Der Vortrag beschreibt numerische Singularitäten bei der h- und p-FEM, wie sie erkannt werden und welche Lösungen möglich sind bzw. was nicht vermieden werden kann.

This thesis deals with optimizing extremal eigenvalues of weighted graph Laplacian matrices. In general, the Laplacian matrix of a (weighted) graph is of particular importance in spectral graph theory and combinatorial optimization (e.g., graph partition like max-cut and graph bipartition). Especially the pioneering work of M. Fiedler investigates extremal eigenvalues of weighted graph Laplacians and provides close connections to the node- and edge-connectivity of a graph. Motivated by Fiedler, Göring et al. were interested in further connections between structural properties of the graph and the eigenspace of the second smallest eigenvalue of weighted graph Laplacians using a semidefinite optimization approach. By redistributing the edge weights of a graph, the following three optimization problems are studied in this thesis: maximizing the second smallest eigenvalue (based on the mentioned work of Göring et al.), minimizing the maximum eigenvalue and minimizing the difference of maximum and second smallest eigenvalue of the weighted Laplacian. In all three problems a semidefinite optimization formulation allows to interpret the corresponding semidefinite dual as a graph realization problem. That is, to each node of the graph a vector in the Euclidean space is assigned, fulfilling some constraints depending on the considered problem. Optimal realizations are investigated and connections to the eigenspaces of corresponding optimized eigenvalues are established. Furthermore, optimal realizations are closely linked to the separator structure of the graph. Depending on this structure, on the one hand folding properties of optimal realizations are characterized and on the other hand the existence of optimal realizations of bounded dimension is proven. The general bounds depend on the tree-width of the graph. In the case of minimizing the maximum eigenvalue, an important family of graphs are bipartite graphs, as an optimal one-dimensional realization may be constructed. Taking the symmetry of the graph into account, a particular optimal edge weighting exists. Considering the coupled problem, i.e., minimizing the difference of maximum and second smallest eigenvalue and the single problems, i.e., minimizing the maximum and maximizing the second smallest eigenvalue, connections between the feasible (optimal) sets are established.

No description available.

Gegeben sei ein lineares Gleichungssystem $Au = f$ mit Koeffizientenmatrix $A$, welche eine spezielle block-tridiagonale Struktur besitzt. Solche lineare Gleichungssysteme entstehen bei der Diskretisierung dreidimensionaler elliptischer Randwertprobleme mit 7- oder 27-Punkte-Stern. In geophysikalischen Anwedungen, insbesondere bei Aufgaben aus der Geoelektrik, haben die Randwertprobleme unstetige Koeffizienten und sind meistens auf nicht-uniformen Gittern diskretisiert. Klassische geometrische Mehrgitterverfahren konvergieren um so langsamer, je stärker die Koeffizientensprünge ausfallen. Außerdem kann die Konvergenz durch die Variation der Gitterabstände beeinträchtigt werden. Zur Lösung wird ein matrix-abhängiges Mehrgitterverfahren vorgestellt. Als Glätter wird eine unvollständige Block LU-Zerlegung verwendet. Die Gittertransferoperationen werden anhand der Einträge der Matrix $A$ ermittelt. Das resultierende Verfahren erweist sich als sehr robust, insbesondere wenn es als Vorkonditionierung für das Verfahren der konjugierten Gradienten eingesetzt wird.

A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated. The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result. As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.

The purpose of this study is to investigate forced convection of air through open cell foams. It can be numerically investigated either by implementing the time efficient macroscopic models or computationally expensive microscopic models. However, during the course of this study, it was observed that the macroscopic models are not sufficient for determining the desired key parameters. Nevertheless, it is still possible that these macroscopic models can be used to design an application accurately with minimum time efforts if the concerned key parameters are already known through other means. Accordingly, in this work, a methodology is developed to determine the desired key parameters by implementing the microscopic models, which are further used into the macroscopic models for designing different applications. To validate the proposed methodology, a set of steady state and transient forced convection experiments were performed for a set of ceramic foams having different pore diameter (10−30 PPI) and porosity (0.79−0.87) for a superficial velocity in the range of 0.5−10 m/s.

