We introduce C-tubes , 3D tubular structures composed of developable surface strips. C-tubes can be understood as a generalization of Monge surfaces—a special class of sweep surfaces—towards the recently introduced conenets. This observation allows formulating a constructive algorithm to create tubular structures that ensures developability of the constituent surfaces, while significantly broadening the design space. Our novel form-finding tool enables design exploration by solving for the input variables of the constructive algorithm so that the C-tube best conforms to user-specified objectives. We discuss several case studies that illustrate the versatility of our approach for the design and fabrication of complex structures, with applications in architecture, furniture, and lighting design.

Surface-based inflatables are composed of two thin layers of nearly inextensible sheet material joined together along carefully selected fusing curves. During inflation, pressure forces separate the two sheets to maximize the enclosed volume. The fusing curves restrict this expansion, leading to a spatially varying in-plane contraction and hence metric frustration. The inflated structure settles into a 3D equilibrium that balances pressure forces with the internal elastic forces of the sheets. We present a computational framework for analyzing and designing surface-based inflatable structures with arbitrary fusing patterns. Our approach employs numerical homogenization to characterize the behavior of parametric families of periodic inflatable patch geometries, which can then be combined to tessellate the sheet with smoothly varying patterns. We propose a novel parametrization of the underlying deformation space that allows accurate, efficient, and systematical analysis of the stretching and bending behavior of inflated patches with potentially open boundaries. We apply our homogenization algorithm to create a database of geometrically diverse fusing patterns spanning a wide range of material properties and deformation characteristics. This database is employed in an inverse design algorithm that solves for fusing curves to best approximate a given input target surface. Local patches are selected and blended to form a global network of curves based on a geometric flattening algorithm. These fusing curves are then further optimized to minimize the distance of the deployed structure to target surface. We show that this approach offers greater flexibility to approximate given target geometries compared to previous work while significantly improving structural performance.

We present a computational inverse design framework for a new class of volumetric deployable structures that have compact rest states and deploy into bending-active 3D target surfaces. Umbrella meshes consist of elastic beams, rigid plates, and hinge joints that can be directly printed or assembled in a zero-energy fabrication state. During deployment, as the elastic beams of varying heights rotate from vertical to horizontal configurations, the entire structure transforms from a compact block into a target curved surface. Umbrella Meshes encode both intrinsic and extrinsic curvature of the target surface and in principle are free from the area expansion ratio bounds of past auxetic material systems. We build a reduced physics-based simulation framework to accurately and efficiently model the complex interaction between the elastically deforming components. To determine the mesh topology and optimal shape parameters for approximating a given target surface, we propose an inverse design optimization algorithm initialized with conformal flattening. Our algorithm minimizes the structure's strain energy in its deployed state and optimizes actuation forces so that the final deployed structure is in stable equilibrium close to the desired surface with few or no external constraints. We validate our approach by fabricating a series of physical models at various scales using different manufacturing techniques.

We present a computational inverse design method for a new class of surface-based inflatable structure. Our deployable structures are fabricated by fusing together two layers of inextensible sheet material along carefully selected curves. The fusing curves form a network of tubular channels that can be inflated with air or other fluids. When fully inflated, the initially flat surface assumes a programmed double-curved shape and becomes stiff and load-bearing. We present a method that solves for the layout of air channels that, when inflated, best approximate a given input design. For this purpose, we integrate a forward simulation method for inflation with a gradient-based optimization algorithm that continuously adapts the geometry of the air channels to improve the design objectives. To initialize this non-linear optimization, we propose a novel surface flattening algorithm. When a channel is inflated, it approximately maintains its length, but contracts transversally to its main direction. Our algorithm approximates this deformation behavior by computing a mapping from the 3D design surface to the plane that allows for anisotropic metric scaling within the bounds realizable by the physical system. We show a wide variety of inflatable designs and fabricate several prototypes to validate our approach and highlight potential applications.

Basket weaving is a traditional craft for creating curved surfaces as an interwoven array of thin, flexible, and initially straight ribbons. The three-dimensional shape of a woven structure emerges through a complex interplay of the elastic bending behavior of the ribbons and the contact forces at their crossings. Curvature can be injected by carefully placing topological singularities in the otherwise regular weaving pattern. However, shape control through topology is highly non-trivial and inherently discrete, which severely limits the range of attainable woven geometries. Here, we demonstrate how to construct arbitrary smooth free-form surface geometries by weaving carefully optimized curved ribbons. We present an optimization-based approach to solving the inverse design problem for such woven structures. Our algorithm computes the ribbons' planar geometry such that their interwoven assembly closely approximates a given target design surface in equilibrium. We systematically validate our approach through a series of physical prototypes to show a broad range of new woven geometries that is not achievable by existing methods. We anticipate our computational approach to significantly enhance the capabilities for the design of new woven structures. Facilitated by modern digital fabrication technology, we see potential applications in material science, bio- and mechanical engineering, art, design, and architecture.

We present X-shells , a new class of deployable structures formed by an ensemble of elastically deforming beams coupled through rotational joints. An X-shell can be assembled conveniently in a flat configuration from standard elastic beam elements and then deployed through force actuation into the desired 3D target state. During deployment, the coupling imposed by the joints will force the beams to twist and buckle out of plane to maintain a state of static equilibrium. This complex interaction of discrete joints and continuously deforming beams allows interesting 3D forms to emerge. Simulating X-shells is challenging, however, due to unstable equilibria at the onset of beam buckling. We propose an optimization-based simulation framework building on a discrete rod model that robustly handles such difficult scenarios by analyzing and appropriately modifying the elastic energy Hessian. This real-time simulation method forms the basis of a computational design tool for X-shells that enables interactive design space exploration by varying and optimizing design parameters to achieve a specific design intent. We jointly optimize the assembly state and the deployed configuration to ensure the geometric and structural integrity of the deployable X-shell. Once a design is finalized, we also optimize for a sparse distribution of actuation forces to efficiently deploy it from its flat assembly state to its 3D target state. We demonstrate the effectiveness of our design approach with a number of design studies that highlight the richness of the X-shell design space, enabling new forms not possible with existing approaches. We validate our computational model with several physical prototypes that show excellent agreement with the optimized digital models.

We study assemblies of convex rigid blocks regularly arranged to approximate a given freeform surface. Our designs rely solely on the geometric arrangement of blocks to form a stable assembly, neither requiring explicit connectors or complex joints, nor relying on friction between blocks. The convexity of the blocks simplifies fabrication, as they can be easily cut from different materials such as stone, wood, or foam. However, designing stable assemblies is challenging, since adjacent pairs of blocks are restricted in their relative motion only in the direction orthogonal to a single common planar interface surface. We show that despite this weak interaction, structurally stable, and in some cases, globally interlocking assemblies can be found for a variety of freeform designs. Our optimization algorithm is based on a theoretical link between static equilibrium conditions and a geometric, global interlocking property of the assembly---that an assembly is globally interlocking if and only if the equilibrium conditions are satisfied for arbitrary external forces and torques. Inspired by this connection, we define a measure of stability that spans from single-load equilibrium to global interlocking, motivated by tilt analysis experiments used in structural engineering. We use this measure to optimize the geometry of blocks to achieve a static equilibrium for a maximal cone of directions, as opposed to considering only self-load scenarios with a single gravity direction. In the limit, this optimization can achieve globally interlocking structures. We show how different geometric patterns give rise to a variety of design options and validate our results with physical prototypes.