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\chapter*{Preface}
\addcontentsline{toc}{chapter}{Preface}
\markboth{Preface}{Preface}

\paragraph{Optimal Design, Structures,  and Composites}

This book discusses problems of structural optimization.  The
problem is to lay out several materials throughout a given domain
to maximize or minimize an integral functional associated with the
conductive or elastic state of an assembled medium.  We assumed
that several materials are available, and one is asked to arrange
them on the volume of the body of a given shape.
  It turns out that the materials in the optimal body are mixed on an
infinitely fine scale:  The finer the scale, the better the
construction. From an engineering point of view, optimization
problems require the use of composites\index{composite} of given
materials rather than materials singly.

As a rule, an optimal design is made of composites.  Physically
speaking, we use composites in designs because we prefer materials
with properties that are not immediately available but can be
obtained by mixing available materials; such a mixture can be more
suitable than any of the individual ingredients.  For example,
composites assembled of isotropic materials can be anisotropic.
Moreover, they can possess such exotic features as a negative
thermal expansion coefficient, or a negative Poisson
ratio.\index{Poisson coefficient} These and similar unusual
features could be useful for solving optimization problems.

Optimal composites correspond to rapidly oscillating state
variables, such as stresses and strains in elasticity or currents
and fields in conductivity. The oscillation of
 optimal
solutions is  well understood in the theory of one-dimensional
control problems.  In some problems, the solution has to zigzag to
satisfy the optimality requirement. The  functional decreases as
the zigzags become more finely scaled. It is not surprising that
such \X{generalized controls} also appear in the multidimensional
problems of optimal layout of materials; here they correspond to
microstructures of  composites. Investigation of multidimensional
optimization problems requires determination of the geometry of
optimal composite structures.  The one-dimensional analogue of the
problem of the best microstructure is relatively simple because
 the only  way to form a mixture in one dimension is to alternate
materials along the line.




\begin{example}
Let us consider the problem of an optimal inhomogeneous conducting
structure that transforms the given boundary potentials to the
desired boundary currents, as  shown in  \fig{f01cond}P.  Suppose
that one has a set of materials of different isotropic
conductivity and  the layout of materials in the designed domain
must be optimized. Clearly, one can control the boundary currents
by varying the materials' layout, because the variation in
conductivity forces the current out of regions of low conductivity
and attracts it into regions of high conductivity. Careful
consideration shows an additional mechanism of control through the
use of anisotropic materials. The current is controlled and sent
in the desired direction by refraction in an anisotropic
composite.  The last mechanism is specific to multidimensional
problems and  has no one-dimensional analogues.  It shows the
usefulness of the anisotropic composite media assembled of
initially isotropic  materials.
\end{example}
\prefacexfgr{f01cond}{The scheme of a composite structure that transforms the
homogeneous boundary potential into an inhomogeneous boundary
current. The horizontal sides are insulated, and the potential on
each vertical side is constant.  The current lines are shown.  The
inhomogeneity of the current   is caused by the inhomogeneity of
the material layout. The good conductor $ A $ attracts the
current, the bad conductor $ C $ pushes the current away, and the
anisotropic composite $ B $ turns the current in a desired
direction.}{An optimal conducting composite structure}{f01cond}
{0.5}

The use of anisotropy to control a process in a medium is well
known. Observe a skier on a slope. The skier can control the
direction of his motion because the resistance to sliding along
the ski is much less than the resistance to sliding in the
orthogonal direction.  This mechanism allows the skier to traverse
across the slope and  make turns.  Anisotropy is also used to
steer a sailboat in a direction different from the direction of
the wind. When a current of passive particles moves in a medium
due to an applied force field, the anisotropy of the medium plays
a similar control role:  It generates a current in a direction
different from the direction of the force.


