Modern digital fabrication allows building complex three-dimensional objects through deformation and assembly of simple components, rather than manufacturing them as a single piece. Elastic rods bent into shape and flat sheets cut and joined into curved surfaces are two such building blocks: lightweight, material-efficient, and well-suited to a wide range of applications. However, the shapes they can achieve are constrained by the physics of elastic deformation and by the geometry of materials that can bend but cannot stretch. Computational design of these systems amounts to searching a parameter space whose feasible subset, shaped by these constraints, is typically too high-dimensional and complex to navigate by hand.

This thesis develops computational inverse design methods that couple physical simulation and geometric modeling with numerical optimization to explore and search this space. We apply these methods to three material systems.

We first present a pipeline to discover and analyze stable equilibria of elastic knots, closed knotted elastic wires with self-contacts. By combining topology-aware randomized initialization with simulation, we discover multiple distinct stable configurations that a given elastic knot can morph and reliably settle into, and support quantitative analysis and identification of recurrent geometric patterns. Inspired by the rich variety of shapes that such elastic knots can assume, we introduce tencers, structures in which a network of tensioning cables pulls a set of elastic beams into a prescribed target geometry. We formulate an inverse design problem that solves for cable layouts producing a desired equilibrium shape, while promoting sparsity in the cable network to facilitate fabrication and assembly. Finally, we present a method for the design and fabrication of C-tubes, piecewise-developable tubular structures assembled from planar strips. We introduce a discrete geometric model that guarantees developability and seam compatibility by construction, and embed it in an optimization framework that lets designers shape free-form tubular geometries against user-defined objectives while remaining fabricable in the process.

Across the three systems, the relationship between the design parameterization and the underlying feasibility constraints takes different forms, ranging from exploration of an unknown feasible set, to constrained optimization that enforces feasibility at every iteration, to a constructive encoding that guarantees it by design. The methods developed in this thesis enable the creation of lightweight and fabrication-ready structures with potential applications in robotics, architecture, product design, and artistic installations. We validate our results extensively with physical prototypes that consistently exhibit the predicted behaviors.