Topological Derivatives in Shape Optimization

ISBN-13: 9783642352447 / Angielski / Miękka / 2012 / 324 str.

Antonio Andr Novotny; Jan Soko Owski

Opis

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.

Recenzje

From the reviews:

"The main aim of the presented book is the presentation of topological derivatives for a wide class of elliptic problems with the applications in numerical methods of shape and topology optimization. This volume is primarily addressed to applied mathematicians working in the field of partial differential equations and their applications, especially those concerned with numerical aspects. However, the book will also be useful for applied scientists from engineering and physics." (Jan Lovísek, zbMATH, Vol. 1276, 2014)
"The book under review concerns new methods of solving a class of shape optimization problems appearing in continuum mechanics, mainly in solid mechanics, composites and plate-like bodies. ... The theoretical results are illustrated by numerical examples. Moreover, carefully selected exercises are provided. This valuable book fills a gap in the literature on topology optimization." (Tomasz Lewinski, Mathematical Reviews, August, 2013)

Spis treści

Domain Derivation in Continuum Mechanics.- Material and Shape Derivatives for Boundary Value Problems.- Singular Perturbations of Energy Functionals.- Configurational Perturbations of Energy Functionals.- Topological Derivative Evaluation with Adjoint States.- Topological Derivative for Steady-State Orthotropic Heat Diffusion Problems.- Topological Derivative for Three-Dimensional Linear Elasticity Problems.- Compound Asymptotic Expansions for Spectral Problems.- Topological Asymptotic Analysis for Semilinear Elliptic Boundary Value Problems.- Topological Derivatives for Unilateral Problems.

Blurb

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, sensitivity analysis in fracture mechanics 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 in the mathematical aspects of topological asymptotic analysis as well as in applications of topological derivatives in computational mechanics.