Zugl.: Bielefeld, Univ., Diss., 2010

Zugl.: Aachen, Techn. Hochsch., Diss., 2008.

Zugl.: Hagen, Fernuniv., Diss., 2009

University, Diss., 2003--Freiburg (Breisgau). / Parallelt.: The method of the nonlocal effective action in higher-dimensional spacetime models.

Partial differential equations describe a number of processes in the physical-technical environment. The equations of the elasticity theory, which can be used to describe the deformations of a sample under application of an outer load, may serve as an example. Among other things, such deformations appear at so-called indentation tests, which are used to determine mechanical properties of thin layers. Since most partial differential equations can not, or only with great difficulty, be solved in an analytical way, numeric attempts to obtain an approximate solution are common. For the solution of elliptical partial differential equations with boundary conditions, the finite element method (FEM) is widely used. A problematic aspect is the growing numeric effort when increasing the accuracy of the approximation. This issue intensifies at higher dimensions. Since the scope of this work is the three-dimensional case, we will investigate possibilities of dimension reduction. Two Fourier approaches, which allow a dimension diminution from three to two, are being examined. If combined with a cylindrical parametrization of the three-dimensional space, the solution can be calculated without loss of information. The application of these approaches is illustrated exemplarily by the modeling of an indentation test with a rotationally symmetric structur and loads without rotational symmetry. / Partielle Differentialgleichungen beschreiben im physikalisch-technischen Umfeld eine Reihe von Prozessen. Ein Beispiel hierfür sind die Gleichungen der Elastizitätstheorie, die genutzt werden können, um die Verformungen einer Probe unter Aufbringung einer äußeren Last zu beschreiben. Solche Verformungen treten unter anderem bei sogenannten Indenterversuchen auf, die eingesetzt werden, um mechanische Größen dünner Schichten zu ermitteln. Da die meisten partiellen Differentialgleichungen auf analytischem Wege nicht, oder nur sehr schwer zu lösen sind, existieren numerischen Ansätze, um eine Lösung auf approximativem Wege zu erzielen. Für die Lösung elliptischer partieller Differentialgleichungen mit Randbedingungen existiert das Verfahren der Finiten-Elemente-Methode (FEM). Ein problematischer Aspekt besteht im wachsenden numerischen Aufwand mit genauer werdender Approximation der Lösung. Mit dem Ansteigen der Dimension der beschriebenen Prozesse verschärft sich dieses Problem. Der Fokus dieser Arbeit liegt auf dreidimensionalen Aufgabenstellungen. Daher ist es ihr Ziel, Möglichkeiten der Dimensionsreduktion zu untersuchen. Betrachtet werden zwei Fourieransätze, die bei einer Parametrisierung eines dreidimensionalen Gebietes mittels Zylinderkoordinaten eine Reduktion von drei auf zwei Freiheiten in der Berechnung der Lösung ermöglichen, ohne dass dabei Informationen verloren gehen. Die Anwendung dieser Ansätze soll beispielhaft durch die Modellierung eines Indenterversuches mit rotationssymmetrischer Anordnung und nichtrotationssymmetrischen Lasten veranschaulicht werden.

In the semiconductor industry the reliability of devices is of paramount importance. Therefore, after removing the defective ones, one wants to detect irregularities in measurement data because corresponding devices have a higher risk of failure early in the product lifetime. The paper presents a method to improve the detection of such suspicious devices where the screening is made on transformed measurement data. Thereby, e.g., dependencies between tests can be taken into account. Additionally, a new dimensionality reduction is performed within the transformation, so that the reduced and transformed data comprises only the informative content from the raw data. This simplifies the complexity of the subsequent screening steps. The new approach will be applied to semiconductor measurement data and it will be shown, by means of examples, how the screening can be improved.

