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This book aims the optimal design of a material (thermic or
electrical) obtained as the mixture of a finite number of original
materials, not necessarily isotropic. The problem is to place these
materials in such a way that the solution of the corresponding
state equation minimizes a certain functional that can depend
nonlinearly on the gradient of the state function. This is the main
novelty in the book. It is well known that this type of problems
has no solution in general and therefore that it is needed to work
with a relaxed formulation. The main results in the book refer to
how to obtain such formulation, the optimality conditions, and the
numerical computation of the solutions. In the case of functionals
that do not depend on the gradient of the state equation, it is
known that a relaxed formulation consists of replacing the original
materials with more general materials obtained via homogenization.
This includes materials with different properties of the originals
but whose behavior can be approximated by microscopic mixtures of
them. In the case of a cost functional depending nonlinearly on the
gradient, it is also necessary to extend the cost functional to the
set of these more general materials. In general, we do not dispose
of an explicit representation, and then, to numerically solve the
problem, it is necessary to design strategies that allow the
functional to be replaced by upper or lower approximations. The
book is divided in four chapters. The first is devoted to recalling
some classical results related to the homogenization of a sequence
of linear elliptic partial differential problems. In the second
one, we define the control problem that we are mainly interested in
solving in the book. We obtain a relaxed formulation and their main
properties, including an explicit representation of the new cost
functional, at least in the boundary of its domain. In the third
chapter, we study the optimality conditions of the relaxed problem,
and we describe some algorithms to numerically solve the problem.
We also provide some numerical experiments carried out using such
algorithms. Finally, the fourth chapter is devoted to briefly
describe some extensions of the results obtained in Chapters 2 and
3 to the case of dealing with several state equations and the case
of evolutive problems. The problems covered in the book are
interesting for mathematicians and engineers whose work is related
to mathematical modeling and the numerical resolution of optimal
design problems in material sciences. The contents extend some
previous results obtained by the author in collaboration with other
colleagues.
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