This considerably enriched new edition provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is the shape or the structure of a geometric object. Shapes and Geometries presents the latest ground-breaking theoretical foundation to shape optimization in a form that can be used by the engineering and scientific communities. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field. A series of generic examples has been added to the introduction and special emphasis has been put on the construction of important metrics. Advanced engineers in various application areas use basic ideas of shape optimization, but often encounter difficulties due to the sophisticated mathematical foundations for the field. This new version of the book challenges these difficulties by showing how the mathematics community has made extraordinary progress in establishing a rational foundation for shape optimization. This area of research is very broad, rich, and fascinating from both theoretical and numerical standpoints. It is applicable in many different areas such as fluid mechanics, elasticity theory, modern theories of optimal design, free and moving boundary problems, shape and geometric identification, image processing, and design of endoprotheses in interventional cardiology.
M. C. Delfour is a Professor of Mathematics and Statistics at the University of Montreal in Canada, a member of the Canadian Academy of Sciences (FRSC), a former Guggenheim and Killam Fellow and a SIAM Fellow. J.-P. Zolesio is Research Director in Mathematics at the CNRS. He is member of the Institut Non Lineaire de Nice (INLN) associated with the Institut National de Recherche en Informatique et Automatique (INRIA) In Sophia Antipolis (France).
List of Figures Preface Chapter 1: Introduction: Examples, Background, and Perspectives Chapter 2: Classical Descriptions of Geometries and Their Properties Chapter 3: Courant Metrics on Images of a Set Chapter 4: Transformations Generated by Velocities Chapter 5: Metrics via Characteristic Functions Chapter 6: Metrics via Distance Functions Chapter 7: Metrics via Oriented Distance Functions Chapter 8: Shape Continuity and Optimization Chapter 9: Shape and Tangential Differential Calculuses Chapter 10: Shape Gradients under a State Equation Constraint Elements of Bibliography Index of Notation Index.