Partial differential equations describe a number of processes in the physical-technical environment. The equations of the elasticity theory, which can be used to describe the deformations of a sample under application of an outer load, may serve as an example. Among other things, such deformations appear at so-called indentation tests, which are used to determine mechanical properties of thin layers. Since most partial differential equations can not, or only with great difficulty, be solved in an analytical way, numeric attempts to obtain an approximate solution are common. For the solution of elliptical partial differential equations with boundary conditions, the finite element method (FEM) is widely used. A problematic aspect is the growing numeric effort when increasing the accuracy of the approximation. This issue intensifies at higher dimensions. Since the scope of this work is the three-dimensional case, we will investigate possibilities of dimension reduction. Two Fourier approaches, which allow a dimension diminution from three to two, are being examined. If combined with a cylindrical parametrization of the three-dimensional space, the solution can be calculated without loss of information. The application of these approaches is illustrated exemplarily by the modeling of an indentation test with a rotationally symmetric structur and loads without rotational symmetry. / Partielle Differentialgleichungen beschreiben im physikalisch-technischen Umfeld eine Reihe von Prozessen. Ein Beispiel hierfür sind die Gleichungen der Elastizitätstheorie, die genutzt werden können, um die Verformungen einer Probe unter Aufbringung einer äußeren Last zu beschreiben. Solche Verformungen treten unter anderem bei sogenannten Indenterversuchen auf, die eingesetzt werden, um mechanische Größen dünner Schichten zu ermitteln. Da die meisten partiellen Differentialgleichungen auf analytischem Wege nicht, oder nur sehr schwer zu lösen sind, existieren numerischen Ansätze, um eine Lösung auf approximativem Wege zu erzielen. Für die Lösung elliptischer partieller Differentialgleichungen mit Randbedingungen existiert das Verfahren der Finiten-Elemente-Methode (FEM). Ein problematischer Aspekt besteht im wachsenden numerischen Aufwand mit genauer werdender Approximation der Lösung. Mit dem Ansteigen der Dimension der beschriebenen Prozesse verschärft sich dieses Problem. Der Fokus dieser Arbeit liegt auf dreidimensionalen Aufgabenstellungen. Daher ist es ihr Ziel, Möglichkeiten der Dimensionsreduktion zu untersuchen. Betrachtet werden zwei Fourieransätze, die bei einer Parametrisierung eines dreidimensionalen Gebietes mittels Zylinderkoordinaten eine Reduktion von drei auf zwei Freiheiten in der Berechnung der Lösung ermöglichen, ohne dass dabei Informationen verloren gehen. Die Anwendung dieser Ansätze soll beispielhaft durch die Modellierung eines Indenterversuches mit rotationssymmetrischer Anordnung und nichtrotationssymmetrischen Lasten veranschaulicht werden.

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functionalmaterials. In this paperwe studymicroheterogeneous, prestrained plates that feature non-flat equilibriumshapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient.By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a Γ-limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

Evaluating group separability is fundamental to pattern recognition. A plethora of dimension reduction (DR) algorithms has been developed to reveal the emergence of geometrical patterns in a lowdimensional space, where high-dimensional sample similarities are approximated by geometrical distances. However, statistical measures to evaluate the group separability attained by DR representations are missing. Traditional cluster validity indices (CVIs) might be applied in this context, but they present multiple limitations because they are not specifically tailored for DR. Here, we introduce a new rationale called projection separability (PS), which provides a methodology expressly designed to assess the group separability of data samples in a DR geometrical space. Using this rationale, we implemented a new class of indices named projection separability indices (PSIs) based on four statistical measures: Mann-Whitney U-test p-value, Area Under the ROC-Curve, Area Under the Precision-Recall Curve, and Matthews Correlation Coeffcient. The PSIs were compared to six representative cluster validity indices and one geometrical separability index using seven nonlinear datasets and six different DR algorithms. The results provide evidence that the implemented statistical-based measures designed on the basis of the PS rationale are more accurate than the other indices and can be adopted not only for evaluating and comparing group separability of DR results but also for fine-tuning DR algorithms' hyperparameters. Finally, we introduce a second methodological innovation termed trustworthiness, a statistical evaluation that accounts for separability uncertainty and associates to the measure of each index a p-value that expresses the significance level in comparison to a null model.

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