Moving shape analysis and control: Applications to fluid structure interactions

Moving Shape Analysis and Control Applications to Fluid Structure Interactions PURE AND APPLIED MATHEMATICS A Program...

Author: Moubachir M. | Zolesio J.-P.

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Moving Shape Analysis and Control Applications to Fluid Structure Interactions

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University Piscataway, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University

Freddy van Oystaeyen University of Antwerp, Belgium Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions (2003) G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative (2003) A. Kelarev, Graph Algebras and Automata (2003) A. H. Siddiqi, Applied Functional Analysis: Numerical Methods, Wavelet Methods, and Image Processing (2004) F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line (2004) G. S. Ladde and M. Sambandham, Stochastic versus Deterministic Systems of Differential Equations (2004) B. J. Gardner and R. Wiegandt, Radical Theory of Rings (2004) J. Haluska, The Mathematical Theory of Tone Systems (2004) C. Menini and F. Van Oystaeyen, Abstract Algebra: A Comprehensive Treatment (2004) E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, Second Edition, Revised and Expanded (2004) M. M. Rao, Measure Theory and Integration, Second Edition, Revised and Expanded (2004) W. J. Wickless, A First Graduate Course in Abstract Algebra (2004) R. P. Agarwal, M. Bohner, and W-T Li, Nonoscillation and Oscillation Theory for Functional Differential Equations (2004) J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (2004) Walter Ferrer and Alvaro Rittatore, Actions and Invariants of Algebraic Groups (2005) Christof Eck, Jiri Jarusek, and Miroslav Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems (2005) M. M. Rao, Conditional Measures and Applications, Second Edition (2005) A. B. Kharazishvili, Strange Functions in Real Analysis, Second Edition (2006) Vincenzo Ancona and Bernard Gaveau, Differential Forms on Singular Varieties: De Rham and Hodge Theory Simplified (2005) Santiago Alves Tavares, Generation of Multivariate Hermite Interpolating Polynomials (2005) Sergio Macías, Topics on Continua (2005) Mircea Sofonea, Weimin Han, and Meir Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage (2006) Marwan Moubachir and Jean-Paul Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions (2006)

Moving Shape Analysis and Control Applications to Fluid Structure Interactions

Marwan Moubachir INRIA, France

Jean-Paul Zolésio CNRS and INRIA, France

Boca Raton London New York

Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-611-0 (Hardcover) International Standard Book Number-13: 978-1-58488-611-2 (Hardcover) Library of Congress Card Number 2005053196 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data Mouchabir, Marwan. Moving shape analysis and control : applications to fluid structure interactions / by Marwan Mouchabir and Jean Paul Zolesio. p. cm. -- (Monographs and textbooks in pure and applied mathematics ; 278) Includes bibliographical references and index. ISBN 1-58488-611-0 (acid-free paper) 1. Shape theory (Topology) 2. Fluid-structure interaction--Mathematics. I. Zolésio, J.P. II. Title. III. Series. QA612.7.M68 2005 514’.24--dc22

2005053196

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc.

and the CRC Press Web site at http://www.crcpress.com

A Val´erie, Olivia et Timoth´e. A Monique.

Contents

1 Introduction 1.1 Classical and moving shape analysis . . . . . . . . . . 1.2 Fluid-Structure interaction problems . . . . . . . . . . 1.3 Plan of the book . . . . . . . . . . . . . . . . . . . . . 1.4 Detailed overview of the book . . . . . . . . . . . . . 1.4.1 Control of a fluid flow around a fixed body . . 1.4.2 Shape design of a fixed solid inside a fluid flow 1.4.3 Dynamical shape design of a solid inside a fluid 1.4.4 Control of an elastic solid inside a fluid flow . . 2 An 2.1 2.2 2.3 2.4 2.5 2.6

. . . . . . . . . . . . . . . . . . flow . . .

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 . . . . .

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1 3 4 6 7 8 13 19 28

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33 33 35 37 43 44 46

3 Weak evolution of sets and tube derivatives 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Weak convection of characteristic functions . . . . . . . . . . 3.2.1 The convection equation . . . . . . . . . . . . . . . . . 3.2.2 The Galerkin approximation . . . . . . . . . . . . . . 3.3 Tube evolution in the context of optimization problems . . . 3.3.1 Penalization using the generalized perimeter’s time integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Parabolic version of Helly’s compactness theorem . . . 3.3.3 Generation of clean open tubes . . . . . . . . . . . . . 3.4 Tube derivative concepts . . . . . . . . . . . . . . . . . . . . 3.4.1 Characteristics versus Eulerian flow setting . . . . . . 3.4.2 Tangential calculus for tubes . . . . . . . . . . . . . . 3.4.3 Classical shape analysis for tubes . . . . . . . . . . . . 3.5 A first example : optimal trajectory problem . . . . . . . . . 3.5.1 Optimality conditions : case of planar parametric curves 76 3.5.2 Optimality conditions : case of the general Eulerian setting . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 50 50 51 54 56 57 60 66 67 69 72 75

78

ix

x 4 Shape differential equation and level set formulation 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Classical shape differential equation setting . . . . . . . . . . 4.3 The shape control problem . . . . . . . . . . . . . . . . . . . 4.3.1 An existence result for the shape differential equation 4.3.2 A constructive algorithm . . . . . . . . . . . . . . . . 4.4 The asymptotic behaviour . . . . . . . . . . . . . . . . . . . 4.5 Shape differential equation for the Laplace equation . . . . . 4.5.1 The Laplace equation . . . . . . . . . . . . . . . . . . 4.5.2 The shape control problem . . . . . . . . . . . . . . . 4.5.3 An asymptotic result in the 2D case . . . . . . . . . . 4.6 Shape differential equation in Rd+1 . . . . . . . . . . . . . . 4.7 The level set formulation . . . . . . . . . . . . . . . . . . . . 4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Solutions without step . . . . . . . . . . . . . . . . . . 4.7.3 Iterative Scheme . . . . . . . . . . . . . . . . . . . . . 4.7.4 An example: the transverse magnetic like inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 86 86 88 89 90 90 94 96 97 101 101 102 103

5 Dynamical shape control of the Navier-Stokes equations 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . 5.3 Elements of non-cylindrical shape calculus . . . . . . . . . 5.3.1 Non-cylindrical speed method . . . . . . . . . . . . . 5.4 Elements of tangential calculus . . . . . . . . . . . . . . . . 5.4.1 Oriented distance function . . . . . . . . . . . . . . . 5.4.2 Intrinsic tangential calculus . . . . . . . . . . . . . . 5.4.3 Tangential Stokes formula . . . . . . . . . . . . . . . 5.5 State derivative strategy . . . . . . . . . . . . . . . . . . . 5.5.1 Weak formulation and solvability . . . . . . . . . . . 5.5.2 The weak Piola material derivative . . . . . . . . . . 5.5.3 Shape derivative . . . . . . . . . . . . . . . . . . . . 5.5.4 Extractor Identity . . . . . . . . . . . . . . . . . . . 5.5.5 Adjoint system and cost function shape derivative . 5.6 Min-Max and function space parametrization . . . . . . . . 5.6.1 Saddle point formulation of the fluid state system . 5.6.2 Function space parametrization . . . . . . . . . . . . 5.6.3 Differentiability of the saddle point problem . . . . . 5.7 Min-Max and function space embedding . . . . . . . . . . . 5.7.1 Saddle point formulation of the fluid state system . 5.7.2 The Lagrange functional . . . . . . . . . . . . . . . . 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 110 114 114 119 119 120 122 122 123 124 133 136 138 140 141 146 147 157 157 159 162

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105

xi 6 Tube derivative in a Lagrangian setting 163 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2 Evolution maps . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2.1 Basic differentiation results . . . . . . . . . . . . . . . 164 6.2.2 Mathematical setting . . . . . . . . . . . . . . . . . . . 165 6.2.3 Elements of shape calculus . . . . . . . . . . . . . . . 165 6.2.4 Unconstraint non-cylindrical shape optimization . . . 171 6.2.5 Shape optimization under state constraints . . . . . . 173 6.2.6 Eulerian vs. Lagrangian non-cylindrical derivative . . 175 6.3 Navier-Stokes equations in moving domain . . . . . . . . . . 178 6.3.1 Transpiration boundary conditions . . . . . . . . . . . 178 6.3.2 Optimal control of the Navier-Stokes system by moving the domain . . . . . . . . . . . . . . . . . . . . . . . . 180 6.3.3 Comparison with the Eulerian derivative version . . . 182 7 Sensitivity analysis for a simple fluid-solid interaction system 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Mathematical settings . . . . . . . . . . . . . . . . . . . . . . 7.3 Well-posedness of the coupled system . . . . . . . . . . . . . 7.4 Inverse problem settings . . . . . . . . . . . . . . . . . . . . . 7.4.1 Analysis strategy . . . . . . . . . . . . . . . . . . . . . 7.4.2 Free divergence and non-homogeneous Dirichlet boundary condition constraints . . . . . . . . . . . . . . . . 7.4.3 Solid reduced order and solid weak state operator . . . 7.4.4 Fluid state operator . . . . . . . . . . . . . . . . . . . 7.4.5 Coupled system operator . . . . . . . . . . . . . . . . 7.4.6 Min-Max problem . . . . . . . . . . . . . . . . . . . . 7.5 KKT Optimality Conditions . . . . . . . . . . . . . . . . . . 7.5.1 Derivatives with respect to state variables . . . . . . . 7.5.2 Solid adjoint system . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 185 189 190 191 191 192 193 194 195 197 197 198 212

8 Sensitivity analysis for a general fluid-structure interaction system 215 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.2 Mechanical problem and main result . . . . . . . . . . . . . . 215 8.2.1 Solid weak state operator . . . . . . . . . . . . . . . . 220 8.2.2 Fluid state operator . . . . . . . . . . . . . . . . . . . 220 8.2.3 Coupled system operator . . . . . . . . . . . . . . . . 221 8.2.4 Min-Max problem . . . . . . . . . . . . . . . . . . . . 222 8.3 KKT optimality conditions . . . . . . . . . . . . . . . . . . . 223 8.3.1 Fluid adjoint system . . . . . . . . . . . . . . . . . . . 223 8.3.2 Solid adjoint system . . . . . . . . . . . . . . . . . . . 224

xii A Functional spaces and regularity of domains A.1 Classical functions . . . . . . . . . . . . . . . . . A.2 Lebesgue spaces . . . . . . . . . . . . . . . . . . A.2.1 Various inequalities . . . . . . . . . . . . . A.2.2 Completeness, separability and reflexivity A.2.3 Convolution . . . . . . . . . . . . . . . . . A.2.4 Density property . . . . . . . . . . . . . . A.3 Smooth domains and boundary measure . . . .

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239 239 241 241 242 242 243 243

B Distribution spaces B.1 The space D(Ω) . . . . . . . . . . . . B.2 The space of distributions D 0 (Ω) . . . B.3 Examples of distributions . . . . . . . B.3.1 Measures on Ω . . . . . . . . . B.3.2 Regular distributions . . . . . . B.3.3 The Dirac measure . . . . . . . B.3.4 Derivative of the Dirac measure B.3.5 The Dirac comb . . . . . . . . B.3.6 Principal value of 1/x . . . . .

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247 247 249 251 251 251 252 252 253 253

C The C.1 C.2 C.3 C.4 C.5 C.6 C.7 C.8

to L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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265 265 266 267 267 269 270 270 271 272 272 274 275 275

Fourier transform The case of L1 functions and extension The space S (Rd ) . . . . . . . . . . . The space of tempered distributions . Fourier transform in S 0 . . . . . . . . Fourier transform and convolution . . Partial Fourier transform . . . . . . . Vector valued distributions . . . . . . Vector valued Lebesgue spaces . . . .

D Sobolev spaces D.1 Spaces H m (Ω) . . . . . . . . . D.2 The space H s (Rd ) . . . . . . . D.3 The topological dual of H s (Rd ) D.4 Sobolev embedding theorems . D.5 Density properties . . . . . . . D.6 Trace theorem for H s (Rd ) . . D.7 Trace theorem for H m (Rd+ ) . . D.8 The space H s (Γ) . . . . . . . . D.9 Trace theorem in H m (Ω) . . . D.10 Extension theorems . . . . . . D.11 The space H −m (Ω) . . . . . . D.12 Compact embeddings . . . . . D.13 Poincar´e inequalities . . . . . .

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xiii References

277

Index

289

List of Figures

1.1 1.2 1.3 1.4 1.5

Fluid flow control . . . . . . . . . . Shape deformation . . . . . . . . . Non-cylindrical space-time domain Shape optimization . . . . . . . . . Rigid solid inside a fluid flow . . .

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2.1 2.2 2.3 2.4

Melting front propagation Non-cylindrical space-time Perturbed tube . . . . . . Transverse map . . . . . .

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7.1

Arbitrary Euler-Lagrange map . . . . . . . . . . . . . . . . .

186

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Geometric description . . . . . . . . . . . . . . . . . . . . . .

216

. . . . . domain . . . . . . . . . .

xv

Preface

Objectives and Scope of the Book This volume intends to provide a mathematical analysis of problems related to the evolution of two or three dimensional domains. This topic includes a number of engineering applications such as free surface flows, phase changes, fracture and contact problems, fluid-structure interaction problems for civil transport vehicules, civil engineering constructions such as stayedcable bridges or tall towers, biomechanical systems, boundary tracking problems, computational vision . . .. The main objective is to furnish various tools to handle the motion of a moving domain on the level of its intrinsic definition, computation, optimization and control. The efficiency of these tools will be illustrated within this volume on different examples in connection with the analysis of non-cylindrical partial differential equations, e.g., the Navier-Stokes equations for incompressible fluids in moving domains. We shall concentrate our analysis on the Eulerian approach. Hence the evolution of a 3D domain is chosen to be described by the flow T t of a nonautonomous Eulerian velocity field V(., t) for t ≥ 0. This flow can be defined in a strong or a weak manner depending on the space and time regularity of the moving domain. The first two chapters give all the details of the existence and uniqueness of the flow for both cases. These results are fundamental in order to deal with the arbitrary Eulerian-Lagrangian formulation or the level-set approach. The sensitivity of the flow mapping with respect to its associated velocity field V in the direction W is completly characterized by the transverse field Z. The evolution of this transverse field is governed by a dynamical transport equation involving the Lie bracket [V, W]. Among other applications, these differentiation results form the basis in order to analyse the first and second-order trajectory variations of any general dynamical system. We furnish different tools that can prove appropriate in order to deal with general variational formulations involving integral functionals using either the tensorial or the joint time-space metric over non-cylindrical time-space domains also called tubes. As an example, we can cite the two ways of handling the tube perimeter constraint that yield to different expressions for the surface tension. We will also revisit the notion of shape differential equations driving the evolution of a domain subjected to certain geometrical constraints. In the context

xvii

xviii of optimal control and inverse problems, this equation corresponds to the first-order optimality conditions associated to the minimization with respect to the velocity field V of a given Eulerian cost function. In the context of the level-set method, this shape differential equation is related to the HamiltonJacobi equation driving the optimal evolution of the level-curves. The next chapters aim at providing illustration of these Eulerian evolution and derivation tools for the control of systems involving fluids and solids. Especially, we shall consider in chapters 5 and 6 the control of a fluid described by the Navier-Stokes equations inside a 3D bounded domain through the dynamical evolution of its boundary. The Eulerian approach is based on the introduction of the transverse field characterizing the sensitivity of the flow with respect to the velocity field perturbations. We state the structure of the optimality conditions using either the fluid state differentiability with respect to V or the min-max derivation principle providing instantly the adjoint state problem. Similar results are obtained using a Lagrangian description of the boundary motion. In this case, the sensitivity analysis is performed using non-cylindrical identity perturbations. An interesting application is the rigorous mathematical justification of transpiration boundary conditions on fixed boundary usually used by engineers as a first-order approximation of no-slip boundary conditions on a moving solid surface. The end of the volume deals with boundary control of fluid-structure interaction systems. The idea is to minimize a given tracking functional related to the solid evolution, with respect to a fluid inflow boundary condition. The case of a rigid solid coupled with an incompressible fluid flow is based on the Eulerian approach for characterizing the partial sensitivity of the fluid with respect to the solid state variables. We obtain a new fluid-structure adjoint problem that allows the computation of a given cost function gradient. In case the rigid solid is replaced by an elastic solid with an arbitrary constitutive law, we use the Lagrangian framework and the non-cylindrical identity perturbations. Again we obtain a new fluid-structure adjoint system.

Intended Audience This book will be useful for researchers or graduate students with a background in applied mathematics who are interested in the control of complex systems involving a moving boundary. This include topics such as free surface flows, phase changes, fracture and contact problems, fluid-structure interaction problems for civil transport vehicules, civil engineering constructions such as stayed-cable bridges or tall towers, biomechanical systems, boundary tracking problems, computational vision . . .. More generally, this book will be of great use for people who are interested

xix in the optimization of complex partial differential systems since it includes various tools that can be used in different contexts. Some parts of the book can require some knowledge of advanced mathematics that are recalled in the appendix at the end of the volume.

xx

Chapter 1 Introduction

Shape Optimization was introduced around 1970 by Jean C´ea [31], who understood, after several engineering studies [127, 12, 35, 110, 102, 83, 84, 7], the future issues in the context of optimization problems. At that time, he proposed a list of open problems at the French National Colloquium in Numerical Analysis. These new problems were formulated in terms of minimization of functionals (referred as open loop control or passive control) governed by partial differential boundary value problems where the control variable was the geometry of a given boundary part [103, 76]. From the beginning, the terminology shape optimization was not connected to the structural mechanical sciences in which elasticity and optimization of the compliance played a central role. Furthermore, these research studies were mainly addressed in the context of the numerical analysis of the finite element methods. At the same time, there was some independent close results concerning fluid mechanics by young researchers such as O. Pironneau [123, 124, 78], Ph. Morice [107] and also several approaches related to perturbation theory by P.R. Garabedian [74, 75] and D.D Joseph [91, 92]. Very soon, it appeared that the shape controlof Boundary Value Problems (BVP) was at the crossroads of several disciplines such as PDE analysis, non-autonomous semi-group theory, numerical approximation (including finite element methods), control and optimization theory, geometry and even physics. Indeed several classical modeling in both structural and fluid mechanics (among other fields) needed to be extended. An illustrative example concerns a very popular problem in the 80’s concerning the thickness optimization of a plate modeled by the classical Kirchoff biharmonic equation. This kind of solid model is based on the assumption that the thickness undergoes only small variations. Therefore, many pioneering works were violating the validity of this assumption, leading to strange results, e.g., the work presented in the Iowa NATO Study [85] stating the existence of optimal beams having zero cross section values. In the branch which followed the passive control approach, we shall mention the work of G. Chavent [32, 34] based on the theory of distributed system control introduced by J-L. Lions [98]. Those results did not address optimization problems related to the domain but instead related to the coefficients inside the PDE. At that time, it was hoped that the solution of elliptic problems would be continuous with respect to the weak convergence of the coefficients.

1

2

Moving Shape Analysis

It appeared that this property was not achieved by this class of problem 1 . At that point a main bifurcation arose with the homogenization approach [10] which up to some point was considered as a part of the Optimal Design theory. The mathematical analysis of shape optimization problems began with the correct definition of derivatives of functionals and functions with respect to the domain, together with the choice of tangential space to the familly of shapes. Following the very powerful theory developed by J. Neˇcas [117], the role of bilipschitzian mapping was emphasized for Sobolev spaces defined in moving domainsbased on the Identity perturbation method [115, 106, 134]. Concerning the large domain deformationviewpoint the previous approach led to the incremental domain evolution methods [143]. After 1975, the second author introduced [145] an asymptotic analysis for domain evolution using classical geometrical flows which are intrinsic tools for manifolds evolutions and gave existence results for the so-called shape differential equation (see also [79]). At that period, applications focused more on sensitivity analysis problems than on asymptotic analysis of domains evolution. In 1972, A.M. Micheletti introduced in parallel [105, 104] a metric based on the Identity perturbation method thanks to the use of differentiable mappings, in order to study eigenvalues perturbation problems. The associated topology was extended by M. Delfour et al. [52] and turns out to be the same as the one induced by the continuity along flow field deformations [147]. The systematic use of flow mapping and intrinsic geometrythrough the fundamental role of the oriented distance function [47, 50] led to the revised analysis of the elastic shell theory [48, 49, 25, 26, 27, 28], of the boundary layer theory [3] or of the manifold derivative tools [53]. The use of both Bounded Variation (BV) analysis and the notion of Cacciapoli sets led to the first compactness method for domain sequences and several extensions to more regular boundaries were done through the use of different concepts such as fractal boundaries, density perimeter [23, 20, 21, 19] or Sobolev domains [50]. At that point, an other important bifurcation point in that theory occurred with the relaxation theory and the Special Bounded Variation (SBV) analysis which was particularly well adapted for image segmentation problems [6]. At the opposite, the capacity constraint for Dirichlet boundary conditions led to a fine analysis initiated in [18] and is still going on for cracks analysis. The method of large evolution based on the flow mapping (known from 1980 as the speed method [150]) turns to be the natural setting for weak evolution of

1 Indeed,

in his thesis [33], G. Chavent referred to such a result to appear in a work by F. Murat [113]. That paper [111] appeared but as a counterexample to the expected continuity property. He showed on a one dimensionnal simple example that with weak oscillating convergence of the coefficients, the associated solution was converging to another problem in which the new coefficients were related to the limit of the inverse coefficients associated to the original problem [112, 114].

Introduction

3

geometry allowing topological changes through the convection of either characteristic functions or oriented distance functions. After this non-exhaustive review of the context in which the shape optimization analysis emerged, we shall concentrate on the particular framework of the present book.

1.1

Classical and moving shape analysis

The object of this book is the mathematical analysis of systems involving the evolution of the geometry. This is motivated by many important applications in physics, engineering or image processing. The classical shape analysis investigates the effects of perturbations of the geometry in terms of continuity, differentiability and optimization of quantities related to the state of a system defined in that geometry. In this case, the geometry is usually perturbed thanks to a map involving a scalar parameter usually referred to as a fictitious time. On the contrary, the moving shape analysis deals with systems that are intrinsically defined on a moving geometry. Hence, we shall deal with sensitivity analysis with respect to a continuous family of shapes over a given time period. In this context, if we consider the geometry in a space-time configuration, the moving shape analysis may also be referred to as a non-cylindrical shape analysis2 . A first issue in this analysis is to model the evolution of the geometry. This is a common topic with the classical shape analysis. There exist many ways to build families of geometries. For instance, a domain can be made variable by considering its image by a family of diffeomorphisms parametrized by the time parameter as it happens frequently in mechanics for the evolution of continuous media. This way of defining the motion of domains avoids a priori the modification of the underlying topology. This change of topology can be allowed by using the characteristic function of families of sets or the level set of a space-time scalar function. We refer the reader to [51] for a complete review on this topic. In chapter 2, we shall deal with the particular problem of defining in a weak manner the convection of a characteristic function in the context of the speed method developed in [147]. In numbers of applications, we shall consider a state variable associated to a system which is a solution of a partial differential equation defined inside the moving domain over a given time period. Hence, we need to analyse the solvability of this non-cylindrical PDE system before going further. Here, again this topic has been already studied since it enters the classical shape analysis

2 The

notion of tube (non-cylindrical evolution domains) was also independently introduced by J.P. Aubin via the concept of abstract mutations [8].

4

Moving Shape Analysis

problem while introducing a perturbed state defined in the moving domain parametrized by the fictitious time parameter. Furthermore, this solvability analysis has been performed in numbers of mathematical problems involving moving domains. Here, we refer the reader to [135, 51] for some particular results in the context of the classical shape analysis. We also refer to the extensive litterature concerning the analysis of PDE systems defined in moving domains, e.g., [96, 126, 132, 62, 130, 88, 128, 70, 100, 11]. Contrary to the last topic, very few references exist for the sensitivity analysis with respect to perturbation of the evolution of the moving geometry. Early studies have been conducted in [90, 151, 158, 141, 120, 43, 142, 2] for specific hyperbolic and parabolic linear problems. An important step was performed in [154, 155] where the second author of the present book established the derivative of integrals over a moving domain with respect to its associated Eulerian velocity. These results were applied in order to study variational principles for an elastic solid under large displacements and for the incompressible Euler equation. This work was generalized in [58, 59].

1.2

Fluid-Structure interaction problems

As already mentioned above, we shall apply the moving shape analysis on systems involving the coupling between fluids and solids. A general fluid-solid model consists of an elastic solid either surrounded by a fluid (aircrafts, automobiles, bridge decks . . .) or surrounding a fluid flow (pipelines, arteries, reservoir tanks . . .). Here the motion of the interface between the fluid and the solid is part of the unknown of the coupled system. It is a free boundary problem that can be solved by imposing continuity properties through the moving interface (e.g., the kinematic continuity of the velocities and the kinetic continuity of the normal stresses). This model has been intensively studied in the last two decades on the level of its mathematical solvability [87, 54, 82, 39, 80, 131, 15, 9, 138, 41], its numerical approximation [89, 55, 119, 118, 122, 95, 67], its stability [73, 38, 64, 65] and more recently on its controllability [66, 109]. In this lecture note, we will restrict ourself to viscous Newtonian incompressible fluid flows described by the Navier-Stokes equations in space dimension two or three. The case of a compressible Newtonian fluid can be incorporated in the present framework with the price of a heavier mathematical analysis (solvability, non-differentiability around shocks . . .). Our goal is to solve inverse or control problems based on the previous general fluid-solid model. As an example, we think to decrease the drag of a car inside the atmospheric air flow by producing specific vibrations on its body using smart materials such as piezoelectrical layers. In this example, the con-

Introduction

5

trol variable can be chosen as the electrical energy input evolution inside the piezoelectrical device and the objective is to decrease the drag which is a function of the coupled fluid-structure state ( the air and the body of the car) and this state depends on the control variable. In order to build a control law for the electrical input, we need to characterize the relationship between the drag function and the control variable on the level of its computation and its variations. As an other example, we can think of the problem of aeroelastic stability of structures. Both authors have been dealing with such a problem in the context of the stability analysis against wind loads of bridge decks. In [108], it has been suggested that such a problem can be set as the inverse problem consisting in recovering the smallest upstream wind speed that leads to the worst bridge deck vibrations. In this example, the decision variable can be chosen as the upstream wind speed and the objective is to increase a functional based on the vibration amplitude history of the bridge deck during a given characteristic time period which is a function of the coupled fluidstructure state ( the wind flow and the bridge deck) which is also a function of the decision variable. Again, in order to recover the wind speed history, we need to characterize the relationship between the objective functional and the decision variable on the level of its computation and its variations. In order to characterize the sensitivity of the objective functional with respect to the control variable, it is obvious that we need to characterize the sensitivity of the coupled fluid-structure state with respect to the control variable. Here we recall that the coupled fluid-structure state is the solution of a system of partial differential equations that are coupled through continuity relations defined on the moving interface (the fluid-structure interface). The key point towards this sensitivity analysis is to investigate the sensitivity of the fluid state, which is an Eulerian quantity, with respect to the motion of the solid, which is a Lagrangian quantity. This task falls inside the moving shape analysis framework described earlier. Indeed the fluid state is the solution of a system of non-linear partial differential equations defined in a moving domain. The boundary of this moving domain is the solid wall. Then using the tools developed in [59], it has been possible to perform in [58] the moving shape sensitivity analysis in the case of a Newtonian incompressible fluid inside a moving domain driven by the non-cylindrical Navier-Stokes equations. All the previous results use a parametrization of the moving domain based on the Lagrangian flow of a given velocity field. Hence, the design variable is the Eulerian velocity of the moving domain, allowing topology changes while using the associated level set formulation. In [13, 14], the author used a noncylindrical identity perturbation technique. It consists in perturbating the space-time identity operator by a family of diffeomorphisms. Then, this family is chosen as the design parameter. It is a Lagrangian description of the moving geometry, which a priori does not allow topology changes but which leads to simpler sensitivity analysis results which are still comparable with the one obtained by the non-cylindrical speed method. In [57], the authors

6

Moving Shape Analysis

came back to the dynamical shape control of the Navier-Stokes and recovered the results obtained in [58] using the Min-Max principle allowing to avoid the state differentiation step with respect to the velocity of the domain. Now, we come back to the original problem consisting in the sensitivity analysis of the coupled fluid-structure state with respect to the control variable. Using the chain rule, the derivative of the coupled state with respect to the control variable involves the partial derivative of the fluid state with respect to the motion of the fluid-structure interface already characterized in [58, 57]. Hence, again using a Lagrangian penalization technique, already used and justified in [45, 46], it has been possible to perform in [109] the sensitivity analysis of a simple fluid-structure interaction problem involving a rigid solid within an incompressible flow of a Newtonian fluid with respect to the upstream velocity field. As already mentioned, this simple model is particularly suited for bridge deck aeroelastic stability analysis [121].

1.3

Plan of the book

The book is divided in eight chapters. Chapter 2 furnishes a simple illustration to some of the moving shape analysis results reported in the core of the lecture note. We deal with a simple inverse problem arising in phase change problems consisting in recovering the moving interface at the isothermal interface between a solid and liquid phase from measurements of the temperature on a insulated fixed part of the solid boundary. We use a least-square approach and we show how to compute the gradient of the least-square functional with respect to the velocity of the moving interface. It involves an ajoint state problem together with an adjoint transverse state, which is the novelty of the moving shape analysis compared to the classical one. In chapter 3, we consider the weak Eulerian evolution of domains through the convection, generated by a non-smooth vector field V, of measurable sets. The introduction of transverse variations enables the derivation of functionals associated to evolution tubes. We also introduce Eulerian variational formulations for the minimal curve problem. These formulations involve a geometrical adjoint state λ which is backward in time and is obtained thanks to the use of the so-called transverse field Z . In chapter 4, we recall the concept of shape differential equation developed in [145],[147]. Here, we present a simplified version and some applications in dimension 2 which enable us to reach the time asymptotic result. Furthermore, we introduce the associated level set formulation whose speed vector version was already contained in [149]. In chapter 5, we deal with a challenging problem in fluid mechanics which consists in the control of a Newtonian fluid flow thanks to the velocity evo-

Introduction

7

lution law of a moving wall. Here, the optimal control problem has to be understood as the open loop version, i.e., it consists in minimizing a given objective functional with respect to the velocity of the moving wall. This study is performed within the non-cylindrical Eulerian moving shape analysis describe in chapters 2 and 3. We focus on the use of a Lagrangian penalization formulation in order to avoid the fluid state differentiation step. In chapter 6, we introduce the Lagrangian moving shape analysis framework. It differs from the Eulerian one from the fact that the design variable is the diffeomorphism that parametrizes the moving geometry. The sensitivity analysis is simpler since it does not involve the transverse velocity field. We apply these tools in order to deal with the control of a Newtonian fluid flow thanks to the displacement evolution law of a moving wall. Chapter 7 moves to inverse problems related to fluid-structure interaction systems. Here, we consider a 2D elastic solid with rigid displacements inside the incompressible flow of a viscous Newtonian fluid. We try to recover informations about the inflow velocity field from the partial measurements of the coupled fluid-structure state. We use a least-square approach together with a Lagrangian penalization technique. We derive the structure of the gradient with respect to the inflow velocity field of a given cost function. Using the Min-Max principle, the cost function gradient reduces to the derivative of the Lagrangian with respect to the inflow velocity at the saddle point. This saddle point is solution of the first order optimality conditions. We use non-cylindrical Eulerian derivatives to compute the partial derivative of the Lagrangian functional with respect to the solid state variables, involved in the optimality system. Finally, in chapter 8 we extend the results of chapter 7, to the case of an elastic solid under large displacements inside an incompressible fluid flow. The main difference with the previous case is the use of a non-cylindrical Lagrangian shape analysis for establishing the KKT system. It forms the adjoint counterpart of the sensitivity analysis conducted in [66].

1.4

Detailed overview of the book

In order for the reader to have a simple overview of the book, we shall describe the different steps encountered while designing a complex fluid-structure interaction system. Indeed, let us consider a mechanical system that consists of a solid and a fluid interacting with each other. We would like to increase the performances of this system. These performances have to be quantitatively translated inside a cost function that we have to optimize with respect to some parameters that we will call the control variables. In the sequel, we will describe different control situations:

8

Moving Shape Analysis 1. Control of a fluid flow around a fixed body: it consists in trying to modify the fluid flow pattern around a fixed body using a boundary control which can act for example by blowing or suctioning the fluid at some part of the solid boundary. The control law will be designed in order to match some efficiency goals using the minimisation of a cost functional. 2. Shape design of a fixed solid inside a fluid flow: in this case, the control is the shape of the body. We would like to find the best shape satisfying some geometrical constraints that will optimize some cost functionals. This problem is somewhat classical in the aeronautical field, but it requires some subtle mathematical tools that we will quickly recall. 3. Dynamical shape design of a solid inside a fluid flow: the novelty compared to the last item is that the shape is moving and we are looking for the best evolution of this shape that both satisfies some geometrical constraints and optimizes some cost functionals. This is a rather natural technique in order to control a fluid flow pattern, but still its design requires some new mathematical tools that will be sketched in this introduction and more detailed in the core of this lecture note. 4. Control of an elastic solid inside a fluid flow: this is the most complex and most realistic situation where both the fluid and the solid have their own dynamics which are coupled through the fluid-solid interface. Then, we would like to control or optimize the behaviour of this coupled system thanks to boundary conditions. The mathematical analysis of this situation uses the whole framework introduced previously. This is a challenging problem, both on the mathematical point of view and on the technological side. The goal of this book is to partially answer to some issues related to this problem.

1.4.1

Control of a fluid flow around a fixed body

Let us consider a fixed solid obstacle Ω ⊂ R3 with boundary Γ surrounded by a viscous Newtonian fluid. We shall restrict the analysis of this system to a control volume D ∈ R3 containing the obstacle Ω (see figure 1.1). The fluid def

domain is denoted by Ωf = D \ Ω . The fluid is described by its velocity u : R+ × Ωf → R3 and its pressure p : R+ × Ωf → R. The flow is supposed to be incompressible and the state variables satisfy the classical Navier-Stokes equations, ∂t u + D u · u − ν ∆u + ∇p = f , (0, τ ) × Ωf (1.1) div u = 0, (0, τ ) × Ωf We must endow this system with boundary conditions such as far-field Dirichlet boundary conditions on ∂D and non-slip boundary conditions on the solid

Introduction boundary Γ

We also add initial conditions,

9

u = u∞ , ∂D u = 0, Γ

(1.2)

u(t = 0) = u0 , Ωf

(1.3)

A classical issue consists in reducing the drag exerted by the fluid on the

Ωf

Γc

D Ω Γ

∂D

FIGURE 1.1: Fluid flow control

solid (e.g., car, aircraft, ship). The first step in this direction is to choose the type of control we would like to apply on this system to reach our objective. The boundary control consists in adding inside the solid a control device that modifies the boundary conditions on a part Γc ⊂ Γ of its boundary, u = g, Γc (1.4) u = 0, Γ \ Γc Here the boundary control g can be either • time-independent and in this case it can be viewed as a passive control strategy consisting in modifying, e.g., the friction of the surface. • or time-dependent and in this case it can be viewed as an active control strategy consisting of a blowing/suction system that can be designed off-line (that means using an open-loop control) or on-line (that means using a feedback control law). Let us describe quickly both situations since it will be useful for the rest of the book. The objective functional A common topic in the optimization and control field of PDE systems is the choice of appropriate cost functionals, i.e., meeting both our objectives and

10

Moving Shape Analysis

the mathematical requirements that guarantee the converge to at least one optimum parameter. This functional can depend both on the state variables (u, p) and on the control parameter g. Let us choose this function as the work needed to overcome the drag exerted by the fluid on the solid boundary, Z Jdrag (u, p) = (u − u∞ ) · σ(u, p) · n dΓ dt (1.5) (0,τ )×Γ

def

def

where σ(u, p) = −p I +ν ε(u) stands for the fluid stress tensor, ε(u) = 1 ∗ 2 (D u + D u) stands for the fluid deformation tensor and n is unit normal field. Another cost function is associated to the objective of steering the fluid along a given pattern ud during a prescribed time, Z 1 Jtrack (u, p) = (u − ud )2 dx dt (1.6) 2 (0,τ )×Ωf More generally, we can consider any cost functionals that are twice-differentiable with respect to their arguments. The control problem Here we choose to work with a control variable defined as a velocity field on the solid boundary and our objective is to minimize a given cost functional, min j(g) g∈U

(1.7)

where j(g) = J[(u, p)(g)], U stands for an appropriate Hilbert space of the Sobolev type, typically H 1 ((0, τ ) × Γc ) and the couple (u, p)(g) stands for a weak solution of the Navier-Stokes equations associated to the control g,  ∂t u + D u · u − ν ∆u + ∇p = f , (0, τ ) × Ωf     div u = 0, (0, τ ) × Ωf    u = u∞ , (0, τ ) × ∂D (1.8) u = g, (0, τ ) × Γc    c  u = 0, (0, τ ) × Γ \ Γ    u(t = 0) = u0 , Ωf

Our goal is now to furnish the first-order optimality conditions associated to the optimization problem 1.7. These conditions are very useful since they are the basis in order to build both a rigourous mathematical analysis and gradient-based optimization algorithms. There exists two main methods in order to derive these conditions: the first one is based on the differentiability of the state variables with respect to the control parameter and the second one relies on the existence of Lagrangian multipliers.

Introduction

11

Sensitivity Let us consider a control point g ∈ U, then the cost functional j(g) is Fr´echet differentiable with respect to g [1, 71] and its directional derivative is given by hj 0 (g), hi = h∂(u,p) J[(u, p)(g)], (u0 , p0 )(g; h)i (1.9) def

d where (u0 , p0 )(g; h) = dg (u, p)(g) · h stands for the directional derivative of (u, p)(g) with respect to g. In the case of the functional (1.6), we have Z 0 hj (g), hi = (u(g) − ud ) · u0 (g; h) dx dt (0,τ )×Ωf

It can be proven by evaluating the differential quotient that (u 0 , p0 )(g; h) exists in an appropriate space and that it is the solution of the linearized Navier-Stokes system,  ∂t u0 + D u0 · u + D u · u0 − ν ∆u0 + ∇p0 = 0, (0, τ ) × Ωf     div u0 = 0, (0, τ ) × Ωf    0 u = 0, (0, τ ) × ∂D (1.10) 0 u = h, (0, τ ) × Γc    0 c  u = 0, (0, τ ) × Γ \ Γ    0 u (t = 0) = 0, Ωf

Then the first-order optimality condition writes hj 0 (g), hi = 0,

∀h ∈ U

(1.11)

That means that the set of optimal controls is contained in the set of critical points for the cost function j(g). However, we would like to obtain an expression of this condition avoiding the direction h ∈ U. To this end, we introduce the adjoint variable (v, π) solution of the adjoint linearized Navier-Stokes system,  −∂t v − D v · u + ∗ D u · v − ν ∆v + ∇π = −(u − ud ), (0, τ ) × Ωf     (0, τ ) × Ωf  div v = 0, v = 0, (0, τ ) × ∂D (1.12)   v = 0, (0, τ) × Γ    v(t = τ ) = 0, Ωf

Consequently, we are able to identify the gradient of the cost function as the trace on Γc of the adjoint normal stress tensor, i.e., ∇j(g) = ∗ γ(0,τ )×Γc σ(v, π) · n (1.13) This formal proof provides the basic steps needed in order to build a gradientbased optimization method associated to the control problem (1.7).

12

Moving Shape Analysis

An alternative approach consists in avoiding the derivation of the fluid state (u, p) with respect to the control g thanks to the introduction of a Lagrangian functional that includes not only the cost functional but also the state equation, L(ψ, r, φ, q; g) = J(ψ, r) + he(ψ, r; g), (φ, q)i (1.14)

where he(u, p; g), (φ, q)i stands for the weak form of the state equations (1.8), e.g., Z he(ψ, r; g), (φ, q)i = [−ψ · ∂t φ + (D ψ · ψ) · φ − ν ψ · ∆φ (0,τ )×Ωf Z +ψ · ∇q − r div φ] + g · (σ(φ, q) · n) dΓ dt (0,τ )×Γc Z Z Z + u∞ · (σ(φ, q) · n) + ψ(τ ) · φ(τ ) − u0 · φ(0) Ωf

(0,τ )×∂D

Ωf

Hence the control problem (1.7) is equivalent to the min-max problem, min min max L(ψ, r, φ, q; g) g∈U (ψ,r) (φ,q)

(1.15)

For every control g ∈ U, it can be proven that the min-max problem min max L(ψ, r, φ, q; g)

(ψ,r) (φ,q)

admits a unique saddle-point (u, p; v, π) which are solutions of the systems (1.8)-(1.12). Finally the first-order optimality for the problem (1.15) writes ∂g L(u, p, v, π; g) = 0

(1.16)

which turns out to be equivalent to (1.13). Then, we can think to solve the optimality condition (1.11), using a continuous iterative method. Indeed let us introduce a scalar parameter s ≥ 0, and a control variable g(s) that is differentiable with respect to s. Hence using the differentiability of J(g), we get Z r J(g(r)) − J(g(0)) = h∇J(g(s)), g0 (s)iU ∗ ,U ds 0

Let us choose the control such that

g0 (s) + A−1 (s)∇J(g(s)) = 0, s ∈ (0, r)

(1.17)

where A stands for an appropriate duality operator, then the functional writes Z r J(g(r)) − J(g(0)) = − h|∇J(g(s)|2 ds 0

That means that the control law (1.17) leads to a functional’s decrease and is referred to as a continuous gradient based optimization method. Using a discretization of the parameter s leads to a standard gradient-based method such as the conjugate-gradient or the quasi-Newton method depending on the choice of A(s).

Introduction

1.4.2

13

Shape design of a fixed solid inside a fluid flow

We again consider the situation where a fixed solid is surrounded by a fluid flow. The shape control consists in finding the optimal shape of the solid that reduces some objective functional (e.g., the drag) under some perimeter, volume or curvature constraints. This optimization is an open-loop control since the shape of the obstacle is time-independent. The speed method In the previous section, we have been dealing with an optimization problem where the control belongs to a linear space. Here the space of shapes is no more a linear space and the associated differential calculus becomes more tricky. Our goal is to build gradient-based methods in order to find the optimal shape, i.e., we would like to solve the following problem, min J(Ω)

Ω∈A

(1.18)

In order to carry out the sensitivity analysis of functionals depending on the shape of the solid Ω , we introduce a family of pertubated domains Ω s ⊂ D parametrized by a scalar parameter 0 ≤ s ≤ ε. These domains are the images ¯ → D, ¯ of the original domain Ω through a given family of smooth maps 3 Ts : D i.e., Ωs = Ts (Ω), Γs = Ts (Γ) Two major classes of such mappings are given by : • the identity perturbation method ([116, 125]), Ts = I +s θ ¯ → D. ¯ where θ : D • the speed method [145, 147], where the transformation is the flow associated to a given velocity field V(s, x), ∂s Ts (x) = V(s, Ts (x)), (s, x) ∈ (0, ε) × D Ts=0 (x) = x, x∈D 3 Typically

we have the following Lipschitz regularity assumptions : T(., x) ∈ C 1 ([0, ε]; R3 ), ∀x ∈ D

kT(., x) − T(., y)kC 0 ([0,ε];R3 ) ≤ Ckx − ykR3 kT where C > 0.

−1

(., x) − T

−1

T−1 (., x) ∈ C 0 ([0, ε]; R3 ), ∀x ∈ D

(., y)kC 0 ([0,ε];R3 ) ≤ Ckx − ykR3

14

Moving Shape Analysis ¯ to be globally invariant under Ts (V) we need to impose In order for D the following viability conditions, V(s, x) · n(x) = 0, x ∈ ∂D

Ts(V) Ω

Ω

D

Γc

D

Γc

Γ

Γ

∂D

∂D

FIGURE 1.2:

Shape deformation

Let us consider the family of functionals J(Ωs ) that depends on the shapes Ωs , e.g., the work to overcome the drag exerted by the fluid on the solid boundary, Z Jdrag (Ω) = (u − u∞ ) · σ(u, p) · ndΓ dt (1.19) (0,τ )×Γ

This functional depends on Ω not only because it is an integral over the boundary Γ, but also because it involves the solution (u, p) of the NavierStokes system,  ∂t u + D u · u − ν ∆u + ∇p = f , (0, τ ) × Ω     (0, τ ) × Ω  div u = 0, u = u∞ , (0, τ ) × ∂D (1.20)   u = 0, (0, τ ) × Γ    u(t = 0) = u0 , Ω that depends on Ω. To perform our sensitivity analysis, we choose to work in the framework of the speed method4 . We define the Eulerian derivative of the shape functional J(Ω) at point Ω in the direction of the vector field V ∈ V as the limit, dJ(Ω; V) = lim

s&0

4 which

J(Ωs (V)) − J(Ω) s

leads, at least for the first order terms, to the same results as the identity perturbation framework [51].

Introduction

15

where V is a linear space5 . If this limit exists and is finite ∀ V ∈ V and the mapping V→R V 7→ dJ(Ω; V) is linear and continuous, then the functional J(Ω) is said to be shape differentiable. Actually if J(Ω) is shape differentiable, then its Eulerian derivative only depends on V(0) and there exists a distribution G(Ω) ∈ D(D; R 3 )0 that we call the shape gradient such that dJ(Ω; V) = hG(Ω), V(0)i,

∀V ∈ V def

In the sequel, we shall use the notation ∇J(Ω) = G(Ω). In the case of smooth domain, the gradient is only supported on the boundary Γ and depends linearly on the normal vector field n. This result, called the structure theorem6 , is recalled as follows, Shape derivative structure theorem: Let J(.) be a differentiable shape functional at every shape Ω of class C k+1 for k ≥ 0 with shape gradient G(Ω) ∈ D(D; R3 )0 . In this case, the shape gradient has the following representation, G(Ω) = ∗ γΓ (g n) where g(Γ) ∈ D −k (Γ) stands for a scalar distribution and ∗ γΓ stands for the adjoint trace operator7 . This result is easier to understand when g(Γ) is integrable over Γ, that is to say, g ∈ L1 (Γ). Indeed in this case, it means that the directional shape derivative can always be written as follows, Z dJ(Ω; V) = g V · n dΓ Γ

Basic shape derivative calculus In the previous paragraph, we have introduced the notion of Eulerian derivative for shape functionals. This notion can be extended to functions defined on Banach or Hilbert spaces built on smooth domains Ω. Hence, a function 5 e.g., def

V = 6 We 7 i.e.,

˘

V ∈ C 0 (0, ε; C 1 (D; R3 )), div V = 0,

in D, hV, ni = 0

on ∂D

refer the reader to [51] Theorem 3.5 for the case of non-smooth domains. h∗ γΓ (g n), ViD(D;R3 )0 ,D(D;R3 ) = hg, V · niD0 (Γ),D(Γ)

¯

16

Moving Shape Analysis

y ∈ H(Ω) admits a material derivative at Ω in the direction V ∈ V if the following limit def

y(Ω; ˙ V) = lim

s→0

1 y(Ωs (V)) ◦ Ts (V) − y(Ω) s

admits a limit in the Hilbert space8 H(Ω). Endowed with the following definition, it is possible to derive the Eulerian shape derivative of the following functionals, Z J(Ω) = y(Ω) dΩ Ω

If y(Ω) is weakly shape differentiable in L1 (Ω), then the functional J(Ω) is shape differentiable and its directional derivate writes, Z dJ(Ω; V) = [y(Ω; ˙ V) + y(Ω) div V(0)] dΩ Ω

In order to apply the structure theorem, it is useful to define the notion of shape derivative for functions. Hence, if y ∈ H(Ω) admits a material derivative y(Ω; ˙ V) ∈ H(Ω) and ∇y · V(0) ∈ H(Ω) for all V ∈ V, we define the shape derivative as y 0 (Ω; V) = y(Ω; ˙ V) − ∇y(Ω) · V(0) In this case, the Eulerian shape derivative of J(Ω) takes the following form, Z dJ(Ω; V) = [y 0 (Ω; V) + div(y(Ω) V(0))] dΩ Ω

If Ω is class C k with k ≥ 1, then using the Stokes formula, we get Z Z y 0 (Ω; V) dΩ + y(Ω)V · n dΓ dJ(Ω; V) = Γ

Ω

REMARK 1.1 In the case where y(Ω) = Y |Ω , where Y ∈ H(D) with Ω ⊂ D, its shape derivative is zero since y(Ω; ˙ V) = ∇Y · V. Hence, Z y(Ω)V · n dΓ dJ(Ω; V) = Γ

This is a simple illustration of the structure theorem. In the case of functionals involving integration over the boundary Γ, we need to introduce the notion of material derivative on Γ. 8 e.g.,

W m,p (Ω) or L2 (0, τ ; W m,p (Ω)).

Introduction

17

Let z ∈ W (Γ) where W (Γ) is an Hilbert space of functions 9 defined over Γ. It is said that it admits a material derivative in the direction V ∈ V, if the following limit, 1 def z(Γ; ˙ V) = lim z(Γs (V)) ◦ Ts (V) − z(Γ) s→0 s admits a limit in the Hilbert space W (Γ). As a consequence of this definition, it is possible to derive the Eulerian shape derivative of the following functional, Z J(Γ) = z(Γ) dΓ Γ

If z(Γ) is weakly shape differentiable in L1 (Γ), then the functional J(Γ) is shape differentiable and its directional derivate writes Z dJ(Γ; V) = [z(Γ; ˙ V) + z(Γ) divΓ V(0)] dΓ Γ

where

def divΓ V = γΓ div V − (D V · n) · n

stands for the tangential divergence. As in the previous case, it is also possible to introduce the notion of shape derivative for z(Γ). Let Ω be of class C k with k ≥ 2. If z ∈ W (γ) admits a material derivative z(Γ; ˙ V) ∈ W (Γ) and ∇Γ y · V(0) ∈ W (Γ) for all V ∈ V, we define the shape derivative as z 0 (Γ; V) = z(Γ; ˙ V) − ∇Γ z(Γ) · V(0) where ∇Γ z = ∇Z|Γ − (∇Z · n) n

stands for the tangential gradient and Z is any smooth extension of z inside Ω. Using the above definition, it is possible to transform the expression of the differential as follows, Z [z 0 (Γ; V) + H z(Γ) V(0) · n] dΓ dJ(Γ; V) = Γ

where H stands for the mean curvature of Γ.

REMARK 1.2 In the case where z(Γ) = y(Ω)|Γ , the Eulerian derivative takes the following form, Z 0 dJ(Γ; V) = y (Ω; V) + (∇y(Ω) · n + H z(Γ) V(0) · n dΓ Γ

9 e.g.,

W m,p (Γ).

18

Moving Shape Analysis

Application to shape design Thanks to the framework introduced previously, it is possible to build a complete sensitivity analysis of shape functionals. Coming back to our optimal shape problem, we can state the following: the shape gradient for the tracking functional Z J(Ω) = (u − ud )2 dx dt (0,τ )×Ω

is given by

∇J(Ω) = ∗ γΓ σ(v, π) · n

(1.21)

where (u, p) is a solution of system (1.20) associated to the shape Ω and (v, π) is solution of the adjoint system,  −∂t v − D v · u + ∗ D u · v − ν ∆v + ∇π = −(u − ud ), (0, τ ) × Ω     (0, τ ) × Ω  div v = 0, v = 0, (0, τ ) × ∂D (1.22)   v = 0, (0, τ) × Γ    v(t = τ ) = 0, Ω The associated shape differential equation

Now as in the previous section, we can choose to solve the first-order optimality equation (1.21) using a continuous gradient-based method. That means that we write Z r J(Ωr (V)) − J(Ω0 ) = h∇J(Ωs (V)), V(s)i ds 0

Then solving the equation ∇J(V(s)) + A−1 (s) · V(s) = 0, s ∈ (0, +∞)

(1.23)

leads to a decrease of the functional J(Ωs (V)). The equation (1.23) is referred to as the shape differential equation and some of its properties are studied in chapter 4 of this book. Notably, we study its solvability in the case of smooth shape functionals. We also prove some results concerning the asymptotic behaviour of the solution of this equation, which hold essentially when the shape gradient has some continuity properties for an ad-hoc shape topology 10 . The level-set framework In chapter 4, we also relate the shape differential equation to the HamiltonJacobi equation involved in the level-set setting. The level-set setting consists 10 The

Hausdorff-complementary topology.

Introduction

19

in parametrizing the perturbed domain Ωs as the positiveness set of a scalar ¯ → R, function Φ : (0, ε) × D def

Ωs = Ωs (Φ) = {x ∈ D,

Φ(s, x) > 0}

and its boundary is the zero-level set, def

Γs = Γs (Φ) = {x ∈ D,

Φ(s, x) = 0}

This parametrization and the one introduced in the speed method can be linked thanks to the following identity, V(s) = −∂s Φ(s)

∇Φ(s) k∇Φ(s)k2

Both frameworks are equivalent if Φ(s) belongs to set of functions without steps, which means that k∇Φ(s)k is different from zero almost everywhere in D. We show how to build without step functions and we study the shape differential equation in this setting.

1.4.3

Dynamical shape design of a solid inside a fluid flow

We return to our model problem, but this time we consider that the shape of the solid is moving. Our goal is to control this motion in order to optimize some objective functionals. Our goal is to build gradient-based methods in order to find the optimal shape dynamic, i.e., we would like to solve the following problem, min J(Q) (1.24) Q∈E

where Q ∈ E is a smooth evolution set, which means [ def Q = {t} × Ωt t∈(0,τ )

where Ωt is a smooth domain of R3 with boundary Γt . The set, [ def Σ = {t} × Γt t∈(0,τ )

stands for the non-cylindrical lateral boundary.We call the set Q a tube. The Rd+1 -approach The optimal control of moving domain is a problem which is relevant of classical (non-linear) control theory as well as of classical shape theory. In fact on the pure theoretical level, the dynamical shape control theory can

20

Moving Shape Analysis t, Tt t=τ

t=τ Q

Q0

Γt

t Ωt Σ0

Σ

t=0

t=0 Ω0 Γ 0

Ω0

Rdx

Γ0

Rdx

FIGURE 1.3: Non-cylindrical space-time domain

be viewed as an application of the shape optimization theory for space-time manifolds. Indeed the dynamical shape control consists in finding the optimal evolution of a spatial domain Ωt in Rd . Let us consider the mapping S : t ∈ R −→ Ωt ∈ P(Rd ) where P(Rd ) stands for the set of parts inside Rd . Usually we would like to minimize some cost functional, Z τ j(S) = J(t, S(t) ) dt 0

Obviously, it is equivalent to the problem of finding the optimal tube [ {t} × Ωt ∈ Rd+1 . Q= 0
In fact the tube Q is the graph in R×P(Rd ) ⊂ Rd+1 of the shape mapping S. As for usual mappings defined from R in some space E, the graph G ⊂ R × E and of course any subset G is not a graph. Now under simple conditions on that set G, it becomes a graph. In the same way, any subset Q ∈ R × P(R d ) will not be a tube. Intuitively we would say that we require some causality in the evolution of the set Ωt . When the boundary of the set Ωt is smooth enough 11 , the idea is to avoid the normal field ν to the lateral boundary Σ of the tube to be strictly vertical. To handle non-smooth situations, we adopt an Eulerian viewpoint that associates 11 say

there exists a tangent space.

Introduction

21

to each tube Q the non-empty closed convex set of speed vector fields V which transport (in a weak sense) the characteristic function of the moving domain. Wen we consider the tube Q as a subset of Rd+1 , the control problem becomes a usual shape optimization problem ( as far as no real time consideration enters). The sensitivity analysis is then classically performed by considering horizontal vector fields ˜ t, x) = (0, Z(s, t, x)) ∈ Rd+1 Z(s, where s is the perturbation parameter of the tube. Then the d+1 dimensional shape optimization analysis fully applies and the so called Shape differential Equation furnishes descent direction, i.e., it furnishes the existence of a vector field Z∗ such that, for some α > 0, Z s J(Qs ) ≤ J(Q) − α kZ∗ (σ)k2 dσ 0

∀s > 0. We show that the existence of that field Z∗ induces the existence of a usual vector field V(t, x) ∈ Rd which builds that tube, i.e., Ωt = T( V )(Ω0 ). The Rd -approach In order to carry out the sensitivity analysis of functionals depending on the def tube Q , we assume that the domains are the images of the domain Ω 0 = Ωt=0 ¯ →D ¯ , i.e., through a given family of smooth maps Tt : D Ωt = Tt (Ω0 ), Γt = Tt (Γ0 ) Two major class of such mappings are given by : • the Lagrangian parametrization, Tt = θ(t, .) ¯ → D. ¯ In this case, the minimization problem (1.24) where θ : (0, τ ) × D can be transformed as min J(Q(θ)) (1.25) θ∈Θ

• the Eulerian parametrization, where the transformation is the flow associated to a given velocity field V(t, x), ∂t Tt (x) = V(t, Tt (x)), (t, x) ∈ (0, τ ) × D Tt=0 (x) = x, x∈D In this case, the minimization problem (1.24) can be transformed as min J(Q(V))

V∈V

(1.26)

22

Moving Shape Analysis

Existence of tubes In the smooth case, the existence of tubes follows the Cauchy-Lipschitz theory on differential equations [147, 51]. In the non-smooth case, the Lipschitz regularity of the velocities V can be weakened using the equations satisfied by the characteristic functions ξ(t, x) associated to the domain Ω t (V), ∂t ξ + ∇ξ · V = 0, (0, τ ) × D (1.27) ξt=0 = χΩ , D We shall consider velocity fields such that V ∈ L1 (0, τ ; L2 (D; Rd )) and the divergence positive part (div V)+ ∈ L1 (0, τ ; L∞ (D)). In this case, using a Galerkin approximation and some energy estimates, we are able to derive an existence result of solutions with initial data given in H −1/2 (D). For the time being, no uniqueness result has been obtained for this smoothness level. Actually, when the field V and its divergence are simply L 1 functions, the notion of weak solutions associated to the convection problems (1.27) does not make sense. In this case, the correct modeling tool for shape evolution is to introduce the product space of elements (ξ = ξ 2 , V) equipped with a parabolic BV like topology for which the constraint (1.27) defines a closed subset TΩ which contains the weak closure of smooth elements def TΩ = (χΩ ◦ T−1 t (V), V)| V ∈ Uad

This approach consists in handling characteristic functions ξ = ξ 2 which belongs to L1 (0, τ ; BV(D)) together with vector fields V ∈ L2 (0, τ ; L2 (D, Rd )) solution of problem (1.27). For a given element (ξ, V) ∈ T Ω , we consider the set of fields W such that (ξ, W) ∈ TΩ . It forms a closed convex set, noted Vξ . Hence, we can define the unique minimal norm energy element V ξ in the convex set Vξ . For a given tube ξ, the element Vξ is the unique (with minimal norm) vector field associated to ξ via the convection equation (1.27). We choose to adopt a different point of view inspired by the optimization problems framework. Indeed, our final goal is to apply the weak set evolution setting to the control problem arising in various fields such as free boundary problems or image processing. The usual situation can be described as follows. Let us consider a given smooth enough functional J(ξ, V). We would like to solve the following optimization problem inf

(ξ,V)∈TΩ

J(ξ, V)

(1.28)

The space Uad is a space of smooth velocities. In most situations, such a problem does not admit solutions and we need to add some regularization terms to ensure its solvability. Consequently, we shall introduce different penalization terms which furnish compactness properties of the minimizing sequences inside an ad-hoc weak topology involving bounded variation constraints. Then, the new problem writes inf

(ξ,V)∈TΩ

J(ξ, V) + F (ξ, V)

(1.29)

Introduction

23

The penalization term F (ξ, V) can be chosen using several approaches: • We can first consider the time-space perimeter of the lateral boundary Σ of the tube, developed in [155]. This approach easily draws part of the variational properties associated to the bounded variation functions space framework. In particular, it uses the compactness properties of tube family with bounded perimeters in R d+1 . Nevertheless, this method leads to heavy variational analysis developments. • We can rather consider the time integral of the spatial perimeter of the moving domain which builds the tube, as introduced in [157]. We shall extend these results to the case of vector fields living in L 2 ((0, τ ) × D; Rd ). In this case, only existence results for solutions of the convection equation can be handled and the uniqueness property is lost. Tube derivative In this paragraph, we are interested in differentiability properties of integrals defined over moving domains, Z J(Q(V)) = f (V) dx dt Q(V)

The transverse map as follows,

Tρt

associated to two vector fields (V, W) ∈ U is defined def

Tρt : Ωt −→ Ωρt = Ωt (V + ρW)

x 7→ Tt (V + ρW) ◦ Tt (V)−1

REMARK 1.3 The transverse map allows us to perform sensitivity analysis on functions defined on the unperturbed domain Ω t (V). The following result states that the transverse map T ρt can be considered as a dynamical flow with respect to the perturbation variable ρ. The transverse map Tρt is the flow of a transverse field Zρt defined as follows: ∂Tρt def Zρt = Z t (ρ, .) = ◦ (Tρt )−1 (1.30) ∂ρ i.e., is the solution of the following dynamical system : Ttρ (Zρt ) : Ωt −→ Ωρt

x 7−→ x(ρ, x) ≡ Ttρ (Zρt )(x)

with

dx(ρ) = Z t (ρ, x(ρ)), ρ ≥ 0 dρ x(ρ = 0) = x, in Ωt (V)

(1.31)

24

Moving Shape Analysis

Since, we will mainly consider derivatives of perturbed functions at point def t ρ = 0, we set Z(t) = Zρ=0 . A fundamental result lies in the fact that Z can be obtained as the solution of a linear time dynamical system depending on the vector fields (V, W) ∈ U. The vector field Z is the unique solution of the following Cauchy problem, ∂t Z + [Z, V] = W, (0, τ ) × D (1.32) Zt=0 = 0, D def

where [Z, V] = D Z · V − D V · Z stands for the Lie bracket of the pair (Z, V). The derivative with respect to ρ at point ρ = 0 of the following composed function, f ρ : [0, ρ0 ] → H(Ωt (V)) ρ 7→ f (V + ρW) ◦ Tρt f˙(V; W) is called the eulerian material derivative of f (V) at point V ∈ U in the direction W ∈ U. We shall use the notation, def d ρ ˙ ˙ f (V) · W = f (V; W) = f dρ ρ=0

With the above definition, we can state the differentiability properties of noncylindrical integrals with respect to their moving support. For a bounded measurable domain Ω0 with boundary Γ0 , let us assume that for any direction W ∈ U the following hypothesis holds, i) f (V) admits an eulerian material derivative f˙(V) · W

then J(V) is Gˆ ateaux differentiable at point V ∈ U and its derivative is given by the following expression, Z h i 0 J (V) · W = f˙(V) · W + f (V) div Z dΩ (1.33) Ωt (V)

Futhermore, if ii) f (V) admits an Eulerian shape derivative given by the following expression, f 0 (V) · W = f˙(V) · W − ∇f (V) · Z (1.34) then J 0 (V) · W =

Z

Ωt (V)

[f 0 (V) · W + div(f (V) Z)] dΩ

(1.35)

Furthermore, if Ω0 is an open domain with a Lipschitzian boundary Γ 0 , then Z Z J 0 (V) · W = f 0 (V) · WdΩ + f (V) Z · n dΓ (1.36) Ωt (V)

Γt (V)

Introduction

25

REMARK 1.4 The last identity will be of great interest while trying to prove a gradient structure result for general non-cylindrical functionals. It is possible to define the solution of the adjoint transverse system.For F ∈ L2 (0, τ ; (H 1 (D))d ), there exists a unique field Λ ∈ C 0 ([0, τ ]; (L2 (D))d )

solution of the backward dynamical system, −∂t Λ − D Λ · V − ∗ D V · Λ − (div V)Λ = F, (0, τ ) Λ(τ ) = 0

(1.37)

REMARK 1.5 The field Λ is the dual variable associated to the transverse field Z and is a solution of the adjoint problem associated to the transverse dynamical system. Actually, we shall deal with specific right-hand sides F of the form F(t) = γΓt (V) (f (t) n). In this case, the adjoint field Λ is supported on the moving boundary Γt (V) and has the following structure,

∗

Λ = ∗ γΣ(V) (λ n)

(1.38)

where λ is the unique solution of the following boundary dynamical system, −∂t λ − ∇Γ λ · V − (div V)λ = f, Σ(V) (1.39) λ(τ ) = 0, Γτ (V) p is the canonical projection on Γt (V) and χΩt (V) is the characteristic function of Ωt (V) inside D. Then in chapter 2 , we will establish the following adjoint identity, Z τZ Z τZ λW · n (1.40) EZ·n=− 0

0

Γt (V)

Γt (V)

where λ is the unique solution of problem (1.39) with f = E. Eulerian dynamical shape control We come back to our design problem. We would like to minimize the following tracking functional with respect to V, Z j(V) = |u − ud |2 (1.41) Q(V)

where (u, p) is a solution of the following system,  ∂t u + D u · u − ν ∆u + ∇p = 0, Q(V)     Q(V)  div u = 0, u = u∞ , (0, τ ) × ∂D   u = V, Σ(V)    u(t = 0) = u0 , Ω0

(1.42)

26

Moving Shape Analysis

Tt(V) Ω0

Ωt

D

D

Γ0

Γt

∂D

∂D

FIGURE 1.4: Shape optimization

Using the Eulerian shape analysis framework rapidly sketched in the previous section, we will prove in chapter 5 that this functional is differentiable and its gradient has the following form, ∇j(V) = ∗ γΣ(V) − λ n − σ(v, π) · n (1.43)

where (v, π) stands for the adjoint fluid state solution of the following system,  −∂t v − D v · u + ∗ D u · v − ν ∆v + ∇π = −(u − ud ), Q(V)     Q(V)  div(v) = 0, v = 0, (0, τ ) × ∂D (1.44)   v = 0, Σ(V)    v(τ ) = 0, Ωτ (V)

and λ is the adjoint transverse boundary field, solution of the tangential dynamical system, −∂t λ − ∇λ · V − (div V) λ = f, Σ(V) (1.45) λ(τ ) = 0, Γτ (V) with

1 f = [−(σ(v, π) · n)] · (D V · n − D u · n) + |u − ud |2 (1.46) 2 This result can be obtained using several techniques. One of them is the use of a min-max formulation involving a Lagrangian functional coupled with a function space embedding, particulary suited for non-homogeneous Dirichlet boundary problems. Let us give here the main steps of this proof. We rewrite the minimization problem as a min-max problem, j(V) =

min

(ψ,r)∈X×P

max

(φ,q)∈Y ×Q

LV (ψ, r; φ, q)

(1.47)

with LV (ψ, r; φ, q) = JV (ψ, r) − eV (ψ, r; φ, q)

(1.48)

Introduction

27

and eV (ψ, r; φ, q) =

Z

Z [∂t ψ + D ψ · ψ − ν∆ψ + ∇r] · φ − q div ψ Q(V) Q(V ) Z − (ψ − V) · σ(φ, q) · n Σ(V)

stands for the weak fluid state operator. The state and multiplier variables are defined on the hold-all domain D, i.e., def

def

X = Y = H 1 (0, τ ; H 2 (D)), P = Q = H 1 (0, τ ; H 1 (D)) The saddle points (u, p; v, π) are solutions of the first-order optimality conditions with respect to the multipliers (φ, q) and the state variables (ψ, r) which leads respectively to the primal system (1.20) and to the adjoint system (1.44). The crucial point concerns the derivation with respect to the design variable V ∈ Uad . We consider a perturbation vector field W ∈ Uad with an increment parameter ρ ≥ 0. Since the state and multiplier variables are defined in the hold-all domain D, the pertubed Lagrangian L ρ only involves perturbed supports. Using the min-max derivation principle [51] recalled in chapter 5, then d ρ def d hj 0 (V), Wi = j(V + ρW) = L (u, p; v, π) (1.49) dρ dρ ρ=0

We can state

hj 0 (V), Wi =

Z

Σ(V)

ρ=0

f Z · n + − σ(v, π) · n · W

with f given by equation (1.46) and where Z stands for the transverse vector field introduced previously. We finally use the transverse adjoint identity, Z Z fZ·n=− λ W · n, ∀ W ∈ Uad Σ(V)

Σ(V)

where λ is solution of equation (1.45) which corresponds to the adjoint system associated to the transverse dynamical system satisfied by Z. This adjoint variable is only supported by the moving boundary Γ t (V) over (0, τ ). Lagrangian dynamical shape control In chapter 6, we use a Lagrangian parametrization of the shape motion. def In this case, we prove that for V = ∂t θ ◦ θ −1 ∈ Uad , the functional j(θ) possesses a gradient ∇j(θ) which is supported on the moving boundary Σ(θ) and can be represented by the following expression, 1 ∇j(θ) = ∗ γΣ(θ) ∂t E + (divΓ V) E + D E · V + E · D u + |u − ud |2 n (1.50) 2

28

Moving Shape Analysis

where E = σ(v, π) · n and (v, π) stands for the adjoint fluid state solution of the following system,  −∂t v − D v · u + ∗ D u · v − ν ∆v + ∇π = −(u − ud ), Q(θ)     Q(θ)  div(v) = 0, v = 0, (0, τ ) × ∂D (1.51)   v = 0, Σ(θ)    v(τ ) = 0, Ωτ

1.4.4

Control of an elastic solid inside a fluid flow

A simple mechanical model We consider our model problem where now the solid is supposed to be a two dimensional elastically supported rigid motion. For the sake of simplicity, we only consider one degree of freedom for the structural motion : the vertical displacement d(t) e2 where e2 is the element of Cartesian basis (e1 , e2 ) in R2 . As previously, we use an Eulerian parametrization, where the transformation is the flow associated to a given velocity field V(t, x),

∂t Tt (x) = V(t, Tt (x)), (t, x) ∈ (0, τ ) × D Tt=0 (x) = x, x∈D

Hence, we have Ωft = Tt (Ωf0 ), Ωst = Tt (Ωs0 ) Since we only consider one degree of freedom motion, we write

Tt(V) Ωf0

Ωft

D

Ωs0

D

Ωst

Γs0

Γst

∂D

∂D

FIGURE 1.5:

Rigid solid inside a fluid flow

Ωst = Ωs0 + d(t) e2

Introduction 29 [ [ We set Σs ≡ ({t} × Γst ), Qf ≡ ({t} × Ωft ) and Σf ≡ (0, τ ) × ∂D 0
0
In our simple case, we can give an example of an appropriate flow vector field :  ˙ x ∈ Ωst  V(t, x) = d(t)e 2, ˙ (1.52) V(t, x) = Ext(d(t)e2 ), x ∈ Ωft  V(t, x) · n = 0, x ∈ ∂D

where Ext is an arbitrary extension operator from Γ s0 into Ωf0 . The map Tt is usually referred to as the Arbitrary Euler-Lagrange map. The solid is described by the evolution of its displacement and its velocity ˙ is the solution of the following ordinary second order and the couple (d, d) differential equation : (

¨ k d = Ff , hm d + i d, d˙ (t = 0) = [d0 , d1 ]

(1.53)

where (m, k) stand for the structural mass and stiffness. F f is the projection of the fluid loads on Γst along the direction of motion e2 . The fluid is assumed to be a viscous incompressible Newtonian fluid. Its evolution is described by its velocity u and its pressure p. The couple (u, p) satisfies the classical Navier-Stokes equations written in non-conservative form  ∂t u + D u · u − ν ∆u + ∇p = 0, Qf (V)    div(u) = 0, Qf (V) Σf Ωf0

u = u∞ ,    u(t = 0) = u0 ,

(1.54)

where ν stands for the kinematic viscosity and u∞ is the farfield velocity field. Hence, the projected fluid loads Ff have the following expression :

Ff = −

Z

Γst

σ(u, p) · n

!