In the optimization of elastic designs, we also find intuitive
reasons for using an anisotropic composite rather than isotropic
materials.
\begin{example}
Let us consider  an elastic material that shows maximal stiffness
under some anisotropic external loading; see \fig{f01elast}P. One
would assume that the larger the stress, the more stiffness is
needed to resist.  Therefore, we anticipate that the structure
tends to be stiffer in the direction of a larger stress, even at
the expense of being weakened in the direction of a smaller
stress. Hence, we expect that anisotropic composites with
controllable degrees of anisotropy are more suitable than
isotropic materials for maximization of the stiffness.
\end{example}
\prefacexfgr{f01elast}{The scheme of an elastic structure with cavities\index{optimal cavities}
showing the maximum stiffness under a given loading.  The
intensity of the loading is anisotropic, and so is the
corresponding optimal structure. Greater intensity of the loading
corresponds to the direction of greater stiffness.}{ Anisotropy of
an optimal elastic structure}{f01elast}{0.5}

Generally, structural optimization determines the structure, that
is best adapted to the object of the design and the loading
conditions. The adaptation implies uniform exploitation of the
material. For example, the stiffness optimization is achieved by a
structure that evenly stresses the material inside the structure.
To keep the stress level constant, the fine-scale geometrical
parameters of an optimal structure vary from one point to another.
 Sometimes, one needs to organize the layout in several  length scales to
optimize a structure, as seen in  the structure of  bones, leaves,
airplane wings, or  domes.

\paragraph{Structural Optimization in Engineering and  Mathematics}

In practice, the process of design always includes a mysterious
element:  The designer chooses the shape and materials for the
construction using intuition and experience.  Since ancient times
this technique has proved
 effective,  and for centuries engineering landmarks such as
aqueducts, cathedrals, and ships were all built without
mathematical or mechanical theories.

However, from the time of Galileo and Hooke, engineers and
mathematicians have developed theories to determine stresses,
deflections,  currents and temperature inside structures.  This
information helps in the selection of a rational choice of
structural elements.  Certain principles of optimality are rooted
in common sense.  For example, one wants to equalize the stresses
in a designed elastic construction by a proper choice of the
layout of materials.  The overstressed parts need more
reinforcement, and the understressed parts can be lightened. These
simple principles form a basis for rational construction of
amazingly complicated mechanical structures, like bridges,
skyscrapers, and cars. Still, knowledge of the stresses in a body
is mostly used as a checking tool, parallel with the design
proper, which remains the responsibility of the design engineer.

In the past few decades, it has become possible to turn the design
process into algorithms  thanks to advances in computer
technology.  Large contemporary projects require the use of
computer-aided design systems.  These systems often incorporate
algorithms that gradually improve the initial design by a suitable
variation of design variables, namely, the materials' cost and
layout.  Optimization techniques are used to effect changes in a
design to make it stronger, lighter, or more reliable.

This progress has stimulated an interest in the mathematical
foundations of structural optimization.  These foundations are the
main topic of this book.  The theory of extremal problems is used
to address problems of design.  A design problem asks for  the
best geometry of layouts of different materials in a given domain.
Of course, this approach simplifies (or, as a mathematician would
say, idealizes) the real engineering problem, because questions
such as convenience or cost of manufacturing are not considered.
Analysis of optimal structures allows us to formulate  general
principles of an optimally designed construction.  In particular,
we can extend the intuitive principle of equally stressed
construction to a multidimensional situation and find optimal
structures that are, in a sense, hybrids of  simple mechanisms.

\paragraph{The Purpose of the Book}
A gap exists between mathematical approaches to variational
problems and the practical use of results in structural
optimization, theory of composites, and other engineering
applications. On the one hand, we shall see how mathematicians
develop advanced theories such as quasiconvexity and
$G$-convergence for this purpose.  On the other hand, the
engineering and numerical community develops software for
numerical optimization of complicated structures and successfully
optimizes constructions of airplanes, bridges, and so on.

