The goal of this research is to develop formulations and solution strategies to generate the shape and topology of mechanical components that are optimal with respect to measures of performance that include functional as well as geometric criteria. The research will emphasize the following: mathematically rigorous formulations, which are posed formally as optimization problems in which the design space includes different topological descriptions of the structure, as well as different structural idealizations of two and three dimensional components; optimization problems that incorporate geometry and topology constraints; formulations that include local functional constraints on the mechanical behavior of the structure, such as measures of stress, deformation, and buckling; and the development of computational procedures suitable for the solution of problems cast in the form outlined above. The investigation seeks to facilitate a systematic search in a space of design configurations that includes shape and topology. The goal of such exploration is not only the generation of improved, optimal or near optimal designs, but also the generation of understanding of this complex design space and of the effect that a meaningful class of design restrictions has on admissible solutions to the design problem. The investigation will produce tools for design capable of suggesting near-optimal configurations for a reasonably large class of mechanical design problems. There are many possible applications of these ideas in industries such as automotive or aerospace. These tools will, if successful, provide one of only a handful of methods that are applicable in the early stages of design.