· e2

(1.55)

where σ(u, p) = −p I +ν (D u + ∗ D u) stands for the fluid stress tensor inside Ωft . We complete the whole system with kinematic continuity conditions at the fluid-structure interface Γst : def u = V = d˙ e2 , on Σ(V)

(1.56)

30

Moving Shape Analysis

To summarize, we get the following coupled system :  ∂t u + D u · u − ν ∆u + ∇p = 0, Qf (V)     div(u) = 0, Qf (V)     u = u∞ , Σf    u = d˙ e2 , Σs (V) ! Z     m d¨ + k d = − σ(u, p) · n · e2 , (0, τ )    Γst  h i    u, d, d˙ (t = 0) = [u0 , d0 , d1 ] , Ωf0 × R2

(1.57)

Optimization problem

Here we choose the control variable as the far-field Dirichlet boundary condition on Σf . The goal is to minimize a functional depending on the state variables of the coupled fluid-structure problem, min j(u∞ ) (1.58) u∞ ∈ U c where j(u∞ ) = Ju∞ [u, p, d] (u∞ ) with [u, p, d] (u∞ ) is a weak solution of problem (1.57) and Ju∞ is a real functional of the following form, Z 1 τ Ju∞ [u, p, d] (u∞ ) = |d − dg |2 dt (1.59) 2 0

This functional is a tracking functional. That means that we would like to control the motion of the solid thanks to the far-field boundary conditions 12 . The purpose is to illustrate how the moving shape analysis presented in the previous section can be applied to the analysis of optimization problems for various coupled systems involving a moving interface. Sensitivity analysis

Using the Eulerian sensitivity analysis introduced in the last section, we prove in chapter 7 that the tracking functional is differentiable and its gradient takes the following form, (1.60) ∇j(u∞ ) = ∗ γΣf σ(v, π) · n with (v, π, b) solutions of the following adjoint system,  −∂t v − D v · u + ∗ D u · v − ν ∆v + ∇π = 0, Qf (V)     Qf (V)  div(v) = 0, v = 0, Σf   v = b e2 , Σs (V)    v(τ ) = 0, Ωfτ (V)

12 It

can also be interpreted as an inverse problem.

(1.61)

Introduction !  Z    σ(v, π) · n · e2  m ¨b + k b = (d − dg ) + ∂t   Γst (V)   Z  h i ˙ 2 (D v · e2 ) · e2 − ν (D v · n) · (D u · n) · n (0, τ ) |d| +    Γst (V)   h i    b, b˙ (τ ) = [0, 0]

31

(1.62)

The proof of this result is based on the Min-Max derivation principle. The minimization problem can be put into a Min-Max formulation involving the Lagrangian functional defined as follows, def

Lu∞ (u, p, d1 , d2 ; v, π, b1 , b2 ) = Ju∞ (u, p, d1 , d2 ) − heu∞ (u, p, d1 , d2 ), (v, π, b1 , b2 )i where eu∞ (u, p, d1 , d2 ) stands for the global weak state operator and where the displacement d1 and the velocity d2 are considered as independent variables. Then derivating first-order optimality conditions with respect to the state variables (u, p, d1 , d2 ) leads to the fluid and solid adjoint systems (1.61)-(1.62), def

where the adjoint state b1 has been eliminated to the benefit of b = b2 . The keystone is the sensitivity of the Navier-Stokes system with respect to pertubations of the solid velocity V = d2 e2 as described in the last section. It involves the transverse adjoint state λ as an intermediate sensitivity multiplier that can be eventually eliminated. Hence the expression of the cost function gradient in equation (1.60) is obtained using a bypass through the Min-Max problem. A general coupled fluid-structure model In chapter 8, we consider a more general model where the solid is supposed to be a 3D non-linear elastic solid. For the sake of simpleness, the reader is referred to this chapter for more details since it requires a heavier analysis.

Chapter 2 An introductory example : the inverse Stefan problem

In this chapter, we shall introduce various concepts that will be further developed in the core of this lecture note. These concepts will be illustrated on a model problem arising in free boundary systems. More precisely, we consider the identification of a moving boundary that represents the isothermal interface between a solid phase and a liquid phase, from measurements on a fixed part of the solid boundary. This problem is referred in the literature as the inverse Stefan problem [61, 144]. We make use of the transverse derivative concepts introduced in [154, 155].

2.1

The mechanical and mathematical settings

We consider a fluid phase located at time t ≥ 0 in a domain Ω f (t) ⊂ D with boundary Γf (t) where D ⊂ Rd is a hold-all domain. It is surrounded def

by a solid phase Ωs (t) = D \ Ωf (t) with boundary ∂Ωs (t) = Γs ∪ Γf (t), as def

described in Figure (2.1) where Γs = ∂D. The motion of the interface Γf (t) is characterized by its scalar Eulerian normal velocity v f (x, t), x ∈ Γf (t). We consider the velocity field V(t, x) : (0, τ ) × D → Rd satisfying the following properties,

   

def

V(t, x) · ns (x) = 0, (t, x) ∈ Σs = (0, τ ) × Γs

f f    V(t, x) · n (t, x) = v ,

def

(t, x) ∈ Σf =

[

t∈(0,τ )

{t} × Γf (t)

(2.1)

33

34

Moving Shape Analysis Tt(V) Ωs0

Ωs(t) Ωf (t)

Ωf0

Γf (t) Γf0

Γs

Γs

FIGURE 2.1: Melting front propagation

We consider an admissible space1 Uad for the velocity field.The map V ∈ Uad can be viewed as a non-autonomous velocity field {V(t) : 0 ≤ t ≤ τ } : V(t) : D → Rd

def

x 7→ V(t)(x) = V(t, x) Let us define the field xV (t; X) : [0, τ ] → Rd solution of the following Cauchy problem, ( dx (t) = V(t, x(t)), t ∈ [0, τ ] (2.2) dt x(t = 0) = X, X ∈ D. We build the corresponding Lagrangian flow transformation, Tt (V) : D → Rd

def

X 7→ Tt (V)(X) = xV (t; X)

1 def

Uad =

n

` ´ V ∈ C 0 [0, τ ]; V0k (D) ,

with k ≥ 0 and where

def

V0k (D) = def

with L ipk (V) =

X

|α|=k

n

∃ c > 0,

L ipk (V(t)) ≤ c, ∀ t ∈ [0, τ ]

o ` ´ ¯ d , V · n∂D = 0, on D V ∈ C k (D) def

L ip(Dα V) for k ≥ 1 and L ip(V) = sup

y6=x

o

|V(y) − V(x)| . |y − x|

An introductory example : the inverse Stefan problem

35

It is shown in [51, 60] that the conditions (2.1) imply that T t (V) is a diffeomorphism2 that maps D into D. In particular Ωf (t) = Tt (V)(Ωf0 ), Ωs (t) = Tt (V)(Ωs0 ) Γf (t) = Tt (V)(Γf0 ), Γs = Tt (V)(Γs ) In the sequel, we shall consider the velocity field V as the main variable def characterizing the evolution of the melting front and define v f = V · nf on Σf . For a given evolution of the boundary Γf (t), we consider the solution y of the heat equation inside the solid phase. We impose a given heat flux f (t, .) on the fixed solid boundary Γs and it is assumed that the temperature is equal to a fixed phase transition temperature yf on the moving melting interface Γf (t) for t ∈ (0, τ ). This leads to the following non-cylindrical system,  [ s  ({t} × Ωs (t)), ∂ y − ∆y = 0, Q ≡  t    0
The space-time domain Qs is called a tube. For a smooth velocity V ∈ Uad and the data (f, y0 ) ∈ L2 (0, τ ); L2 (Γs ) × L2 (Ωs0 ), there exists a unique solution [59] y ∈ L2 (0, τ ); Hy1f ,Γf (t) (Ωs (t)) ∩ L∞ (0, τ ); L2 (Ωs (t)) (2.4)

where

def

1 (D) = Hϕ,Γ

2.2

φ ∈ H 1 (D),

φ = ϕ on Γ

The inverse problem setting

For a given evolution of the melting interface, we consider the solution of the heat equation (2.3) and we consider its trace on the fixed boundary Σ s . 2

` ` ´ ´ ` ` ´ ´ ¯ d ∩ C 0 [0, τ ]; W k+1,∞ (D) d T(V) ∈ C 1 [0, τ ]; C k (D) ` ` ´d ´ V ◦ T(V) ∈ L∞ (0, τ ); W k+1,∞ (D) ` ` ´ ´ ¯ d T−1 (V) ∈ C 1 [0, τ ]; C k (D)

36

Moving Shape Analysis Tt(V) t=τ

t=τ Qs0

Qs Qf0

Qf Γf (t)

t Σf0

Σf

t=0

t=0 Γs Ωs0

Ωf0 Γf0

Γs Ωs0

Rdx

Ωf0

Γf0

Rdx

FIGURE 2.2: Non-cylindrical space-time domain

On the mathematical viewpoint, we introduce the observation space def O = L2 (0, τ ); L2 (Γs ) and the observation operator,

O : Uad → O

def

V 7→ O(V) = γΣs (y(V))

(2.5)

where y(V) stands for the solution of equation (2.3) and γ Σs is the zero order trace operator on Σs . The inverse Stefan problem consists in recovering the evolution of the melting front Γf (t) from the knowledge of the temperature on the fixed solid boundary Γs . This means that for a given temperature yd ∈ L2 (0, τ ); L2 (Γs ) , we look for V ∈ Uad such that O(V) = yd , in O (2.6)

It is a non-linear ill-posed inverse problem that can be solved using a leastsquare minimization problem regularized thanks to a Tikhonov zero order term.Hence we look for the solution V of the following optimization problem, min

V∈Uad

α 1 kO(V) − yd k2O + kVk2Uad 2 2

(2.7)

with α > 0. In the sequel, we will prove the following result, THEOREM 2.1 For V ∈ Uad , yd ∈ O, we consider the functional Z Z 1 τ j(V) = |y(V) − yd |2 ds dt 2 0 Γs

(2.8)

An introductory example : the inverse Stefan problem where y(V) is the solution of the following system  ∂t y − ∆y = f, Q(V),    ∂n y = 0, Σs , Σ(V),  y = yf ,   y(t = 0) = y0 , Ω0

37

(2.9)

j(V) is differentiable with respect to V and its gradient has the following structure ∗ ∇j(V) = −∗ γΣ(V) λ n , in Uad (2.10) 0 1 where the transverse adjoint state λ ∈ C [0, τ ]; H (Γt (V)) is the solution of the backward dynamical system, −∂t λ − ∇Γ λ · V − (div V) λ = ∂n y ∂n ϕ, Σ(V) (2.11) λ(t = τ ) = 0, Γτ (V) and where the adjoint state ϕ is the solution of the following backward system,  ∂t ϕ + ∆ϕ = 0, Q(V)       ∂n ϕ = y(V) − yd , Σs (2.12)  ϕ = 0, Σ(V)      ϕ(t = τ ) = 0, Ωτ (V)

2.3

The eulerian derivative and the transverse field

A possible choice in order to solve the above minimization problem is to use a gradient based method such as the conjugate gradient method. Hence, we need to evaluate the gradient with respect to V of the functional 1 kO(V) − yd k2O (2.13) 2 where, for the sake of simpleness, we have dropped the regularizing term α 2 2 kVkUad . Let us choose a perturbation direction W ∈ U ad . We would like to compute the directional derivative of j, 1 def [DV [j](V)] · W = lim j(V + ρ W) − j(V) (2.14) ρ→0 ρ def

j(V) =

Then the goal is to evaluate the directional derivative of the element y(V) which is a solution of the moving heat equation (2.3). In order to do so, we write the associated variational formulation satisfied y(V), Z τZ [∂t y(V) φ(V) + ∇y(V) · ∇φ(V)] dx dt = 0, 0 Ωt (V) (2.15) 2 1 ∀ φ(V) ∈ L (0, τ ); H0,Γt (V) (Ωt (V))

38

Moving Shape Analysis

where we have set without loss of generality, (f, y d , y0 ) = (0, 0, 0) together def

def

with Ωt (V) = Ωs (t) and Γt (V) = Γf (t). Looking at equation (2.15), it is clear that we need to establish how to differentiate the generic term Z τZ J(V) = f (V) dx dt 0

Ωt (V)

with respect to V. To this end, we introduce the perturbated moving domain def

Ωt (V + ρ W) = Tt (V + ρ W)(Ω0 ) This family generates a perturbed tube def

Q(V + ρ W) =

[

0≤t≤τ

{t} × Ωt (V + ρ W)

as described in Figure (2.3). Since the function f (V) is defined on the non-

Tt(V + ρ W)

V(t)

V(t) + ρ W(t)

t

Γt(V + ρ W) Q0

Tt(V)

Γt(V) Q(v + ρ W)

Ω 0

Ω0

FIGURE 2.3: Perturbed tube

cylindrical reference tube Q(V), it is natural to introduce the transformation between Q(V) and Q(V + ρ W). A canonical choice is furnished by T t (ρ; x) : Ωt (V) → Ωt (V + ρ W) def x 7→ T t (ρ; x) = Tt (V + ρ W)] ◦ Tt (V)−1 (x)

Hence, the perturbated functional can be written as follows,

An introductory example : the inverse Stefan problem Tρt

39

V(t)

V(t) + ρ W(t)

t

t Q(V) Q(V + ρ W)

Γ t (V + ρ W)

Γt(V)

Ω

0

FIGURE 2.4: Transverse map

J(V + ρ W) = = =

Z

Z

Z

τ 0 τ 0 τ 0

Z

Z

Z

f (V + ρ W) dx dt Ωt (V+ρ W)

Tρt Ωt (V)

Ωt (V)

f (V + ρ W) dx dt

det D Tρt f (V + ρ W) ◦ Tρt dx dt

where we have performed a transport into the moving reference domain Ω t (V). Now we shall need to differentiate the terms inside the integral with respect to ρ at point ρ = 0. The easiest way to do so is to connect this problem to the classical shape derivative calculus handled inside the speed method framework [147, 135]. This means that we need to identify a transverse velocity field that may generate the transverse map T t (ρ; x) as the solution of a dynamical system with respect to the parameter ρ ∈ [0, ρ0 ]. Actually, it can be proven that T(V + ρ W) is continuously differentiable3 with respect to ρ and that the transverse map Tρt can be considered as the flow with respect to ρ of the transverse vector field def Z(ρ; (t, x)) = ∂ρ T t (ρ) ◦ T t (ρ)−1 (x) = ∂ρ T(V + ρ W) ◦ T(V + ρ W)−1 (x) Indeed the map Z ∈ Zad can be viewed as a non-autonomous velocity field : Z(ρ) : (0, τ ) × D → Rd

def

(t, x) 7→ Z(ρ)(t, x) = Z(ρ; (t, x)) 3 in

` ´ def ¯ d . Zad = C 0 [0, τ ]; (C k−1 (D))

40

Moving Shape Analysis

with 0 ≤ ρ ≤ ρ0 . We define the field XZt (ρ; x) : [0, ρ0 ] → Rd solution of the following Cauchy problem,   dX (ρ) = Z(ρ; (t, X (ρ))), ρ ∈ [0, τ ] (2.16) dρ  X (ρ = 0) = x, x ∈ Ωt (V)

We build the corresponding Lagrangian flow transformation, Tρ (Z) : (0, τ ) × D → Rd

def

(t, x) 7→ Tρ (Z)(t, x) = XZ (ρ; (t, x)) and we identify the transverse map T (ρ) with Tρ (Z). This allows writing the perturbated functional as follows, J(V + ρ W) =

Z

τ 0

Z

Ωt (V)

det D Tρ (Zρt ) f (V + ρ W) ◦ Tρ (Zρt ) dx dt

Compared with the classical speed method, the parameter ρ plays the role of the peuso-time t, the field Z t (ρ) plays the role of the speed V(t) and the def

transverse field Z(t) = Z t (ρ = 0) has the same role as V(0). Hence, the non-cylindrical shape derivative calculus reduces to apply the classical shape derivative calculus with the following identification : V(0) ↔ Z(t). Following this idea, the following identity holds LEMMA 2.1

∂ρ det D Tρ (Zρt ) |ρ=0 = div Z(t)

(2.17)

In the classical shape derivative calculus, the notion of material derivative has been introduced. Here we define its non-cylindrical counterpart. Let Y(V) be a given functional space defined on Q(V). An example is furnished 1 by Y(V) = L2 (0, τ ); H0,Γ (Ω (V)) . t t (V) DEFINITION 2.1 We say that a function f (V) ∈ Y(V) for V ∈ Uad admits an Eulerian material derivative f˙(V; W) in the direction W ∈ Uad if the following limit f (V + ρ W) ◦ Tρt (V) − f (V) ρ→0 ρ

def f˙(V; W) = lim

exists in Y(V). Using this definition, we easily establish the following result,

(2.18)

An introductory example : the inverse Stefan problem

41

LEMMA 2.2 Let V ∈ Uad and f (V) ∈ Y(V) such that it admits an Eulerian material derivative f˙(V; W) in the direction W ∈ Uad . Then J(V) is differentiable with respect to V and its derivative is furnished by the following expression, Z τZ h i DV [J(V)] · W = (div Z) f (V) + f˙(V; W) dx dt (2.19) 0

Ωt (V)

It remains to precise a specific rule to compute the transverse Z starting from the knowledge of V, W ∈ Uad . This is realized thanks to the following result, THEOREM 2.2 [154, 59] The transverse field Z is the unique vector in Zad such that ∂t Z + D Z · V ∈ Zad and is the solution of the following Cauchy problem, ∂t Z + D Z · V − D V · Z = W, (0, τ ) × D (2.20) Z(t = 0, .) = 0, D REMARK 2.1

In the sequel, we shall use the notation def

[Z, V] = D Z · V − D V · Z for the Lie bracket associated to (Z, V) ∈ Zad × Uad . PROOF

It consists of 4 steps :

1. We first prove the differentiability of the flow T(V + ρ W) with respect to ρ, LEMMA 2.3 Let us consider (V, W) ∈ Uad × Uad , then there exists ρ0 > 0 such that the application ¯ d) ]0, ρ0 [ −→ C 0 ([0, τ ]; C k−1 (D) ρ 7→ T(V + ρ W)

is continuously differentiable and its derivative satisfies for any t ∈ [0, τ ], ∂ρ Tt (V + ρ W) = Z t D(V + ρ W)(µ, Tµ V + ρ W ) · ∂ρ Tµ (V + ρ W) dµ 0 Z t + W µ, Tµ (V + ρ W) dµ. 0

42

Moving Shape Analysis def 2. Let us set S(ρ) = ∂ρ T(V + ρ W) , then the transverse field writes Z(ρ) = S(ρ) ◦ T(V + ρ W)−1

At ρ = 0, we set S = Z ◦ T(V) We can state the following result

LEMMA 2.4 def

¯ d ), The function S is the unique vector field, in Zad = C 1 ([0, τ ]; (C k−1 (D)) satisfying S(t) =

Z

t

W(µ, Tµ (V))dµ + 0

Z

t 0

D V(µ, Tµ (V)) · S(µ) dµ.

(2.21)

3. Then we prove the equivalence between equation (2.20) and equation (2.21),

LEMMA 2.5 (i) Let us consider Z ∈ Zad such that (a) the following regularity holds ¯ d) ∂t Z + D Z · V ∈ C 0 ([0, τ ]; (C k−1 (D)) (b) and it satisfies equation (2.20) then Z ◦ T(V) belongs to Zad and satisfies equation (2.21). (ii) Let us consider S ∈ Zad solution of equation (2.21), then S ◦ T(V)−1 ∈ Zad is such that (a) the following regularity holds ¯ d) ∂t S ◦ T(V)−1 + D S ◦ T(V)−1 · V ∈ C 0 ([0, τ ]; (C k−1 (D))

(b) and it satisfies equation (2.20).

4. Since we already have proven the unique solvability of equation (2.21), we use the above lemma to conclude.

An introductory example : the inverse Stefan problem

2.4

43

The Eulerian material derivative of the state

With the previous framework, we are ready to differentiate the variational formulation (2.15) with respect to V. We set yρ = y(V + ρ W), Fρ = D Tρ and Jρ = det(Fρ ). The perturbated formulation writes Z τZ Jρ (∂t yρ ) ◦ Tρ φρ ◦ Tρ + (∇yρ ) ◦ Tρ · (∇φρ ) ◦ Tρ dx dt = 0, 0

Ωt (V)

∀ φρ ∈ Θ(ρ)

where def

Θ(ρ) =

n

1 φ ◦ (Tρ )−1 | φ ∈ L2 (0, τ ); H0,Γ (Ωt (V)) t (V)

The following identities are easy to check, (∇yρ ) ◦ Tρ = ∇(yρ ◦ Tρ )Fρ−1

(∂t yρ ) ◦ Tρ = ∂t (yρ ◦ Tρ ) − ∇(yρ ◦ Tρ )Fρ−1 ∂t Tρ

o

(2.22)

∂t Tρ = −Fρ · V + (V + ρ W) ◦ Tρ

def

We set y ρ = yρ ◦ Tρ , then we get Z τZ Jρ ∂t y ρ − ∇y ρ Fρ−1 ∂t Tρ φ + ∇y ρ Fρ · ∇φFρ dx dt = 0, 0

Ωt (V)

∀φ ∈ Θ

def 1 where Θ = L2 (0, τ ); H0,Γ (Ωt (V)) . t (V) The following identities are easy to check, ∂ρ Jρ |ρ=0 ∂ρ Fρ |ρ=0 −1 ∂ρ Fρ |ρ=0 ∂ρ ∂t Tρ |ρ=0

= div Z = DZ = −DZ

(2.23)

= W − [Z, V]

def

Recalling that Tρ |ρ=0 = I and y˙ = ∂ρ y ρ |ρ=0 , the derivative of the variational formulation writes Z τZ div Z [∂t y φ + ∇y · ∇φ] + ∂t y˙ − ∇y · W − [Z, V] φ 0

Ωt (V)

+ ∇y˙ · ∇φ − ∇y D Z · ∇φ − ∇y · ∇φ D Z dx dt = 0, ∀φ ∈ Θ

44

Moving Shape Analysis

Using the transverse dynamical system, we get Z τZ Z τZ [∂t y˙ φ + ∇y˙ · ∇φ] dx dt + [div Z ∂t y − ∇y · (∂t Z)] φ dx dt 0

−

Z

0

Ωt (V) τ Z

Ωt (V)

∀φ ∈ Θ

0

Ωt (V)

∇y − (div Z) I + D Z + ∗ D Z · ∇φ dx dt = 0,

Performing integration by parts, we shall obtain the strong formulation satisfied by the material Eulerian derivative y, ˙  ∂t y˙ − ∆y˙ = −(div Z)∂t y + ∇y · (∂t Z)      + div ∇y (div Z) I −(D Z + ∗ D Z) Q(V)    ∂n y˙ = ∇y − (div Z) I + D Z + ∗ D Z · n, Σs (2.24)     y˙ = 0, Σ(V)     y(t ˙ = 0) = 0, Ω0 together with

(

∂t Z + D Z · V − D V · Z = W, (0, τ ) × D Z(t = 0, .) = 0,

D

(2.25)

Finally using the fact that Γs does not move, the directional derivative of j(V) is given by the following expression Z τZ y(V) − yd y(V; ˙ W) dx dt (2.26) [DV [j](V)] · W = 0

2.5

Γs

The Eulerian partial derivative of the state

Now, we would like to identify the gradient ∇j(V). To this end, we need to introduce several adjoint problems. Before proceeding, it may be convenient to define the notion of Eulerian partial derivative. DEFINITION 2.2 We say that a function f (V) ∈ Y(V) for V ∈ Uad admits an Eulerian partial derivative f 0 (V; W) in the direction W ∈ Uad if it admits an Eulerian material derivative f˙(V; W) and def f 0 (V; W) = f˙(V; W) − ∇f · Z

(2.27)

exists in Y(V) and depends linearly on W, where Z is the transverse field solution of the transverse dynamical system (2.20).

An introductory example : the inverse Stefan problem

45

Going back to the derivation of the generic functional J(V), we are able to quote the following result, LEMMA 2.6 Let V ∈ Uad and f (V) ∈ Y(V) such that it admits an Eulerian partial derivative f 0 (V; W) in the direction W ∈ Uad . Then J(V) is differentiable with respect to V and its derivative is furnished by the following expression, Z τZ DV [J(V)] · W = [f 0 (V; W) + div(f Z)] dx dt (2.28) 0

Ωt (V)

Furthermore if the boundary Γt (V) is Lipschitz, then Z τZ Z τZ DV [J(V)] · W = f 0 (V; W) dx dt + 0

Ωt (V)

0

f (V)hZ, ni ds dt Γt (V)

(2.29)

The main interest in the use of the Eulerian partial derivative is that it commutes with time and space derivatives. In order to obtain, the system satisfied by y 0 , we consider the variational formulation (2.15) with φ ∈ L 2 (0, τ ) × D , Z τZ [−y(V) ∂t φ + ∇y(V) · ∇φ] dx dt = 0, 0

Ωt (V)

∀ φ ∈ L2 (0, τ ) × D

Differentiation with respect to V leads to Z τZ [−y 0 ∂t φ + ∇y 0 · ∇φ] dx dt 0

+

Z

0

Ωt (V) τ Z

Γs ∪Γt (V) 2

[−y(V) ∂t φ + ∇y(V) · ∇φ] hZ, ni

∀ φ ∈ L (0, τ ) × D

Integration by parts inside the last equation with φ ∈ D(Q(V)) leads to ∂t y 0 − ∆y 0 = 0, Q(V) y 0 (t = 0) = 0, Ω0 Using that hZ, ni = 0 on Γs , we get ∂n y 0 = 0, Γs The Dirichlet boundary condition, y = yf , Γt (V)

46

Moving Shape Analysis

differentiates as follows, y 0 = −∂n y hZ, ni, Γt (V) Hence, the Eulerian partial derivative of y(V) is the solution of the following linearized system,  0 0   ∂t y − ∆y = 0, Q(V)     ∂n y 0 = 0, Σs (2.30) 0  y = −∂ y hZ, ni, Σ(V)  n     y 0 (t = 0) = 0, Ω0 Finally, the derivative of j(V) writes Z τZ [DV [j](V)] · W = y(V) − yd y 0 (V; W) ds dt

2.6

(2.31)

Γs

0

The adjoint state and the adjoint transverse field

In order to identify the gradient ∇j(V), we introduce the following adjoint problem,  ∂t ϕ + ∆ϕ = 0, Q(V)       ∂n ϕ = y(V) − yd , Σs (2.32)  ϕ = 0, Σ(V)      ϕ(t = τ ) = 0, Ωτ (V) Using several integrations by parts, it turns that the derivative of j(V) can be written Z τZ [DV [j](V)] · W = ∂n y ∂n ϕ hZ, ni ds dt (2.33) 0

Γt (V)

This expression does not allow to identify ∇j(V) since it does not explicitly involve the vector field W. Let us consider a distributed term involving the transverse field Z, Z Z F · Z dx dt 0,τ

D

with F ∈ L2 (0, τ ; H(D)) where H(D) is an ad-hoc4 Hilbert space of functions defined in the hold-all domain D. Using several integrations by parts the 4 stable

¯ by multiplication of functions in the space C k−1 (D).

An introductory example : the inverse Stefan problem following identity holds, Z τZ Z

0

τ Z

0

D

D

∂t Z + D Z · V − D V · Z · Λ =

− ∂t Λ − D Λ · V − ∗ D V · Λ − (div V) Λ · Z =

Z

τ

47

Z

Z 0τ Z D 0

D

W·Λ W·Λ

for any Λ ∈ D (0, τ ) × D . Hence, we introduce the adjoint transverse field Λ as the unique solution in an ad-hoc space5 of the following adjoint transverse dynamical problem, −∂t Λ − D Λ · V − ∗ D V · Λ − (div V) Λ = F, (0, τ ) × D (2.34) Λ(t = τ ) = 0, D Hence the following adjoint identity holds, Z τZ Z τZ F · Z dx dt = Λ · W dx dt 0

D

0

D

In the case where F = ∗ γΓt (V) (f n), where f ∈ L2 (Σ(V)), the distributed term turns out to be supported by the lateral boundary, Z τZ Z τZ ∗ γΓt (V) f n · Z dx dt = fZ·n 0

D

0

Γt (V)

Then in this case, the adjoint field is also supported by Σ(V) and it can be expressed as Λ = −∗ γΓt (V) (λ n)

where λ is the unique solution6 of the following adjoint tangential transverse problem −∂t λ − ∇Γ λ · V − (div V) λ = f, Σ(V) (2.35) λ(t = τ ) = 0, Γτ (V) This means that we have Z Z τZ f Z · n ds dt = − 0

Γt (V)

τ 0

Z

Γt (V)

λ n · W ds dt

(2.36)

This identity allows to identify the gradient of j(V) with respect to V by setting f = ∂n y ∂n ϕ.

5Λ

` ´ ¯ Λ ∈ C 0 [0, τ ]; H(D) , ∂t Λ + D Λ · V ∈ L2 (0, τ ; H(D)) . ˘ ` ´ ¯ def = λ ∈ C 0 [0, τ ]; L2 (Γt (V)) , ∂t λ + ∇Γ λ · V ∈ L2 (0, τ ; H(D)) .

def ad

6 in

=

λad

˘

Chapter 3 Weak evolution of sets and tube derivatives

As a first reading, it is kindly advised for someone new to this field, to be focused on Section 4 and 5. The Sections 2 and 3 furnish the necessary mathematical framework that justify the different calculus tools described later on. These tools are usually built when the dynamic of the domains are smooth enough, which means that the velocity field is Lipschitz. However in most cases such as the tube deformations or other intrinsic objects, this regularity is not necessary. This is why we will concentrate on the weak evolution of sets.

3.1

Introduction

The weak convection of measurable sets has been introduced in [157], [155] to deal with the shape differential equation and its applications. We enlarge this approach to variational problems related to the tubes evolution ( this terminology is borrowed from [8] where it is defined thanks to mutational concepts). We focus on compactness results using different kinds of boundedness assumptions. This leads to existence results connected to the “parabolic version ” of the compact inclusion property from the bounded variation functions space into the integrable functions space, known as the Helly’s compactness theorem. One of the boundedness hypotheses is based on the use of the “Density Perimeter” properties [22], [21]. The so-called “Speed Method” has been developed in relation with the shape optimization of systems governed by Partial Differential Equations (PDE) in e.g., [147], [145], [146], [150]. In the strong version, we considered the flow mapping Tt (V) of a smooth vector field V ∈ C 0 ([0, ∞[, C k (Rd , Rd ) ∩ def

L∞ (Rd , Rd )). Then for any set Ω0 ⊂ Rd , the transported set Ωt (V) = Tt (V)(Ω0 ) is defined at all times. Hence the shape differential equation has been studied in [145],[150],[79] for shape functional governed by several linear classical boundary value problems (with use of the extractor estimate for the shape gradient [24]), and in [155] for the case of a non-linear viscous flow. In order to weaken the above regularity requirement, we can notice that the

49

50

Moving Shape Analysis def

characteristic function of the evolution domain, ξ t = ξΩt (V) , can be written as ξt = ξ0 ◦ [Tt (V)]−1 and is the solution of the following convection equation, ∂t ξ + ∇ξ · V = 0, ξ(t = 0) = ξΩ0 This problem admits solutions when the vector field satisfies V ∈ L 2 , div V ∈ L2 and some growth assumption on the positive part (div V) + . The incompressible situation was already introduced in [24], [155]. We shall add several results about the continuity and the compactness of the solutions.

3.2 3.2.1

Weak convection of characteristic functions The convection equation

Let V be a smooth vector field in Rd and Tt (V) its associated flow mapping. def

For any measurable set Ω ⊂ Rd , we consider the perturbed set Ωt (V) = Tt (V)(Ω). Its characteristic function satisfies the following strong transport property, (3.1) ξΩt = ξΩ ◦ [Tt (V)]−1 def

and the function ξ(t, x) = ξΩt (x) satifies, in a weak sense, the following convection equation, ∂t ξ + ∇ξ · V = 0, (0, τ ) × D (3.2) ξ(t = 0) = ξΩ , D REMARK 3.1 Notice that the term ∇ξ.V = div(ξV)−ξ div V does make sense as a distribution, whenever ξ ∈ L∞ ((0, τ ) × D), V ∈ L1 ((0, τ ) × D; Rd ) and div V ∈ L1 ((0, τ ) × D). When V is a smooth vector field, e.g., V ∈ L1 (0, τ ; W 1,∞ (D, Rd )) where D stands for a bounded hold-all domain1 containing the perturbed sets for any time t ≥ 0, the associated flow mapping is well defined for time t ∈ [0, τ ] and the identities (3.2) - (3.1) are equivalent. Many questions arise concerning the convection equation (3.2) when the vector field V is non-smooth. Nevertheless, to our mind, this is the correct 1 With

the following boundary condition associated to the vector field V, hV, ni = 0, on ∂D

or any viability condition in the case of a non-smooth boundary [51].

Weak evolution of sets and tube derivatives

51

approach for many shape problems such as shape evolution, shape identification or optimization, free boundary problems, fluid-structure interaction problems, computer vision or problems where the topology of the set Ω t may be changing. Actually, the problem (3.2) is equivalent to the following one: Z τZ Z ξ (∂t φ + hV, ∇φi + φ div V)dx dt + φ(0, x) dx = 0 (3.3) 0 D Ω ∀ φ ∈ C ∞ ([0, τ ] × D), φ(τ, .) = 0 We shall consider velocity fields such that V ∈ L1 (0, τ ; L2 (D; Rd )) and the divergence positive part2 (div V)+ ∈ L1 (0, τ ; L∞ (D)). In this case, using a Galerkin approximation and some energy estimate, we are able to derive an existence result of solutions with initial data given in H −1/2 (D). For the time being, any uniqueness result has been obtained for this smoothness level, but there exists an estimate (3.5) for such variational weak solutions. Actually, when the field V and its divergence are simply L 1 functions, the notion of weak solutions associated to the convection problems (3.2)-(3.3) does not make sense. In this case, the correct modeling tool for shape evolution is to introduce the product space of elements (ξ = ξ 2 , V) equipped with a parabolic BV like topology for which the constraint (3.3) defines a closed subset TΩ which contains the weak closure of smooth elements ¯ Rd ) (ξΩ ◦ [Tt (V)]−1 , V), V ∈ C ∞ ([0, τ ] × D;

This approach consists in handling characteristic functions ξ = ξ 2 which belongs to L1 (0, τ ; BV(D)) together with vector fields V ∈ L2 (0, τ ; L2 (D, Rd )) solution of problem (3.3). For a given element (ξ, V) ∈ T Ω , we consider the set of fields W such that (ξ, W) ∈ TΩ . It forms a closed convex set, noted Vξ . Hence, we can define the unique minimal norm energy element V ξ in the convex set Vξ . For a given tube3 ξ, the element Vξ is the unique (with minimal norm) vector field associated to ξ via the convection equation (3.2).

3.2.2

The Galerkin approximation

Let us consider the following evolution problem, ∂t u + ∇u · V = f, (0, τ ) × D u(t = 0) = u0 , D

(3.4)

We shall state the following existence result, 2 To

deal with the dual evolution problem (3.14), we shall assume (div V)− ∈ L1 (0, τ ; L∞ (D)).

3 We

say that ξ ∈ TΩ is a tube, i.e., a measurable non-cylindrical subset in (0, τ )×D, defined up to a zero measure set with bottom Ω.

52

Moving Shape Analysis

PROPOSITION 3.1 Let V ∈ L1 (0, τ ; L2 (D, R3 )) with div V ∈ L1 (0, τ ; L2 (D, R3 )), k(div V(t))+ kL∞ (D) ∈ L1 (0, τ ),

1

hV, ni = 0, in L1 (0, τ ; H − 2 (∂D))

and f ∈ L1 0, τ ; L2 (D) , u0 ∈ L2 (D). Then, equation (3.4) admits at least one weak solution u ∈ L∞ (0, τ ; L2 (D)) ∩ C 0 ([0, τ ]; W −1,1 (D)) together with ∂t u ∈ L1 (0, τ ; W −1,1 (D)) Moreover, there exists a constant M > 0 such that the following estimate holds true, kukL∞ (0,τ ;L2 (D)) ≤ M ku0 kL2 (D) + kf kL1 (0,τ ;L2 (D)) [ 1+

Z  + 

τ

0

k(div V(s))+ kL∞ (D,R3 ) + kf (s)kL2 (D,R3 )

REMARK 3.2

Z

τ

s

(k(div V(σ))+ kL∞ (D) +

i kf (σ)kL2 (D) )dσ ds (3.5) div V = (div V)+ − (div V)− .

PROOF Let us consider a dense family {ek }k≥1 in L2 (D) with ek ∈ Cc∞ (D), ∀ k ≥ 1. Consider the element m

u (t, x) =

m X

um i (t) ei (x)

i=1

solution of the following linear ordinary differential system: Z Z m m [∂t u (t, x) + hV(t, x), ∇u (t, x)i] ej (x) dx = f (t, x) ej (x) dx, D

D

1 ≤ j ≤ m (3.6)

def

Using the vector notation (U m )1≤i≤m = (um i )1≤i≤m , the last system is equivalent to the following one, ∂t U m (t) + M −1 · A(t) · U m (t) = F (t)

(3.7)

Weak evolution of sets and tube derivatives where [Mi,j ]1≤i,j≤m = [Ai,j ]1≤i,j≤m (t) =

Z

D

Z

53

ei (x) ej (x) dx D

hV(t, x), ∇ei (x)iej (x) dx

Classically, this linear differential system admits a global solution when V ∈ L1 (0, τ ; L2 (D, R3 )). We shall derive classical energy estimates, by multiplying the system (3.6) by um (t) and by using the following identity, Z Z 1 hV(t, x), ∇um (t, x)ium (t, x) dx = − |um (t, x)|2 div V(t, x) dx (3.8) 2 D D Then we get Z Z Z 1 1 ∂t |um (t)|2 ≤ |um (t)|2 (div V(t))+ + f (t) um (t) 2 D 2 D D

(3.9)

Integrating over (0, τ ), we obtain ku

m

(τ )k2L2 (D)

≤ ku

m

(0)k2L2 (D)

+

Z

τ 0

Z

+2

D

Z

|um (s, x)|2 (div V(s, x))+ dx ds

0

τ

Z

f (s, x) um (s, x) dx ds

(3.10)

D

Now we set ψ(s) = k(div V(s, .))+ kL∞ (D) We have kum (τ )k2L2 (D) ≤ kum (0)k2L2 (D) +

Z

τ

ψ(s)kum (s)k2L2 (D) ds + 2

0

Z

τ 0

Z

f (s) u(s) D

Using Cauchy-Schwartz inequality, we have Z 2 f (s) u(s) dx ≤ 2kf (s)kL2 (D) kum (s)kL2 (D) D

And using Young inequality, we get kum (s)kL2 (D) ≤

i 1h 1 + kum k2L2 (D) 2

from which we deduce h i kum (τ )k2L2 (D) ≤ kum (0)k2L2 (D) + kf kL1 (0,τ ;L2 (D)) Z τ + ψ(s) + kf (s)kL2 (D) kum (s)k2L2 (D) ds 0

54

Moving Shape Analysis

Using that u0 ∈ L2 (D), we get h i kum (τ )k2L2 (D) ≤ M ku0 k2L2 (D) + kf kL1 (0,τ ;L2 (D)) Z τ + ψ(s) + kf (s)kL2 (D) kum (s)k2L2 (D) ds 0

From Gronwall’s inequality, we derive: kum (τ )kL2 (D) ≤ M ku0 kL2 (D) + kf kL1 (0,τ ;L2 (D)) 1+ Z

τ

0

ψ(s) + kf (s)kL2 (D) exp

Z

τ

s

ψ(σ) + kf (σ)kL2 (D) dσ ds

(3.11)

Hence um remains bounded in L∞ (0, τ ; L2 (D)) and there exists an element u ∈ L∞ (0, τ ; L2 (D)) and a subsequence still denoted um which converges to u for the weak-* topology. In the limit case, the element u satisfies itself to the previous estimate and it can be checked that u is the solution of equation (3.2) in the distribution sense, i.e, Z τZ Z Z τZ − u ∂t φ + div(φ V) dx dt = φ(0) u0 dx + f φ dx dt 0

D

D

0

∀ φ ∈ H01 (0, τ ; L2 (D)) ∩ L2 (0, τ ; H01 (D)),

D

φ(τ ) = 0

(3.12)

REMARK 3.3 Notice that the identity (3.8) can also be written in the following form Z Z 1 div(V(t, x) um (t, x)) um (t, x) dx = + hum (t, x), um (t, x)i div V(t, x) dx 2 D D (3.13) Hence when (div V)+ is turned into (div V)− , the previous existence results apply for the following evolution problem, ∂t p(t) + div(p(t) V(t)) = g, (0, τ ) × D (3.14) p(t = 0) = p0 , D

3.3

Tube evolution in the context of optimization problems

In the previous section, we have stated an existence result for weak solutions of the convection equation (3.2). Unfortunately, this result does not guaran-

Weak evolution of sets and tube derivatives

55

tee neither uniqueness nor the stability of characteristic functions. A possible strategy is to assume different regularity levels for the velocity field in order to recover uniqueness and stability properties for initial characteristic functions [5]. This makes the mathematical setting even more complex. Here, we chose to adopt a different point of view inspired by the optimization problems framework. Indeed, our final goal is to apply the weak set evolution setting to the control problem arising in various fields such as free boundary problems or image processing. The usual situation can be described as follows: let us consider a given smooth enough functional J(ξ, V). We would like to solve the following optimization problem inf

(ξ,V)∈TΩL ip

where

def

TΩL ip =

J(ξ, V)

(3.15)

(χΩ ◦ T−1 t (V), V)| V ∈ Uad

stands for the space of smooth admissible couples. The space U ad is a space of smooth velocities4 . In most situations, such a problem does not admit solutions and we need to add some regularization terms to ensure its solvability. Consequently, we shall introduce different penalization terms which furnish compactness properties of the minimizing sequences inside an ad-hoc weak topology involving bounded variation constraints. Then, the new problem writes inf

(ξ,V)∈TΩL ip

J(ξ, V) + F (ξ, V)

(3.16)

The penalization term F (ξ, V) can be chosen using several approaches: • We can first consider the time-space perimeter of the lateral boundary Σ of the tube, developed in [155]. This approach easily draws part of the variational properties associated to the bounded variation functions space framework. In particular, it uses the compactness properties of tubes family with bounded perimeters in Rd+1 . Nevertheless, this method leads to heavy variational analysis developments. • We can rather consider the time integral of the spatial perimeter of the moving domain which builds the tube, as introduced in [157]. We 4 def

Uad =

n

` ´ V ∈ C 0 [0, τ ]; V0k (D) ,

with k ≥ 0 and where

def

V0k (D) = def

with L ipk (V) =

X

|α|=k

n

∃c > 0,

L ipk (V(t)) ≤ c, ∀ t ∈ [0, τ ]

o ` ´ ¯ d , V · n∂D = 0, on D V ∈ C k (D) def

L ip(Dα V) for k ≥ 1 and L ip(V) = sup

y6=x

o

|V(y) − V(x)| . |y − x|

56

Moving Shape Analysis shall extend these results to the case of vector fields living in L 2 ((0, τ ) × D; Rd ). In this case, only existence results for solutions of the convection equation can be handled and the uniqueness property is lost. The main idea, here, is to consider the set of pairs (ξ, V) solving the convection equation with a given initial set Ω inside D.

3.3.1

Penalization using the generalized perimeter’s time integral

In this section, we shall consider the optimization problem (3.16) with a particular penalization term of the following type, Z τ Z τh i F (ξ, V) = α kξ(t)kBV(D) dt+β kV(t)k2L2 (D;Rd ) + k div V(t)k2L2 (D) dt 0

0

(3.17) This penalization term, involving the L1 norm of the generalized perimeter in D associated to the set characterized by ξ, will furnish the required compactness properties for the minimizing sequences of problem (3.16) in order to get an existence result in some ad-hoc functional spaces endowed with specific weak topologies. REMARK 3.4 However, with the above penalization, we can not guarantee the generated solution set to be an open set. This drawback will be solved in the section 3.3.3.

To be more precise, given a measurable subset Ω ⊂ D, we consider the following sets equipped with their respective weak topologies: DEFINITION 3.1 A = ξ 2 = ξ, ξ ∈ L∞ (0, τ ; L∞ (D)),

∇ξ ∈ L1 (0, τ ; M 1 (D, Rd ))

where M 1 (D; Rd ) stands for the sets of Radon5 measures on D with values in Rd . DEFINITION 3.2 The sequence ξn → ξ in A if and only if the following conditions hold: Z τZ (ξn − ξ) φ → 0, ∀ φ ∈ L1 ((0, τ ) × D), 0 D Z τZ (∇ξn − ∇ξ) · g → 0, ∀ g ∈ L1 (0, τ ; Cc0 (D, Rd )) 0

D

5 This space can be defined as the dual space associated to the space (C 0 (D; Rd ), k.k ∞ ). L c A useful property says that the bounded sets inside L1 (D, Rd ) are relatively compact inside M 1 for the weak-* σ(M 1 , Cc0 ) topology.

Weak evolution of sets and tube derivatives DEFINITION 3.3 B = V ∈ L2 (0, τ ; L2 (D, Rd )),

57

div V ∈ L2 (0, τ ; L2 (D))

Let us consider the following set:

TΩ = {(ξ, V) ∈ A × B, with ∂t ξ + ∇ξ · V = 0,

ξ(t = 0) = χΩ }

(3.18)

THEOREM 3.1 The set TΩ is closed in A × B. PROOF Let (ξn , Vn ) be a sequence inside TΩ converging towards (ξ, V). From the Banach-Steinhaus theorem,we get the boundedness of the following L1 (0, τ ; BV(D)) norm: Z τ ||∇ξn ||M 1 (D) dt ≤ M 0

Using the next Proposition6 3.2, we deduce that there exists a subsequence still denoted ξn which converges in L1 for the strong topology. Then, the limit element satisfies ξ 2 = ξ. Finally, in the limit case, the weak formulation of problem (3.3), Z

τ 0

Z

D

ξn (∂t φ + hVn , ∇φi − φ div Vn ) +

Z

φ(0, x) = 0, Ω

∀ φ ∈ C ∞ ([0, τ ] × D),

φ(τ, .) = 0

is satisfied by (ξ, V) since Vn and div Vn weakly converge in L2 . Hence, the limit pair (ξ, V) turns out to be an element of TΩ .

3.3.2

Parabolic version of Helly’s compactness theorem

Let us define the space, def

W =

f ∈ L1 0, τ ; BV(D) ,

∂t f ∈ L2 0, τ ; H −2 (D)

The following compactness result holds, 6 with

and

fn = ξn ∈ L1 (0, τ ; BV(D)) ` ´ 0 fn0 = ξn = − div(ξn Vn ) + ξn div Vn ∈ L2 0, τ ; H −2 (D) .

(3.19)

58

Moving Shape Analysis

PROPOSITION 3.2 Uniformly bounded sets in W are sequentially relatively compact in L1 0, τ ; L1 (D) PROOF

For d ≤ 3 the following continuous injection holds, L1 (D) ,→ H −2 (D).

Furthermore using Helly’s theorem (Theorem 1.19 in [77] page 17) the following continous, compact injection holds, BV(D) ,→c L1 (D). Then, we have the following embedding chain, BV(D) ,→c L1 (D) ,→ H −2 (D) We adapt the proof of the compactness result (Theorem 5.1 page 58 in [97]) to the non-reflexive situation. Let us consider {fn }n≥0 a uniformly bounded sequence in W. Using the result proven in the next Lemma 3.1, we deduce for f ∈ L1 0, τ ; BV(D) , ∀ n > 0, there exists a constant dn > 0 such that kf k

L1 0,τ ;L1 (D)

≤ nkf k

L1 0,τ ;BV(D) def

+ dn kf k

L1 0,τ ;H −2 (D)

We apply this inequality with f = fn,m = fn − fm , for m > n. Let us fix ε > 0, since there exists M > 0 such that

kfn,m kL1 (0,τ ;BV(D)) ≤ M We can choose n such that n M ≤ kfn,m kL1 (0,τ ;L1 (D)) ≤

ε 2

and we get

ε + dn kfn,m kL1 (0,τ ;H −2 (D)) 2

At this stage, if we establish the strong convergence towards zero of f n,m in the space L1 (0, τ ; H −2 (D)), then the proof is complete. As L1 (D) ⊂ H −2 (D) (for d ≤ 3), using Lemma 1.2 page 7 in [97], we have fn,m ∈ W 1,1 (0, τ ; H −2 (D)) ,→ C 0 ([0, τ ], H −2 (D)) Then, we have the following uniform estimate kfn,m (t0 ))kH −2 (D) ≤ M,

∀ t0 ∈ [0, τ ]

Using the Lebesgue dominated convergence theorem,it will be sufficient to prove the pointwise convergence of fn,m (t0 ) strongly to zero in H −2 (D). To

Weak evolution of sets and tube derivatives

59

this end, we use a technique introduced by R.Temam as reported in [97] (footnote of the proof of Theorem 5.1 page 59, see also [139] page 273). It is easy to show that the following expression holds true, fn,m (t0 ) = an,m + bn,m with an,m = 1/s

Z

t0 +s

t0

fn,m (t) dt, bn,m = −

1 s

Z

t0 +s t0

0 (s − t) fn,m (t) dt

Let us choose s such that dn kbn,m kH −2 (D) ≤

Z

t0 +s t0

≤ dn s1/2 ε ≤ 4

0 kfn,m (t)kH −2 (D) dt ≤

Z

t0 +s t0

0 kfn,m (t)k2H −2 (D)

1/2

Finally, we have the estimate kan,m kBV(D) ≤

1 M kfn,m kL1 (0,τ ;BV(D)) ≤ s s

Again using Helly’s compactness result, we deduce that a n converges strongly in H −2 (D) towards an element a ∈ H −2 (D) . This limit is actually zero R t +s since by the same arguments we can show that 1/s t00 fn (t) dt converges in H −2 (D) for the strong topology. Then it is a Cauchy sequence and we deduce that a = 0. Then, there exists N ≤ 0, such that for m > n ≥ N , dn kan,m kH −2 (D) ≤

ε 4

We conclude that ∀ ε > 0, ∃ N ≥ 0, ∀ m > n ≥ N, ε ε ε kfn,m kL1 (0,τ ;L1 (D)) ≤ + + ≤ ε 2 4 4 Then the sequence {fn }n≥0 is a Cauchy sequence in the Banach space L1 (0, τ ; L1 (D)). Then it is a convergent sequence for the strong topology.

LEMMA 3.1 ∀ η > 0, there exists a constant cη with ∀ φ ∈ BV(D), kφkL1 (D) ≤ ηkφkBV(D) + cη kφkH −2 (D)

60

Moving Shape Analysis

PROOF Let us assume that the above estimate does not hold. Then, ∀ η > 0, there exists φn ∈ BV(D) and cn → ∞ such that kφn kL1 (D) ≥ ηkφn kBV(D) + cn kφn kH −2 (D) We introduce ψn = φn /kφn kBV(D) , and we derive: kψn kL1 (D) ≥ η + cn kψn kH −2 (D) ≥ η But also kψn kL1 (D) ≤ c kψn kBV(D) = c , for some constant c. Then, we have kψn kH −2 (D) → 0 But as kψn kBV(D) = 1, there exists a subsequence strongly convergent in L1 (D) ⊂ H −2 (D), which turns to be strongly convergent to zero. This is in contradiction with the fact that kψn kL1 (D)) ≥ η.

