Applications of the topological derivative method
The present book deals with some aspects of numerical solutions for problems with unknown shapes. The book consists of 11 chapters and references with 259 entities. In the first chapter Introduction, the authors present the theoretical foundations of the procedures used in the book. They give definition and examples of topological derivative together with open problems for the case when topological derivatives are used as optimization method. In the next Chapter 2 the theory in singularly perturbed geometrical domains and the mathematical preliminaries for the optimization procedure is explained. In Chapter 3 the theory of Steklov-Poincaré theory for Helmholtz equation is briefly treated. The role of topological derivative, the main tool of the optimization procedure presented in the book, is explained in Chapters 4 and 5. Gradient type method with various applications is presented in Chapter 6 while the specific problems for thermomechanical actuators are presented in Chapter 7. The same type of problems is treated in Chapter 8 where piezomechanical actuators are analysed and shape optimization of such devices is presented. Asymptotic analysis of variational inequalities is presented in Chapter 9 together with concrete examples of shape optimization. In Chapter 10 the second-order topological derivatives are introduced and used for numerical solution of some representative problems. Finally, in Chapter 11 another type of problem is analysed. In the so-called electrical impedance tomography problem, the shape of the hidden anomaly has to be determined from the measurement of electrical potential on the boundary of the body. This problem is stated within the framework of topology optimization procedure, and some specific examples are treated. 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.
scientific article; zbMATH DE number 7024833