Shape Optimization by the Homogenization Method

469 Pages·2002·11.659 MB·English

by  Grégoire Allaire (auth.)

Applied Mathematical Sciences
Volume 146
Editors
S.S. Antman J.E. Marsden L. Sirovich
Advisors
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J. Keener J. Keller
BJ. Matkowsky A. Mielke
C.S. Peskin K.R.S. Sreenivasan
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Gregoire Allaire
Shape Optimization by the
Homogenization Method
With 54 Illustrations
Springer
Gregoire Allaire
Centre of Applied Mathematics
Ecole Polytechnique
Paliseau Cedex 91128
France
[email protected]
Editors
Stuart S. Antman J. E. Marsden
Department of Mathematics Control and Dynamical Systems
and Mai! Code 107-81
Institute for Physical Science and Technology Califomia Institute of Technology
University of Maryland Pasadena, CA 91125
College Park, MD 20742-4015 USA
USA
L. Sirovich
Division of Applied Mathematics
Brown University
Providence, RI 02912
USA
Mathematies Subjeet Classifieation (2000): 35B27, 49J45, 49K20, 65K10, 74Q20
Library of Congress Cataloging-in-Publieation Data
Allaire, Gregoire.
Shape optimization by the homogenization methodlGregoire Allaire.
p. em. - (Applied mathematieal seienees; v. 146)
Includes bibliographical referenees and index.
ISBN 978-1-4419-2942-6 ISBN 978-1-4684-9286-6 (eBook)
DOI 10.1007/978-1-4684-9286-6
I. Struetural optimization. 2. Homogenization (Differential equations) I. Tide.
II. Applied mathematical seienees (Springer-Verlag New York, Inc.); v. 146.
QA1.A647
[TA658.8]
624.l-de21 2001032845
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© 2002 Springer-Verlag New York, Inc.
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Preface
The topic of this book is homogenization theory and its applications to
optimal design in the conductivity and elasticity settings. Its purpose is to
give a self-contained account of homogenization theory and explain how it
applies to solving optimal design problems, from both a theoretical and a
numerical point of view. The application of greatest practical interest tar
geted by this book is shape and topology optimization in structural design,
where this approach is known as the homogenization method.
Shape optimization amounts to finding the optimal shape of a domain
that, for example, would be of maximal conductivity or rigidity under some
specified loading conditions (possibly with a volume or weight constraint).
Such a criterion is embodied by an objective function and is computed
through the solution of astate equation that is a partial differential equa
tion (modeling the conductivity or the elasticity of the structure). Apart
from those areas where the loads are applied, the shape boundary is al
ways assumed to support Neumann boundary conditions (i.e., isolating or
traction-free conditions). In such a setting, shape optimization has a long
history and has been studied by many different methods. There is, therefore,
a vast literat ure in this field, and we refer the reader to the following short
list of books, and references therein [39], [42], [130], [135], [149], [203], [220],
[225], [237], [245], [258]. We shall not consider the case of Dirichlet boundary
conditions (for which the homogenization theory is somehow very different),
referring the reader only to [62], [63], [64], [267], and references therein. Nor
shall we discuss any other types of optimal design problems (like material
properties optimization), even in the cases where homogenization could be
relevant. We note that optimal design may be seen as a special branch of
optimal control, where the control is the domain itself or the coefficients in
the domain (the material properties). There are also many relevant refer
ences in optimal control, and more specifically in controllability, of so-called
distributed systems (i.e., for which the state equation is a partial differential
equation). In addition to the pioneering work of Pontryagin [223], we simply
mention the classical books of Lions [164], [165].
In order to shed light on the originality of the homogenization method,
we first briefly recall what can be called the classical method for shape op-
VI Preface
timization. The main idea goes, at least, as far back as Hadamard [129],
and has been furt her developed by many authors, e.g., [39], [72], [203], [220],
[255], [258]. Given an initial guess of the shape, one allows the boundary to
vary along its normal (or, more generally, along any smooth velo city field).
From both a theoretical and numerical pont of view, this approach gives
a convenient framework for applying the calculus of variations. Neverthe
less, the method suffers from two main drawbacks: It requires a smooth
parametrization of the boundary, and it never changes the topology of the
original domain (i.e., its number of holes or connected components). In struc
tural design, this is often a serious drawback, since it is widely acknowledged
that a porous structure of the same weight as a bulky one may be drastically
more efficient. There have been some recent attempts to change the topology
in this framework of shape sensitivity (e.g., the buble method [103], or the
topology gradient [68], [257]), but they are either ad hoc methods, or very
close in spirit to the homogenization method. Let us mention in passing that
topology optimization can also be studied by stochastic approaches, like, for
example, genetic algorithms, simulated annealing, or neural networks (e.g.,
[114], [143]).
