Moving Shape Analysis and Control:

Problems involving the evolution of two- and three-dimensional domains arise in many areas of science and engineering. These range from free surface flows, phase changes, and fracture and contact problems to applications in civil engineering constructions, biomechanical systems, and computer vision. Emphasizing an Eulerian approach, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions presents valuable tools for the mathematical analysis of evolving domains.This book illustrates the efficiency of the tools presented through different examples connected to the analysis of noncylindrical partial differential equations, such as the Navier-Stokes equations for incompressible fluids in moving domains. The authors begin by providing 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 boundary control of fluid-structure interaction systems, followed by helpful appendices that review some of the advanced mathematics used throughout the text.Whether you are investigating the control of complex systems involving a moving boundary or interested in the optimization of partial differential equations, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions is an authoritative source for useful computational tools.

Marwan Moubachir is a research scientist for Project Opale at INRIA, Sophia Antipolis, France Jean-Paul Zolesio is a research scientist for Project Opale at INRIA, Sophia Antipolis, France

Introduction

1

(32)

Classical and moving shape analysis

3

(1)

Fluid-Structure interaction problems

4

(2)

Plan of the book

6

(1)

Detailed overview of the book

7

(26)

Control of a fluid flow around a fixed body

8

(5)

Shape design of a fixed solid inside a fluid flow

13

(6)

Dynamical shape design of a solid inside a fluid flow

19

(9)

Control of an elastic solid inside a fluid flow

28

(5)

An introductory example : the inverse Stefan problem

33

(16)

The mechanical and mathematical settings

33

(2)

The inverse problem setting

35

(2)

The eulerian derivative and the transverse field

37

(6)

The Eulerian material derivative of the state

43

(1)

The Eulerian partial derivative of the state

44

(2)

The adjoint state and the adjoint transverse field

46

(3)

Weak evolution of sets and tube derivatives

49

(34)

Introduction

49

(1)

Weak convection of characteristic functions

50

(4)

The convection equation

50

(1)

The Galerkin approximation

51

(3)

Tube evolution in the context of optimization problems

54

(12)

Penalization using the generalized perimeter's time integral

56

(1)

Parabolic version of Helly's compactness theorem

57

(3)

Generation of clean open tubes

60

(6)

Tube derivative concepts

66

(9)

Characteristics versus Eulerian flow setting

67

(2)

Tangential calculus for tubes

69

(3)

Classical shape analysis for tubes

72

(3)

A first example : optimal trajectory problem

75

(8)

Optimality conditions : case of planar parametric curves

76

(2)

Optimality conditions : case of the general Eulerian setting

78

(5)

Shape differential equation and level set formulation

83

(26)

Introduction

83

(1)

Classical shape differential equation setting

84

(2)

The shape control problem

86

(3)

An existence result for the shape differential equation

86

(2)

A constructive algorithm

88

(1)

The asymptotic behaviour

89

(1)

Shape differential equation for the Laplace equation

90

(7)

The Laplace equation

90

(4)

The shape control problem

94

(2)

An asymptotic result in the 2D case

96

(1)

Shape differential equation in Rd+1

97

(4)

The level set formulation

101

(8)

Introduction

101

(1)

Solutions without step

102

(1)

Iterative Scheme

103

(2)

An example: the transverse magnetic like inverse problem

105

(4)

Dynamical shape control of the Navier-Stokes equations

109

(54)

Introduction

109

(1)

Problem statement

110

(4)

Elements of non-cylindrical shape calculus

114

(5)

Non-cylindrical speed method

114

(5)

Elements of tangential calculus

119

(3)

Oriented distance function

119

(1)

Intrinsic tangential calculus

120

(2)

Tangential Stokes formula

122

(1)

State derivative strategy

122

(18)

Weak formulation and solvability

123

(1)

The weak Piola material derivative

124

(9)

Shape derivative

133

(3)

Extractor Identity

136

(2)

Adjoint system and cost function shape derivative

138

(2)

Min-Max and function space parametrization

140

(17)

