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Applications of the Topological Derivative Method

  • Book
  • © 2019

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. Institut Élie Cartan de Nancy, UMR 7502, Université de Lorraine, CNRS, Vandœuvre-Lès-Nancy, France

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  3. Antoni Żochowski

    1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

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  • Presents new results and applications of the topological derivative method in control theory, topology optimization, and inverse problems
  • Introduces the theory of singularly perturbed geometrical domains using selected examples
  • Describes the first order topology design algorithm with its applications in topology optimization
  • Shares the second order Newton-type reconstruction algorithm of based on higher order topological derivatives for solving inverse reconstruction problems

Part of the book series: Studies in Systems, Decision and Control (SSDC, volume 188)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Introduction

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 1-12

  3. Theory in Singularly Perturbed Geometrical Domains

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 13-39

  4. Steklov–Poincaré Operator for Helmholtz Equation

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 41-50

  5. Topological Derivatives for Optimal Control Problems

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 51-60

  6. Optimality Conditions with Topological Derivatives

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 61-84

  7. A Gradient-Type Method and Applications

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 85-107

  8. Synthesis of Compliant Thermomechanical Actuators

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 109-128

  9. Synthesis of Compliant Piezomechanical Actuators

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 129-148

  10. Asymptotic Analysis of Variational Inequalities

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 149-164

  11. A Newton-Type Method and Applications

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 165-181

  12. The Electrical Impedance Tomography Problem

    • Antonio André Novotny, Jan Sokołowski, Antoni Żochowski
    Pages 183-200

  13. Back Matter

    Pages 201-212
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Keywords

About this book

The book presents new results and applications of the topological derivative method in control theory, topology optimization and inverse problems. It also introduces the theory in singularly perturbed geometrical domains using selected examples. Recognized as a robust numerical technique in engineering applications, such as topology optimization, inverse problems, imaging processing, multi-scale material design and mechanical modeling including damage and fracture evolution phenomena, the topological derivative method is based on the asymptotic approximations of solutions to elliptic boundary value problems combined with mathematical programming tools. The book presents the first order topology design algorithm and its applications in topology optimization, and introduces the second order Newton-type reconstruction algorithm based on higher order topological derivatives for solving inverse reconstruction problems. It is intended for researchers and students in applied mathematics and computational mechanics interested in the mathematical aspects of the topological derivative method as well as its applications in computational mechanics.

Reviews

“The book is well written, and the examples treated are carefully motivated. The references are up-to-date. The material presented in the book requires good mathematical background, so that the book could be used, as usefull reference, for researchers and graduate students that are working in the optimization theory.”(Teodor Atanacković, zbMATH 1460.74001, 2021)

“The book under review provides an excellent overview of historical and current developments in the field and, at the same time, is a very good introduction to readers familiar with linear elliptic equations and variational inequalities. … In all chapters, a thorough literature discussion is provided along with a detailed discussion of open problems.” (Guenter Leugering, Mathematical Reviews, October, 2019)

Authors and Affiliations

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

    Antonio André Novotny

  • Institut Élie Cartan de Nancy, UMR 7502, Université de Lorraine, CNRS, Vandœuvre-Lès-Nancy, France

    Jan Sokołowski

  • Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Antoni Żochowski

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