The topological derivative is defined as the first term
(correction) of the asymptotic expansion of a given shape
functional with respect to a small parameter that measures the size
of singular domain perturbations, such as holes, inclusions,
defects, source-terms and cracks. Over the last decade, topological
asymptotic analysis has become a broad, rich and fascinating
research area from both theoretical and numerical standpoints. It
has applications in many different fields such as shape and
topology optimization, inverse problems, imaging processing and
mechanical modeling including synthesis and/or optimal design of
microstructures, fracture mechanics sensitivity analysis and damage
evolution modeling. Since there is no monograph on the subject at
present, the authors provide here the first account of the theory
which combines classical sensitivity analysis in shape optimization
with asymptotic analysis by means of compound asymptotic expansions
for elliptic boundary value problems. This book is intended for
researchers and graduate students in applied mathematics and
computational mechanics interested in any aspect of topological
asymptotic analysis. In particular, it can be adopted as a textbook
in advanced courses on the subject and shall be useful for readers
interested on the mathematical aspects of topological asymptotic
analysis as well as on applications of topological derivatives in
computation mechanics.
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