Progress in the area of numerical approaches\index{numerical
methods} is often ahead of mathematical methods required for an
adequate formulation and rigorous solution to corresponding
optimization problems.  Mathematics deals with its own objectives:
Standards of rigor are higher and models are simpler.  This tends
to make mathematical papers not too exciting for engineers.  As
usual, mathematicians use advanced methods to solve simple
equations, and engineers use simple methods but work with
complicated models. As a result, many practically oriented
researchers are skeptical about the usefulness of refined
mathematical theories.  An opposite tendency, to interpret
abstract mathematical results as  prophecy, is no less risky.

These two approaches should be used in concert, each highlighting
supplementary  ideas of optimal design. I hope to present the
foundations of structural optimization in a sufficiently simple
form to make them available for practical use and to allow their
critical appraisal for improving and adapting these results to
specific models. I also hope that the reader will enjoy the beauty
and elegance of the presented mathematical methods.


Often, mathematical analysis of an optimization problem leads to
``unusual'' solutions that are characterized by fractal geometries
and are hardly suitable for manufacturing. This is acceptable in
the framework of the chosen approach:  We are looking for a
mathematically correct solution, and we accept its features. From
a practical point of view, the emergence of ``strange'' solutions
reveals  certain hidden features of optimality. These solutions
should not be rejected as mathematical
extravagance\index{mathematical extravagance}, but rather should
be understood and interpreted in depth; often, they point to
better solutions that may be approximated with available
resources.

\paragraph{The Contents of the Book}
Let us outline the contents of the five parts of the book.


 \subparagraph{Preliminaries}
 The exposition starts with an introductory
Chapter 1  that discusses instabilities in one-dimensional
variational problems.  Specifically, we study variational problems
with rapidly oscillating solutions and  ways to describe these
solutions.  We also introduce the concept of relaxation of a
nonstable variational problem by replacing the Lagrangian with its
convex envelope.

Chapter 2 introduces the subject of optimization. We discuss
conductivity of inhomogeneous materials and composites. The
properties of a composite significantly depend on  its
microstructure. We introduce homogenization methods to describe
the effective behavior of structures and  calculate  effective
properties of special structures. Homogenization theory, in  turn,
puts forward the so-called $ G $-closure%
\index{G-closure@$ G $-closure} problem (Chapter 3) that asks for
bounds of effective properties of composites assembled from given
materials. Bounds of $ G $-closures correspond to composites of
\X{extreme effective properties} that arise in optimal design.


 \subparagraph{Optimization of Conducting Composites}

A large class of optimization problems of conducting composites
requires only the simplest laminate structures for solution.
These problems are used in the book as the testing ground for
methods of structural optimization.  We introduce all the control
methods, including sufficient and necessary conditions of
optimality and minimizing sequences.
 Chapter 4 deals with the
optimization of the total conductivity of a domain. This problem
does not have a classical solution; the optimal layout is a
fine-scale mixture or a composite. We reformulate (relax) the
problem, replacing the layout of available materials with the
layout of optimal composites made of them. We also investigate the
fields in optimal structures. Chapter 5 treats the problems of
minimization of a large class of functionals associated with the
solution to the conductivity problem, such as the minimization of
the mean temperature in a part of the domain or the maximization
of the boundary current.

 \subparagraph{Quasiconvexity and Relaxation}

The second part deals with the relaxation technique of
multidimensional variational problems with nonconvex integrands.
This part contains  most of the new mathematical results.
 In Chapter 6, we briefly discuss instabilities,
the Weierstrass test\index{Weierstrass test}, and we introduce the
main tool for relaxation--the quasiconvex envelope.

In Chapter 7 we obtain upper bounds of the quasiconvex envelope by
constructing some special minimizing sequences.  The optimal
layouts are represented by alternating  materials
 in   laminate  microstructures.
We introduce special layouts with hierarchical  geometries  called
``laminates of a high rank'' and we derive their properties.

