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This thesis is concerned with theoretical concepts of phenomenological and statistical thermodynamics and their computational realization. The main goal of this thesis is to provide efficient workflows for an accurate description of thermodynamic properties of molecules and solid state materials. The Cp-MD workflow developed within this thesis is applied to characterize binary battery materials, such as lithium silicides.This workflow enables a numerically efficient description of macroscopic thermodynamic properties. For battery materials and metal-organic frameworks, it is shown that some macroscopic properties are dominantly controlled by microscopic properties. These microscopic properties are well described by respective small clusters or molecules.Given their reduced size, these systems can be calculated using more accurate and numerically more demanding methods. Standard density functional theory (DFT) and the so called Fermi-Löwdin orbital self-interaction correction (FLO-SIC) method are used for further investigations. It will be shown that SIC is able to overcome some of the problems of DFT. Given further workflows, it is demonstrated how a combination of different computational methods can speed up thermodynamic calculations and is able to deepen the understanding of the driving forces of macroscopic thermodynamic properties.:1 Introduction 2 Open-source and open-science I Theoretical basics 3 Computational methods 4 Computation of thermodynamic properties II Thermodynamics of solid state systems 5 Methodical developments 6 Lithium silicides 7 Metal-organic frameworks III Thermodynamics of nuclei and electrons 8 Electrons and bonding information 9 Thermodynamic properties IV Summary 10 Conclusion 11 Outlook V Appendix / Diese Arbeit befasst sich mit theoretischen Konzepten der phänomenologischen und statistischen Thermodynamik und deren numerischer Umsetzung. Das Hauptziel dieser Arbeit ist es, Arbeitsabläufe für die akurate Beschreibung von thermodynamischen Eigenschaften von Molekülen und Festkörpern zur Verfügung zu stellen. Der während dieser Arbeit entwickelte Cp -MD Arbeitsablauf wird angewandt um binäre Batteriemateralien, wie Lithiumsilizide, zu charakterisieren. Dieser Arbeitsablauf ermöglicht eine numerisch effiziente Beschreibung von makroskopischen thermodynamischen Eigenschaften. Für Batteriemateralien und metallorganische Gerüstverbindungen wird gezeigt, dass einige makroskopische Eigenschaften hauptsächlich von mikroskopischen Eigenschaften kontrolliert sind. Diese mikroskopischen Eigenschaften können mittels zugehöriger Cluster oder Moleküle beschrieben werden. Aufgrund ihrer reduzierten Größe können diese Systeme mit genaueren und numerisch aufwendigeren Methoden berechnet werden. Standard Dichtefunktionaltheorie (DFT) und die Fermi-Löwdin-Orbital Selbstwechselwirkungskorrektur (FLO-SWK) werden für weitere Untersuchungen verwendet. Es wird gezeigt, dass die SWK einige Probleme der DFT überwinden kann. Anhand weiterer Arbeitsabläufe wird gezeigt, wie eine Kombination von verschiedenen numerischen Methoden thermodynamische Berechnungen beschleunigen kann und in der Lage ist das Verständnis der Triebkräfte von makroskopischen thermodynamischen Eigenschaften zu vertiefen.:1 Introduction 2 Open-source and open-science I Theoretical basics 3 Computational methods 4 Computation of thermodynamic properties II Thermodynamics of solid state systems 5 Methodical developments 6 Lithium silicides 7 Metal-organic frameworks III Thermodynamics of nuclei and electrons 8 Electrons and bonding information 9 Thermodynamic properties IV Summary 10 Conclusion 11 Outlook V Appendix

An der TU Chemnitz haben sich seit über zwei Jahrzehnten die Gebiete der rechnergestützten Wissenschaften (Computational Science) sowie des parallelen und verteilten Hochleistungsrechnens mit zunehmender Verzahnung entwickelt. Die Koordinierung und Bündelung entsprechender Forschungsarbeiten in der Profillinie 6 “Modellierung, Simulation, Hochleistungsrechnen” wird es ermöglichen, im internationalen Wettbewerb des Wissens mitzuhalten.

Lightweight structures became more and more important over the last years. One special class of such structures are short fibre reinforced composites, produced by injection moulding. To avoid expensive experiments for testing the mechanical behaviour of these composites proper material models are needed. Thereby, the stochastic nature of the fibre orientation is the main problem. In this thesis it is looked onto the simulation of such materials in a linear thermoelastic setting. This means the material is described by its heat conduction tensor κ(p), its thermal expansion tensor T(p), and its stiffness tensor C(p). Due to the production process the internal fibre orientation p has to been understood as random variable. As a consequence the previously mentioned material quantities also become random. The classical approach is to average these quantities and solve the linear hermoelastic deformation problem with the averaged expressions. Within this thesis the incorpora- tion of this approach in a time and memory efficient manner in an existing finite element software is shown. Especially for the time and memory efficient improvement several implementation aspects of the underlying software are highlighted. For both - the classical material simulation as well as the time efficient improvement of the software - numerical results are shown. Furthermore, the aforementioned classical approach is extended within this thesis for the simulation of the thermal stresses by using the stochastic nature of the heat conduc tion. This is done by developing it into a series w.r.t. the underlying stochastic. For this series known results from uncertainty quantification are applied. With the help of these results the temperature is developed in a Taylor series. For this Taylor series a suitable expansion point is chosen. Afterwards, this series is incorporated into the computation of the thermal stresses. The advantage of this approach is shown in numerical experiments.

