Shape Optimization by the Homogenization Method

This book is about the homogenization method and its applications to optimal design in the conductivity and elasticity setting. It explains how homogenization theory may be applied to solve optimal design problems, both from a theoretical and a numerical point of view. It is intended for an audience of researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.

From the reviews:

"The book is a research monograph, but the structure and completeness of the presentation means that the book constitutes a good basis for a graduate course in applied mathematics. The rigorous mathematical presentation is supplemented with numerous remarks and comments which discuss the subject in broader terms, greatly simplifying the reading process. The book is a welcome and up-to-date addition to the literature in the area and it is necessary reading for any researcher and student " (M.P. Bendsøe, Structural Multidisciplinary Optimization, 5, 2002)

"The book is very well structured, very clearly written, very well motivated, and complete in its treatment of modelling, analysis and simulation. It will be a basic reference for whoever wants to deeply understand homogenization from the point of view of its application to optimal design. The treatment is right to the point, a quality that is very much appreciated by readers. In summary, I believe this text may become a main source for the subject of optimal design and shape optimization." (Pablo Pedregal, Mathematical Reviews, 2002 h)

"The book under review presents a comprehensive introduction to the homogenisation method applied to optimal design, including many proofs which were hitherto only scattered throughout the literature this one provides the most complete treatment of numerical methods A number of realistic examples, mostly for elasticity, has been developed in detail. In summary, we would like to warmly recommend this book to anybody working in optimal shape design, composites and homogenisation, as well to those who wish to enter these fields." (Nenad Antonic and Marko Vrdoljak, Zentralblatt MATH, 990:15, 2002)

Contenu
1 Homogenization.- 1.1 Introduction to Periodic Homogenization.- 1.2 Definition of H-convergence.- 1.3 Proofs and Further Results.- 1.4 Generalization to the Elasticity System.- 2 The Mathematical Modeling of Composite Materials.- 2.1 Homogenized Properties of Composite Materials.- 2.2 Conductivity.- 2.3 Elasticity.- 3 Optimal Design in Conductivity.- 3.1 Setting of Optimal Shape Design.- 3.2 Relaxation by the Homogenization Method.- 4 Optimal Design in Elasticity.- 4.1 Two-phase Optimal Design.- 4.2 Shape Optimization.- 5 Numerical Algorithms.- 5.1 Algorithms for Optimal Design in Conductivity.- 5.2 Algorithms for Structural Optimization.