##
**Optimal shape design for elliptic systems.**
*(English)*
Zbl 0534.49001

Springer Series in Computational Physics. New York etc.: Springer-Verlag. XII, 168 p., 57 figs. DM 80.00; $ 29.90 (1984).

Optimal Shape Design (OSD) is a special family of control or identification problems in which the variable to be controlled or identified is a geometrical domain of \({\mathbb{R}}^ n\). Being a recent branch of the calculus of variations, the subject is strongly related to other fields of applicable analysis, which are briefly reviewed in this book, namely: elliptic partial differential equations (Ch. 1: Sobolev spaces, variational formulations); optimization (Ch. 4: classical algorithms like the method of steepest descent Newton’s and conjugate gradient methods); optimal control (Ch. 5; which techniques are reviewed in three examples); and numerical analysis (Ch. 1 and 4). The problem statement, short historical comments and informal arguments for solving some OSD problems arising in engineering and applied science are given in Ch. 2.

The introduction to the mathematical techniques begins in Ch. 3 with some results on the existence of solutions to OSD problems. Here the crucial tools are the compactness of the Hausdorff distance on the compact subsets of \({\mathbb{R}}^ n\) and the uniform boundary regularity of the domains (the cone or Lipschitz conditions). Interesting counterexamples are given to show that without some restrictions on the boundaries certain problems are ill posed. In between a large field remains open. The optimality conditions with respect to the domain in a number of typical situations are derived, in Ch. 6, by the method of normal variations of Hadamard. For the numerical computations, the discretization by the finite element method (FEM) is presented in Ch. 7 for the Neumann, the Dirichlet and the transmission problems. Two other different methods (the methods of mappings and of characteristic functions) together with numerical algorithms for the finite difference and the boundary element methods are given in Ch. 8. Finally, in Ch. 9, two industrial examples (the design of an electromagnet and of an airfoil) are presented in detail with numerical data based on the FEM.

This book does not treat OSD problems in the theory of elasticity where a substantial literature already exists, and merely refers to an example of a free boundary problem (p. 23), while this is an important and closely related field, which perhaps deserves a larger treatment in this context. One must refer a small imprecision in (34) of p. 35 and (114) of p. 120, where the convergence of the characteristic functions is in \(L^{\infty}\)-weak* (and also in \(L^ p\)-strong, for all \(p<\infty)\). The book presents the approach and some of the recent research results of what the author calls ”the French School of Applied Mathematics”. As a whole it is an interesting and valuable text for graduate students and researchers in the field.

The introduction to the mathematical techniques begins in Ch. 3 with some results on the existence of solutions to OSD problems. Here the crucial tools are the compactness of the Hausdorff distance on the compact subsets of \({\mathbb{R}}^ n\) and the uniform boundary regularity of the domains (the cone or Lipschitz conditions). Interesting counterexamples are given to show that without some restrictions on the boundaries certain problems are ill posed. In between a large field remains open. The optimality conditions with respect to the domain in a number of typical situations are derived, in Ch. 6, by the method of normal variations of Hadamard. For the numerical computations, the discretization by the finite element method (FEM) is presented in Ch. 7 for the Neumann, the Dirichlet and the transmission problems. Two other different methods (the methods of mappings and of characteristic functions) together with numerical algorithms for the finite difference and the boundary element methods are given in Ch. 8. Finally, in Ch. 9, two industrial examples (the design of an electromagnet and of an airfoil) are presented in detail with numerical data based on the FEM.

This book does not treat OSD problems in the theory of elasticity where a substantial literature already exists, and merely refers to an example of a free boundary problem (p. 23), while this is an important and closely related field, which perhaps deserves a larger treatment in this context. One must refer a small imprecision in (34) of p. 35 and (114) of p. 120, where the convergence of the characteristic functions is in \(L^{\infty}\)-weak* (and also in \(L^ p\)-strong, for all \(p<\infty)\). The book presents the approach and some of the recent research results of what the author calls ”the French School of Applied Mathematics”. As a whole it is an interesting and valuable text for graduate students and researchers in the field.

Reviewer: J.F.Rodrigues

### MSC:

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

35J50 | Variational methods for elliptic systems |

49K20 | Optimality conditions for problems involving partial differential equations |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49M15 | Newton-type methods |

49M05 | Numerical methods based on necessary conditions |

93B99 | Controllability, observability, and system structure |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

35R30 | Inverse problems for PDEs |

65K10 | Numerical optimization and variational techniques |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

93C20 | Control/observation systems governed by partial differential equations |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

93B30 | System identification |