The homogenization method was developed in order to remove smooth
ness and topological constraints inherent in the classical method of shape
optimization. Another major difference of the homogenization method is
the class of numerical algorithms it yields. As classical numerical methods
are shape-tracking algorithms (where the shape fits to the mesh, which is
deformed during the iterations), the homogenization-based numerical meth
ods are rat her shape-capturing algorithms (where the shape is captured on
a fixed mesh). Before going to numerical issues, we now describe the main
features of the homogenization approach.
In a first step, we rephrase a shape optimization problem as a two-phase
optimization problem. Thanks to the homogeneous Neumann boundary con
ditions that they support, the holes in the domain may be filled by a very
weak material (a poor conductor or a compliant elastic phase). In the limit
where the properties of this ersatz material go to zero, we recover holes with
Neumann boundary conditions (note that such a limit process is not always
easy to justify rigorously). In other words, shape optimization is just the
degenerate limit of two-phase optimal design, where the interface between
the two phases replaces the boundary of the shape. A precise definition of
two-phase optimization is as follows. In the conductivity setting, given two
conducting materials, the goal is to find the best arrangement of the ma
terials in a fixed domain that maximizes, for example, overall conductivity.
Preface Vll
Similarly, in the elasticity setting, the goal is to find the best way of mixing
two elastic materials yielding the most rigid structure. In both cases, there is
a volume constraint on the best component phase, which may be seen as the
extra cost on the good conductor or the stiff material (which are expected
to be more expensive). Two-phase optimization is also an optimal design
problem of interest, and we study it in great detail.
A key feature of two-phase optimal design problems is that they usually
do not admit a solution in the absence of any smoothness or topological con
straint on the interface or on the shape boundary. The root of this pathology
is that it is often more advantageous having many tiny inclusions of one phase
into the other than just a few big ones. Therefore, any proposed design can
be improved by making small variations of the phases arrangement, and can
not be optimal. This process of making increasingly fine inclusions (or holes)
shows that optimality can be achieved only in the limit of infinitely fine scale
mixt ure of the two phases. Such mixtures are called composite materials and
allow one to define optimal generalized designs. Homogenization is precisely
the right mathematical tool for studying composite materials and defining
generalized optimal designs.
Therefore, in a second step we introduce generalized designs for which
one can prove existence of an optimum and derive necessary conditions of
optimality. This process of enlarging the space of admissible designs in order
to make the problem weIl-posed is called relaxation. A generalized design
is often equivalently called a relaxed, or composite, design. Homogenization
theary is the main ingredient for performing the relaxation of two-phase
optimal design problems. As such, homogenization appears as just a trick
far proving existence theorems, but its importance goes far beyond that,
since it yields new numerical algorithms. EventuaIly, it remains to justify
this relaxation process in the degenerate limit of shape optimization, and to
post-process numerically these optimal composite designs to recover classical
shapes.
The homogenization method in optimal design was initiated by Murat
and Tartar in the late 1970's. Many joined their efforts in the development
of that theory, and the first relevant references are the works of Murat and
Tartar [205], [206], [269], [271] in France; Cherkaev, Lurie and their cowork
ers [115], [172], [178], [177], as weIl as Raitum [227] in Russia; and Kohn and
Strang [152] in the United States. Surprisingly enough, the numerical al
gorithms based on the homogenization method have matured slowly. After
the early contributions of Gibiansky and Cherkaev [115], Glowinski [117],
Goodman, Kohn, and Reyna [119], and Lavrov, Lurie, and Cherkaev [160],
viii Preface
which were mostly restricted to academic problems, it was the paper of Bend
soe and Kikuchi [47] in the late 1980's, which was the first to demonstrate
the efficiency of such methods in shape optimization. After that pioneer
ing contribution, many other works appeared that contributed to make the
homogenization method one of the most popular and efficient methods for
shape and topology optimization (see Chapter 5 and the References).
Although the main motivation of this book is shape optimization, while
homogenization is, to some extent, just a technical tool for implementing op
timization methods, we somehow proceed in reverse order. Indeed, the first
chapter is concerned solely with homogenization theory. Mathematically,
it can be defined as a theory for averaging partial differential equations.
Homogenization has many potential applications, but we consider it only
as a tool for deriving macroscopic or effective properties of microscopically
heterogeneous media. As such it provides a firm basis to the notion of com
posite material obtained by mixing, on a very fine scale, several phase com
ponents. Although this quest ion of averaging and finding effective properties
is very old in physics or mechanics, the mathematical theory of homogeniza
tion is quite recent, going back to the 1970's. We expose the most general
framework, known as the H-convergence, or G-convergence, introdueed by
Spagnolo [260], [261], and generalized by Tartar and Murat [204], [270].
