(Georgia Institute of Technology, 2023-08-23)
Gloyd, James Todd
Discrete lattices use individually manufactured unit cell building blocks, which are then assembled to form large lattice structures, the quality of which is not significantly influenced by the scale of the structure. Furthermore, the size of the final structure is not bounded by the footprint of the manufacturing equipment. Tuning the elastic behavior of materials and structures provides the possibility for significant improvements to overall performance, as demonstrated by structural and topology optimization studies. Similar improvements can be made in discrete lattice applications, as shown by the Coded Structures Laboratory at NASA. Improvements made to performance of discrete lattice structures are, so far, limited by the lack of a systematic, direct method to dictate the behavior---that is, prescribe the deformation---of the final structure. Here we present a direct method of prescribed structural behavior integrating structural and topology optimization, for both discrete lattice structures and general structures. Also presented are formulas and methods for calculating the determinant and inverse of a linear combination of matrices, which originally stemmed from the development of prescribed behavior methods however, while applicable to prescribed deformation problems, are much more useful in other situations. The direct methods of prescribed deformation presented here automatically produce dictated behavior from the candidate structure when possible and produce an approximation when the desired behavior is impossible. These methods are shown to move towards a minimizer with quadratic convergence, with improved results in situations with fewer limits on the prescribed behavior. Additionally, the presented formula for calculation of the determinant of a linear combination of matrices provides exact results in as little as one tenth of the time of traditional approximation methods, and the exact inverse of the linear combination is calculated in as little as one quarter of the time of traditional exact methods. We show these formulas provide significant computational and conceptual improvement to current methods and provide unmatched performance in parallel computing settings.