Techn. University, Diss., 2007--Darmstadt.

University, Diss., 2005--Münster (Westfalen).

In der vorliegenden Arbeit werden lineare Systeme elliptischer partieller Differentialgleichungen in schwacher Formulierung auf konischen Gebieten untersucht. Auf einem zunächst unbeschränkten Kegelgebiet betrachten wir den Fall beschränkter und nur von den Winkelvariablen abhängiger Koeffizientenfunktionen. Die durch selbige definierte Bilinearform genüge einer Gårdingschen Ungleichung. In gewichteten Sobolevräumen werden Existenz- und Eindeutigkeitsfragen geklärt, wobei das Problem mittels Fouriertransformation auf eine von einem komplexen Parameter abhängige Familie T(·) von Fredholmoperatoren zurückgeführt wird. Unter Anwendung des Residuenkalküls gewinnen wir eine Darstellung der Lösung in Form einer Zerlegung in einen glatten Anteil einerseits sowie eine endliche Summe von Singulärfunktionen andererseits. Durch Abschneidetechniken werden die gewonnenen Erkenntnisse auf den Fall schwach formulierter elliptischer Systeme auf beschränkten Kegelgebieten unter Formulierung in gewöhnlichen, nicht-gewichteten Sobolevräumen angewendet. Die für Regularitätsfragen maßgeblichen Eigenwerte der Operatorfunktion T mit minimalem positiven Imaginärteil werden im letzten Kapitel der Arbeit am Beispiel der ebenen elastischen Gleichungen numerisch bestimmt. / In the present PhD thesis we investigate systems of linear partial elliptic equations in weak formulation on conical domains. For an unbounded cone, first, we study the case of bounded and radially constant coefficient functions. The so defined bilinear form is supposed to satisfy a (local) Gårding inequality. In weighted Sobolev spaces we study questions of existence and uniqueness of solutions. In this context the problem is Fourier-transformed onto a set of smaller problems, represented by Fredholm operators T(·) that holomorphically depend on a complex parameter. Via the residual theorem we yield a decomposition of the solution into a regular part and a finite sum of singular functions. Using cut-off techniques we are able to transfer the preceeding results onto the case of weak formulated linear elliptic systems on bounded cones under restriction to usual, non weighted Sobolev spaces. In the last chapter, the eigenvalues of T with minimal positive imaginary part, which are responsible for regularity properties, are numeriaclly determined for the example of the plane Elastic Equations.

Zugl.: Ilmenau, Techn. Univ., Diss., 2009

The thesis is concerned with group theoretical properties of unitary groups, mainly of II_1 factors. The author gives a new and elementary proof of an result on extreme amenability, defines the bounded normal generation property and invariant automatic continuity property and proves these for various unitary groups of functional analytic types.

The thesis is concerned with group theoretical properties of unitary groups, mainly of II_1 factors. The author gives a new and elementary proof of an result on extreme amenability, defines the bounded normal generation property and invariant automatic continuity property and proves these for various unitary groups of functional analytic types.

In this thesis, the structure of pure measures is investigated. These are elements of the dual of the space of essentially bounded functions. A more precise representation of the dual space of the space of essentially bounded functions is given, leading to the definition and analysis of density measurs which constitute a new large class and yield numerous new examples of pure measures which are well-suited for applications in very general Divergence Theorems. The existence of pure normal measures for sets of finite perimeter is demonstrated. These yield Gauß formulas for essentially bounded vector fields having divergence measure. Furthermore, a result of Silhavy is extended. In particular, it is shown that a Gauß-Green Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field.

In this thesis, the structure of pure measures is investigated. These are elements of the dual of the space of essentially bounded functions. A more precise representation of the dual space of the space of essentially bounded functions is given, leading to the definition and analysis of density measurs which constitute a new large class and yield numerous new examples of pure measures which are well-suited for applications in very general Divergence Theorems. The existence of pure normal measures for sets of finite perimeter is demonstrated. These yield Gauß formulas for essentially bounded vector fields having divergence measure. Furthermore, a result of Silhavy is extended. In particular, it is shown that a Gauß-Green Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field.