Saddle point formulation of the fluid state system

141

(5)

Function space parametrization

146

(1)

Differentiability of the saddle point problem

147

(10)

Min-Max and function space embedding

157

(5)

Saddle point formulation of the fluid state system

157

(2)

The Lagrange functional

159

(3)

Conclusion

162

(1)

Tube derivative in a Lagrangian setting

163

(22)

Introduction

163

(1)

Evolution maps

163

(15)

Basic differentiation results

164

(1)

Mathematical setting

165

(1)

Elements of shape calculus

165

(6)

Unconstraint non-cylindrical shape optimization

171

(2)

Shape optimization under state constraints

173

(2)

Eulerian vs. Lagrangian non-cylindrical derivative

175

(3)

Navier-Stokes equations in moving domain

178

(7)

Transpiration boundary conditions

178

(2)

Optimal control of the Navier-Stokes system by moving the domain

180

(2)

Comparison with the Eulerian derivative version

182

(3)

Sensitivity analysis for a simple fluid-solid interaction system

185

(30)

Introduction

185

(1)

Mathematical settings

185

(4)

Well-posedness of the coupled system

189

(1)

Inverse problem settings

190

(7)

Analysis strategy

191

(1)

Free divergence and non-homogeneous Dirichlet boundary condition constraints

191

(1)

Solid reduced order and solid weak state operator

192

(1)

Fluid state operator

193

(1)

Coupled system operator

194

(1)

Min-Max problem

195

(2)

KKT Optimality Conditions

197

(15)

Derivatives with respect to state variables

197

(1)

Solid adjoint system

198

(14)

Conclusion

212

(3)

Sensitivity analysis for a general fluid-structure interaction system

215

(24)

Introduction

215

(1)

Mechanical problem and main result

215

(8)

Solid weak state operator

220

(1)

Fluid state operator

220

(1)

Coupled system operator

221

(1)

Min-Max problem

222

(1)

KKT optimality conditions

223

(16)

Fluid adjoint system

223

(1)

Solid adjoint system

224

(15)

A. Functional spaces and regularity of domains

239

(8)

Classical functions

239

(2)

Lebesgue spaces

241

(2)

Various inequalities

241

(1)

Completeness, separability and reflexivity

242

(1)

Convolution

242

(1)

Density property

243

(1)

Smooth domains and boundary measure

243

(4)

B. Distribution spaces

247

(8)

The space D(Ω)

247

(2)

The space of distributions D'(Ω)

249

(2)

Examples of distributions

251

(4)

Measures on Ω

251

(1)

Regular distributions

251

(1)

The Dirac measure

252

(1)

Derivative of the Dirac measure

252

(1)

The Dirac comb

253

(1)

Principal value of 1/x

253

(2)

C. The Fourier transform

255

(10)

The case of L1 functions and extension to L2

255

(2)

The space J(Rd)

257

(1)

The space of tempered distributions

258

(1)

Fourier transform in J'

259

(1)

Fourier transform and convolution

260

(1)

Partial Fourier transform

261

(1)

Vector valued distributions

261

(2)

Vector valued Lebesgue spaces

263

(2)

D. Sobolev spaces

265

(12)

Spaces Hm(Ω)

265

(1)

The space Hs (Rd)

266

(1)

The topological dual of Hs (Rd)

267

(1)

Sobolev embedding theorems

267

(2)

Density properties

269

(1)

Trace theorem for Hs (Rd)

270

(1)

Trace theorem for Hm (Rd+)

270

(1)

The space Hs (Γ)

271

(1)

Trace theorem in Hm(Ω)

272

(1)

Extension theorems

272

(2)

The space H-m (Ω)

274

(1)

Compact embeddings

275

(1)

Poincare inequalities

275

(2)

References

277

(12)

Index

289

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Moving Shape Analysis and Control: Applications to Fluid Structure Interactions

Summary

Author Biography

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Moving Shape Analysis and Control: Applications to Fluid Structure Interactions

Summary

Author Biography

Table of Contents

Supplemental Materials

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