3.3.3

Generation of clean open tubes

Clean open tubes DEFINITION 3.4 for a.e. t > 0, the set

˜ in ]0, τ [×D such that A clean open tube is a set Q n ˜ t = x ∈ D, Ω

˜ (t, x) ∈ Q

o

(3.20)

is an open set in D satisfying for a.e. t ∈ (0, τ ) the following cleanness property: ˜ t ) = 0, Ω ˜ t = int(cl(Ω ˜ t )). meas(∂ Ω

(3.21)

˜ t is an open set in D for a.e. t ∈ (0, τ ), the set REMARK 3.5 If the set Ω ˜ Q is not necessarily an open [ subset in ]0, τ [×D. Nethertheless when the field V is smooth, the tube {t} × Tt (V)(Ω0 ) is open (resp. open and clean 0
open) when the initial set Ω0 is open (resp. open and clean open) in D. REMARK 3.6

Two tubes Q, Q0 are said to be equivalent if χQ = χQ0 , in L2 (0, τ ; L2 (D))

This means that Ωt and Ω0t are the same up to zero-measure sets. LEMMA 3.2 ˜ Let Q be a measurable set in ]0, τ [×D. If there exists a clean open tube Q such that χQ = χQ˜

Weak evolution of sets and tube derivatives

61

Then, this clean tube is unique. ˜ and Q ˜ 0. PROOF Assume that there exists two equivalent clean tubes Q 0 ˜ ˜ Then a.e. t ∈ (0, τ ), we have Ωt = Ωt up to a measurable set Et with meas(Et ) = 0. As these two open sets satisfy the cleanness property (3.21), they must be equals. DEFINITION 3.5 Let Ω0 be a clean open set in D and Q a clean open def tube in ]0, τ [×D with ξ = χQ ∈ C 0 ([0, τ ]; H −1/2 (D)). If there exists a divergence free field V ∈ B such that: ∂t ξ + ∇ξ · V = 0, ξ(t = 0) = χΩ0

(3.22)

we say that V builds the tube and we note Q = QV . DEFINITION 3.6 tube Q,

We define the set of fields that build the clean open VQ = {V ∈ B,

QV = Q}

(3.23)

PROPOSITION 3.3 If the set VQ is non-empty, then the following minimization problem, def

eD (QV ) = min kVkL2 (0,τ ;L2 (D;Rd )) V∈VQ

(3.24)

admits a unique solution VQ ∈ B ∩ L2 (0, τ ; L2 (D; Rd )). PROOF It is easy to prove that if the set VQ is non-empty, then it is closed and convex in B. Hence, we use the projection theorem in the Hilbert space L2 (0, τ ; L2 (D; Rd )). Properties of the density perimeter In this section, we introduce the notion of density perimeter [22], [21] for a closed set A inside D. This notion has been introduced in order to have a control on the boundary measure of the limit set associated to a convergent family of open sets. This allow us to build a penalization term that will generate open set solutions to the optimization problem (3.16). DEFINITION 3.7 set,

The ε-dilatation of the set A is given by the following def

Aε =

[

x∈A

B(x, ε)

(3.25)

62

Moving Shape Analysis

DEFINITION 3.8 Let γ > 0, the density perimeter of the set A is given by the following expression, meas(Aε ) Pγ (A) = sup (3.26) 2ε ε∈(0,γ) Property 3.3.1 ([22]) The mapping Ω −→ Pγ (∂Ω) is lower semi-continuous for the Hausdorff7 complementary topology H c . Property 3.3.2 ([22]) If the boundary set ∂Ω is such that Pγ (∂Ω) < ∞ then meas(∂Ω) = 0 and Ω \ ∂Ω is open in D Property 3.3.3 ([22]) Let us consider a family {Ωn }n≥0 of open sets in D. Let us suppose that this family converges to some subset Ω ⊂ D for the H c topology and that there exists M > 0 such that Pγ (∂Ωn ) ≤ M . Then the convergence holds for the char-topology8 . Penalization using the parabolic generalized density perimeter In this section, we shall describe a particular type of clean tubes construction for which the set VQ is non-empty. Actually we shall combine the compactness properties of the generalized perimeter with the openness control properties of the density perimeter. For any smooth free divergence vector field, V ∈ C 0 ([0, τ ]; W01,∞ (D, Rd )), we consider the following minimization problem, Θγ (V, Ω0 ) = 7 The

min

µ∈Mγ (V,Ω0 )

kµk ˙ 2L2 (0,τ )

(3.27)

Hausdorff complementary topology is given by the metric, dH c (Ω1 , Ω2 ) = dH d (Ωc1 , Ωc2 )

where Ω1 , Ω2 are open subsets of D and Ωc1 , Ωc2 their associated complementary sets in D. Furthermore dH d stands for the Hausdorff metric defined by dH d (F1 , F2 ) = max(ρ(F1 , F2 ), ρ(F2 , F1 )) for any closed sets F1 , F2 in D and where ρ(X, Y ) = sup inf kx − yk x∈X y∈Y

8 The

char-topology is defined on the family of measurable subsets of Rd by the L2 -metric, Z dchar (A1 , A2 ) = |χA1 − χA2 | Rd

Weak evolution of sets and tube derivatives

63

where Mγ (V, Ω0 ) =

  

µ ∈ H 1 (0, τ ),

 Pγ (∂Ωt (V)) ≤ µ(t) a.e. t ∈ (0, τ ),  µ(t = 0) ≤ (1 + γ) Pγ (∂Ω0 )

When this set is empty, we choose to set Θγ (V, Ω0 ) = +∞. REMARK 3.7



(3.28)

When the mapping p : (0, τ ) −→ R

t 7→ Pγ (∂Ωt (V))

is an element of H 1 (0, τ ) , i.e., p ∈ Mγ (V, Ω0 ), we have Θ(V, Ω0 ) < kpk ˙ 2L2 (0,τ ) as the minimizer will escape to possible variations of the function p. PROPOSITION 3.4 For any V ∈ C 0 ([0, τ ]; W01,∞ (D, Rd )) with free divergence, the following inequality holds, √ Pγ (∂Ωt (V)) ≤ 2 Pγ (∂Ω0 ) + τ Θ(V, Ω0 )1/2 (3.29) PROOF

By definition, Pγ (∂Ωt (V)) ≤ µ(t) ≤ 2 Pγ (∂Ω0 ) +

Z

τ

µ(t) ˙ dt 0

Then, using Cauchy-Schwartz inequality, we get √ Pγ (∂Ωt (Vn )) ≤ µ(t) ≤ 2 Pγ (∂Ω0 ) + τ

Z

τ 2

0

|µ(t)| ˙ dt

1/2

Minimizing over µ ∈ Mγ (V, Ω0 ) furnishes the correct estimation. PROPOSITION 3.5 Let Vn ∈ C 0 ([0, τ ]; W01,∞ (D, Rd )) with the following convergence property, Vn −→ V in L2 (0, τ ; L2 (D, Rd ))

(3.30)

and the uniform boundedness, ∃M > 0, Θ(Vn , Ω0 ) ≤ M

(3.31)

64

Moving Shape Analysis

Then Θ(V, Ω0 ) ≤ lim inf Θ(Vn , Ω0 )

(3.32)

n→∞

PROOF Let µn be the unique minimizer in H 1 (0, τ ) associated with Θ(Vn , Ω0 ). Using the boundedness assumption and the estimation (3.29), we get √ Pγ (∂Ωt (Vn )) ≤ 2Pγ (∂Ω0 ) + τ M 1/2 | {z } C

We deduce that there exists a subsequence, still denoted µ n , which weakly converges to an element µ ∈ H 1 (0, τ ). Furthermore, this convergence holds in L2 (0, τ ) for the strong topology, and as a consequence a.e. t ∈ (0.τ ). By definition, we have Pγ (∂Ωt (Vn )) ≤ µn

a.e. t ∈ (0, τ )

Using the l-s-c property of the density perimeter (Theorem 1.9 page 7 in [77]), we have Pγ (∂Ωt (V)) ≤ lim inf Pγ (∂Ωt (Vn )) ≤ µ(t) n→∞

a.e. t ∈ (0, τ )

But the square of the norm being also w-l-s-c, we have Z τ Z τ 2 |µ˙n (t)|2 dt |µ(t)| ˙ dt ≤ lim inf n→∞

0

This leads to Θ(V, Ω0 ) ≤

Z

τ 0

0

2 |µ(t)| ˙ dt ≤ lim inf Θ(Vn , Ω0 ) n→∞

PROPOSITION 3.6 Let Vn ∈ C 0 ([0, τ ]; W01,∞ (D, Rd )) with convergence property (3.30) and bound estimate (3.31). We assume that Ω0 is an open subset in D verifying Ω0 = int(cl(Ω0 )) Then, there exists a clean open tube [ ˜= ˜t Q {t} × Ω 0
such that, for a.e. t ∈ (0, τ ) in L2 (D), χΩt (Vn ) → χΩ˜ t , ˜ t , in H c -topology Ωt (Vn ) → Ω

Weak evolution of sets and tube derivatives

65

˜ t is the single open set such that cleanness condition (3.21) is Moreover Ω satisfied and whose characteristic function χΩ˜ t solves the convection problem: ∂t ξ + ∇ξ · V(t) = 0, ξ(t = 0) = χΩ0 ˜ is the unique clean tube equivalent to the limit tube QV This means that Q built by V. PROOF We have χQVn → ξQV in L2 ((0, τ ) × D). Then, for almost every t ∈ (0, τ ), we have χΩt (Vn ) → χΩt (V) in L2 (D). At each time t ∈ (0, τ ), there exists a subsequence (depending on t) which converges, for the H c -topology, to an open set ωt . Meanwhile, we also have Ωt (Vn ) → Ωt (V) in the char-topology for a.e. t ∈ (0, τ ). Then, χωt = χΩt (V) for a.e. t ∈ (0, τ ). From the boundedness of Pγ (Ωt (Vn )) and using property (3.3.2), we derive that ωt (V) is an open set in D and meas(∂ωt (V)) = 0. ˜ t = cl(ωt ) − ∂ωt . Hence, we set Ω DEFINITION 3.9 For smooth vector fields V, we define the parabolic generalized density perimeter of the moving domain Ωt (V) over the time period (0, τ ), Z τ PD (Ωt (V)) dt + γ Θγ (V, Ω0 ) (3.33) pγ (V, Ω0 ) = 0

where PD (Ωt (V)) stands for the generalized perimeter9 of Ωt (V). THEOREM 3.2 Let Vn ∈ C 0 ([0, τ ]; W01,∞ (D, Rd )) which weakly 2 2 d L (0, τ ; L (D, R )) to V with the boundedness condition:

converges

pγ (Vn , Ω0 ) ≤ M

in

(3.34)

˜ V built by V with the following converThen there exists a clean open tube Q gence: for a.e. t ∈ (0, τ ), in L2 (D), ξVn (t) → ξV (t), ˜ t , in H c -topology Ωt (Vn ) → Ω 9 The

generalized perimeter of a measurable set A in D is given by Z PD (A) = k∇χA kM 1 (D,Rd ) = sup − div ϕ dx A ϕ ∈ Cc∞ (D, Rd ), kϕk∞ ≤ 1

66

Moving Shape Analysis

Furthermore, pγ (V) ≤ lim inf pγ (Vn ) n→∞

Finally, if we define the following penalization term Z τh i F (ξ, V) = α pγ (V, Ω0 )+β kV(t)k2L2 (D;Rd ) + k div V(t)k2L2 (D) dt (3.35) 0

we can obtain an existence result for the optimization problem (3.16) with the property that the associated tube solution is clean open if Ω 0 is open.

3.4

Tube derivative concepts

In this section, we introduce alternative tools to handle the sensivity analysis of tube functionals. The Eulerian setting has been presented in Chapter 2 and it is based on the introduction of a velocity field V ∈ U ad which is smooth enough 10 for its Lagrangian flow to be defined in a classical manner 11 . Hence we have a parametrization of the moving set based on the couple V, T(V) . Introducing the transverse field which builds the flow mapping the reference moving domain into the perturbed one, allowed us to differentiate the following Eulerian integral Z τZ F (V)(x, t) dx dt 0

Ωt (V)

with respect to V. A second choice consists in parametrizing the moving set thanks to the couple V, χ(V) where χ(V) is the characteristic function of the moving set Ω t (V) and is the solution of the transport equation (3.2). In this setting, it is possible to perform a sensitivity analysis of the following integral, Z τZ χ(x, t) F (χ)(x, t) dx dt. 0

10 def

Uad =

n

` ´ V ∈ C 0 [0, τ ]; V0k (D) ,

with k ≥ 0 and where

def

V0k (D) = def

with L ipk (V) = 11

D

X

|α|=k

n

∃c > 0,

L ipk (V(t)) ≤ c, ∀ t ∈ [0, τ ]

o ` ´ ¯ d , V · n∂D = 0, on D V ∈ C k (D) def

L ip(Dα V) for k ≥ 1 and L ip(V) = sup

y6=x

o

|V(y) − V(x)| . |y − x|

` ` ´ ´ ` ` ´ ´ ¯ d ∩ C 0 [0, τ ]; W k+1,∞ (D) d T(V) ∈ C 1 [0, τ ]; C k (D)

Weak evolution of sets and tube derivatives

67

A third choice consists in working directly on the tube inside the time-space def Euclidian space Rd+1 = Rt × Rx . Hence it is possible to work in the classical ˜ def Eulerian setting with extended velocity field V = ∗ (0, V). This allows dealing with tube integrals of the following form, Z Z ˜ Q+ ˜ Σ F (V)d f (V)d ˜ Q(V)

˜ Σ(V)

where Σ stands for the boundary of the tube Q. A motivation for this setting is the introduction of perimeter constraints inside optimization problems for tube evolution. As recalled in section 3.3, we can either define the time integral of the space perimeter, Z Z τ

dΓ dt

0

or the time-space perimeter,

Ω(t)

Z

dΣ Σ

The boundedness of the latter characterizes the BV compactness of the tube Q inside Rd+1 . Furthermore it has been proved in section 3.3 that the boundedness of the first perimeter furnishes the same compactness result. These perimeter constraints are associated in a variational way to the notion of surface tension which is highly required while studying free boundary problems as the one described in [153] concerning the hydrodynamical model of water waves.

3.4.1

Characteristics versus Eulerian flow setting

Let (ξ, V) ∈ TΩ and a vector field W such that for all s, |s| ≤ s1 there exists ξ s with (ξ s , V + sW) ∈ TΩ . First, let us consider heuristically the term (if it exists) ξ˙ = ∂s ξ s |s=0 as a measure over (0, τ )×D. This measure should solve the evolution problem (3.4) with a measure term as right hand side: ˙ = 0) = 0 ∂t ξ˙ + ∇ξ˙ · V = −∇ξ · W, ξ(t

(3.36)

We consider a non-cylindrical functional depending on the vector field V, Z τZ Z τZ Z j(V) = FΩt (V) + fΓt (V) + gΩτ (V) (3.37) 0

Ωt (V)

0

Γt (V)

Ωτ

We first consider the case where the functional only involves a distributed def term defined on Q(V), i.e;, fΓt (V) = 0 and gΩτ (V) = 0. We set ξ = χΩt (V) , and we can write Z τZ ξ Fξ

j(ξ, V) =

0

D

68

Moving Shape Analysis

where FΩt (V) = Fξ |Ωt (V) and Fξ is defined in (0, τ ) × D. We can define the derivative of the functional j(V) with respect to V in the direction W as follows, def j 0 (V, W) = ∂s j(ξ s , V + s W)|s=0 (3.38) Here, it leads to 0

j (V, W) =

Z

τ 0

def

Z h

ξ˙ Fξ + ξ F 0

D

i

(3.39)

where F 0 = ∂s F (ξ s )|s=0 . We introduce the following non-cylindrical adjoint state λ: −∂t λ − div(λ V) = Fξ , λ(τ ) = 0

(3.40)

For the sake of simplicity, we assume that F 0 = 0, and we get Z τZ j 0 (V, W) = −ξ˙ [∂t λ + div(λ V)] 0

D

Using integration by parts and equation (3.36), we have Z τZ Z τZ i h 0 ˙ ˙ λ ∇ξ · W j (V, W) = λ ∂t ξ + ∇ ξ · V = − 0

0

D

D

Integrating again by parts and using Stokes formula, we get Z τZ ξ div(λ W), j 0 (V, W) = 0 D Z τZ λ hW, nt i = 0

(3.41)

Γt (V)

In [154], [59], we have introduced the transverse field Z such that "Z Z # Z τZ τ ∂s F = F hZ, nt i dΓt dt 0 Ωt (V+sW) 0 ∂Ωt (V)

(3.42)

s=0

We deduce that, as a measure on ]0, τ [×D, the element ξ˙ satisfies the following identity, Z τZ Z τZ ξ˙ F = F hZ, nt i dΓt dt, 0 ∂Ωt (V) 0 D Z = F hZ, nt i(1 + v 2 )−1/2 dΣ (3.43) Σ(V)

def

where v = hV, nt i. We introduce γΣ ∈ L(C 0 (]0, τ [×D), C 0 (Σ)), the trace operator on the lateral

Weak evolution of sets and tube derivatives

69

∗ boundary Σ of the tube Q(V) and γΣ ∈ L(M(Σ), M(]0, τ [×D)) its adjoint operator. Hence, we deduce ! hZ, n i t ∗ ξ˙ = γΣ · p (3.44) (1 + v 2 )

Comparing equation (3.41) and (3.43), we derive the following result, Z τZ Z τZ F hZ, nt i = λ hW, nt i (3.45) 0

Γt (V)

0

Γt (V)

In the case where we keep the final time term gΩτ , we set, gΩτ = g|Ωτ where g is defined on D. Introducing the adjoint problem, ¯ − div(λ ¯ V) = Fξ , λ(τ ¯ )=g −∂t λ We have the following duality identity, Z Z τZ g hZ(τ ), nτ i dΓτ (V) = Γτ (V)

0

which leads to the functional derivative, Z τZ 0 j (V, W) = 0

3.4.2

Γt (V)

Γt (V)

¯ hW(t), nt i dΓt dt λ

(3.46)

(3.47)

¯ hW, nt i λ

Tangential calculus for tubes

In the last section, we have proven that the directional derivative of distributed non-cylindrical functionals involves the normal trace def

zt = hZ, nt i, on Σ(V)

(3.48)

Let us recall the following characterization of z t , LEMMA 3.3 [58],[59]

zt ◦ Tt (V) = Z t Z t hW(σ), nσ i ◦ Tσ (V) exp hD V(σ) · nr , nr i ◦ Tr (V) dr dσ, 0

σ

on Γ0

70

Moving Shape Analysis

REMARK 3.8

When the vector field V is chosen in the canonical form V = V ◦ pt

the following identity holds, D V · nt = 0,

on Γt (V)

and we get zt ◦ Tt (V) =

DEFINITION 3.10

Z

t 0

hW(σ), nσ i ◦ Tσ (V) dσ, on Γ0

Let us consider the noncylindrical gradient operator, h∇Σ λ, ντ i = ∂t λ + ∇λ · V

def

(3.49)

where the vector ντ = (1, V),

(3.50)

on Σ.

REMARK 3.9 The vector ντ is tangent to the lateral surface Σ since the outgoing normal field is given by 1 def ν = √ (−v, nt ) 1 + v2

(3.51)

def

with v = hV, nt i. The duality identity (3.45) writes Z Z zt hW, nt i F √ λ√ dΣ = dΣ 2 1+v 1 + v2 Σ Σ

(3.52)

Using the operator ∇Σ and the equation (3.40) satisfied by λ , we get Z Z zt hW, nt i [∇Σ λ · ντ + λ div V] √ dΣ = − λ√ dΣ 2 1+v 1 + v2 Σ Σ as λ zt = 0, on ∂Σ Performing an integration by parts on the manifold Σ, we get the following identity, Z zt zt −λ divΣ √ ντ + λ div V √ dΣ 1 + v2 1 + v2 Σ Z hW, nt i =− λ√ dΣ (3.53) 1 + v2 Σ

Weak evolution of sets and tube derivatives

71

So we can prove the following proposition, PROPOSITION 3.7 p zt 2 − div V(t) zt + 1 + v divΣ √ ντ = hW(t), nt i 1 + v2

(3.54)

PROPOSITION 3.8 The normal transverse field zt satisfies the following identity, 1 [v (∂nt e + h∂t (e V), nt − v∂t ei)] 1 + v2 1 1 − hD(e V) · nt , nt i = √ hW, nt i (3.55) 1 + v2 1 + v2

∂t e + div(e V) − (div V) e +

zt where e = √ . 1 + v2 PROOF

We shall set E = ∗ (E1 , E2 ) = e ∗ (1, V)

Hence, using classical differential operators defined on (0, τ ) × D ˜ − hDQ E ˜ · ν, νi divΣ E = divQ E

(3.56)

def

˜ stands for an arbitrary extension of E = ∗ (E1 , E2 ) in (0, τ ) × D and where E divQ E = ∂t E1 + div E2 DQ E = Recalling that ν =

∗ √ 1 (−v, nt ), 1+v 2

∂t E 1 D E 1 ∂t E 2 D E 2

we get

1 hDQ ∗ (E1 , E2 ) · ∗ (−v, nt ), ∗ (−v, nt )i 1 + v2 1 = ∂t E1 + div E2 − h∗ (−v∂t E1 + ∂nt E1 , −v∂t E2 1 + v2 + D E2 · nt ) , ∗ (−v, nt )i 1 2 = ∂t E1 + div E2 − v ∂t E1 − v∂nt E1 − v ∂t E2 · nt 1 + v2 +hD E2 · nt , nt i]

divΣ E = divQ ∗ (E1 , E2 ) −

72

Moving Shape Analysis

3.4.3

Classical shape analysis for tubes

Let us consider a tube Q with lateral boundary Σ, and an horizontal perturbation leading to the perturbed tube Qs given in the following form: ˜ def • We consider an horizontal field Z = ∗ (0, Z) defined in Rd+1 = Rt × Rdx and its associated flow mapping, ˜ : Rt × Rd −→ Rt × Rd Ts (Z) x x (t, x) 7→ (t, Ts (Z)(x) ) ˜ Σ the tangential trace of the vector field Z ˜ on the lateral We designate by Z surface Σ. From the expression of the normal field ν, we easily derive ˜Σ = Z

1 ∗ (v zt , Z − zt nt ) 1 + v2

(3.57)

Here ν is to be understood as any extension of the normal field to a neighbourhood of the lateral boundary of the tube. For example, we can choose nt = ∇bΩt where bΩt stands for the oriented distance to the section of the tube at time t and V can be understood as V ◦ pt where pt is the Rd projection on Γt = ∂Ωt . Mean curvature of the lateral boundary of tubes

PROPOSITION 3.9 Assume that the field V verifies for each t ∈ (0, τ ): V(t) = V(t) ◦ pt Then, on the boundary Γt (V), we can define the time-space mean curvature of the lateral boundary Σ, def

HΣ = divQ ν

1 Ht = − √ [h∂t V, nt i − h[∇Γt hV, nt i] , V|Γt i] + √ 2 3 ( 1+v ) 1 + v2 (3.58)

PROOF

We choose v = hV, ∇bΩt (V) i. It is obvious that v 1 ∂t √ = √ ∂t v 2 1+v ( 1 + v 2 )3

But ∂t v = h∂t V, ∇bi + h∂t ∇b, Vi

Weak evolution of sets and tube derivatives

73

From the definition of the oriented distance, we got ∂t bΩt (V) = −hV, nt i ◦ pt

(3.59)

and ∂t ∇bΩt (V) = − [∇Γt hV, nt i] ◦ pt Then we get ∂t v = h∂t V, nt i − h[∇Γt hV, nt i] , V|Γt i On the other hand, nt 1 1 div √ = −h∇ √ , nt i + √ div nt 1 + v2 1 + v2 1 + v2 1 Ht =− √ hε(V) · nt , nt i + √ ( 1 + v 2 )3 1 + v2 where ε(V) = 21 (D V + ∗ D V) is the deformation tensor. We consider the situation in which the field V verifies the following property: V = V ◦ pt

(3.60)

with pt = I −bΩt (V) ∇bΩt (V) from which we deduce ∂t pt = −∂t bΩt (V) ∇bΩt (V) − bΩt (V) ∇(∂t bΩt (V) ) The restriction to the boundary Γt leads to the distance bΩt (V) = 0, so the expression simplifies as follows, ∂t pt |Γt = hV, nt i nt and D V · nt = 0,

on Γt (V)

DEFINITION 3.11 Consider f (Σ) a function defined on the lateral boundary Σ. We call the Eulerian material derivative of f (Σ) in the direction ˜ the following quantity, Z h i ˜ ˜ def (3.61) f˙Σ (Z) = ∂s f (Σs ) ◦ Ts (Z) s=0

and its associated partial Eulerian derivative is given by the following expression, ˜ def ˜ − h∇Σ f (σ), Z ˜ Σi fΣ0 (Z) = f˙Σ (Z) (3.62)

74

Moving Shape Analysis Using the expression of the tangential operator ∇ Σ , we get ˜ = f˙Σ (Z) ˜ − v zt ∂t f − h∇Γ f, Z − zt nt i fΣ0 (Z) 1 + v2 1 + v2

REMARK 3.10 When f (Σ) is the restriction to the lateral boundary Σ of a function F defined over Rd+1 , we get ˜ = F 0 (Z)| ˜ Σ + ∂ν F hZ, ˜ νi f 0 (Z) Σ

Q

˜ νi = ∂ν F hZ,

˜ = 0 since F does not depend on Σ. as FQ0 (Z) Shape derivative on the lateral boundary of tubes ¯ First, we assume that Let us now consider a function F ∈ C 1 ([0, τ ] × D). F is zero in the neighbourhood of t = τ so that the following derivative of the lateral boundary integral could be considered as the derivative of integral on the total boundary of the tube (as it will generate no term on the top t = τ of the tube). We set Σs = {(t, Tt (V + sW)(x)),

x ∈ ∂Ω0 }

PROPOSITION 3.10 Assume the vector field V in the canonical form V(t) = V(t) ◦ pt in a neighbourhood of the lateral boundary Σ and set v = hV(t), nt i on Γt . The following identity holds, Z Z τZ 1 s √ = ∂s F dΣ [−v ∂t F + ∂nt F ] s 1 + v2 0 Γt Σ s=0 1 +[− √ [h∂t V, nt i − h∇Γt v, V|Γt i] ( 1 + v 2 )3 Ht +√ F hZ, nt i dΓt dt (3.63) 1 + v2 PROOF

The classical shape calculus can be applied in R t × Rd , Z Z s ˜ νi dΣ = (∂ν F + HΣ F ) hZ, F dΣ ∂s Σs

We have

s=0

Σ

˜ νi = √ zt hZ, 1 + v2

and ∂ν F = D Q F · ν 1 = √ [−v ∂t F + ∂nt F ] 1 + v2

Weak evolution of sets and tube derivatives

75

Using Proposition (3.9), we get Z 1 √ = F dΣ s [−v ∂t F + ∂nt F ] 1 + v2 Σ Σs s=0 1 − √ [h∂t V, nt i − h∇Γt v, V|Γt i] ( 1 + v 2 )3 Ht 1 +√ F hZ, nt i √ dΣ 1 + v2 1 + v2 √ We conclude using dΣ = 1 + v 2 dΓt dt. ∂s

Z

REMARK 3.11 We can easily obtain the optimality condition for a minimal surface tube, by setting F = 1 in equation (3.63), ∂s

Z

dΣ s s

Σ

= s=0

Z

τ 0

Z

Γt

1 − √ [h∂t V, nt i − h∇Γt v, V|Γt i] ( 1 + v 2 )3 Ht +√ hZ, nt i dΓt dt 1 + v2

We introduce the adjoint field λ such that 1 Ht −∂t λ − div(λ V) = − √ [h∂t V, nt i − h∇Γt v, V|Γt i] + √ 2 3 ( 1+v ) 1 + v2 with λ(t = τ ) = 0. Then, using the duality identity (3.45), we deduce that the optimality condition writes Z λ hW, nt i dΣ = 0, ∀ W Σ

from which we deduce λ = 0 on Σ. This implies that the right-hand side of the backward equation is zero, i.e., h∂t V, nt i − h∇Γt v, V|Γt i = (1 + v 2 ) Ht

3.5

A first example : optimal trajectory problem

In this section, we investigate a problem related to optimal trajectories. Let D be a compact domain in Rd with a smooth boundary ∂D. Let a be a point

76

Moving Shape Analysis

inside D and for any point x ∈ ∂D, we consider the family of C 1 curves which join these two points in D. We set Ca,x = C 1 curves Ca,x ⊂ D, with extreme points a and x (3.64)

For any curve Ca,x ∈ Ca,x , there exists an injective mapping γ ∈ C 1 ([0, 1]; D) satisfying the following properties, Ca,x = γ([0, 1]), γ(0) = a, γ(1) = x Let gC ∈ L1 (Ca,x ). Our objective is to minimize with respect to the curves Ca,x the integral quantity, Z J(Ca,x ) = gC dC Ca,x

¯ Rd )) with hV, ni = 0 on the boundary Let V be a vector field in L1 (0, 1; C 1 (D, ∂D, then we define the perturbed curve def

V 0 Ca,x = T1 (V)(Ca,x ) ∈ Ca,x 0 V The claim is that from any element Ca,x in Ca,x the element Ca,x furnishes all 1 1 ¯ d the curves when V described the linear space L ((0, 1); C (D, R )), i.e., V ¯ Rd )), hV, ni = 0 on ∂D , V ∈ L1 (0, 1; C 1 (D, Ca,x = Ca,x

Hence, the minimization problem can be written in the following terms, min j(V)

(3.65)

V∈E

where j(V) = with

3.5.1

Z

V Ca,x

V V dCa,x gCa,x

¯ Rd )), E = V ∈ L1 (0, 1; C 1 (D,

hV, ni = 0 on ∂D

Optimality conditions : case of planar parametric curves

In order to solve the above problem, we need to compute the following cost function derivative, j 0 (V, W) = ∂s j(V + sW)|s=0 ,

∀ (V, W) ∈ E × E

First, we consider the simple case in which the density function g C is the ¯ that is gC = G|C . In that case, restriction of a given function G ∈ C 1 (D),

Weak evolution of sets and tube derivatives

77

the parameter s of perturbations only occurs in the measure element on the 0 curves. Let γ be a parametrization of the reference curve C a,x . We have j(V + sW) =

Z

1 0

G ◦ T1 (V + sW) ◦ γ(σ) k [D T1 (V + sW) ◦ γ · γ 0 ] (σ)k dσ

We introduce St (V; W) = ∂s Tt (V + sW)|s=0 We have ∂s k [D T1 (V + sW) ◦ γ · γ 0 ] (σ)k2 |s=0 = 2 hD S1 (V; W) ◦ γ · γ 0 , D T1 (V) ◦ γ · γ 0 i from which we deduce ∂s k [D T1 (V + sW) ◦ γ · γ 0 ] (σ)k|s=0 = hD S1 (V; W) ◦ γ · γ 0 (σ), τ (σ)i =

hD S1 (V; W) · τ, τ1 ◦ T1 (V)i ◦ γ(σ) kγ 0 k

V where τ1 is the unitary tangential vector on the curve Ca,x while τ is the 0 unitary tangential vector to the reference curve C a,x . For 0 ≤ t ≤ 1, St is the solution of the following dynamical system [154, 59],

∂t St − D V(t) ◦ Tt (V) · St = W(t) ◦ Tt (V), St (t = 0) = 0

(3.66)

This leads to the following derivative expression, j 0 (V, W) =

Z

1 0

[h(∇G) ◦ T1 (V)), S1 i(γ(σ))

+G(T1 (V)(γ(σ)) hD S1 (γ(σ)) · τ (γ(σ)), τ1 (T1 (γ(σ)))i] kγ 0 (σ)k dσ Eliminating the parametrization γ, we get Z j 0 (V, W) = [h ∇G(T1 (V)), S1 i + G(T1 (V)) hD S1 · τ, τ1 (T1 (V)) i] dC 0 Ca,x

(3.67) Obviously j 0 (V, W) depends linearly on W through the term S1 . The Jacobian matrix D S1 is itself a solution of the following dynamical system, 12 2  ∂ D S − D V ◦ T (V) · D T (V) · St − D V ◦ Tt (V) · D St  t t t t   = D(W ◦ Tt (V)), (3.68)    D S(t = 0) = 0 12 We

shall assume in this case V ∈ C 0 ([0, 1]; C 2 (D, Rd )).

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Moving Shape Analysis

The couple (St , DSt ) is the solution of the dynamical system (3.66), (3.68). We shall now introduce the backward adjoint dynamical system. Its solution (θ, A) allows to clarify the linear contribution of the field W in the expression of j 0 (V, W). 0 The vector θ and the matrix A are defined on [0, 1] × C a,x with variables 13 (t, x = γ(σ)) where the parameter σ ∈ [σ0 , σ1 ]. Let θ solve the following backward dynamical system,  ∗  −∂t θ − D V(t) ◦ Tt (V) · θ = 0, (3.69)  θ(1, γ(σ)) = ∇G(T1 (V)(γ(σ))) and the matrix A solves the following backward dynamical system, 2  ∗ ∗  −∂t A − D V ◦ Tt (V) · D Tt (V) · A − D V ◦ Tt (V) · A = 0, 

(3.70)

A(1, γ(σ)) = G(T1 (V))(γ(σ)) τ (γ(σ)) · ∗ τ1 (T1 (V)(γ(σ)))

In this setting, the derivative of the functional takes the following form, 0

j (V, W) = = =

Z

Z

Z

1 0

(hθ(1), S1 i + A(1) · · D S1 ) ◦ γ(σ) kγ 0 (σ)k dσ

o Ca,x

(hθ(1), S1 i + A(1) · · D S1 ) dC

1

dt 0

Z

1 0

(θ(t) W(t) ◦ Tt (V)

+ D θ(t) · · D(W ◦ Tt (V))) ◦ γ(σ) dσ]

(3.71)

Then we get a backward calculus for the optimal field V along the reference o trajectory Ca,x parametrized by γ. Actually the previous Eulerian approach, developed for planar curves, leads to a more explicit expression in the general setting of functional derivatives, as we shall see in the next section.

3.5.2

Optimality conditions : case of the general Eulerian setting

We consider now a calculus method which shall never refer to the reference V V +sW 0 . and Ca,x curve Ca,x but only to the moving curves Ca,x Let us consider the following transverse mapping, Tst = Tt (V + sW) ◦ Tt (V)−1

(3.72)

13 Of course σ could be chosen as the arc length of the reference curve C 0 . In this case, a,x we would have kγ 00 (σ)k = 1.

Weak evolution of sets and tube derivatives

79

def

V V +sW This map sends the curve Ca,x onto C s = Ca,x . Then we can write

j(V + sW) =

Z

gC s dC s Cs

Here s is understood as the shape perturbation parameter in the classical setting. Hence, we consider the associated speed vector Zt (s, x) =

−1 ∂ t Ts ◦ Tst ∂s

(3.73)

Hence, the moving curve C s is obtained from the reference curve through the flow mapping associated to the vector field Z t . This flow evolves with respect to the parameter s for a fixed time t = 1. We introduced the terminology transverse flow, V C s = Ts (Z t )(Ca,x ) Using classical differentiation results for boundary integrals 14 , we obtain Z 0 V j 0 (V, W) = gC;Z + H hZ(1), ni dCa,x (3.74) V Ca,x

with def

Z(t, x) = Z t (0, x), Z(1) = Z(1, .) V where H stands for the curvature of Ca,x . 0 The term gC;Z is the so-called boundary shape derivative of the function g(C) V on Ca,x in the direction of the vector field Z(1). We recall here the very definition, 0 V , Z(1)i (3.75) gC;Z = ∂s gC s ◦ Ts1 s=0 − h∇τ gCa,x

Here ∇τ g = ∇G − ∂n G n stands for the tangential derivative of gC along the curve and is independent on the choice of the extension G of g C outside of the curve. We shall now study two specific situations, where g C is either the trace of a distributed function defined over (0, τ ) × D or depends on the curvature V of the curve Ca,x . Distributed density In the very simple case where the function gC is the restriction to the curve C of a smooth function: gC = G|C , we get 0 gC;Z = ∂n G hZ(1), ni

V on Ca,x

14 Here, we assume that the dimension is d = 2, so that the curves can be considered as a part of the boundary of a moving set.

80

Moving Shape Analysis

which leads to the following derivative, Z V j 0 (V, W) = [∂n G + H] hZ(1), ni dCa,x

(3.76)

V Ca,x

We recall that the transverse field Z solves the following dynamical system, ∂t Z + [Z, V] = W, (0, 1) × D (3.77) Z(t = 0) = 0 D def

with [Z, V] = D Z · V − D V · Z. We introduce the adjoint state Λ solution of the following backward dynamical system, ∗ Λ(1) = γC · [(∂n G + H) n] , D V a,x (3.78) ∗ −∂t Λ − D V · Λ − D Λ · V + (div V) Λ = 0, (0, 1) × D ¯ C V ) is the trace operator on the curve C V . This V where γCa,x ∈ L(C 0 (D), a,x a,x leads to the following formula, j 0 (V, W) = hΛ(1), Z(1)iM(D)×C 0 (D) We may use the following integration by parts formula, Z

1 0

hΛ, ∂t Z + [Z, V]iM(D)×C 0 (D) dt =

Z

1 0

h−∂t Λ − D V · Λ − ∗ D Λ · V + (div V) Λ, ZiM(D)×C 0 (D) dt + hΛ(1), Z(1)iM(D)×C 0 (D) − hΛ(0), Z(0)iM(D)×C 0 (D)

(3.79)

As Z(0) = 0 and Λ is the solution of the adjoint system (3.78), we get the following functional derivative, Z 1 j 0 (V, W) = hΛ, WiM(D)×C 0 (D) dt (3.80) 0

Actually, it can be proven that the measure Λ is of the following form, ∗ ˜ nt ), (λ Λ(t) = γT o t (V)(Ca,x )

t ∈ (0, 1)

We can also observe that directly from (3.76), using (3.47), we have Z 1Z ¯ hW(t), nt i dC V dt j 0 (V, W) = λ 0

V Ca,x

¯ solves the backward problem, where λ ¯ ˜ λ(1) = k, D ¯ ¯ V) = 0, (0, 1) × D ∂t λ + div(λ

a,x

(3.81)

(3.82)

Weak evolution of sets and tube derivatives

81

def

with k˜ any extension of the function k = ∂n G + H to the domain D. If div V = 0, then the equation for λ turns to be a backward convection of ˜ = λ(1 ¯ − t) and V(t) ˜ the ending term. We set λ(t) = −V(1 − t), then we get ˜ + ∇λ ˜·V ˜ = 0, (0, 1) × D ∂t λ ˜ ˜ λ(0) = k, D and we get

˜ = k ◦ Tt (V) ˜ −1 λ(t)

Using the fact that Tt (V)−1 = Tt (Wt ) with Wt (s, y) = −V(t − s, y), it leads to ¯ = λ(1 ˜ − t) = k ◦ T1−t (V) λ(t) Then, we get

j 0 (V, W) = and

Z

1 0

Z

V Ca,x

V k ◦ T1−t (V) hW(t), nt i dCa,x dt

V ∇j(V) = ∗ γCa,x k ◦ T1−t (V) nt

Curvature dependent density

(3.83)

Now, we assume that the density function g depends on the curvature of the curves. Here, we choose g(y) = g¯(H(y), y),

V y ∈ V (Ca,x )

def

where H(y) = ∆b(x). Then, we get 0 0 (y) + h∇y g¯(H(y), y), n(y)i hZ(1), ni = ∂H g¯(H(y), y) HC,Z(1) gC;Z(1)

Furthermore, we have 0 = −∆C (hZ(1), ni) HC,Z(1)

This leads to 0

j (V, W) =

Z

V Ca,x

V [−∂H g¯ ∆C (hZ(1), ni) + (H g + h∇¯ g , ni) hZ(1), ni] dCa,x

V Using tangential Stokes formula [53] on Ca,x , we have Z V j 0 (V, W) = [−∆C (∂H g¯) + H g + h∇¯ g , ni] hZ(1), ni dCa,x V Ca,x

+ ∂τ (∂H g¯(x)) hZ(1)(x), n(x)i − ∂τ (∂H g¯(a)) hZ(1)(a), n(a)i − ∂H g¯(x) ∂τ hZ(1)(x), n(x)i + ∂H g¯(a) ∂τ hZ(1)(a), n(a)i (3.84)

82

Moving Shape Analysis

Since the extreme points x and a are fixed, their velocities are zero, i.e., V(a) = V(x) = W(a) = W(x) = 0 which implies Z(1)(a) = Z(1)(x) = 0 and the previous derivative becomes Z 0 V j (V, W) = [−∆C (∂H g¯) + H g + h∇¯ g , ni] hZ(1), ni dCa,x (3.85) V Ca,x

Eventually, we get the gradient of j(V) using the expression furnished by equation (3.83) with k = −∆C (∂H g¯) + H g¯ + h∇¯ g , ni

(3.86)

Chapter 4 Shape differential equation and level set formulation

4.1

Introduction

In this chapter, we recall the concept of shape differential equation developed in [145],[147]. Here, we present a simplified version and some applications in dimension 2 which enable us to reach the time asymptotic result. We consider a shape functional J which is shape differentiable in O k with respect to Vk to be specified later on. We denote ∇J(Ω) its gradient, considered as a distribution in A∗k . For any Ω0 in Ok and V in Vk , the absolute continuity of J writes Z s h∇J(Ωt (V)) , V(t)iA∗ × Ak dt, ∀ s > 0 (4.1) J(Ωs (V)) − J(Ω0 ) = k

0

Classical gradient based methods allow us to control the variations of J with respect to the domain. Considering the problem, min J(Ω(V))

V∈Vk

we would like to elaborate a constructive method to decrease the functional following the gradient. This may be done by solving the non-linear equation for large evolution of the domain, ∇J(Ωt (V)) + A(V(t)) = 0,

∀t > 0

(4.2)

where A is an ad-hoc duality operator.This corresponds to the well-known steepest descent method. From the structure theorem for shape gradient [135], we have 1 , ∇J(Ω) = γΓ∗ · (g n) where the shape density gradient g is a distribution on the boundary Γ. Usually it is a function on Γ so that we consider any extension G of g defined 1 Under

some regularity assumptions which are satisfied for a large class of problems.

83

84

Moving Shape Analysis

in a neighbourhood of the boundary Γ. In this case the term γ Γ∗ · (g n) can be identified with G ∇χ. Hence, the shape differential equation turns into an Hamilton-Jacobi equation for the characteristic function χ, ∂t χ + h∇χ, A−1 · (G(χ)∇χ)i = 0, (0, τ ) (4.3) χ(0) = χΩ0 , Ω0 ⊂ D We shall see in the sequel that the previous equation can be weakened using the level set formulation where we shall solve the following equation, ∂t Φ − h∇Φ, A−1 · (G(χt (Φ))∇χt (Φ))i = 0, (0, τ ) (4.4) Φ(0) = Φ0 , where χt (Φ) = {x ∈ D, Φ(t, x) > 0}. We are going to recall the constructive proof of the existence of a V satisfying (4.2) and investigate the asymptotic behaviour of the method. The existence of a solution for this so-called shape differential equation has been proven in [147] inside a larger setting2 .

4.2

Classical shape differential equation setting

In this section, we recall the material introduced in [145, 146, 147] while solving the so-called shape differential equation. We denote T k the subset of ¯ Rd ) whose elements are C k -diffeomorphism of D. ¯ It is endowed with C k (D, 3 the Courant metric dk which is defined on the family of images of a given domain. We fix a smooth bounded hold-all D in Rd and a non-negative integer k. We denote by Ok the set of all open C k -submanifold of D, and Olip the set of Lipschitz open subset of D. We are going to use the following spaces o n Ak = V ∈ C k (D, Rd ) hV , νiRd = 0, on ∂D , Vk (I) = V ∈ C 0 (I, Ak ) (4.5) where I is an interval of R+ which contains 0. For I = R+ , we simply denote Vk = Vk (R+ ). The operator A∗k stands for the dual space of distributions of order less than

2 It holds for shape differentiable functional whose gradient is continuous and bounded on Ok , endowed with the Courant’s metric topology, ranging in a Sobolev space of Distributions. 3 We refer the reader to the book [51].

Shape differential equation and level set formulation k. In the case k = 0, it corresponds to the space of Radon measures. For any fixed domain Ω0 ∈ Ok , we set n o Ok (Ω0 ) = Ω ∈ Ok ∃ T ∈ Tk , Ω = T (Ω0 ) REMARK 4.1

(Ok (Ω0 ), dk ) is a complete metric space.

For a bounded universe D, the following compactness result holds, PROPOSITION 4.1 [51] The inclusion (Ok+1 (Ω0 ), dk+1 ) ,→ (Ok (Ω0 ), dk ) is compact. Also for a bounded universe D, the following continuity result holds, THEOREM 4.1 [147],[150] The mapping Vk −→ C 0 (I, Ok (Ω0 )) V 7→ [t 7→ Ωt (V) = Tt (V)(Ω0 )] is continuous and maps bounded subsets on equicontinuous parts. LEMMA 4.1 The mappings ¯ Rd )) Vk (I) → C 1 (I, C k (D, V 7→ [t → 7 Tt (V)]

and

¯ Rd )) Vk (I) → C 1 (I, C k (D, −1 V 7→ [t 7→ Tt (V)]

are continuous. Although Tk is not a vector-space, we will write, for shortness, n o C 1 (I, Tk ) = T ∈ C 0 (I, Tk ) T 0 ∈ C 0 (I, Tk )

This space is endowed with the canonical norm

kT kC 1 (I,Tk ) = sup kT (s)kTk + sup kT 0 (s)kTk s∈I

s∈I

For (Tk , dk ), we have a result similar to theorem 4.1. THEOREM 4.2 The mapping Vk (I) → C 1 (I, Tk ) V 7→ [t → 7 Tt (V)]

85

86

Moving Shape Analysis

is surjective, continuous and maps bounded subsets on equicontinuous parts. We have the following characterization of the shape continuity, COROLLARY 4.1 Let G be a shape function defined on Ok with values in a fixed Banach space B. The followings properties are equivalent i) G is shape continuous with respect to Vk (I): for any initial domain Ω0 , for all V ∈ Vk (I), s 7→ G(Ωs (V)) belongs to C 0 (I, B). ii) for any initial domain Ω0 , for any T in C 1 (I, Tk ), s 7→ G T (s)(Ω0 ) belongs to C 0 (I, B). It is important to notice how easy it is to characterize the shape continuity via the space Tk . A characterization involving Ok would be more elegant, since the real objects are the domains, not the diffeomorphism. It is known (see [147, 51] for instance) that a shape functional G defined on O k with values in a fixed Banach space B is shape continuous (in the usual sense) as soon as [s 7→ G(Ω(s))] is continuous (i.e., belongs to C 0 (I, B)) for any [s 7→ Ω(s)] ∈ C 0 (I, Ok ).

4.3 4.3.1

The shape control problem An existence result for the shape differential equation

This section aims at proving the following theorem, using a solution of equation (4.2). THEOREM 4.3 Let J be a shape functional, differentiable in Ok with respect to Vk . Let us assume the following conditions: (i) Both J and ∇J are uniformly bounded on Ok+1 (respectively in R and A∗k+1 ), (ii) ∇J is shape-continuous on Ok+1 , in A∗k , with respect to Vk+1 . Then, there exists V ∈ Vk+1 ∩ L2 (R+ ; Ak+1 ) and c > 0 such that, for any s > 0, Z s Z s 2 2 J(Ωs (V)) − J(Ω0 (V)) = − kV(t)k dt = −c k∇J(Ωt (V))k dt 0

0

Shape differential equation and level set formulation

87

PROOF Provided the duality operator A of equation (4.2) exists, a solution of this equation is convenient for the theorem. We are going to use a Sobolev space embedded in Ak to ensure the existence (and good properties) of the duality operator A, and give a constructive proof of the existence of a solution of (4.2). We fix κ > 1 such that the following space n o H = V ∈ Hκ (D, Rd ) hV , niRd = 0 on ∂D satisfies the following embedding chain rule,

H,→Ak+1 ,→Ak

(4.6)

We denote A the linear and continuous duality operator from H to its dual H∗ . We consider, the domain Ω0 being fixed in Ok+1 , an arbitrary interval I of R+ which contains 0. Let GI be the mapping defined by GI (V) : I −→ H s 7→ −A−1 ∇J(Ωs (V))

(4.7)

for V ∈ C 0 (I, H) ⊂ Vk+1 . Since we assumed the shape continuity of ∇J, GI (V) ∈ C 0 (I, H). We are going to prove that GI has a fixed point, i.e., that equation (4.2) admits at least one solution. LEMMA 4.2 There exists m > 0 such that Bk,m = V ∈ Vk+1 sup kV(s)kAk < m ⊃ GI (Bk,m ) s∈I

PROOF Due to the boundedness of ∇J, there exists m1 (which may depend on Ω0 ) such that for any Ω ∈ Ok (Ω0 ), k∇J(Ω)kA∗ ≤ m1 k

It follows that kGI (V)(s)kH ≤ m1 kA−1 kL(H∗ ,H) = m1 The choice m = m1 is convenient. LEMMA 4.3 The mapping GI is continuous. Provided I is compact, G is compact.

88

Moving Shape Analysis

PROOF

GI can be split using the following chain rule GI = G 3 ◦ G 2 ◦ G 1

where G1 : C 0 (I, H) −→ C 0 (I, Tk ) V 7→ [s 7→ Ts (V)] G2 : C 0 (I, Tk ) −→ C 0 (I, A∗k ) T 7→ ∇J(T (.)(Ω0 )) G3 : C 0 (I, A∗k ) −→ C 0 (I, H∗ ) −→ C 0 (I, H) g 7→ g 7→ −A−1 g Theorem 4.2 provides the continuity of G1 . Using Corollary 4.1, the continuity of G2 is equivalent to the shape continuity of the ∇J and since the continuity of G3 is clear, we deduce that GI is continuous. We suppose I is compact. By Theorem 4.2, a bounded subset B ⊂ C 0 (I, Ak+1 ) is mapped by G1 on a equicontinuous part of C 0 (I, Tk+1 (Ω0 )). By Ascoli’s theorem and the compactness of the inclusion of T k+1 (Ω0 ) in Tk (Ω0 ) (Theorem 4.1), the image of B is para-compact in C 0 (I, Tk ). Accordingly, G1 is a compact mapping, and so is GI . Applying Leray-Schauder’s fixed point theorem,we infer that for any initial domain Ω0 there exists V in C 0 ([0, 1], H) with G[0,1] (V) = V.

4.3.2

A constructive algorithm

Now, we shall build a sequence of converging domains. Let us define V ∈ C 0 (R+ , H) and (Ωn )n∈N ⊂ Ok+1 (Ω0 ) by ∀n ∈ N ,

 G[n,n+1] (V ) = V on D  [n,n+1] [n,n+1]   n Ωn (Vn ) = Ω n+1  Ω (V ) = Ω  n+1 n  Ω0 (V0 ) = Ω0

The continuity at integer points comes from equation (4.2). We have, for any s > 0, and any n, Z n+1 Z n+1 2 2 k∇J(Ωt (V)kH∗ dt (4.8) kV(t)kH dt = − J(Ωn+1 ) − J(Ωn ) = − n

n

This so-built field V satisfies Theorem 4.3. Since we assumed ZJ is bounded, s 2 kV(t)kH dt, s 7→ J(Ωn ) is bounded decreasing, hence has a limit and so does which proves V ∈ L2 (R+ , Ak+1 ).

0

Shape differential equation and level set formulation

4.4

89

The asymptotic behaviour

If V is given by Theorem 4.3, there exists a non-decreasing sequence (s n )n>0 such that V(sn ) → 0 since V ∈ L2 (R+ , Ak+1 ). We denote Ωn = Ωsn (V) ⊂ Ok+1 (Ω0 ). The sequence (Ωn ) may not be bounded in Ok (Ω0 ) or Ok+1 (Ω0 ), since the L2 convergence of the speed given by this method is not sufficient in general. Nevertheless, we can use a weaker topology on the space of domains. We denote Oop the family of all open subsets of D. In [51] it is proven to be a compact metric space for the Hausdorff-complementary metric

d(Ω1 , Ω2 ) = max

(

sup

inf

x1 ∈D\Ω1 x2 ∈D \Ω2

|x1 − x2 | ,

sup

inf

x2 ∈D\Ω2 x1 ∈D \Ω1

|x1 − x2 |

)

(4.9)

LEMMA 4.4 Assume that (i) the shape functional J verifies the assumptions of Theorem 4.3, (ii) the shape functional J is defined and continuous in Oop , (iii) the shape functional gradient ∇J is continuous for the Hausdorff complementary topology on Ok (Ω0 ). Then (Ωn ) has cluster points in Oop and if Ω∗ is one of them, then Ωn → Ω∗ in Oop and ∇J(Ωn ) → 0 and J(Ωn ) → J(Ω∗ ) PROOF The sequence (Ωn ) may be regarded as a sequence in the compact space Oop . Hence passing to a subsequence, it converges towards an open subset Ω∗ of D. The gradient ∇J(Ω∗ ) is not a priori defined, since the limit set has not enough regularity. Nevertheless since V satisfies (4.2), kV(n)k H = k∇J(Ωn )k hence ∇J(Ωn ) → 0. In the next section, we shall furnish an example of a functional where the continuity hypothesis of Lemma (4.4) is satisfied.

90

4.5 4.5.1

Moving Shape Analysis

Shape differential equation for the Laplace equation The Laplace equation

In this section, we are given a family g = (gΩ )Ω∈Olip such that for any Ω ∈ Olip , gΩ ∈ H−1 (Ω). We consider the Dirichlet problem P(Ω, g)

−∆y = gΩ in Ω y = 0 on Γ = ∂Ω

which has a unique solution y(Ω, g) in the space H10 (Ω) endowed with the Z 2 norm kzkΩ = |∇z|2 . Ω

A priori estimates An a priori estimate for solution y(Ω, g) of P(Ω, g) is derived from the variational formulation of the problem : y(Ω, g) is the unique minimum of the functional EΩ,g defined on H10 (Ω) by Z 1 EΩ,g (z) = − hgΩ , ziH−1 (Ω)×H1 (Ω) + |∇z|2 (4.10) 0 Ω 2 where |.| denotes the euclidean norm in Rd . Accordingly, EΩ,g (y(Ω, g)) ≤ 0 1 2 ky(Ω, g)kΩ ≤ kgΩ kΩ,∗ ky(Ω, g)kΩ where k.kΩ,∗ denotes the 2 norm in the dual space H−1 (Ω). This yields to Thus it leads to

ky(Ω)kΩ ≤ 2kgΩ kΩ,∗

(4.11)

A mere consequence of this estimate is the following uniform boundedness result. LEMMA 4.5 Let O be a subset of Olip such that {kgΩ kΩ,∗ |Ω ∈ O} is bounded. Then {ky(Ω, g)kΩ |Ω ∈ O} is bounded.

Shape differential equation and level set formulation

91

In the sequel, we shall use the family (f|Ω )Ω∈Olip with f ∈ L2 (D). Since kf|Ω kΩ,∗ ≤ kf kL2 (D)

sup z∈H1 0 (Ω) kzk 2 ≤1 L (Ω)

kzkΩ ≤ cP (Ω)kf kL2 (D)

where cP (Ω) is the Poincar´e’s constant4 for the domain Ω. Therefore, the uniform boundedness property of the solutions of P(Ω, f ) 5 will arise from the following uniform boundedness of Poincar´e’s constant. LEMMA 4.6 There exist a constant cP > 0 such that ∀ Ω ∈ Olip , cp (Ω) ≤ cP PROOF It is classical that for any Ω ∈ Olip , there exists zΩ in H10 (Ω) with cP (Ω)−1 = k∇zΩ kL2 (Ω)d . Extending zΩ by 0 provides a z˜Ω in H10 (D) such that cP (D)−1 ≤ k∇z˜Ω kL2 (Ω)d . Thus cP (D) > cP (Ω) so cP = cP (D) is convenient.

Eventually, we have proven the uniform boundedness of the solutions of P(Ω, f ) with respect to the domain. Strong shape continuity properties Let Ω0 be a fixed initial domain in Olip and let us assume the following, i) The family gΩ is shape continuous, i.e. for any V in Vk , the mapping6 s 7→ gΩs (V) ? Ts is continuous from R+ to H−1 (Ω0 ) ii) For any V in Vk ,

4 It

s 7→ kgΩs (V) kH−1 (Ω

s)

+ ∈ L∞ loc (R )

(4.12)

may be defined via Rayleigh-quotient, cP (Ω)−1 =

inf

z∈H1 0 (Ω) z6=0

k∇zkL2 (Ω)d kzkL2 (Ω)

5 The

accurate notation for this is P(Ω, (f|Ω )Ω∈Olip ). mapping is defined as follows, For any z in H10 (Ω0 ), ˙ ¸ ˙ ¸ gΩs (V) ? Ts , z H−1 (Ω )×H1 (Ω ) = gΩs (V) , γs−1 z ◦ Ts−1 H−1 (Ω

6 This

0

where γs = det D Ts .

0

0

1 s (V))×H0 (Ωs (V))

92

Moving Shape Analysis

THEOREM 4.4 Under the last hypothesis, for k > 2, the following properties hold, i) the transported solution of P(Ω0 , g) is shape continuous: s→0

∀ V ∈ Vk , y(Ωs (V), g) ◦ Ts −→ y(Ω0 , g) in H10 (Ω0 ) ii) the energy functional, E(Ω0 , g) = EΩ0 ,g (y(Ω0 , g)) is continuous, iii) the extended solution of P(Ω0 , g) is shape continuous: s→0

∀ V ∈ Vk , y˜(Ωs (V), g) −→ y˜(Ω0 , g) in H10 (D) Moreover, these properties are still valid for k = 1 provided gΩ ∈ L2 (Ω) for any Ω ∈ Olip . PROOF For the sake of simplicity, we denote ys = y(Ωs (V), g) and y s = ys ◦ Ts . We have y0 = y 0 . Due to the local boundedness property (4.12), there exists ε such that (kys kΩs )0≤s≤ε is uniformly bounded. Since 2

ky s kΩ0 ≤ kγ(s)−1 | D(Ts−1 )−1 |kL∞ (D) kys kΩs the family (y s )0≤s≤ε is uniformly bounded in H10 (Ω0 ). Hence we can extract a subsequence (y sn )n>0 where 0 ≤ sn ≤ ε and sn → 0 that converges towards an element y ∗ , weakly in H10 (Ω0 ). We denote E Ωs ,g the functional defined on H10 (Ω0 ) by the following identity, def

E Ωs ,g (z) = EΩs ,g (z ◦ Ts−1 ) = − hgΩs ? Ts , γs ziH−1 (Ω0 )×H1 (Ω0 ) + 0

Z

Ω0

1∗ | D Ts−1 · ∇z|2 γs 2

and we have min E Ωs ,g = E Ωs ,g (y s ) = EΩs ,g (ys ) = min EΩs ,g

(4.13)

Since k > 2 the Jacobian γsn converges towards γ(0) ≡ 1 in C (D, R ). This is sufficient for

s→0

gΩsn ? Tsn , γ(sn )y sn H−1 (Ω0 )×H1 (Ω0 ) −→ gΩ0 , y 0 H−1 (Ω0 )×H1 (Ω0 ) (4.14) 1

d

0

0

Furthermore, the convergence of γsn (D Tsn ) towards I in C (D, Rd ) induces ) · ∇y sn towards ∇y ∗ in L2 (Ω0 )d . Using the weak convergence of (γsn ∗ D Ts−1 n the weak-lower semi-continuity of the L2 -norm, we have Z Z 1 1∗ |∇y ∗ |2 ≤ lim inf | D Ts−1 · ∇y sn |2 γsn n n→∞ 2 2 Ω0 Ω0 −1

0

Shape differential equation and level set formulation

93

Thus, we have proven the weak-lower semi-continuity of the mapping (s, z) 7→ E Ωs ,g (z) on R+ × H10 (Ω0 ) at (0, z) for any z. This proves that for any z E Ω0 ,g (y ∗ ) ≤ lim inf E Ωsn ,g (y sn ) ≤ lim inf E Ωsn ,g (z) n→∞

n→∞

But s 7→ E Ωs ,g (z) is continuous for any z. Hence y ∗ = y 0 = y0 , which proves a weak shape-continuity for the transported solutions of P(Ω, g). The strong continuity will arise from a continuity of the norms, via the socalled compliance equality, and a compactness argument for the y s . For any Ω in Olip , the necessary (and sufficient) condition of optimality for E Ω,g is written Z 1 ∀ z ∈ H0 (Ω), h∇y(Ω, g) , ∇ziRd = hgΩ , ziH−1 (Ω)×H1 (Ω) 0

Ω

Choosing z = y(Ω, g) in the above equation, we come to 1 1 2 EΩ,g (y(Ω, g)) = − ky(Ω, g)kΩ = − hgΩ , y(Ω, g)iH−1 (Ω)×H1 (Ω) 0 2 2

(4.15)

which leads together with equation (4.14) and the weak continuity of s 7→ y s , that s 7→ EΩs ,g (y(Ωs , g)) is continuous. Since the sequence (kysn kΩ ) = (ky˜sn kD ), where ˜. denote the extension to D with 0, there exists a y∗ in H10 (D) such that y˜sn *y∗ in H10 (D). Since the sequence (Ωsn ) converges towards Ω0 for the Hausdorff complementary topology, y∗ has support in Ω0 and may be written y˜] with y] ∈ H10 (Ω0 ), since the boundary ∂Ω0 has non-zero capacity. Using arguments similar to the ones which established (4.14), we have y˜sn ◦ Ts−1 = ysn^ ◦ Ts−1 ˜] in H10 (D) n *y n Accordingly, y sn *y] in H10 (Ω0 ) and y] = y0 . This eventually proves that y˜sn *y˜0 in H10 (D) But due to the continuity of s 7→ EΩs ,g (y(Ωs , g)) and equation (4.15), ky˜sn kD = kysn kΩsn → ky˜0 kD = ky0 kΩ0 and this is sufficient for y˜sn → y˜0 (strongly) in H10 (D) and y sn → y 0 (strongly) in H10 (Ω0 )

94

Moving Shape Analysis

REMARK 4.2 In this proof, the assumption k > 2 is needed to prove the convergence result (4.14). In the case where g Ω ∈ L2 (Ω) for any Ω ∈ Olip , this assumption is not needed anymore. The following differentiability result is well known when the right-hand side is fixed in H1 (D). It may easily be extended to a domain-dependent right-hand side [135]. THEOREM 4.5 Assume that for any Ω0 in Ok , gΩ0 ∈ L2 (Ω0 ) and for any V inVk the mapping s 7→ gΩs (V) ◦ Ts is strongly differentiable at s = 0 in H−1 (Ω0 ) with derivative g˙ (Ω0 ;V) . Then the solution y(Ω) of problem P(Ω, f ) has a material derivative y(Ω; ˙ V) in H10 (Ω) for any velocity field V ∈ Vk . Moreover, for any φ ∈ H10 (Ω), Z

h∇y(Ω; ˙ V) , ∇φi = Z 1 − [ div V(0) I −ε(V(0))] · ∇y(Ω) , ∇φ + [gΩ div(V(0)) + g˙ Ω0 ;V ] φ 2 Ω Rd (4.16) Ω

4.5.2

The shape control problem

We would like to minimize the following shape functional, J(Ω) =

Z

Ω

(y(Ω) − yd )2

(4.17)

where yd ∈ H10 (D) is a given target. The shape gradient PROPOSITION 4.2 For any domain Ω in Olip , for any field V ∈ Vk (k ≥ 1), the functional J admits an Eulerian derivative, dJ(Ω; V) =

Z

Ω

(y − yd ) [div V(0)(y − yd ) − D yd · V(0)] + 2 hA0 ∇y , ∇pi − 2 hdiv(f V) , pi (4.18)

˜ − yd ). where p is the solution of the adjoint problem P(Ω, y(Ω)

Shape differential equation and level set formulation

95

PROOF If Ts is the flow-mapping associated to the velocity field V, a change of variable gives Z (ys − yd )2 J(Ωs ) = Ωs Z = γs (y s − yd ◦ Ts )2 Ω

Since s 7→ Ts is of class C 1 , we get Z ∂s J(Ωs ) = γ 0 (s)(y s − yd ◦ Ts )2 − 2γ(s)(y s − yd ◦ Ts )(∂s y s − D y ◦ Ts ∂s Ts ) Ω

Therefore, Z

dJ(Ω; V) =

Ω

[div V(0)] (y − yd )2 − 2(y − yd ) (y˙ − D yd · V(0))

We consider the adjoint problem, ˜ − yd ) P(Ω, y(Ω)

−∆p = y − yd on Ω p=0 on Γ = ∂Ω

(4.19)

Thanks to Theorem 4.5, the strong material derivative of y(Ω) satisfies Z Z ∀ φ ∈ H10 (Ω) , h∇y˙ , ∇φi = − hA0 (V).∇y , ∇φi + hdiv(f V(0)) , φi Ω

with

Ω

def

A0 (V(0)) = div V(0) I −2 ε(V(0))

Then, we deduce dJ(Ω; V) = = =

Z

Z

Ω

ZΩ Ω

div V(0)(y − yd )2 + 2(∆p) (y˙ − D yd · V(0)) div V(0)(y − yd )2 − D yd · V(0) − 2∇p · ∇y˙ (y − yd )[div V(0)(y − yd ) − D yd · V(0)]

+2 hA0 (V(0))∇y , ∇pi − 2 hdiv(f V) , pi

When the domain Ω is fixed, the mapping V 7→ dJ(Ω; V) is linear and continuous. We consider the element ∇J(Ω) of A∗k given by Z h∇J(Ω) , ViA∗ ×Ak = (y(Ω) − yd )[div V(y(Ω) − yd ) − D yd · V] k Ω Z 0 + 2 hA (V)∇y(Ω) , ∇pi − 2 hdiv(f V) , p(Ω)i (4.20) Ω

96

Moving Shape Analysis

Uniform boundedness PROPOSITION 4.3 There exists a constant M > 0 such that k∇J(Ω)kA∗ ≤ M, k

∀ Ω ∈ Olip

PROOF k∇J(Ω)kA∗ = sup h∇J(Ω) , ViA∗ ×Ak k

k

V∈Ak

2 y d k L2

≤ kdiv VkL∞ ky − + kD yd · (y − yd )kL1 kVkL∞ +2kA0 (V)kL∞ kh∇y , ∇pikL1 + 2 kf |∇p|kL1 kVkL∞ Then, there exists a constant m > 0 such that k∇J(Ω)kA∗ ≤ mkykΩ kpkΩ kVkAk k

and the uniform boundedness of kykΩ and kpkΩ gives the correct result. Shape continuity properties of the gradient THEOREM 4.6 The gradient distribution ∇J is continuous in A∗k . PROOF The gradient ∇J(Ω) is a continuous function G(χ Ω , y˜(Ω), p˜(Ω)) which is continuous from L2 (D) × H10 (D) × H10 (D) to A∗k with hG(χΩ , y, p) , ViA∗ ×Ak = k Z χΩ (y − yd )[div V · (y − yd ) − D yd .V] + 2χΩ hA0 (V)∇y , ∇pi D

− 2χΩ hdiv(f V) , pi

Theorem 4.4 provides that the mapping s 7→ y˜(Ωs (V)), s 7→ p˜(Ωs (V)) are continuous for any V in Vk . Consequently, s 7→ ∇J(Ωs (V)) is continuous.

4.5.3

An asymptotic result in the 2D case

We shall apply theorem 4.3 to the Laplace equation P(Ω, g). PROPOSITION 4.4 For any Ω0 in Ok+1 , there exists a V ∈ Vk+1 and an open subset Ω∗ of D ⊂ R2 such that,

Shape differential equation and level set formulation Z s 2 i) J(Ωs (V)) − J(Ω0 ) = kV(t)k dt

97

0

ii) for any sequence (sn )n>0 with sn → ∞, Ωsn (V) → Ω∗ for Hausdorff complementary topology, J(Ωsn (V)) → J(Ω∗ ) and ∇J(Ωsn (V)) → 0. The general asymptotic behaviour of section 4.4 may be developed in the case d = 2 . Indeed, the continuity of Ω 7→ y˜(Ω) for Hausdorff-complementary topology does not hold in general. Nevertheless, this continuity holds under capacity constraints( [23], [19]). In the 2-dimensional case, Sverak has proven in [137],[136] the convergence of y˜(Ωn ) towards y˜(Ω), provided ({Ωn )n converges to {Ω for Hausdorff topology, with ]{Ωn uniformly bounded.