In the framework of homogenization theory, the second chapter focuses
on the study of two-phase composite materials. These composite materials
(obtained as fine mixt ure limits of the two original phases) playa key role in
the homogenization method used for shape optimization: N amely, as gener
alized designs. It is therefore erucial to find the range of all possible effective
properties of such composites. This is called the G-closure problem, and it
was solved in the conductivity setting for two isotropie phases by Murat and
Tartar [205], [274], and Lurie and Cherkaev [175], [176]. Unfortunately, a
similar answer is yet unknown in the elasticity setting. In this latter case,
one can only obtain bounds that must be satisfied by the effective properties.
In their most general form such bounds are called Hashin-Shtrikman energy
bounds since they are derived by using the famous variational prineiple in
troduced by Hashin and Shtrikman [133]. They turn out to be optimal,
i.e., the values of these bounds are exactly attained by special choices of
composites. In particular, optimal composites can be chosen in the dass of
so-called sequential laminates, which will play an important part in shape
optimization.
The third ehapter is devoted to the application of homogenization to
problems of two-phase optimization in conductivity. After introdueing a
Preface IX
precise mathematical framework, we give explicit examples of the nonex
istence of optimal designs in the original space of "classical" designs. It is
therefore necessary to relax the problem by enlarging the space of admissible
designs, i.e., by allowing for generalized designs that are made up of com
posite materials. The key tool of this relaxation process is homogenization
theory. As an additional advantage, the relaxed formulation allows one to
derive optimality conditions that are at the root of numerical algorithms.
Our exposition follows the original work of Murat and Tartar [205J.
The fourth chapter is also concerned with the homogenization method
for optimal design, but in the elasticity setting. Since the G-closure set of
two elastic isotropie phases is still unknown, a rigorous relaxation procedure
is available only for special objective functions including compliance (i.e.,
the work done by the loads). In such a case, the optimization problem
being self-adjoint, the relaxation requires only the knowledge of optimal
energy bounds instead of the full G-closure. Those bounds (called Hashin
Shtrikman bounds) are precisely studied in Chapter 2, where they are shown
to be attained by sequentiallaminates. Therefore, in the relaxation process
the unknown full G-closure set can be replaced by its explicit subset of
sequentiallaminates. EventuaIly, a complete relaxation procedure for shape
optimization (i.e., when one of the phases degenerates to holes) is rigorously
established. Here we follow mainly our work [15], [21], that of Gibiansky
and Cherkaev [115], [116], and that of Kohn and Strang [152J.
FinaIly, the fifth chapter is devoted to numerical issues for the homoge
nization method in optimal design. We mostly focus on shape optimization
for elastic structures, since it is by far the most important application of
the homogenization method. Building upon our knowledge of the relaxed,
or homogenized, formulation, we discuss the two main types of algorithms,
the optimality criteria and gradient methods, as weIl as several numerical
technicalities. In particular, we explain how a numerical procedure, called
penalization, allow one to post-process the optimal generalized designs in or
der to recover classical shapes. This chapter is illustrated by several results
of two-dimensional and three-dimensional computations. Our exposition is
complementary to that of Bendsoe in his book [42], which is a very good
introduction to the numerical aspects of the homogenization method for a
more practically inclined reader.
In the sequel, we do not point out open problems, but it is clear that
the range of applications covered by this book, although very important, is
somehow narrow. Of course, there are many other types of optimal design
problems, apart from shape optimization, that have not yet been attacked
x Preface
with the homogenization method (or that cannot be treated by this method).
It is our hope that this book can serve as a basis for furt her developments
in new directions. The references are not exhaustive, by any means, and so
I apologize to those inadvertently left off of the list.
To conclude this introduction, I want to thank all my coworkers, with
whom I learned so much about shape optimization: S. Aubry, Z. Belhachmi,
E. Bonnetier, G. Francfort, F. Jouve, R. Kohn, V. Lods, and F. Murat. I
benefited from many stimulating discussions with G. Milton and L. Tartar,
that I acknowledge with much pleasure. Special thanks are due to F. Jouve,
who was the corner stone of the development of a three-dimensional shape
optimization code, and to G. Francfort for his comments on an earlier version
of the manuscript. Last but not least, I owe a lot to F. Murat, who had so
much influence on my understanding of homogenization, and to R. Kohn,
who introduced me to optimal design.
Paris, France G REGOIRE ALLAIRE
May 2001