A numerical treatment of non-linear higher-order geometric evolution equations with the level set and the finite element method is presented. The isotropic, weak anisotropic and strong anisotropic situation is discussed. Most of the equations considered in this work arise from the field of thin film growth. A short introduction to the subject is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an introduction to the numerical schemes, first the isotropic and weak anisotropic situation is considered. Then strong anisotropies (non-convex anisotropies) are used to simulate the phenomena of faceting and coarsening. The experimentally observed effect of corner and edge roundings is reached in the simulation through the regularization of the strong anisotropy with a higher-order curvature term. The curvature regularization leads to an increase by two in the order of the equations, which results in highly non-linear equations of up to 6th order. For the numerical solution, the equations are transformed into systems of second order equations, which are solved with a Schur complement approach. The adatom model constitutes a diffusion equation on a moving surface. An operator splitting approach is used for the numerical solution. In difference to other works, which restrict to the isotropic situation, also the anisotropic situation is discussed and solved numerically. Furthermore, a treatment of geometric evolution equations on implicitly given curved surfaces with the level set method is given. In particular, the numerical solution of surface diffusion on curved surfaces is presented. The equations are discretized in space by standard linear finite elements. For the time discretization a semi-implicit discretization scheme is employed. The derivation of the numerical schemes is presented in detail, and numerous computational results are given for the 2D and 3D situation. To keep computational costs low, the finite element grid is adaptively refined near the moving curves and surfaces resp. A redistancing algorithm based on a local Hopf-Lax formula is used. The algorithm has been extended by the authors to the 3D case. A detailed description of the algorithm in 3D is presented in this work. / In der Arbeit geht es um die numerische Behandlung nicht-linearer geometrischer Evolutionsgleichungen höherer Ordnung mit Levelset- und Finite-Elemente-Verfahren. Der isotrope, schwach anisotrope und stark anisotrope Fall wird diskutiert. Die meisten in dieser Arbeit betrachteten Gleichungen entstammen dem Gebiet des Dünnschicht-Wachstums. Eine kurze Einführung in dieses Gebiet wird gegeben. Es werden vier verschiedene Modelle diskutiert: mittlerer Krümmungsfluss, Oberflächendiffusion, ein kinetisches Modell, welches die Effekte des mittleren Krümmungsflusses und der Oberflächendiffusion kombiniert und zusätzlich eine kinetische Komponente beinhaltet, und ein Adatom-Modell, welches außerdem freie Adatome berücksichtigt. Als Einführung in die numerischen Schemata, wird zuerst der isotrope und schwach anisotrope Fall betrachtet. Anschließend werden starke Anisotropien (nicht-konvexe Anisotropien) benutzt, um Facettierungs- und Vergröberungsphänomene zu simulieren. Der in Experimenten beobachtete Effekt der Ecken- und Kanten-Abrundung wird in der Simulation durch die Regularisierung der starken Anisotropie durch einen Krümmungsterm höherer Ordnung erreicht. Die Krümmungsregularisierung führt zu einer Erhöhung der Ordnung der Gleichung um zwei, was hochgradig nicht-lineare Gleichungen von bis zu sechster Ordnung ergibt. Für die numerische Lösung werden die Gleichungen auf Systeme zweiter Ordnungsgleichungen transformiert, welche mit einem Schurkomplement-Ansatz gelöst werden. Das Adatom-Modell bildet eine Diffusionsgleichung auf einer bewegten Fläche. Zur numerischen Lösung wird ein Operatorsplitting-Ansatz verwendet. Im Unterschied zu anderen Arbeiten, die sich auf den isotropen Fall beschränken, wird auch der anisotrope Fall diskutiert und numerisch gelöst. Außerdem werden geometrische Evolutionsgleichungen auf implizit gegebenen gekrümmten Flächen mit Levelset-Verfahren behandelt. Insbesondere wird die numerische Lösung von Oberflächendiffusion auf gekrümmten Flächen dargestellt. Die Gleichungen werden im Ort mit linearen Standard-Finiten-Elementen diskretisiert. Als Zeitdiskretisierung wird ein semi-implizites Diskretisierungsschema verwendet. Die Herleitung der numerischen Schemata wird detailliert dargestellt, und zahlreiche numerische Ergebnisse für den 2D und 3D Fall sind gegeben. Um den Rechenaufwand gering zu halten, wird das Finite-Elemente-Gitter adaptiv an den bewegten Kurven bzw. den bewegten Flächen verfeinert. Es wird ein Redistancing-Algorithmus basierend auf einer lokalen Hopf-Lax Formel benutzt. Der Algorithmus wurde von den Autoren auf den 3D Fall erweitert. In dieser Arbeit wird der Algorithmus für den 3D Fall detailliert beschrieben.

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It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.

This thesis consists of two main parts. First of all, the condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is studied. The first time quantitative bounds for the extreme singular values are proven in the multivariate setting and when nodes of the Vandermonde matrix form clusters. In the second part, an optimized presentation of the deterministic stability analysis of the subspace method ESPRIT is given and results from the first part are applied.