GENERALIZED SHAPE OPTIMIZATION OF 3-DIMENSIONAL STRUCTURES USING MATERIALS WITH OPTIMUM MICROSTRUCTURES

This paper deals with generalized shape optimization of linearly elast ic, three-dimensional continuum structures, i.e. we consider the probl em of determining the structural topology (or layout) such that the sh ape of external as well as internal boundaries and the number of inner holes are optimized simultaneously. For prescribed static loading and given boundary conditions, the optimum solution is sought from the co ndition of maximum integral stiffness (minimum elastic compliance) sub ject to a specified amount of structural material within a given three -dimensional design domain. This generalized shape optimization proble m requires relaxation which leads to the introduction of microstructur es. A class of optimum three-dimensional microstructures and explicit analytical expressions for their optimum effective stiffness propertie s have been developed by Gibiansky and Cherkaev (1987) [Gibiansky, L.V ., Cherkaev, A.V., 1987. Microstructures of composites of extremal rig idity and exact estimates of provided energy density tin Russian). Rep ort (1987) No. 1155. A.F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad. English translation in: Kohn, R.V. , Cherkaev, A.V. (Eds.), Topics in the Mathematical Modelling of Compo site Materials. Birkhauser, New York. 1997]. The present paper gives a brief account of the results in Gibiansky and Cherkaev (1987) which w ill be utilized for our microlevel problem analysis. It is a character istic feature that the use of optimum microstructures renders the glob al problem convex if an appropriate parametrization is applied. Hereby local optima can be avoided and we can construct a simple gradient ba sed numerical method of mathematical programming for solution of the c omplete optimization problem. Illustrative examples of optimum layout and topology designs of three-dimensional structures are presented at the end of the paper. (C) 1998 Elsevier Science Ltd. All rights reserv ed.