionalen exzeptionellen irreduziblen symplektischen Mannigfaltigkeit, die von O Grady konstruiert wurde. Die Idee besteht darin, zunächst eine Lagrangefaserung zu konstruieren und dann die Eulerzahlen der Fasern zu berechnen. Es stellt sich heraus, dass fast alle Fasern die Eulerzahl 0 haben, und deswegen reduziert sich das Problem auf die Berechnung der Eulerzahlen der übrigen Fasern. Diese Fasern sind Modulräume von halbstabilen Garben auf singulären Kurven. Der Hauptteil dieser Dissertation ist der Berechnung der Eulerzahlen dieser Modulräume gewidmet. Diese Resultate sind von unabhängigem Interesse.

Finite-element discretization of 3D energy-transport equations for semiconductors

In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and demonstrated in detail. We compare these new iterations with a standard method that is complemented by a feature to fit in the current context. A further innovation is the computation of solutions in three-dimensional domains, which are still rare. Special attention is paid to applicability of the 3D simulation tools. The programs are designed to have justifiable working complexity. The simulation results of some models of contemporary semiconductor devices are shown and detailed comments on the results are given. Eventually, we make a prospect on future development and enhancements of the models and of the algorithms that we used.

Multilevel Domain Decomposition Algorithms for Monolithic Fluid-Structure Interaction Problems with Application to Haemodynamics

Finite element techniques for solving the problem of fluid-structure interaction of an elastic solid material in a laminar incompressible viscous flow are described. The mathematical problem consists of the Navier-Stokes equations in the Arbitrary Lagrangian-Eulerian formulation coupled with a non-linear structure model, considering the problem as one continuum. The coupling between the structure and the fluid is enforced inside a monolithic framework which computes simultaneously for the fluid and the structure unknowns within a unique solver. We used the well-known Crouzeix-Raviart finite element pair for discretization in space and the method of lines for discretization in time. A stability result using the Backward-Euler time-stepping scheme for both fluid and solid part and the finite element method for the space discretization has been proved. The resulting linear system has been solved by multilevel domain decomposition techniques. Our strategy is to solve several local subproblems over subdomain patches using the Schur-complement or GMRES smoother within a multigrid iterative solver. For validation and evaluation of the accuracy of the proposed methodology, we present corresponding results for a set of two FSI benchmark configurations which describe the self-induced elastic deformation of a beam attached to a cylinder in a laminar channel flow, allowing stationary as well as periodically oscillating deformations, and for a benchmark proposed by COMSOL multiphysics where a narrow vertical structure attached to the bottom wall of a channel bends under the force due to both viscous drag and pressure. Then, as an example of fluid-structure interaction in biomedical problems, we considered the academic numerical test which consists in simulating the pressure wave propagation through a straight compliant vessel. All the tests show the applicability and the numerical efficiency of our approach to both two-dimensional and three-dimensional problems.

Transformer modeling ATP : internal faults & high-frequency discretization

Power transformers are key components of the power grid and are also one of the most subjected to a variety of power system transients. The failure of a large transformer can cause severe monetary losses to a utility, thus adequate protection schemes are of great importance to avoid transformer damage and maximize the continuity of service. Computer modeling can be used as an efficient tool to improve the reliability of a transformer protective relay application. Unfortunately, transformer models presently available in commercial software lack completeness in the representation of several aspects such as internal winding faults, which is a common cause of transformer failure. It is also important to adequately represent the transformer at frequencies higher than the power frequency for a more accurate simulation of switching transients since these are a well known cause for the unwanted tripping of protective relays. This work develops new capabilities for the Hybrid Transformer Model (XFMR) implemented in ATPDraw to allow the representation of internal winding faults and slow-front transients up to 10 kHz. The new model can be developed using any of two sources of information: 1) test report data and 2) design data. When only test-report data is available, a higher-order leakage inductance matrix is created from standard measurements. If design information is available, a Finite Element Model is created to calculate the leakage parameters for the higher-order model. An analytical model is also implemented as an alternative to FEM modeling. Measurements on 15-kVA 240?/208Y V and 500-kVA 11430Y/235Y V distribution transformers were performed to validate the model. A transformer model that is valid for simulations for frequencies above the power frequency was developed after continuing the division of windings into multiple sections and including a higher-order capacitance matrix. Frequency-scan laboratory measurements were used to benchmark the simulations. Finally, a stability analysis of the higher-order model was made by analyzing the trapezoidal rule for numerical integration as used in ATP. Numerical damping was also added to suppress oscillations locally when discontinuities occurred in the solution. A maximum error magnitude of 7.84% was encountered in the simulated currents for different turn-to-ground and turn-to-turn faults. The FEM approach provided the most accurate means to determine the leakage parameters for the ATP model. The higher-order model was found to reproduce the short-circuit impedance acceptably up to about 10 kHz and the behavior at the first anti-resonant frequency was better matched with the measurements.

Leonhard Euler, Brustbild

Signatur des Originals: S 36/G00802

Leonhard Euler, Brustbild

Signatur des Originals: S 36/G00803

Techniques for Area Discretization and Coverage in Aerial Photography for Precision Agriculture employing mini quad-rotors

Remote sensed imagery acquired with mini aerial vehicles, in conjunction with GIS technology enable a meticulous analysis from surveyed agricultural sites. This paper sums up the ongoing work in area discretization and coverage with mini quad-­?rotors applied to Precision Agriculture practices under the project RHEA.

Automatic mesh generation processor for 3-D parabolic boundary discretization

An automatic Mesh Generation Preprocessor for BE Programs with a considerable of capabilities has been developed. This program allows almost any kind of geometry and tipology to be defined with a small amount of external data, and with an important approximation of the boundary geometry. Also the error checking possibility is very important for a fast comprobation of the results.

Elementos finitos con acciones repartidas equivalentes de cualquier orden. Aplicación a los modelos de vigas de Timoshenko y Bernoulli-Euler

En este trabajo se introducen, en el contexto del Método de Elementos Finitos, dos alternativas posibles en relación con el concepto de acción repartida equivalente. La primera consiste en emplear pocos elementos, elevando el orden de dicha acción, mientras que la segunda se basa en emplear un mayor número de elementos dejando la acción en el orden más bajo posible. Se ilustran ambas situaciones mediante aplicaciones a los modelos de vigas de Timoshenko y Bernoulli-Euler, empleando estas acciones con diferentes órdenes, las cuales aproximan a la acción original, mediante polinomios ortogonales de Legendre en cada elemento. Como conclusión destacable, se indica que cuando se considera el menor número posible de elementos, es decir uno, para los casos de carga poco regular, ha bastado con utilizar acciones repartidas equivalentes de orden ligeramente superior al mínimo (orden cuatro), para obtener una excelente aproximación en los desplazamientos, giros y esfuerzos en el interior de los elementos.