Deployable structures that transform from a planar assembly-friendly compact state to an expansive freeform surface state have diverse applications in robotics, medical devices, temporary installations, and architecture. Umbrella Meshes are a new class of volumetric deployable structures with extensive shape expression capabilities compared to existing plane-to-surface deployables. They are modular, made of Umbrella cells consisting of identical rigid plates and rotational joints connected by elastic beams of varying heights. Deployment is actuated by pushing the cells orthogonal to the plane, rotating the elastic beams from vertical to horizontal configurations, thus redistributing material from out of the plane into it. In contrast to rigid scissor mechanisms, the beams deform elastically, making the deployed equilibrium bending-active. Assembled in a stress-free planar configuration, an Umbrella Mesh can be programmed to deploy to a desired target shape by virtue of the optimized heights of the constituent cells. The rich design space facilitates programming a large range of target shapes, controlling the structural stiffness, and encoding extrinsic curvature.

This thesis contributes a comprehensive computational framework for the design and optimization of Umbrella Meshes. To facilitate design exploration of the deployed structure, we develop a physics-based simulation modeling the deployment process under actuation forces. We abstract the deployment transformation of an umbrella mesh using conformal geometry, providing intuitive design initializations for a specific target surface. Our inverse design algorithm leverages the simulation pipeline and numerical optimization to iteratively refine a design to approximate a target surface while minimizing the elastic energy and actuation forces involved. We build optimized physical prototypes through digital fabrication and validate our computational pipeline.

The inverse design framework exemplifies a design-driven approach to fabricating optimized physical structures. The latter half of this thesis focuses on fabrication-driven design. We develop a computational framework to rationalize bending-active structures into a sparse kit of parts, allowing cost-effective fabrication. Our method can either find an optimal kit of parts for multiple input designs or rationalize existing designs to use a pre-fabricated kit of parts. To tackle the non-trivial coupling of components in bending-active systems, we propose a relaxed continuous formulation of the combinatorial problem of grouping components to a sparse part set, allowing us to incorporate physics-based simulation that tracks multiple bending-active equilibria. We demonstrate our approach on Umbrella Meshes, C-shells, and orthogonal gridshells.

The thesis culminates with Reconfigurable Umbrella Meshes (RUMs) consisting of identical reconfigurable cells. Each reconfigurable cell can assume the form of a continuous range of parts, thus combining the benefits of pre-fabrication and precisely inverse-designed heights. Assembled from these identical mass-producible cells, the same RUM can deploy into several shapes over multiple deployment cycles. Our inverse design enables precise reconfiguration of the compact state and opens up multiple research avenues for high-fidelity shape morphing control with applications in soft robotics and sustainable architecture.