In Chapter 8 we derive lower bounds for the relaxed functional
that
 correspond to sufficient conditions of optimality.
The lower bound is built by a so-called translation method.
 We develop this method using the theories of quasiconvexity and
compensated compactness.

In Chapter 9 we develop a technique of minimal
extensions\index{minimal extension}  based on \X{necessary
conditions of the Weierstrass type}. The extension we obtain gives
an upper bound for the functional but avoids the explicit
consideration of minimizing  sequences.

All of these three approaches are illustrated by the solution of
an optimization problem of a conducting structure that minimizes a
sum of energies caused by several external sources.


 \subparagraph{$ G $-Closures}

To find  the optimal structure of a composite, one first describes
the set of effective properties of all possible microstructures.
This set is called the $ G $-closure of the properties of
initially given materials. The fourth part  discusses the knotty
problems of $ G $-closures. Chapter 10 deals with techniques used
to describe the boundaries of the closures, i.e., the extreme
effective properties of composites. The techniques are based  on
the variational methods introduced in Part III.

In Chapter 11 several examples of $ G $-closures are constructed.
These  include the $ G $-closures of conducting materials, the
exact coupled bounds for conducting properties of composites,  and
bounds for properties of polycrystals.

 Chapter 12 discusses  multimaterial composites. The methods for these problems are less developed and  more diverse. In particular,
 the technique of necessary conditions
allows us to address the problem of bounds for a three-material
composite.

Chapter 13 deals with the problem of complex conductivity. We
suggest a variational principle for this problem, and we apply the
variational technique to find coupled bounds on  the real and
imaginary parts of conductivity tensor.

 \subparagraph{Optimization of Elastic Structures}

The last part of the book deals with optimal design of elastic
structures.  We begin with a discussion of the equations and
variational principles for elasticity of inhomogeneous media and
the algebra of fourth-rank tensors of elastic moduli (Chapter 14).
In this chapter we also derive effective properties of elastic
composites.

In Chapter 15 we consider the problem of minimization of the
compliance of an elastic body, exploiting its similarity to the
problems discussed in earlier chapters; we obtain elastic
structures of extreme stiffness. We also discuss optimization of
the shapes of cavities.

In Chapter 16 we survey the results regarding bounds for elastic
moduli.  Specifically, we consider an isotropic composite of two
isotropic materials (plane problem),
 and we describe the bounds on its shear and bulk moduli. These bounds are coupled.
We also consider the problem of isotropic polycrystals with
extreme properties and describe the fractal geometry of optimal
polycrystals. These examples demonstrate advanced applications of
the variational technique described in  Part III.

 Chapter 17 discusses new formulations of a number of  problems of
structural optimization.  We consider the minimization of the sum
of elastic energies of different processes, the optimization of a
periodic composite, the optimization of a nonenergetic functional,
and  the optimization in an unknown class of loadings.  This last
problem is formulated as a min-max game between the applied
loadings and the responding structure.

\paragraph{Mathematical Methods}
Mathematically, the book considers one type of problems in
different settings. We describe  optimal solutions to unstable
variational problems. The goal is to define a solution that is
 reasonably smooth; particularly, it should not depend on the mesh in a
discretization scheme.
 However, it often turns out that the optimal solution is
characterized instead by infinitely fine oscillations.  Special
tests are developed to distinguish variational problems with
smooth and nonsmooth solutions, and suitable frameworks for
describing the solution with fine oscillations are worked out.

Both aspects deal with a special property of  Lagrangians of the
variational problem called quasiconvexity.  Variational problems
with quasiconvex Lagrangians possess stable solutions and problems
with
 nonquasiconvex Lagrangians may not.
Therefore, the test for oscillatory solutions requires
consideration of the quasiconvexity of the Lagrangian.  For
one-dimensional variational problems and for some multidimensional
problems, quasiconvexity degenerates to convexity, which makes the
determination easy. Generally, however,  the property of
quasiconvexity is not geometric, and we need more refined tools to
determine that a Lagrangian is quasiconvex.

