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 Non-Cylindrical 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