Table Of ContentISNM
International Series of Numerical Mathematics
Vol.147
Managing Editors:
K.-H. Hoffmann, Bonn
D. Mittelmann, Tempe
Associate Editors:
R. E. Bank, La Jolla
H. Kawarada, Chiba
R. J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J. Todd, Pasadena
Free Boundary Problems
Theory and Applications
Pierluigi Colli
Claudio Verdi
Augusto Visintin
Editors
Springer Basel AG
Editors:
Pierluigi Colli Augusto Visintin
Dipartimento di Matematica Dipartimento di Matematica
Universitä di Pavia Via Sommarive, 14
Via Ferrata, 1 38050 Povo di Trento
27100 Pavia Italy
Italy e-mail: [email protected]
e-mail: [email protected]
Claudio Verdi
Dipartimento di Matematica
Universitä di Milano
Via Saldini, 50
20133 Milano
Italy
e-mail: [email protected]
2000 Mathematics Subject Classification 35R35, 35Qxx, 35Dxx
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
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in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-0348-9613-9
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© 2004 Springer Basel AG
Originally published by Birkhäuser Verlag,Basel in 2004
Softcover reprint of the hardcover 1st edition 2004
Printed on acid-free paper produced of chlorine-free pulp. TCF °°
ISBN 978-3-0348-9613-9 ISBN 978-3-0348-7893-7 (eBook)
DOI 10.1007/978-3-0348-7893-7
98765432 1 www.birkhasuer- science. com
Contents
Preface ...................................................................... vii
Gregoire Allaire, Frantyois Jouve and Anca-Maria Toader
Structural Optimization by the Level-Set Method ........................ 1
Luigi Ambrosio and Simon Masnou
On a Variational Problem Arising in Image Reconstruction ............. 17
Daniele Andreucci, Giovanni Caruso and Emmanuele DiBenedetto
Ill-Posed Hele-Shaw Flows ............................................. 27
Eberhard Biinsch, Pedro Morin and Ricardo H. Nochetto
Finite Element Methods for Surface Diffusion ........................... 53
Martin Burger
Crystal Growth and Impingement in Polymer Melts .................... 65
Vincenzo Capasso, Ramon Escobedo and Claudia Salani
Moving Bands and Moving Boundaries in an Hybrid Model
for the Crystallization of Polymers ..................................... 75
Zhiming Chen and Xingye Vue
Upscaling of Well Singularities in the Flow Transport
through Heterogeneous Porous Media ................................... 87
Pierre Degond, Celine Parzani and Marie-Helene Vignal
On plasma expansion in vacuum ...................................... 103
Wolfgang Dreyer and Frank Duderstadt
Towards the thermodynamic modeling of nucleation and growth
of liquid droplets in single crystals ..................................... 113
Joachim Escher and Kazuo Ito
On the Intermediate Surface Diffusion Flow ........................... 131
Antonio Fasano, Alberto Mancini and Riccardo Ricci
Solid Core Revisited ................................................... 139
Takesi Fukao, Nobuyuki Kenmochi and Irena Pawlow
Transmission-Stefan Problems Arising in Czochralski Process
of Crystal Growth ..................................................... 151
Martin E. Glicksman, Afina Lupulescu and Matthew B. Koss
Quasi-static Melting of Crystals: Experiments and Analysis ............ 167
VI
Reiner Henseler, Barbara Niethammer and Felix Otto
A reduced model for simulating grain growth .......................... 177
Danielle Hilhorst, Masayasu Mimura and Remi Weidenfeld
On a Reaction-Diffusion System for a Population of Hunters
and Farmers .......................................................... 189
Alessandra Micheletti and Vincenzo Capasso
The Stochastic Geometry of the Crystallization Process of Polymers ... 197
Ricardo H. N ochetto
A Posteriori Error Control of Free Boundary Problems ................ 207
Matteo N ovaga
The Total Variation Flow ............................................. 225
Oliver Penrose and J. W. Cahn
A Mathematical Model for Diffusion-induced Grain Boundary Motion .237
Mario Primicerio and Boris Zaltzman
Continuation of the Solution to the Chemotaxis Problem
Beyond its Blow-up ................................................... 255
Fernando Reitich
Shape Deformations and Analytic Continuation in
Free Boundary Problems .............................................. 265
Giuseppe Savare
Error Estimates for Dissipative Evolution Problems ................... 281
Alfred Schmidt
A Multi-mesh Finite Element Method for 3D Phase Field Simulations . 293
Andres Sole, Vicent Caselles, Guillermo Sapiro, and Francisco Anindiga
Morse Description and Geometric Encoding of Digital Elevation Maps . 303
Victor N. Starovoitov
Behavior of a Rigid Body in an Incompressible Viscous Fluid
Near a Boundary ...................................................... 313
Stephen J. Watson
Crystal Growth, Coarsening and the Convective
Cahn-Hilliard Equation ............................................... 329
List of Participants ......................................................... 343
Preface
This volume collects the proceedings of a conference on free boundary problems
that took place in Trento (Italy) in June 2002.