If the Lagrangian lacks quasiconvexity, the minimizers generally
are replaced with oscillating minimizing sequences.  We perform
the relaxation of the problem, also called the minimal extension,
by averaging the solution over an infinitesimal volume.  This
corresponds to  replacing the original nonquasiconvex Lagrangian
with its quasiconvex envelope.  In this way we obtain a new
variational problem that possesses the same cost as the original
one, but its solution is smooth and equal to the mean value of the
fast oscillatory solution.

If quasiconvexity degenerates to convexity, the convex envelope
can be built by systematic geometrical methods.  There is no
systematic universal
method for constructing quasiconvex envelopes, so we instead build
two extensions of the original Lagrangian, one above and one below
the quasiconvex envelope (Chapters 7--9). Sometimes, these
extensions coincide, in which case the quasiconvex envelope is
determined.

The technique of bounds is addressed three times:  first, in the
context of one-dimensional variational problems (Chapter 1), then
for the simplest multidimensional problems with a scalar potential
(Chapter 3), and then in the  general case (Chapters 6--9) of
multidimensional problems with several state variables. This
technique is used many times to solve various problems of
$G$-closure (Part IV) and  optimal elastic structures (Part V).



\paragraph{Related Topics}


The theory of structural optimization lies at a busy intersection
of several mathematical disciplines---optimal control, calculus of
variations, homogenization, convex analysis---and is strongly
influenced by materials science.   Its applications include
traditional optimal design, theory of composites, phase transition
in solids, ``smart'' materials, nondestructive testing,
self-organization in physics, biomaterials, and so on.  Each of
these fields has its own philosophy, its history, and a huge
literature. Here we mention several of the related fields in
mathematics and engineering.  Each field could probably be
identified by a representative, but not complete, list of the
contributors. Specific references are placed in the body of the
text.

The variational problems and  problems of optimal control require
methods of selecting and describing solutions with infinitely fast
oscillations.  It is known in control theory that minimization is
generally achieved by an infinitely rapid oscillating control
function, called the chattering control.\index{chattering control}
This theory was originated by  Pontryagin and Young and  developed
by
  Gamkrelidze,   Krotov, Rozonoer, Varga,  and others.
The variational methods for nonconvex problems were introduced in
works by Carath\'{e}odory,  Morrey, and Young and
 developed in the works
by  Dacorogna, Ekeland, Kohn, Lions, Lurie,  M{\"u}ller, Murat,
 Raitum, Rockafellar, Strang, Tartar, Temam, and many others.
\nocite{Bakhvalov:1989:HAP}%
\nocite{Bendsoe:1992:TDS}%
\nocite{Bensoussan:1978:AAP}%
\nocite{Bruggemann:1935:BVP}%
\nocite{Castaneda:1988:OPN}%
\nocite{Castaneda:1992:BEP}%
\nocite{Castaneda:1992:NVPa}%
\nocite{Castaneda:1996:ESO}%
\nocite{Castaneda:1997:NCM}%
\nocite{Fonseca:1989:PTE}%
\nocite{Fonseca:1994:EFD}%
\nocite{Giorgi:1994:CSS}%
\nocite{Giorgi:1994:NIC}%
\nocite{Hashin:1962:VATa}%
\nocite{Hashin:1962:VATb}%
\nocite{Hill:1963:EPR}%
\nocite{Hill:1964:TMP}%
\nocite{Jager:1993:APL}%
\nocite{Jikov:1994:HDO}%
\nocite{Kaganova:1987:ESI}%
\nocite{Keller:1964:TCC}%
\nocite{Khachaturyan:1983:TST}%
\nocite{Olhoff:1979:OSR}%
\nocite{Pedersen:1993:ODB}%
\nocite{Sanchez-Palencia:1980:NMV}%
\nocite{Schulgasser:1976:RBS}%
\nocite{Talbot:1992:SSE}%
\nocite{Talbot:1995:IHS}%
\nocite{Talbot:1996:TPB}%
\nocite{Talbot:1997:BTO}%
\nocite{Taylor:1993:GEP}%
\nocite{Taylor:1999:OME}%
\nocite{Voigt:1928:LK}%
\nocite{Wiener:1912:AMP}%
\nocite{Willis:1981:VRM}%
\nocite{Zhikov:1991:EAM}%

