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**Introduction to shape optimization: shape sensitivity analysis.**
*(English)*
Zbl 0761.73003

Springer Series in Computational Mathematics. 16. Berlin etc.: Springer- Verlag. 250 p. (1992).

The book under review, written by two active specialists in the domain, is an introduction to shape optimization. The first chapter describes the problem of shape optimization and its motivation. Briefly such a problem can be formulated as follows: (1) \(\Omega^*\in U_{ad}\): \(J(\Omega^*)=\inf_{\Omega\in U_{ad}}J(\Omega)\), where \(U_{ad}\) is the set of admissible domains in \(\mathbb{R}^ N\), \(N=1,2,\dots\) . Usually the cost functional \(J(\Omega)\) takes the form \(J(\Omega)=h(\Omega,y(\Omega))\), whre \(y(\Omega)\) is given as the solution of a well posed boundary value problem in the domain \(\Omega\). The existence of a solution to (1) is ensured, provided the set \(U_{ad}\) is endowed with a topology such that the mapping \(\Omega\to J(\Omega)\) is lower semicontinuous and the set \(U_{ad}\) is compact. Much of the book is concerned with the shape sensitivity analysis for unilateral problems describing physical phenomena as contact problems in elasticity, elasto-plastic torsion problems and the obstacle problem.

The second chapter is concerned with mathematical methods used in the shape sensitivity analysis. The concepts of material derivative and the speed method are introduced. There are described mathematical tools that can be used to prove the existence of solutions to related shape optimization problem. Several examples of the second order and the fourth order elliptic problems related to applications are given.

Chapter 3 states necessary optimality conditions for problem (1). Moreover, related results on the shape sensitivity analysis for linear problems including systems of equations of linear elasticity, the Kirchhoff plate, multiple eigenvalue problems, heat transfer equations, and wave equations are presented. The last chapter is concerned with the shape sensitivity analysis of variational inequalities. There are presented, among others, results on the differential stability of the metric projection in Hilbert spaces, the obstacle problem in \(H^ 1(\Omega)\), the shape controllability of the free boundary.

The book is written in a clear and precise style. It has a systematic and natural mainstream development of all the four chapters. This book should be a valuable contribution to an important field in the theory of optimization.

The second chapter is concerned with mathematical methods used in the shape sensitivity analysis. The concepts of material derivative and the speed method are introduced. There are described mathematical tools that can be used to prove the existence of solutions to related shape optimization problem. Several examples of the second order and the fourth order elliptic problems related to applications are given.

Chapter 3 states necessary optimality conditions for problem (1). Moreover, related results on the shape sensitivity analysis for linear problems including systems of equations of linear elasticity, the Kirchhoff plate, multiple eigenvalue problems, heat transfer equations, and wave equations are presented. The last chapter is concerned with the shape sensitivity analysis of variational inequalities. There are presented, among others, results on the differential stability of the metric projection in Hilbert spaces, the obstacle problem in \(H^ 1(\Omega)\), the shape controllability of the free boundary.

The book is written in a clear and precise style. It has a systematic and natural mainstream development of all the four chapters. This book should be a valuable contribution to an important field in the theory of optimization.

Reviewer: G.Aniculaesei (Iaşi)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74P99 | Optimization problems in solid mechanics |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |