Ebook: Shape Optimization by the Homogenization Method

The topic of this book is homogenization theory and its applications to optimal design in the conductivity and elasticity settings. Its purpose is to give a self-contained account of homogenization theory and explain how it applies to solving optimal design problems, from both a theoretical and a numerical point of view. The application of greatest practical interest tar­ geted by this book is shape and topology optimization in structural design, where this approach is known as the homogenization method. Shape optimization amounts to finding the optimal shape of a domain that, for example, would be of maximal conductivity or rigidity under some specified loading conditions (possibly with a volume or weight constraint). Such a criterion is embodied by an objective function and is computed through the solution of astate equation that is a partial differential equa­ tion (modeling the conductivity or the elasticity of the structure). Apart from those areas where the loads are applied, the shape boundary is al­ ways assumed to support Neumann boundary conditions (i. e. , isolating or traction-free conditions). In such a setting, shape optimization has a long history and has been studied by many different methods. There is, therefore, a vast literat ure in this field, and we refer the reader to the following short list of books, and references therein [39], [42], [130], [135], [149], [203], [220], [225], [237], [245], [258].

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This book provides an introduction to the theory and numerical developments of the homogenization method. Its main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials;a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.

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This book provides an introduction to the theory and numerical developments of the homogenization method. Its main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials;a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.
Content:
Front Matter....Pages i-xv
Homogenization....Pages 1-89
The mathematical modeling of composite materials....Pages 91-188
Optimal Design in Conductivity....Pages 189-257
Optimal Design in Elasticity....Pages 259-342
Numerical Algorithms....Pages 343-425
Back Matter....Pages 427-458

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This book provides an introduction to the theory and numerical developments of the homogenization method. Its main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials;a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.
Content:
Front Matter....Pages i-xv
Homogenization....Pages 1-89
The mathematical modeling of composite materials....Pages 91-188
Optimal Design in Conductivity....Pages 189-257
Optimal Design in Elasticity....Pages 259-342
Numerical Algorithms....Pages 343-425
Back Matter....Pages 427-458
....