An average description of the layout  for the highly oscillatory
materials is the subject of the theory of homogenization.  It was
originated  in the works by Babu\v{s}ka, Bakhvalov, Bensoussan,
Hashin, Keller, Khruslov, Lions, Olejnik, Papanicolaou,
Sanchez-Palencia, and Shtrikman, and developed  in many respects
in the works by Benveniste, Bergman, Bruno, Golden, Kohn, Kozlov,
Markov, Milton, Norris, Panasenko, Telega, Torquato, Vigdergauz,
Vogelius, Zhikov, and others. The advanced theories of solution to
differential equations with rapidly oscillating coefficients can
be found in the papers by Berlyand, Buttazzo, Cioranescu, Dal
Maso, de Georgi, Fonseca, Francfort, Kinderlehrer, Kohn,
M{\"u}ller, Pedregal, Sukey, and Tartar, among others.

Approaches for bounds on the effective properties of composites
are especially useful for our goals.  This area, initiated around
the beginning of the twentieth century by Rayleigh, Reuss, Voigt,
and Wiener, was  developed by Bruggeman, Hill, Hashin, Shtrikman,
and Walpole and recently updated by Avellaneda, Benveniste,
Beran,  Francfort, Gibiansky,  Kohn, Lurie, Markov, Milton, Murat,
Nesi, Ponte Cas\-ta\-{\~n}e\-da, Schulgasser, Talbot, Tartar,
Torquato, Willis, and Zhikov, among others.

The physical side of the picture was highlighted  by the
mechanicians and applied mathematicians who formulated and solved
structural optimization problems for several decades, starting
from the works by Prager.  We mention here the works of Armand,
Arora, Banichuk, Bends{\o}e, Diaz, Eshenauer, Fuchs, Haber,
Haftka, Kikuchi, Kirsch, Litvinov, Lipton, Mota Soares, Mroz,
Olhoff, Pedersen, Rasmussen, Rozvany, Sigmund, Taylor, Tortorelli,
and Zowe.

Computational techniques of structural optimization deserve
special considerations, yet we feel that it does not fit the scope
of this book, which is devoted exclusively to mathematical
foundations of structural optimization.  A detailed discussion of
the computational techniques can be found, for example, in the
books by Bends{\o}e,  Haftka and  G{\"u}rdal, Rozvany, and
Papalambros and Wilde.
\nocite{Bendsoe:1995:OST,Haftka:1992:ESO,Papalambros:1988:POD,Rozvany:1989:SDO}

\subparagraph{Natural Phenomena}

Natural \X{phase transition}s, \X{shape memory alloys}, and
naturally optimal \X{biomaterials} form a novel area of
application of the discussed mathematical techniques. These
problems, involving complicated materials, are in many respects
similar to structural optimization. In both cases one deals with
several materials or solid phases that are distributed in a domain
in a specific way. The optimality requirement posed by a designer
is parallel to a \X{natural variational principle} of minimization
of the total energy of the system (the \X{Gibbs principle}).
 The transformation from one phase to another  is parallel to the use of different materials in a design.
In minimizing its energy, a natural system exhibits \X{phase
separation} and forms a sort of natural composite that possesses
optimal microstructure.

These similarities suggest that corresponding approaches could be
applied to describe \X{natural mixtures with minimal energy}. This
concept was put forward in the works of  Ericksen, Khachaturyan,
and Kinderlehrer and developed in the  works of  Ball,
Bhattarcharya, Kohn, Fonseca, Grinfield, James, Luskin, Roitburd,
Rosakis, Truskinovsky, and others. The methods of quasiconvexity
are successively implemented for an explanation of structures
arriving at some natural phase transitions; we refer to the works
of
 the above-mentioned authors.

