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Description
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: Metrics, Analysis, Differential Calculus, and Optimization 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.
Excerpt
1 Objectives and Scope of the Book
The objective of this book is to give a comprehensive presentation of mathematical constructions and tools that can be used to study problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. In that context, a good analytical framework and good modeling techniques must be able to handle the occurrence of singular behaviors whenever they are compatible with the mechanics or the physics of the problems at hand. In some optimization problems, the natural intuitive notion of a geometric domain undergoes mutations into relaxed entities such as microstructures. So the objects under consideration need not be smooth open domains, or even sets, as long as they still makes sense mathematically.
This book covers the basic mathematical ideas, constructions, and methods that come from different fields of mathematical activities and areas of applications that have often evolved in parallel directions. The scope of research is frighteningly broad because it touches on areas that include classical geometry, modern partial differential equations, geometric measure theory, topological groups, and constrained optimization, with applications to classical mechanics of continuous media such as fluid mechanics, elasticity theory, fracture theory, modern theories of optimal design, optimal location and shape of geometric objects, free and moving boundary problems, and image processing. Innovative modeling or new issues raised in some applications force a new look at the fundamentals of well-established mathematical areas such as geometry, to relax basic notions of volume, perimeter, and curvature or boundary value problems, and to find suitable relaxations of solutions. In that spirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notion of volume to the one of measure on an equivalence class of measurable sets in 1907. He was followed in that endeavor in the early 1950s by the celebrated work of E. De Giorgi, who used the relaxed notion of perimeter defined on the class of Caccioppoli sets to solve Plateau's problem of minimal surfaces.
The material that is pertinent to the study of geometric objects and the entities and functions that are defined on them would necessitate an encyclopedic investment to bring together the basic theories and their fields of applications. This objective is obviously beyond the scope of a single book and two authors.
©2011 SIAM
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