Optimal Reliability for Components under Thermomechanical Cyclic Loading

We consider the existence of optimal shapes in the context of the thermomechanical system of partial differential equations (PDE) using the recent approach based on elliptic regularity theory. We give an extended and improved definition of the set of admissible shapes based on a class of sufficiently differentiable deformation maps applied to a baseline shape. The obtained set of admissible shapes again allows one to prove a uniform Schauder estimate for the elasticity PDE. In order to deal with thermal stress, a related uniform Schauder estimate is also given for the heat equation. Special emphasis is put on Robin boundary conditions, which are motivated from convective heat transfer. It is shown that these thermal Schauder estimates can serve as an input to the Schauder estimates for the elasticity equation. This is needed to prove the compactness of the (suitably extended) solutions of the entire PDE system in some state space that carries a c2-H\"older topology for the temperature field and a C3-H\"older topology for the displacement. From this one obtains he property of graph compactness, which is the essential tool in an proof of the existence of optimal shapes. Due to the topologies employed, the method works for objective functionals that depend on the displacement and its derivatives up to third order and on the temperature field and its derivatives up to second order. This general result in shape optimization is then applied to the problem of optimal reliability, i.e. the problem of finding shapes that have minimal failure probability under cyclic thermomechanical loading.Comment: 32 pages 1 figur

Adaptive Multi-Fidelity Modeling for Efficient Design Exploration Under Uncertainty

This thesis work introduces a novel multi-fidelity modeling framework, which is designed to address the practical challenges encountered in Aerospace vehicle design when 1) multiple low-fidelity models exist, 2) each low-fidelity model may only be correlated with the high-fidelity model in part of the design domain, and 3) models may contain noise or uncertainty. The proposed approach approximates a high-fidelity model by consolidating multiple low-fidelity models using the localized Galerkin formulation. Also, two adaptive sampling methods are developed to efficiently construct an accurate model. The first acquisition formulation, expected effectiveness, searches for the global optimum and is useful for modeling engineering objectives. The second acquisition formulation, expected usefulness, identifies feasible design domains and is useful for constrained design exploration. The proposed methods can be applied to any engineering systems with complex and demanding simulation models

Risk averse shape optimization - risk measures and stochastic orders

Thema der vorliegenden Arbeit ist risikoaverse Formoptimierung. Gewöhnliche Formoptimierung umfasst jene Art von Problemen, bei denen die zu optimierende Variable die Form eines Objekts ist. Hier ist das Objekt ein elastischer Körper auf den eine Kraft einwirkt. Basierend auf den Konzepten der Sensitivitätsanalyse bezüglich Form und Topologie wird eine Algorithmus konstruiert, der den elastischen Körper bezüglich seiner elastischen Eigenschaften und dem dabei insgesamt gebrauchten Volumen sukzessiv verbesssert. Das Resultat ist eine Struktur, die auf die wirkende Kraft so stabil wie möglich reagiert und dabei möglichst geringes Volumen hat. Diese Art von Problemen benötigen effiziente numerische Löser für die zugrundeliegenden partiellen Differentialgleichungen (hier ist dies das Modell der linearen Elastizität). Dazu werden unterschiedliche Finite Elemente Methoden sowie Ansätze zur Gittergenerierung benutzt. Mit den sogenannten Levelset Methoden wird die Entwicklung der Struktur numerisch beschrieben. Unsicherheit kommt dann ins Spiel, wenn die berücksichtigten Kräfte als zufällig angenommen werden. So kann nun das elastische Verhalten des Körpers als Zufallsvariable angesehen werden, die von der Form abhängt. In einem ersten Ansatz wird die Form bezüglich verschiedener Risikomaße optimiert, die vor allem in Modellen der (Finanz-) Okonomie zum Einsatz kommen. Solche Risikomaße definieren unterschiedliche Bewertungen von Risiko, welches einer Zufallsvariable anhängt. Risikoneutrale und risikoaverse Modelle werden zur Formoptimierung benutzt. Eine neue Perspektive eröffnet sich, wenn 'stochastic dominance relations' zur Risikobewertung herangezogen werden. Diese definieren eine Halbordnung auf dem Raum der Zufallsvariablen und ermöglichen es, diese in Relation zu stellen. Ausgehend von einem Benchmark, welches eine gewisse Güte für das Verhalten unter Einwirkung der zufälligen Kräfte beschreibt, kann eine Menge von Formen identifiziert werden, deren Verhalten unter Einwirkung der Kräfte nicht schlechter als das Benchmark ist. Aus dieser Menge werden dann Formen nach einem weiteren Kriterium, z.B. möglichst geringes Volumen, ausgewählt. Auf diese Art und Weise werden Formen mit geringem Volumen gefunden, die aber dennoch die zuvor gestellten Anforderungen erfüllen.In this thesis, risk averse shape optimization of elastic strucures is at issue. Shape optimization, in general, deals with the type of problems where the variable to be optimized is the geometry or the shape of a domain. Here, the domain represents an elastic body which is subjected to a force applied. Relying on the concepts of shape and topology sensitivity analysis an algorithm is implemented which successively improves the elastic body concerning the elastic response and the total volume. Eventually, this procedure results in a body which is, on the one hand, as stiff as possible regarding the force applied and, on the other hand, whose volume is as small as possible. Solving such problems numerically requires an efficient solver for the underlying partial differential equation (here the linearized elasticity model). To this end, different finite element methods and mesh generation approaches are applied. Level set methods are employed to realize the evolution of the elastic body in the discrete setting. Uncertainty is then introduced by considering the force applied to be random. Thus, the elastic body can be interpreted as a parameter defining a random variable. In a first approach risk measures, which are well-known in economics, are proposed to assess random variables. Risk measures give a notion of risk associated with random variables. Risk neutral and risk averse models are discussed and used to optimize over a class of shapes. A new perspective arises when stochastic dominance relations are employed for the assessment of risk. They define an order on the space of random variables and allow to compare these to each other directly. Taking a benchmark random variable associated with a required behavior under uncertainty, a set of acceptable shapes can be identified by comparision to this benchmark. An additional criterion, e.g. minimal volume, is used to select shapes from this set. In that way, shapes with low volume are found which still meet the prescribed requirements