A.A. Novotny, J. Sokołowski, Topological Derivatives in Shape Optimization

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▶ First monograph describing in details the new developments in shape optimization for elliptic boundary value problems
▶ Presents a wide spectrum of examples and techniques for
▶ learning how to use the modern mathematics in applied shape optimization of structures
▶ Makes this important field of research accessible for the students of mathematics and of mechanics

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.

¹ National Laboratory for Scientific Computing, Petrópolis, Brazil;
² University of Henri Poincare Nancy I, Vandoeuvre-lès-Nancy, France

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P. Plotnikov, J. Sokołowski, Compressible Navier-Stokes Equations

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▶ First monograph on the mathematical theory of shape optimization for compressible Navier-Stokes equations
▶ Clear explanation of the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field
▶ New concepts and results are presented
The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with particular emphasis on applications to aerodynamics. The topics covered include: modeling of compressible viscous flows; modern mathematical theory of nonhomogeneous boundary value problems for viscous gas dynamics equations; applications to optimal shape design in aerodynamics; kinetic theory for equations with oscillating data; new approach to the boundary value problems for transport equations.
The monograph offers a comprehensive and self-contained introduction to recent mathematical tools designed to handle the problems arising in the theory.
¹ Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia;
² Institut Elie Cartan, Vandoeuvre Lès Nancy, France

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