Die Arbeit abstrahiert bekannte Konzepte der endlichdimensionalen Optimierung im Hinblick auf deren Anwendung in Banachräumen. Hierfür werden zunächst grundlegende Elemente der Funktionalanalysis wie schwache Konvergenz, Dualräume und Reflexivität vorgestellt. Anschließend erfolgt eine kurze Einführung in die Thematik der Fréchet-Differenzierbarkeit und eine Abstraktion des Begriffs der partiellen Ordnungsrelation in normierten Räumen. Nach der Formulierung eines allgemeinen Existenzsatzes für globale Optimallösungen von abstrakten Optimierungsaufgaben werden notwendige Optimalitätsbedingungen vom Karush-Kuhn-Tucker-Typ hergeleitet. Abschließend wird eine hinreichende Optimalitätsbedingung vom Karush-Kuhn-Tucker-Typ unter verallgemeinerten Konvexitätsvoraussetzungen verifiziert.

The aim of this thesis is the fast and exact simulation of modern materials like fibre reinforced thermoplastics and fibre reinforced elastomers. These simulations are in the scope of large strain deformations and contain anisotropic and viscoelastic behaviour. The chapter Differential geometry outlines the necessary tensor analysis and differential geometry. We present the weak formulation in the undeformed domain and use Newton’s method to approximate the solution of this formulation, cf. Section 3.1 and Chapter 4, respectively. For the viscoelasticity we use a special ansatz for the internal variable. Next, we compute all necessary derivations for the Newton system, cf. Sections 4.2 and 4.3. We also investigate the symmetry of the material tensors in Section 4.4. Further, we present three methods to improve the convergence of Newton’s method, cf. Section 4.5. With these three methods we are able to consider more problems, compute them faster and in a more robust way. In Chapter 5 we concisely discuss the FEM and show the appearing matrices in detail. The aim of Chapter 6 is the application of the a posteriori error estimator to this complex material behaviour. We present some numerical examples in Chapter 7. In Chapter 8 the problems that arise in the simulation of fibre-reinforced elastomers are analysed and tackled with help of mixed formulations. We derive a symmetric mixed formulation from a reduced form of the energy density. Also, we reformulate the mixed variable for inextensibility to avoid the numerical cancellation in Section 8.3. The Section 8.4 is about a joined mixed formulation to solve problems with inextensible fibres in an incompressible matrix, like fibre-reinforced rubber. The succeeding section Section 8.5 deals with the arising indefinite block matrix system.:Contents Glossary 5 1 Introduction – motivation 13 2 Differential geometry 15 2.1 From parametrisations to the Lagrangian strain 15 2.2 Derivatives of tensors 20 3 Physical foundations 25 3.1 Large Deformation 25 3.1.1 Balance of forces 25 3.1.2 Energy minimisation 28 3.2 Anisotropic energy density 29 3.3 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Newton’s method 37 4.1 Newton system 37 4.2 Anisotropic material tensor 40 4.3 Viscoelastic material tensor 41 4.4 Symmetry of the material tensor 44 4.5 Load steps and line-search 47 4.5.1 Load steps – time steps 47 4.5.2 Backtracking for det ℱ > 0 48 4.5.3 Line search for energy minimisation 49 5 Implementation 53 5.1 Numerical Integration 53 5.2 Finite element discretisation 54 5.3 Voigt notation 56 6 Mesh control 65 7 Numerical results 69 7.1 Semi-analytical example 69 7.2 Cook’s membrane 71 7.2.1 Viscoelastic example 72 7.3 Chemnitz hook – Chemnitzer Haken 72 8 Mixed formulation 75 8.1 Motivation 75 8.2 General considerations 78 8.3 Smooth square root 81 8.4 Joined mixed formulation 84 8.5 Matrix representation 86 9 Conclusion 91 10 Theses 93 11 Appendix 95 11.1 Derivatives of the distortion-invariants with respect to the pseudo invariants 95 Bibliography 101