4.6

Shape differential equation in Rd+1

We come back to the optimal dynamical evolution of a geometrical domain, a moving domain, say Ωt , where t is the time parameter.That optimality is built in the context of the minimization of some cost functional. For each tube [ Q= {t} × Ωt ⊂ Rd+1 0≤t≤τ

with lateral boundary Σ=

[

0≤t≤τ

{t} × ∂Ωt ⊂ Rd+1

We define as previously the unitary normal field to Σ pointing outside Q, ν(t, x) = (1 + v(t, x)2 )−1/2 − v(t, x), nt (x) def

where nt (x) ⊂ Rd is the unitary normal field to Γt = ∂Ωt , pointing outside the moving domain Ωt . Let us consider a cost functional of the tracking type, Z 1 J(Q) = |yQ − Yg |2 dx dt 2 Q where Yg is a given element, e.g., Yg ∈ H 1 ((0, τ ) × D) where D is a given bounded open domain in Rd+1 such that the cylindrical domain ]0, τ [×D will contain the potential evolution of the moving tubes Q s . The element yQ ∈ W (Q) is a state variable which is defined as an element of some functional space built on the tube Q. As an example of such a state variable, we can consider the non-cylindrical

98

Moving Shape Analysis

wave equation, where for a given rhs F ∈ L2 0, τ ; L2 (D) , the state variable yQ ∈ L2 (0, τ, H01 (Ωt )) satisfies the following equation, ∂tt y − ∆y = F, in Q

(4.21)

The basic “Optimal Moving Domain” problem consists in finding the “best” dynamical time evolution for the moving domain Ω t in order to minimize the functional J. In other words, we look for the best tube Q ⊂ R d+1 which minimizes the functional J. For that purpose, we just consider the tube Q as a moving domain in Rd+1 and we consider an “evolution” parameter s to “follow” the virtual evolution of the tube Qs . Given a smooth enough vector field ˜ t, x) def Z(s; = 0, Z(s; t, x) ∈ Rt × Rdx

we consider the associated flow mapping def

˜ Ts (Z)(t, x) = t, Ts (Z)(t, x)

The specific structure of that flow is obviously deriving from the “horizontal” ˜ itself. That choice does not produce any genercharacter of the vector field Z 7 ality loss in the potential deformations of the cylinder Q. A special role will be played by the d dimensional vector field Z(t)(s, x) := Z(s; t, x) From the classical shape analysis, we derive the R d+1 -shape gradient associated to the shape functional J(Q) as follows: J(Qs ) =

min

max

φ∈L2 (0,τ,H01 (Ωt )) ψ∈L2 (0,τ,H01 (Ωt ))

L(s; φ, ψ)

where the Lagrangian functional is given by the following expression, L(s; φ, ψ) =

1 2

Z

Qs

+

Z

|φ(t) ◦ Ts (Z)−1 − Yg |2 dx dt Qs

∂t (φ(t) ◦ Ts (Z)−1 ) ∂t (ψ(t)oTs (Z)−1 )

+ ∇(φ(t) ◦ Ts (Z)−1 ) · ∇(ψ(t) ◦ Ts (Z)−1 ) dx dt Z Z y1 ψ(0) dx F ψ(t) ◦ Ts (Z)−1 , dx dt − − Qs

Ω0

7 Indeed a general form for the vector field Z ˜ could induce a change in the time variable t which would create a “time extension” of the tube.

Shape differential equation and level set formulation

99

Using the structure theorem, we can state that the eulerian directional derivative of J is only supported by the lateral boundary Σ and is given by the following expression, Z ∂ ˜ def ˜ · ν dΣ dJ(Q, Z) = J(Qs ) = gZ ∂s Σ s=0 Notice that Z Moreover

So that

f dΣ = Σ

Z

τ 0

Z

f (t, x) Γt

p 1 + v(t, x)2 dt dΓ(x)

1 ˜ x), ν(t, x)i = p hZ(t, hZ(t, x), nt (x)i 1 + v(t, x)2 ˜ = dJ(Q, Z)

Z

τ

0

Z

Γt

g(t, x) Z(t, x) · nt (x) dΓt (x) dt

That is ˜ = hG(Q), Z(0)i ˜ dJ(Q, Z) D 0 (]0,τ [×D,Rd+1 )×D(]0,τ [×D,Rd+1 ) Thus the “time-space” vectorial distribution is given by the following expression, def ∗

G(Q) =

γΣ (g ν )

where the density gradient g is an element of L1 (Σ) (in general a (d+1)- vector measure on Σ with zero transverse order) and its expression involves, in the non-cylindrical wave equation case, the normal derivatives at the boundary of the state and co-state solution. Now we apply, in the (d+1)-dimensional case, the Shape Differential Equation framework developed in the previous section of this chapter. As a consequence, ˜ = (0, Z) to the shape differential we derive the existence of a solution Z equation, ˜ = 0, A−1 .G(Ts (Z) ˜ Z (4.22)

where A stands for an ad hoc duality positive operator as described previously. So we obtain Z s0 2 ˜ kZ(s)k ds J(Qs ) ≤ J(Q) − α 0

0

H(D)

for some “pivot” Hilbertian Sobolev vector space H(D) over the hold-all domain D. def ˜ ˜ solution The construction of the tube Qs0 = Ts (Z)(Q) from the field Z d+1 to the previous R -shape differential equation consists in finding the R d dimensional vector field V(t, x) whose flow mapping T t (V) builds the tube

100

Moving Shape Analysis

Qs0 from the initial domain Ω0 . Let us assume that the initial tube Q is the cylindrical domain Q =]0, τ [×Ω0 That tube is obviously built by the vector speed V = 0. Now we look for a vector field Vs0 (t, x) such that its flow mapping Tt (Vs0 ) builds the “optimal” tubes Qs0 from the initial domain Ω0 . This means that the following identity holds true, Tt (Vs0 ) = Ts0 (Z(t))(Ω0 ), ∀0 < t < τ (4.23) where Z(t) stands for the Rd+1 -dimensional vector field Z(t)(s, x) = Z(s, t, x). Since, by definition, we have Z s0 ˜ ˜ x) = t, x + Z(σ; t, Tσ (Z)(t, x)))dσ Ts0 (Z)(t, 0

we get

∂ ˜ (t, x) = 1, Ts0 (Z) Z∂ts0 ∂ ∂ ˜ ˜ ˜ Z(t, Tσ (Z)(t, x)) + Dx Z(σ; t, Tσ (Z)(t, x)) · (Tσ (Z)(t, x)) dσ ∂t ∂t 0

By definition, the Rd -dimensional speed vector V s0 is furnished by the following expression, def ∂ ˜ ˜ −1 (t, x)) 1, Vs0 (t, x) = Ts (Z)(t, Ts0 (Z) ∂t 0

(4.24)

Now, since the Rd+1 -dimensional flow is horizontal, i.e., ˜ Ts0 (Z)(t, x) = (t, Ts0 (Z(t))(x))

(4.25)

then we obtain Vs0 (t, x) =

∂ Ts0 (Z(t)) Ts0 (Z(t))−1 (x) ∂t

(4.26)

Also the previous differential equation can be rewritten in term of Z(t) as follows, Z s0 Z(t)(σ, Tσ (Z(t))(x)) dσ (4.27) Ts0 (Z(t))(x) = x + 0

so that

∂ [Ts0 (Z(t))](x) = ∂t

Z

s0 0

{

∂ Z(t)(σ, Tσ (Z(t)))(x) ∂t + D Z(t)(σ, Tσ (Z(t)))(x) ·

∂ Tσ (Z(t))(x)} dσ ∂t

Shape differential equation and level set formulation

101

Then, Z

s0

∂ Z(t)(σ, Tσ (Z(t)) ◦ Ts0 (Z(t))−1 )(x) ∂t 0 +D Z(t)(σ, Tσ (Z(t))◦Ts0 (Z(t))−1 )(x))·Vσ (t, Tσ (Z(t))◦Ts0 (Z(t))−1 (x)) dσ s0

V (t, x) =

4.7 4.7.1

{

The level set formulation Introduction

In [149],[135],[51],[93] we have considered domains family parametrized as follows: Ωt = Ωt (Φ) = {x ∈ D | Φ(t, x) > 0 }, Γt = {x ∈ D | Φ(t, x) = 0 } (4.28) where Φ(t, .) is a function defined on D verifying a negative condition at the boundary of D, say Φ = −1 on the boundary ∂D. The singular points are those in D at which the gradient of Φ(t, .) vanishes. We assume that no such points lie on Γt so that in a neighbourhood of the lateral boundary Σ of the tube, the following vector field def

V(t, x) = −∂t Φ(t, x)

∇Φ(t, x) k∇Φ(t, x)k2

(4.29)

is always defined. If V is smooth enough we get Ωt = Tt (V)(Ω0 ) In this case, the shape differential equation writes A−1 .∇J(Ωt (V)) − ∂t Φ(t, x)

∇Φ(t, x) =0 k∇Φ(t, x)k2

(4.30)

which would imply, with the notation Ωt (Φ) = Ωt (V) (V previously defined), the following Hamilton-Jacobi equation for the function Φ:

∂t Φ(t, x) − ∇Φ(t, x) , A−1 · ∇J(Ωt (Φ)) = 0 (4.31)

Notice that the Hamilton-Jacobi versions (4.30) and (4.31) are not equivalent. They would be merely equivalent if A−1 · ∇J(Ωt (Φ)) was proportional to ∇Φ(t, x). In order to bypass that point, we consider the scalar shape differential equation. From the general structure theorem for shape gradient [135], we have G = ∇J(Ω) = γΓ∗ .(g n) = G(Ω) ∇χΩ

102

Moving Shape Analysis

where n is the normal field to the boundary Γ, while g is a scalar distribution on the boundary. In almost classical regular problems g turns to be a function defined on the boundary with g ∈ L2 (Γ). We choose to work with the normal scalar shape differential equation obtained by setting def

v = hV(t, .), nt i = − A−1 Γt · g(Γt ) on Γt

(4.32)

∇Φ(t.) In terms of level set modeling, with nt = k∇Φ(t,.)k on Γt , it leads to the following normal-level shape differential equation :

∂t Φ = −A−1 Γt · g(Γt ) k∇Φk

(4.33)

It leads to the following normal Hamilton-Jacobi level set equation in the whole domain D : ∂t Φ + A−1 (4.34) Γt · G k∇Φk = 0 Assuming Φ is a solution to the previous normal Hamilton-Jacobi, in order to derive a solution to the previous normal scalar shape differential equation, we need to divide by k∇Φk. For that reason we now focus on a class of functions Φ without step so that k∇Φk is different from zero almost everywhere in D.

4.7.2

Solutions without step

We are interested in functions Φ(t, .) without steps. DEFINITION 4.1 step t if

We say that a function f defined on a set D has a meas({x ∈ D | f (x) = t }) > 0

Now we shall describe a construction of function without step, derived from a method, we have introduced for the modeling of free boundary value problems arising in plasma physics 8 . Consider, for any ε > 0 and for g ∈ H −1 (D), the variational problem z ∈ arg min 1

u∈H0 (D)

Z D

Z Z 1 ε k∇uk2 + (ε|D| − g) u dx − (u(x) − u(y))+ dx dy 2 2 D D

The associated Euler-Lagrange equation writes −∆z = ε β(z) + g, 8 The

in D

(4.35)

so-called Harold Grad Adiabatic equation of plasmas at equilibrium in the Tokomak [148], [149].

Shape differential equation and level set formulation with β(z)(x) = meas ({y ∈ D step condition :

|

meas ({x ∈ D

103

z(y) < z(x)}), together with the extra no

| z(x) = t}) = 0,

∀ t ∈ R+

(4.36)

REMARK 4.3

This method can be limited to zero step functions : Z Z 1 ε 2 z ∈ arg min k∇uk − g u dx − (u(x))+ dx 2 D u∈H01 (D) D 2

The associated Euler-Lagrange equation writes −∆z = ε β0 (z) + g,

in D

(4.37)

with β0 (z)(x) = χ{x∈D|z(x)>0} , together with the extra no step condition : meas ({x ∈ D

4.7.3

| z(x) = 0}) = 0

(4.38)

Iterative Scheme

Obviously the function χ must satisfy the constraint χ 2 = χ while the level set function Φ does not. Then, immediately from the previous study, we understand that in the second Hamilton-Jacobi equation (4.31), the velocity vector field V = A−1 ·∇J(Ωt (Φ)) only needs to be in L2 (0, τ, L2 (D)3 ) together with its divergence too. Indeed, in order to perform a fixed point argument inside the following iterative approximation scheme, ∂t Φn (t, x) − h∇Φn (t, x), Vn−1 i = 0 with

def

Vn−1 = A−1 · ∇J(Ωt (Φn−1 ))

we only need Vn ∈ L2 (0, τ ; L2 (D)3 ) and div Vn ∈ L2 (0, τ ; L∞ (D)). The idea is to choose a possibly non-linear operator A powerfull enough so that its inverse A−1 would map compactly and continuously the functional space where lies the cost function gradient into the previous velocity space. In almost all smooth problems, i.e., following the general assumptions of the structure derivative theorem, the cost gradient takes the following form, G = ∇J(Ωt (Φ)) = G(χt (Φ))∇χt (Φ) So that G is a distribution, with support on the boundary Γ, zero transverse order and lying in some negative Sobolev space over D. Furthermore, G ∈ W 1,1 (D) and is non-uniquely determined. Actually, only its trace 9 g on the 9 g is called the density gradient. It is a scalar distribution on the manifold Γ, when Γ is smooth enough in order to make sense.

104

Moving Shape Analysis

boundary Γ, if smooth enough, is intrinsically determined. We consider the following family of without step functions, (Aε )−1 (g) = { z solutions to (4.35), (4.36) } (A0ε )−1 (g) = { z solutions to (4.37), (4.38) } LEMMA 4.7 Let gn −→ g in H −1 (D) and zn ∈ (Aε )−1 (gn ) (resp. zn ∈ (A0ε )−1 (gn )). Then there exists a subsequence and a limit element z such that: znk −→ z weakly in H01 (D) Moreover any such element z verifies z ∈ (Aε )−1 (g) (resp. z ∈ (A0ε )−1 (g). The most important result concerning such elements without zero step is the following, LEMMA 4.8 Let Φn , Φ ∈ L2 (D). Assume that Φn −→ Φ strongly in L2 (D) and Φ is without zero step : meas({x ∈ D | Φ(x) = 0 }) = 0 Then let Ωn = {x ∈ D | Φn (x) > 0 }, Ω = {x ∈ D | Φ(x) > 0 }. We denote by χn and χ the respective characteristic functions of those subsets. Then χn −→ χ strongly in L2 (D). PROOF

Obviously, we have χn Φn = (Φn )+

There exists a subsequence and an element ζ, 0 ≤ ζ ≤ 1 such that χ n weakly converges to ζ in L2 (D). In the limit case, as |(Φn )+ (x) − (Φ)+ (x)| ≤ |Φn (x) − Φ(x)|, we get : ζ Φ = (Φ)+ So that ζ = 1 a.e. in Ω, and ζ = 0 a.e. in D − Ω. Since Φ has no zero step, we conclude that ζ = χ, so that the sequence converges strongly in L 2 (D). At that point, it is obvious that with some continuity assumption on the gradient G(χ) (that hypothesis will be satisfied in the following example), with respect to the characteristic function, the previous iterative construction Vn will converge and we will derive the existence of solutions to the Hamilton-Jacobi equation for the Level set function associated to the multivalued operators Aε and A0ε .

Shape differential equation and level set formulation

4.7.4

105

An example: the transverse magnetic like inverse problem

As an illustration for the construction of solution to the Hamilton-Jacobi equation, we consider a simplified version of the famous transverse magnetic inverse problem. The analysis is greatly simplified since we consider a bounded def universe D. Let Ω ⊂ D and y = y(χ) ∈ H01 (D) be the solution to the following problem, −∆y + k χ y = f def

where k is the contrast parameter while f is given in L 2 (D) and χ = χΩ . We introduce the observability functional, Z J(χ) = 1/2 (y − yd )2 dx E

where E ⊂ D. The classical shape derivative Here, we would like to compute the derivative of J(χ) with respect to χ. It is given by G(Ω) = γΓ∗ (y p n) (4.39) where p ∈ H01 (D) is the solution of the adjoint equation : −∆p + k χ p = χE ( y − yd )

(4.40)

The associated density gradient is given by g(Γ) = (y p)|Γ Let us consider the extension G of g inside D, G(χ) = y p ∈ W 2,1 (D) We easily check the continuity of the following mapping, χ ∈ L2 (D) −→ G(χ) ∈ W 2,1 (D) Topological derivative versus set derivative At that point, it is very interesting, on that example, to understand that the shape derivative of that functional J(Ω) can be relaxed to a set derivative setting which coincide in a smooth situation with the classical shape derivative and shape gradient analysis. We introduce the usual Lagrangian functional, Z (∇φ.∇ψ + k χΩ φ ψ − f ψ + χE 1/2 (φ − yd )2 ) dx (4.41) L(Ω, φ, ψ) = D

106

Moving Shape Analysis

Then J(Ω) =

min

max L(Ω, φ, ψ)

φ∈H 2 (D) ψ∈H 2 (D)

with unique saddle point (y, p). The Lagrangian L being partially concaveconvex, weakly lower semi-continuous and upper weakly semi-continuous, considering any evolutive characteristic function χ(t), we get (see([42]) : ∂ ∂ J(χ)|{t=0} = L(χ(t), y0 , p0 ))|{t=0} ∂t ∂t where y0 , p0 are the solution at t = 0. That derivative does make sense as soon as the right hand side derivative does. This technique enables us to manage holes creation inside D. Assume for example that Ωt is a vanishing sequence, as t → 0 of measurable subsets in D ⊂ Rd verifying: Ωn → {x0 } in Hausdorff topology (meas(Ωt ) / td ) → 0 as t → 0 ¯ we get : Then, assuming d = 2 so that y, p ∈ H 2 (D) ⊂ C 0 (D), ∂ J(χ(t)){t=0} = k (yp)(x0 ) ∂t This set derivative setting works thanks to the simple fact that χ(t) → δ x0 t (Dirac measure at point x0 ∈ D), as t → 0. Hence, we understand that the important concept in that situation is the derivative, in a measure space of the mapping t → χ(t) at t = 0. Following this line, we shall consider, when it exists, the following element: χ(t) ˙ = lim

ε→0

(χ(t + ε) − χ(t)) , ε

in M(D)

if the saddle point (yt , pt ) satisfies the following continuity property: ¯ t −→ (yt , pt ) continuous in H(D) ⊂ C 0 (D) We get the notion of set derivative for the set functional J(Ω), defined as follows, ∂ J(χ(t)) = < χ(t), ˙ G(χ(t)) > (4.42) ∂t In the previous example, we had G(χ) = y p. Such an example is considered in ([129]) for a transverse magnetic inverse shape problem in buried obstacle reconstruction.

Shape differential equation and level set formulation

107

Hole-differentiability Here we shall illustrate the various dependencies of the topological derivative in the case of the Laplace equation with Dirichlet boundary conditions. Let us consider the domain Ω = x ∈ R2 | kxk < 1

together with the perforated subdomain Ωε = {x ∈ Ω

| kxk > ε}

Let yε be the solution to the following Dirichlet problem,   ∆yε = 0, Ωε , yε (x) = 0, kxk = ε,  yε (x) = 1, kxk = 1

Obviously, we get

yε (x) = −

ln r + 1, ln ε

and y0 (x) = 1. Let us first consider the functional Z j 1 (ε) =

(

kxk=1

We get j 1 (ε) − j 1 (0) =

(4.43)

kxk = r

∂ yε (s))2 ds ∂n

2π → 0, ε → 0 (ln ε)2

Setting a1ε = ( ln1ε )2 , we get j 1 (ε) − j 1 (0) = 2π ε→0 a1ε lim

This limit is usually referred to as the topological derivative of the functional j 1 (ε). Now, let us choose a different functional, Z |∇yε (x)|2 dx j 2 (ε) = Ωε

We find j 2 (ε) − j 2 (0) = −

2π ln ε

Setting a2ε = −( ln1ε ), we get j 2 (ε) − j 2 (0) = 2π ε→0 a2ε lim

Hence, the choice of the scaling parameter aε depends not only on the type of state equation, but also on the type of functional.

Chapter 5 Dynamical shape control of the Navier-Stokes equations

5.1

Introduction

This chapter deals with the analysis of an inverse dynamical shape problem involving a fluid inside a moving domain. This type of inverse problem happens frequently in the design and the control of many industrial devices such as aircraft wings, cable-stayed bridges, automobile shapes, satellite reservoir tanks and more generally of systems involving fluid-solid interactions. The control variable is the shape of the moving domain, and the objective is to minimize a given cost functional that may be chosen by the designer. On the theoretical level, early works concerning optimal control problems for general parabolic equations written in non-cylindrical domains have been considered in [43], [29], [30], [142], [2]. In [140], [151], [152], the stabilization of structures using the variation of the domain has been addressed. The basic principle is to define a map sending the non-cylindrical domain into a cylindrical one. This process leads to the mathematical analysis of non-autonomous PDE’s systems. Recently, a new methodology to obtain Eulerian derivatives for non-cylindrical functionals has been introduced in [157], [156], [58]. This methodology was applied in [59] to perform dynamical shape control of the non-cylindrical NavierStokes equations where the evolution of the domain is the control variable. Hence the classical optimal shape optimization theory has been extended to deal with non-cylindrical domains. The aim of this chapter is to review several results on the dynamical shape control of the Navier-Stokes system and suggest an alternative treatment using the Min-Max principle [45, 46]. Despite its lack of rigorous mathematical justification in the case where the Lagrangian functional is not convex, we shall show how this principle allows, at least formally, to bypass the tedious computation of the state differentiability with respect to the shape of the moving domain.

109

110

5.2

Moving Shape Analysis

Problem statement

Let us consider a moving domain Ωt ∈ Rd . We introduce a diffeomorphic map sending a fixed reference domain Ω0 into the physical configuration Ωt at time t ≥ 0. Without loss of generality, we choose the reference configuration to be the physical configuration at initial time Ωt=0 . Hence we define a map Tt ∈ C 1 (Ω0 ) such that Ωt = Tt (Ω0 ),

We set Σ ≡

[

0
Γt = Tt (Γ0 ) [ ({t}×Γt ), Q ≡ ({t}×Ωt ). The map Tt can be actually 0
defined as the flow of a particular vector field, as described in the following lemma : THEOREM 5.1 [147] Ωt = Tt (V )(Ω0 ), Γt = Tt (V )(Γ0 ) where Tt (V ) is solution the of the following dynamical system : Tt (V ) : Ω0 −→ Ω

x0 7−→ x(t, x0 ) ≡ Tt (V )(x0 )

with

dx = V (τ, x(τ )), τ ∈ [0, T ] dτ x(τ = 0) = x0 , in Ω0

(5.1)

The fluid filling Ωt is assumed to be a viscous incompressible newtonian fluid. Its evolution is described by its velocity u and its pressure p. The couple (u, p) satisfies the classical Navier-Stokes equations written in nonconservative form,  ∂t u + D u · u − ν∆u + ∇p = 0, Q(V )    div(u) = 0, Q(V ) (5.2) u = V, Σ(V )    u(t = 0) = u0 , Ω0 where ν stands for the kinematic viscosity. The quantity σ(u, p) = −p I +ν(D u + ∗ D u) stands for the fluid stress tensor

Dynamical shape control of the Navier-Stokes equations

111

inside Ωt , with (D u)i,j = ∂j ui . We are interested in solving the following minimization problem : min j(V )

(5.3)

V ∈U

where j(V ) = JV (u(V ), p(V )) with (u(V ), p(V )) is a weak solution of problem (5.2) and JV (u, p) is a real functional of the following form : JV (u, p) =

α γ kB uk2Q(V ) + kK V k2Σ(V ) 2 2

(5.4)

where B ∈ L(H, H∗ ) is a general linear differential operator satisfying the following identity, hB u, vi + hu, B ∗ vi = hBΣ u, viL2 (Σ) (5.5) where H = v ∈ L2 (0, T ; (H01 (div, Ωt (V )))d ) and K ∈ L(U, L2 (Σ(V ))) is a general linear differential operator satisfying the following identity, hK u, viL2 (Σ) + hu, K∗ viL2 (Σ) = hKΣ u, viL2 (Σ)

(5.6)

The main difficulty in dealing with such a minimization problem is related to the fact that integrals over the domain Ωt (V ) depend on the control variable V . This point will be solved by using the Arbitrary Lagrange-Euler (ALE) map Tt (V ) introduced previously. The purpose of this chapter is to prove using several methods the following result, MAIN RESULT: For V ∈ U and Ω0 of class C 2 , the functional j(V ) possesses a gradient ∇j(V ) which is supported on the moving boundary Γ t (V ) and can be represented by the following expression, ∇j(V ) = −λ n − σ(ϕ, π) · n + α BΣ B u + γ [−K∗ K V + KΣ K V ]

(5.7)

where (ϕ, π) stands for the adjoint fluid state solution of the following system,  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = −α B ∗ B u, Q(V )    div(ϕ) = 0, Q(V ) (5.8) ϕ = 0, Σ(V )    ϕ(T ) = 0, ΩT and λ is the adjoint transverse boundary field, solution of the tangential dynamical system, −∂t λ − ∇Γ λ · V − (div V ) λ = f, (0, T ) (5.9) λ(T ) = 0, ΓT (V )

with f = [−(σ(ϕ, π) · n) + α BΣ B u] · (D V · n − D u · n) +

1 α|B u|2 + γ H|K V |2 2 (5.10)

112

Moving Shape Analysis

Example 5.1 We set (B, B ∗ , BΣ ) = (I, − I, 0) (K, K∗ , KΣ ) = (I, − I, 0) This means that we consider the cost functional, JV (u, p) =

α γ kuk2L2 (Q(V )) + kV k2L2 (Σ(V )) 2 2

(5.11)

Then its gradient is given by ∇j(V ) = −λ n − σ(ϕ, π) · n + γ V

(5.12)

where (ϕ, π) stands for the adjoint fluid state solution of the following system,  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = α u, Q(V )    div(ϕ) = 0, Q(V ) (5.13) ϕ = 0, Σ(V )    ϕ(T ) = 0, ΩT

and λ is the adjoint transverse boundary field, solution of the tangential dynamical system, −∂t λ − ∇Γ λ · V − (div V ) λ = f, (0, T ) (5.14) λ(T ) = 0, ΓT (V ) with 1 f = −ν(D ϕ · n) · (D V · n − D u · n) + (α + γ H)|V |2 2

(5.15)

Example 5.2 We set (B, B ∗ , BΣ ) = (curl, curl, ∧n)

(K, K∗ , KΣ ) = (I, − I, 0) JV (u, p) =

α γ k curl uk2L2 (Q(V )) + kV k2L2 (Σ(V )) 2 2

(5.16)

Dynamical shape control of the Navier-Stokes equations

113

Then its gradient is given by ∇j(V ) = −λ n − σ(ϕ, π) · n + α (curl u) ∧ n + γ V

(5.17)

where (ϕ, π) stands for the adjoint fluid state solution of the following system,  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = −α ∆u, Q(V )    div(ϕ) = 0, Q(V ) (5.18) ϕ = 0, Σ(V )    ϕ(T ) = 0, ΩT

and λ is the adjoint transverse boundary field, solution of the tangential dynamical system, −∂t λ − ∇Γ λ · V − (div V ) λ = f, (0, T ) (5.19) λ(T ) = 0, ΓT (V ) with f = [−ν D ϕ · n + α (curl u) ∧ n] · (D V · n − D u · n) 1 + α| curl u|2 + γ H|V |2 2

In the next section, we introduce several concepts closely related to shape optimization tools for moving domain problems. We also recall elements of tangential calculus that will be used through this chapter. Then we treat successively the following points, 1. In section (5.5), we choose to prove the differentiability of the fluid state (u, p) with respect to the design variable V . The directional shape derivative (u0 , p0 )(V ) · W is then used to compute the directional derivative j 0 (V ) · W of the cost functional j(V ). Using the adjoint state (ϕ, π)(V ) associated to (u0 , p0 )(V ) and the adjoint field Λ associated to the transverse field Zt introduced in section (5.3), we are able to furnish an expression of the gradient ∇j(V ) which is a distribution supported by the moving boundary Γt (V ). 2. In section (5.6), we choose to bypass the computation of the state shape derivative (u0 , p0 )(V ) · W , by using a Min-Max formulation of problem (5.3) and a transport technique. The state and multiplier spaces are chosen in order to be independent on the scalar perturbation parameter used in the computation of the derivative of the Lagrangian functional with respect to V . This method directly furnishes the fluid state and transverse field adjoint systems and the resulting gradient ∇j(V ).

114

Moving Shape Analysis

3. In section (5.7), we again use a Min-Max strategy coupled with a state and multiplier functional space embedding. This means that the state and multiplier variables live in the hold-all domain D. Hence the derivative of the Lagrangian functional with respect to V only involves terms coming from the flux variation through the moving boundary Γ t (V ). This again leads to the direct computation of the fluid state and transverse field adjoints and consequently to the gradient ∇j(V ).

5.3

Elements of non-cylindrical shape calculus

This section introduces several concepts that will be intensively used through this chapter. It concerns the differential calculus of integrals defined on moving domains or boundaries with respect to their support.

5.3.1

Non-cylindrical speed method

In this paragraph, we are interested in differentiability properties of integrals defined over moving domains, Z J1 (Ωt ) = f (Ωt ) dΩ Ωt

J2 (Γt ) =

Z

g(Γt ) dΓ Γt

The behaviour of J1 and J2 while perturbing their moving support highly depends on the regularity in space and time of the domains. In this work, we choose to work with domains Ωt that are images of a fixed domain Ω0 through an ALE map Tt (V ) as introduced in the first section. Hence, the design parameter is no more the support Ωt but rather the velocity field def V ∈ U = C 0 ([0, T ]; (W k,∞ (D))d ) that builds the support. This technique has the advantage to transform shape calculus into classical differential calculus on vector spaces [157],[59]. For an other choice based on the non-cylindrical identity perturbation, the reader is referred to the next chapter. Before stating the main result of this section, we recall the notion of transverse field. Transverse applications DEFINITION 5.1 The transverse map Tρt associated to two vector fields (V, W ) ∈ U is defined as follows, def

Tρt : Ωt −→ Ωρt = Ωt (V + ρW )

x 7→ Tt (V + ρW ) ◦ Tt (V )−1

Dynamical shape control of the Navier-Stokes equations

115

REMARK 5.1 The transverse map allows us to perform sensitivity analysis on functions defined on the unperturbed domain Ω t (V ). The following result states that the transverse map T ρt can be considered as a dynamical flow with respect to the perturbation variable ρ, THEOREM 5.2 [156] The Transverse map Tρt is the flow of a transverse field Zρt defined as follows ∂Tρt def Zρt = Z t (ρ, .) = ◦ (Tρt )−1 (5.20) ∂ρ i.e., is the solution of the following dynamical system : Ttρ (Zρt ) : Ωt −→ Ωρt

x 7−→ x(ρ, x) ≡ Ttρ (Zρt )(x)

with

d x(ρ) = Z t (ρ, x(ρ)), ρ ≥ 0 dρ x(ρ = 0) = x, in Ωt (V )

(5.21)

Since, we will mainly consider derivatives of perturbed functions at point def t ρ = 0, we set Zt = Zρ=0 . A fundamental result lies in the fact that Zt can be obtained as the solution of a linear time dynamical system depending on the vector fields (V, W ) ∈ U, THEOREM 5.3 [59] The vector field Zt is the unique solution of the following Cauchy problem, ∂t Zt + [Zt , V ] = W, D × (0, T ) (5.22) Zt=0 = 0, D def

where [Zt , V ] = DZt · V − DV · Zt stands for the Lie bracket of the pair (Zt , V ). Shape derivative of non-cylindrical functionals The main theorem of this section uses the notion of a non-cylindrical material derivative that we recall here, DEFINITION 5.2 The derivative with respect to ρ at point ρ = 0 of the following composed function, f ρ : [0, ρ0 ] → H(Ωt (V )) ρ 7→ f (V + ρW ) ◦ Tρt

116

Moving Shape Analysis

f˙(V ; W ) is called the non-cylindrical material derivative of f (V ) at point V ∈ U in the direction W ∈ U. We shall use the notation, def d ρ f˙(V ) · W = f˙(V ; W ) = f dρ ρ=0

With the above definition, we can state the differentiability properties of non-cylindrical integrals with respect to their moving support,

THEOREM 5.4 [59] For a bounded measurable domain Ω0 with boundary Γ0 , let us assume that for any direction W ∈ U the following hypothesis holds, i) f (V ) admits a non-cylindrical material derivative f˙(V ) · W

then J1 (.) is Gˆ ateaux differentiable at point V ∈ U and its derivative is given by the following expression, Z h i f˙(V ) · W + f (V ) div Zt dΩ (5.23) J10 (V ) · W = Ωt (V )

Futhermore, if ii) f (V ) admits a non-cylindrical shape derivative given by the following expression, f 0 (V ) · W = f˙(V ) · W − ∇f (V ) · Zt (5.24) then J10 (V

)·W =

Z

Ωt (V )

[f 0 (V ) · W + div(f (V ) Zt )] dΩ

(5.25)

Furthermore, if Ω0 is an open domain with a Lipschitzian boundary Γ0 , then Z Z 0 0 J1 (V ) · W = f (V ) · W dΩ + f (V ) hZt , nidΓ (5.26) Ωt (V )

Γt (V )

REMARK 5.2 The last identity will be of great interest while trying to prove a gradient structure result for general non-cylindrical functionals. It is also possible to establish a similar result for integrals over moving boundaries. For that purpose, we need to define the non-cylindrical tangential material derivative, DEFINITION 5.3 The derivative with respect to ρ at point ρ = 0 of the following composed function, g ρ : [0, ρ0 ] → H(Γt (V )) ρ 7→ g(V + ρW ) ◦ Tρt

Dynamical shape control of the Navier-Stokes equations

117

is called the non-cylindrical material derivative of the function g(V ) ∈ H(Γt (V )) in the direction W ∈ U. We shall use the notation, def d ρ g(V ˙ ) · W = g(V ˙ ;W) = g dρ ρ=0

This concept is involved in the differentiability property of boundary integrals, THEOREM 5.5 For a bounded measurable domain Ω0 with boundary Γ0 , let us assume that for any direction W ∈ U the following hypothesis holds, i) g(V ) admits a non-cylindrical material derivative g(V ˙ )·W then J2 (.) is Gˆ ateaux differentiable at point V ∈ U and its derivative is given by the following expression, Z J20 (V ) · W = [g(V ˙ ) · W + g(V ) divΓ Zt ] dΓ (5.27) Γt (V )

Futhermore, if ii) g(V ) admits a non-cylindrical shape derivative given by the following expression, g 0 (V ) · W = g(V ˙ ) · W − ∇Γ g(V ) · Zt (5.28) then J20 (V ) · W =

Z

Γt (V )

[˜ g 0 (V ) · W + H g(V )hZt , ni] dΓ

(5.29)

where H stands for the additive curvature (Def. (5.4)). Furthermore, if g(V ) = g˜(V )|Γt (V ) with g˜ ∈ H(Ωt (V )), then Z J20 (V ) · W = [g 0 (V ) · W + (∇˜ g (V ) · n + H g(V )) hZt , ni] dΓ (5.30) Γt (V )

Adjoint transverse field It is possible to define the solution of the adjoint transverse system, THEOREM 5.6 [58] For F ∈ L2 (0, T ; (H 1 (D))d ), there exists a unique field Λ ∈ C 0 ([0, T ]; (L2 (D))d ) solution of the backward dynamical system, −∂t Λ − D Λ · V − ∗ D V · Λ − (div V )Λ = F, (0, T ) Λ(T ) = 0

(5.31)

118

Moving Shape Analysis

REMARK 5.3 The field Λ is the dual variable associated to the transverse field Zt and is the solution of the adjoint problem associated to the transverse dynamical system. In this chapter, we shall deal with a specific right-hand side F of the form F (t) = ∗ γΓt (V ) (f (t)n). In this case, the adjoint field Λ is supported on the moving boundary Γt (V ) and has the following structure, THEOREM 5.7 [59] For F (t) = ∗ γΓt (V ) (f (t)n), with f ∈ L2 (0, T ; L2 (Γt (V ))), the unique solution Λ of the problem is given by the following identity, Λ = (λ ◦ p) ∇ χΩt (V ) ∈ C 0 ([0, T ]; (H 1 (Γt ))d )

(5.32)

where λ ∈ C 0 ([0, T ]; H 1 (Γt )) is the unique solution of the following boundary dynamical system,

−∂t λ − ∇Γ λ · V − (div V )λ = f, (0, T ) λ(T ) = 0, Γt (V )

(5.33)

p is the canonical projection on Γt (V ) and χΩt (V ) is the characteristic function of Ωt (V ) inside D. Gradient of non-cylindrical functionals In the next sections, we will often deal with boundary integrals of the following forms, Z TZ E hZt , ni K= 0

Γt (V )

with E ∈ L2 (0, T ; Γt (V )) and Zt is the solution of the transverse equation (5.22). The following result allows us to eliminate the auxiliary variable Z t inside the functional K, THEOREM 5.8 [59] For any E ∈ L2 (0, T ; Γt (V )) and (V, W ) ∈ U, the following identity holds, Z

T 0

Z

Γt (V )

E hZt , ni = −

Z

T 0

Z

Γt (V )

λ hW, ni

(5.34)

where λ ∈ C 0 ([0, T ]; H 1 (Γt )) is the unique solution of problem (5.33) with f = E.

Dynamical shape control of the Navier-Stokes equations

5.4

119

Elements of tangential calculus

In this section, we review basic elements of differential calculus on a C k submanifold with k ≥ 2 of codimension one in Rd . The following approach avoids the use of local bases and coordinates by using the intrinsic tangential derivative.

5.4.1

Oriented distance function

Let Ω be an open domain of class C k in Rd with compact boundary Γ. We define the oriented distance function to be as follows, dΓ (x), x ∈ Rd \ Ω bΩ (x) = −dΓ (x), x ∈ Ω where dΓ (x) = min |y − x|. y∈Ω

PROPOSITION 5.1 [51] Let Ω be an open domain of class C k for k ≥ 2 in Rd with compact boundary Γ. There exists a neighbourhood U (Γ) of Γ, such that b ∈ C k (U (Γ)). Furthermore, we have the following properties, i) ∇b|Γ = n, where n stands for the unit exterior normal on Γ, ii) D2 b : Tp(x) Γ → Tp(x) Γ coincides with the second fundamental form on Γ, where p : U (Γ) → Γ x 7→ x − b(x) · ∇b(x) stands for the projection mapping and Tp(x) Γ stands for the tangent plane. iii) (0, β1 , . . . , βd−1 ) are the eigenvalues of D2 b associated to the eigenfunctions (n, µ1 , . . . , µd−1 ) where (βi , µi )1≤i≤d−1 are the mean curvatures and principal direction of curvatures of Γ. PROPOSITION 5.2 [51] For Γ of class C 2 , the projection mapping p is differentiable and its derivative

120

Moving Shape Analysis

has the following properties, ∗

D p = D p = I −∇b · ∗ ∇b − b D2 b D p · τ = τ, on Γ, D p · n = 0, on Γ

(5.35)

DEFINITION 5.4 [51] For Γ of class C 2 , the additive curvature H of Γ is defined as the trace of the second order fundamental form : ¯ H = Tr D2 b = ∆b = (d − 1)H,

on Γ

(5.36)

¯ stands for the mean curvature of Γ. and H

5.4.2

Intrinsic tangential calculus

Using arbitrary smooth extensions of functions defined on Γ to Ω ∈ R d is the most classical way of defining tangential operators. Hence the differential calculus on manifolds can be reduced to classical differential calculus in R d . In this section we recall standard formulas for differential tangential operators using arbitrary extensions. We also emphasize the particular case where the extension is of the canonical type (f ◦ p). This is the basis of a simple differential calculus in the neighbourhood of Γ. DEFINITION 5.5 For Γ of class C 2 , given any extension F ∈ C 1 (U (Γ)) of f ∈ C 1 (Γ), the tangential gradient of f is defined as def

∇Γ f = ∇F |Γ − (∂n F ) n

(5.37)

where ∂n F = ∇F · n. PROPOSITION 5.3 [51] Assume that Γ of class C 2 is compact and f ∈ C 1 (Γ), then i) ∇Γ f = (P ∇F )|Γ n · ∇Γ f = ∇b · ∇Γ f = 0

(5.38)

def

where P = I −∇b∗ ∇b is the orthogonal projection operator onto the tangent plane Tp(x) Γ. ii) ∇(f ◦ p) = I −b D2 b ∇Γ f ◦ p ∇(f ◦ p)|Γ = ∇Γ f

(5.39)

Dynamical shape control of the Navier-Stokes equations

121

Hence (f ◦ p) plays the role of a canonical extension in the neighbourhood U (Γ) and its gradient is tangent to the level sets of b. Consequently, we can define in an intrinsic way the tangential gradient, DEFINITION 5.6 For Γ of class C 2 and f ∈ C 1 (Γ), the tangential gradient of f is defined as ∇Γ f = ∇(f ◦ p)|Γ

(5.40)

In the sequel, we shall use the above definition for the tangential gradient whenever the function under derivation is intrinsically defined on Γ. We now define the other classical tangential operators, DEFINITION 5.7

For Γ of class C 2 ,

i) for v ∈ (C 1 (Γ))d , and v˜ ∈ (C 1 (U (Γ)))d an arbitrary extension, the tangential jacobian is defined as follows, def

DΓ v = D v˜|Γ − (D v˜ · n)∗ n = D v˜|Γ − D v˜ · (n ⊗ n)

(5.41)

Furthermore, D(v ◦ p) = DΓ v ◦ p I −b D2 b

DΓ v = D(v ◦ p)|Γ

(5.42)

ii) for v ∈ (C 1 (Γ))d , and v˜ ∈ (C 1 (U (Γ)))d an arbitrary extension, the tangential divergence is defined as follows, def

divΓ v = div v˜|Γ − (D v˜ · n) · n

(5.43)

divΓ v = div(v ◦ p)|Γ = Tr(DΓ v)

(5.44)

Furthermore,

iii) for f ∈ C 2 (Γ), and F ∈ C 2 (U (Γ)) an arbitrary extension, the tangential Laplace-Beltrami operator is defined as follows, ∆Γ f = ∆F |Γ − H∂n F − ∂n2 F

(5.45)

with ∂n2 F = (D2 F · n) · n. Furthermore, ∆Γ f = divΓ (∇Γ f ) = ∆(f ◦ p)|Γ

(5.46)

122

Moving Shape Analysis

In some cases, it may be interesting to use a splitting of the function v onto a normal and a tangential component, DEFINITION 5.8 For v ∈ (C 1 (Γ))d , we define the tangential component vΓ ∈ (C 1 (Γ))d and the normal component vn ∈ C 1 (Γ) such that v = v Γ + vn n

(5.47)

Using the above definition, we obtain the following identities, PROPOSITION 5.4 For v ∈ (C 1 (Γ))d , we have DΓ v = D Γ v Γ + v n D 2 b + n · ∗ ∇Γ v n

(5.48)

∇Γ v n = ∗ DΓ v · n + D 2 b · v Γ

(5.49)

divΓ v = divΓ vΓ + H vn

5.4.3

(5.50)

Tangential Stokes formula

In order to perform integration by parts on Γ, we will use the following tangential Stokes identity, PROPOSITION 5.5 Let Γ be a C 2 -submanifold in Rd ; for E ∈ H 1 (Γ; Rd ) and ψ ∈ H 1 (Γ; R) the following identity holds Z Z Z (5.51) H ψ hE, niRd hE, ∇Γ ψiRd + (divΓ E)ψ = Γ

5.5

Γ

Γ

State derivative strategy

In this section, we shall prove the main theorem of this chapter using an approach based on the differentiability of the solution of the Navier-Stokes system (Eq. (5.2)) with respect to the velocity field V . First, we introduce a weak formulation for Eq. (5.2) and recall the associated classical solvability result. Then, using the weak implicit function theorem, we will prove the existence of a weak material derivative. Finally, introducing the adjoint equations associated to the linearized fluid and transverse systems, we will be able to express the gradient of the functional j(V ). For the sake of simplicity, we shall only prove the main theorem in the case of example (5.1)-1 with free divergence control velocity fields.

Dynamical shape control of the Navier-Stokes equations

5.5.1

123

Weak formulation and solvability

In order to take into account the non-homogeneous Dirichlet boundary condition on Γt (V ), we use the following change of variable u ˜ = u − V , where u ˜ satisfies the following homogeneous Dirichlet Navier-Stokes system,  ∂t u ˜ + Du ˜·u ˜ + Du ˜ · V + DV · u ˜ − ν∆˜ u + ∇p = F (V ), Q(V )    div(˜ u) = 0, Q(V ) (5.52) u ˜ = 0, Σ(V )    u ˜(0) = u0 − V (0), Ω0

with F (V ) = −∂t V − D V · V + ν∆V . We consider the following classical functional spaces [97], [139], H(D) = v ∈ (L2 (D))d , div v = 0, in D, v · n = 0 on ∂D H01 (div, D) = v ∈ (H01 (D))d , div v = 0, in D H = v ∈ L2 (0, T ; (H01 (div, Ωt (V )))d ) V = v ∈ H, ∂t v ∈ L2 (0, T ; (H01 (Ωt (V )))d )

In the sequel, we shall use the notation u instead of u ˜, keeping in mind that the original variable is obtained by translation. DEFINITION 5.9 The function u ∈ V is called a weak solution of problem (5.52), if it satisfies the following identity, heV (u), vi = h[e1V (u), e2V (u)], vi = [0, 0],

∀v ∈ H

(5.53)

with Z

T 0

Z

he1V (u), vi = Ωt (V )

[(∂t u + D u · u + D u · V + D V · u) · v + ν D u · · D v] − he2V

(u), vi =

Z

We set U = V ∈ H 1 (0, T ; (H m (D))d ),

with m > 5/2.

Ω0

Z

T 0

Z

Ωt (V )

(u(0) − u ˜0 ) · v(0)

div V = 0 in D,

(5.54)

F (V ) · v (5.55)

V · n = 0 on ∂D

THEOREM 5.9 [58] We assume the domain Ω0 to be of class C 1 . For V ∈ U and u0 ∈ H(D) such that u0 |Ω0 ∈ H(Ω0 ),

124

Moving Shape Analysis

1. it exists at least a weak solution of problem (5.52) with u ∈ H ∪ L∞ (0, T ; H), 2. if u0 ∈ (H 2 (D))d ∪ H01 (div, D) and ν is large or u0 is a small data, then the uniqueness of a weak solution is guaranteed, and we have ∂t u ∈ H ∪ L∞ (0, T ; H(Ωt )), 3. if Ω is of class C 2 , u ∈ L∞ (0, T ; (H 2 (Ωt ))d ∪ H01 (div, Ωt )).

5.5.2

The weak Piola material derivative

We are interested in solving the following minimization problem : min j(V )

(5.56)

V ∈U

with j(V ) =

α 2

Z

T 0

Z

Ωt (V )

|u(V )|2 +

γ 2

Z

T 0

Z

Ωt (V )

|V |2

(5.57)

In order to derive first-order optimality conditions for problem (5.56), we need to analyse the derivability of the state u(V ) with respect to V ∈ U. There exist at least two methods in order to establish such a differentiability result: • Limit analysis of the differential quotient, u(V ˙ ; W ) = lim

ρ→0

1 (u(V + ρW ) ◦ Tt (V + ρW ) − u(V ) ◦ Tt (V )) ρ

• Application of the weak implicit function theorem and deduction of the local differentiability of the solution u(V ) associated to the implicit equation heV (u), vi = 0, ∀ v ∈ H. We recall here how the weak implicit function theorem can be applied following the result obtained in [59]. In order to work with divergence free functions, we need to introduce the Piola transform that preserves the free divergence condition. LEMMA 5.1 [13] The Piola transform, PI : H01 (div, Ωt (V )) −→ H01 (div, Ωρt ) v 7→ (D Tρt · v) ◦ (Tρt )−1 is an isomorphism.

Dynamical shape control of the Navier-Stokes equations

125

We consider the solution uρ = u(V + ρW ) defined on Ωρt of the following implicit equation, he(V +ρW ) (u), vi = 0, ∀ v ∈ Hρ with

Hρ = v ∈ L2 (0, T ; (H01 (div, Ωt (V + ρ W )))d )

We introduce the element u ˆρ = (D Tρt )−1 · (uρ ◦ Tρt ) defined on Ωt (V ) (i.e., uρ = PI (ˆ uρ ). LEMMA 5.2 The element uρ satisfies the following identity, he(V +ρW ) (u), vi = 0,

∀ v ∈ Hρ

if and only if u ˆρ satisfies the following identity, heρ (ˆ uρ ), vˆi = 0,

∀ vˆ ∈ H

with

heρ1 (v), wi = Z

T 0

Z

Ωt (V )

(∂t (D Tρt · v)) · (D Tρt · w)

−(D(D Tρt · v) · (D Tρt )−1 · (∂t Tρt )) · (D Tρt · w) + (D(D Tρt · v) · v) · (D Tρt · w) +(D(D Tρt · v) · (D Tρt )−1 · ((V + ρW ) ◦ Tρt )) · (D Tρt · w) +(D((V + ρW ) ◦ Tρt ) · v) · (D Tρt · w) +ν(D(D Tρt · v) · (D Tρt )−1 ) · ·(D(D Tρt · w) · (D Tρt )−1 ) −(F (V + ρW ) ◦ Tρt ) · (D Tρt · w) heρ2 (v), wi

=

Z

Ω

(v(0) − u ˆ0 ) · w

with F (V ) = −∂t V − D V · V + ν∆V

126

Moving Shape Analysis

and ∂t Tρt = (V + ρW ) ◦ Tρt − D Tρt · V PROOF We consider the solution uρ of the perturbed state equation e(V +ρW ) = 0, with he(V +ρW ) (u), vi =

Z

T 0

Z

Ωρ t

∂t u + D u · u + D u · (V + ρW )

+ D(V + ρW ) · u · v + ν D u · · D v − F (V + ρW ) · v

with v ∈ Hρ . We introduce the variables (ˆ u, vˆ) defined in Ω t (V ) such that [u, v] = (D Tρt · u ˆ) ◦ (Tρt )−1 , (D Tρt · vˆ) ◦ (Tρt )−1

We replace this new representation inside the state equation and we use a back transport in Ωt (V ). This leads to the following identity, heρ1 (ˆ u), vˆi =

Z

Q(V )

∂t ((D Tρt · u ˆ) ◦ (Tρt )−1 )

+ D((D Tρt · u ˆ) ◦ (Tρt )−1 ) · (D Tρt · u ˆ) ◦ (Tρt )−1

+ D((D Tρt · u ˆ) ◦ (Tρt )−1 ) · (V + ρW ) + D(V + ρW ) · (D Tρt · u ˆ) ◦ (Tρt )−1 ◦ Tρt · (D Tρt · vˆ)

+ν D((D Tρt · u ˆ) ◦ (Tρt )−1 ) ◦ Tρt · ·(D((D Tρt · vˆ) ◦ (Tρt )−1 )) ◦ Tρt

−F (V + ρW ) ◦ Tρt · (D Tρt · vˆ)

LEMMA 5.3

D((Tρt )−1 ) ◦ Tρt = (D Tρt )−1

∂t ((Tρt )−1 ) ◦ Tρt = −(D Tρt )−1 · ∂t Tρt

∂t ((D Tρt · u ˆ)◦(Tρt )−1 )◦Tρt PROOF

=

∂t (D Tρt ◦ u ˆ)−D(D Tρt ◦ u ˆ)·(D Tρt )−1 ·∂t Tρt

Using the identity, (Tρt )−1 ◦ Tρt = I

we get D((Tρt )−1 ◦ Tρt ) = I

D((Tρt )−1 ) ◦ Tρt · D Tρt = I

(5.58) (5.59) (5.60)

Dynamical shape control of the Navier-Stokes equations

127

By differentiation with respect to time t, we also get ∂t ((Tρt )−1 ◦ Tρt ) = 0

∂t ((Tρt )−1 ) ◦ Tρt + D((Tρt )−1 ) ◦ Tρt · ∂t Tρt = 0

∂t ((Tρt )−1 ) ◦ Tρt + (D Tρt )−1 · ∂t Tρt = 0

Using the chain rule, we deduce [∂t ((D Tρt · u ˆ) ◦ (Tρt )−1 )] ◦ Tρt = [∂t (D Tρt · u ˆ) ◦ (Tρt )−1

+ D(D Tρt · u ˆ) ◦ (Tρt )−1 · ∂t ((Tρt )−1 )] ◦ Tρt

= ∂t (D Tρt · u ˆ) + D(D Tρt · u ˆ) · ∂t ((Tρt )−1 ) ◦ Tρt = ∂t (D Tρt · u ˆ) − D(D Tρt · u ˆ) · (D Tρt )−1 · ∂t Tρt

In order to get the correct state operator, we need also the following identities, LEMMA 5.4

D(φ ◦ (Tρt )−1 ) ◦ Tρt = D(φ) · (D Tρt )−1

(5.61)

D(V + ρW ) ◦ Tρt · (D Tρt ) = D((V + ρW ) ◦ Tρt )

(5.62)

We shall apply the first identity with φ = (D Tρt · u ˆ). Finally, using all the identities proven above, we deduce the expression of e 1ρ (ˆ u, vˆ). Now, we simply need to prove the following lemma in order to conclude the proof, LEMMA 5.5 ∂t Tρt = (V + ρW ) ◦ Tρt − D Tρt · V PROOF

(5.63)

We use the definition of the Transverse map,

∂t (Tρt ) = ∂t (Tt (V + ρW ) ◦ Tt (V )−1 )

= ∂t (Tt (V + ρW )) ◦ Tt (V )−1 + D(Tt (V + ρW )) ◦ Tt (V )−1 · ∂t (Tt (V )−1 ) = ((V + ρW ) ◦ Tt (V + ρW )) ◦ Tt (V )−1

− D(Tt (V + ρW )) ◦ Tt (V )−1 · (D Tt−1 (V )) · ∂t (Tt (V )) ◦ Tt−1 (V ) = (V + ρW ) ◦ Tρt − D(Tt (V + ρW ) ◦ Tt (V )−1 ) · ∂t (Tt (V )) ◦ Tt−1 (V )

= (V + ρW ) ◦ Tρt − D(Tρt ) · V

128

Moving Shape Analysis

We now consider the application, [0, ρ0 ] × V → H∗ × H01 (div, Ω0 ) (ρ, v) 7→ eρ (v)

(5.64)

[0, ρ0 ] → H ρ 7→ u ˆρ = (D Tρt )−1 · (uρ ◦ Tρt )

(5.65)

and

where u ˆρ ∈ V is the solution of the state equation, heρ (v), wi = 0,

∀w ∈ H

(5.66)

LEMMA 5.6 [135] For any F ∈ H s (D), with s ≥ 1, 1 ρ→0 (F ◦ Tρt − F ) −→ ∇ F · Zt ρ

(5.67)

strongly in H s−1 (D). In the case s < 1, the convergence only holds weakly in H s−1 (D). In order to prove the differentiability of u ˆ ρ with respect to ρ in a neighbourhood of ρ = 0, we cannot use the classical implicit function theorem, since it requires strong differentiability results in H −1 for our application. Then we shall use the weak implicit function theorem, recalled below, THEOREM 5.10 [147] Let X, Y ∗ be two Banach spaces, I an open bounded set in R, and consider the following mapping, e:I ×X → Y∗

(ρ, x) 7→ e(ρ, x)

Let us assume the following hypothesis, a)

(i) the application ρ 7→ he(ρ, x), yi is continuously differentiable for any y ∈ Y ,

(ii) the application (ρ, x) 7→ h∂ρ e(ρ, x), yi is continuous. b) It exists u ∈ X such that u ∈ C 0,1 (I; X) e(ρ, u(ρ)) = 0, ∀ ρ ∈ I

Dynamical shape control of the Navier-Stokes equations

129

c) x 7→ e(ρ, x) is differentiable and (ρ, x) 7→ ∂x e(ρ, x) is continuous.

d) It exists ρ0 ∈ I such that ∂x e(ρ, x)|(ρ0 ,x(ρ0 )) ∈ ISOM(X, Y ∗ ).

Then the mapping

u(.) : I → X ρ 7→ u(ρ) is differentiable at point ρ = ρ0 for the weak topology in X and its weak derivative u(ρ) ˙ is the solution of the following linearized equation, h∂x e(ρ0 , u(ρ0 )) · u(ρ ˙ 0 ), yi + h∂ρ e(ρ0 , u(ρ0 )), yi = 0,

∀y ∈ Y

(5.68)

In order to apply the above theorem to Eq. (5.66), we need to state the following properties, LEMMA 5.7 The mapping, [0, ρ0 ] → R ρ 7→ heρ (v), wi

(5.69)

is C 1 for any (v, w) ∈ V ×H and its derivative is given by the following expression, h∂ρ eρ1 (v), wi = Z

Q(V )

(∂t (D(Zρt · Tρt ) · v)) + (D(D(Zρt · Tρt ) · v) · V + (D(D(Zρt · Tρt ) · v) · v)

+ D (D(V + ρW ) · Zρt ) ◦ Tρt + W ◦ Tρt · v − ∂ρ (F (V + ρW ) ◦ Tρt ) ·(D Tρt ·w) + (∂t (D Tρt · v)) + (D(D Tρt · v) · V + (D(D Tρt · v) · v)

+(D((V + ρW ) ◦ Tρt ) · v) − (F (V + ρW ) ◦ Tρt · (D(Zρt · Tρt ) · w) + ν(D(D(Zρt ◦ Tρt ) · v) · (D Tρt )−1 ) · ·(D(D Tρt · w) · (D Tρt )−1 )

− ν(D(D Tρt · v) · (D Tρt )−1 ) · D(Zρt ◦ Tρt ) · (D Tρt )−1 ) · ·(D(D Tρt · w) · (D Tρt )−1 ) + ν(D(D Tρt · v) · (D Tρt )−1 ) · ·(D(D(Zρt ◦ Tρt ) · w) · (D Tρt )−1 ) − ν(D(D Tρt · v) · (D Tρt )−1 ) · ·(D(D Tρt · w) · (D Tρt )−1 ) · D(Zρt ◦ Tρt ) · (D Tρt )−1 )

130

Moving Shape Analysis

PROOF that

We first simplify the expression of the weak state operator, using

∂t Tρt = (V + ρW ) ◦ Tρt − D Tρt · V and we get heρ1 (v), wi = Z TZ 0

Ωt (V )

(∂t (D Tρt · v)) + (D(D Tρt · v) · V + (D(D Tρt · v) · v)

+(D((V + ρW ) ◦ Tρt ) · v) − (F (V + ρW ) ◦ Tρt · (D Tρt · w)

+ ν(D(D Tρt · v) · (D Tρt )−1 ) · ·(D(D Tρt · w) · (D Tρt )−1 )

We use the expression of the weak state operator and the following identities, ∂ρ Tρt = Zρt ◦ Tρt

∂ρ (D Tρt )−1 = −(D Tρt )−1 · D(Zρt ◦ Tρt ) · (D Tρt )−1

LEMMA 5.8 The mapping, [0, ρ0 ] × V → H∗ (ρ, v) 7→ ∂ρ eρ (v)

(5.70)

is weakly continuous. PROOF We can prove that for (V, W ) ∈ V, the associated flow T ρt ∈ C 1 ([0, ρ0 [; C 2 (D, R3 )), and the weak continuity follows easily. In order to apply the implicit function derivative identity, we need to express the derivative ∂ρ eρ (v) at point ρ = 0,

Dynamical shape control of the Navier-Stokes equations

131

LEMMA 5.9 h∂ρ eρ1 |ρ=0 (v), wi = Z

Q

[∂t (D Zt · v) + D(D Zt · v) · V + D(D Zt · v) · v + D [D V · Zt + W ] · v] · w + [∂t v + D v · V + D v · v + D V · v] · (D Zt · w) + ν D(D Zt · v) · · D w − ν(D v · D Zt ) · · D w + ν D v · · D(D Zt · w) − ν D v · ·(D w · D Zt ) + [∂t W + D W · V + D V · W − ν∆W ] · w

+ (D [∂t V + D V · V − ν∆V ] · Zt ) · w + [∂t V + D V · V − ν∆V ] · (D Zt · w) PROOF We set ρ = 0 in the expression of h∂ρ eρ (v), wi and we use the following identities, t Tρ=0 =I def

Zρt |ρ=0 = Zt

LEMMA 5.10 The mapping, V → H∗

v 7→ eρ (v)

(5.71)

is differentiable for any ρ ∈ [0, ρ0 ] and its derivative is given by the following expression, h∂v eρ1 (v) · δv, wi = Z

Q(V )

(∂t (D Tρt · δv)) + D(D Tρt · δv) · V + D(D Tρt · δv) · v + D(D Tρt · v) · δv

+(D((V + ρW ) ◦ Tρt ) · δv) +ν(D(D Tρt ·δv)·(D Tρt )−1 )··(D(D Tρt ·w)·(D Tρt )−1 )

and the mapping,

[0, ρ0 ] × V → L(V; H∗ ) (ρ, v) 7→ ∂v eρ (v)

(5.72)

132

Moving Shape Analysis

is continuous. LEMMA 5.11 The mapping, V→F δv 7→ ∂v eρ=0 (v) · δv

(5.73)

is an isomorphism and its expression is furnished by the following identity, h∂v eρ=0 (v) · δv, wi = 1 Z [(∂t δv) · w + (D δv · v) · w + (D v · δv) · w + (D δv · V ) · w Q(V )

+(D V · δv) · w + ν D δv · · D w]

PROOF This result follows from the uniqueness result for the NavierStokes system under regularity and smallness assumptions (see Th. (5.9) and [139]). Indeed, for u1 and u2 solutions of the Navier-Stokes equations, it is proven that the element y = u1 − u2 satisfying the following identity,

Z

Q(V )

[(∂t y) · w + (D y · u1 ) · w + (D u2 · y) · w + (D y · V ) · w + (D V · y) · w +ν D y · · D w] = 0,

∀w ∈ H

exists and is identically equal to the null function. Similar a priori estimates hold for δv and the unique solvability of the linearized system is established.

