Topology optimization of three-dimensional structures using optimum microstructures

In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic compliance or minimum total elastic energy at equilibrium. The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem. Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.