Topology optimization of cables, cloaks, and embedded lattices

Materials play a critical role in the behavior and functionality of natural and engineered systems. For example, the use of cast-iron and steel led to dramatically increased bridge spans per material volume with the move from compression-dominant arch bridges to tensile-capable truss, suspension, and cable-stayed bridges; materials underlie many of the major technological advancements in the auto and aerospace industries that have made cars and airplanes increasingly light, strong, and damage tolerant; and the great diversity of biological materials and bio-composites enable complex biological and mechanical functions in nature. Topology optimization is a computational design method that simultaneously enhances efficiency and design freedom of engineered parts, but is often limited to a single, solid, isotropic, linear-elastic material. To understand how the material space can be tailored to enhance design freedom and/or promote desired mechanical behavior, several topology optimization problems are explored in this dissertation in which the space of available materials is either relaxed or restricted. Specifically, in a discrete topology optimization setting defined by 1D (truss) elements, tension-only systems are targeted by restricting the material space to that of a tension-only material and tailoring a formulation to handle the associated nonlinear mechanics. The discrete setting is then enhanced to handle 2D (beam) elements in pursuit of cloaking devices that hide the effect of a hole or defect on the elastostatic response of lattice systems. In this case the material space is relaxed to allow for a continuous range of stiffness and the objective is formulated as a weighted least squares function in which the physically-motivated weights promote global stiffness matching between the cloaked and undisturbed systems. Continuous 2D and 3D structures are also explored in a density-based topology optimization setting in which the material space is relaxed to accommodate an arbitrary number of candidate materials in a general continuum mechanics framework that can handle material anisotropy. The theoretical and physical relevance of such framework is highlighted via a continuous embedding scheme that enables manufacturing in the relaxed (or restricted) design space of lattice-based microstructural-materials. Implications of varying the material design space on the mechanics, mathematics, and computations needed for topology optimization are discussed in detail.Ph.D