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Optimal treatment of structures. (Conception optimale de structures.) (French) Zbl 1132.49033

Mathématiques & Applications (Berlin) 58. Berlin: Springer (ISBN 3-540-36710-1/pbk). xii, 278 p. (2007).
This book corresponds to lectures given by the author at École Polytechnique, France. It gives a wide overview of problems and methods in shape optimization. The main objective of the author is not necessarily to give all the mathematical details of proofs, but rather to highlight essential tools and results (precise references are given when the proofs are not complete). It is an excellent introduction in this field and it may concern a large audience going from engineers dealing with problems in structural mechanics to researchers interested by numerical calculations in shape optimization.
In chapter 1, dealing with particular problems in elasticity and in aerodynamics, the author introduces in a nice way different paradigms in shape optimization and he presents the methods which will be developed in the body of the book. From chapter 2 to chapter 4, preliminary tools are briefly recalled. The finite element method is presented in chapter 2. Optimality conditions for optimization problems and the main algorithms used in shape optimization are introduced in chapter 3. A brief introduction to optimal control problems governed by partial differential equations is given in chapter 4 (in particular the notion of adjoint state is introduced). Chapter 5 deals with parametric optimization. The parameter to be optimized is here a geometrical parameter (the thickness, the curbature\(\dots\)) or a physical parameter (a diffusion coefficient, an elasticity coefficient). The different aspects are treated: existence of solutions (for the continuous problems and the discrete ones), optimality conditions, and numerical experiments based on gradient algorithms. Several numerical results are presented.
Chapter 6 is devoted to the geometrical optimization. The derivatives of functionals with respect to variations of the domain is the main concept introduced in this chapter. Topological optimization and homogenization are introduced in chapter 7. The method of two-scale convergence is presented. The Hashin-Shtrikman bounds are given for composite materials in elasticity problems. The last chapter, written by Marc Schoenauer, deals with evolutionary algorithms. The main ingredients of this approach are first presented. Numerical tests in structural mechanics are next given.
This book is nicely written and its reading is very pleasant. I hope it will be translated soon in English since no other text book in this field gives a so wide overview in less than 300 pages.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
65K10 Numerical optimization and variational techniques
74P05 Compliance or weight optimization in solid mechanics
74P15 Topological methods for optimization problems in solid mechanics
74P20 Geometrical methods for optimization problems in solid mechanics
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
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