No description available.

This dissertation describes the improvement of signal analysis in ultrasonic microscopy for nondestructive testing. Specimens with many thin layers, like modern electronic components, pose a particular challenge for identifying and localizing defects. In this thesis, new evaluation algorithms have been developed which enable analysis of highly complex layer-stacks. This is achieved by a specific evaluation of multiple reflections, a newly developed iterative reconstruction and deconvolution algorithm, and the use of classification algorithms with a highly optimized simulation algorithm. Deep delaminations inside a 19-layer component can now not only be detected, but also localized. The new analysis methods also enable precise determination of elastic material parameters, sound velocities, thicknesses, and densities of multiple layers. The highly improved precision of determined reflections parameters with deconvolution also provides better and more conclusive results with common analysis methods.:Kurzfassung......................................................................................................................II Abstract.............................................................................................................................V List ob abbreviations........................................................................................................X 1 Introduction.......................................................................................................................1 1.1 Motivation.....................................................................................................................2 1.2 System theoretical description.....................................................................................3 1.3 Structure of the thesis..................................................................................................6 2 Sound field.........................................................................................................................8 2.1 Sound field measurement............................................................................................8 2.2 Sound field modeling..................................................................................................11 2.2.1 Reflection and transmission coefficients.........................................................11 2.2.2 Sound field modeling with plane waves..........................................................13 2.2.3 Generalized sound field position.....................................................................19 2.3 Receiving transducer signal.......................................................................................20 2.3.1 Calculation of the transducer signal from the sound field...............................20 2.3.2 Received signal amplitude..............................................................................21 2.3.3 Measurement of reference signals..................................................................24 3 Ultrasonic Simulation......................................................................................................27 3.1 State of the art............................................................................................................27 3.2 Simulation approach..................................................................................................28 3.2.1 Sound field measurement based simulation...................................................28 3.2.2 Reference signal based simulation.................................................................30 3.3 Determination of the impulse response.....................................................................31 3.3.1 1D ray-trace algorithm....................................................................................31 3.3.2 2D ray-trace algorithm....................................................................................33 3.3.3 Complexity reduction – optimizations.............................................................35 4 Deconvolution – Determination of reflection parameters............................................38 4.1 State of the art............................................................................................................39 4.1.1 Decomposition techniques..............................................................................39 4.1.2 Deconvolution.................................................................................................41 4.2 Analytic signal investigations for deconvolution.........................................................42 4.3 Single reference pulse deconvolution........................................................................44 4.4 Multi-pulse deconvolution..........................................................................................47 4.4.1 Homogeneous multi-pulse deconvolution.......................................................48 4.4.2 Multi-pulse deconvolution with simulated GSP profile....................................49 5 Reconstruction.................................................................................................................50 5.1 State of the art............................................................................................................50 5.2 Reconstruction approach...........................................................................................51 5.3 Direct material parameter estimation.........................................................................52 5.3.1 Sound velocities and layer thickness..............................................................52 5.3.2 Density, elastic modules and acoustic attenuation.........................................54 5.4 Iterative material parameter determination of a single layer......................................56 5.5 Reconstruction of complex specimens......................................................................60 5.5.1 Material characterization of multiple layers ....................................................60 5.5.2 Iterative simulation parameter optimization with correlation...........................62 5.5.3 Pattern recognition reconstruction of specimens with known base structure. 66 6 Applications and results.................................................................................................71 6.1 Analysis of stacked components................................................................................71 6.2 Time-of-flight and material analysis...........................................................................74 7 Conclusions and perspectives.......................................................................................78 References.......................................................................................................................82 Figures.............................................................................................................................86 Tables...............................................................................................................................88 Appendix..........................................................................................................................89 Acknowledgments.........................................................................................................100 Danksagung...................................................................................................................101 / Die vorgelegte Dissertation befasst sich mit der Verbesserung der Signalauswertung für die Ultraschallmikroskopie in der zerstörungsfreien Prüfung. Insbesondere bei Proben mit vielen dünnen Schichten, wie bei modernen