Moubachir / Zolesio Moving Shape Analysis and Control

Applications to Fluid Structure Interactions
1. Auflage 2010
ISBN: 978-1-4200-0324-6
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 312 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

Problems involving the evolution of two- and three-dimensional domains arise in many areas of science and engineering. Emphasizing an Eulerian approach, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions presents valuable tools for the mathematical analysis of evolving domains. The book illustrates the efficiency of the tools presented through different examples connected to the analysis of noncylindrical partial differential equations (PDEs), such as Navier–Stokes equations for incompressible fluids in moving domains. The authors first provide all of the details of existence and uniqueness of the flow in both strong and weak cases. After establishing several important principles and methods, they devote several chapters to demonstrating Eulerian evolution and derivation tools for the control of systems involving fluids and solids. The book concludes with the boundary control of fluid–structure interaction systems, followed by helpful appendices that review some of the advanced mathematics used throughout the text. This authoritative resource supplies the computational tools needed to optimize PDEs and investigate the control of complex systems involving a moving boundary.

Zielgruppe

Engineers, applied mathematicians, students, and researchers, especially those with a focus on fluid mechanics.

Autoren/Hrsg.

Moubachir, Marwan

Zolesio, Jean-Paul

Fachgebiete

• Mathematik | Informatik Mathematik Topologie Algebraische Topologie

Inhaltsverzeichnis

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