LEMMA 5.12 The solution u ˆρ ∈ V of the implicit equation, heρ (v), wi = 0,

∀w ∈ H

(5.74)

is Lipschitz with respect to ρ. PROOF We need the identity satisfied by u ˆ ρ1 − u ˆρ2 and we shall follow the same steps described in [56] (pp. 31). Hence the hypothesis of Th. (5.10) is satisfied by the Eq. (5.66) and we can state the following differentiability result,

Dynamical shape control of the Navier-Stokes equations

133

THEOREM 5.11 The Piola material derivative u˙ P = ∂ρ (ˆ uρ )|ρ=0 exists and is characterized by the linear tangent equation, h∂v eρ=0 (v)|v=ˆu · u˙P , wi + h∂ρ eρ (ˆ u)|ρ=0 , wi = 0,

∀w ∈ H

(5.75)

which possesses the following structure, Z

Q(V )

(∂t u˙ P ) · w + (D u˙ P · u) · w + (D u · u˙ P ) · w + (D u˙ P · V ) · w

+(D V · u˙ P ) · w + ν D u˙ P · · D w = hL(u, Zt , V, W ), wi

with

hL(u, Zt , V, W ), wi = −

Z

Q

[∂t (D Zt · u) + D(D Zt · u) · V + D(D Zt · u) · u + D [D V · Zt + W ] · u]·w

− [∂t u + D u · V + D u · u + D V · u] · (D Zt · w) − ν D(D Zt · u) · · D w + ν(D u · D Zt ) · · D w − ν D u · · D(D Zt · w) + ν D u · ·(D w · D Zt ) + [−∂t W − D W · V − D V · W + ν∆W ] · w − (D [∂t V + D V · V − ν∆V ] · Zt ) · w − [∂t V + D V · V − ν∆V ] · (D Zt · w)

5.5.3

Shape derivative

In the last section, we have proven that the solution u(V ) of the moving Navier-Stokes system is differentiable with respect to the velocity V . We have also characterized the linearized system satisfied by the Piola material derivative u˙ P (V ) · W . In this paragraph, we will identify the shape derivative u0 (V ) · W under some regularity assumptions. Let us consider the weak solution u ˜ of Eq. (5.52), i.e., heV (˜ u), vi = h[e1V (u), e2V (u)], vi = [0, 0],

∀v ∈ H

(5.76)

with he1V (˜ u), vi =

Z

Q(V )

[(∂t u ˜ + Du ˜·u ˜ + Du ˜ · V + DV · u ˜) · v +2νε(˜ u) · ·ε(v) − F (V ) · v]

134

Moving Shape Analysis Z 2 heV (˜ u), vi = (˜ u(0) − u0 ) · v(0)

(5.77)

Ω0

where ε(v) = 21 (D v + ∗ D v) stands for the symmetrical deviation tensor. This definition is motivated by the following lemma, LEMMA 5.13 Z Z −ν ∆u · v = 2ν Ωt

Ωt

ε(u) · ·ε(v) − 2ν

Z

Γt

hε(u) · n, vi,

∀ v ∈ H 1 (div, Ωt ) (5.78)

THEOREM 5.12 For Ω0 of class C 2 , the shape derivative u ˜0 = u ˜˙ − D u ˜ · Zt exists and is characterized as the solution of the following linearized system,  ∂t u ˜0 + D u ˜0 · u ˜ + Du ˜·u ˜0 + D u ˜0 · V + D V · u ˜0 − ν∆˜ u0 + ∇p0 = L(V, W ), Q    0 div(˜ u ) = 0, Q 0 u ˜ = −(D u ˜ · n)hZ , ni, Σ  t   0 u ˜ (0) = 0, Ω0 (5.79) with L(V, W ) = −∂t W − D W · V − D V · W + ν∆W − D u ˜ · W − DW · u ˜ (5.80) PROOF d dV

Z

In order to state such a result, we use Th. (5.4) and we get ! Z Z G(V )dx · W = G0 (V ) · W dx + G hZt , ni (5.81)

Ωt (V )

Ωt (V )

Γt (V )

where G0 (V ) · W stands for non-cylindrical shape derivative of G and Z t is the transverse vector field solution of the Transverse Equation (Eq. (5.22)) with G = [(∂t u ˜ + Du ˜·u ˜ + Du ˜ ·V +DV ·u ˜) · v + νε(˜ u) · ·ε(v) − F (V ) · v] We assume that v has a compact support, then G|Γt (V ) = 0. LEMMA 5.14 G0 (V ) · W = [(∂t u ˜0 + D u ˜0 · u ˜ + Du ˜·u ˜0 + D u ˜0 · V + D u ˜·W 0 0 +DW · u ˜ + DV · u ˜ ) · v + νε(˜ u ) · ·ε(v) − F 0 (V ) · W · v] with F 0 (V ) · W = −∂t W − D W · V − D V · W + ν∆W

Dynamical shape control of the Navier-Stokes equations

135

Finally we obtain d he1 (˜ u) · W, ·vi = dV

Z

Q(V )

[(∂t u ˜0 + D u ˜0 · u ˜ + Du ˜·u ˜0 + D u ˜0 · V + D u ˜·W

+DW · u ˜ + DV · u ˜0 ) · v + νε(˜ u0 ) · ·ε(v) − F 0 (V ) · W · v]

for any v ∈ H with compact support.

Using integration by parts for the term

Z

Q(V )

νε(˜ u0 ) · ·ε(v), we recover the

correct strong formulation of the linearized equation (Eq. (5.79)) satisfied by the shape derivative u0 (V ) · W . The boundary condition comes from the fact that the shape derivative of the condition u ˜ = 0 on Γt (V ) is given by u ˜0 = − D u · Zt , on Γt (V ) Since u = 0 on Γt (V ), we have D u|Γt = D u · (n ⊗ n) which gives u ˜0 = −(D u · n)hZt , ni, on Γt (V )

The shape derivative u0 (V )·W of the solution u of the original non-homogeneous Dirichlet boundary problem (Eq. (5.2)) is given by the expression u0 (V ) · W = u ˜0 (V ) · W + W

(5.82)

COROLLARY 5.1 The shape derivative u0 (V )·W of the solution u of Eq. (5.2) exists and satisfies the following linearized problem,  ∂t u0 + D u0 · u + D u · u0 − ν∆u0 + ∇p0 = 0, Q    div(u0 ) = 0, Q (5.83) 0 u = W + (D V · n − D u · n)hZ , ni, Σ  t   0 u (0) = 0, Ω0 PROOF

We simply set in Eq. (5.79), u ˜0 = u0 − W and u ˜=u−V.

REMARK 5.4 If we choose V = (V ◦ p) the canonical extension of V in Eq. (5.83), then we get the simpler boundary condition, u0 = W − (D u · n)hZt , ni,

on Γt (V )

(5.84)

136

Moving Shape Analysis

5.5.4

Extractor Identity

In the last section, we have established the structure of the system satisfied by the non-cylindrical shape derivative u0 (V ) · W of the solution u(V ) of the Navier-Stokes problem in the moving domain Ωt (V ). This linearized system has been obtained independently of the system satisfied by the non-cylindrical material derivative u ˜˙ P (V ) · W . However, there exists an explicit relation between the original shape u0 and the Piola material derivative u ˜˙ P (V ) · W of the shift state u ˜=u−V. LEMMA 5.15 Let u(V ) stand for the weak solution of the non-homogeneous Navier-Stokes equations (Eq. (5.2)) in moving domain, u0 (V )·W stands for its shape derivative and u ˜˙ P (V ) · W stands for the Piola material derivative of the shift flow u ˜ = u(V ) − V in the direction W . Then the following identity holds, u ˜˙ P (V ) · W = u ˜0 (V ) · W + [˜ u(V ), Zt ]

(5.85)

= u0 (V ) · W + [u(V ), Zt ] − [V, Zt ] − W

(5.86)

where [X, Y ] = D X · Y − D Y · X. This relation can be fruitful in order to obtain an identity concerning the solution u ˜(V ) inside Ωt (V ).

PROPOSITION 5.6 We consider Ω0 of class C 2 , u ˜ solution of the homogeneous Navier-Stokes equations Eq. (5.52) and Zt solution of Eq. (5.22), the following identity holds for all (V, W ) ∈ U, Z

Q(V )

{ ∂t (D u ˜ · Zt ) + D(D u ˜ · Zt ) · u ˜ + Du ˜ · (D u ˜ · Zt ) + D(D u ˜ · Zt ) · V

+ D V · (D u ˜ · Zt ) · w − D u ˜ · (D Zt · u ˜) + D V · (D Zt · u ˜) − D(D V · Zt ) · u ˜ ·w + ∂t u ˜ + Du ˜ · V + Du ˜·u ˜ + DV · u ˜ · (D Zt · w) − D u ˜·W ·w

+ν D(D u ˜ ·Zt )·· D w−ν(D u ˜ ·D Zt )·· D w+ν D u ˜ ·· D(D Zt ·w)−ν D u ˜ ··(D w·D Zt ) + (D ∂t V + D V · V − ν∆V · Zt ) · w + ∂t V + D V · V − ν∆V · (D Zt · w)} = 0, ∀w ∈ H

Dynamical shape control of the Navier-Stokes equations

137

PROOF We recall that the shape derivative u ˜ 0 satisfies the following identity, Z [∂t u ˜0 + D u ˜0 · u ˜ + Du ˜·u ˜0 + D u ˜0 · V + D V · u ˜0 ]·w+ν D u ˜0 ·· D w = h`1 , wi Q(V )

with h`1 , wi =

Z

Q(V )

[−∂t W − D W · V − D V · W + ν∆W − D u ˜ · W − DW · u ˜]·w

Then we set u ˜0 (V ) = u ˜˙ P − [˜ u, Zt ] = u ˜˙ P − D u ˜ · Z t + D Zt · u ˜. This leads to the following identity, Z (∂t u˙ P ) · w + (D u˙ P · u) · w + (D u · u˙ P ) · w + (D u˙ P · V ) · w Q(V )

with h`2 , wi = Z Q(V )

+(D V · u˙ P ) · w + ν D u˙ P · · D w = h`2 , wi

∂t (D u ˜ · Zt ) + D(D u ˜ · Zt ) · u ˜ + Du ˜ · (D u ˜ · Zt ) + D(D u ˜ · Zt ) · V

+ D V · (D u ˜ · Zt ) · w − ∂t (D Zt · u ˜) + D(D Zt · u ˜) · u ˜ + Du ˜ · (D Zt · u ˜) +D(D Zt · u ˜)·V +D V ·(D Zt · u ˜) ·w + ν D(D u ˜ ·Zt )·· D w −ν D(D Zt · u ˜)·· D w + − ∂t W − D W · V − D V · W + ν∆W − D u ˜ · W − DW · u ˜ ·w

Using Theorem (5.11), we deduce that

h`2 , wi = hL, wi,

∀w ∈ H

(5.87)

with hL, wi = Z − ∂t (D Zt · u ˜) + D(D Zt · u ˜) · u ˜ + D(D Zt · u ˜) · V + D(D V · Zt ) · u ˜ Q(V )

+ DW · u ˜ · w − [∂t u ˜ + Du ˜ · V + Du ˜·u ˜ + DV · u ˜] · (D Zt · w)

− ν D(D Zt · u ˜) · · D w + ν(D u ˜ · D Zt ) · · D w − ν D u ˜ · · D(D Zt · w) + ν Du ˜ · ·(D w · D Zt ) + [−∂t W − D W · V − D V · W + ν∆W ] · w − (D [∂t V + D V · V − ν∆V ] · Zt ) · w − [∂t V + D V · V − ν∆V ] · (D Zt · w)

138

Moving Shape Analysis

The sequence [−∂t (D Zt · u ˜) − D(D Zt · u ˜) · u ˜ − D(D Zt · u ˜) · V − D W · u ˜] · w

− ν D(D Zt · u ˜) · · D w + [−∂t W − D W · V − D V · W + ν∆W ] · w

cancels and it remains the following terms, Z [∂t (D u ˜ · Zt ) + D(D u ˜ · Zt ) · u ˜ + Du ˜ · (D u ˜ · Zt ) + D(D u ˜ · Zt ) · V Q(V )

+ D V · (D u ˜ · Zt )] · w − [D u ˜ · (D Zt · u ˜) + D V · (D Zt · u ˜) − D(D V · Zt ) · u ˜] · w + [∂t u ˜ + Du ˜ · V + Du ˜·u ˜ + DV · u ˜] · (D Zt · w) − D u ˜·W ·w +ν D(D u ˜ ·Zt )·· D w−ν(D u ˜ ·D Zt )·· D w+ν D u ˜ ·· D(D Zt ·w)−ν D u ˜ ··(D w·D Zt ) +(D [∂t V + D V · V − ν∆V ]·Zt )·w+[∂t V + D V · V − ν∆V ]·(D Zt ·w) = 0 REMARK 5.5 If we set u ˜ = u − V , we can obtain an identity only involving the 4-uplet (u, Zt , V, W ).

5.5.5

Adjoint system and cost function shape derivative

We are now coming back to the original problem of computing the gradient of the cost function j(V ). Let us first state a differentiability property, PROPOSITION 5.7 For Ω0 of class C 2 , the functional j(V ) is Gˆ ateaux differentiable at point V ∈ U and its directional derivative has the following expression, Z 0 α u(V ) · u0 (V ) · W hj (V ), W i = +

Z

Σ(V )

Q(V )

1 2 γ V · W + (α + γ H)|V | hZt , ni , 2

∀W ∈ U

(5.88)

where u0 (V ) · W is the solution of the shape derivative system (Eq. (5.83)). PROOF

We recall that Z Z Z Z α T γ T j(V ) = |u(V )|2 + |V |2 2 0 Ωt (V ) 2 0 Γt (V )

(5.89)

Dynamical shape control of the Navier-Stokes equations

139

The differentiability property is an easy consequence of the differentiability of JV (u) with respect to (u, V ) and the shape differentiability of u(V ) with respect to V . The expression of the directional derivative is a direct consequence of Th. (5.4) and Th. (5.5). Using the fluid adjoint state and the adjoint transverse field, it is possible to identify the gradient distribution associated to the functional j(V ), THEOREM 5.13 For V ∈ U and Ω0 of class C 2 , the functional j(V ) possesses a gradient ∇j(V ) which is supported on the moving boundary Γt (V ) and can be represented by the following expression, ∇j(V ) = −λ n − σ(ϕ, π) · n + γ V

(5.90)

where (ϕ, π) stands for the adjoint fluid state solution of the following system,  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = α u, Q(V )    div(ϕ) = 0, Q(V ) (5.91) ϕ = 0, Σ(V )    ϕ(T ) = 0, ΩT

and λ is the adjoint transverse boundary field and is the solution of the tangential dynamical system, −∂t λ − ∇Γ λ · V = f, (0, T ) (5.92) λ(T ) = 0, ΓT (V ) with f = −(σ(ϕ, π) · n) · (D V · n − D u · n) + 21 (α + γ H)|V |2 . PROOF Z TZ =

Z

0

T

0

Z

Ωt (V )

We need the following identity, [∂t u0 + D u0 · u + D u · u0 − ν∆u0 + ∇ p0 ] v − ∗

Ωt (V )

0

[−∂t v − D v · u + D u · v − ν∆v + ∇ q] u − +

Z

T 0

Z

Γt (V )

Z

T 0

Z

T 0

Z

q div u0 Ωt (V )

Z

p0 div v Ωt (V )

[p0 n · v − νv · ∂n u0 + νu0 · ∂n v − u0 · q n] (5.93)

We define (ϕ, π) to be the solution of the adjoint system (Eq. (5.8)), and we set (v, q) = (ϕ, π) in Eq. (5.93). We get Z TZ Z TZ hσ(ϕ, π) · n, u0 i (5.94) α u · u0 = − 0

Ωt (V )

0

Γt (V )

140

Moving Shape Analysis

We use the boundary condition on Γt (V ) for the linearized state u0 , i.e., u0 = W + (D V · n − D u · n)hZt , ni,

on Γt (V )

(5.95)

Thus, 0

hj (V ), W i =

Z

T 0

Z

Γt (V )

− (σ(ϕ, π) · n) · (D V · n − D u · n)

1 + (α + γ H)|V |2 hZt , ni + 2

Z

T 0

Z

Γt (V )

[−σ(ϕ, π) · n + γV ] · W

Then we use Th. (5.8) with

1 E = −(σ(ϕ, π) · n) · (D V · n − D u · n) + (α + γ H)|V |2 2 and we get the correct result. REMARK 5.6 the formula,

Actually, we have π(D(V − u) · n) · n = π div V |Γt using

(D(V − u) · n) · n = div(V − u)|Γt − divΓ (V − u) and the fact that V − u = 0 on Γt . Furthermore, we have considered free divergence field V ; then this term is null and we get that 1 f = −ν(D ϕ · n) · (D V · n − D u · n) + (α + γ H)|V |2 2

5.6

Min-Max and function space parametrization

In the previous section, we have been using the differentiability of the fluid state with respect to the Eulerian velocity V as a sufficient condition in order to derive first-order optimality conditions, involving the adjoint of the linearized state. Actually, the tedious computation of the state differentiability is not necessary in many cases, and even if the state is not differentiable, it can happen that first-order optimality conditions still hold. This is a consequence of a fundamental result in optimal control theory, the so-called Maximum Principle. Avoiding the differentiation of the state equations with respect to the design variable V is of great interest for shape optimization problems, especially if

Dynamical shape control of the Navier-Stokes equations

141

we deal with a moving domain system. In this section, we are concerned with the function space parametrization, which consists in transporting the different quantities defined in the perturbed moving domain back into the reference moving domain that does not depend on the perturbation parameter. Thus, differential calculus can be performed since the functions involved are defined in a fixed domain with respect to the perturbations. In the first part, we define the saddle point formulation of the fluid state equations and the Lagrangian functional associated to the cost functional. Then, we perform a sensitivity analysis of the Lagrangian thanks to the transverse field and the fundamental Min-Max principle. This allows us to derive the expression of the cost function gradient involving the fluid and transverse field adjoints.

5.6.1

Saddle point formulation of the fluid state system

In the next paragraphs, we shall describe how to build an appropriate Lagrangian functional that can take into account all the constraints imposed by the mechanical problem, such as the divergence free condition or the nonhomogeneous Dirichlet boundary conditions. Null divergence condition The divergence free condition coming from the fact that the fluid has an homogeneous density and evolves as an incompressible flow is difficult to impose on the mathematical and numerical point of view. We suggest at least 3 possible choices to handle this condition in our Min-Max formulation, 1. It can be taken into account in the state and multipliers spaces. In this case, the divergence free condition must be invariant with respect to the use of transport map during the derivation of optimality condition for the Lagrangian functional. This reduces the choice of appropriate maps and indeed the ALE map Tt does not satisfy this invariance condition. It is well known that the Piola transform does preserve the divergence quantity. Indeed we have the following property: LEMMA 5.16 [13] The Piola transform Pt : H01 (div, Ω0 ) −→ H01 (div, Ωt ) ϕ 7−→ ((Jt )−1 D Tt · ϕ) ◦ Tt−1

(5.96)

is an isomorphism.

This new transform introduces additional mathematical and computational efforts, but it seems to be the best approach in order to get rig-

142

Moving Shape Analysis orous mathematical justifications of the Lagrangian framework in the context of non-cylindrical and free boundary problems.

2. One way to avoid the use of this transform is the penalization of the divergence free condition inside the Navier-Stokes system. Let ε > 0 be a small parameter. We may consider the new penalized system :  1   ∂t u + D u · u − ν∆u − ∇(div u) = 0, Q ε (5.97) u = V, Σ   2 u(t = 0) = u0 , Ω0 × R with σ ε (u) = 1ε div(u) I +ν(D u + ∗ D u). We may work with such a modified system, derive the optimality conditions of the penalized Lagrangian functional and finally perform an asymptotic analysis on the adjoint and primal system. For the time being, it is not clear if such a procedure may actually work, since even for non-moving Navier-Stokes problem, the convergence of the penalized adjoint is not established.

3. A third choice is to include the divergence free condition directly into the Lagrangian functional thanks to a multiplier that may play the role of the adjoint variable associated to the primal pressure variable. This leads in a certain sense to a saddle point formulation or mixed formulation of the Navier-Stokes system. It is well known that the wellposedness of such formulations is only established for the Stokes system, and that the Navier-Stokes suffers from a lack of convexity while taken into account in the Lagrangian functional. But still, it seems to be the easiest way, at least on the mathematical computation point of view, to deal with divergence free conditions in a sensitivity analysis of the moving system. In the sequel, we adopt such a strategy, keeping in minds, its lack of rigorous mathematical justification. Non-homogeneous boundary conditions The Navier-Stokes system (Eq. (5.2)) involves an essential non-homogeneous Dirichlet boundary condition, u = V,

on Γt (V )

(5.98)

Again, there exists different methods to take into account this boundary condition in a Min-Max formulation, 1. We can use a lifting of the boundary conditions inside the fluid domain and define a change of variable inside the coupled system, as done in Section (5.5). It has the drawback to put additional terms inside the Lagrangian functionals and to impose more regularity on the boundary conditions.

Dynamical shape control of the Navier-Stokes equations

143

2. We can use a very weak formulation of the state equation, consisting in totally transposing the Laplacian operator, Z Z Z −ν∆u · φ = −ν∆φ · u + ν [u · ∂n φ − φ · ∂n u] dΓ (5.99) Ωt

Ωt

Γt

Then we shall substitute inside this identity the desirable boundary conditions. We recover the boundary constraints in performing an integration by parts in the optimality conditions corresponding to the sensitivity with respect to the multipliers. This procedure has been already used in [135] to perform shape optimization problems for elliptic equations using Min-Max principles. REMARK 5.7 This method has been popularized in [99] as a systematic way to study non-homogeneous linear partial differential equations. These formulations are usually called very weak formulations or transposed formulations. We shall notice that these methods are still valid in the non-linear case to obtain regularity or existence results. We refer to [4] for recent applications to the Navier-Stokes system. Fluid state operator In this section we shall summarize the different options that we have chosen for the Lagrangian framework and define the variational state operator constraint. In the sequel, we will need to define precise state and multiplier spaces in order to endow our problem with a Lagrangian functional framework. Following the existence result stated previously, we introduce the fluid state spaces: def

X(Ωt ) =

u ∈ H 2 (0, T ; (H 2 (Ωt ))d ∩ (H 1 (Ωt ))d )

def

Z =

p ∈ H 1 (0, T ; (H 1 (D))d )

We also need test function spaces that will be useful to define Lagrange multipliers: def

v ∈ L2 (0, T ; (H 2 (Ωt ))2 ∩ (H01 (Ωt ))d ) def Q = q ∈ H 1 (0, T ; (H 1 (D))2 )

Y (Ωt ) =

We define the fluid weak state operator, eV : X × Z −→ (Y × Q)∗

144

Moving Shape Analysis

whose action is defined by : heV (u, p), (v, q)i =

Z

T 0

Z

Ωt (V )

− u · ∂t v + (D u · u) · v − νu · ∆v

Z TZ + u · ∇ q − p div v + V · (σ(v, q) · n) 0 Γt (V ) Z Z + u(T ) · v(T ) − u0 · v(t = 0), ∀ (v, q) ∈ Y × Q ΩT

Ω0

Min-Max problem In this section, we introduce the Lagrangian functional associated with Eq. (5.2) and Eq. (5.3) : def

LV (u, p; v, q) = JV (u, p) − heV (u, p), (v, q)i with JV (u, p) =

α 2

Z

T 0

Z

Ωt (V )

|u|2 +

γ 2

Z

T 0

Z

Γt (V )

(5.100)

|V |2

(5.101)

Using this functional, the optimal control problem Eq. (5.3) can be put in the following form: min V ∈ U

min (u, p) ∈ X(Ωt (V )) × Z

max (v, q) ∈ Y (Ωt (V )) × Q

LV (u, p; v, q)

(5.102) By using the Min-Max framework, we avoid the computation of the state derivative with respect to V . First-order optimality conditions will furnish the gradient of the original cost functional using the solution of an adjoint problem. Let us first study the saddle point problem, min

(u,p)∈X×Z

max

(v,q)∈Y ×Q

LV (u, p; v, q)

(5.103)

Optimality Conditions In this section, we are interested in establishing the first order optimality condition for problem Eq. (5.103), better known as Karusch-Kuhn-Tucker optimality conditions. This step is crucial, because it leads to the formulation of the adjoint problem satisfied by the Lagrange multipliers (ϕ(V ), π(V )). The KKT system will have the following structure : ∂(v,q) LV (u, p; v, q) · (δv, δq) = 0, ∀ (δv, δq) ∈ Y × Q → State Equations ∂(u,p) LV (u, p; v, q) · (δu, δp) = 0, ∀ (δu, δp) ∈ X × Z → Adjoint Equations

Dynamical shape control of the Navier-Stokes equations

145

LEMMA 5.17 For V ∈ U, (p, v, q) ∈ Z × Y × Q, LV (u, p; v, q) is differentiable with respect to u ∈ X and we have Z

h∂u LV (u, p; v, q), δui = T Z [α u · δu + δu · ∂t v − [D δu · u + D u · δu] · v + νδu · ∆v − δu · ∇ q] 0 Ωt (V ) Z δu(T ) · v(T ), ∀ δu ∈ X + ΩT

In order to obtain a strong formulation of the fluid adjoint problem, we perform some integration by parts : LEMMA 5.18 Z Z (D δu · u) · v = − Q(V )

Q(V )

[D v · u + div(u) · v] · δu +

Z

Σ(V )

(δu · v)(u · n)

It leads to the following identity : h∂uˆ LV (u, p; ϕ, π), δui = Z [−∂t ϕ + (∗ Du) · ϕ − (Dϕ) · u − div(u) · ϕ − ν∆ϕ + ∇π − αu] · δu − Q(V ) Z Z −ν (∂n δu) · ϕ − ϕ(T ) · δu(T ) Σ(V )

ΩT

LEMMA 5.19 For V ∈ U, (u, v, q) ∈ X × Y × Q, LV (u, p; v, q) is differentiable with respect to p ∈ Z and we have Z TZ (δp) div ϕ, ∀ δp ∈ Z (5.104) h∂p LV (u, p; ϕ, π), δpi = 0

Ωt

This leads to the following fluid adjoint strong formulation,  −∂t ϕ − D ϕ · u + (∗ Du) · ϕ − ν∆ϕ + ∇q = αu, Q(V )    div(ϕ) = 0, Q(V ) ϕ = 0, Σ(V )    ϕ(T ) = 0, ΩT

(5.105)

REMARK 5.8 Existence and regularity results for the linearized NavierStokes adjoint problem can be found in [1, 86] for the 2D case. These results can be easily adapted for the moving domain case. There is a lack of results for the 3D case.

146

5.6.2

Moving Shape Analysis

Function space parametrization

To compute the first-order derivative of j(V ), we perturb the moving domain Ωt (V ) by a velocity field W which generates the family of transformation def

Ttρ = Tt (V +ρW ), with ρ ≥ 0 and the family of domains and their boundaries, def

Ωρt = Tt (V + ρW )(Ω0 ) def

Γρt = Tt (V + ρW )(Γ0 ) We set g(ρ) = j(V + ρW ) =

min

(u,p)∈X(Ωρ t )×Z

max

(v,q)∈Y (Ωρ t )×Q

L(V +ρW ) (u, p; v, q) (5.106)

The objective of this section is to compute the following derivative : lim

ρ&0

1 (g(ρ) − g(0)) ρ

(5.107)

We need a theorem that would give the derivative of a Min-Max function with respect to a real parameter ρ ≥ 0. In our case, it is not trivial since the state and multiplier spaces X(Ωρt ) × Y (Ωρt ) depend on the perturbation parameter ρ. This point can be solved using particular parametrization of the functional spaces. To this aim, we use the transverse map introduced in Section (5.3), Tρt : Ωt −→ Ωρt

x 7→ Tt (V + ρW ) ◦ Tt (V )−1

and we define the following parametrization, X(Ωρt ) = u ◦ (Ttρ )−1 , u ∈ X(Ωt (V )) Y (Ωρt ) = v ◦ (Ttρ )−1 , u ∈ Y (Ωt (V ))

(5.108) (5.109)

This parametrization does not affect the value of the saddle point functional g(ρ), but changes the parametrization of the Lagrangian functional,

min (u, p) ∈ X(Ωt ) × Z

max (v, q) ∈ Y (Ωt ) × Q

g(ρ) = j(V + ρW ) = L(V +ρW ) (u ◦ Rρt , p; v ◦ Rρt , q) (5.110)

Dynamical shape control of the Navier-Stokes equations

147

def

with Rρt = (Ttρ )−1 . We set LρV,W (u, p; v, q) = JV +ρW (u ◦ Rρt , p) = Z TZ − − u ◦ Rρt · ∂t (v ◦ Rρt ) + (D u ◦ Rρt · u ◦ Rρt ) · v ◦ Rρt 0

Ωρ t

− νu ◦ Rρt · ∆(v ◦ Rρt ) + u ◦ Rρt · ∇ q − p div(v ◦ Rρt ) Z TZ − (V + ρW ) · (σ(v ◦ Rρt , q) · nρ ) Γρ t

0

−

Z

ΩT

u(T ) · v(T ) +

Z

Ω0

u0 · v(t = 0) ∀ (v, q) ∈ Y (Ωt (V )) × Q

where nρ stands for unit exterior normal of the perturbed boundary Γ ρt .

5.6.3

Differentiability of the saddle point problem

In this section, we first state a general theorem concerning the differentiability of a Min-Max problem with respect to a scalar parameter. Then we apply it to our case of study. Finally, using a fundamental identity, we are able to express the gradient ∇ j(V ) as stated in the main theorem of this chapter. General theorem We a consider a functional, G : [0, ρ0 ] × X × Y → R

(5.111) def

with ρ0 ≥ 0 and two topological spaces (X, Y ). For each ρ ∈ I = [0, ρ0 ], we define g(ρ) = inf sup G(ρ, x, y) (5.112) x∈X

y∈Y

and the sets X(ρ) = Y (ρ, x) =

ρ

xρ ∈ X,

y ∈ Y,

G(ρ, xρ , y) = g(ρ)

sup y∈Y ρ

G(ρ, x, y ) = sup

G(ρ, x, y)

y∈Y

(5.113) (5.114)

In a similar way, we define dual functions and sets, h(ρ) = sup y∈Y

inf

x∈X

G(ρ, x, y)

(5.115)

148

Moving Shape Analysis

and the sets

ρ

ρ

Y (ρ) = y ∈ Y, inf G(ρ, x, y ) = h(ρ) x∈X ρ ρ X(ρ, y) = x ∈ X, G(ρ, x , y) = inf G(ρ, x, y) x∈X

(5.116) (5.117)

Finally we define the sets of saddle points, S(ρ) = {(x, y) ∈ X × Y,

g(ρ) = G(ρ, x, y) = h(ρ)}

(5.118)

THEOREM 5.14 [40] Assume that the following hypothesis hold, (H1) The set S(ρ) 6= ∅, ρ ∈ I. (H2) The partial derivative ∂ρ G(ρ, x, y) exists in I for all     [ [ [ X(ρ) × Y (0) X(0) × Y (ρ) (x, y) ∈  ρ∈I

ρ∈I

(H3) There exists a topology TX on X such that, for any sequence (ρn )n≥0 ∈ I with lim ρn = 0, there exists x0 ∈ X(0) n%∞

and a subsequence ρnk and for each k ≥ 1, there exists xnk ∈ X(ρnk ) such that, i) lim xnk = x0 for the TX topology, n%∞

ii) lim inf

(ρ,k)&%(0,∞)

∂ρ G(ρ, xnk , y) ≥ ∂ρ G(0, x0 , y)

∀ y ∈ Y (0). (H4) There exists a topology TY on Y such that, for any sequence (ρn )n≥0 ∈ I with lim ρn = 0, there exists y 0 ∈ Y (0) n%∞

and a subsequence ρnk and for each k ≥ 1, there exists ynk ∈ Y (ρnk ) such that, i) lim ynk = y 0 for the TY topology, n%∞

ii) lim inf

(ρ,k)&%(0,∞)

∀ x ∈ X(0).

∂ρ G(ρ, x, ynk ) ≤ ∂ρ G(0, x, y 0 )

Dynamical shape control of the Navier-Stokes equations

149

Then there exists (x0 , y 0 ) ∈ X(0) × Y (0) such that g(ρ) − g(0) = ρ&0 ρ

dg(0) = lim

inf

x∈X(0)

sup

∂ρ G(0, x, y)

y∈Y (0)

= ∂ρ G(0, x0 , y 0 ) = sup y∈Y (0)

inf

x∈X(0)

∂ρ G(0, x, y) (5.119)

This means that (x0 , y 0 ) ∈ X(0) × Y (0) is a saddle point of ∂ρ G(0, x, y). Derivative of the perturbed Lagrangian Following Th. (5.14), we need to differentiate the perturbed Lagrangian functional L(ρ). We shall successively differentiate the distributed and the boundary integrals involved in the perturbed Lagrangian: a) Distributed terms: We set G(ρ, .) = −u ◦ Rtρ · ∂t (v ◦ Rtρ ) + D(u ◦ Rtρ ) · (u ◦ Rtρ ) · v ◦ Rtρ

−ν(u ◦ Rtρ ) · ∆(v ◦ Rtρ ) + (u ◦ Rtρ ) · ∇q − p div(v ◦ Rtρ )

def

with Rtρ = (Tρt )−1 . We shall need the following lemmas in order to derivate G(ρ, .) with respect to ρ,

LEMMA 5.20

dTρt = Zt dρ ρ=0

dRtρ = −Zt dρ ρ=0

LEMMA 5.21 d u ◦ Rtρ dρ

!

ρ=0

= − D u · Zt

150

Moving Shape Analysis PROOF

Using the chain rule we get ! D u ◦ Rtρ D Rtρ = D u ◦ Rtρ · Dρ D ρ ρ=0 ρ=0 = − D u ◦ Rtρ · Z t (ρ, .) ρ=0 = − D u · Zt

LEMMA 5.22 Then, we have the following result ∂ρ G(ρ, .)|ρ=0 = [(D u · Zt ) · ∂t v + u · (∂t (D v · Zt )) − [(D(D u · Zt )) · u + D u · (D u · Zt )] · v − (D u · u) · (D v · Zt )

+ν(D u · Zt ) · ∆v + νu · (∆(D v · Zt )) + p div(D v · Zt ) − (D u · Zt ) · ∇q] PROOF (5.21).

It comes easily using definition of G(ρ, .) and Lem. (5.20)-

Then we have an expression of the derivative of distributed terms coming from the Lagrangian with respect to ρ, d dρ

Z

! G(ρ, x)dx ρ Ωt

= ρ=0

Z

Ωt

[(D u · Zt ) · ∂t v + u · (∂t (D v · Zt ))

− [(D(D u · Zt )) · u + D u · (D u · Zt )] · v − (D u · u) · (D v · Zt ) +ν(D u · Zt ) · ∆v + νu · (∆(D v · Zt )) + p div(D v · Zt ) − (D u · Zt ) · ∇q] Z + [−u · ∂t v + (Du · u) · v − νu · ∆v + u · ∇q − p div(v)] hZt , ni Γt

b) Boundary terms : We must now take into account the terms coming from the moving boundary Γρt . Then we set φ(ρ, .) = (V + ρW ) · −q I +ν D(v ◦ Rtρ ) · nρ = E(ρ) · nρ

(5.120)

Since φ(ρ, .) is defined on the boundary Γρt , we need some extra identities corresponding to boundary shape derivates of terms involved in φ(ρ, .).

Dynamical shape control of the Navier-Stokes equations

151

LEMMA 5.23 [53] 0

∂ρ nρ |ρ=0 = nΓ = −∇Γ (Zt · n) LEMMA 5.24 Z

d dρ

PROOF

! hE(ρ), n idΓ Γρ t ρ

= ρ=0

=

Z

Z

Γt

hE 0 |Γt , ni + (div E)hZt , ni

Γt

hEΓ0 t , ni + (divΓ E)hZt , ni

First, we use that Z Z hE(ρ), nρ i = Γρ t

(5.121)

Ωρ t

div E(ρ)

Then we derive this quantity using Th. (5.4), ! Z Z Z d 0 div E(ρ) = div E + (div E)hZt , ni dρ Ωρ Ωt Γt t ρ=0

We conclude using

Z

div E 0 =

Ωt

using the Th. (5.5), we have ! Z d t hE(ρ), (n ◦ Rρ )idΓ ρ dρ Γt

Z

Γt

= ρ=0

hE 0 , ni. For the second identity,

Z

0

Γt

hEΓ0 , ni+hE, nΓ i+HhE, nihZt , ni

Using Lem. (5.23), we get d dρ

Z

Γρ t

hE(ρ), (n ◦

Rtρ )idΓ

!

= ρ=0

Z

Γt

hEΓ0 , ni − hE, ∇Γ (Zt · n)i + HhE, nihZt , ni

Then using the tangential Stokes identity from Lem. (5.5), we obtain the correct result. Hence, we only need to compute the quantity EΓ0 . To this end, we need the following identities,

152

Moving Shape Analysis LEMMA 5.25 v ◦ Rtρ PROOF

0 Γ

ρ=0

= − DΓ v · Zt

. Since v ◦ Rtρ ρ=0 = ∂ρ v ◦ Rtρ ◦ Tρt ρ=0 = ∂ρ v|ρ=0 = 0.

LEMMA 5.26 D v ◦ Rtρ PROOF

0 Γ

ρ=0

= − D v · D Zt − (DΓ (D v)) · Zt

By definition we have 0 . D v ◦ Rtρ Γ = D v ◦ Rtρ Γ − DΓ D(v ◦ Rtρ ) · Zρt ρ=0 ρ=0 ρ=0 t = ∂ρ D v ◦ Rρ ◦ Tρt ρ=0 − (DΓ (D v)) · Zt = ∂ρ (D v) ◦ Rtρ · D Rtρ ◦ Tρt ρ=0 − (DΓ (D v)) · Zt = ∂ρ (D v) · D Rtρ ◦ Tρt ρ=0 − (DΓ (D v)) · Zt = − D v · D Zt + D v · D(D Rtρ ) · ∂ρ (Tρt ) ρ=0

− (DΓ (D v)) · Zt

= − D v · D Zt − (DΓ (D v)) · Zt

Using these results, we can state the following : LEMMA 5.27 EΓ0 = W · [−q I +ν D v] + ν V · [− D v · D Zt − DΓ (D v) · Zt ] This means that we have ! Z d φ(ρ, x)dΓ ρ dρ Γt

ρ=0

=

Z

Γt (V )

(5.122)

W · [−q n + ν D v · n]

+ν V ·[−(D v · D Zt ) · n − (DΓ (D v) · Zt ) · n]+divΓ (V ·[−q I +ν D v])hZt , ni We have also, LEMMA 5.28 E 0 |Γ = W · [−q I +ν D v] − ν V · [D(D v) · Zt ]

(5.123)

Dynamical shape control of the Navier-Stokes equations

153

Hence, we have d dρ

Z

! φ(ρ, x)dΓ Γρ t

= ρ=0

Z

Γt (V )

W · [−q n + ν D v · n]

− ν V · [D(D v) · Zt · n] + div(V · [−q I +ν ∗ D v])hZt , ni

REMARK 5.9 Z

We recall that Z div(∗ D v · V ) V · (D v · n) = Γt Ωt Z D v · · D V + V · ∆v =

(5.124)

Ωt

We shall use this expression in the sequel. We recall that the perturbed Lagrangian has the following form, ρ LρV,W = JV,W −

Z

−

T 0

Z

Z

ΩT

Ωρ t

G(ρ) −

Z

T 0

u(T ) · v(T ) +

Z

Z

Γρ t

Ω0

φ(ρ) u0 · v(t = 0) ∀ (v, q) ∈ Y (Ωt ) × Q

Hence its derivative with respect to ρ at point ρ = 0 has the following expression, ! Z T Z d ρ d ρ d = LV,W − JV,W G(ρ) ρ dρ dρ dρ 0 Ωt ρ=0 ρ=0 ρ=0 ! Z T Z d − φ(ρ) ∀ (v, q) ∈ Y (Ωt ) × Q ρ dρ 0 Γt ρ=0

Furthermore we have, LEMMA 5.29

Z TZ Z TZ d ρ JV,W = −α u · (D u · Zt ) + γV ·W dρ 0 Ωt (V ) 0 Γt (V ) ρ=0 Z TZ hα i γ + |u|2 + H|V |2 hZt , ni 2 0 Γt (V ) 2

154

Moving Shape Analysis Using the last identities concerning the derivative of the distributed and the boundary terms with respect to ρ, we shall get the following expression, d ρ = −AZt − BZt − CW (5.125) LV,W dρ ρ=0

with

AZ t =

Z

T 0

Z

Ωt (V )

αu · (D u · Zt ) + (D u · Zt ) · ∂t v −

+ D u · D u · Zt

D(D u · Zt ) · u

· v + ν(D u · Zt ) · ∆v − (D u · Zt ) · ∇q

+ u·(∂t (D v·Zt ))−(D u·u)·(D v·Zt )+νu·(∆(D v·Zt ))+p div(D v·Zt )

B Zt =

Z

T 0

Z

Γt (V )

− u · ∂t v + (Du · u) · v − νu · ∆v + u · ∇q

− p div(v) (Zt · n) − ν V · [(D(D v) · Zt ) · n] + div(V · [−q I +ν D v])hZt , ni −

CW =

Z

T 0

Z

Γt (V )

hα 2

|u|2 +

i γ H|V |2 hZt , ni 2

[W · [−q n + ν D v · n] − γ V · W ]

The shape derivative kernel identity We shall now assume that (u, p, v, q) = (u, p, ϕ, π) is a saddle point of the Lagrangian functional LV . This will help us to simplify several terms involved in the derivative of LV with respect to V . Indeed, we would like to express the distributed term A Zt as a boundary quantity defined on the moving boundary Γt . THEOREM 5.15 For (u, p, ϕ, π) saddle points of the Lagrangian functional (Eq. (5.100)), the

Dynamical shape control of the Navier-Stokes equations

155

following identity holds, Z TZ [α u · (D u · Zt ) + (D u · Zt ) · ∂t v − [(D(D u · Zt )) · u 0

Ωt (V )

+ D u · (D u · Zt )] · v + ν(D u · Zt ) · ∆v − (D u · Zt ) · ∇q] + u · (∂t (D v · Zt )) − (D u · u) · (D v · Zt ) + νu · (∆(D v · Zt )) + p div(D v · Zt ) Z TZ − [ν V · (D(D ϕ · Zt ) · n) − (D ϕ · Zt ) · (−p n + ν(D u · n))] = 0, 0

Γt (V )

∀W ∈ U

PROOF We shall use extremal conditions associated to variations with respect to (u, v) in the Lagrangian functional where we add a boundary integral since we consider test functions v that do not vanish on the boundary Γt (V ), i.e., L2V (u, p; v, q) = Z TZ JV (u, p) − 0

−

Z

T

0

Ωt (V )

Z

[−u · ∂t v + (D u · u) · v − νu · ∆v + u · ∇ q − p div v]

Z TZ V · (σ(v, q) · n) + v · (σ(u, p) · n) Γt (V ) 0 Γt (V ) Z Z u0 · v(t = 0), ∀ (v, q) ∈ Y × Q u(T ) · v(T ) + − Ω0

ΩT

This leads to the following perturbation identity,

∂(u,v) L2V · (δu, δv) = Z − [−αu · δu − δu · ∂t v − u · ∂t δv + D(δu · u) · v + D(u · δu) · v Q(V )

+

Z

T 0

+D(u · u) · δv − ν(δu · ∆v) − ν(u · ∆δv) + δu · ∇q − p div(δv)] Z − ν V · (D δv · n) + ν v · (D δu · n) + δv · (−p n + ν(D u · n)) Γt (V ) Z [δu(T )v(T ) + u(T )δv(T )] , ∀ (δu, δv) ∈ X(Ωt ) × Y (Ωt ) − ΩT

We choose specific perturbation directions, i.e., δu = D u · Zt

δv = D v · Zt

with δu(T ) = δv(T ) = δu(0) = δv(0) = 0, where (u, v) are saddle points of the Lagrangian, i.e., solutions of respectively the primal and adjoint fluid problem. We recognize immediately the distributed and boundary terms involved in the shape derivative kernel identity.

156

Moving Shape Analysis

Cost functional gradient Now, we set (u, v) = (u, ϕ) and we use the fact that u = V, on Γ t and ϕ = 0, on Γt to simplify the remaining terms. AZ t =

Z

T 0

Z

Γt (V )

[ν V · (D(D ϕ · Zt ) · n) − ν(D ϕ · Zt ) · (D u · n)]

(5.126)

REMARK 5.10 We have used that (D ϕ · Zt ) · (p n) = (D ϕ · (n ⊗ n) · Zt ) · (p n) = p ((D ϕ · n) · n) · hZt , ni = (p div ϕ)hZt , ni = 0.

B Zt =

Z

T 0

Z

Γt (V )

[−νV · ∆ϕ + V · ∇π] hZt , ni −

1 [α + γ H] |V |2 hZt , ni 2

−ν V ·[(D(D ϕ) · Zt ) · n]+[−π div V − V · ∇ π + ν D ϕ · · D V + ν V · ∆ϕ])hZt , ni CW =

Z

T 0

Z

Γt (V )

[W · [−π n + ν D ϕ · n] − γ V · W ]

We need to establish the following identity, LEMMA Z 5.30 Z (D ϕ · Zt ) · (D u · n) = (D ϕ · n) · (D u · n)hZt , ni Γt

(5.127)

Γt

Then −

d ρ LV,W = dρ ρ=0 Z ν V · (D(D ϕ · Zt ) · n) + [−ν(D ϕ · n) · (D u · n) − νV · ∆ϕ Σ(V )

+V · ∇π] hZt , ni + [−π div V − V · ∇ π + ν D ϕ · · D V + ν V · ∆ϕ])hZt , ni 1 [α + γ H] |V |2 hZt , ni 2 + [W · [−π n + ν D ϕ · n] − γ V · W ]

− ν V · [(D(D ϕ) · Zt ) · n] −

This allows us to derive the expression of the cost function directional derivative,

Dynamical shape control of the Navier-Stokes equations

157

PROPOSITION 5.8 Z

Σ(V )

dg(0) = 1 2 −ν(D ϕ · n) · (D V · n − D u · n) + π div V + (α + γ H)|V | hZt , ni 2 Z + [−σ(ϕ, q) · n + γV ] · W Σ(V )

(5.128)

Then we use Th. (5.8) with, 1 E = −ν(D ϕ · n) · (D V · n − D u · n) + π div V + (α + γ H)|V |2 2 and we get the correct result.

5.7

Min-Max and function space embedding

In the previous section, we have used a function space parametrization in order to get the gradient of a given functional related to the solution of the Navier-Stokes system in moving domain, with respect to the speed of the moving domain. In this section, we use a different method based on function space embedding particulary suited for non-homogeneous Dirichlet boundary problems. It means that the state and multiplier variables are defined in a hold-all domain D that contains the moving domain Ω t (V ) for t ∈ (0, T ) and ∀ V ∈ U.

5.7.1

Saddle point formulation of the fluid state system

We recall that we are dealing with the Navier-Stokes in a moving domain Ωt (V ) which is driven by an Eulerian velocity field V ∈ U,

and

 ∂t u + D u · u − ν∆u + ∇p = 0, Q(V )    div(u) = 0, Q(V ) u = V, Σ(V )    u(t = 0) = u0 , Ω0

U = V ∈ H 1 (0, T ; (H m (D))d ),

div V = 0 in D,

(5.129)

V · n = 0 on ∂D (5.130)

158

Moving Shape Analysis

with m > 5/2. We introduce a Lagrange multiplier µ and a functional, Z

EV (u, p; v, q, µ) =

T 0

Z

Ωt (V )

[∂t u + D u · u − ν∆u + ∇ p] · v −

Z

T 0

Z

Ωt (V )

q div u −

Z

T 0

Z

Γt (V )

(u − V ) · µ

for (u, p) ∈ X × P , (v, q) ∈ Y × Q and µ ∈ M with def

def

def

def

X = Y = H 1 (0, T ; H 2 (D)) P = Q = H 1 (0, T ; H 1 (D)) M = H 1 (0, T ; H 3/2 (Γt )) We are interested in the following Min-Max problem, min

(u,p)∈X×P

max

(v,q,µ)∈Y ×Q×M

EV (u, p; v, q, µ)

(5.131)

The solution (y, p, ϕ, π, λ) of this problem is characterized by the following optimality system, • The primal state (y, p) is a solution of the Navier-Stokes system,  ∂t y + D y · y − ν∆y + ∇p = 0, Q(V )    div(y) = 0, Q(V ) (5.132) y = V, Σ(V )    y(t = 0) = y0 , Ω0

• The dual state (ϕ, π) is the solution of the fluid adjoint system,  −∂t ϕ − D ϕ · u + (∗ Du) · ϕ − ν∆ϕ + ∇π = 0, Q(V )    div(ϕ) = 0, Q(V ) (5.133) ϕ = 0, Σ(V )    ϕ(t = T ) = 0, ΩT

• The multiplier satisfies the following identity, µ = −q n + ν(D ϕ · n),

on Γt (V )

(5.134)

Then we can choose the above particular representation of the boundary Lagrange multiplier µ. This yields to the following functional, EV (u, p; v, q) =

Z

T 0

Z

Ωt (V )

[∂t u + D u · u − ν∆u + ∇ p]·v− −

Z

T 0

Z

Γt (V )

Z

T 0

Z

q div u Ωt (V )

(u − V ) · σ(v, q) · n

Dynamical shape control of the Navier-Stokes equations

159

for (u, p) ∈ X × P , (v, q) ∈ Y × Q, with σ(v, q) · n = −q n + ν(D ϕ · n),

on Γt (V )

The following identities hold true, LEMMA 5.31 Z Z (u − V ) · (D v · n) = div [∗ D v · (u − V )] Γt (V ) Ωt (V ) Z = [D(u − V ) · · D v + (u − V ) · ∆v] Ωt (V )

and Z

Γt (V )

(u − V ) · q n = =

Z

Z

Ωt (V )

div [q(u − V )]

Ωt (V )

[(u − V ) · ∇q + q div(u − V )]

Using this identity, we may get the final expression of our saddle functional, EV (u, p; v, q) = +

Z

T 0

Z

Ωt (V )

Z

T 0

Z

Ωt (V )

[∂t u + D u · u − ν∆u + ∇ p]·v−

Z

T 0

Z

q div u Ωt (V )

[(u − V ) · ∇q + q div(u − V ) − ν D(u − V ) · · D v − ν(u − V ) · ∆v]

for (u, p) ∈ X × P , (v, q) ∈ Y × Q. REMARK 5.11 The above expression of the Lagrange functional has the advantage to include only distributed terms. This will be useful for its differentiation with respect to V .

5.7.2

The Lagrange functional

We are interested in the following minimization problem, min j(V )

(5.135)

V ∈U

where j(V ) = JV (u(V ), p(V )) with (u(V ), p(V )) is a weak solution of problem (5.2) and JV (u, p) is a real functional of the following form : JV (u, p) =

α 2

Z

T 0

Z

Ωt (V )

|u|2 +

γ 2

Z

T 0

Z

Γt (V )

|V |2

(5.136)

160

Moving Shape Analysis

We may solve this problem by studying the equivalent Min-Max problem, min

min

V∈U

(u,p)∈X×P

max

(v,q)∈Y ×Q

LV (u, p; v, q)

(5.137)

with LV (u, p; v, q) = JV (u, p) − EV (u, p; v, q)

(5.138)

Our main concern is the differentiation of the above functional with respect to V ∈ U. As in the previous section we perturb the tubes using a vector field W ∈ U with an increment parameter ρ ≥ 0. Since the functions are embedded in the hold-all domain D, the perturbed Lagrangian has the following form, Lρ (u, p; v, q) = JV +ρW (u, p) − EV +ρW (u, p; v, q)

(5.139)

The set of saddle points, S(ρ) = X(ρ) × P × Y (ρ) × Q ∈ X × P × Y × Q is not a singleton since X(ρ) = u ∈ X, u|Ωρt = y(ρ) Y (ρ) = v ∈ Y, v|Ωρt = ϕ(ρ)

We make the conjecture that we can bypass the min-max, and state d d ρ j(V + ρW ) = min max L (u, p; v, q) (5.140) dρ dρ (u,p)∈X×P (v,q)∈Y ×Q ρ=0 ρ=0

Using non-cylindrical shape derivative framework, we can state LEMMA 5.32 ∂V LV (u, p; v, q)·W = −

Z

T 0

Z

Γt (V )

[(∂t u + D u · u − ν∆u + ∇ p) · v − q div u

+(u − V ) · ∇q + q div(u − V ) − ν D(u − V ) · · D v − ν(u − V ) · ∆v − i γ −H |V |2 hZt , ni− 2

Z

T 0

Z

Ωt (V )

α 2 |u| 2

[−W · ∇q − q div W + ν D W · · D v + νW · ∆v]

+

Z

T 0

Z

Γt (V )

γV · W

Dynamical shape control of the Navier-Stokes equations Then we set (u, p) = (y, p) and (v, q) = (ϕ, π) with  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = α u, Q(V )    div(ϕ) = 0, Q(V ) ϕ = 0, Σ(V )    ϕ(T ) = 0, ΩT

161

(5.141)

and we use that

(y, ϕ) = (V, 0) on Γt (V ) and Z

Ωt (V )

[−W · ∇q − q div W + ν D W · · D v + νW · ∆v] =

Z

Γt (V )

W · σ(v, q) · n

Then, ∂V j(V ) · W = −

Z

T 0

Z

Γt (V )

− π div y + π div(y − V ) − ν D(y − V ) · · D ϕ

1 − (α + Hγ)|V |2 hZt , ni + (σ(ϕ, π) · n − γV ) · W 2

Using regularity assumptions on y and the free divergence condition on y, we may state that div y|Γt = 0. LEMMA 5.33 D y · · D ϕ|Γt (V ) = (D y · n) · (D ϕ · n) PROOF

(5.142)

Using that ϕ = 0 on Γt (V ) yields to D ϕ|Γt = D ϕ · (n ⊗ n)|Γt

then, we get D y · · D ϕ = D y · ·(D ϕ · (n ⊗ n)) = (D y · n) · (D ϕ · n) 2 Consequently we get Z TZ hj 0 (V ), W i = 0

Γt (V )

− ν(D ϕ · n) · (D V · n − D u · n) + π div V

1 + (α + γ H)|V |2 hZt , ni + 2

Z

T 0

Z

Γt (V )

[−σ(ϕ, q) · n + γV ] · W (5.143)

162

Moving Shape Analysis

We then use theorem (5.8) with 1 E = −ν(D ϕ · n) · (D V · n − D u · n) + π div V + (α + γ H)|V |2 2 and we get the correct result. 2

5.8

Conclusion

In this chapter, we have been dealing with a particular shape optimization problem involving the Navier-Stokes equations. Its originality lies in the fact that the domain containing the fluid is moving. We have introduced an open loop control problem based on the velocity of the moving domain with the goal of reaching a given objective related to the behaviour of the fluid. Our main concern was to show how the gradient of the cost functional involved in the optimal control problem can be obtained by using non-cylindrical shape optimization concepts. In addition to the classical method based on the state derivative with respect to shape motions, we have introduced two different methods based on the Min-Max principle. Even if for the time being these methods lack from a rigorous mathematical framework, they allow more flexible computations which can be very useful for practical purposes. On the numerical point of view, an implementation of the open loop control is under study in the 2D case. We believe that the concepts introduced in this chapter will prove large efficiency for coupled problems involving a moving boundary, as it will be shown in the next chapters.

Chapter 6 Tube derivative in a Lagrangian setting

6.1

Introduction

In this chapter, we state similar results as the previous chapter but with a method based directly on the Lagrangian flows which do not define a linear space, but has the advantage to first avoid the introduction of transverse fields and could prove appropriate to deal with fluid-solid interaction problems. The chapter is organised as follows, 1. In section 6.2, we introduce the general framework of non-cylindrical Lagrangian shape derivative and we state the structure of the gradient for general functionals under or without state constraints. We also establish an equivalence result between Eulerian and Lagrangian derivatives. 2. In section 6.3, we apply the general framework to mechanical systems involving a fluid in a moving domain. The fluid is described by the Navier-Stokes equations. We deal with the state derivative with respect to the Lagrangian flow and an optimal control with a general cost function. We finally recover results obtained in [57] using Eulerian derivative concepts.

6.2

Evolution maps

Let us consider a smooth map θˆ : (0, T ) × Rd −→ Rd ˆ x) (t, x) 7→ θ(t, and the image Ωθˆ by θˆ of a fixed domain Ω0 ⊂ Rd with a Lipschitzian boundary Γ0 .

163

164

Moving Shape Analysis def

We set C k = (C k (Rd ))d and we define the space of evolution map, def

Θ =

n

ˆ (θ, ˆ θˆ−1 ) ∈ (C 1 ([0, T ]; C k ))2 θ|

o

(6.1)

ˆ 0 ) is still an open set and its REMARK 6.1 For θˆ ∈ Θ, the set Ωθˆ = θ(Ω boundary Γθˆ is Lipschitzian.

6.2.1

Basic differentiation results

Given a function f defined on (0, T ) × Rd , we would like to perform the ˆ differentiation of the function θˆ 7→ f ◦ θ. LEMMA 6.1 Let us consider the scalar function f ∈ L2 (0, T ; W m,p (Rd )), m ≥ 1, 1 ≤ p ≤ ∞, k ≥ m − 1 f ∈ C m (Rd ), if p = ∞ Then the application Θ −→ L2 (0, T ; W m,p (Rd )) θˆ 7→ f ◦ θˆ is differentiable at point θˆ and its derivative in the direction δ θˆ ∈ Θ is given by ∂ ˆ · δ θˆ = ∇f · δ θ, ˆ ∈ L2 (0, T ; W m−1,p (Rd )) [ (f ◦ θ)] (6.2) ∂ θˆ ˆ which depends on the evolution map Now, let us consider a function φ(θ) ˆ θ. We shall define the derivative of this function in the following way, LEMMA 6.2 Let us consider the scalar function φ ∈ L2 (0, T ; W m,p (Rd )), m ≥ 1, 1 ≤ p ≤ ∞, k ≥ m − 1 f ∈ C m (Rd ), if p = ∞ Let us assume that the function Θ −→ L2 (0, T ; W m,p (Rd )) ˆ ◦ θˆ θˆ 7→ φ(θ)

Tube derivative in a Lagrangian setting

165

is differentiable, then the application Θ −→ L2 (0, T ; W m,p (Rd )) ˆ θˆ 7→ φ(θ) is differentiable and its derivative in the direction δ θˆ ∈ Θ is given by ∂ ˆ ∂ ˆ ◦ θ)] ˆ · δ θˆ ◦ θˆ−1 − ∇φ(θ) ˆ · (δ θˆ ◦ θˆ−1 ), [ φ(θ)] · δ θˆ = [ (φ(θ) ∂ θˆ ∂ θˆ ∈ L2 (0, T ; W m−1,p (Rd ))

In this section, we would like to perform a sensitivity analysis with respect to the shape of integrals defined on evolution sets Q ∈ E T with Q ⊂ D × (0, T ). This analysis will be performed using the Lagrangian horizontal mapping ˆ (t, θ).

6.2.2

Mathematical setting

Generally speaking, we would like to solve the following problem, min J(Q)

(6.3)

Q∈A

with

A ≡ Q ∈ ET ,

and

Q⊂D

J : A −→ R Q 7→ J(Q)

(6.4)

(6.5)

In order to work with standard differential calculus, we define the functional j = J ◦ θt such that j : Θad −→ R (6.6) ˆ = J(Q ˆ) θˆ 7→ j(θ) θ [ ˆ 0 ). with Qθˆ = ({t} × Ωt ) = (t, θ)(Q 0
ˆ Then, derivatives in the sense of Gˆ ateaux or Fr´echet could make sense for j( θ).

6.2.3

Elements of shape calculus

Using extension of classical elements of cylindrical shape calculus using the perturbation of identity ([116], [101],[135]), it is possible to define derivation ˆ concepts of functions defined in Qθˆ or Σθˆ with respect to θ.

166

Moving Shape Analysis

Derivation of moving domain integrals In this paragraph, we are interested in the differentiability properties of integrals defined on the non-cylindrical set Q θˆ, J1 (Qθˆ) =

Z

Qθˆ

f (Qθˆ)dQθˆ

(6.7)

ˆ We We are trying to characterize the variations of J 1 (Qθˆ) with respect to θ. shall state a differentiability result that uses the notion of non-cylindrical material derivative, ˆ ∈ H(Q ˆ) admits a non-cylindrical DEFINITION 6.1 A function f (θ) θ ˆ · δθ defined over Q ˆ at point θˆ ∈ Θad in the direction material derivative f˙(θ) θ def δ θˆ = δθ ◦ θˆ ∈ Θ, if the following composed function, f ρ : [0, ρ0 [ → H(Qθˆ)

ˆ ◦ (I +ρδθ) ρ 7→ f ((I +ρδθ) ◦ θ)

ˆ · δθ = is differentiable at point ρ = 0, a.e. (x, t) ∈ Qθˆ and f˙(θ)

d ρ dρ f

ρ=0

.

Under regularity conditions, it is possible to define a different derivative that will be useful in the context of Eulerian state equations. ˆ ∈ H(D) admits a non-cylindrical DEFINITION 6.2 A function f (θ) 0 ˆ shape derivative f (θ) · δθ defined over Qθˆ at point θˆ ∈ Θad in the direction def δ θˆ = δθ ◦ θˆ ∈ Θ, if the following composed function, fρ : [0, ρ0 [ → H(D)

ˆ ρ 7→ f ((I +ρδθ) ◦ θ)

ˆ · δθ = is differentiable at point ρ = 0, a.e. (x, t) ∈ Qθˆ and f 0 (θ) REMARK 6.2

d dρ fρ

ρ=0

.

It can be proven that the following identity holds, ˆ · δθ = f 0 (θ) ˆ · δθ + ∇f (θ) ˆ · δθ f˙(θ)

(6.8)

We can now state the differentiability properties of non-cylindrical integrals with respect to their moving supports,

Tube derivative in a Lagrangian setting

167

THEOREM 6.1 For a bounded measurable domain Ω0 with boundary Γ0 , let us assume that def for any direction δ θˆ = δθ ◦ θˆ ∈ Θ the following hypothesis holds, ˆ admits a non-cylindrical material derivative f˙(θ) ˆ · δθ then J1 (.) is i) f (θ) ˆ Gˆ ateaux differentiable at point θ ∈ Θad and its derivative is given by the following expression, Z h i d ˆ ˆ · δθ + f (θ) ˆ div δθ dQ ˆ (6.9) J1 (θ) · δθ = f˙(θ) θ dθˆ Qθˆ Futhermore, if ˆ admits a non-cylindrical shape derivative f 0 (θ) ˆ · δθ ∈ H(Q ˆ), then ii) f (θ) θ d ˆ · δθ = J1 (θ) dθˆ

Z

Qθˆ

h

i ˆ · δθ + div(f (θ) ˆ δθ) dQ ˆ f 0 (θ) θ

(6.10)

Furthermore, if Q0 is an open domain with a Lipschitzian boundary Σ0 , then Z Z d ˆ · δθ = ˆ · δθdQ ˆ + ˆ J1 (θ) f 0 (θ) f (θ)hδθ, νidΣθˆ (6.11) θ dθˆ Qˆ Σˆ θ

θ

REMARK 6.3 Since we deal with horizontal transformations of the type ˆ it means that perturbations are of the type (0, δ θ), ˆ then (t, θ), hδθ, νi = (1 + vν2 )−1/2 (h0, −vν i + hδθ, nt i) using that dΣθˆ = (1 + vν2 )1/2 dt dΓθˆ, we get Z

Σθˆ

ˆ f (θ)hδθ, νidΣθˆ =

Z

T 0

Z

Γθˆ

ˆ f (θ)hδθ, nθˆidΓθˆ dt

(6.12)

Derivation of moving boundary integrals It is also possible to establish a similar result for integrals over moving boundaries, Z (6.13) g(Σθˆ)dΣθˆ J2 (Σθˆ) = Σθˆ

For that purpose, we need to define the non-cylindrical tangential material derivative,

168

Moving Shape Analysis

ˆ ∈ H(Σ ˆ) admits a non-cylindrical DEFINITION 6.3 A function g(θ) θ ˆ · δθ at point θˆ ∈ Θad in the direction δ θˆ def material derivative g( ˙ θ) = δθ ◦ θˆ ∈ Θ if the following composed function, g ρ : [0, ρ0 ] → H(Σθˆ)

ˆ ◦ (I +ρδθ) ρ 7→ g((I +ρδθ) ◦ θ)

ˆ · δθ = is differentiable at point ρ = 0, a.e. (t, x) ∈ Σθˆ and g( ˙ θ)

d ρ dρ g

ρ=0

.

