Generalized shape optimization of three-dimensional structures using materials with optimum microstructures

• Published in: Mechanics of materials 1998, Vol.28 (1), p.207-225
• Main Authors: Jacobsen, Jørgen Bay, Olhoff, Niels, Rønholt, Erik
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Homogenization; Mathematical programming; Optimum material microstructures; Optimum structural design; Physics; Solid mechanics; Static elasticity; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Three-dimensional layout and topology optimization
• ISSN: 0167-6636; 1872-7743
• DOI: 10.1016/S0167-6636(97)00056-2

This paper deals with generalized shape optimization of linearly elastic, three-dimensional continuum structures, i.e. we consider the problem of determining the structural topology (or layout) such that the shape 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 condition of maximum integral stiffness (minimum elastic compliance) subject to a specified amount of structural material within a given three-dimensional design domain. This generalized shape optimization problem requires relaxation which leads to the introduction of microstructures. A class of optimum three-dimensional microstructures and explicit analytical expressions for their optimum effective stiffness properties have been developed by Gibiansky and Cherkaev (1987) [Gibiansky, L.V., Cherkaev, A.V., 1987. Microstructures of composites of extremal rigidity and exact estimates of provided energy density (in Russian). Report (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 Composite Materials. Birkhaüser, New York. 1997]. The present paper gives a brief account of the results in Gibiansky and Cherkaev (1987) which will be utilized for our microlevel problem analysis. It is a characteristic feature that the use of optimum microstructures renders the global problem convex if an appropriate parametrization is applied. Hereby local optima can be avoided and we can construct a simple gradient based numerical method of mathematical programming for solution of the complete optimization problem. Illustrative examples of optimum layout and topology designs of three-dimensional structures are presented at the end of the paper.