Variational methods for Structural Optimization

by Andrej Cherkaev

Springer Verlag, NY 2000.

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The book describes mathematical foundations of structural optimization. Optimal varying structure adapts the construction to its goal and to loading conditions. Mathematically, the problem is formulated as a variational problem. It possesses non-stable solutions: the optimization of materials collocations results in a system of patterns or in optimal microstructures. These occur if the Lagrangian lacks the convexity or the quasiconvexity. To relax the problem, a quasiconvex envelope of the Lagrangian must be built; the book discusses methods for obtaining quasiconvex envelopes.
Special chapters introduce conductivity and elasticity of inhomogeneous media and variational problems with unstable solutions. Then the structural optimization problems are formulated. The tools for bounding of the quasiconvex envelope are described. Those are special necessary conditions of optimality, minimal extensions, minimizing sequences, and the Translation method of sufficient conditions. Special attention is paid to the problem of G-closure, that is to description of all possible material characteristics of a composites with arbitrary geometry. The methods are described to determine geometry of optimal structures and its homogenized behavior. Other topics include homogenization, min-max formulation, and stability of optimal design problems. Many examples of structural optimization are discussed. Special chapters introduce conductivity and elasticity of inhomogeneous media and variational problems with unstable solutions.

The book is addressed to students and researchers is applied mathematics, mechanics, and material sciences. Its challenge is to expose the foundations of structural optimization in simple terms to make them available for practical use and adaptable to specific models. A number of problems is listed in the end of the chapters.

The style is informal: the usual conflict between mathematical rigor and the physical clarity usually is solved in favor of the last one. The simplest possible mathematical tools are used. New mathematical theories are not introduced if the goals can be achieved using a simpler representation. The upper bound of the complexity is given by the Ockham's Razor Principle: "Pluralitas non est ponenda sine neccesitate" -- entities must not be multiplied beyond what is necessary. The lower bound comes from the Einstein's warning " Everything should be made as simple as possible, but not simpler."