Motion planning for the two-phase Stefan problem in level set formulation [Elektronische Ressource] / author: Martin Bernauer

Motion Planning for the Two-Phase StefanProblem in Level Set FormulationS~ρLV·n = [k∇y] ·n on Γ (t)IFu≤u≤u on ΓC~φ +V·∇φ = 0tφ(0) =φ0min J(y,φ,u)y,φ,uy =y on Γ (t)M IR R δφ ∂gDJ gds,δφK =− +κg ds|∇φ| ∂nΓ (t) Γ (t)I IR RTγ 2max{0,φ − 0.02} dsdtd2 0 Γ (t)IDissertation submitted to theFaculty of MathematicsatChemnitz University of Technologyin accordance with the requirements for the degree Dr. rer. nat.Author: Dipl.-Ing. Martin BernauerAdvisor: Prof. Dr. Roland HerzogChemnitz, July 14, 2010Fur Veronika und JonasContentsStatutory Declarations viiiAcknowledgments ixAbstract xZusammenfassung xiChapter 1. Introduction 11.1 Motivation 11.2 Outline 4Chapter 2. The Two-Phase Stefan Problem 52.1 Mathematical Modeling 52.1.1. The Heat Equation 62.1.2. Modeling the Phase Change 72.1.3. Representing the Moving Interface 102.2 The Level Set Method 112.3 Extensions and Applications of the Stefan Problem 152.4 Weak and Strong Solutions 172.5 Overview of Existing Optimal Control Approaches 21Chapter 3. Motion Planning Subject to Control Constraints 263.1 Principles of PDE-Constrained Optimization 263.2 Motion Planning as an Optimal Control Problem 283.3 Derivation of the Optimality System 333.3.1. The Adjoint Temperature 343.3.2. The Level Set Function 373.3.3. Summary and Interpretation of the Adjoint Stefan Problem 433.3.4. The First-Order Optimality System 443.4 Optimization Methods 453.4.1.