Preface
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 intend to provide here the first detailed account of the theory that combines classical sensitivity analysis in shape optimization with asymptotic analysis by means of compound asymptotic expansions for elliptic boundary value problems. The presented theory of topological derivatives is a natural continuation of the shape sensitivity analysis techniques developed in the monograph Introduction to shape optimization – shape sensitivity analysis, Springer-Verlag (1992), by Sokołowski and Zol´esio. We show that the velocity method of shape optimization can be combined with the asymptotic analysis in singularly perturbed domains and as a result the new properties of shape functionals are derived for the purposes of optimality conditions and numerical solutions in shape and topology optimization problems. The explicit formulae for the topological derivatives are well suited for the numerical algorithms of shape optimization and are already used in many research papers.
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