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**Variational methods for structural optimization.**
*(English)*
Zbl 0956.74001

Applied Mathematical Sciences. 140. New York, NY: Springer. xxvi, 545 p. DM 149.00; öS 1088.00; sFr 136.00; £51.50; $ 79.95 (2000).

An applied problem of composition of different materials with the aim to achieve the required properties of a body is formulated as a variational problem of structural optimization, i.e. as a problem of optimization of an integral functional with respect to boundary conditions. Such variational problems normally are non-convex and multidimensional. Recently, special mathematical methods were developed to attack these difficult problems. The book intends to close a gap between theoretical achievements in multidimensional non-convex variational calculus and practical methods of structural design.

The book is divided into 5 parts and 17 chapters. The introductory material necessary to analyze the above problems is presented in part 1. Here the author gives a brief review of one-dimensional variational problems, including chattering controls, Weierstrass test, convex envelopes of non-convex Lagrangians, and relaxation. A model of conducting composites is presented by describing conductivity of an inhomogeneous medium, differential constraints and potentials for fields and currents, and jump conditions on boundaries between different materials. Then a dual variational principle is introduced, homogenization procedure is proposed, and \(G\)-closure problem is discussed.

Part 2 (“Optimization of conducting composites”) covers optimal design problems for two-component conducting body of maximal conductivity, and stationary conductivity problem. Different approaches are developed which lead to similar results.

In part 3 (“Quasiconvexity and relaxation”) the author considers the multidimensional variational problem with non-convex integrand. The quasiconvex envelope is constructed to justify the relaxation procedure, and a technique based on necessary conditions is developed. Part 4 is devoted to a comprehensive analysis of \(G\)-closures and to their applications to optimal design of different structures including multi-material composites. A special case of \(G\)-closures appearing in the study of linear processes in dissipative media is discussed.

The last part of the book (“Optimization of elastic structures”) deals with, seemingly, most difficult optimal design problems. The author presents here the equations and variational principles for elasticity of inhomogeneous media and considers the problem of minimization of the compliance of an elastic body, including structures of extreme stiffness and optimal shapes of cavities. The results on bounds of improvement in properties of a composite (e.g., isotropic composites of two isotropic materials) due to variation of its structure are reviewed. The author formulates several new problems of structural optimization, including optimization of non-energetic functions and min-max problem appearing in the case of unknown loading.

The book is divided into 5 parts and 17 chapters. The introductory material necessary to analyze the above problems is presented in part 1. Here the author gives a brief review of one-dimensional variational problems, including chattering controls, Weierstrass test, convex envelopes of non-convex Lagrangians, and relaxation. A model of conducting composites is presented by describing conductivity of an inhomogeneous medium, differential constraints and potentials for fields and currents, and jump conditions on boundaries between different materials. Then a dual variational principle is introduced, homogenization procedure is proposed, and \(G\)-closure problem is discussed.

Part 2 (“Optimization of conducting composites”) covers optimal design problems for two-component conducting body of maximal conductivity, and stationary conductivity problem. Different approaches are developed which lead to similar results.

In part 3 (“Quasiconvexity and relaxation”) the author considers the multidimensional variational problem with non-convex integrand. The quasiconvex envelope is constructed to justify the relaxation procedure, and a technique based on necessary conditions is developed. Part 4 is devoted to a comprehensive analysis of \(G\)-closures and to their applications to optimal design of different structures including multi-material composites. A special case of \(G\)-closures appearing in the study of linear processes in dissipative media is discussed.

The last part of the book (“Optimization of elastic structures”) deals with, seemingly, most difficult optimal design problems. The author presents here the equations and variational principles for elasticity of inhomogeneous media and considers the problem of minimization of the compliance of an elastic body, including structures of extreme stiffness and optimal shapes of cavities. The results on bounds of improvement in properties of a composite (e.g., isotropic composites of two isotropic materials) due to variation of its structure are reviewed. The author formulates several new problems of structural optimization, including optimization of non-energetic functions and min-max problem appearing in the case of unknown loading.

Reviewer: A.Žilinskas (Vilnius)

### MSC:

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

74Pxx | Optimization problems in solid mechanics |

74E30 | Composite and mixture properties |

74Q05 | Homogenization in equilibrium problems of solid mechanics |

74Q20 | Bounds on effective properties in solid mechanics |