Under regularity conditions, it is possible to define a different derivative that will be useful in the context of Eulerian state equations. ˆ ∈ H(Σ ˆ) admits a non-cylindrical DEFINITION 6.4 A function g(θ) θ 0 ˆ shape derivative g (θ) · δθ defined over Σθˆ at point θˆ ∈ Θad in the direction def δ θˆ = δθ ◦ θ ∈ Θ, if the following composed function, gρ : [0, ρ0 [ → H(D)

ˆ ◦ θ) ◦ p ρ 7→ g((I +ρδ θ)

ˆ ·δ = is differentiable at point ρ = 0, a.e. (x, t) ∈ Qθˆ and g 0 (θ)

p stands for the projection mapping on Γθˆ ([51]). REMARK 6.4

d dρ gρ

ρ=0

. Here

It can be proven that the following identity holds, ˆ · δθ = g 0 (θ) ˆ · δθ + ∇Γ g(θ) ˆ · δθ g( ˙ θ)

(6.14)

These concepts are involved in the differentiability property of boundary integrals, THEOREM 6.2 For a bounded measurable domain Ω0 with boundary Γ0 , let us assume that def for any direction δ θˆ = δθ ◦ θˆ ∈ Θ the following hypothesis holds,

ˆ admits a non-cylindrical material derivative g( ˆ · δθ then J2 (.) is i) g(θ) ˙ θ) Gˆ ateaux differentiable at point θˆ ∈ Θad and its derivative is given by the following expression, Z h i d ˆ · δθ + g(θ) ˆ divΓ δθ dΣ ˆ ˆ g( ˙ θ) (6.15) J2 (θ) · δθ = θ dθˆ Σθˆ and if,

Tube derivative in a Lagrangian setting

169

ˆ admits a non-cylindrical shape derivative , g 0 (θ) ˆ · δθ, then ii) g(θ) Z h i d ˆ · δθ = ˆ · δθ + H g(θ)hδθ, ˆ J2 (θ) g 0 (θ) νi dΣθˆ dθˆ Σθˆ

(6.16)

ˆ = g˜(θ)| ˆ Σ with g˜ ∈ H(Q ˆ), then Furthermore, if g(θ) ˆ θ θ Z h i d ˆ · δθ = ˆ · δθ + (∇Γ g˜(V ) · n ˆ + H g(θ)) ˆ hδθ, νi dΣ ˆ J2 (θ) g˜0 (θ) θ θ dθˆ Σθˆ (6.17) REMARK 6.5 Using the fact that the transformations θˆ on Qθˆ are ˆ on (0, T ) × Ω0 , we get horizontal, i.e., of the type (t, θ) d ˆ · δθ = J2 (θ) dθˆ

Z

T 0

Z

Γθˆ

h

i ˆ · δθ(1 + v 2 )1/2 + H g(θ)hδθ, ˆ g 0 (θ) nt i dt d Γθˆ (6.18) ν

An adjoint identity In the remaining part of this chapter, we will try to obtain several functionals with respect to the Lagrangian mapping we will need to use the following fundamental integration by

the structure of ˆ To this end, θ. parts formula,

THEOREM 6.3 def For any E ∈ L2 (Σθˆ) and δ θˆ = δθ ◦ θˆ ∈ Θ, the following identity holds, Z

T 0

Z

Γθˆ

ˆ ◦ θˆ−1 = E∂t (δθ ◦ θ)

Z

T 0

Z

Γθˆ

−∂t E − (divΓ Vθˆ) E − D E · Vθˆ · δθ

(6.19)

ˆ ◦ θˆ−1 . with Vθˆ = (∂t θ) PROOF

We first need to prove the following lemma,

LEMMA 6.3 Z

T 0

Z

Γθˆ

∂t f · g =

"Z

Γθˆ(τ )

f ·g

#T 0

−

Z

T 0

Z

Γθˆ

f · ∂t g + H(f · g) (Vθˆ · nθˆ) + ∇(f · g) · nθˆ (Vθˆ · nθˆ) (6.20)

170

Z

Moving Shape Analysis

Using classical integral derivatives formulas, we get ! Z Z Z T T h i ∂t f ·g = (f · g)0Γθˆ + H(f · g) (Vθˆ · nθˆ)

0

Γθˆ

0

= =

Z

Z

T

0 T 0

Z

Z

Γθˆ

Γθˆ

Γθˆ

∂t (f · g)|Γθˆ + ∇(f · g) · nθˆ + H(f · g) (Vθˆ · nθˆ)

∂t f · g + f · ∂t g + ∇(f · g) · nθˆ

+H(f · g) (Vθˆ · nθˆ)

2 ˆ ◦ In order to use this lemma, we need to modify the expression of E∂ t (δθ ◦ θ) −1 ˆ θ . For that purpose, we use the following identities, LEMMA 6.4

(∂t h) ◦ θˆ−1 = ∂t (h ◦ θˆ−1 ) − (D h) ◦ θˆ−1 ∂t (θˆ−1 ) ∂t (θˆ−1 ) = − D(θˆ−1 ) · V ˆ

(6.21) (6.22)

θ

(∂t h) ◦ θˆ−1 = ∂t (h ◦ θˆ−1 ) + D(h ◦ θˆ−1 ) · Vθˆ

(6.23)

We apply the last identity with h = δθ ◦ θˆ and we obtain ˆ ◦ θˆ−1 = ∂t δθ + D δθ · V ˆ ∂t (δθ ◦ θ)

(6.24)

θ

This allows us to state the following, Z Z TZ Z TZ ˆ ◦ θˆ−1 = E · (∂t δθ) + E∂t (δθ ◦ θ) 0

0

Γθˆ

Γθˆ

T 0

Z

Γθˆ

E · (D δθ · Vθˆ) (6.25)

Using lemma (6.3) with f = δθ and g = E, we obtain Z TZ Z TZ − δθ · ∂t E − ∇(E · δθ) · nθˆ (Vθˆ · nθˆ) E · (∂t δθ) = 0

Γθˆ

0

Γθˆ

− H(E · δθ) (Vθˆ · nθˆ)

For the second right hand side we need to establish the following lemma, LEMMA 6.5 Z TZ Z E · (D δθ · Vθˆ) = 0

Γθˆ

T 0

Z

Γθˆ

−δθ · (D E · Vθˆ) − (divΓ Vθˆ) δθ · E

+ ∇(E · δθ) · nθˆ (Vθˆ · nθˆ) + H (E · δθ) (Vθˆ · nθˆ) Combining these two identities, we obtain the expected result.

(6.26)

Tube derivative in a Lagrangian setting

6.2.4

171

Unconstraint non-cylindrical shape optimization

We would like to minimize over θ ∈ Θad the following functional, Z Z ˆ = ˆ y(θ))dxdt ˆ ˆ y(θ))dΣ ˆ j(θ) F (θ, + S(θ, θˆ with Qθˆ =

[

0
Qθˆ

(6.27)

Σθˆ

ˆ 0 )) and y(θ) ˆ ∈ H(Q ˆ). Using the shape derivative ({t} × θ(Ω θ

calculus tools introduced in the previous section, it is possible to obtain the ˆ derivative with respect to θˆ of the functional j(θ), PROPOSITION 6.1 def Let us assume that for θˆ ∈ Θad and for any direction δ θˆ = δθ ◦ θˆ ∈ Θ, ˆ admits a material derivative y( ˆ · δθ ∈ H(Q ˆ), 1. y(θ) ˙ θ) θ 0

ˆ ˆ · δθ − ∇y(θ) ˆ · δθ exists in H(Q ˆ), 2. the partial derivative y (θ)δθ = y( ˙ θ) θ ˆ is differentiable at τ = 0, 3. τ 7→ j((I +τ δθ) ◦ θ)

ˆ ∈ then j(.) is differentiable at point θˆ and there exists a distribution G(θ) 0 2 d L (0, T ; D (R )) with support inside Qθˆ such that Z T d ˆ ˆ δθi 0 dt j(θ) · δθ = hG(θ), (6.28) D ,D dθˆ 0 ˆ admits the two following representations, Furthermore G(θ) • using material derivative, Z

+

Z

Qθˆ

Σθˆ

h

h

ˆ δθi = hG(θ), i ˆ y(θ))δθ ˆ ˆ y(θ)) ˆ · y( ˆ ˆ y(θ)) ˆ · div δθ dQ ˆ F˙ (θ, + ∇y F (θ, ˙ θ)δθ + F (θ, θ

i ˆ y(θ))δθ ˆ ˆ y(θ)) ˆ · y( ˆ ˆ y(θ)) ˆ · divΓ δθ dΣ ˆ ˙ θ, S( + ∇y S(θ, ˙ θ)δθ + S(θ, θ

(6.29)

• using shape partial derivative, Z h i ˆ δ θi ˆ = ˆ y(θ))δθ ˆ ˆ y(θ)) ˆ · y 0 (θ) ˆ · δθ dQ ˆ hG(θ), F 0 (θ, + ∇y F (θ, θ + +

Z

Z

Qθˆ

Σθˆ T 0

h

i ˆ y(θ)) ˆ · δθ + ∇y S(θ, ˆ y(θ)) ˆ · y 0 (θ)δθ ˆ S 0 (θ, dΣθˆ

Z

Γθˆ

h

ˆ y(θ)) ˆ + ∂n S(θ, ˆ y(θ)) ˆ + ∇y S(θ, ˆ y(θ))∂ ˆ n y(θ) ˆ F (θ,

i ˆ y(θ)) ˆ hδθ, n ˆi dt d Γ ˆ +HS(θ, θ θ

(6.30)

172

Moving Shape Analysis

PROOF We perform the same steps as in [135] and we use the noncylindrical calculus introduced previously. 0

ˆ From now on, let us assume that the partial shape derivative y (θ)δθ is the unique solution of the following linear tangent problem, ˆ · δθ, φi = L(δθ, φ), hAy 0 (θ)

∀ φ ∈ H(Qθˆ)

(6.31)

with A ∈ L(H(Qθˆ, H(Qθˆ)∗ )) and L(δθ, .) ∈ H(Qθˆ)∗ . We define the element ˆ ∈ H(Q ˆ)∗ as the solution of the following adjoint problem, p(θ) θ ∗

hA p, ψi =

Z

Qθˆ

h

Z i ˆ ˆ ∇y F (θ, y(θ)) · ψ d Qθˆ +

Σθˆ

h

i ˆ y(θ)) ˆ · ψ dΣ ˆ, ∇y S(θ, θ

∀ ψ ∈ H(Qθˆ)

We then use the following identity ˆ y 0 (θ) ˆ · δθi = hAy 0 (θ) ˆ · δθ, p(θ)i ˆ = L(δθ, p(θ)) ˆ hA∗ p(θ),

(6.32)

ˆ with respect to θˆ has This allows us to conclude that the derivative of j( θ) the following structure, 0

ˆ · δθ = L(δ, p(θ)) ˆ + j (θ) +

Z

Σθˆ

Z

Qθˆ

ˆ y(θ)) ˆ · δθ d Q ˆ + F 0 (θ, θ

Z

Σθˆ

ˆ y(θ))hδθ, ˆ F (θ, nθˆi

ˆ y(θ)) ˆ · δθ + ∂n S(θ, ˆ y(θ)) ˆ + ∇y S(θ, ˆ y(θ))∂ ˆ n y(θ) ˆ S 0 (θ,

ˆ y(θ)) ˆ hδθ, n ˆi + HS(θ, θ

REMARK 6.6 Particular case : We assume the following hypothesis: Z ˆ ˆ hδθ, n ˆi dt dΓ ˆ, 1. L(δθ, p(θ)) = `(p(θ)) θ θ Σθˆ

2.

3.

Z Z

Qθˆ

ˆ y(θ)) ˆ · δθ d Q ˆ = 0, F 0 (θ, θ

Σθˆ

ˆ y(θ)) ˆ · δθ d Σ ˆ = S 0 (θ, θ

Z

Σθˆ

ˆ y(θ)) ˆ hδθ, n ˆi dt d Γ ˆ. s(θ, θ θ

ˆ is only supported on Γ ˆ with a density g ˆ In this case the gradient of j(θ) θ θ

Tube derivative in a Lagrangian setting

173

such that the following identity holds, Z T 0 ˆ · δθ = ˆ δθ · n ˆi j (θ) hg(θ), θ =

Z

0

T

0

Z

Γθˆ

h

ˆ + s(θ, ˆ y(θ)) ˆ + ∂n S(θ, ˆ y(θ)) ˆ `(p(θ))

i ˆ y(θ))∂ ˆ n y(θ) ˆ + HS(θ, ˆ y(θ)) ˆ hδθ, n ˆi dt d Γ ˆ +∇y S(θ, θ θ

6.2.5

(6.33)

Shape optimization under state constraints

In the previous section, we have established for a general functional, the structure of its gradient with respect to the moving domain. This general functional involved distributed and boundary terms and a state function ˆ ∈ H(Q ˆ) that depends on the moving domain Q ˆ and its parametrization y(θ) θ θ ˆ This gradient will be involved in the first-order optimality conditions of θ. ˆ the minimization problem for j(θ). In this paragraph, we are interested by optimization problems involving state constraints. The design variables are the Lagrange mapping and a state variˆ y) ∈ θˆad × H(D) related by an abstract state variable able (θ, ˆ y) = 0, e(θ,

in H(D)∗

(6.34)

This setting allows to avoid the derivation of the state variable y(θ) solution of the implicit equation (6.34) with respect to the mapping θˆ ∈ Θad . Hence we would like to solve the following minimization problem, ˆ y) min j(θ, ˆ ∈ θad × H(D), : e(θ, ˆ y) = 0, in H(D)∗ 8 < (θ, ˆ y)

with

ˆ y) = j(θ,

Z

Qθˆ

ˆ y) d Q ˆ + F (θ, θ

Z

Σθˆ

ˆ y) d Σ ˆ S(θ, θ

(6.35)

(6.36)

and ˆ y), φi = he(θ,

Z

Qθˆ

ˆ y, φ) d Q ˆ + EΩ (θ, θ

Z

Σθˆ

ˆ y, φ) d Σ ˆ, EΓ (θ, θ

∀ φ ∈ H(D)

(6.37) EΩ (., ., φ) where we suppose that the applications (., ., φ) −→ are linear conEΓ (., ., φ) tinuous on H(D).

174

Moving Shape Analysis

In order to derive the first-order optimality conditions for problem (6.35), we introduce the Lagrangian functional, ˆ y; φ) = j(θ, ˆ y) − he(θ, ˆ y), φiH(D)∗ ,H(D) L(θ,

(6.38)

Consequently, solving problem (6.35) is equivalent to finding the saddle point ¯ˆ ¯ y¯, φ) ∈ Θad × H(Qθ¯ˆ) × H(Qθ¯ˆ)∗ of the following Min-Max problem, (θ, min ˆ y) ∈ Θad × H(D) (θ,

ˆ y; φ) max L(θ, φ ∈ H(D)∗

(6.39)

PROPOSITION 6.2 Let us assume the following points, ¯ ˆ y¯) and the functional j(θ, ˆ y) is 1. Problem (6.35) admits a local solution (θ, ¯ ˆ Fr´echet differentiable in a neighborhood of point (θ, y¯) ∈ Θad × H(Qθ¯ˆ). ¯ˆ 2. The local solution (θ, y¯) is a regular point, i.e., that the linear tangent ¯ 0 ˆ application e ((θ, y¯)) is surjective. Then there exists a Lagrange multiplier φ¯ ∈ H(Qθ¯ˆ)∗ , such that the following first-order optimality conditions hold,

0

L(θ,y) ˆ

¯ ˆ y¯) = 0, e(θ, ¯ ˆ y¯; φ) ¯ =0 (θ,

(6.40) (6.41)

We shall now derive the explicit form of the optimality conditions for problem (6.39). Using differentiability results of j(., .), the following identity comes easily, Z h i 0 ˆ ˆ y) · δy + ∇y EΩ (θ, ˆ y, φ) · δy dQ ˆ + hLy (θ, y; φ), δyi = ∇y F (θ, θ Qθˆ

Z

Σθˆ

h

i ˆ y) · δy + ∇y EΓ (θ, ˆ y, φ) · δy dΣ ˆ (6.42) ∇y S(θ, θ

and ˆ = hL0θˆ, δ θi

Z

Qθˆ

h

i ˆ y) · δθ + E 0 (θ, ˆ y, φ) · δθ d Q ˆ F 0 (θ, Ω θ +

+

Z

Σθˆ

h

Z

Σθˆ

h

ˆ y, φ) · δθ ˆ y) · δθ + E 0 (θ, S 0 (θ, Γ

i

i ˆ y) + EΩ (y, θ; ˆ φ) + (H I +∂n ) · (S(θ, ˆ y) + EΓ (θ, ˆ y, φ)) ·hδθ, n ˆi d Σ ˆ F (θ, θ θ

Tube derivative in a Lagrangian setting

175

¯ˆ At a regular point (θ, y¯), there exists a Lagrange multiplier φ¯ solution of the following linear tangent adjoint equation, Z h i ¯ˆ ¯ ˆ y¯, φ) ¯ · ψ dQ ˆ ∇y F (θ, y¯) · ψ + ∇y EΩ (θ, θ Qθ¯ Z h i ¯ ¯ ˆ y¯) · ψ + ∇y EΓ (θ, ˆ y¯, φ) ¯ · ψ dΣ ˆ = 0, + ∇y S(θ, θ Σθ¯

∀ψ ∈ H(D)

Then the optimality condition reduces to Z h i 0 ¯ ¯ ¯ ˆ ˆ ˆ y¯) · δθ + E 0 (θ, ˆ y¯, φ) ¯ · δθ dQ ˆ j (θ; δ θ) = F 0 (θ, Ω θ Q¯ Z θh i ¯ ¯ ˆ y¯) · δθ + E 0 (θ, ˆ y¯, φ) ¯ · δθ + S 0 (θ, Γ Σ¯ Z θ ¯ ¯ ¯ ˆ y¯) + EΩ (θ, ˆ y¯; φ) ¯ + (H I +∂n ) · (S(θ, ˆ y¯) + F (θ,

¯ˆ ¯ y¯, φ)) · hδθ, nθˆi dΣθˆ +EΓ (θ,

Σθ¯

=0

(6.43)

REMARK 6.7 Let us add the following hypothesis, Z h i ˆ y) · δθ + E 0 (θ, ˆ y, φ) · δθ dQ ˆ = 0, F 0 (θ, 1. Ω θ 2.

Z

Qθˆ

Σθˆ

h

ˆ y) · δθ + E 0 (θ, ˆ y, φ) · δθ S 0 (θ, Γ

hδθ, nθˆi dΣθˆ.

i

dΣθˆ =

Z

Σθˆ

h

i ˆ y) + eΓ (θ, ˆ y, φ) · s(θ,

In this case, the gradient is only supported on Σ θ¯ˆ, and its representation has the following form, Z h 0 ¯ ¯ ¯ ¯ˆ ¯ ˆ y¯, φ) ¯ + F (θ, ˆ y¯) + EΩ (θ, ˆ y¯, φ) ¯ ˆ s(θ, y¯) + eΓ (θ, j (θ) · δθ = Σθ¯

i ¯ ¯ ˆ y¯) + EΓ (θ, ˆ y¯, φ)) ¯ hδθ, nθˆi dΣθˆ = 0 + (H I +∂n ) · (S(θ,

6.2.6

(6.44)

Eulerian vs. Lagrangian non-cylindrical derivative

We shall now ask a fundamental question concerning the link between the derivative using perturbations of the identity and Eulerian derivative introduced in [59].

176

Moving Shape Analysis

Let us consider a functional J, J : A −→ R Q 7→ J(Q) with

A ≡ Q ∈ ET ,

(6.45)

Q⊂D

(6.46)

Let us assume that this functional can be parametrized by either the Lagrangian mapping parametrizing the tubes in A with j ` = J ◦ θˆ such that j` : Θad −→ R ˆ = J(Q ˆ) θˆ 7→ j` (θ) θ

(6.47)

or the velocity field V ∈ V building the tubes in A with j V = J ◦ Tt (V ) such that je : V −→ R (6.48) V 7→ je (V ) = J(Q(V )) The following result holds true, THEOREM 6.4 The differentiability of the functional j` at point θˆ ∈ Θad in the direction def δ θˆ = δθ ◦ θˆ ∈ Θ is equivalent to the differentiability of je with respect to V ∈ Vad in the direction W ∈ V . Furthermore the respective functional derivatives are related by the following relation, ˆ δθi = hj 0 (V ˆ), W ˆi, hj`0 (θ), e θ θ

def

δ θˆ = δθ ◦ θˆ ∈ Θ

(6.49)

ˆ ◦ θˆ−1 − D V ˆ · δθ and the transverse field with Vθˆ = ∂t θˆ ◦ θˆ−1 , Wθˆ = ∂t (δθ ◦ θ) θ involved in the Eulerian derivative is given by Zt = δθ. PROOF As shown here, in the Lagrangian mapping case, we deal with a first-order perturbation of the identity inside Ω θˆ. Ω0 θˆ + ρθˆ

↓

θˆ

−→

Ωθˆ

. I +ρδθ

Ωθ+ρδ ˆ θˆ The structure of the perturbation rules is similar in the case of an Eulerian

Tube derivative in a Lagrangian setting

177

description using velocity fields, Ω0 Tt (V + ρW )

↓

Tt (V )

−→

Ω(V ) def

. Tρt = Tt (V + ρW ) ◦ Tt (V )−1

Ω(V + ρW ) As proved in [59], the transverse map Tρt is the flow of a transverse velocity field Zρt . This velocity field is written Zt for ρ = 0 and it satisfies the following Cauchy problem, ∂t Zt + D Zt · V − D V · Zt = W, (0, T ) (6.50) Zt=0 = 0 Using chain rule it is possible to deduce the following identity, je0 (V ) · W =

d d d j` (Tt (V )) · W = j` (Tt (V )) · ( Tt (V ) · W ) dV dV dθˆ

(6.51)

As recalled above, the derivative of the mapping T t (V ) with respect to V is d Tt (V ) · W = Zt ◦ Tt (V ) dV

(6.52)

def We then set δ θˆ = δθ ◦ θˆ = Zt ◦ Tt (V ), which means that δθ = Zt . For a given Lagrangian map θˆ ∈ Θad , there exists Vθˆ = ∂t θˆ ◦ θˆ1 ∈ V such that θˆ = Tt (Vθˆ). def ˆ we Hence if we perturbed the Lagrangian map in the direction δ θˆ = δθ ◦ θ, generate a velocity field perturbation Wθˆ given by the following expression,

ˆ ◦ θˆ−1 − D V ˆ ◦ δθ Wθˆ = ∂t (δθ ◦ θ) θ

(6.53)

Indeed, using the chain rule, we have ˆ ◦ θˆ−1 = ∂t δθ + D δθ · ∂t θˆ ◦ θˆ−1 ∂t (δθ ◦ θ) = ∂t δθ + D δθ · Vθˆ As stated before, we can identify the transverse velocity field Z t with δθ. Using the transverse equation we define Wθˆ such that ∂t δθ + D δθ · Vθˆ = Wθˆ + D Vθˆ · δθ

(6.54)

from which we deduce that ˆ ◦ θˆ−1 − D V ˆ · δθ Wθˆ = ∂t (δθ ◦ θ) θ

(6.55)

178

6.3

Moving Shape Analysis

Navier-Stokes equations in moving domain

In this section, we shall apply the previous non-cylindrical shape derivative concepts, in order to solve two questions related to the Navier-Stokes equations in a moving domain: 1. We will first obtain a differentiability result concerning the sensitivity of the solution of Navier-Stokes equations in a moving domain with respect to the Lagrangian mapping θˆ ∈ Θad . This result allows us to justify in a rigorous manner transpiration boundary conditions [63] in fixed domain as a first order approximation of the Navier-Stokes system in moving domain. This approximation is valid around a fixed domain configuration, i.e., around θˆ = I. 2. Then, we deal with an optimal control problem for the Navier-Stokes ˆ system where the control is the displacement of the moving domain θ. Using results obtained in the first section, we derive the structure of the gradient for a general cost functional. We recover early results obtained using Eulerian field derivative methods.

6.3.1

Transpiration boundary conditions

We consider a viscous incompressible newtonian fluid inside a moving domain Ωθˆ. Its evolution is described by its velocity u and its pressure p. The couple (u, p) satisfies the classical Navier-Stokes equations written in nonconservative form,  ∂t u + D u · u − ν∆u + ∇p = 0, Qθˆ    div(u) = 0, Qθˆ (6.56) u = V , Σθˆ  ˆ θ   u(t = 0) = u0 , Ω0

A kinematic continuity boundary condition is satisfied on Γ θˆ; this means that ˆ θˆ−1 the fluid velocity is equal to the velocity of the moving boundary V θˆ = ∂t θ◦ on Γθˆ. The main result of this paragraph is the following, THEOREM 6.5 Let us assume that

• The mapping θˆ ∈ Θ is such that Vθˆ = ∂t θˆ ◦ θˆ−1 ∈ H 1 (0, T ; H0m (D)) with m ≥ 5/2, then the solution (uθˆ, pθˆ) of the Navier-Stokes system (6.56) admits a material shape derivative in L2 (0, T ; H 2 (Ωθˆ)) ∩ L∞ (0, T ; H 1 (Ωθˆ)). Furthermore,

Tube derivative in a Lagrangian setting

179

ˆ · δθ, p0 (θ) ˆ · δθ) solution of the (uθˆ, pθˆ) admits a partial shape derivative (u0 (θ) following linear tangent system,  ∂t u0 + D u0 · u + D u · u0 − ν∆u0 + ∇p0 = 0, Qθˆ    div(u0 ) = 0, Qθˆ 0 −1 ˆ ˆ  u = ∂ (δθ ◦ θ) ◦ θ − D u · δθ, Σ t  θˆ  0 u (0) = 0, Ω0

(6.57)

PROOF We follow the proof used in [59],[57]. The only difference is that we do not need to use the transerve field Zt . We refer to [13] for the details in case of homogeneous Dirichlet boundary conditions. We shall detail the computation of the linearized boundary conditions. We perturbed the Lagrangian mapping with the increment ρδ θˆ and we get the following perturbed boundary conditions written on the fixed boundary Γ 0 , d ˆ d ˆ ◦ (θˆ + ρδ θ) ˆ ˆ ˆ = u(θ + ρδ θ) , on Γ0 (∂ ( θ + ρδ θ)) t dρ dρ ρ=0 ρ=0 d ˆ ˆ ˆ ◦ θˆ · δ θˆ = ∂t δ θˆ u(θ + ρδ θ) ◦ θˆ + (D u(θ)) dρ ρ=0 h i ˆ · δθ) ◦ θˆ + D u(θ) ˆ · δθ ◦ θˆ = ∂t (δθ ◦ θ) ˆ (u0 (θ) then we get ˆ · δθ = ∂t (δθ ◦ θ) ˆ ◦ θˆ−1 − D u(θ) ˆ · δθ, u0 (θ)

on Γθˆ

(6.58)

Applying this theorem in the particular case, where the reference flow is considered in a fixed domain, i.e., with θˆ = I, leads to the derivation of transpiration boundary condition for linearized system, COROLLARY 6.1 Considering a reference flow (u0 , p0 ) defined in a fixed domain Ω0 × (0, T ), i.e., satisfying the following Navier-Stokes system,  ∂t u0 + D u0 · u0 − ν∆u0 + ∇p0 = 0, Ω0 × (0, T )    div(u0 ) = 0, Ω0 × (0, T ) 0 u = 0, Γ0 × (0, T )    0 0 u (t = 0) = u0 , Ω0

(6.59)

The first order variation (δu, δp) = (D u0 · δθ, D p0 · δθ) + (u0 , p0 ) around this flow of the solution (u0 , p0 ) of problem (6.59) exists and (u0 , p0 ) satisfies the

180

Moving Shape Analysis

following linear tangent system written in the reference fixed domain,  ∂t u0 + D u0 · u0 + D u0 · u0 − ν∆u0 + ∇p0 = 0, Ω0 × (0, T )    div(u0 ) = 0, Ω × (0, T ) 0

ˆ ◦ θˆ−1 − (D u0 · n) hδθ, ni,  u0 = ∂t (δθ ◦ θ)   0 u (0) = 0,

Γ0 × (0, T ) Ω0

(6.60)

PROOF This a direct consequence of theorem (6.5). We only need to check the boundary conditions. In the general case, we have ˆ · δθ = ∂t (δθ ◦ θ) ˆ ◦ θˆ−1 − D u(θ) ˆ · δθ, u0 (θ)

on Γθˆ

(6.61)

We set θˆ = I and we get ˆ ◦ θˆ−1 − D u0 · δθ, u0 = ∂t (δθ ◦ θ)

Γ0 × (0, T )

We then use D u0 · δθ = (DΓ u0 + D u0 n ⊗ n) · δθ But u0 = 0 on Γ0 , then DΓ u0 = 0 on Γ0 , from which we deduce ˆ ◦ θˆ−1 − (D u0 · n) hδθ, ni, u0 = ∂t (δθ ◦ θ)

6.3.2

Γ0 × (0, T )

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

In this paragraph, we shall deal with an optimal control for the NavierStokes system, where the control is the displacement of the domain boundary ˆ u ˆ, p ˆ). We Γθˆ. We choose to minimize a general cost functional based on ( θ, θ θ are interested in solving the following minimization problem : ˆ min j(θ) ˆ θ∈U

(6.62)

ˆ = J ˆ(u(θ), ˆ p(θ)) ˆ with (u(θ), ˆ p(θ)) ˆ is a weak solution of problem where j(θ) θ (6.56) and Jθˆ(u, p) is a real functional of the following form : Jθˆ(u, p) =

α γ kB uk2L2 (Q ˆ) + kK Vθˆk2L2 (Σ ˆ) θ θ 2 2

(6.63)

where B ∈ L(H, L2 (Qθˆ)) is a general linear differential operator satisfying the following identity, hB u, viL2 (Qθˆ) + hu, B ∗ viL2 (Qθˆ) = hBΣ u, viL2 (Σθˆ)

(6.64)

Tube derivative in a Lagrangian setting

181

and K ∈ L(U, L2 (Σθˆ)) is a general linear differential operator satisfying the following identity, hK u, viL2 (Σθˆ) + hu, K∗ viL2 (Σθˆ) = hKΣ u, viL2 (Σθˆ)

(6.65)

The main difficulty in dealing with such a minimization problem is related to ˆ the fact that integrals over the domain Ωθˆ depend on the control variable θ. THEOREM 6.6 def ˆ possesses For Vθˆ = ∂t θˆ ◦ θˆ−1 ∈ U and Ω0 of class C 2 , the functional j(θ) ˆ a gradient ∇j(θ) which is supported on the moving boundary Γθˆ and can be represented by the following expression, ˆ = − ∂t E ˆ − (divΓ V ˆ) E ˆ − D E ˆ · V ˆ ∇j(θ) θ θ θ θ θ − (−σ(ϕ, π) · nθˆ + αBΣ B u) · D u − γ(−K∗ K + KΣ K) Vθˆ · DΓ Vθˆ 1 + (6.66) α |Bu|2 + γ |KVθˆ|2 nθˆ 2

where (ϕ, π) stands for the adjoint fluid state solution of the following system,  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = −α B ∗ B u, Qθˆ    div(ϕ) = 0, Qθˆ (6.67) ϕ = 0, Σθˆ    ϕ(T ) = 0, ΩT and

Eθˆ = −σ(ϕ, π) · nθˆ + γ [−K∗ K + KΣ K] Vθˆ + αBΣ B Vθˆ

(6.68)

ˆ in the Using chain rule, we obtain the differentiability of j( θ) def perturbation direction δ θˆ = δθ ◦ θˆ ∈ Θ. Using theorems (6.1) and (6.2), we can state PROOF

ˆ δθi = αhBu, Bu0 iL2 (Q ) + γhKV ˆ, K(∂t (δ θ) ˆ ◦ θˆ−1 − DΓ V ˆ · δθ)iL2 (Σ ) hj 0 (θ), ˆ ˆ θ θ θ θ 1 2 2 + h α |Bu| + γ |KVθˆ| nθˆ, δθiL2 (Σθˆ) 2

Using the adjoint operators, we perform an integration by parts, ˆ δθi = hj 0 (θ),

ˆ ◦ θˆ−1 − DΓ V ˆ · δθiL2 (Σ ) dis − αhB ∗ B u, u0 iL2 (Qθˆ) + γh(−K∗ K + KΣ K) Vθˆ, ∂t (δ θ) ˆ θ θ 1 + hαBΣ B u, u0 iL2 (Σθˆ) + h α |Bu|2 + γ |KVθˆ|2 nθˆ, δθiL2 (Σθˆ) 2

182

Moving Shape Analysis

We now define the fluid adjoint state (ϕ, π) solution of the adjoint problem (5.8) and we deduce the following identity,

=

Z

Z

0

Qθˆ

0

0

0

[∂t u + D u · u + D u · u − ν∆u + ∇ p ] ϕ − ∗

Qθˆ

0

0

[−∂t ϕ − D ϕ · u + D u · ϕ − ν∆ϕ + ∇ π] u − +

Z

Σθˆ

Z

π div u0 Qθˆ

Z

p0 div ϕ Qθˆ

[p0 n · ϕ − νϕ · ∂n u0 + νu0 · ∂n ϕ − u0 · π n] (6.69)

Hence we get −αhB ∗ B u, u0 iL2 (Qθˆ) = −hσ(ϕ, π) · nθˆ, u0 iL2 (Σθˆ)

(6.70)

Using the linearized system (5.83) satisfied by u 0 , we deduce that ˆ ◦ θˆ−1 − D u · δθ, u0 = ∂t (δθ ◦ θ)

on Σθˆ

then, ˆ ◦ θˆ−1 iL2 (Σ ) ˆ δθi = h−σ(ϕ, π) · n ˆ + αBΣ B u + γ(−K∗ K + KΣ K) V ˆ, ∂t (δ θ) hj 0 (θ), ˆ θ θ θ − h−σ(ϕ, π) · nθˆ + αBΣ B u, D u · δθiL2 (Σθˆ)

− hγ(−K∗ K + KΣ K) Vθˆ, DΓ Vθˆ · δθiL2 (Σθˆ) 1 + h α |Bu|2 + γ |KVθˆ|2 nθˆ, δθiL2 (Σθˆ) 2

We now set Eθˆ = −σ(ϕ, π) · nθˆ + αBΣ B u + γ (−K∗ K + KΣ K) Vθˆ and we use theorem (6.3) and we obtain ˆ δθi = h −∂t E ˆ − (divΓ V ˆ) E ˆ − D E ˆ · V ˆ hj 0 (θ), θ θ θ θ θ − (−σ(ϕ, π) · nθˆ + αBΣ B u) · D u − γ(−K∗ K + KΣ K) Vθˆ · DΓ Vθˆ 1 + α |Bu|2 + γ |KVθˆ|2 nθˆ, δθiL2 (Σθˆ) 2

ˆ from which we deduce the expression of the gradient ∇ j( θ).

6.3.3

Comparison with the Eulerian derivative version

Using theorem (6.4), we can reciprocally deduce the expression of one of the directional derivatives from another. Let us apply this equivalence on the optimal control treaten previously. We have established the structure of the Lagrangian derivative of the func-

Tube derivative in a Lagrangian setting

183

ˆ tional j(θ), ˆ δθi = h−σ(ϕ, π) · n ˆ + αBΣ B u + γ(−K∗ K + KΣ K) V ˆ, ∂t (δ θ) ˆ ◦ θˆ−1 iL2 (Σ ) hj 0 (θ), ˆ θ θ θ − h−σ(ϕ, π) · nθˆ + αBΣ B u, D u · δθiL2 (Σθˆ)

− hγ(−K∗ K + KΣ K) Vθˆ, DΓ Vθˆ · δθiL2 (Σθˆ) 1 + h α |Bu|2 + γ |KVθˆ|2 nθˆ, δθiL2 (Σθˆ) 2

ˆ ◦ θˆ−1 = W ˆ + D V ˆ · δZt ; we get Let us set θˆ = Tt (Vθˆ), δθ = Zt and ∂t (δθ ◦ θ) θ θ hj 0 (Tt (Vθˆ)), Zt i = h−σ(ϕ, π) · nθˆ + αBΣ B u + γ(−K∗ K + KΣ K) Vθˆ, Wθˆ + D Vθˆ · Zt iL2 (Σ(Vθˆ)) − h−σ(ϕ, π) · nθˆ + αBΣ B u, D u · Zt iL2 (Σθˆ)

We recall that

− hγ(−K∗ K + KΣ K) Vθˆ, DΓ Vθˆ · Zt iL2 (Σ(Vθˆ)) 1 + h α |Bu|2 + γ |KVθˆ|2 , hZt , nθˆiiL2 (Σ(Vθˆ)) 2

D Vθˆ · Zt = DΓ Vθˆ · Zt + (D Vθˆ · nθˆ) hZt , nθˆi D u · Zt = DΓ u · Zt + (D u · nθˆ) hZt , nθˆi

Using u = Vθˆ, on Γθˆ we deduce that D u·Zt = DΓ Vθˆ ·Zt +(D u·nθˆ) hZt , nθˆi, then we deduce hj 0 (Tt (Vθˆ)), Zt i = h−σ(ϕ, π) · nθˆ + αBΣ B u + γ(−K∗ K + KΣ K) Vθˆ, WθˆiL2 (Σ(Vθˆ)) h−σ(ϕ, π) · nθˆ + αBΣ B u, D(Vθˆ − u) · nθˆ hZt , nθˆiiL2 (Σθˆ) + hγ(−K∗ K + KΣ K) Vθˆ, D Vθˆ · nθˆ hZt , nθˆiiL2 (Σ(Vθˆ)) 1 + h α |Bu|2 + γ |KVθˆ|2 , hZt , nθˆiiL2 (Σ(Vθˆ)) 2

We choose a canonical extension Vθˆ◦p. This means that D Vθˆ·nθˆ = 0, then hje0 (Vθˆ), Wθˆi = hj 0 (Tt (Vθˆ)), Zt i = h−σ(ϕ, π) · nθˆ + αBΣ B u + γ(−K∗ K + KΣ K) Vθˆ, WθˆiL2 (Σ(Vθˆ)) + h σ(ϕ, π) · nθˆ − αBΣ B u · D u · nθˆ +

1 α |Bu|2 + γ |KVθˆ|2 , hZt , nθˆiiL2 (Σθˆ) 2

on Γθˆ,

184

Moving Shape Analysis

This is exactly the expression obtained in [57], using Eulerian methods. Let us define the adjoint transverse boundary field λθˆ solution of the following tangential dynamical system, −∂t λθˆ − ∇Γ λθˆ · Vθˆ − div(Vθˆ) λθˆ = f, (0, T ) (6.71) λθˆ(T ) = 0, ΓT (V ) with

then

1 f = σ(ϕ, π) · nθˆ − αBΣ B u · D u · nθˆ + α |Bu|2 + γ |KVθˆ|2 2

(6.72)

∇je (Vθˆ) = −λθˆ nθˆ − σ(ϕ, π) · nθˆ + α BΣ B u + γ −K∗ K Vθˆ + KΣ K Vθˆ (6.73)

where (ϕ, π) stands for the adjoint fluid state.

Chapter 7 Sensitivity analysis for a simple fluid-solid interaction system

7.1

Introduction

This chapter deals with the analysis of an inverse boundary problem arising in the study of bridge deck aeroelastic stability. The aeroelastic system consists of an elastically supported rigid solid moving inside an incompressible fluid flow in 2-D. As described in the introduction of this lecture note, the keystone of the sensitivity analysis of coupled fluid-structure model is based on the moving shape analysis introduced in the previous chapters. The chapter is organized in four parts : • In section 7.2, we introduce the notations and the mechanical system we shall deal with and we state the main result of this chapter, namely the structure of a cost functional gradient with respect to inflow boundary conditions pertubations. The proof is given in the remaining part of the chapter. • In section 7.3, we recall early results concerning the well-posedness of the coupled fluid-structure system. • In section 7.4, we introduce the minimization problem and its associated Lagrangian functional. • In section 7.5, we perform derivation of the Lagrangian with respect to state variables. This allows us to obtain the structure of the adjoint variables involved in the cost functional gradient.

7.2

Mathematical settings

We consider a two dimensionnal flexible structure in rigid motion. For the sake of simplicity, we only consider one degree of freedom for the structural motion : the vertical displacement d(t)e2 where e2 is the element of cartesian

185

186

Moving Shape Analysis Tt(V )(.) Γo

Γo

Ωf0

Ωft Ωst

Ωs0 Γs0

Γst

V∞

V∞

FIGURE 7.1:

Arbitrary Euler-Lagrange map

basis (e1 , e2 , e3 ) in R3 . The structure is surrounded by a viscous fluid in the plane (e 1 , e2 ). We consider a control volume Ω ⊂ R2 containing the solid for every time t ≥ 0. Hence, the analysis of the coupled problem is set in Ω × (0, T ) where T > 0 is an arbitrary time. In order to deal with such a coupled system, we introduce a diffeomorphic map sending a fix reference domain Ω0 into the physical configuration Ω at time t ≥ 0. Without loss of generality, we choose the reference configuration to be the physical configuration at initial time Ω(t = 0). Hence, we define a map Tt ∈ C 1 (Ω0 ) such that Ωft = Tt (Ωf0 ), Ωst = Tt (Ωs0 ) Since we only consider one degree of freedom motion, we write

We set Σs ≡

[

0
Ωst = Ωs0 + d(t)e2 [ ({t} × Γst ), Qf ≡ ({t} × Ωft ) and Σf∞ ≡ Γf∞ × (0, T ) 0
The map Tt can be actually defined as the flow of a particular vector field, as described in the following lemma : LEMMA 7.1 [147] Assuming that d(.) is smooth enough, there exists a vector field V that builds Q , i.e, Ωft = Tt (V )(Ωf0 ), Ωst = Tt (V )(Ωs0 ) where Tt (V ) is the solution of the following dynamical system : Tt (V ) : Ω0 −→ Ω x0 7−→ x(t, x0 ) ≡ Tt (V )(x0 )

Sensitivity analysis for a simple fluid-solid interaction system

187

with dx = V (τ, x(τ )), τ ∈ [0, T ] dτ x(τ = 0) = x0 , in Ω0

(7.1)

In our simple case, we can give an example of an appropriate flow vector field :  ˙ x ∈ Ωst  V (x, t) = d(t)e 2, ˙ (7.2) V (x, t) = Ext(d(t)e2 ), x ∈ Ωft  V (x, t) · n = 0, x ∈ Γf∞

where Ext is an arbitrary extension operator from Γ s0 into Ωf0 . The map Tt is usually referred to as the Arbitrary Euler-Lagrange map. The solid is described by the evolution of its displacement and its velocity ˙ is the solution of the following ordinary second order and the couple (d, d) differential equation : (

¨ k d = Ff , hm d + i d, d˙ (t = 0) = [d0 , d1 ]

(7.3)

where (m, k) stand for the structural mass and stiffness. F f is the projection of the fluid loads on Γst along the direction of motion e2 . The fluid is assumed to be a viscous incompressible newtonian fluid. Its evolution is described by its velocity u and its pressure p. The couple (u, p) satisfies the classical Navier-Stokes equations written in non-conservative form:  ∂t u + D u · u − ν∆u + ∇p = 0, Qf (V )    div(u) = 0, Qf (V ) Σf∞ Ωf0

u = u∞ ,    u(t = 0) = u0 ,

(7.4)

where ν stands for the kinematic viscosity and u∞ is the farfield velocity field. Hence, the projected fluid loads Ff have the following expression : Ff = −

Z

Γst

σ(u, p) · n

!

· e2

(7.5)

where σ(u, p) = −p I +ν(D u + ∗ D u) stands for the fluid stress tensor inside Ωft , with (D u)ij = ∂j ui = ui,j . We complete the whole system with kinematic continuity conditions at the fluid-structure interface Γst : u = V = d˙ e2 , sur Σs (V )

(7.6)

188

Moving Shape Analysis

To summarize, we get the following coupled system :  ∂t u + D u · u − ν∆u + ∇p = 0, Qf (V )     div(u) = 0, Qf (V )     u = u∞ , Σf∞    ˙ u = d e2 , Σs (V ) ! Z     σ(u, p) · n · e2 , (0, T ) m d¨ + k d = −    Γst  h i    u, d, d˙ (t = 0) = [u0 , d0 , d1 ] , Ωf0 × R2

(7.7)

Main Result: For u∞ ∈ Uc the following minimization problem, min j(u∞ ) (7.8) u∞ ∈ U c h i h i where j(u∞ ) = Ju∞ ( u, p, d, d˙ (u∞ )) with u, p, d, d˙ (u∞ ) is a weak solution

of problem (7.7) and Ju∞ is a real functional of the following form : Z h i i γ α Th Ju∞ ( u, p, d, d˙ ) = |d − d1g |2 + |d˙ − d2g |2 + ku∞ k2Uc 2 0 2

(7.9)

admits at least a solution u∗∞ which satisfies the following necessary first-order optimality conditions, ∇j(u∗∞ ) = (σ(ϕu∗∞ , πu∗∞ ) · n)|Σf∞ + γ u∗∞ = 0

(7.10)

with (ϕ, π, b1 , b2 ) are solutions of the following adjoint system,  −∂t ϕ − D ϕ · u + (∗ Du) · ϕ − ν∆ϕ + ∇q = 0, Qf (V )     Qf (V )  div(ϕ) = 0, ϕ = b 2 e2 , Σs (V )   ϕ = 0, Σf∞    ϕ(T ) = 0, ΩfT

(7.11)

−b˙1 + k b2 = α(d − d1g ), (0, T ) b1 (T ) = 0,

(7.12)

−∂t λ − ∇Γ λ · V = f, Σs (d˙ e2 ) λ(T ) = 0, ΓT (d˙ e2 )

(7.13)

i h ˙ 2 (D ϕ · e2 ) · e2 and with f = −d˙ b˙2 + ν (D ϕ · n) · (D u · n) − |d| Z

Γst (V )

Z h i λ n = −b1 − m b˙2 − α(d˙ − d2g ) e2 +

Γst (V )

σ(φ, π) · n

(7.14)

Sensitivity analysis for a simple fluid-solid interaction system

189

REMARK 7.1 We can eliminate the auxiliary adjoint variables (λ, b 1 ), in order to get a system only involving the adjoint variables (ϕ, π, b 2 ) . We then replace equations (7.2), (7.13) , (7.14) by the following second order ODE,  ! Z    m ¨b2 + k b2 = α(d − d1g ) − α(d˙ − d˙2g ) + ∂t σ(φ, π) · n · e2    Γst (V )    Z h i ˙ 2 (D ϕ · e2 ) · e2 − ν (D ϕ · n) · (D u · n) · n + |d| (0, T )    Γst (V )    h i    b2 , b˙ 2 (T ) = [0, 0] (7.15)

7.3

Well-posedness of the coupled system

We are interested in recalling global existence results for weak solutions to the initial boundary value problem (7.7). One should expect some restrictions on the existence time for the solutions since depending on the data, the solid may vanish outside the control volume Ω. This problem was recently investigated by several authors [131], [39], [54], [68], [81], [133]. We refer to [80] and [81] for a complete review. In our case, we only have one degree of freedom for the solid motion and we are dealing with two dimensional Navier-Stokes equations. Nevertheless, we deal with nonhomogeneous Dirichlet boundary conditions at the farfield fluid boundary. THEOREM 7.1 Assume the following hypothesis : i) Ωs0 , Ω0 are of class C 2 ,

ii) a = dist(Γf∞ , Ωs0 ) > 0, iii)

u0 ∈ (L2 (Ωf0 ))2 u∞ ∈ (H m (Σf∞ ))2 m > div(u0 ) = 0, dans Ωf0 u 0 = d 1 e2 , sur Γs0

3 4

(7.16)

then there exists a positive real time T0 = T0 (u0 , a, Ωs0 , Ω0 ) such that for any T ∈ (0, T0 ), there exists at least one weak solution to the initial-boundary value problem (7.7), with d ∈ W 1,∞ (0, T ; R) ∩ C 0 ([0, T ]; R) ˙ ∈ L2 (0, T ; Vd(.) ) ∩ L∞ (0, T ; Hd(.) ) (u, d)

190

Moving Shape Analysis

with Hd(t) ≡ (v, `) ∈ (L2 (Ω0 ))2 × R, v · n = 0,

div(v) = 0, Γf∞ ,

on

v|Ωst = ` · e2 , supp(v) ⊂ Ωft

and Vd(t) ≡ (v, `) ∈ (H 1 (Ω0 ))2 × R, v = 0,

div(v) = 0, on

Γf∞ ,

v|Ωst = ` · e2 , supp(v) ⊂ Ωft

and satisfies the following identity : # Z T "Z − u · ∂t v + md˙ `˙ − k d ` 0

+

Z

T 0

Z

Ωft

Ωft

[(D u · u) · v + ν D u · · D v] = md1 `(0) +

˙ )=0 ∀ (v, `) ∈ C ([0, T ]; Vd(.) ) with v(T ) = `(T 1

Z

Ωf0

u0 · v(0)

o

o

(7.17)

REMARK 7.2 Using results from Fursikov et al [72], we can relax the regularity assumption for u∞ and ask : iii) 1

1

u∞ · τ ∈ L2 (0, T ; (H 2 (Γf∞ ))2 ) ∩ H 4 (0, T ; (L2 (Γf∞ ))2 ) 1

3

u∞ · n ∈ L2 (0, T ; (H 2 (Γf∞ ))2 ) ∩ H 4 (0, T ; (H −1 (Γf∞ ))2 )

7.4

Inverse problem settings

We are interested in solving the following minimization problem : min j(u∞ ) (7.18) u∞ ∈ U c h i h i where j(u∞ ) = Ju∞ ( u, p, d, d˙ (u∞ )) with u, p, d, d˙ (u∞ ) is a weak solution of problem (7.7) and Ju∞ is a real functional of the following form : Z h i i γ α Th ˙ Ju∞ ( u, p, d, d ) = |d − d1g |2 + |d˙ − d2g |2 + ku∞ k2Uc (7.19) 2 0 2

Sensitivity analysis for a simple fluid-solid interaction system

191

The main difficulty in dealing with such a minimization problem is related to the dependence of integrals on the unknown domain Ω ft which depends also on the control variable u∞ . This point will be solved by using the ALE map Tt introduced previously.

7.4.1

Analysis strategy

We shall focus our efforts on the derivation of first-order optimality conditions for problem (7.18). This involves the computation of the gradient with respect to the inflow condition u∞ of the cost function Ju∞ . There exist several methods to handle such a question, D • Following [58], it is possible to handle the derivative Du (u, p, d) · δu∞ ∞ using a back transport map into a fix domain and use the weak implicit function theorem to justify and obtain the linearized system. Once the full linear tangent system is defined, it is possible to define an adjoint system whose solution may be involved in the computation of the objective function gradient.

• An other possible choice is to try to pass through the computation of a linear tangent system and directly get the adjoint system. This may be realized using a Min-Max formulation of the minimization problem (7.18) taking into account the coupled system constraint through Lagrange multipliers. In this chapter, we shall use the latter choice coupled with the introduction of a transverse map in order to handle the sensitivity analysis of the Lagrangian functional with respect to variation of the fluid domain.

7.4.2

Free divergence and non-homogeneous Dirichlet boundary condition constraints

We shall now describe the way to take into account inside the Lagrangian functional several constraints associated to the coupled system. The divergence free condition coming from the fact that the fluid has an homogeneous density and evolves as an incompressible flow is difficult to impose on the mathematical and numerical point of view. We choose to include the divergence free condition directly into the Lagrangian functional thanks to a multiplier that may play the role of the adjoint variable associated to the primal pressure variable. This leads in a certain sense to a saddle point formulation or mixed formulation of the Navier-Stokes subsystem. We shall use the following identity, Z Z Z Z u · (q n) (7.20) u · ∇q − u · (q n) − q div u = − Ωft

Ωft

Γf∞

Γst

192

Moving Shape Analysis

The coupled system (7.7) involves two essential non-homogeneous Dirichlet boundary conditions, u = u∞ , Σf∞ u = d˙ e2 , Σs

(7.21) (7.22)

0

We use a very weak formulation of the state equation, consisting in totally transposing the Laplacian operator, Z Z Z −ν∆u · φ = −ν∆φ · u + ν [u · ∂n φ − φ · ∂n u] dΓ (7.23) Ωft

Ωft

Γf∞ ∪Γst

We shall substitute inside this identity the desirable boundary conditions and we will recover the boundary constraints in performing an integration by parts inside the optimality conditions corresponding to the sensitivity with respect to the multipliers. This procedure has been already used in [135] and [45, 46] to perform shape optimization problems for elliptic equations using Min-Max principles. We shall also choose to transpose the time operator inside the weak formulation. This has to be performed very carefully, since we are dealing with a moving domain, Z TZ Z TZ Z TZ (u · v) hV, ni (7.24) u · ∂t v − ∂t u · v = − 0

Ωft

+

Z

0

ΩfT

Ωft

u(T ) · v(T ) −

∂Ωft

0

Z

Ωf0

u(0) · v(0)

(7.25)

REMARK 7.3 This kind of technique has been popularized in [99] as a systematic way to study non-homogeneous linear partial differential equations. These formulations are usually called very weak formulations or transposition formulations. We shall notice that these methods are still valid in the nonlinear case to obtain regularity or existence results. We refer to [4] for recent applications on the Navier-Stokes system.

7.4.3

Solid reduced order and solid weak state operator

For the sake of simplicity, we reduce the order of the solid governing equation by defining the global solid variable, d d1 ˜ d= = ˙ (7.26) d2 d leading to the first order ordinary differential equation, ˙ M d˜ + K d˜ = F

(7.27)

Sensitivity analysis for a simple fluid-solid interaction system with

193

1 0 M= 0m 0 −1 K= k 0

and

F =

0 Ff

In the case of the coupled fluid-solid system, the loads depend on the fluid state variable, ! Z Ff (u, p) = − σ(u, p) · n · e2 Γst

We introduce the state and multiplier spaces in order to define a proper solid weak state operator, X1s = d1 ∈ C 1 ([0, T ]) (7.28) s 1 X2 = d2 ∈ C ([0, T ]) (7.29) s 1 Y1 = b1 ∈ C ([0, T ]) (7.30) Y2s = b2 ∈ C 1 ([0, T ]) (7.31) and the load space,

L = F ∈ C 1 ([0, T ])

This allows us to define a solid state operator,

es : X1s × X2s × L −→ (Y1s × Y2s )∗ whose action is defined by the following identity, Z Th i hes (d1 , d2 , F ), (b1 , b2 )i = −d1 b˙1 − d2 b1 − d01 b1 (0) + d1 (T )b1 (T ) +

7.4.4

Z

T

0

h

0

Z i −m d2 b˙2 + k d1 b2 − m d02 b2 (0) + m d2 (T ) b2 (T ) −

T

F b2 0

Fluid state operator

In this section, we summarize the different options that we have chosen for the Lagrangian framework and define the variational state operator constraint. In order to deal with rigid displacement vector fields, we introduce the following spaces : Rigid displacement spaces: def H1 = φ ∈ (H 1 (Ω0 ))2 ; ∇φ = 0, in Ωs0 def H01 = φ ∈ (H 1 (Ω0 ))2 ; ∇φ = 0, in Ωs0 , φ = 0, on Γf∞

194

Moving Shape Analysis

LEMMA 7.2 For ϕ ∈ H1 , assuming that Ωs0 is connected, there exists C ∈ R2 such that ϕ|Ωs0 = C In the sequel, we will need to define precise state and multiplier spaces in order to endow our problem with a Lagrangian functional framework. Following the existence result stated previously, we introduce the fluid state space : o n def X f = u ∈ H 2 (0, T ; (H 2 (Ωft ))2 ∩ H1 ) def

Zf =

p ∈ H 1 (0, T ; (H 1 (D))2 )

We also need test function spaces that will be useful to define Lagrange multipliers : n o def Y f = v ∈ L2 (0, T ; (H 2 (Ωft ))2 ∩ H01 ) def V f = q ∈ H 1 (0, T ; (H 1 (D))2 ) def Wsf = (v, b2 ) ∈ Y f × Y2s , v|Γst = b2 · e2 We define the fluid weak state operator,

efu∞ : X f × Z f × U s −→ (Y f × V f )∗ whose action is defined by : hefu∞ (u, p, us ), (v, q)i = Z TZ [−u · ∂t v + (D u · u) · v − νu · ∆v + u · ∇ q − p div v] +

Z

0

−

7.4.5

Ωft

0

T

Z

Γf∞

Z

T

0

u∞ · (σ(v, q) · n) + Z

Γst

Z

v · (σ(u, p) · n) +

T

0

Z

Z

ΩT

Γst

[us · (σ(v, q) · n) − (u · v) hus , ni]

u(T ) · v(T ) −

Z

Ω0

u0 · v(t = 0)

∀ (v, q) ∈ Y f × V f

Coupled system operator

Our mechanical system consists of a solid part and a fluid part. These subsystems have been represented thanks to a solid and a fluid state operator. It is now possible to couple these two operators in order to build an ad-hoc

Sensitivity analysis for a simple fluid-solid interaction system

195

coupled system operator. The major point here is to notice that the fluid load F f appears in the fluid state operator and then can be coupled with the solid part thanks to the input load F of the solid operator. To achieve this coupling, we need to decide whether or not the fluid and the solid multipliers match at the fluidsolid interface. If not, we have to work with the fluid constraint tensor at the fluid-solid boundary, which may be not convenient due to a regularity requirement. Hence, we choose to work with continuous test functions on Γ st . This means that we shall choose the fluid and second solid multiplier spaces to be the space Wsf . We define the coupled system weak state operator as follows, eu∞ : Y f × Z f × X1s × X2s −→ (Wsf × V f × Y1s )∗ whose action is defined by the following identity, heu∞ (u, p, d1 , d2 ), (v, q, b1 , b2 )i =

+

Z

T 0

hes (d1 , d2 , Ff ), (b1 , b2 )i + hefu∞ (u, p, d2 e2 ), (v, q, d1 · e2 )i Z TZ = [−u · ∂t v + (D u · u) · v − νu · ∆v + u · ∇ q − p div v]

Z

0

Γf∞

Ωft

u∞ · (σ(v, q) · n) + +

Z

T 0

Z

T

0

Z

Γst

[(d2 e2 ) · (σ(v, q) · n) − (u · v) hd2 e2 , ni]

i i Z Th ˙ −m d2 b˙2 + k d1 b2 −d1 b1 − d2 b1 + 0 Z Z + u(T ) · v(T ) − u0 · v(t = 0)

h

ΩT

Ω0

− d01 b1 (0) + d1 (T ) b1 (T ) − md02 b2 (0) + m d2 (T ) b2 (T ),

∀ (v, b2 , q, b1 ) ∈ Wsf × V f × Y1s

7.4.6

Min-Max problem

In this section, we introduce the Lagrangian functional associated with problem (7.7) and problem (7.18) : def

Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) = Ju∞ (u, p, d1 , d2 )−heu∞ (u, p, d1 , d2 ), (v, q, b1 , b2 )i (7.32) with α Ju∞ (u, p, d1 , d2 ) = 2

"Z

T 0

|d1 −

d1g |2

+ |d2 −

d2g |2

#

+

γ ku∞ k2Uc 2

196

Moving Shape Analysis

Using this functional, the optimal control problem (7.18) can be put in the following form : min

u∞ ∈ Uc

min (u, p, d1 , d2 ) ∈

X f ×Z f ×X1s ×X2s

max (v, b2 , q, b1 ) ∈

Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 )

Wsf ×V f ×Y1s

(7.33) By using the Min-Max framework, we avoid the computation of the state derivative with respect to u∞ . First-order optimality conditions will furnish the gradient of the original cost function using the solution of an adjoint problem. We would like to apply min-max differentiability results to problem (7.33). The main issue is to prove that the min-max subproblem min

(u,p,d1 ,d2 )∈X f ×Z f ×X1s ×X2s

max

(v,b2 ,q,b1 )∈Wsf ×V f ×Y1s

Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) (7.34)

admits at least one saddle point for u∞ ∈ Uc . Reduced Gradient We assume that the conditions to apply the Min-Max principle [40] are fulfilled so we can bypass the derivation with respect to the control variable u∞ through the min-max subproblem (7.34). It leads to the following result : THEOREM 7.2 For u∞ ∈ Uc , and (uu∞ , pu∞ , d˜u∞ , ϕu∞ , πu∞ , ˜bu∞ ) the unique saddle point of problem (7.34), the gradient of the cost function j at point u∞ ∈ Uc is given by the following expression : ∇j(u∞ ) = (σ(ϕu∞ , πu∞ ) · n)|Σf∞ + γ u∞

(7.35)

PROOF Using theorem (3) from [45, 46], we bypass the derivation with respect to u∞ inside the min-max subproblem (7.34) : 0

< j (u∞ ), δu∞ >= D Lu (uu∞ , pu∞ , d1u∞ , d2u∞ ; φu∞ , πu∞ , b1u∞ , b2u∞ ), δu∞ i h Du∞ ∞ D = min max h Lu (u, p, d1 , d2 ; v, q, b1 , b2 ), δu∞ i Du∞ ∞ (u, p, d1 , d2 ) ∈ (v, b2 , q, b1 ) ∈ X f ×Z f ×X1s ×X2s

=

Z

Wsf ×V f ×Y1s T

0

Z

Γf∞

(ν ∂n φu∞ − πu∞ n) · δu∞ + γ

Z

T 0

Z

Γf∞

u∞ · δu∞

Sensitivity analysis for a simple fluid-solid interaction system

7.5

197

KKT Optimality Conditions

In this section, we are interested in establishing the first order optimality condition for problem (7.34), better known as Karusch-Kuhn-Tucker optimality conditions. This step is crucial, because it leads to the formulation of the adjoint problem satisfied by the Lagrange multipliers (ϕ u∞ , πu∞ , b1u∞ , b2u∞ ). We recall the expression of the Lagrangian, def

Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) = Ju∞ (u, p, d1 , d2 )−heu∞ (u, p, d1 , d2 ), (v, q, b1 , b2 )i (7.36) The KKT system will have the following structure : ∂(v,q,b1 ,b2 ) Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) · (δv, δq, δb1 , δb2 ) = 0,

∀ (δv, δb2 , δq, δb1 ) ∈ Wsf × V f × Y1s → State Equations ∂(u,p,d1 ,d2 ) Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) · (δu, δp, δd1 , δd2 ) = 0, ∀ (δu, δp, δd1 , δd2 ) ∈ X f × Z f × X1s × X2s →

7.5.1

Adjoint Equations

Derivatives with respect to state variables

Fluid adjoint system LEMMA 7.3 For u∞ ∈ Uc , (p, b1 , b2 , v, b2 , q, b1 ) ∈ Z f × X1s × X2s × Wsf × V f × Y1s , Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) is differentiable with respect to u ∈ Y f and we have h∂u Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ), δui = Z TZ [−δu · ∂t v + [D δu · u + D u · δu] · v − νδu · ∆v + δu · ∇ q] − 0

Ωft

+

Z

T

0

Z

Γst

(δu · v) hd2 e2 , ni −

Z

ΩT

δu(T ) · v(T )

∀δu ∈ X f

In order to obtain a strong formulation of the fluid adjoint problem, we perform some integration by parts : LEMMA 7.4 Z Z (D δu · u) · v = − Ωft

Ωft

[D v · u + div(u) v] · δu +

Z

Γf∞ ∪Γst

(δu · v) hu, ni

198

Moving Shape Analysis

It leads to the following identity : h∂uˆ Lu∞ (u, p, d1 , d2 ; ϕ, π, b1 , b2 ), δui = Z − [−∂t ϕ + (∗ Du) · ϕ − (Dϕ) · u − div(u) ϕ − ν∆ϕ + ∇π] · δu Qf Z − ϕ(T ) · δu(T ) ΩfT

LEMMA 7.5 For u∞ ∈ Uc , (u, b1 , b2 , v, b2 , q, b1 ) ∈ X f × X1s × X2s × Wsf × V f × Y1s , Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) is differentiable with respect to p ∈ Z f and we have h∂p Lu∞ (ˆ u, p, d1 , d2 ; ϕ, q, b1 , b2 ), δpi =

Z

T 0

Z

Ωft

δp div ϕ,

∀δp ∈ Z f

This leads to the following fluid adjoint strong formulation,  −∂t ϕ − D ϕ · u + (∗ Du) · ϕ − ν∆ϕ + ∇q = 0, Qf     div(ϕ) = 0, Qf  ϕ = b2 · e2 , Σs   ϕ = 0, Σf∞    ϕ(T ) = 0, ΩfT

7.5.2

(7.37)

Solid adjoint system

We recall that the ALE map is built as the flow of a vector field V that matches the solid velocity at the fluid-solid interface and is arbitrary inside the fluid domain, i.e., using the reduced oder model,  x ∈ Ωst  V (x, t) = d2 · e2 , V (x, t) = Ext(d2 · e2 ), x ∈ Ωft  V (x, t) · n = 0, x ∈ Γf∞

(7.38)

Hence, the ALE map depends on d2 through V . Furthermore, the derivative with respect to d1 may be simpler since it does not involve a derivative with respect to the geometry. Then, we have the following result, LEMMA 7.6 For u∞ ∈ Uc and (u, p, d1 , d2 , v, b2 , q, b1 ) ∈ X f ×Z f ×X1s ×X2s ×Wsf ×V f ×Y1s , the Lagrangian Lu∞ (u, p, d1 , d2 ; v, q, b1 , b2 ) is differentiable with respect to d1 ∈

Sensitivity analysis for a simple fluid-solid interaction system

199

X1s and we have h∂d1 Lu∞ (u, p, d1 , d2 ; ϕ, π, b1 , b2 ), δd1 i = Z Th i α (d1 − d1g ) δd1 − k δd1 b2 + δd1 b˙ 1 − δd1 (T ) b1 (T ) 0

From which we deduce the following adjoint ODE, −b˙1 + k b2 = α(d1 − d1g ), (0, T ) b1 (T ) = 0

(7.39)

The derivative of the Lagrangian with respect to the solid velocity variable d 2 involves shape derivatives of domain integrals. This point has been investigated previously by Zol´esio in [157], [156] and in [59], [57]. Then we need to introduce the concept of Transverse Field associated to a perturbation of the solid velocity. We introduce a perturbation flow W associated to the perturbation δd 2 e2 . For example, W = Ext(δd2 · e2 ) This flow generates new fluid and solid domains through the ALE map, T t (V + ρW ), with ρ ≥ 0. We set def

Ωf,ρ = Tt (V + ρW )(Ωf0 ) t def

Ωs,ρ = Tt (V + ρW )(Ωs0 ) t

For the sake of simplicity, we set def

Ttρ = Tt (V + ρW ) The objective of this paragraph is to compute the following derivative : d L(ˆ u, p, d1 , d2 + ρδd2 ; vˆ, q, b1 , b2 ) dρ ρ=0

Transverse map and vector field

Since we would like to differentiate the Lagrangian functional with respect to ρ at point ρ = 0, it is convenient to work with functions already defined in def Ωf,ρ=0 = Ωft as we proceed to the limit ρ → 0. t To this end, we introduce a new map as in [157] : def

Tρt = Ttρ ◦ Tt−1 : Ωft −→ Ωf,ρ t Ωst −→ Ωs,ρ t

(7.40)

200

Moving Shape Analysis

This map is actually the flow of a vector field following for ρ ∈ (0, ρ 0 ), THEOREM 7.3 The Transverse map Ttρ is the flow of a transverse field Zρt defined as follows: def

Zρt = Z t (ρ, .) =

∂Tρt ∂ρ

◦ (Tρt )−1

(7.41)

i.e., is the solution of the following dynamical system: Ttρ (Zρt ) : Ω −→ Ω

x 7−→ x(ρ, x) ≡ Ttρ (Zρt )(x)

with d x(ρ) = (ρ, x(ρ)), ρ ≥ 0 dρ x(ρ = 0) = x, in Ω

(7.42)

def

t Since, we only consider derivatives at point ρ = 0, we set Z t = Zρ=0 . We recall a result from [58] which might be useful for the sequel,

THEOREM 7.4 The mapping, [0, ρ0 ] −→ C 0 ([0, T ]; W k−1,∞ (Ω)) ρ 7−→ Tt (V + ρW ) is continuously differentiable and def

S t (ρ, .) = ∂ρ [Tt (V + ρW )] =

Z

t 0

[D(V + ρW ) ◦ Tτ (V + ρW ) · S τ (ρ, .)