Halbleiterbauelementen, ist das Auffinden und die Bestimmung der Lage von Fehlstellen eine große Herausforderung. In dieser Arbeit wurden neue Auswertealgorithmen entwickelt, die eine Analyse hochkomplexer Schichtabfolgen ermöglichen. Erreicht wird dies durch die gezielte Auswertung von Mehrfachreflexionen, einen neu entwickelten iterativen Rekonstruktions- und Entfaltungsalgorithmus und die Nutzung von Klassifikationsalgorithmen im Zusammenspiel mit einem hoch optimierten neu entwickelten Simulationsalgorithmus. Dadurch ist es erstmals möglich, tief liegende Delaminationen in einem 19-schichtigem Halbleiterbauelement nicht nur zu detektieren, sondern auch zu lokalisieren. Die neuen Analysemethoden ermöglichen des Weiteren eine genaue Bestimmung von elastischen Materialparametern, Schallgeschwindigkeiten, Dicken und Dichten mehrschichtiger Proben. Durch die stark verbesserte Genauigkeit der Reflexionsparameterbestimmung mittels Signalentfaltung lassen sich auch mit klassischen Analysemethoden deutlich bessere und aussagekräftigere Ergebnisse erzielen. Aus den Erkenntnissen dieser Dissertation wurde ein Ultraschall-Analyseprogramm entwickelt, das diese komplexen Funktionen auf einer gut bedienbaren Oberfläche bereitstellt und bereits praktisch genutzt wird.:Kurzfassung......................................................................................................................II Abstract.............................................................................................................................V List ob abbreviations........................................................................................................X 1 Introduction.......................................................................................................................1 1.1 Motivation.....................................................................................................................2 1.2 System theoretical description.....................................................................................3 1.3 Structure of the thesis..................................................................................................6 2 Sound field.........................................................................................................................8 2.1 Sound field measurement............................................................................................8 2.2 Sound field modeling..................................................................................................11 2.2.1 Reflection and transmission coefficients.........................................................11 2.2.2 Sound field modeling with plane waves..........................................................13 2.2.3 Generalized sound field position.....................................................................19 2.3 Receiving transducer signal.......................................................................................20 2.3.1 Calculation of the transducer signal from the sound field...............................20 2.3.2 Received signal amplitude..............................................................................21 2.3.3 Measurement of reference signals..................................................................24 3 Ultrasonic Simulation......................................................................................................27 3.1 State of the art............................................................................................................27 3.2 Simulation approach..................................................................................................28 3.2.1 Sound field measurement based simulation...................................................28 3.2.2 Reference signal based simulation.................................................................30 3.3 Determination of the impulse response.....................................................................31 3.3.1 1D ray-trace algorithm....................................................................................31 3.3.2 2D ray-trace algorithm....................................................................................33 3.3.3 Complexity reduction – optimizations.............................................................35 4 Deconvolution – Determination of reflection parameters............................................38 4.1 State of the art............................................................................................................39 4.1.1 Decomposition techniques..............................................................................39 4.1.2 Deconvolution.................................................................................................41 4.2 Analytic signal investigations for deconvolution.........................................................42 4.3 Single reference pulse deconvolution........................................................................44 4.4 Multi-pulse deconvolution..........................................................................................47 4.4.1 Homogeneous multi-pulse deconvolution.......................................................48 4.4.2 Multi-pulse deconvolution with simulated GSP profile....................................49 5 Reconstruction.................................................................................................................50 5.1 State of the art............................................................................................................50 5.2 Reconstruction approach...........................................................................................51 5.3 Direct material parameter estimation.........................................................................52 5.3.1 Sound velocities and layer thickness..............................................................52 5.3.2 Density, elastic modules and acoustic attenuation.........................................54 5.4 Iterative material parameter determination of a single layer......................................56 5.5 Reconstruction of complex specimens......................................................................60 5.5.1 Material characterization of multiple layers ....................................................60 5.5.2 Iterative simulation parameter optimization with correlation...........................62 5.5.3 Pattern recognition reconstruction of specimens with known base structure. 66 6 Applications and results.................................................................................................71 6.1 Analysis of stacked components................................................................................71 6.2 Time-of-flight and material analysis...........................................................................74 7 Conclusions and perspectives.......................................................................................78 References.......................................................................................................................82 Figures.............................................................................................................................86 Tables...............................................................................................................................88 Appendix..........................................................................................................................89 Acknowledgments.........................................................................................................100 Danksagung...................................................................................................................101

