Moving Shape Analysis and Control: Applications to Fluid Structure Interactions

Introduction Classical and Moving Shape Analysis Fluid-Structure Interaction Problems Plan of the Book Detailed Overview of the Book An Introductory Example: The Inverse Stefan Problem The Mechanical and Mathematical Settings The Inverse Problem Setting The Eulerian Derivative and the Transverse Field The Eulerian Material Derivative of the State The Eulerian Partial Derivative of the State The Adjoint State and the Adjoint Transverse Field Weak Evolution of Sets and Tube Derivatives Introduction Weak Convection of Characteristic Functions Tube Evolution in the Context of Optimization Problems Tube Derivative Concepts A First Example: Optimal Trajectory Problem Shape Differential Equation and Level Set Formulation Introduction Classical Shape Differential Equation Setting The Shape Control Problem The Asymptotic Behavior Shape Differential Equation for the Laplace Equation Shape Differential Equation in Rd+1 The Level Set Formulation Dynamical Shape Control of the Navier-Stokes Equations Introduction Problem Statement Elements of Noncylindrical Shape Calculus Elements of Tangential Calculus State Derivative Strategy Min-Max and Function Space Parameterization Min-Max and Function Space Embedding Conclusion Tube Derivative in a Lagrangian Setting Introduction Evolution Maps Navier-Stokes Equations in Moving Domain Sensitivity Analysis for a Simple Fluid-Solid Interaction System Introduction Mathematical Settings Well-Posedness of the Coupled System Inverse Problem Settings KKT Optimality Conditions Conclusion Sensitivity Analysis for a General Fluid-Structure Interaction System Introduction Mechanical Problem and Main Result KKT Optimality Conditions Appendix A: Functional Spaces and Regularity of Domains Appendix B: Distribution Spaces Appendix C: The Fourier Transform Appendix D: Sobolev Spaces References Index