+W ◦ Tτ (V + ρW )dτ ]

COROLLARY 7.1 def St (.) = S t (ρ = 0, .) is the unique solution of the following Cauchy problem,

∂t St − (D V ◦ Tt ) · St = W ◦ Tt , Ω0 × (0, T ) St=0 = 0, Ω0

(7.43)

A fundamental result furnishes the dynamical system satisfied by the vector field Zt related to the vector fields (V, W ),

Sensitivity analysis for a simple fluid-solid interaction system

201

THEOREM 7.5 [59] The vector field Zt is the unique solution of the following Cauchy problem, ∂t Zt + [Zt , V ] = W, Ω0 × (0, T ) (7.44) Zt=0 = 0, Ω0 def

where [Zt , V ] = DZt · V − DV · Zt stands for the Lie bracket of the pair (Zt , V ). Shape derivatives In the sequel, we will need general results concerning shape derivatives of integrals over domains or boundaries. We will use the framework developed in Sokolowski-Zol´esio [135]. LEMMA 7.7 d dρ

Z

! G(ρ) dΩ f,ρ Ωt

= ρ=0

Z

Ωft

LEMMA 7.8 Z ! d φ(ρ) dΓ dρ Γs,ρ t

∂ρ G(ρ) dΩ +

= ρ=0

Z

Γst

h

Z

Γst

G(ρ = 0)hZt , ni dΓ (7.45)

0

φΓ + Hφ hZt , ni

i

dΓ

(7.46)

0

where φΓ stands for the tangential shape derivative of φ(ρ, .) ∈ L1 (Γst ). We recall classical definitions of shape derivative functions : DEFINITION 7.1 For φ(ρ, x) ∈ C 0 (0, ρ0 ; C 1 (Γs,ρ t )), the material derivative is given by the following expression : d φ˙ = φ(ρ, .) ◦ Tρt dρ ρ=0

then the tangential shape derivative of φ is given by the following expression, 0 def φΓ = φ˙ − ∇Γ φ · Zt

REMARK 7.4 we have

0 with φ˜ |Γst

If φ is the trace of a vector field φ˜ defined over Ω, then 0

0 = φ˜ |Γst + ∂n φ˜ hZt , ni . ρ=0 s

φΓ def d ˜ .) = φ(ρ, dρ

def

Γt

202

Moving Shape Analysis

Following this remark, we have LEMMA 7.9 ! Z d ˜ x)da φ(ρ, dρ Γs,ρ t

= ρ=0

Z

Γst

h

h i i 0 φ˜ + H φ˜ + ∂n φ˜ hZt , ni dΓ

(7.47)

Derivation of the perturbed Lagrangian Thanks to the introduction of the transverse map, it is now possible to work with functions (u, v) that are defined on Ωft . This leads to the following perturbed Lagrangian : def

Lρu∞ = L(u, p, d1 , d2 + ρδd2 ; v, q, b1 , b2 ) = Ju∞ (u, p, d1 , d2 + ρδd2 ) Z TZ − −(u ◦ Rtρ ) · ∂t (v ◦ Rtρ ) + (D u ◦ Rtρ · u ◦ Rtρ ) · v ◦ Rtρ 0

−νu ◦

Rtρ

Ωf,ρ t

· ∆v ◦ +

Rtρ Z

+u◦

T 0

Z

Γs,ρ t

Rtρ

· ∇ q − p div v ◦

Rtρ

+

Z

T

0

Z

Γf∞

u∞ · (σ(v, q) · n)

((d2 + ρδd2 ) e2 ) · (σ(v ◦ Rtρ , q) · nρ )

− u ◦ Rtρ · v ◦ Rtρ · ((d2 + ρδd2 ) e2 ) · nρ Z Th i Z Th i ˙ + −d1 b1 − (d2 + ρδd2 ) b1 + −m (d2 + ρδd2 ) b˙2 + k d1 b2 0 Z Z 0 u0 · v(t = 0) − d01 b1 (0) + d1 (T ) b1 (T ) − m d02 b2 (0) u(T ) · v(T ) − + Ω0

ΩT

+ m (d2 + ρδd2 )(T ) b2 (T ),

∀ (v, b2 , q, b1 ) ∈ Wsf × V f × Y1s

def

with Rtρ = (Tρt )−1 . We apply the previous results to the perturbed Lagrangian functional L. a) Distributed terms: We set G(ρ, .) = −u ◦ Rtρ · ∂t (v ◦ Rtρ ) + D(u ◦ Rtρ ) · (u ◦ Rtρ ) · v ◦ Rtρ

−ν(u ◦ Rtρ ) · ∆(v ◦ Rtρ ) + (u ◦ Rtρ ) · ∇q − p div(v ◦ Rtρ )

def

with Rtρ = (Tρt )−1 . We need the following lemmas in order to derivate G(ρ, .) with respect to ρ,

Sensitivity analysis for a simple fluid-solid interaction system

203

LEMMA 7.10

dTρt = Zt dρ ρ=0

dRtρ = −Zt dρ ρ=0

LEMMA 7.11 d u ◦ Rtρ dρ

PROOF

!

ρ=0

= − D u · Zt

Using the chain rule we get

D Rtρ d u ◦ Rtρ = D u ◦ Rtρ · dρ D ρ ρ=0 ρ=0 = − D u ◦ Rtρ · Z t (ρ, .) ρ=0 = − D u · Zt

LEMMA 7.12 Then, we have the following result ∂ρ G(ρ, .)|ρ=0 = [(D u · Zt ) · ∂t v + u · (∂t (D v · Zt )) − [(D(D u · Zt )) · u + D u · (D u · Zt )] · v − (D u · u) · (D v · Zt )

+ν(D u · Zt ) · ∆v + νu · (∆(D v · Zt )) + p div(D v · Zt ) − (D u · Zt ) · ∇q] PROOF (7.11).

It comes easily using definition of G(ρ, .) and lemma (7.10)-

Then we have an expression of the derivative of distributed terms coming

204

Moving Shape Analysis from the Lagrangian with respect to ρ, ! Z Z d G(ρ)dΩ = [(D u · Zt ) · ∂t v + u · (∂t (D v · Zt )) dρ Ωf,ρ Ωft t ρ=0

− [(D(D u · Zt )) · u + D u · (D u · Zt )] · v − (D u · u) · (D v · Zt ) +ν(D u · Zt ) · ∆v + νu · (∆(D v · Zt )) + p div(D v · Zt ) − (D u · Zt ) · ∇q] Z [−u · ∂t v + (Du · u) · v − νu · ∆v + u · ∇q − p div(v)] hZt , ni + Γt

b) Boundary terms : We must now take into account the terms coming from the moving boundary Γs,ρ t . Then we set

φ(ρ, .) =

(d2 + ρδd2 ) e2 ) · (σ(v ◦ Rtρ , q) − u ◦ Rtρ · v ◦ Rtρ · ((d2 + ρδd2 ) e2 ) ·nρ

We set V = d2 e2 and W = δd2 e2 , then φ(ρ, .) = (V + ρW ) · σ(v ◦ Rtρ , q) − u ◦ Rtρ · v ◦ Rtρ · nρ

Since φ(ρ, .) is defined on the boundary Γs,ρ t , we need some extra identities corresponding to boundary shape derivates of terms involved in φ(ρ, .). LEMMA 7.13 [53] 0

∂ρ nρ |ρ=0 = nΓ = −∇Γ (Zt · n) LEMMA 7.14 d dρ

PROOF

Z

! hE(ρ), n idΓ s,ρ Γt ρ

= ρ=0

Z

Γst

First, we use that Z Z hE(ρ), nρ i = Γs,ρ t

hE 0 |Γt , ni + (div E)hZt , ni

Ωf,ρ t

div E(ρ)

then we derive this quantity using lemma (7.7), ! Z Z Z d div E(ρ) = div E 0 + (div E)hZt , ni dρ Ωf,ρ Ωft Γst t ρ=0

Sensitivity analysis for a simple fluid-solid interaction system Z Z 0 We conclude using div E = hE 0 , ni. Ωft

205

Γst

Then we use LEMMA 7.15 E 0 | Γt = W ·[−q I +ν D v − u · v]+V ·[−ν D(D v · Zt ) + (D u · Zt ) · v + u · (D v · Zt )] Hence, we have d dρ

+

Z Z

! φ(ρ) dΓ Γs,ρ t

Γst (V )

= ρ=0

Z

Γst (V )

W · [−q I +ν D v − u · v] · n

V · [−ν D(D v · Zt ) + (D u · Zt ) · v + u · (D v · Zt )] · n

+

REMARK 7.5 Z

Γt

Z

Γst (V

div(V · [−q I +ν ∗ D v − u · v])hZt , ni

)

We recall that Z V · (D v · n) = =

Z

Ωft

div(∗ D v · V )

Ωft

D v · · D V + V · ∆v

(7.48)

We use this expression in the sequel. We recall that the perturbed Lagrangian has the following form, LρV,W

=

ρ JV,W

−

Z

T 0

−

Z

Ωf,ρ t

Z

ΩT

G(ρ) −

Z

T 0

Z

u(T ) · v(T ) +

Γs,ρ

Zt

Ω0

φ(ρ) u0 · v(t = 0) ∀ (v, q) ∈ Y (Ωft ) × V f

206

Moving Shape Analysis

Hence, its derivative with respect to ρ at point ρ = 0 has the following expression, ! Z T Z d ρ d ρ d LV,W = JV,W − G(ρ) f,ρ dρ dρ dρ 0 Ωt ρ=0 ρ=0 ρ=0 ! Z T Z d − φ(ρ) , ∀ (v, q) ∈ Y (Ωft ) × V f s,ρ dρ 0 Γt ρ=0

Furthermore we have,

LEMMA 7.16

Z T d ρ J = α(d2 − d2g ) e2 · W dρ V,W ρ=0 0

(7.49)

Using the last identities concerning the derivative of the distributed and the boundary terms with respect to ρ, we get the following expression, d ρ = −AZt − BZt − CW (7.50) LV,W dρ ρ=0

with

Z

AZ t =

T 0

Z

Ωft (V )

[(D u · Zt ) · ∂t v − [(D(D u · Zt )) · u

+ D u · (D u · Zt )] · v + ν(D u · Zt ) · ∆v − (D u · Zt ) · ∇q] + u · (∂t (D v · Zt )) − (D u · u) · (D v · Zt ) + νu · (∆(D v · Zt ))

+ p div(D v · Zt )

B Zt = Z TZ 0

Γst (V )

[−u · ∂t v + (Du · u) · v − νu · ∆v + u · ∇q − p div(v)] (Zt · n)

+ V · [−ν D(D v · Zt ) + (D u · Zt ) · v + u · (D v · Zt )] · n + div(V · [−q I +ν ∗ D v − u · v])hZt , ni −

Z

T 0

Z

Γf∞

ν u∞ · D(D v · Zt ) · n

Sensitivity analysis for a simple fluid-solid interaction system

CW =

Z

T 0

Z

Γst (V )

207

h i W · −b1 e2 − m b˙2 e2 + σ(v, q) · n − (u · v) n − α(d2 − d2g ) e2

The shape derivative kernel identity We shall now assume that (u, p, ϕ, π) is a saddle point of the Lagrangian functional L. This will help us to simplify several terms involved in the derivative of L with respect to V . Indeed, we would like to express the distributed term A Zt as a boundary quantity defined on the fluid moving boundary Γst and the fixed boundary Γf∞ . THEOREM 7.6 For (u, p, ϕ, π) saddle points of the Lagrangian functional (7.32), the following identity holds, Z

T 0

Z

Ωft (V )

[(D u · Zt ) · ∂t ϕ − [(D(D u · Zt )) · u

+ D u · (D u · Zt )] · ϕ + ν(D u · Zt ) · ∆ϕ − (D u · Zt ) · ∇π] + u · (∂t (D ϕ · Zt )) − (D u · u) · (D ϕ · Zt ) + νu · (∆(D ϕ · Zt )) + p div(D ϕ · Zt ) −

+

Z

T 0

Z

Z

T 0

Γst (V )

Z

Γst (V )

V · [ν D(D ϕ · Zt ) − (D u · Zt ) · ϕ − u · (D ϕ · Zt )] · n

[ν (ϕ − b2 e2 ) · (D(D u · Zt ) · n) + (D ϕ · Zt ) · (−p n + ν(D u · n))]

−

Z

T 0

Z

Γf∞

ν u∞ · (D(D ϕ · Zt ) · n) = 0,

def

∀ W = δd2 e2

PROOF We use extremal conditions associated to variations with respect to (u, v) in the Lagrangian functional where we have added a boundary since we consider test functions v that do not vanish on Γ f∞ and do not match the

208

Moving Shape Analysis

solid test functions at the fluid-solid interface Γ st (V ) , i.e., L2 (u, p, d1 , d2 ; v, q, b1 , b2 ) = Ju∞ (u, p, d1 , d2 ) Z TZ − [−u · ∂t v + (D u · u) · v − νu · ∆v + u · ∇ q − p div v] Ωft (V )

0

−

Z

Z

T

0

Γf∞

u∞ · σ(v, q) · n − +

Z

−

Z

T 0

Z

Γst (V )

T

0

Z

Γst (V )

V · [σ(v, q) · n − (u · v) n]

(v − b2 e2 ) · (σ(u, p) · n)

i Z Th i −d1 b˙1 − d2 b1 − −md2 b˙2 + kd1 b2 0 0 Z u(T )v(T ) + u0 v(t = 0) + d01 b1 (0) − d1 (T )b1 (T )

− ΩT

T

h

Ω0

md02 b2 (0)

+

Z

Z

− md2 (T )b2 (T )∀ (v, q, b1 , b2 ) ∈ Y f × V f × Y1s × Y2s

This leads to the following perturbation identity, ∂(u,v) L2 · (δu, δv) = Z TZ [−δu · ∂t v − u · ∂t δv + D(δu · u) · v + D(u · δu) · v − Ωft

0

−

Z

+D(u · u) · δv − ν(δu · ∆v) − ν(u · ∆δv) + δu · ∇q − p div(δv)] Z TZ [ν (v − b2 e2 ) · (D δu · n) + δv · (−p n + ν(D u · n))] + T

0

Z

Γst (V )

0

Γst (V )

−

Z

V · [ν D(δv) − δu · v − u · δv] · n − [δu(T )v(T ) + u(T )δv(T )] ,

ΩT

Z

T

0

Z

Γf∞

ν u∞ · (D δv · n)

∀ (δu, δv) ∈ X(Ωft ) × Y (Ωft )

We choose specific perturbation directions, i.e., δu = D u · Zt

δv = D v · Zt

with δu(T ) = δv(T ) = δu(0) = δv(0) = 0, where (u, v) are saddle points of the Lagrangian, i.e., solutions of respectively the primal and adjoint fluid problem. We recognize immediately the distributed and boundary terms involved in the shape derivative kernel identity. Solid and ALE adjoint problem Using the shape derivative kernel identity, we simplify the Lagrangian derivative at the saddle point (u, p, d1 , d2 , ϕ, b1 , b2 ). We set (u, v) = (u, ϕ)

Sensitivity analysis for a simple fluid-solid interaction system and we get Z TZ AZ t = −

Z

Z

T

0

0

Γst (V )

Γst (V )

209

V · [ν D(D ϕ · Zt ) − (D u · Zt ) · ϕ − u · (D ϕ · Zt )] · n

[ν (ϕ − b2 e2 ) · (D(D u · Zt ) · n) + (D ϕ · Zt ) · (−p n + ν(D u · n))] +

Z

T 0

Z

Γf∞

ν u∞ · (D(D ϕ · Zt ) · n)

We use that ϕ = b2 e2 on Γst (V ) and the following identities, (D ϕ · Zt ) · (p n) = (p div ϕ)hZt , ni (D ϕ · Zt ) · (D u · n) = (D ϕ · n) · (D u · n)hZt , ni

(7.51) (7.52)

then, AZ t =

Z

T 0

Z

−

Γst (V )

Z

T 0

V · [ν D(D ϕ · Zt ) − (D u · Zt ) · ϕ − u · (D ϕ · Zt )] · n

Z

Γst (V )

[−p div ϕ + ν (D ϕ · n) · (D u · n)] hZt , ni +

Z

T 0

Z

Γf∞

ν u∞ · (D(D ϕ · Zt ) · n)

Using the following identity, div(V · [−π I +ν ∗ D φ − u · ϕ]) = −π div V − V · ∇ π + ν D ϕ · · D V + ν V ∆ϕ − (div V ) u · ϕ − V · ∇(u · ϕ) (7.53) we get B Zt = Z TZ 0

Γst (V )

[−u · ∂t ϕ + (Du · u) · ϕ − νu · ∆ϕ + u · ∇π − p div(ϕ)] hZt , ni

+ V · [−ν D(D ϕ · Zt ) + (D u · Zt ) · ϕ + u · (D ϕ · Zt )] · n + − π div V − V · ∇ π + ν D ϕ · · D V + ν V ∆ϕ Z TZ − (div V ) u · ϕ − V · ∇(u · ϕ) hZt , ni − ν u∞ · D(D ϕ · Zt ) · n 0

Γf∞

then we get,

AZ t + B Z t =

Z

T 0

Z

Γst (V )

[ν (D ϕ · n) · (D(V − u) · n) − u · ∂t ϕ + (Du · u) · ϕ −π div V − (div V ) u · ϕ − V · ∇(u · ϕ)] hZt , ni

210

Moving Shape Analysis

We use the following identity, V · ∇(u · ϕ) = ϕ · D u · V + V · D ϕ · u and the boundary conditions, u = d2 e2 on Γst , ϕ = b2 e2 on Γst . This leads to AZ t + B Z t =

Z

T 0

Z

Γst (V )

ν (D ϕ · n) · (D(V − u) · n) − d2 b˙2 − π div V

− (div V ) u · ϕ − d2 e2 · D ϕ · d2 e2 hZt , ni

We choose the velocity field V = Ext(d2 e2 ) ◦ p, then (D V · n) · n|Γst = 0 and

div V |Γst = divΓ V + (D V · n) · n =0 Finally, we have AZ t + B Z t = Z TZ h 0

Γst (V )

i −d2 b˙2 + ν (D ϕ · n) · (D u · n) − |d2 |2 (D ϕ · e2 ) · e2 hZt , ni

and CW =

Z

T 0

Z

Γst (V )

i h W · −b1 e2 − m b˙2 e2 + σ(φ, π) · n − α(d2 − d2g ) e2

where we have used that Z

Γst (V )

d2 e2 · b 2 e2 · n = 0

We introduce the adjoint field λ solution of the following system, −∂t λ − ∇Γ λ · V − λ divΓ V = f, (0, T ) λ(T ) = 0, ΓT (V )

(7.54)

with h i f = −d2 b˙2 + ν (D ϕ · n) · (D u · n) − |d2 |2 (D ϕ · e2 ) · e2 REMARK 7.6

In our case divΓ V = 0.

(7.55)

Sensitivity analysis for a simple fluid-solid interaction system

211

We recall the following property, LEMMA 7.17 [59],[57] For any E(V ) ∈ L2 (Σs (V )) and (V, W ) ∈ U, the following identity holds, Z

T 0

Z

Γst (V )

E(V ) hZt , ni = −

Z

T 0

Z

λhW, ni

(7.56)

Γst (V )

where λ ∈ C 0 ([0, T ]; H 1 (Γst )) is the unique solution of problem (7.54) with f = E. Then we have AZ t + B Z t =

Z

T 0

=−

Z

Z

Γst (V )

T

0

Z

f hZt , ni

Γst (V )

hλ n, W i

However, using the optimality condition for the Lagrangian functional, we obtain −(AZt + BZt ) = CW . This leads to Z

T 0

Z

Γst (V )

dis

Z

T

0

λn · W = Z

Γst (V )

h

i −b1 e2 − m b˙2 e2 + σ(φ, π) · n − α(d2 − d2g ) e2 · W, def

∀ W = δd2 e2 from which we deduce the following identity, Z

Γst (V )

Z h i λ n = −b1 − m b˙2 − α(d2 − d2g ) e2 +

Γst (V )

σ(φ, π) · n

(7.57)

We now use the following lemma, LEMMA 7.18 ∂t

Z

λn Γst (V )

!

=

Z

Γst (V )

[∂t λ + ∇Γ λ · V ] n

(7.58)

212

Moving Shape Analysis

We get ∂t

h

−b1 − m b˙2 − α(d2 − Z

Γst (V )

h

d2g )

i

e2 +

Z

Γst (V )

σ(φ, π) · n

!

=

i −d2 b˙2 + ν (D ϕ · n) · (D u · n) − |d2 |2 (D ϕ · e2 ) · e2 n

This can be written as b˙ 1 + m ¨b2 = −α(d˙2 − d˙2g ) + ∂t Z

Γst (V

)

Z

Γst (V )

σ(φ, π) · n

!

· e2 +

|d2 |2 (D ϕ · e2 ) · e2 − ν (D ϕ · n) · (D u · n) · n

and we recall the other solid adjoint equation, −b˙1 + k b2 = α(d − d1g ), (0, T ) b1 (T ) = 0

(7.59)

Let us inject b˙1 inside the first one, m ¨b2 + k b2 = α(d − d1g ) − α(d˙2 − d˙2g ) + ∂t Z

Γst (V )

Z

Γst (V )

σ(φ, π) · n

!

· e2 +

|d2 |2 (D ϕ · e2 ) · e2 − ν (D ϕ · n) · (D u · n) · n

This concludes the proof of the main result.

7.6

Conclusion

In this chapter, we have investigated sensitivity analysis for a simple 2D coupled fluid-structure system. This analysis was performed using a Lagrangian functional and non-cylindrical shape derivative tools to handle perturbation with respect to the velocity of the solid. This led to the derivation of first-order optimality conditions for an optimal control problem related to a tracking functional. This optimality system can be numerically approximated and included inside a gradient based optimization procedure. This point is under investigation following the strategy adopted for Navier-Stokes optimal control problems as in [69].

Sensitivity analysis for a simple fluid-solid interaction system

213

The methodology used in this chapter can be generalized to more complex fluid-structure interaction problems. We can either change the fluid model and handle compressibility as in [37], or change the solid equations and use a real 3D non-linear elastic model as described in the next chapter.

Chapter 8 Sensitivity analysis for a general fluid-structure interaction system

8.1

Introduction

In the previous chapter, we have performed a sensitivity analysis for a coupled fluid-solid model, where the solid was driven by a basic second order differential equation. This has allowed us to simply reduce the differential order by expanding the solid state and use its velocity as an explicit state variable. Then we have been able to use the framework developed in Chapter 5 concerning the derivative of the Navier-Stokes system with respect to the velocity of the moving domain. Here, the situation is quite different since we are considering a more general solid model with a non-linear constitutive law. In this case, the parametrization of the solid by its velocity is more challenging and the straightforward use of the results obtained in Chapter 7 is not obvious. As a consequence, we choose to work with the solid displacement state variable. Then, we need to use the results obtained in Chapter 6 concerning the derivative of the Navier-Stokes using the non-cylindrical identity perturbations. The chapter is organized as follows: first of all, we introduce the mechanical problem and its mathematical description. We use the classical arbitrary Euler-Lagrange (ALE) formulation, particularly suited for problems involving moving boundaries. Then, in the second part, we state the main result of this chapter, namely the cost function gradient computation involving the solution of a linear adjoint problem. Finally, its proof is fully developed.

8.2

Mechanical problem and main result

In this section we introduce a fluid-structure interaction model. We consider a solid located at time t ≥ 0 in a domain Ωs (t) ⊂ R3 with boundary Γs (t). It is surrounded by a fluid in R3 . We introduce a control volume Ω ⊂ R3 containing 215

216

Moving Shape Analysis

the solid at each time t ≥ 0. The notation ∂Ω stands for the boundary of Ω. s Hence, the fluid evolution is restricted to the domain Ω f (t) = Ω − Ω (t). In f the sequel we set Γ = ∂Ω and ∂Ωf (t) = Γf ∪ Γs (t) stands for the fluid domain boundary; see figure 8.1. We assume the fluid

ˆ t) θ(·,

Γf Ωs0

Γf Ωf (t) Ωs (t)

Γs0

Γs (t)

Ωf0 Ω

Ω FIGURE 8.1:

Geometric description

to be Newtonian viscous, homogeneous and incompressible. Its behavior is described by its velocity and pressure. The elastic solid under large displacements is described by its velocity and its stress tensor. The classical conservation laws of the continuum mechanics drive the evolution of these unknowns. The fluid state satisfies the following incompressible Navier-Stokes equations written in Eulerian conservative formulation: ∂t u + D u · u − ν∆u + ∇ p = 0, in Ωf (t),

(8.1)

div u = 0, in Ωf (t) where (u, p) stand, respectively, for the fluid velocity and pressure. In addition, the fluid stress tensor is given by σ(u, p) = −p I +ν [D u + ∗ D u] with ν the kinematic viscosity of the fluid. In a fluid-structure interaction framework the evolution of the fluid domain Ωf (t) is induced by the structural deformation through the fluid-structure ins terface Γs (t). Indeed, by definition Ωf (t) = Ω − Ω (t). It leads us to describe Ωf (t) according to a map acting in a fixed reference domain. This approach is usually used for the solid domain Ωs (t), by means of the Lagrangian formulation [94]. Given a material reference configuration for the solid Ω s0 ⊂ Ω with boundary

Sensitivity analysis for a general fluid-structure interaction system

217

s

Γs0 , we take Ωf0 = Ω − Ω0 as the reference fluid configuration. Then, the s control volume Ω = Ωf (t) ∪ Ω (t) is described by a smooth and injective map: θˆ : Ω × R+ −→ Ω ˆ ˆ (x0 , t) 7−→ θ = θ(x0 , t) We set θˆf = θˆ|Ωf and θˆs = θˆ|Ωs0 , such that [95]: 0

• for x0 ∈ Ωs0 , θˆs (x0 , t) represents the position at time t ≥ 0 of the material point x0 . This corresponds to the classical Lagrangian flow, s • the map θˆf is defined from θˆ|Γ s , as an arbitrary extension over domain 0

f

Ω0 , which preserves Γf0 = Γf = ∂Ω.

In short, the ALE map θˆ is given by f

ˆ 0 , t) = Ext(θˆs s )(x0 , t), θ(x |Γ0

ˆ 0 , t) = θˆs (x0 , t), θ(x

∀x0 ∈ Ω0 , s

∀x0 ∈ Ω0 . f

Here, Ext represents an extension operator from Γs0 to Ω0 such that s Ext(θˆ|Γ s) f = I f Γ 0 |Γ 0

0

s

REMARK 8.1 We set Ω0 = Ωf0 ∪ Ω0 as the reference domain, and s Ω = Ωf (t) ∪ Ω (t) stands for the actual configuration at time t > 0. REMARK 8.2 We set Σθˆ ≡ We set

[

The operator Ext is arbitrary defined inside Ω f0 .

0
({t} × Γsθˆ), Q ≡

[

0
({t} × Ωfθˆ ) and Σf∞ ≡ Γf∞ × (0, T )

ˆ = det(D θ) ˆ >0 J(θ) ˆ keeping in mind that it correIn the sequel, we shall only use the notation θ, s sponds to the solid displacement in Ω0 . ˆ The solid evolution is given by its motion θˆ and the stress tensor field S(θ) ˆ ˆ (second Piola-Kirchoff tensor [36]). The field S( θ) is related to θ through an ˆ S(θ)) ˆ satisfies the non-linear appropriate constitutive law. Then, the pair ( θ, elastodynamic equations [36]: ˆ = f, ∂tt θˆ − div0 [D θˆ · S(θ)]

in

Ωs0

(8.2)

218

Moving Shape Analysis

REMARK 8.3 The constitutive law for a St. Venant-Kirchhoff material is given by the following expression, ˆ = λ (Tr E(θ)) ˆ I +2 µ E(θ) ˆ S(θ)

ˆ is the strain tensor, where λ, µ > 0 stands for the Lam´e coefficients and E( θ)

ˆ = E(θ)

i 1 h∗ ˆ D θ · D θˆ − I 2

(8.3)

This kind of constitutive law is often referred to as “ large displacement small strain ” model. We shall also assume that there exists two operators T A defined on Ωs0 and ˆ def ˆ · δ θˆ in the TΓ defined on Γs0 such that the linear tangent tensor T (δ θ) = S 0 (θ) direction δ θˆ satisfies the following adjoint identity, Z

Ωs0

ˆ · ·B + T (δ θ)

Z

Ωs0

δ θˆ · TA (B) =

Z

Γs0

TΓ (B) · δ θˆ

(8.4)

for any smooth second order tensor B. The coupling between the solid and the fluid is realized through standard boundary conditions at the fluid-structure interface Γ s0 , namely, the kinematic continuity of the velocity and the kinetic continuity of the stress [95]:

def u = Vθˆ = ∂t θˆ ◦ θˆ−1 , on Γsθˆ,

(8.5)

ˆ · n0 = J(θ)σ(u, ˆ ˆ −1 · n0 , on Γs D θˆ · S(θ) p)(∗ D θ) 0 where n0 stands for the unit normal vector on Γs0 pointing inside Ωs0 . Moreover, we endow the fluid equations with Dirichlet boundary condition on the farfield boundary Γf∞ : u = u∞ ,

on Γf∞

Sensitivity analysis for a general fluid-structure interaction system

219

In summary, the strong coupled problem, with a ALE formulation for the fluid, is given by:                          

∂t u + D u · u − ν∆u + ∇ p = 0, in Ωfθˆ , div u = 0, in Ωfθˆ , u = u∞ , on Γf∞ ,

u = ∂t θˆ ◦ θˆ−1 , on Γsθˆ,      ˆ · n0 = J(θ)σ(u, ˆ ˆ −1 · n0 , on Γs ,  D θˆ · S(θ) p)(∗ D θ)  0         ˆ = f, in Ωs , ∂tt θˆ − div0 [D θˆ · S(θ)]  0         ˆ ∂t θˆ  u, θ, = (u0 , θˆ0 , θˆ1 ), in Ωf0 × (Ωs0 )2

(8.6)

|t=0

Here, f represents the applied body force, u0 the initial fluid velocity and θˆ0 , θˆ1 the initial displacement and solid velocity, respectively. The principal result of this chapter reads, Main Result: For smooth u∞ ∈ Uc , the following cost functional, ˆ ∞ )) j(u∞ ) = Ju∞ (u(u∞ ), p(u∞ ), θ(u

(8.7)

ˆ ∞ )) is a weak solution of problem (8.6) and where with (u(u∞ ), p(u∞ ), θ(u ˆ J(u, p, θ) is a real functional of the following form ˆ = α Ju∞ (u, p, θ) 2

Z

T 0

Z

Ωs0

|θˆ − θˆd |2

(8.8)

admits a gradient given by the following expression, ∇j(u∞ ) = σ(ϕ, π) · n|(0,T )×Γf∞

(8.9)

with (ϕ, π) being a solution of the following backward fluid adjoint system,  ∗ f   −∂t ϕ − D ϕ · u + D u · ϕ − ν∆ϕ + ∇π = 0, Qf   div ϕ = 0, Q  ϕ = ψˆ ◦ θˆ−1 , Σs   ϕ = 0, Σf∞    ϕ(T ) = 0, ΩfT

(8.10)

220

Moving Shape Analysis

and ψˆ being a solution of the following backward solid adjoint system,  ˆ − D2 ψˆ · ·S(θ) ˆ − TA (∗ D θˆ · D ψ) ˆ = ∂tt ψˆ − D ψˆ · div S(θ)       α (θˆ − θˆd ), (0, T ) × Ωs0       ˆ ˆ |t=T = (0, 0), Ωs0 (ψ, ∂t ψ) (8.11) together with the following condition, h i ˆ ·n ˆ ◦ θˆ−1 [J(θ)| ˆ ∗ (D θ) ˆ −1 · n D ψˆ · ∗ S(θ) ˆ + TΓ (∗ D θˆ · D ψ) ˆ |]−1 ◦ θˆ−1 =

− Vθˆ · ∂t ϕ − p div ϕ − π div Vθˆ + ν D ϕ · · D Vθˆ − (div Vθˆ) Vθˆ · ϕ

− (D ϕ · Vθˆ) · ϕ · n − ∗ D ϕ · σ(u, p) · n + ∗ D Vθˆ · σ(ϕ, π) − Vθˆ · ϕ · n

+ −(∂t E) · n + E · ∇Γ (Vθˆ · n) − (divΓ Vθˆ) E · n − D E · Vθˆ · n − E · DΓ n · Vθˆ on Γθˆ (8.12) with E = σ(ϕ, π) − (Vθˆ · ϕ) I.

8.2.1

Solid weak state operator

We define the solid state operator, ˆ s −→ (Yˆ s )∗ es : X whose action is defined by the following identity, ˆ vˆs i = hes (θ), Z −

Z

T 0

T 0

Z

8.2.2

Ωs0

Z

Ωs0

ˆ · · D vˆs ] + [θˆ · ∂tt vˆs + D θˆ · S(θ)

f ·ˆ vs+

Z

Ωs0

Z

T 0

Z

ˆ )·ˆ ˆ )·∂t vˆs (T )]+ v s (T )−θ(T [∂t θ(T

Γs0

Z

ˆ · n0 ] · vˆs [D θˆ · S(θ)

Ωs0

[−θˆ1 ·ˆ v s (0)+θˆ0 ·∂t vˆs (0)]

Fluid state operator

We define the fluid state spaces, n o def X f = u ∈ H 2 (0, T ; (H 2 (Ωft ))

Sensitivity analysis for a general fluid-structure interaction system def

Zf =

p ∈ H 1 (0, T ; H 1 (D))

221

We also need test function spaces that will be useful to define Lagrange multipliers : def

Yf =

n

o

q ∈ H 1 (0, T ; (H 1 (D))2 ) n o def Wsf = (v f , vˆs ) ∈ Y f × Yˆ s , vˆf = vˆs , on Γs0 def

Vf =

v f ∈ L2 (0, T ; H 2 (Ωft ))

We define the fluid weak state operator, efu∞ : X f × Z f × U s × X s −→ (Y f × V f )∗ whose action is defined by : ˆ (v f , q)i = hefu∞ (u, p, us , θ), Z TZ −u · ∂t v f + (D u · u) · v f − νu · ∆v f + u · ∇ q − p div v f +

Z

0

T

0

Z

ˆ Ωft (θ)

f

Γf∞

−

Z

u∞ · (σ(v , q) · n) + T

0

Z

f

Γsˆ θ

Z

T

0

v · (σ(u, p) · n) +

Z

Z

Γsˆ θ

us · (σ(v f , q) · n) − (u · v f ) hus , ni f

ΩT

u(T ) · v (T ) −

∀ (v f , q) ∈ Y f × V f ,

8.2.3

Z

def

Ω0

u0 · v f (0)

with v f = vˆf ◦ θ,

vˆf ∈ Yˆ f

Coupled system operator

Our mechanical system consists of a solid part and a fluid part. These subsystems have been represented thanks to a solid and a fluid state operator. It is now possible to couple these two operators in order to build an ad-hoc coupled system operator. The major point here is to notice that the kinetic continuity of the stress holds on the fluid-structure interface. To achieve the coupling, we need to decide whether or not the fluid and the solid multipliers match at the fluid-solid interface. If not, we have to work with the fluid constraint tensor at the fluidsolid boundary, which may be not convenient due to regularity requirement. Hence, we choose to work with continuous test functions on Γ sθˆ. This means that we shall choose the fluid and solid multipliers inside the space W sf . We define the coupled system weak state operator as follows, ˆ s −→ (W f × V f )∗ eu∞ : Y f × Z f × X s

222

Moving Shape Analysis

whose action is defined by the following identity, ˆ (v f , q, vˆs )i = hes (θ), ˆ vˆs )i + hef (u, p, ∂t θˆ ◦ θˆ−1 , θ), ˆ (v f , q)i heu∞ (u, p, θ), u∞ =

Z

T 0

+

+

Z

T 0

Z

Z Z

+

T 0

Z

Γsˆ θ

−u · ∂t v f + (D u · u) · v f − νu · ∆v f + u · ∇ q − p div v f

h

(∂t θˆ ◦ θˆ−1 ) · (σ(v f , q) · n) − (u · v f ) h(∂t θˆ ◦ θˆ−1 ), ni

u∞ ·(σ(v f , q)·n)+

Γf∞

Z

ˆ Ωft (θ)

Ωs0

Z

T

0

Z

Ωs0

ˆ tt vˆs +D θ·S( ˆ θ)·· ˆ D vˆs ]− [θ·∂

Z

T 0

i

Z

Ωs0

f ·ˆ vs

ˆ ) · vˆs (T ) − θ(T ˆ ) · ∂t vˆs (T ) − θˆ1 · vˆs (0) + θˆ0 · ∂t vˆs (0)] [∂t θ(T

+

Z

ΩT

u(T ) · v f (T ) −

Z

Ω0

u0 · v f (0) ∀ (v f , vˆs , q) ∈ Wsf × V f

8.2.4

Min-Max problem

In this section, we introduce the Lagrangian functional associated with problem (8.6) and problem (7.18) : ˆ (v f , q, vˆs )i ˆ − heu (u, p, θ), ˆ v f , q, vˆs ) def = Ju∞ (u, p, θ) Lu∞ (u, p, θ; ∞ with ˆ = Ju∞ (u, p, θ)

α 2

Z

T 0

Z

Ωs0

(8.13)

|θˆ − θˆd |2

Using this functional, the cost function can be put in the following form : j(u∞ ) =

min

f ×Z f ×X s ˆ (u,p,θ)∈X

max

(v f ,ˆ v s ,q)∈Wsf ×V f

ˆ v f , q, vˆs ) (8.14) Lu∞ (u, p, θ;

Reduced Gradient We assume that the conditions to apply the Min-Max principle [40] are fulfilled so we can bypass the derivation with respect to the control variable u∞ through the min-max subproblem (8.14). It leads to the following result :

Sensitivity analysis for a general fluid-structure interaction system

223

THEOREM 8.1 ˆ ϕ, π, ψ) ˆ the unique saddle point of problem (8.14), For u∞ ∈ Uc , and (u, p, θ, the gradient of the cost function j at point u∞ ∈ Uc is given by the following expression : ∇j(u∞ ) = σ(ϕ, π) · n|(0,T )×Γf∞ + γ [∗ K K] · u∞

(8.15)

PROOF Using theorem (3) from [45, 46], we bypass the derivation with respect to u∞ inside the min-max problem (8.14).

8.3

KKT optimality conditions

In this section, we are interested in establishing the first order optimality condition for problem (8.14), better known as Karusch-Kuhn-Tucker optimality conditions. This step is crucial, because it leads to the formulation of the ˆ We recall the adjoint problem satisfied by the Lagrange multipliers (ϕ, π, ψ). expression of the Lagrangian, ˆ (v f , q, vˆs )i ˆ v f , q, vˆs ) def ˆ − heu (u, p, θ), Lu∞ (u, p, θ; = Ju∞ (u, p, θ) ∞

(8.16)

The KKT system will have the following structure : ˆ v f , q, vˆs ) · (δv f , δq, δˆ ∂(vf ,q,ˆvs ) Lu∞ (u, p, θ; v s ) = 0,

∀ (δv f , δˆ v s , δq) ∈ Wsf × V f → State Equations ˆ f ˆ = 0, ˆs ) · (δu, δp, δ θ) ∂(u,p,θ) ˆ Lu∞ (u, p, θ; v , q, v ˆ ∈ Xf × Zf × Xs → ∀ (δu, δp, δ θ)

8.3.1

Adjoint Equations

Fluid adjoint system

LEMMA 8.1 ˆ v f , q, vˆs ), δui = h∂u Lu∞ (u, p, θ; Z TZ −δu · ∂t v f + [D δu · u + D u · δu] · v f − νδu · ∆v f + δu · ∇ q − 0

Ωfˆ θ

+

Z

T

0

Z

Γsˆ θ

(δu · v f ) h∂t θˆ ◦ θˆ−1 , ni −

Z

ΩT

δu(T ) · v f (T )

∀δu ∈ X f

224

Moving Shape Analysis

In order to obtain a strong formulation of the fluid adjoint problem, we perform some integration by parts : LEMMA 8.2 Z Z (D δu · u) · v f = − Ωfˆ θ

Ωfˆ θ

D v f · u + div(u) v f · δu +

Z

Γf∞ ∪Γsˆ θ

(δu · v f ) hu, ni

It leads to the following identity : ˆ ϕ, π, ψ), ˆ δui = h∂uˆ Lu∞ (u, p, θ; Z − [−∂t ϕ + ∗ D u · ϕ − D ϕ · u − (div u) ϕ − ν∆ϕ + ∇π] · δu Qf Z − ϕ(T ) · δu(T ) ΩfT

LEMMA 8.3 ˆ ϕ, π, ψ), ˆ δpi = h∂p Lu∞ (u, p, θ;

Z

T 0

Z

Ωfˆ

δp div ϕ,

θ

∀δp ∈ Z f

This leads to the following fluid adjoint strong formulation,  −∂t ϕ − D ϕ · u + ∗ D u · ϕ − ν∆ϕ + ∇π = 0, Qf     Qf  div ϕ = 0, ϕ = ψˆ ◦ θˆ−1 , Σs   ϕ = 0, Σf∞    ϕ(T ) = 0, ΩfT

8.3.2

(8.17)

Solid adjoint system

Now we shall perform the differentiation of the Lagrangian with respect to ˆ We introduce a perturbation map δ θˆ together with a scalar ρ ≥ 0 and the θ. following perturbed sets, h i h i def ˆ ˆ (Ωf ) = (I +ρδθ) ◦ θˆ (Ωf ) Ωfθ+ρδ = θ + ρδ θ 0 0 ˆ θˆ h i h i def Γsθ+ρδ = θˆ + ρδ θˆ (Γs0 ) = (I +ρδθ) ◦ θˆ (Γs0 ) ˆ θˆ def with δθ = δ θˆ ◦ θˆ−1 .

Shape derivative tools We shall need the following shape derivative formulas,

Sensitivity analysis for a general fluid-structure interaction system

225

LEMMA 8.4 d dρ

Z

Ωfˆ

ˆ θ+ρδ θ

! G(ρ) dΩ

LEMMA 8.5 Z d dρ Γsˆ

ˆ θ+ρδ θ

0

= ρ=0

Z

Ωfˆ

∂ρ G(ρ)|ρ=0 dΩ +

θ

Z

G(ρ = 0)hδθ, ni dΓ Γsˆ θ

(8.18)

! φ(ρ) dΓ

= ρ=0

Z

Γsˆ θ

h

0

φΓ + Hφ hδθ, ni

i

dΓ

(8.19)

where φΓ stands for the tangential shape derivative of φ(ρ, .) ∈ L1 (Γsθˆ) We recall classical definitions of shape derivative functions : DEFINITION 8.1 For φ(ρ, x) ∈ C 0 ((0, ρ0 ; C 1 (Γsθ+ρδ )), the material ˆ θˆ derivative is given by the following expression : d φ˙ = [φ(ρ, .) ◦ (I +ρδθ)] dρ ρ=0

then the tangential shape derivative of φ is given by the following expression, 0 def φΓ = φ˙ − ∇Γ φ · δθ

REMARK 8.4 we have

0 with φ˜ |Γsˆ θ

If φ is the trace of a vector field φ˜ defined over Ω, then 0 = φ˜ |Γsˆ + ∂n φ˜ hδθ, ni θ . ρ=0 s 0

φΓ def d ˜ = φ(ρ, .) dρ

def

Γˆ θ

Following this remark, we have LEMMA ! Z 8.6 d ˜ φ(ρ, x)da dρ Γsˆ ˆ θ+ρδ θ

= ρ=0

Z

Γsˆ θ

h

h i i 0 φ˜ + H φ˜ + ∂n φ˜ hδθ, ni dΓ

(8.20)

Derivation of the perturbed Lagrangian f The map I +ρδθ sends respectively the sets (Ωfθˆ , Γsθˆ) into (Ωθ+ρδ ). , Γsθ+ρδ ˆ ˆ θˆ θˆ def

Then defining its inverse Rtρ = (I +ρδθ)−1 we can perturb the integral support inside the Lagrangian and keep functions defined on Ω fθˆ and Γsθˆ. This

226

Moving Shape Analysis

leads to the following perturbed Lagrangian, def ˆ v f , q, vˆs ) = Ju (u, p, θˆ + ρδ θ) ˆ Lρu∞ = L(u, p, θ; ∞

−

Z

T 0

Z

Ωfˆ ˆ θ+ρδ θ

− (u ◦ Rtρ ) · ∂t (v f ◦ Rtρ ) + [D(u ◦ Rtρ ) · u ◦ Rtρ ] · (v f ◦ Rtρ )

− ν(u ◦ Rtρ ) · ∆(v f ◦ Rtρ ) + (u ◦ Rtρ ) · ∇ q − p div(v f ◦ Rtρ ) −

Z

Z

T 0

Γsˆ

ˆ θ+ρδ θ

ˆ ◦ (θˆ + ρ δ θ) ˆ −1 ] · (σ(v f ◦ Rt , q) · nρ ) [∂t (θˆ + ρδ θ) ρ

ˆ ˆ θ+ρδ ˆ ˆ −1 , nρ i − −(u◦Rtρ )·(v f ◦Rtρ ) h∂t (θ+ρδ θ)◦( θ) −

Z

Z

T 0

Ωs0

−

Z

T 0

Z

Γf∞

u∞ ·(σ(v f , q)·n)

ˆ · ∂tt vˆs + D(θˆ + ρδ θ) ˆ · S(θˆ + ρδ θ) ˆ · · D vˆs (θˆ + ρδ θ)

Z

Ωs0

ˆ ) · vˆs (T ) − (θˆ + ρδ θ)(T ˆ ) · ∂t vˆs (T ) ∂t (θˆ + ρδ θ)(T

− θˆ1 · vˆs (0) + θˆ0 · ∂t vˆs (0) − +

Z

Ω0

u0 · v f (0) +

Z

T 0

Z

Ωf,ρ T

Z

Ωs0

(u ◦ Rtρ )(T ) · (v f ◦ Rtρ )(T )

f · vˆs ,

∀ (v f , vˆs , q) ∈ Wsf × V f

Fluid distributed terms We set G(ρ, .) = − (u ◦ Rtρ ) · ∂t (v f ◦ Rtρ ) + [D(u ◦ Rtρ ) · u ◦ Rtρ ] · (v f ◦ Rtρ ) − ν(u ◦ Rtρ ) · ∆(v f ◦ Rtρ ) + (u ◦ Rtρ ) · ∇ q − p div(v f ◦ Rtρ ) LEMMA 8.7

d Rtρ = −δθ dρ ρ=0

Sensitivity analysis for a general fluid-structure interaction system

227

Then we have the following derivative, ∂ρ G(ρ, .)|ρ=0 = (D u · δθ) · ∂t v f + u · ∂t (D v f · δθ)

− [(D(D u · δθ)) · u + D u · (D u · δθ)] · v f − (D u · u) · (D v f · δθ)

+ν(D u · δθ) · ∆v f + νu · ∆(D v f · δθ) + p div(D v f · δθ) − (D u · δθ) · ∇q

Then we have an expression for the derivative of fluid distributed terms coming from the Lagrangian with respect to ρ, ! Z Z d G(ρ)dΩ = (D u · δθ) · ∂t v f + u · ∂t (D v f · δθ) f f dρ Ω Ω ˆ ˆ θ+ρδ θ

ˆ θ

ρ=0

− [(D(D u · δθ)) · u + D u · (D u · δθ)] · v f − (D u · u) · (D v f · δθ)

+ν(D u · δθ) · ∆v f + νu · ∆(D v f · δθ) + p div(D v f · δθ) − (D u · δθ) · ∇q Z + −u · ∂t v f + (Du · u) · v f − νu · ∆v f + u · ∇q − p div v f hδθ, ni Γsˆ θ

Fluid boundary terms We must now take into account the terms coming from the moving boundary Γsθ+ρδ . Then we set ˆ θˆ ˆ ˆ θ+ρ ˆ ˆ −1 · σ(v f ◦Rt , q)−(u◦Rt )·(v f ◦Rt ) ·nρ φ(ρ, .) = ∂t (θ+ρδ θ)◦( δ θ) ρ ρ ρ LEMMA 8.8 [53] 0

∂ρ nρ |ρ=0 = nΓ = −∇Γ (δθ · n) LEMMA 8.9 Z d dρ Γsˆ

ˆ θ+ρδ θ

PROOF

! hE(ρ), n idΓ ρ

= ρ=0

Z

First, we use that Z Z hE(ρ), nρ i = Γsˆ

ˆ θ+ρδ θ

Γsˆ θ

Ωfˆ

hE 0 |Γt , ni + (div E)hδθ, ni

div E(ρ)

ˆ θ+ρδ θ

then we derive this quantity using lemma (8.4), ! Z Z Z d (div E)hδθ, ni div E 0 + div E(ρ) = dρ Γs Ωf Ωf ˆ ˆ θ+ρδ θ

ρ=0

ˆ θ

ˆ θ

228 We conclude using

Z

Ωfˆ θ

Moving Shape Analysis Z 0 div E = hE 0 , ni. Γsˆ θ

LEMMA 8.10 def d −1 ˆ ˆ ˆ ˆ ˆ ◦ θˆ−1 − D V ˆ · δθ Wδθ = ∂t (θ + ρδ θ) ◦ (θ + ρ δ θ) = ∂t (δθ ◦ θ) θ dρ ρ=0 (8.21) def −1 ˆ ˆ where we recall that V ˆ = (∂t θ) ◦ θ . θ

Using the last identities, we shall obtain, LEMMA 8.11 E 0 |Γsˆ = Wδθ · − q I +ν D v f − u · v f + Vθˆ · − ν D(D v f · δθ) + (D u · δθ) · v f θ + u · (D v f · δθ) It leads to the final expression, d dρ

Z +

Γsˆ

ˆ θ+ρδ θ

Z

Γsˆ θ

! φ(ρ) dΓ

= ρ=0

Z

Wδθ · −q I +ν D v f − u · v f · n

Γsˆ θ

Vθˆ · −ν D(D v f · δθ) + (D u · δθ) · v f + u · (D v f · δθ) · n +

REMARK 8.5 Z

Γsˆ θ

Z

Γsˆ θ

div(Vθˆ · −q I +ν ∗ D v f − u · v f )hδθ, ni

We recall that Vθˆ · (D v f · n) = =

Z

Z

Ωfˆ θ

Ωfˆ θ

div(∗ D v f · Vθˆ) D v f · · D Vθˆ + Vθˆ · ∆v f

Distributed solid terms We set ˆ · ∂tt vˆs + D(θˆ + ρδ θ) ˆ S(θˆ + ρδ θ) ˆ · · D vˆs m(ρ) = (θˆ + ρδ θ)

(8.22)

Sensitivity analysis for a general fluid-structure interaction system

229

LEMMA 8.12 ! Z TZ d m(ρ) = dρ 0 Ωs0 ρ=0 Z TZ ˆ + D θˆ · S 0 (θ) ˆ · δ θˆ · · D vˆs δ θˆ · ∂tt vˆs + D δ θˆ · S(θ) Ωs0

0

Final time terms We set $(ρ) =

Z

Ωs0

ˆ ) · vˆs (T ) − (θˆ + ρδ θ)(T ˆ ) · ∂t vˆs (T ) ∂t (θˆ + ρδ θ)(T +

Z

Ωf,ρ T

(u ◦ Rtρ )(T ) · (v f ◦ Rtρ )(T )

LEMMA 8.13 Z d ˆ ) · vˆs (T ) − δ θ(T ˆ ) · ∂t vˆs (T ) $(ρ) = (∂t δ θ)(T dρ Ωs0 ρ=0 +

Z

ΩfT

f

f

−(D u·δθ)(T )·v (T )−u(T )·(D v ·δθ)(T ) +

Z

u(T )·v f (T )hδθ(T ), ni ΓsT

Complete derivative We set f

s

`(v , vˆ ) =

Z

Ωs0

− θˆ1 · vˆs (0) + θˆ0 · ∂t vˆs (0) −

Z

f

Ω0

u0 · v (0) −

Z

T 0

Z

Ωs0

f · vˆs

The perturbed Lagrangian has the following form, Lρ = J ρ "Z Z T − 0

Then,

Ωfˆ

G(ρ) +

ˆ θ+ρδ θ

Z

T 0

Z

φ(ρ) + Γsˆ

ˆ θ+ρδ θ

Z

T 0

Z

m(ρ) Ωs0

Z TZ d ρ J =α (θˆ − θˆd ) · δ θˆ dρ ρ=0 0 Ωs0

!

+ $(ρ) + `

#

Using the previous identities, we shall furnish the derivative of the perturbed Lagrangian Lρ with respect to ρ at point ρ = 0, i h d ρ Lθ,δ (8.23) = − A f + B Γ s + C Γ s + D Ωs + F f ˆ θˆ Γ∞ Ωˆ 0 ˆ ˆ θ θ dρ θ ρ=0

230

Moving Shape Analysis

with A Ωf = ˆ θ

Z

Z

T 0

Ωfˆ θ

(D u · δθ) · ∂t v f − [(D(D u · δθ)) · u

+ D u · (D u · δθ)] · v f + ν(D u · δθ) · ∆v f − (D u · δθ) · ∇q +

Z

Z

T 0

Ωfˆ θ

u · ∂t (D v f · δθ) − (D u · u) · (D v f · δθ) + νu · ∆(D v f · δθ)

+ p div(D v f · δθ) + BΓsˆ = θ

Z

0

+

+

+

+

CΓsˆ = θ

DΩs0 =

Z Z

Z

T

Z Z Z Z

Γsˆ θ

Z

T 0

Z

T 0

Z

T 0

0

T 0

−α

Z Z

Γsˆ θ

FΓf∞ = −

− (D u · δθ)(T ) · v f (T ) − u(T ) · (D v f · δθ)(T )

− u · ∂t v f + (Du · u) · v f − νu · ∆v f + u · ∇q − p div v f hδθ, ni

Γsˆ θ

(D Vθˆ · δθ) · −q I +ν D v f − u · v f · n

Γsˆ θ

Γsˆ θ

T 0

Z

Vθˆ · −ν D(D v f · δθ) + (D u · δθ) · v f + u · (D v f · δθ) · n div(Vθˆ · −q I +ν ∗ D v f − u · v f )hδθ, ni

ˆ ◦ θˆ−1 · −q I +ν D v f − u · v f · n (∂t δ θ)

Ωs0

Z

ΩfT

u(T ) · v f (T )hδθ(T ), ni

ΓsT

T

Z

T 0

ˆ + D θˆ · S 0 (θ) ˆ · δ θˆ · · D vˆs δ θˆ · ∂tt vˆs + D δ θˆ · S(θ)

Z

Ωs0

Z

(θˆ − θˆd ) · δ θˆ +

Γf∞

Z

Ωs0

ˆ ) · vˆs (T ) − δ θ(T ˆ ) · ∂t vˆs (T ) (∂t δ θ)(T

ν u∞ · D(D v f · δθ) · n

Sensitivity analysis for a general fluid-structure interaction system

231

The shape derivative kernel identity

We shall now assume that (u, p, ϕ, π) is a saddle point of the Lagrangian functional L. This will help us to simplify several terms involved in the derivative of L with respect to ρ.