Im ersten Teil wird eine neue Konstruktion der parabolischen Induktion für pro-p Iwahori-Heckemoduln gegeben. Dabei taucht eine neue Klasse von Algebren auf, die in gewisser Weise als Interpolation zwischen der pro-p Iwahori-Heckealgebra einer p-adischen reduktiven Gruppe $G$ und derjenigen einer Leviuntergruppe $M$ von $G$ gedacht werden kann. Für diese Algebren wird ein Induktionsfunktor definiert und eine Transitivitätseigenschaft bewiesen. Dies liefert einen neuen Beweis für die Transitivität der parabolischen Induktion für Moduln über der pro-p Iwahori-Heckealgebra. Ferner wird eine Funktion auf einer parabolischen Untergruppe untersucht, die als Werte nur p-Potenzen annimmt. Es wird gezeigt, dass sie eine Funktion auf der (pro-p) Iwahori-Weylgruppe von $M$ definiert, und dass die so definierte Funktion monoton steigend bzgl. der Bruhat-Ordnung ist und einen Vergleich der Längenfunktionen zwischen der Iwahori-Weylgruppe von $M$ und derjenigen der Iwahori-Weylgruppe von $G$ erlaubt. Im zweiten Teil wird ein allgemeiner Zerlegungssatz für Polynome über der sphärischen (parahorischen) Heckealgebra einer p-adischen reduktiven Gruppe $G$ bewiesen. Diese Zerlegung findet über einer parabolischen Heckealgebra statt, die die Heckealgebra von $G$ enthält. Für den Beweis des Zerlegungssatzes wird vorausgesetzt, dass die gewählte parabolische Untergruppe in einer nichtstumpfen enthalten ist. Des Weiteren werden die nichtstumpfen parabolischen Untergruppen von $G$ klassifiziert. / The first part deals with a new construction of parabolic induction for modules over the pro-p Iwahori-Hecke algebra. This construction exhibits a new class of algebras that can be thought of as an interpolation between the pro-p Iwahori-Hecke algebra of a p-adic reductive group $G$ and the corresponding algebra of a Levi subgroup $M$ of $G$. For these algebras we define a new induction functor and prove a transitivity property. This gives a new proof of the transitivity of parabolic induction for modules over the pro-p Iwahori-Hecke algebra. Further, a function on a parabolic subgroup with p-power values is studied. We show that it induces a function on the (pro-p) Iwahori-Weyl group of $M$, that it is monotonically increasing with respect to the Bruhat order, and that it allows to compare the length function on the Iwahori-Weyl group of $M$ with the one on the Iwahori-Weyl group of $G$. In the second part a general decomposition theorem for polynomials over the spherical (parahoric) Hecke algebra of a p-adic reductive group $G$ is proved. The proof requires that the chosen parabolic subgroup is contained in a non-obtuse one. Moreover, we give a classification of non-obtuse parabolic subgroups of $G$.

