A homogenization method for shape and topology optimization

Structural optimization has recently received wide range attention in computer aided design. One of the reasons is rapid development of so vhisticated but inexpensive engineering workstations with graphic capabilities, multi-windows and fast computation. The other reason is acceptance by people in industry of various computer aided engineering systems including reliable analysis capability such as finite element, finite difference and boundary element methods which are supported by geometric modeling capability with automatic mesh generation methods. The notion of structural optimization was widely discussed by structural engineers in the early 70s right after the rapid development of finite element methods. However, it could not provide practical means to design structures except in aeronautical engineering, because of lack of flexible geometric modeling and handy interactive graphic display capabilities at that time. Since most of structural design of machine parts involves heavily their geometric representation, in other words, their shape, structural design must deal with shape optimization, while most of aeronautical applications are based on frame structures with shell reinforcements which need only the concept of sizing optimization dealing with, for example, the sizes of the cross section of frames as well as the thickness of shells. It is clear that geometric representation of the structural configuration is not required in sizing optimization, and thus, its development may not be bounded by the capability of geometric modeling. A modern theory of structural optimization based on mathematical programmings and sensitivity analysis was developed by Schmit [1] and Fox [2] in the early 60s, although the concept of fully stressed design was widely applied in design practice without solid mathematical justification but the engineers' intuition. Prager and Taylor [3] made justification of the fully stressed design for a class of structural optimization problems by deriving their optimality criteria, whose direct use in constructing optimization algorithms leads the so-called optimality criteria method in contrast to the mathematical programming method, using variational