LEMMA 8.14

Z

T 0

Z

Ωfˆ θ

[(D u · δθ) · ∂t ϕ − [(D(D u · δθ) · u)

+ D u · (D u · δθ)] · ϕ + ν(D u · δθ) · ∆ϕ − (D u · δθ) · ∇ π] + [u · ∂t (D ϕ · δθ) − (D u · u) · D ϕ · δθ + νu · ∆(D ϕ · δθ) + p div(D ϕ · δθ)] +

Z

T 0

Z

Γsˆ θ

−

−

Z

T 0

Z

Γf∞

Z

ν(ϕ − vˆs ◦ θˆ−1 ) · (D(D u · δθ) · n) + (D ϕ · δθ) · (−p n + ν D u · n) T 0

Z

Γsˆ θ

Vθˆ · [ν D(D ϕ · δθ) − (D u · δθ) · ϕ − u · (D ϕ · δθ)] · n

νu∞ ·(D(D ϕ·δθ)·n)−

Z

(D u·δθ)(T )·ϕ(T )+u(T )·(D ϕ·δθ)(T ) = 0,

ΩT

∀δ θˆ ∈ Yˆ s

PROOF We define a new Lagrangian, where we take into account the normal stress on the moving interface by choosing test functions discontinuous

232

Moving Shape Analysis

at the fluid-structure interface, i.e., (v f , vˆs ) ∈ Y f × Yˆ s ,

ˆ v f , q, vˆs ) = Ju (u, p, θ) ˆ Ldu∞ (u, p, θ; ∞

−

Z

−

Z

−

Z

T 0

Z

Z

T 0

T 0

Γf∞

−

Z

Ωfˆ θ

Z

Γsˆ θ

−u · ∂t v f + (D u · u) · v f − νu · ∆v f + u · ∇ q − p div v f

Vθˆ · σ(v f , q) − (u · v f ) · n + (v f − vˆs ◦ θˆ−1 ) · (σ(u, p) · n)

u∞ ·(σ(v f , q)·n)−

Ωs0

Z

T

0

Z

Ωs0

ˆ tt vˆs +D θ)·S( ˆ ˆ D vˆs ]+ [θ·∂ θ)··

Z

T 0

Z

Ωs0

f ·ˆ vs

ˆ ) · vˆs (T ) − θ(T ˆ ) · ∂t vˆs (T ) − θˆ1 · vˆs (0) + θˆ0 · ∂t vˆs (0)] [∂t θ(T

−

Z

f

ΩT

u(T ) · v (T ) +

Z

Ω0

u0 · v f (0)

∀ (v f , vˆs , q) ∈ Y f × Yˆ s × V f

Let us differentiate the above functional with respect to (u, v),

∂(u,vf ) Ldu∞ · (δu, δv f ) Z TZ = δu · ∂t v f − [(D δu · u) + (D u · δu)] · v f + νδu · ∆v f − δu · ∇ q Ωfˆ

0

θ

+

−

Z

T 0

Z

Γf∞

Z

T 0

Z

Ωfˆ θ

u · ∂t δv f − (D u · u) · δv f + νu · ∆δv f + p div δv f

νu∞ ·(D δv f ·n)+

Z

0

T

Z

Γsˆ θ

ν(v f −ˆ v s ◦θˆ−1 )·(D δu·n)+δv f ·(−p n+ν D u·n)

Sensitivity analysis for a general fluid-structure interaction system Z TZ − Vθˆ · ν D δv f − δu · v f − u · δv f · n

233

Γsˆ

0

θ

−

Z

ΩT

δu(T ) · v f (T ) + u(T ) · δv f (T ) ∀ (v f , vˆs , q) ∈ Y f × Yˆ s × V f

The first order optimality conditions corresponding to the saddle point formulation with respect to (u, v f ) writes f ∂(u,vf ) Ld | ∀ (v f , vˆs , q) ∈ Y f × Yˆ s × V f ˆ · (δu, δv ) = 0, u∞ (u,ϕ,θ)

We choose specific perturbations, δu = D u · δθ,

δv f = D v f · δθ

Solid adjoint equation Using the shape derivative kernel identity, we simplify the Lagrangian ˆ and the distributed A f only inderivative at the saddle point (u, p, ϕ, π, ψ) Ω ˆ θ

volves boundary terms, A Ωf = ˆ θ

Z −

Z

T 0

Z

T 0

Γsˆ θ

Z

Γsˆ θ

Vθˆ · [ν D(D ϕ · δθ) − (D u · δθ) · ϕ − u · (D ϕ · δθ)] · n

ν(ϕ − ψˆ ◦ θˆ−1 ) · (D(D u · δθ) · n) + (D ϕ · δθ) · (−p n + ν D u · n) +

Z

T 0

Z

Γf∞

νu∞ · (D(D ϕ · δθ) · n)

Using the fluid adjoint system (8.17), we have ϕ = ψˆ ◦ θˆ−1 , We have AΩf + FΓf∞ = ˆ θ

Z

T 0

−

Z

Z

Γsˆ

T 0

θ

Z

on Γsθˆ

Vθˆ · [ν D(D ϕ · δθ) − (D u · δθ) · ϕ − u · (D ϕ · δθ)] · n

Γsˆ θ

(D ϕ · δθ) · σ(u, p) · n

234

Moving Shape Analysis

LEMMA 8.15 The following identity holds,

div(Vθˆ · [−π I +ν ∗ D ϕ − u · ϕ]) = −π div Vθˆ − Vθˆ · ∇ π + ν D ϕ · · D Vθˆ + ν Vθˆ∆ϕ − (div Vθˆ) u · ϕ − (D u · Vθˆ) · ϕ − (D ϕ · u) · ϕ

Using the above identity, we get

BΓsˆ = θ

Z

T 0

Z

Γsˆ θ

− Vθˆ · ∂t ϕ + (Du · Vθˆ) · ϕ − νVθˆ · ∆ϕ + Vθˆ · ∇π − p div ϕ hδθ, ni +

+

+

Z

T 0

Z

Z

T 0

Γsˆ θ

Z

Γsˆ θ

Z

T 0

Z

Γsˆ θ

(D Vθˆ · δθ) · [−π I +ν D ϕ − u · ϕ] · n

Vθˆ · [−ν D(D ϕ · δθ) + (D u · δθ) · ϕ + u · (D ϕ · δθ)] · n

− π div Vθˆ − Vθˆ · ∇ π + ν D ϕ · · D Vθˆ + ν Vθˆ∆ϕ − (div Vθˆ) u · ϕ

− (D u · Vθˆ) · ϕ − (D ϕ · Vθˆ) · ϕ hδθ, ni +

Z

ΓsT

u(T ) · ϕ(T )hδθ(T ), ni

Several terms can be eliminated inside the above expressions. Furthermore,

Sensitivity analysis for a general fluid-structure interaction system

235

adding the terms AΩf + FΓf∞ , we obtain ˆ θ

AΩf + BΓsˆ + FΓf∞ = θ

ˆ θ

Z

T 0

Z

Γsˆ θ

− Vθˆ · ∂t ϕ − p div ϕ − π div Vθˆ + ν D ϕ · · D Vθˆ

− (div Vθˆ) u · ϕ − (D ϕ · Vθˆ) · ϕ hδθ, ni − (D ϕ · δθ) · (−p n + ν D u · n) +

Z

T

0

Z

Γsˆ θ

(D Vθˆ · δθ) · [−π I +ν D ϕ − u · ϕ] · n

+

Z

ΓsT

u(T ) · ϕ(T )hδθ(T ), ni

We now turn to the analysis of the term CΓsˆ , θ

CΓsˆ = θ

Z

T 0

Z

Γsˆ θ

ˆ ◦ θˆ−1 · σ(ϕ, π) − V ˆ · ϕ · n (∂t δ θ) θ

We shall use the following fundamental adjoint identity,

LEMMA 8.16 For any smooth E defined in the hold-all domain D,

Z

T 0

Z

Γθˆ

ˆ ◦ θˆ−1 · E = ∂t (δ θ)

Z

T 0

+

Z

"Z

Γθˆ

Γsˆ θ

−∂t E − (divΓ Vθˆ) E − D E · Vθˆ · δθ

E · δθ

#T 0

(8.24)

236

Moving Shape Analysis

Let us apply this lemma, with (E)i = Ei,j nj Z

T 0

Z

Γθˆ

ˆ ◦ θˆ−1 · (E · n) = ∂t (δ θ) Z

T 0

Z

Γθˆ

−(∂t E) · n + E · ∇Γ (Vθˆ · n) − (divΓ Vθˆ) E · n

− D E · Vθˆ · n − E · DΓ n · Vθˆ · δθ +

"Z

Γsˆ θ

(E · n) · δθ

#T 0

Using the previous identity, we get Z TZ CΓsˆ = −(∂t E) · n + E · ∇Γ (Vθˆ · n) − (divΓ Vθˆ) E · n θ

0

Γθˆ

− D E · Vθˆ · n − E · DΓ n · Vθˆ · δθ +

Z

Γsˆ θ

(E · n)(T ) · δθ(T )

with E = σ(ϕ, π) − (Vθˆ · ϕ) I and δθ(0) = 0. Let us now analyse the solid distributed term, DΩs0 = Z

T 0

−α

Z

Z

Ωs0

T 0

Z

ˆ + D θˆ · S 0 (θ) ˆ · δ θˆ · · D ψˆ δ θˆ · ∂tt ψˆ + D δ θˆ · S(θ)

Ωs0

(θˆ − θˆd ) · δ θˆ +

Z

Ωs0

ˆ ) ˆ ) · ψ(T ˆ ) − δ θ(T ˆ ) · ∂t ψ(T (∂t δ θ)(T

The Green formula leads to the following identity, Z Z h i ˆ · · D ψˆ = − ˆ + D2 ψˆ · ·S(θ) ˆ · δ θˆ [D δ θˆ · S(θ)] D ψˆ · div S(θ) Ωs0

+

Z

Ωs0

Γs0

h

i ˆ · n · δ θˆ D ψˆ · ∗ S(θ)

We use the adjoint identity (8.4) with B = ∗ D θˆ · D ψˆ and we get Z Z Z ∗ ∗ ˆ ˆ ˆ ˆ ˆ · δ θˆ T (δθ) · ·( D θ · D ψ) = − δθ · TA ( D θ · D ψ) + TΓ (∗ D θˆ · D ψ) Ωs0

Ωs0

Γs0

(8.25)

Sensitivity analysis for a general fluid-structure interaction system

237

This leads to the following expression, DΩs0 = Z

T 0

Z

+

Ωs0

Z

T 0

ˆ − D2 ψˆ · ·S(θ) ˆ − TA (∗ D θˆ · D ψ) ˆ − α (θˆ − θˆd ) · δ θˆ ∂tt ψˆ − D ψˆ · div S(θ)

Z

Γs0

Z h i ∗ ∗ ˆ ˆ ˆ ˆ ˆ D ψ · S(θ) · n + TΓ ( D θ · D ψ) · δ θ +

Ωs0

ˆ ) · ψ(T ˆ ) (∂t δ θ)(T

ˆ ) · ∂t ψ(T ˆ ) − δ θ(T

Using the optimality condition (8.23), we shall deduce that the adjoint solid state is the solution of the following backward second order system,  ˆ − D2 ψˆ · ·S(θ) ˆ − TA (∗ D θˆ · D ψ) ˆ = ∂tt ψˆ − D ψˆ · div S(θ)       α (θˆ − θˆd ), (0, T ) × Ωs0       ˆ ˆ |t=T = (0, 0), (ψ, ∂t ψ) Ωs0 (8.26) Furthermore, the following identity holds on the moving boundary Γ θˆ, 0 = − Vθˆ · ∂t ϕ − p div ϕ − π div Vθˆ + ν D ϕ · · D Vθˆ − (div Vθˆ) u · ϕ

− (D ϕ · Vθˆ) · ϕ · n − ∗ D ϕ · σ(u, p) · n + ∗ D Vθˆ · [σ(ϕ, π) − u · ϕ] · n

+ −(∂t E) · n + E · ∇Γ (Vθˆ · n) − (divΓ Vθˆ) E · n − D E · Vθˆ · n − E · DΓ n · Vθˆ h i ˆ ·n ˆ ◦ θˆ−1 [J(θ)| ˆ ∗ (D θ) ˆ −1 · n + D ψˆ · ∗ S(θ) ˆ + TΓ (∗ D θˆ · D ψ) ˆ |]−1 ◦ θˆ−1

with E = σ(ϕ, π) − (Vθˆ · ϕ) I. This last identity ends the proof of the main result of this chapter.

Appendix A Functional spaces and regularity of domains

A.1

Classical functions

Let Ω be an open subset of Rd , and we denote by C 0 (Ω) the space of continuous functions from Ω to R. For k ∈ N∗ ,

def

C k (Ω) =

f ∈ C k−1 (Ω) : Dα f ∈ C 0 (Ω),

where α = (α1 , . . . , αd ) ∈ Nd is a multi-index. |α| = derivative and

Y

def

Dα f =

i Dα i f =

1≤i≤d

∀α, |α| = k X

(A.1)

αi is the order of

1≤i≤d

∂ |α| f d . . . xα d

1 ∂xα 1

(A.2)

REMARK A.1 The spaces C 0 (Ω) and C k (Ω) for k ∈ N∗ are not Banach1 spaces. However if {Kn }n∈N is a sequence of compact sets of Ω such that Ω=

[

Kn ,

n∈N

we can define a sequence of semi-norms dn (x) = supx∈Kn |u(x)| 1A

complete normed space. A norm on a vector space E is a real-valued functional m on E such that • m(x) ≥ 0,

∀ x ∈ E with equality iff x = 0,

• m(c x) = |c| m(x),

∀ x ∈ E,

• m(x + y) ≤ m(x) + m(y),

c ∈ C,

∀ x, y ∈ E

A normed space is a vector space which is provided with a norm. A Cauchy sequence in a metric space is a sequence {xn }n≥0 which satisfies limn,p→∞ d(xn , xp ) = 0. A metric space is said to be complete in a metric space iff every Cauchy squence is convergent.

239

240

Moving Shape Analysis

and a distance d(u, v) = with an ≥ 0 such that

X

X

n∈N

an

dn (u − v) 1 + dn (u − v)

an = 1. Then C 0 (Ω), d(., .) is a Fr´echet space.2

n∈N In a similar way, we can define on C k (Ω) a Fr´echet space structure with the following sequence of semi-norms X dkn (u) = dn (Dα u) |α|≤k

Since Ω is open, functions in C k (Ω) can be unbounded. Thus, let us denote by B 0 (Ω) the space of bounded continuous functions. For k ∈ N ∗ , we define B k (Ω) = f ∈ B k−1 (Ω) : Dα f ∈ B 0 (Ω), ∀ α, |α| = k Endowed with the norm

def

kf kC k (Ω) =

max sup | Dα f (x)|

0≤|α|≤k x∈Ω

(A.3)

B k (Ω) is a Banach space. If a function f is bounded and uniformly continuous 3 on Ω, it has a unique, continuous extension to the closure Ω. Hence we introduce the space C k (Ω) = f ∈ B k (Ω) : Dα f uniformly continuous on Ω, ∀ 0 ≤ |α| ≤ k

The space (C k (Ω), k.kC k (Ω) ) is a Banach space. Given 0 < ` ≤ 1, a function f is (0, `)-H¨ older continuous in Ω if ∃ c > 0,

|f (y) − f (x)| ≤ c|y − x|` ,

∀x, y ∈ Ω

When ` = 1, we say that f is Lipschitz. Similarly for k ≥ 1, f is (k, `)-H¨ older continuous in Ω if ∀α, 0 ≤ |α| ≤ k, ∃ c > 0, | Dα f (y) − Dα f (x)| ≤ c|y − x|` ,

∀x, y ∈ Ω

2 A Fr´ echet space is complete metrizable locally convex topological vector space. E is a topological vector space if E is furnished with a topology compatible with the structure of the vector space, i.e., such that the vector space operations of addition and scalar multiplication are continuous. It is locally convex if each neighborhood of the origin in E contains a convex neighborhood. A topological space is said to be metrizable if its topology can be defined by a metric. 3 A function f : Ω → R is uniformly continuous if for each ε > 0 there exists δ > 0 such that

|f (x) − f (y)| < ε,

∀x, y ∈ Ω with |x − y| < δ

Functional spaces and regularity of domains

241

We denote by C k,` (Ω) the space of all (k, `)-H¨ older continuous functions in Ω. def

We also define C k,` (Ω) = C k,` (Ω) ∪ C k (Ω). Endowed with the norm def

kf kC k,` (Ω) = kf kC k (Ω) + max

0≤|α|≤k

sup x, y ∈ Ω x 6= y

| Dα f (y) − Dα f (x)| |y − x|`

(A.4)

C k,` (Ω) is a Banach space. The spaces D k (Ω) for k ∈ N∗ and D(Ω) contain respectively all the functions of class C k and infinitely continuously differentiable with compact support.

A.2

Lebesgue spaces

Let Ω be any open set in Rd . The space Lp (Ω), 1 ≤ p < +∞ is the set of real functions defined on Ω that have finite norm Z 1/p def p kukLp (Ω) = |u(x)| dx (A.5) Ω

We note that we identify functions that are different only on a set of Lebesgue measure zero. The space L∞ (Ω) is the set of functions on Ω that are measurable with respect to the Lebesgue measure dx and have a finite norm kukL∞ (Ω) =

A.2.1

inf

dx −meas{x∈Ω:|u(x)|>C}=0

C

(A.6)

Various inequalities

LEMMA A.1 H¨ older inequality Z u(x) v(x) dx ≤ kukLp (Ω) kvkLq (Ω) ,

1/p + 1/q = 1,

(A.7)

Ω

for u ∈ Lp (Ω) , v ∈ Lq (Ω). REMARK A.2

This identity is a consequence of the Young inequality, a b ≤ 1/p ap + 1/q bq ,

for a ≥ 0, b ≥ 0.

1/p + 1/q = 1,

(A.8)

242

Moving Shape Analysis

REMARK A.3 inequality.

For p = q = 2, this inequality is referred to as the Cauchy

LEMMA A.2 Interpolation inequality 1−α kukLr (Ω) ≤ kukα Lp (Ω) kukLq (Ω) ,

1/r = α/p + (1 − α)/q,

(A.9)

for u ∈ Lp (Ω) ∩ Lq (Ω).

A.2.2

Completeness, separability and reflexivity

LEMMA A.3 For 1 ≤ p ≤ +∞, the space Lp (Ω) is a Banach space4 . For 1 < p < +∞ the space Lp (Ω) is a separable reflexive space and its dual space can be identified with Lq (Ω) with 1/p + 1/q = 1. The space L1 (Ω) is separable and its dual can be identified with L∞ (Ω).

A.2.3

Convolution

THEOREM A.1 Let u ∈ L1 (Rd ) and v ∈ Lp (Rd ) with 1 ≤ p ≤ ∞. Then y 7→ u(x − y) v(y) is summable over Rd for a.e. x ∈ Rd . We define their convolution product by Z u(x − y) v(y) dy (A.10) u ? v(x) = Rd which satisfies u ? v ∈ Lp (Rd ) and ku ? vkLp ≤ kukL1 kvkLp

(A.11)

The convolution can be extended to a function u ∈ L p (Rd ) thanks to the following theorem, THEOREM A.2 Young inequality Let u ∈ Lp (Rd ) and v ∈ Lq (Rd ) with 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, 1/r = 1/p + 1/q − 1 ≥ 0

(A.12)

u ? v ∈ Lr (Rd ) and ku ? vkLr ≤ kukLp kvkLq

(A.13)

then

4 Fischer-Riez

theorem [17] Theorem IV. 8, p. 57

Functional spaces and regularity of domains

243

The following regularizing property will be useful to state a density property of regular functions space in Lp , THEOREM A.3 Let g ∈ D k (Rd ) for k ∈ N and u ∈ L1loc (Rd ), then g ? u ∈ C k (Rd ) and Dα (g ? u) = (Dα g) ? u,

|α| ≤ k

(A.14)

REMARK A.4 As an application of the previous theorem, we deduce that if g ∈ D(Rd ) and u ∈ L1loc (Rd ) then g ? u ∈ C ∞ (Rd ).

A.2.4

Density property

THEOREM A.4 [44] The space D(Ω) is dense in Lp (Ω) for 1 ≤ p < +∞ and every convergent sequence in D(Ω) converges in Lp (Ω). REMARK A.5 This theorem can be established using the truncation + regularization procedure introduced by Leray and Friedrichs that will be introduced in the next section.

A.3

Smooth domains and boundary measure def

Let Ω be an open set of Rd and let its associated boundary Γ = ∂Ω be a manifold in Rd of codimension 1. The smoothness of Γ is classically characterized by introducing at each point of Γ a local diffeomorphism that locally flattens the boundary. In order to make the purpose more precise, let us introduce some notations and definitions. Let {ei }1≤i≤d be the canonical unit orthogonal basis in Rd . We use the notation ζ = (ζ 0 , ζd ) for a point ζ ∈ Rd with ζ 0 = (ζ1 , . . . , ζd−1 ). Let B be the unit open ball in Rd and let us define the following sets, def

B0 = {ζ ∈ B : ζd = 0} def

(A.15)

def

B+ = {ζ ∈ B : ζd > 0} , B− = {ζ ∈ B : ζd < 0} DEFINITION A.1

def

(A.16)

Let Ω be a subset of Rd such that Γ = ∂Ω 6= ∅.

244

Moving Shape Analysis

a) Ω is said to be of class C k , 0 ≤ k ≤ +∞, if for each point x ∈ Γ there exist – a neighbourhood U (x) ⊂ Rd of x,

– a bijective map gx : U (x) → B with the following properties : def

gx ∈ C k (U (x), B), hx = gx−1 ∈ C k (B, U (x))

(A.17)

int[Ω ∩ U (x)] = hx (B + )

(A.18)

Γx = Γ ∩ U (x) = hx (B0 ), B0 = gx (Γx )

(A.19)

def

b) Ω is said to be of class C k,` , k ≥ 0, 0 < ` ≤ 1, if the above conditions are satisfied with a map gx ∈ C k,` (U (x), B) together with its inverse hx ∈ C k,` (B, U (x)). For sets of class C 1 , the unit exterior normal n to the boundary Γ can be characterized through the Jacobian matrices of g x and hx . LEMMA A.4 The outward unit normal field n(y) at point y ∈ Γx is given by the following expressions, n(y) = − =

∗

(D hx )−1 (ζ 0 , 0) · ed , ∗ | (D hx )−1 (ζ 0 , 0)ed | −

∗

D gx (y)ed , |∗ D gx (y) · ed |

∀ hx (ζ 0 , 0) = y ∈ Γx

(A.20)

∀ y ∈ Γx

(A.21)

with (D g)i,j = ∂j gi . The family of neighbourhoods U (x) associated with all the points x of Γ is an open cover of Γ. LEMMA A.5 Let Ω ⊂ Rd be a bounded domain with compact boundary Γ. There exists a finite subcover, i.e., there exists a finite sequence of points {x i }1≤i≤m of Γ such that Γ ⊂ U1 ∪ . . . ∪ Um , where Ui = U (xi ). Furthermore there exists a partition of unity {ri }1≤i≤m associated to the family of open neighbourhoods {Ui }1≤i≤m of Γ such that : ri ∈ D(Ui ), 0 ≤ ri (x) ≤ 1, P (A.22) r (x) = 1, ∀ x∈U i 1≤i≤m

Functional spaces and regularity of domains ¯⊂ with U

[

245

Ui .

1≤i≤m

Let us define the boundary integral of a continuous function f on Γ : LEMMA A.6 For f ∈ C 0 (Γ), its integral over Γ is defined as follows Z X Z def f dΓ = f ri dΓi Γ

where

Z

1≤i≤m

def

f ri dΓi = Γi

Z

B0

Γi

(f ri ) ◦ hi (ζ 0 , 0)ωi (ζ 0 )dζ 0

with Γi = U (xi ) ∩ Γ and ωi = Ωxi and ωx (ζ 0 ) = |mx (hx (ζ 0 , 0))|| det D hx (ζ 0 , 0)| with

mx (y) = −∗ D gx (y)ed

(A.23)

Appendix B Distribution spaces

B.1

The space D(Ω)

Let Ω be an open set in Rd . The spaces D k (Ω) for k ∈ N∗ and D(Ω) contain respectively all the functions of class C k and infinitely continuously differentiable with compact support1 in Ω. For any open set O ⊂ Rd such that Ω ⊂ O, we define D(Ω) = {φ|Ω : φ ∈ D(O)}

(B.1)

The space D(Ω) does not reduce to the null function. Indeed let ω : R d → R defined as follows : 2 e1/(|x| −1) , |x| < 1 ω(x) = 0, |x| ≥ 1 Let Br (a) be an open ball contained in Ω, then the function ϕa,r (x) = ω(

x−a ) r

belongs to D(Ω). There exists a general procedure for the construction of functions in D(Ω) based on the previous particular function and the notion of convolution product. DEFINITION B.1 mollifier mollifier if

We call a family of function {ρε }ε≥0 a

• ρε ∈ D(Rd ), • Supp ρε ⊂ B(0, ε), Z • ρε ≥ 0 and ρε = 1. Rd 1 For

f : Ω → R, the support of f is defined as follows Supp f = {x ∈ Ω : f (x) 6= 0}

247

248

Moving Shape Analysis

An example of a mollifier is furnished by ρε (x) = ε−d ϕ0,1 (x/ε)

(B.2)

LEMMA B.1 Let u be a function locally integrable in Ω with compact support subset of Ω. Then there exists ε0 > 0 such that uε = ρε ? u ∈ D(Ω) for ε < ε0 . This lemma shows that the space D(Ω) is sufficiently rich and we shall see that it has density properties in numerous common functional spaces. The space D(Ω) is neither a Banach space nor a Fr´echet space, but the inductive limit of Fr´echet spaces which makes it a locally convex topological vector space. But the corresponding inductive limit topology is not easy to manipulate. Fortunately, the notion of convergence of sequences will be sufficient for our purpose. DEFINITION B.2 Let {ϕk }k≥0 be sequence of elements of D(Ω). It converges to ϕ ∈ D(Ω) if there exists a compact subset K ⊂ Ω such that Supp ϕk ⊂ K, ∀k ≥ 0 Dα ϕk converges uniformly on K to D α ϕ as k → ∞ LEMMA B.2 Generalized Urysohn Let K be a compact set in Rd and ε > 0 arbitrary. Then there exists a function ψε ∈ D(Rd ) such that • ψε (x) = 1,

∀ x ∈ K and ψε ≥ 0,

• Supp ψε ⊂ Kε = ∪x∈K B(x, ε). With the above definition, we can state a useful density property for D(R d ), THEOREM B.1 The space D(Rd ) is contained and dense in C k (Rd ) for 0 ≤ k ≤ ∞ and every convergent sequence in D(Rd ) converges in C k (Rd ). REMARK B.1

The proof consists of two steps :

• Truncation : We prove that D k is dense in C k using the family up (x) = ψ(x/p) u(x) ∈ Dk with ψ given by lemma B.2 with K = B(0, 1).

Distribution spaces

249

• Regularization : We show that D is dense in D k using the sequence up = u ? ρp ∈ D with ρp = ρε=1/p and ρε a mollifying family.

B.2

The space of distributions D 0 (Ω)

DEFINITION B.3 A continuous linear form on D(Ω) is called a distribution on Ω. We denote by D 0 (Ω) the space of distributions on Ω. If T ∈ D 0 (Ω), we note hT, ϕi = T (ϕ), ∀ ϕ ∈ D(Ω) Here T continuous2 on D(Ω) means lim hT, ϕk i = hT, ϕi,

k→∞

∀ {ϕk }k∈N ∈ D(Ω) with lim ϕk = ϕ in D(Ω) k→∞

0

The space D (Ω) is a vector space and is in fact the topological dual space of D(Ω). As a dual space, D 0 (Ω) is equipped with the weak-star topology σ(D 0 (Ω), D(Ω)), defined by the family of semi-norms θϕ : D0 (Ω) → R T 7→ θϕ (T ) = T (ϕ) = hT, ϕi

0

for ϕ ∈ D(Ω) and where h., .i stands for the duality pairing between D (Ω) and D(Ω). In this topology, which is again not metrizable, a sequence of distributions T k converges to T if and only if hTk , ϕi →k→∞ hT, ϕi,

∀ϕ ∈ D(Ω)

THEOREM B.2 If {Tk }k∈N is a sequence of elements in D 0 (Ω) such that hTk , ϕi is convergent for all ϕ ∈ D(Ω) as k → ∞, then the linear form ϕ 7→ lim hTk , ϕi k→∞

2 Equivalently,

given any compact subset K ⊂ Ω, there exist a constant C(K) and an integer m(K) ≥ 0 such that |T (ϕ)| ≤ C(K)

max

sup | Dα ϕ(x)|,

|α|≤m(K) x∈Ω

∀ ϕ ∈ D(Ω) with Supp ϕ ⊂ K

When the integer m can be chosen independently on K, the distribution T is said to be of finite order and its order is defined as the smallest suitable integer m.

250

Moving Shape Analysis

belongs to D 0 (Ω). When f is locally integrable on Ω, then f can be identified with a distribution by Z hf, ϕi = f (x) ϕ(x) dx , ∀ ϕ ∈ D(Ω) Ω

Hence the duality pairing h., .i is an extension of the scalar product of the Hilbert space (L2 (Ω), k.kL2 (Ω) ) . The differentiation of distributions (which coincides with the usual differentiation when applied to continuously differentiable functions) is the essential property which justifies the introduction of distributions. 0

DEFINITION B.4 Letting T ∈ D (Ω) , the derivative Dα T , α ∈ Nd is the distribution on Ω defined by hDα T, ϕi = (−1)|α| hT, Dα ϕi,

∀ ϕ ∈ D(Ω)

THEOREM B.3 0 Let {Tk }k∈N ∈ D (Ω) be a convergent sequence of distributions towards T ∈ 0 D (Ω). Then the derivative of T has the following property 0

lim Dα Tk = Dα T in D (Ω),

k→∞

∀ α ∈ Nd

THEOREM B.4 Let Ω be an open set of Rd and E a locally convex topological vector space of functions on Ω satisfying : • D(Ω) is contained and dense in E, • every convergent sequence in D(Ω) converges in E. Then the continuous linear forms on E are identified with distributions ( i.e., elements of D 0 (Ω)). REMARK B.2 The identification is made in the following manner : let T be a continuous linear form on E, then the restriction of T to D(Ω) is the distribution we are looking for.

Distribution spaces

B.3 B.3.1

251

Examples of distributions Measures on Ω

DEFINITION B.5 We call every continuous linear form on the space D0 (Ω) of continuous functions with compact support a measure on Ω. The space of measures is denoted M (Ω). Using that D(Ω) ,→ D 0 (Ω), we can state that µ ∈ M (Ω) defines an element of D0 (Ω). We usually note µ(ϕ) = hµ, ϕi the value of the linear form µ on ϕ ∈ D 0 (Ω). We also say that µ(ϕ) is the integral of the function ϕ relative to the measure µ and we write Z µ(ϕ) =

ϕ(x)dµ

Ω

We can extend the definition of µ(ϕ) for measurable functions that do not belong to D 0 (Ω). These are called summable functions.

B.3.2

Regular distributions

We recall that L1loc (Ω) is the set of functions integrable on every compact set in Ω. DEFINITION B.6 µf by

Let f ∈ L1loc (Ω); we define a corresponding measure µf (ϕ) =

Z

f (x)ϕ(x) dx Ω

The application [.] : L1loc (Ω) → M (Ω) f 7→ [f ] = µf is an injection and its image is called the set of absolutely continuous measures in Ω. f is called the density of µf with respect to the Lebesgue measure dx ( dµf = f (x) dx ). It is usual to identify µf with its density f . Furthermore µf defines a distribution and we say that this distribution is a function f . REMARK B.3 As Lp (Ω) ,→ L1loc (Ω) with 1 ≤ p ≤ +∞, every element of Lp (Ω) defines a distribution on Ω ( which is a measure on Ω).

252

B.3.3

Moving Shape Analysis

The Dirac measure Let x0 ∈ Ω. The mapping

DEFINITION B.7 Dirac measure

δx0 : D0 (Ω) → R ϕ 7→ ϕ(x0 ) defines a measure in M (Ω) called the Dirac measure at point x0 . REMARK B.4 The Dirac measure is the simplest example of measure which is not a function. However we shall abuse the notation and write sometimes δ(x − x0 ) for δx0 . Using integral notation, we have the following properties, Z δx0 (ϕ) = ϕ(x0 ) = Z

Rd

Rd

ϕ(x)δ(x − x0 ) dx

δ(x − x0 ) dx = 1

The Dirac function can also be characterized as the derivative in the sense of distributions of a particular function, DEFINITION B.8 Heaviside function Y :R→R x 7→ Y (x) :

Let

+1, x > 0 0, x < 0

Obviously Y ∈ L1loc (R). Then it defines a distribution. We can then compute its derivation in D 0 (R), hD Y, ϕi = −

Z

+∞

ϕ0 (x) dx = ϕ(0)

0

then D Y = δ,

B.3.4

in D 0 (R)

Derivative of the Dirac measure

Let us compute the derivative of the Dirac measure in the sense of distributions, hDm δ, ϕi = (−1)m hδ, Dm ϕi = (−1)m Dm ϕ(0)

Distribution spaces

253

Hence to define the distributions Dm δ, we need to consider test functions which belong at least to the class C m . We thus conclude that Dm δ is not a measure for m ≥ 1. REMARK B.5 In general, a derivative in the sense of distributions of a measure on Ω defines a distribution which is not a measure except in the case where the measure is a sufficiently regular distribution.

B.3.5

The Dirac comb

DEFINITION B.9

The application S : D(R) → R X ϕ 7→ ϕ(k) k∈Z

defines a distribution called Dirac comb.

B.3.6

Principal value of 1/x

Let us consider the function, R→R x 7→ log(|x|) By a direction computation, it can be stated that its derivative in the sense of distributions is given by the following identity, Z +∞ ϕ(x) − ϕ(−x) hD log(|x|), ϕi = lim dx ε→0 ε x The right hand side is referred to as the Cauchy principal value of R +∞ and is denoted v. p −∞ ϕ(x) x dx . It can be checked that the mapping D(R) → R ϕ 7→ v. p

Z

+∞ −∞

R +∞ −∞

ϕ(x) x

dx

ϕ(x) dx x

is linear and continuous on D(R) (but not in D 0 (R)). Then it defines a distribution (which is not a measure) which we denote by v. p

1 x

254

Moving Shape Analysis

It is sometimes called the pseudo-function log(|x|) in the sense of distributions. REMARK B.6 regular distribution.

1 x

1 x

and it is the derivative of the

is not locally summable and there exists no associated

Appendix C The Fourier transform

C.1

The case of L1 functions and extension to L2

DEFINITION C.1 follows

Let u ∈ L1 (Rd ). We define the Fourier transform as def

u ˆ(ξ) = (F u)(ξ) = (2π)−d/2

Z

Rd

e−i x·ξ u(x) dx ,

∀ ξ ∈ Rd

(C.1)

THEOREM C.1 The Fourier transform belongs to L (L1 (Rd ), B0 (Rdξ )). REMARK C.1 The space B0 (Rd ) stands for the space of bounded continuous functions on Rd which tend to zero at infinity. Provided with the norm kuk∞ = supx∈Rd |u(x)|, B0 (Rd ) is a Banach space. DEFINITION C.2 as follows

For u ∈ L1 (Rd ), we define the Fourier co-transform

¯ˆ(ξ) = (F u)(ξ) def u = (2π)−d/2

DEFINITION C.3 is defined by

Z

Rd

ei x·ξ u(x) dx ,

∀ ξ ∈ Rd

(C.2)

The translate of the function f of amplitude a ∈ Rd τa f (x) = f (x − a),

∀ x ∈ Rd

(C.3)

and its symmetriser by ςf (x) = f (−x),

∀ x ∈ Rd

(C.4)

THEOREM C.2 For u, v ∈ L1 (Rd ), the following properties hold 255

256

Moving Shape Analysis

¯ˆ = F (u) = F (¯ i) u u), F (ς u) = F u = u ˆ, Z Z ii) u ˆ(x) v(x) dx = u(y) vˆ(y) dy, Rd Rd iii) F (τa u)(ξ) = e−i a·ξ u ˆ(ξ),

ς(F u) = F u ¯,

F (e−i a·x u) = τ−a u ˆ.

REMARK C.2 The property (iii) expresses that the Fourier transform exchanges translation and multiplication by a complex exponential. THEOREM C.3 i) If xα u ∈ L1 (Rd ) for |α| ≤ k, then u ˆ ∈ C k (Rd ) and we have Dα u ˆ = (−i)|α| F (xα u)

(C.5)

ii) If u ∈ C k (Rd ) with Dβ u ∈ L1 (Rd ) for |β| ≤ k, then ξ β u ˆ ∈ L∞ and i|β| ξ β u ˆ = F (Dβ u)

(C.6)

REMARK C.3 The two last properties assert that the Fourier transform exchanges reciprocally differentiation of order |α| with multiplication by x α which suggests a kind of stability that will be established for a subspace of L1 . THEOREM C.4 Let u ∈ L1 (Rd ) and real number λ > 0. We have F (f (λ.))(x) = λ−d fˆ(ξ/λ)

(C.7)

THEOREM C.5 Let u, v ∈ L1 (Rd ). Then the Fourier transform of the convolution product is given by the following identity F (u ? v) = (F u) (F v)

(C.8)

The next theorem is a fundamental result justifying the power of the Fourier transform, THEOREM C.6 Fourier inversion Let u ∈ L1 (Rd ) such that fˆ ∈ L1 (Rdξ ). Then f (x) = F (fˆ)(x)

(C.9)

The Fourier transform REMARK C.4

257

This result is a consequence of both the following formula 2

F (e−a|.| )(ξ) = (2 a)−d/2 e−|ξ|

2

/4a

(C.10)

and the regularizing property of convolution with Gaussian functions. REMARK C.5 The inversion formula gives f = F (g) with g = fˆ ∈ L1 . This implies that f ∈ B0 (Rd ) which is not convenient for the applications. A great improvement is the extension of the Fourier transform to square summable functions furnished by the next theorem. THEOREM C.7 Fourier-Plancherel The Fourier transform F : L1 (Rd ) ∪ L2 (Rd ) → L2 (Rd ) can be continuously extended into a surjective isometry F : L2 (Rd ) → L2 (Rd ). REMARK C.6 This result is a consequence of the Fourier inversion Theorem and the identity (ii) in Theorem (C.2).

C.2

The space S (Rd )

We define the space of C ∞ functions with rapid decay at infinity1 , def S (Rd ) = u ∈ C ∞ (Rd ), lim xα Dβ u(x) = 0, ∀ α, β ∈ Nd (C.11) |x|→+∞

REMARK C.7 S (Rd ) can be endowed with the structure of a Fr´echet space with the family of semi-norms, X Nk (ϕ) = kxα Dβ ϕkL∞ |α|,|β|≤k

∀ k ∈ N. THEOREM C.8 The space D(Rd ) is contained with dense and continuous injection in the Fr´echet space S (Rd ). 1 It

can be equivalently defined as follows n def S (Rd ) = u ∈ C ∞ (Rd ), xα Dβ u(x) ∈ L2 (Rd ),

∀ α, β ∈ Nd

o

258

Moving Shape Analysis

THEOREM C.9 The Fourier transform F is an isometry from S into itself when S is endowed with the topology induced by L2 (Rd ). Its inverse is given by F −1 = F and the following identities hold true, (u, v)L2 = (ˆ u, vˆ)L2 , Parseval kukL2 = kˆ ukL2 , Plancherel

(C.12) (C.13)

REMARK C.8 Using the density of S in L2 , the Fourier transform F ∈ ISOM(S ) can be extended in a unique manner as F ∈ ISOM(L 2 ). This furnishes an alternative way of defining the Fourier transform in L 2 .

C.3

The space of tempered distributions

DEFINITION C.4 The space S 0 (Rd ) is defined as the space of continuous linear forms on the Fr´echet space S (Rd ). REMARK C.9 Using the continuous and dense injection of D(R d ) ,→ S (Rd ) (Theorem (C.8)) and Theorem (B.4), we can identifiy elements of S 0 (Rd ) with distributions in D 0 (Rd ) . The distributions in S 0 (Rd ) are called tempered distributions. REMARK C.10 The elements of Lp for 1 ≤ p ≤ ∞ can be identified with tempered distributions. A sequence {Tk }k∈N of tempered distributions is convergent in S 0 (Rd ) if ∀ ϕ ∈ S (Rd )

lim hTk , ϕi = hT, ϕi,

k→∞

THEOREM C.10 If T ∈ S 0 (Rd ), all its derivatives belong to S 0 (Rd ). Furthermore, let {Tk }k∈N ∈ S 0 (Rd ) be a convergent sequence of tempered distributions towards T ∈ S 0 (Ω). Then the derivative of Tk has the following property 0

lim Dα Tk = Dα T in S (Ω),

k→∞

∀ α ∈ Nd

The Fourier transform

259

DEFINITION C.5 Let OM (Rd ) be the space of infinitely differentiable functions of slow growth in Rd , as well as all their derivatives, i.e., n o def OM (Rd ) = ϕ ∈ E (Rd ) : ∀β ∈ Nd ∃Cβ , mβ > 0; | Dβ ϕ| ≤ Cβ (1 + |x|)mβ THEOREM C.11 Let f ∈ OM (Rd ), then we have (i) f ϕ ∈ S (Rd ), (ii) f T ∈ S 0 (Rd ),

∀ ϕ ∈ S (Rd ). ∀ T ∈ S 0 (Rd ).

(ii) limk→∞ f Tk = f T, S 0 (Rd ) if limk→∞ Tk = T, S 0 (Rd ). REMARK C.11 Hence the space OM leaves stable by multiplication the space S and S 0 , whence the notation 00 M 00 for multiplicator.

C.4

Fourier transform in S 0

DEFINITION C.6 Let T ∈ S 0 (Rd ). Its Fourier transform F T ∈ d 0 S (R ) is the tempered distribution defined by def

hF T, ϕi = hT, F ϕi,

∀ ϕ ∈ S (Rd )

(C.14)

THEOREM C.12 The Fourier transform is such that F ∈ ISOM(S 0 (Rd )) with inverse F −1 = F and the following properties, i) F (τa T ) = e−i a·. F T ,

F (e−i a·. T ) = τ−a F T .

ii) For α ∈ Nd , F (Dα T ) = (i · .)α F T . iii) For β ∈ Nd , Dβ F (T ) = F ((−i · .)β T ). Let us consider an automorphism θ of Rd . We denote by T θ its transpose, i.e., T θ(x) · y = x · θ(y), ∀ x, y ∈ Rd

260

Moving Shape Analysis

For T ∈ S 0 (Rd ), we define its direct image θ∗ T ∈ S 0 (Rd ) by θ, hθ∗ T, ϕi = hT, ϕ ◦ θi,

∀ ϕ ∈ S (Rd )

(C.15)

def

and its inverse image T θ∗ T = | det D T θ|−1 (T θ−1 )∗ T . THEOREM C.13 Let T ∈ S 0 (Rd ), then we have F (θ∗ T ) = T θ∗ F (T )

(C.16)

In the special case of distributions of compact support, we have the following result, THEOREM C.14 Let T ∈ E 0 (Rd ). Then its Fourier transform F T ∈ OM (Rd ) is given by the following identity, (F T )(ξ) = hT (x), e−i x·ξ i (C.17) REMARK C.12 This theorem furnishes an explicit calculus of the Fourier transform of a distribution, when its support is compact. As an application, we can easily compute the Fourier transform of the Dirac measure, since its support is reduced to a singleton F (δa ) = ei a·.

C.5

Fourier transform and convolution

THEOREM C.15 Let ϕ, ψ ∈ S (Rd ). We define their convolution product ϕ ? ψ ∈ S (Rd ) by Z (ϕ ? ψ)(y) = ϕ(y − x) ψ(x) dx (C.18) Rd and whose Fourier transform satisfies F (ϕ ? ψ) = (2π)d/2 F (ϕ) F (ψ)

(C.19)

The Fourier transform

261

DEFINITION C.7 We introduce the space of convolutors of S (Rd ), n o O 0 (Rd ) = T ∈ S 0 (Rd ) : T ? ϕ ∈ S (Rd ), ∀ ϕ ∈ S (Rd ) REMARK C.13 This space can be characterized as the space of distributions with rapid decay at infinity. THEOREM C.16 0 0 (i) F (OM (Rd )) = Oc0 (Rd ) and F (Oc0 (Rd )) = OM (Rd ). 0 (ii) If S ∈ S 0 (Rd ), T ∈ Oc0 (Rd ) and α ∈ OM (Rd ), we have

C.6

F (T ? S) = (2π)d/2 F (T ) F (S)

(C.20)

F (α S) = (2π)d/2 F (α) ? F (S)

(C.21)

Partial Fourier transform

THEOREM C.17 Let ϕ ∈ S (R × Rd ). Its partial Fourier transform Fx with respect to x ∈ Rd is defined by Z (Fx ϕ)(t, ξ) = e−ix·ξ ϕ(t, x) dx (C.22) d R Furthermore Fx ∈ ISOM(S (R × Rd )) with inverse Fx−1 = Fx .

REMARK C.14 By transposition we define Fx in S 0 (Rd ) and we have Fx ∈ ISOM(S 0 (R × Rd )).

C.7

Vector valued distributions

Let X be a Banach space and ]a, b[ an open set of R. The measure dt stands for the Lebesgue measure on ]a, b[. DEFINITION C.8 The space of linear continuous mapping of D(]a, b[) into X, is called the space of vectorial distributions over ]a, b[ with values in

262

Moving Shape Analysis

X and we note def

D0 (]a, b[; X) = L (D(]a, b[); X)

(C.23)

It is endowed with the topology of the uniform convergence on bounded sets in D(]a, b[). Hence, if f ∈ D 0 (]a, b[; X), then the application D(]a, b[) → X

ϕ 7→ hf, ϕi

is continuous. REMARK C.15 Rd .

This definition can be extended to any open set Ω of

As in the case of scalar distributions, we can define derivatives in the sense of distributions, DEFINITION C.9 For f ∈ D 0 (]a, b[; X), we define its first order derivad tive in the sense of distributions as the unique element dt f ∈ D0 (]a, b[; X) such that d d h f, ϕi = −hf, ϕi, in X (C.24) dt dt ∀ ϕ ∈ D(]a, b[) REMARK C.16

The application D0 (]a, b[; X) → D 0 (]a, b[; X) d f 7→ f dt

is continuous. REMARK C.17 derivatives, with

This definition can be extended to the higher order h

di di f, ϕi = −hf, i ϕi, i dt dt

∀ ϕ ∈ D(]a, b[), and i ∈ N.

in X

(C.25)

The Fourier transform

C.8

263

Vector valued Lebesgue spaces

Let X be a Banach space and ]a, b[ an open set of R. The measure dt stands for the Lebesgue measure on ]a, b[. DEFINITION C.10 of functions

We denote by Lp (a, b; X) for 1 ≤ p < +∞ the space ]a, b[ → X t 7→ f (t)

such that (i) f is (strongly) dt-measurable on ]a, b[, (ii) def

kf kLp (a,b;X) = DEFINITION C.11

Z

b a

kf (t)kpX

dt

!1/p

< +∞

(C.26)

We denote by L∞ (a, b; X), the space of functions ]a, b[ → X t 7→ f (t)

such that (i) f is (strongly) dt-measurable on ]a, b[, (ii) f is bounded almost everywhere over ]a, b[ and we set kf kL∞ (a,b;X) =

inf

dt−meas{t∈]a,b[:kf (t)kX >C}=0

C

(C.27)

THEOREM C.18 [16] For 1 ≤ p ≤ ∞, the space Lp (a, b; X) is a Banach space if X is a Banach space. It is a Hilbert space if X is a Hilbert space. In the applications, the following lemmas will be of great use, LEMMA C.1 Let X, Y be two Banach spaces and u ∈ L1 (a, b; X) and A ∈ L (X, Y ), then 1. A u ∈ L1 (a, b; Y ),

264 2.

Rb a

A u(t) dt = A

Moving Shape Analysis b u(t) dt . a

R

LEMMA C.2 1. Let ]a, b[ be an open set of R, u ∈ L1 (a, b; X) and φ ∈ X 0 , then hφ,

Z

a

b

u(t) dti =

Z

b a

hφ, u(t)i dt

(C.28)

2. For |a|, |b| < ∞, the equality (C.28) is valid for u ∈ Lp (a, b; X) and p ≥ 1.

Appendix D Sobolev spaces

D.1

Spaces H m (Ω)

Let Ω be an open set in Rd . We introduce the Sobolev space of integer regularity m ∈ N, def H m (Ω) = u ∈ D 0 (Ω) : Dα u ∈ L2 (Ω), |α| ≤ m (D.1) THEOREM D.1 The space H m (Ω) endowed with the scalar product X Z def Dα u(x)Dα v(x) dx (u, v)H m (Ω) = |α|≤m

(D.2)

Ω

and the associated norm 

kukH m (Ω) = 

X

|α|≤m

is a Hilbert space.

1/2

k Dα uk2L2 (Ω) 

(D.3)

THEOREM D.2 If m1 > m2 the space H m1 (Ω) ,→ H m2 (Ω) with continuous injection. In the case Ω = Rd , it is possible to define equivalently the space H m (Rd ) thanks to the Fourier transform. THEOREM D.3 Let m ∈ N, then

(i) H m (Rd ) ⊂ S 0 (Rd ).

(ii) The space H m (Rd ) coincides with following space n o u ∈ S 0 (Rd ) : (1 + |ξ|2 )m/2 u ˆ ∈ L2 (Rd ) 265

266

Moving Shape Analysis where u ˆ is the Fourier transform of u defined in section 1.4.4.

(iii) The norm k.kH m (Rd ) is equivalent to the following norm, Z

2 m

Rd

2

(1 + |ξ| ) |ˆ u(ξ)| d ξ

1/2

REMARK D.1 The proof is based on the identity F (Dα u) = (iξ)α F u and the following inequality that holds ∀ 0 ≤ |α| ≤ m,  m   X X X Y |ξj |2αj ≤ 1 + |ξj |2  ≤ C 1 + |ξj |2αj  1≤j≤m

1≤j≤m

0<|α|≤m 1≤j≤m

This theorem furnishes a new definition of the space H m (Rd ) that we shall extend to real parameter m in the next section.

D.2

The space H s (Rd )

For s ∈ R we define the following Sobolev space, n o def H s (Rd ) = u ∈ S 0 (Rd ) : (1 + |ξ|2 )s/2 u ˆ(ξ) ∈ L2 (Rdξ )

endowed with the following scalar product Z def (1 + |ξ|2 )s u ˆ(ξ) vˆ(ξ) d ξ (u, v)H s (Rd ) = d R

(D.4)

(D.5)

and the associated norm, kukH s (Rd ) =

Z

2 s

Rd

2

(1 + |ξ| ) |ˆ u(ξ)| d xi

1/2

(D.6)

THEOREM D.4 For s ∈ R, the space (H s (Rd ), (., .)H s (Rd ) ) is a Hilbert space. Furthermore the continuous injection H s1 (Rd ) ,→ H s2 (Rd ) holds for s1 ≥ s2 .

REMARK D.2 The proof of completeness uses the fact that F ∈ ISOM(S 0 (Rd )) and the completeness of L2 (Rd ). The injection property is

Sobolev spaces

267

an easy consequence of the inequality (1 + kξk2 )s2 ≤ (1 + kξk2 )s1 ,

s1 ≥ s2

THEOREM D.5 The space H s (Rd ) is a normal space of distributions1 .

D.3

The topological dual of H s (Rd )

In this section we shall characterize the space (H s (Rd ))0 of continuous linear forms on H s (Rd ). We already have the following inclusions (H s (Rd ))0 ⊂ S 0 (Rd ) ⊂ D 0 (Rd ) since D(Rd ) is dense in S (Rd ) which is dense in H s (Rd ). THEOREM D.6 ∀ s ∈ R, the dual space of H s (Rd ) coincides (algebraically and topologically ) in D0 (Rd ) with H −s (Rd ).

D.4

Sobolev embedding theorems

THEOREM D.7 Let s ∈ R and k ∈ N satisfying s > d/2 + k. Then the continuous injection H s (Rd ) ,→ B0 (Rd )

(D.7)

holds true. REMARK D.3 The proof is based on Theorem (C.1) and (1 + |ξ| 2 )s/2 ∈ L2 (Rdξ ) for s > d/2. REMARK D.4 1 i.e.,

The condition s > d/2 + k is optimal.

D(Rd ) is dense in H s (Rd ).

268

Moving Shape Analysis

THEOREM D.8 For s > d/2, the Hilbert space H s (Rd ) is an algebra. REMARK D.5

Using Theorem(A.1) we have

F (u v) = F u ? F v and using w(ξ) = (1 + |ξ|2 )s ≤ 4s (1 + |ξ − η|2 )s + (1 + |η|2 )s , for s ≥ 0, we get (1 + |ξ|2 )s |F (u v)(ξ)| ≤ C |(w |F u|) ? v| + C |F u| ? (w F v) ∈ L2 (Rd ) for s ≥ d/2. THEOREM D.9 For s ∈ R, if u ∈ H s (Rd ) and ϕ ∈ S (Rd ) then ϕ u ∈ H s (Rd ). REMARK D.6

Again we use F (u ϕ) = F u ? F ϕ

and the inequality (1 + |ξ|2 )s ≤ 2|s| (1 + |η|2 )s (1 + |ξ − η|2 )|s| . THEOREM D.10 The continuous injection H s (Rd ) ,→ Lp (Rd ),

∀2 ≤ p ≤ 2d/(d − 2s)

(D.8)

holds for s < d/2. REMARK D.7 The Lebesgue indice can be deduced from homogeneity arguments. We set vλ (x) = v(λ x) and we have kvλ kLp = λ−d/p kvkLp

with |v|H s

|vλ |H s = λ(−d+2s)/2 |v|H s 1/2 def R 2s 2 |ˆ v (ξ)| d ξ , from which we deduce 1/p = 1/2−s/d. = d |ξ| R

REMARK D.8 u ∈ H s as

The proof is based on a decomposition of the function u = uL,A + uH,A

Sobolev spaces

269

with the low frequency part uL,A = F −1 (χB(0,A) u ˆ) and the high frequency part uH,A = F −1 (χ{B(0,A) u ˆ). We use also the characterization kukpLp = p

Z

∞

λp−1 meas(|u| > λ) dλ

0

and the Bienaym´e-Tchebytchev inequality meas(|u| > λ) ≤ λ−1 kuk2L2

THEOREM D.11 The following continuous injection H s (Rd ) ,→ Lp (Rd ),

2 ≤ p < 2d/(d − 2)

(D.9)

holds for s = d/2.

D.5

Density properties

We already saw in Theorem (D.5) that D(Rd ) is dense in H s (Rd ) for s ∈ R. In this section, we deal with the case where Ω R d . THEOREM D.12 Let k ∈ N∗ and Ω a domain of class C k . The space D(Ω) is dense in H k (Rd ). REMARK D.9 The proof is first performed for Rd+ by truncation and regularization. In the general case, we may assume that Ω is compact and use local maps (Definition(A.1)) and partition of unity. We define the space H0m (Ω) = D(Ω)

H m (Ω)

(D.10)

270

D.6

Moving Shape Analysis

Trace theorem for H s (Rd )

DEFINITION D.1

The linear application

γ0 : S (Rd ) → S (Rd−1 ) u 7→ x0 7→ γ0 u(x0 ) = u(x0 , xd = 0) stands for the trace operator of order 0. THEOREM D.13 For s > 1/2, (i) The linear mapping γ0 is continuous from S (Rd ) endowed with the topology of H s (Rd ) into S (Rd−1 ) endowed with the topology of H s−1/2 (Rd−1 ). (ii) This mapping can be extended to a continuous linear mapping from H s (Rd ) into H s−1/2 (Rd−1 ), again denoted γ0 . Furthermore γ0 is surjective. REMARK D.10 We use the partial Fourier transform (Fx0 u)(ξ 0 , xd ) of u(x0 , xd ) in the direction x0 ∈ Rd−1 .

D.7

Trace theorem for H m (Rd+ )

DEFINITION D.2 γj :

The linear application,

D(Rd+ )

→ D(Rd−1 ) u 7→ x0 7→ (γj u)(x0 ) = ∂xj u(x0 , xd )|xd =0 d

stands for the trace operator of order j. THEOREM D.14 The following trace mapping can be defined on D(Rd+ ) and extended by density to a continuous linear surjective application. Y H m−j−1/2 (Rd−1 ) γm : H m (Rd+ ) → 0≤j≤m−1

u 7→ (γj u)1≤j≤m−1

Sobolev spaces

271

and ker γm = H0m (Rd+ ).

D.8

The space H s (Γ)

There exists different methods to define Sobolev spaces on submanifolds of codimension 1 in Rd . Here we may consider a bounded domain Ω ⊂ Rd with compact boundary Γ of class C ∞ . • Definition by localisation and diffeomorphism : We use local maps (Uj , hj )1≤j≤m and partition of unity {rj }1≤j≤m to define functional spaces on Γ. Using the result of Lemma (A.6), we define the function (rj u) ◦ hj (ζ 0 , 0), ζ 0 ∈ B0 (θj u)(ζ 0 ) = (D.11) 0, ζ 0 ∈ {B0 DEFINITION D.3 We define the Sobolev space of fractional order on the C ∞ manifold Γ by n o def H s (Γ) = u ∈ D 0 (Γ) : θj u ∈ H s (Rd−1 ), 1 ≤ j ≤ m (D.12) THEOREM D.15 1. The space H s (Γ) provided with the norm, 

kukH s (Γ) = 

X

1≤j≤m

kθj uk2

H s (R

d−1

is a Hilbert space.

)

1/2 

(D.13)

2. The space D(Γ) is dense in H s (Γ) for s ≥ 0. def

3. Using the space L2 (Γ) = H 0 (Γ) as the pivot space, we have the following identification, (H s (Γ))0 ≡ H −s (Γ)

(D.14)

REMARK D.11 The definition of the space H s (Γ) depends neither on the atlas (Uj , hj )1≤j≤m nor on the associated partition of unity (rj )1≤j≤m . The associated norm does depend on the choice of the atlas, but defines a class of equivalent norms.

272

Moving Shape Analysis

• Definition as the domain of tangential operator : We consider the Laplace-Beltrami operator ∆Γ on the manifold Γ. Considered as a positive definite self-adjoint operator in L 2 (Ω) with domain D(−∆Γ ), we define the space def

H s (Γ) = D((−∆Γ )s/2 ),

s∈R

(D.15)

provided with the norm of the graph kvkH s (Γ) = (kvk2L2 (Γ) + k(−∆Γ )s/2 vk2L2 (Γ) )1/2

D.9

(D.16)

Trace theorem in H m (Ω)

THEOREM D.16 Let Ω be a bounded domain in Rd with compact boundary Γ of class C k . Let m ≤ k, then the mapping γm defined on D(Ω) with values in [D(Γ)]m can be extended by density to a continuous linear surjection : Y γm : H m (Ω) → H m−j−1/2 (Γ) 0≤j≤m−1

u 7→ (γj u)1≤j≤m−1

and ker γm = H0m (Ω).

D.10

Extension theorems

DEFINITION D.4 An open set Ω ⊂ Rd has the m-extension property for m ∈ N, if there exists an operator P such that (i) P ∈ L (H m (Ω), H m (Rd )),

(ii) P u = u,

a.e. in Ω,

∀ u ∈ H m (Ω).

We may use a general procedure that may allow to define m-extension operators by density, LEMMA D.1 Let V ⊂ H m (Ω) be a dense subspace and an operator P such that

Sobolev spaces

273

(i) P ∈ L (V, H m (Rd )), (ii) P u = u,

a.e. in Ω,

∀u ∈ V .

Then P has a unique extension into an operator of m-extension from H m (Ω) into H m (Rd ). LEMMA D.2 def Let Ω, Ω1 , Ω2 ⊂ Rd and S = Ω1 ∩ Ω2 of class C m+1 such that Ω = Ω1 ∪ S ∪ Ω 2 Let u ∈ C m (Ω) such that u|Ωi ∈ D(Ωi ),

i ∈ {1, 2}

then u ∈ H m+1 (Ω) REMARK D.12 If we set vi |Ωj = ∂xi (u|Ωj ) for 1 ≤ i ≤ d and j ∈ {1, 2}, the following identity holds Z Z hvi , ϕi = [u] ϕ · ni − u ∂xi ϕ, ∀ ϕ ∈ D(Ω) S

Ω

where [u] is the saltus of u across S. THEOREM D.17 The open set Rd+ possesses the m-extension property for m ∈ N. REMARK D.13 For m = 0, we only use the extension by zero. For m ≥ 1, we use the extension by reflexion (Babitch extension), u(x0 , xd ), xd ≥ 0 0 (P u)(x , xd ) = P 0 α u(x , −k x ), x d d <0 1≤k≤m k

where we choose (αk )1≤k≤m such that P u ∈ C m−1 (Ω) which impose continuity X of derivatives with respect to xd up to order m − 1. This implies (−k)j αk = 1, 0 ≤ j ≤ m − 1 which is a unique solvable Van der 1≤k≤m

Monde system. We conclude using Lemma (D.2).

THEOREM D.18 Let Ω ⊂ Rd be a bounded open set with boundary Γ of class C k . Then Ω possesses the m-extension property with 0 ≤ m ≤ k.

274

Moving Shape Analysis

DEFINITION D.5 For s ∈ R, we define the space n o def H s (Ω) = u ∈ D 0 (Ω), ∃U ∈ H s (Rd )such thatu = U |Ω

(D.17)

provided with the quotient norm

def

kukH s (Ω) =

inf kU kH s (Rd ) U ∈ H s (Rd ) U |Ω = u

(D.18)

THEOREM D.19 If Ω has the m-extension property with m ∈ N, then H s (Ω) ≡ H m (Ω)

D.11

The space H −m (Ω)

DEFINITION D.6 space of H0m (Ω).

For m ∈ N, we define the space H −m (Ω) as the dual

THEOREM D.20 (i) The space H −m (Ω) endowed with the norm def

kF kH −m (Ω) =

sup kukH m (Ω) ≤ 1 u 6= 0

|hF, ui| kukH m (Ω)

is a Hilbert space. (ii) For m1 > m2 , H −m2 (Ω) ,→ H −m1 (Ω). DEFINITION D.7 For m ∈ N, we define the space of distributions     X def Em = T ∈ D0 (Ω) : T = Dα fα , fα ∈ L2 (Ω) (D.19)   |α|≤m

endowed with the topology induced by kT k2 =

inf 2 fP α ∈ L (Ω) T = |α|≤m Dα fα

X

|α|≤m

kfα k2

Sobolev spaces

275

THEOREM D.21 The space Em coincides algebraically and topologically with H −m (Ω). REMARK D.14 This is a consequence of the Hahn-Banach theorem allowing to extend linear forms on a sub-space into the whole Banach space and the Riesz representation theorem in Hilbert spaces. THEOREM D.22 The operator −∆ + I ∈ ISOM(H01 (Ω), H −1 (Ω)).

D.12

Compact embeddings

THEOREM D.23 Rellich Let Ω ⊂ Rd be a bounded open set. Then for m ∈ N, the injection H0m+1 (Ω) ,→ H0m (Ω) is compact. THEOREM D.24 Let Ω ⊂ Rd be a bounded open set possessing the (m+1)-extension property. Then the injection H m+1 (Ω) ,→ H m (Ω) is compact.

D.13

Poincar´ e inequalities

DEFINITION D.8 Let ζ ∈ Rd with |ζ| = 1, ]a, b[⊂ R and 0 < e < ∞ such that meas(]a, b[) = e. We define the band of Rd of width e in the direction ζ by n o def Be (ζ) = x ∈ Rd : a < x · ζ < b (D.20) THEOREM D.25 Poincar´ e inequality in H01 (Ω) d Let Ω ⊂ R be an open set bounded in some direction, i.e., such that there

276

Moving Shape Analysis

exists a band Be (ζ) with Ω ⊂ Be (ζ) Then kuk2L2 (Ω) ≤ e2 /2k ∇ uk2L2 (Ω) ,

∀ u ∈ H01 (Ω)

(D.21)

THEOREM D.26 Poincar´ e inequality in H 1 (Ω) Let Ω ⊂ Rd be a bounded open set such that the following compact injection H 1 (Ω) ,→ L2 (Ω) holds. Then we have 2 Z kuk2L2 (Ω) − (meas Ω)−1 u(x) dx ≤ P (Ω)k ∇ uk2L2 (Ω)

(D.22)

Ω

REMARK D.15 If Ω has the 1-extension property then the compact injection holds. We call Nikodym open sets the sets for which the Poincar´e inequality holds. REMARK D.16 The Poincar´e constant P (Ω) = 1/λ2 where λ2 is the first non-zero eigenvalue of the Neumann problem, −∆u = λu, Ω ∂n u = 0, Γ

THEOREM D.27 Let Ω ⊂ Rd be a bounded open set with regular boundary Γ. Then there exists C > 0 such that kuk2L2 (Ω) ≤ C k ∇ uk2L2 (Ω) + kγ0 uk2L2 (Γ) , ∀ u ∈ H 1 (Ω) (D.23)

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Index

Navier-Stokes, 4, 8, 14, 110, 178, 187, 216 noncylindrical heat, 35 noncylindrical wave, 98 nonlinear elastodynamic, 217 ODE, 187 partial differential, 3 strong convection, 50 weak convection, 51

Aeroelastic stability, 5 Boundary control, 9 Dirichlet, 8 free, 4, 33 kinematic continuity, 187, 218 kinetic continuity, 218 noncylindrical, 19 value problem, 1

Fluid drag, 9, 14 incompressible, 4, 110 Function characteristic, 3, 50 cost, 10 cost gradient, 11 Gˆ ateaux differentiable, 24 minimization, 10 oriented distance, 2, 119 without steps, 102 Functional lagrangian, 12, 144, 159, 195, 222 tracking, 18

Condition transpiration, 178 viability, 14, 50 Control active, 9 feedback law, 9 optimal, 180 passive, 1, 9 piezoelectrical device, 4 Domain deformation, 2 moving, 2, 3 perturbation, 13, 38 tube, 19, 35, 38, 60 weak convection, 49

Geometry additive curvature, 120 curve, 77 density perimeter, 61 intrinsic, 2 mean curvature, 17, 72, 119 moving, 4 perimeter, 23, 55 perturbation, 3 tangential calculus, 119

Equation adjoint elastodynamic, 220 adjoint Navier-Stokes, 11, 139, 188, 198, 219 adjoint ODE, 199 Euler-Lagrange, 103 Hamilton-Jacobi, 18, 101 Laplace, 90 linearized Navier-Stokes, 11

Interface motion, 4, 5, 33, 216

289

290 Mapping diffeomorphism, 3 lagrangian, 34 projection, 119 transverse, 23, 39 Method Galerkin, 51 identity perturbation, 2, 5, 13, 165 level set, 18, 101 speed, 3, 13, 39 Model fluid-solid, 4, 215 fluid-structure, 5, 185 St Venant-Kirchoff, 217 Stefan, 33 transverse magnetic wave, 105 wave, 98

Index

asymptotic limit, 89 calculus, 114, 165 continuity, 85 control, 1 derivative, 17 derivative structure theorem, 15, 16 differentiable, 16 differentiable equation, 84 differential equation, 2, 18, 97 eulerian derivative, 14, 16 eulerian material derivative, 24, 40, 44, 73 eulerian partial derivative, 44, 46, 73, 133 eulerian shape derivative, 24 extractor identity, 136 kernel identity, 154, 207 lagrangian material derivative, 168 Operator lagrangian partial derivative, 168 adjoint trace, 15 material derivative, 16 duality, 83 min-max, 140, 195, 222 Lie bracket, 24 moving analysis, 3–5 noncylindrical gradient, 70 noncylindrical analysis, 3 observation, 36 optimization, 1, 2 tangential divergence, 17, 121 space embedding, 157 tangential gradient, 17, 120 space parametrization, 146 tangential Laplace-Beltrami, 121 topological derivative, 105 trace, 68 transerve field, 199 Optimization transverse equation, 24, 41 first-order optimality, 10, 144, 197 transverse field, 23, 39, 68, 175 ill-posed, 36 tube derivative, 66 inverse problem, 5, 36, 190 velocity field, 34 lagrangian multiplier, 10, 158, 194, weak Piola derivative, 124 221 Space regularization, 22, 55 Hilbert, 10, 46 Tikhonov, 36 Sobolev, 2 special bounded variation, 2 Regularity lipschitz, 13 Theorem Ascoli, 88 Shape Banach-Steinhaus, 57 adjoint transverse equation, 25, Helly, 58 47 adjoint transverse field, 25, 47, Lebesgue dominated convergence, 69, 188 58

Index Leray-Schauder, 88 Min-Max, 147 weak implicit function, 128 Topology, 3 BV, 22, 51 change, 5 Hausdorff, 18, 62, 89

291