However,  natural phenomena are much deeper than the problems of
structural optimization. Indeed, the best engineering system
should reach the global minimum of the minimizing functional
 that represents the quality of the system. On the contrary, an equilibrium state of a natural system corresponds
to any local minimum of the energy. The energy of complicated
natural systems is  typically characterized by a large class of
metastable \X{local minima}.

There are other differences, too. Contrary to an optimal
engineering  construction,
 a realizable equilibrium state of a natural system corresponds
to a  dynamical  process that has
 led to it.  Finally, natural composites usually are
a random mixture of the states that correspond
 to local minima.
The search for a distribution of \X{local minima} requires
different techniques from those discussed here; we
 do not touch on this subject in the book.

\subparagraph{Biomaterials} The amazing rationality of biological
``constructions'' also calls for the use of  mathematical methods
of structural optimization  to model them. Consider, for example,
the  problem of the structure of a bone. A bone is a mechanical
structure made of composites with variable parameters that adapts
itself to its working conditions. It performs the clear mechanical
task of supporting the organism. These features are similar to
such man-made composite structures as masts, bridges, and towers.
Therefore, it would be natural to apply  optimization methods
developed for engineering constructions to bone structures.

However, the two problems are not the same. In addition to the
problems of local minima, stable evolutionary dynamics, and
randomness  already mentioned, it is not clear what quantity is
minimized in  natural evolutionary \X{biomaterials} (we mean the
explicit \X{optimality criterion of a  natural structure}, not  a
general reference to the \X{evolution} that perfects organisms).
 In engineering
problems, the goal is the minimization of a given functional that
is not the subject of a search or even a discussion. The problem
is to find the structure that minimizes a functional prescribed by
a designer. On the other hand, the structure of a bone is known,
but it is not clear in what sense (if  any)
 the bone structure is optimal.

The corresponding problem is  the search for the cost functional
of an optimization problem with a known  solution. This problem
has  not been sufficiently investigated, to our knowledge.



\paragraph{Indexes, References, etc.}

The electronic version of the manuscript for the book was prepared
with the help of Professor Nelson Beebe using special \BibTeX{}
and \LaTeX{} macros that he
 developed. In
addition to the detailed table of contents, it contains the list
of figures, references, the author\slash editor index, and the
index of topics.  Each item in the references points to the pages
on which the source was referred to. The references section is
ordered alphabetically by the name of primary author.

The author\slash editor index refers to the pages that contain the
reference.  Boldface author names  indicate primary authors, while
names in Roman text are nonprimary authors.

The book's Web site, \path=http://www.math.utah.edu/books/vmso=,
contains an \X{expanded bibliography} in \BibTeX{} form, an
\X{errata list}, and other related resources. Please email your
comments to \path=cherk@math.utah.edu=.

\paragraph{Use in the Classroom}%
\index{classroom use} The book is an extended and edited version
of the author's lecture notes for courses delivered at the
University of Utah. The contents of the book may be used for a
year-long graduate course for students in applied mathematics,
science, and engineering. We do our best to keep the exposition
simple and do not hesitate to sacrifice rigor in favor of
vividness, and generality in favor of vigorous illustrations.  The
references point to more rigorous formulations. The problems for
discussion are in the end of chapters. Some of them are simple
exercises; the others require more serious analysis and can be
used for course projects.

A \X{course in calculus of variations} may be based on the
classical material
\cite{Gelfand:1963:CV,Ewing:1969:CVA,Weinstock:1974:CV},
supplemented by Chapter 1 (nonconvex one-dimensional problems),
Chapter 4 (an example of a variational problem for a distributed
system), and Chapters 6--9 (relaxation of nonconvex multivariable
problems), with examples from Chapters 10--12 ($ G $-closures).

