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An Introduction to the Topological Derivative Method

  • Book
  • © 2020

Overview

Authors:
  1. Antonio André Novotny

    1. Coordenação de Métodos, Matemáticos e Computacionais, Laboratório Nacional de Computação, Científica LNCC/MCTIC, Petrópolis, Brazil

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  2. Jan Sokołowski

    1. Systems Research Institute Polish Academy of Sciences Warsaw, Poland, Universidade Federal da Paraiba, Centro de informatica João Pessoa, PB, Brazil, Institut Élie Cartan de Nancy, UMR 7502 Université de Lorraine, CNRS, Vandoeuvre-Lès-Nancy, France

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  • Introduces the concept of topological derivative in a simple and pedagogical manner using a direct approach based on calculus of variations combined with compound asymptotic analysis
  • Offers numerical methods in shape optimization, including algorithms and applications in the context of compliance structural topology optimization and topology design of compliant mechanisms
  • Explores the mathematical aspects of topological asymptotic analysis as well as on applications of the topological derivative in computational mechanics, including shape and topology optimization

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-x
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  2. Introduction

    • Antonio André Novotny, Jan Sokołowski
    Pages 1-16

  3. Singular Domain Perturbation

    • Antonio André Novotny, Jan Sokołowski
    Pages 17-34

  4. Regular Domain Perturbation

    • Antonio André Novotny, Jan Sokołowski
    Pages 35-52

  5. Domain Truncation Method

    • Antonio André Novotny, Jan Sokołowski
    Pages 53-66

  6. Topology Design Optimization

    • Antonio André Novotny, Jan Sokołowski
    Pages 67-93

  7. Back Matter

    Pages 95-114
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Keywords

About this book

This book presents the topological derivative method through selected examples, using a direct approach based on calculus of variations combined with compound asymptotic analysis. This new concept in shape optimization has applications in many different fields such as topology optimization, inverse problems, imaging processing, multi-scale material design and mechanical modeling including damage and fracture evolution phenomena. In particular, the topological derivative is used here in numerical methods of shape optimization, with applications in the context of compliance structural topology optimization and topology design of compliant mechanisms. Some exercises are offered at the end of each chapter, helping the reader to better understand the involved concepts.




Authors and Affiliations

  • Coordenação de Métodos, Matemáticos e Computacionais, Laboratório Nacional de Computação, Científica LNCC/MCTIC, Petrópolis, Brazil

    Antonio André Novotny

  • Systems Research Institute Polish Academy of Sciences Warsaw, Poland, Universidade Federal da Paraiba, Centro de informatica João Pessoa, PB, Brazil, Institut Élie Cartan de Nancy, UMR 7502 Université de Lorraine, CNRS, Vandoeuvre-Lès-Nancy, France

    Jan Sokołowski

About the authors

Antonio André Novotny is a Senior Researcher at the National Laboratory for Scientific Computing, Petrópolis, Brazil. His research topics include the theoretical development and applications of the topological derivative method to shape and topology optimization; inverse problems; imaging processing;  multi-scale material design; and mechanical modeling, including damage and fracture phenomena.



Jan Sokolowski is a Full Professor at the Institute of Mathematics (IECL) at the Université de Lorraine in Nancy, France, and at the Polish Academy of Sciences’ Systems Research Institute. He has published five monographs with Springer and Birkhauser, and over 200 research papers in international journals. His research focuses on shape and topology optimization for the systems described by partial differential equations.

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