Magnetoelastic coupling describes the mutual dependence of the elastic and magnetic fields and can be observed in certain types of materials, among which are the so-called "magnetostrictive materials". They belong to the large class of "smart materials", which change their shape, dimensions or material properties under the influence of an external field. The mechanical strain or deformation a material experiences due to an externally applied magnetic field is referred to as magnetostriction; the reciprocal effect, i.e. the change of the magnetization of a body subjected to mechanical stress is called inverse magnetostriction. The coupling of mechanical and electromagnetic fields is particularly observed in "giant magnetostrictive materials", alloys of ferromagnetic materials that can exhibit several thousand times greater magnitudes of magnetostriction (measured as the ratio of the change in length of the material to its original length) than the common magnetostrictive materials. These materials have wide applications areas: They are used as variable-stiffness devices, as sensors and actuators in mechanical systems or as artificial muscles. Possible application fields also include robotics, vibration control, hydraulics and sonar systems. Although the computational treatment of coupled problems has seen great advances over the last decade, the underlying problem structure is often not fully understood nor taken into account when using black box simulation codes. A thorough analysis of the properties of coupled systems is thus an important task. The thesis focuses on the mathematical modeling and analysis of the coupling effects in magnetostrictive materials. Under the assumption of linear and reversible material behavior with no magnetic hysteresis effects, a coupled magnetoelastic problem is set up using two different approaches: the magnetic scalar potential and vector potential formulations. On the basis of a minimum energy principle, a system of partial differential equations is derived and analyzed for both approaches. While the scalar potential model involves only stationary elastic and magnetic fields, the model using the magnetic vector potential accounts for different settings such as the eddy current approximation or the full Maxwell system in the frequency domain. The distinctive feature of this work is the analysis of the obtained coupled magnetoelastic problems with regard to their structure, strong and weak formulations, the corresponding function spaces and the existence and uniqueness of the solutions. We show that the model based on the magnetic scalar potential constitutes a coupled saddle point problem with a penalty term. The main focus in proving the unique solvability of this problem lies on the verification of an inf-sup condition in the continuous and discrete cases. Furthermore, we discuss the impact of the reformulation of the coupled constitutive equations on the structure of the coupled problem and show that in contrast to the scalar potential approach, the vector potential formulation yields a symmetric system of PDEs. The dependence of the problem structure on the chosen formulation of the constitutive equations arises from the distinction of the energy and coenergy terms in the Lagrangian of the system. While certain combinations of the elastic and magnetic variables lead to a coupled magnetoelastic energy function yielding a symmetric problem, the use of their dual variables results in a coupled coenergy function for which a mixed problem is obtained. The presented models are supplemented with numerical simulations carried out with MATLAB for different examples including a 1D Euler-Bernoulli beam under magnetic influence and a 2D magnetostrictive plate in the state of plane stress. The simulations are based on material data of Terfenol-D, a giant magnetostrictive materials used in many industrial applications.

In this thesis we consider the directional analysis of stationary point processes. We focus on three non-parametric methods based on second order analysis which we have defined as Integral method, Ellipsoid method, and Projection method. We present the methods in a general setting and then focus on their application in the 2D and 3D case of a particular type of anisotropy mechanism called geometric anisotropy. We mainly consider regular point patterns motivated by our application to real 3D data coming from glaciology. Note that directional analysis of 3D data is not so prominent in the literature. We compare the performance of the methods, which depends on the relative parameters, in a simulation study both in 2D and 3D. Based on the results we give recommendations on how to choose the methods´ parameters in practice. We apply the directional analysis to the 3D data coming from glaciology, which consist in the locations of air-bubbles in polar ice cores. The aim of this study is to provide information about the deformation rate in the ice and the corresponding thinning of ice layers at different depths. This information is substantial for the glaciologists in order to build ice dating models and consequently to give a correct interpretation of the climate information which can be found by analyzing ice cores. In this thesis we consider data coming from three different ice cores: the Talos Dome core, the EDML core and the Renland core. Motivated by the ice application, we study how isotropic and stationary noise influences the directional analysis. In fact, due to the relaxation of the ice after drilling, noise bubbles can form within the ice samples. In this context we take two classification algorithms into consideration, which aim to classify points in a superposition of a regular isotropic and stationary point process with Poisson noise. We introduce two methods to visualize anisotropy, which are particularly useful in 3D and apply them to the ice data. Finally, we consider the problem of testing anisotropy and the limiting behavior of the geometric anisotropy transform.

Wir zeigen an einigen Beispielen, wie man numerische Simulationen in Tabellenkalkulationsprogrammen (hier speziell in Excel) erzeugen kann. Diese können beispielsweise im Kontext von mathematischer Modellierung verwendet werden. Die Beispiele umfassen ein Modell zur Ausbreitung von Krankheiten, die Flugkurve eines Fußballs unter Berücksichtigung von Luftreibung, eine Monte-Carlo-Simulation zur experimentellen Bestimmung von pi, eine Monte-Carlo-Simulation eines gemischten Kartenstapels und die Modellierung von Benzinpreisen mit einem Preistrend und Rauschen

Dieser Beitrag beschreibt eine Lernumgebung für Schülerinnen und Schüler der Unter- und Mittelstufe mit einem Schwerpunkt im Fach Mathematik. Das Thema dieser Lernumgebung ist die Simulation von Entfluchtungsprozessen im Rahmen von Gebäudeevakuierungen. Dabei wird das Konzept eines zellulären Automaten vermittelt, ohne dabei Programmierkenntnisse vorauszusetzen oder anzuwenden. Anhand dieses speziellen Simulationswerkzeugs des zellulären Automaten werden Eigenschaften, Kenngrößen sowie Vor- und Nachteile von Simulationen im Allgemeinen thematisiert. Dazu gehören unter anderem die experimentelle Datengewinnung, die Festlegung von Modellparametern, die Diskretisierung des zeitlichen und räumlichen Betrachtungshorizonts sowie die zwangsläufig auftretenden (Diskretisierungs-)Fehler, die algorithmischen Abläufe einer Simulation in Form elementarer Handlungsanweisungen, die Speicherung und Visualisierung von Daten aus einer Simulation sowie die Interpretation und kritische Diskussion von Simulationsergebnissen. Die vorgestellte Lernumgebung ermöglicht etliche Variationen zu weiteren Aspekten des Themas „Evakuierungssimulation“ und bietet dadurch auch vielfältige Differenzierungsmöglichkeiten.