Boreale Wälder in Permafrostregionen sind ein wesentlicher Bestandteil regionaler und globaler Klimamuster und machen etwa ein Drittel der weltweiten Waldfläche aus. Die Entwicklung der Waldbedeckung hat einen wichtigen Einfluss auf den Permafrost, da dieser durch die Vegetation geschützt wird. Der direkte Einfluss des Klimawandels auf die Wälder und der indirekte Effekt durch eine Veränderung der Permafrostdynamik können zu weitreichenden Ökosystemverschiebungen führen, die wiederum die Persistenz des Permafrosts beeinträchtigen und wichtige Ökosystemfunktionen destabilisieren könnten. Ziel dieser Dissertation ist es zu verstehen, wie sich die komplexen Wechselwirkungen zwischen der Vegetation, dem Permafrost und der Atmosphäre auf die Wälder und den darunterliegenden Permafrost auswirken. Im Rahmen dieser Dissertation habe ich ein eindimensionales, numerisches Landoberflächenmodell (CryoGrid), das zur Simulation der physikalischen Prozesse in Permafrostgebieten verwendet werden kann, für die Anwendung in bewaldetem Gebieten angepasst. Dazu habe ich ein detailliertes, mehrschichtiges Kronendachmodell (CLM-ml v0) und ein dynamisches Lärchenbestandsmodell gekoppelt. Dies ermöglichte den Energietransfer und das Wärmeregime welche für die komplexe Wald-Permafrost-Dynamik verantwortlich sind an verschiedenen Untersuchungsstandorten in gemischten und lärchendominierten Wäldern in Ostsibirien zu reproduzieren. Die numerischen Simulationen ergaben, dass die Wälder den thermischen und hydrologischen Zustand des Permafrosts hauptsächlich durch die Veränderung der Strahlungsbilanz und der Phänologie der Schneedecke beeinflussen und so eine stabilisierende Wirkung haben. Die Untersuchung der unterschiedlichen isolierenden Wirkung verschiedener Waldtypen und Walddichten sowie die Rückkopplungsmechanismen nach Störungen zeigen Veränderungen der thermischen und hydrologischen Bedingungen und der Tiefe der Auftauschicht. Zusammenfassend legen die Ergebnisse nahe, dass lokale, detaillierte und spezifische Landoberflächenmodelle erforderlich sind, um die komplexe Dynamik in borealen Permafrostökosystemen vollständig zu erfassen. Veränderungen der Rückkopplungen zwischen Permafrost, Klima, Wald und Störungen werden die eng gekoppelten Ökosystemfunktionen destabilisieren. Die induzierten Bodenveränderungen werden sich auf wichtige Wald- und Permafrostfunktionen, wie beispielsweise die Isolation des Permafrostbodens oder die Kohlenstoffspeicherung, und Rückkopplungsmechanismen wie Überschwemmung, Dürren, Brände, und Waldverlust, auswirken. / Boreal forests in permafrost regions make up around one-third of the global forest cover and are an essential component of regional and global climate patterns. The forests efficiently protect the underlying permafrost but the exact processes are not well understood. The direct influence of climatic change on forests and the indirect effect through a change in permafrost dynamics can lead to extensive ecosystem shifts, which will, in turn, affect permafrost persistence and potentially destabilize various ecosystem functions. The aim of this dissertation is to understand how complex interactions between the vegetation, permafrost, and the atmosphere stabilize the forests and the underlying permafrost. Within this dissertation, I have adapted a one-dimensional, numerical land surface model (CryoGrid), which can be used to simulate the physical processes in permafrost regions, for the application in vegetated areas by coupling a detailed multilayer canopy model (CLM-ml v0), and a dynamic larch stand model. An intensive validation of the model setup has allowed for the precise quantification of the heat- and water transfer processes responsible for the complex permafrost dynamics under boreal forest covers. At a variety of study sites throughout eastern Siberia, the numerical simulations revealed that the forests exert a strong control on the thermal and hydrological state of permafrost through changing the radiation balance and snow cover phenology. The forest cover has a net stabilizing effect on the permafrost ground below. The detailed physical model has furthermore enabled me to study the variation in insulation effect between different forest types and densities as well as the feedback mechanisms occurring after disturbances. In summary, the results suggest that local, detailed, and specific land surface models are required to fully comprehend the complex dynamics in boreal permafrost ecosystems. The research revealed that the feedbacks between permafrost, climate, boreal forest, and disturbances will destabilize tightly coupled ecosystem functions. The induced changes will affect key forest and permafrost functions, such as the forest's insulation capacity or the carbon budget, as well as feedback mechanisms like swamping, droughts, fires, or forest loss.