A \X{course in homogenization} may use chapters from the
``homogenization'' books \cite{Bensoussan:1978:AAP,Jikov:1994:HDO}
and  Chapters 2 and 3 (conductivity, homogenization, $G$-closure),
Chapter 7 ( laminates, various structures of laminates of high
rank), Chapter 12 (multiphase structures), and Chapter 14
(elasticity, homogenization and matrix laminates). Chapters 4 and
5 (optimization by laminates) may be used as examples of the use
of composites.

A \X{course in structural optimization} may use Chapters 4 and 5
(optimization of conducting bodies), Chapters 6--9 (relaxation of
nonconvex multivariable problems), and Chapters 14--17
(elasticity, optimization of elastic structures).




\paragraph{Credits}

The author has been tempted to present his view of the history of
structural optimization and nonconvex variational methods, but he
has given that up because the topic is too awesome,
 controversial, and sensitive. The very length of the preceding list of names and topics testifies to this.
 We give credit here  to authors in a specific context of themes discussed.
It is hardly possible  to give a complete survey of even the
recent development of related topics:
 New branches of the theory are constantly appearing. Instead, we concentrate our attention on underlying ideas
 and methods that should help to find solutions to new problems.



Most of the results and opinions presented are based on or related
to the author's research, conducted for the most part in
collaboration with Tim Burns, Elena Cherkaeva, Andrey Fedorov,
Leonid Gibiansky, Lars Krog, Konstantin Lurie, Graeme Milton,
Robert Palais, and Shmul Vigdergauz.

Many new results obtained by Gr{\'e}goire Allaire, Marco
Avellaneda, Martin Bends{\o}e, John Ball, Gilles Francfort, Leonid
Gibiansky, Zvi Hashin, Robert Kohn, Robert Lipton, Konstantin
Lurie, Graeme Milton, Fran{\c{c}}ois Murat, Vincenzo Nesi, Niels
Olhoff, Ole Sigmund, Gil Strang, Vladim{\'\i}r {\v{S}}ver{\'a}k,
Luc Tartar, Salvatore Torquato, Smul Vigdergauz, Vasily Zhikov,
and others are explicitly used in the text.

\paragraph{Acknowledgments}




The author gratefully acknowledges support and encouragement from
colleagues and students in the Department of Mathematics,
University of Utah, as well from  friends and family. My special
thanks are to Elena Cherkaev, Leonid Gibiansky, Konstantin Lurie,
and Graeme Milton, who discussed different topics of the book and
made valuable suggestions. Other useful suggestions were made by
anonymous referees, whom the author wants to thank.  Many
suggestions were made by the members of my classes, especially  O.
Hamed, I. Kucuk, and T. Robbins.  The author is indepted to
colleagues: Martin Bends{\o}e,
 Robert Kohn, Robert Lipton, Konstantin Lurie,
Graeme Milton, Robert Palais, Ole Sigmund, and John Taylor who
looked through the manuscript, suggested improvements of the text,
and  provided additional references. The electronic manuscript was
produced with the help of Nelson Beebe, whose expertise in
\LaTeX{} problems and various suggestions greatly
 helped  improve the text.

My special thanks go to the staff at Springer-Verlag. The book was
copyedited by David Kramer and Shari Chappell, who greatly amended
the English as well as the presentation style.  Achi Dosanjh,
senior editor,  Lesley Poliner, senior production editor, and
Christina Torster, production associate, were extremely effective
and encouraging.

\begin{flushright}
%    Andrej Cherkaev \hfill Salt Lake City, Utah, November 1998
%    Andrej Cherkaev \hfill Salt Lake City, Utah, July 1999
%    Andrej Cherkaev \hfill Salt Lake City, Utah, 1 March 2000
    Salt Lake City, Utah \hfill Andrej Cherkaev \\
    \hfill 1 March 2000
\end{flushright}