Many phenomena of interest for applications are represented by differential
equations which are defined in a domain whose boundary is a priori unknown, and
is accordingly named a "free boundary"; a further quantitative condition is then
provided in order to exclude indeterminacy. Free boundaries are often related to
discontinuities in constitutive relations; this raises a number of interesting math
ematical issues: existence of solutions in function spaces, uniqueness, regularity
properties, numerical approximation procedures, and other questions have been
extensively investigated.
Examples of free boundary problems include the classical Stefan problem and
more general models of phase transitions: here the free boundary is represented
by the moving interface between the phases. Free boundaries also occur as fronts
between saturated and unsaturated regions in filtration through porous media,
between plastic and elastic phases in continuous mechanics, between magnetic
domains in ferromagnetism, just to mention a few cases. Relevant examples also
come from reaction-diffusion, fluid dynamics, biomathematics, and so on.
Several of these problems are of interest for applications in industry and offer
opportunities of collaboration among mathematicians, physicists, engineers, ma
terial scientists, biologists, and other researchers. A large community of applied
scientists spread over the world has been formulating and studying these prob
lems for many years. They have also been meeting in major conferences every
three years or so; the first appointment was held at Montecatini in 1981, and was
followed by Maubuissons, Irsee, Montreal, Toledo, Zakopane, Crete, Chiba. The
next conference is planned for 2005 in Portugal. The proceedings of these con
ferences provide a comprehensive picture of the development of research on free
boundary problems in the last decades.
The Trento conference saw the participation of more than 150 scientists,
coming from a long list of countries. Talks included twenty plenary addresses, and
seven focus sessions were devoted to selected topics: free boundary problems in
polymer science, image processing, grain boundary motion, numerical aspects of
free boundary problems, free boundary problems in biomathematics, modelling in
crystal growth, and transitions in anisotropic materials. A poster session open to
the contribution of all participants was also organized. Post-dinner entertainment
included two concerts performed by professional and semi-professional musicians.
Following the tradition of the series, the contributions concern problems
which are either directly related to free boundaries, or may be so in perspec
tive. Special emphasis was put on interdisciplinarity and on issues of applicative
relevance. This variety is obviously reflected in these proceedings, which are likely
to meet the interests of a large spectrum of readers.
Vlll Preface
The editors thank the co-organizers C. Dziuk, A. Fasano, K.-H. Hoffmann,
and J. Sprekels. Scientific assistance by the other members of the scientific com
mittee, M. Fremond, A. Friedman, S. Luckhaus, M. Niezg6dka, J. Ockendon, M.
Primicerio, J.F. Rodrigues, and the collaboration of M. Paolini, are also grate
fully acknowledged; R. Rossi also carefully contributed to the revision of these
proceedings.
The conference was financially supported by the Istituto Nazionale di Alta
Matematica (through C.N.A.M.P.A., C.N.F.M., C.N.C.S.), by three national re
search projects of the Italian Ministry for University, by the University of Trento,
by the C.A.E.S.A.R. Institut (Bonn), and by the Weiestrass Institut (Berlin).
