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**Optimization of structural topology, shape, and material.**
*(English)*
Zbl 0822.73001

Berlin: Springer-Verlag. xi, 271 p. (1995).

The book deals with the so-called “homogenization method” used in structural optimization. The book consists of six chapters. Chapter 1 contains a description of the homogenization approach to topology design, ranging over basic modelling and existence issues, derivation of effective moduli of periodic composites, derivation of optimality criteria, and computational as well as implementation issues. Chapter 2 shows the close connection between material science and generalized optimal shape design using the fundamental extremal energy principles of mechanics, i.e., optimizing the potential or complementary energies of a structure with respect to microstructure.

Chapter 3 deals with the extremal energy functionals for a free parametrization of material, shape and topology. Material properties are represented in the most general form possible for a linear elastic continuum. Employing this description of the material, the author demonstrates in analytical form the local properties associated with materials that are optimal with respect to the global compliance objective. Chapter 4 contains the topology design of trusses with the purpose of devising efficient numerical algorithms. As in chapter 2 and 3, the fundamental approach revolves around the derivation of optimal energy functionals and equivalent problem statements, relating recent developments to classical truss topology design formulations. Chapter 5 describes various extensions of topology and material design including the plate optimization problem. Other subjects cover design for vibration and stability problems as well as hierarchical design of topology and geometry of the ground structure. Chapter 6 contains bibliographical notes covering the main subjects of optimization problems as well as related background material useful for the reader.

This is a first book devoted to description of homogenization method and its application to optimum structural design including topology, shape, and material optimization. The book is addressed to engineers and researches, but it may as well be used as an auxiliary text in teaching the graduate courses in structural optimization.

Chapter 3 deals with the extremal energy functionals for a free parametrization of material, shape and topology. Material properties are represented in the most general form possible for a linear elastic continuum. Employing this description of the material, the author demonstrates in analytical form the local properties associated with materials that are optimal with respect to the global compliance objective. Chapter 4 contains the topology design of trusses with the purpose of devising efficient numerical algorithms. As in chapter 2 and 3, the fundamental approach revolves around the derivation of optimal energy functionals and equivalent problem statements, relating recent developments to classical truss topology design formulations. Chapter 5 describes various extensions of topology and material design including the plate optimization problem. Other subjects cover design for vibration and stability problems as well as hierarchical design of topology and geometry of the ground structure. Chapter 6 contains bibliographical notes covering the main subjects of optimization problems as well as related background material useful for the reader.

This is a first book devoted to description of homogenization method and its application to optimum structural design including topology, shape, and material optimization. The book is addressed to engineers and researches, but it may as well be used as an auxiliary text in teaching the graduate courses in structural optimization.

Reviewer: St.Jendo (Warszawa)

### MSC:

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

74P99 | Optimization problems in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

74E05 | Inhomogeneity in solid mechanics |