This book provides theories on non-parametric shape optimization
problems, systematically keeping in mind readers with an engineering
background. Non-parametric shape optimization problems are defined as
problems of finding the shapes of domains in which boundary value
problems of partial differential equations are defined. In these
problems, optimum shapes are obtained from an arbitrary form without any
geometrical parameters previously assigned. In particular, problems in
which the optimum shape is sought by making a hole in domain are called
topology optimization problems. Moreover, a problem in which the optimum
shape is obtained based on domain variation is referred to as a shape
optimization problem of domain variation type, or a shape optimization
problem in a limited sense. Software has been developed to solve these
problems, and it is being used to seek practical optimum shapes.
However, there are no books explaining such theories beginning with
their foundations.
The structure of the book is shown in the Preface. The theorems are
built up using mathematical results. Therefore, a mathematical style is
introduced, consisting of definitions and theorems to summarize the key
points. This method of expression is advanced as provable facts are
clearly shown. If something to be investigated is contained in the
framework of mathematics, setting up a theory using theorems prepared by
great mathematicians is thought to be an extremely effective approach.
However, mathematics attempts to heighten the level of abstraction in
order to understand many things in a unified fashion. This
characteristic may baffle readers with an engineering background. Hence
in this book, an attempt has been made to provide explanations in
engineering terms, with examples from mechanics, after accurately
denoting the provable facts using definitions and theorems.