Pavia, Milano, Trento, May 2003 P. Colli, C. Verdi, A. Visintin
International Series of Numerical Mathematics, Vol. 147, 1-15
© 2003 Birkhauser Verlag Basel/Switzerland
Structural Optimization by the
Level-Set Method
Gregoire Allaire, Fran<;ois Jouve, and Anca-Maria Toader
Abstract. In the context of structural optimization, we describe a new numer
ical method based on a combination of the classical shape derivative and of
the level-set method for front propagation. We have implemented this method
in two and three space dimensions for models of linear or non-linear elastic
ity, with various objective functions and constraints on the volume or on the
perimeter. The shape derivative is computed by an adjoint method. The cost
of our numerical algorithm is moderate since the shape is captured on a fixed
Eulerian mesh. Although this method is not specifically designed for topology
optimization, it can easily handle topology changes.
1. Introduction
Shape optimization of elastic structures is a very important and popular field.
The classical method of shape sensitivity (or boundary variation) has been much
studied (see e.g. [13], [16], [20], [21]). It is a very general method which can han
dle any type of objective functions and structural models, but it has two main
drawbacks: its computational cost (because of remeshing) and its tendency to fall
into local minima far away from global ones. The homogenization method (see
e.g. [1], [2], [5], [7], [8], [12]) is an adequate remedy to these drawbacks but it is
mainly restricted to linear elasticity and particular objective functions (compli
ance, eigenfrequency, or compliant mechanism). Yet, another method has recently
been proposed in [14], [18], [4J relying on the level-set method which has been
devised by Osher and Sethian [15], [17J for numerically tracking fronts and free
boundaries. The level-set method is versatile and computationally very efficient:
it is by now a classical tool in many fields such as motion by mean curvature, fluid
mechanics, image processing, etc.
The paper [14J studied a two-phase optimization of a membrane (modelled by
a linear scalar partial differential equation), i.e. the free boundary was the interface
between two constituents occupying a given domain. It combined the level-set
method with the shape sensitivity analysis framework. On the other hand, the work
[18J focused on structural optimization within the context of two-dimensional linear
elasticity. In that setting, the free boundary was the shape of the structure, which
was captured on a fixed mesh using the immersed interface method. However,
2 G. Allaire, F. Jouve, and A. Toader
[18] did not rely on shape sensitivity analysis: rather, the structural rigidity was
improved by using an ad hoc criterion based on the Von Mises equivalent stress.
In [4], we have generalized these two previous works in several aspects. We
give here a brief review of our approach, based on a systematic implementation of
the level-set method in which the front velocity is derived from a shape sensitivity
analysis. We focus on shape optimization rather than two-phase optimization,
and we replace the immersed interface method by the simpler "ersatz material"
approach which amounts to filling the holes by a weak phase. This is a well
known approach in topology optimization, and it can be rigorously justified in
some cases [1]. We compute a shape derivative by using an adjoint problem. Then,
the shape derivative is used as the normal velocity of the free boundary which is
moved during the optimization process. Front propagation is performed by solving
a Hamilton-Jacobi equation for a level-set function.
Acknowledgements: This work has been supported by the grant CNRSjICCTI n°
2002-12163 of the Centre National de la Recherche Scientifique (France) and the
Instituto de Coopera<;ao Cientffica e Tecnologica Internacional (Portugal).
2. Setting of the problem
We start by describing a model problem in linearized elasticity. There is no concep
tual difficulty in choosing another model, and in particular a nonlinear elasticity
problem. Let n c JR.d (d = 2 or 3) be a bounded open set occupied by a li
near isotropic elastic material with Hooke's law A. Recall that, for any symmetric
matrix ~, A is defined by
A~ = 2Jl~ + )'(Tr~) Id,
where Jl and), are the Lame moduli of the material. The boundary of n is made
up of two disjoint parts
an = rN ur
D,
with Dirichlet boundary conditions on rD, and Neumann boundary conditions on
r N. The two boundary parts r D and r N are allowed to vary in the optimization
process, although it is possible to fix some portion of it (see the numerical examples
below).
We denote by ! (g) the vector-valued function of the volume forces (of the
n
surface loads, respectively). The displacement field u in is the solution of the
linearized elasticity system
n
-div(Ae(u))=! in
{ u=O on rD (1)
r
(Ae(u))n = g on N.
Since n is varying during the optimization process, ! and g must be known for
every possible configuration of n. We therefore introduce a working domain D (a
bounded open set of JR.d) which contains all the admissible shapes n.
