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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.
List of contents
Introduction.- Theory in Singularly Perturbed Geometrical Domains.- Steklov-Poincare´ Operator for Helmholtz Equation.- Topological Derivatives for Optimal Control Problems.- Optimality Conditions with Topological Derivatives.- A Gradient-Type Method and Applications.- Synthesis of Compliant Thermomechanical Actuators.- Synthesis of Compliant Piezomechanical Actuators.- Asymptotic Analysis of Variational Inequalities.- A Newton-Type Method and Applications.- The Electrical Impedance Tomography Problem.
Summary
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.
Additional text
“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)
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"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 Atanackovic, 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)
