Figure 1: Six basic elastic textures are used to obtain a large range of homogenized isotropic material properties. A 3 × 3 × 1 tiling of each pattern is shown, along with rendered (left) and fabricated (right) cell geometry below. The naming convention is explained in Section 4. AbstractWe introduce elastic textures: a set of parametric, tileable, printable, cubic patterns achieving a broad range of elastic material properties: the softest pattern is over a thousand times softer than the stiffest, and the Poisson's ratios range from below zero to nearly 0.5. Using a combinatorial search over topologies followed by shape optimization, we explore a wide space of truss-like, symmetric 3D patterns to obtain a small family. This pattern family can be printed without internal support structure on a single-material 3D printer and can be used to fabricate objects with prescribed mechanical behavior. The family can be extended easily to create anisotropic patterns with target orthotropic properties. We demonstrate that our elastic textures are able to achieve a user-supplied varying material property distribution. We also present a material optimization algorithm to choose material properties at each point within an object to best fit a target deformation under a prescribed scenario. We show that, by fabricating these spatially varying materials with elastic textures, the desired behavior is achieved.

Direct digital manufacturing is a set of rapidly evolving technologies that provide easy ways to manufacture highly customized and unique products. The development pipeline for such products is radically different from the conventional manufacturing pipeline: 3D geometric models are designed by users often with little or no manufacturing experience, and sent directly to the printer. Structural analysis on the user side with conventional tools is often unfeasible as it requires specialized training and software. Trial-anderror, the most common approach, is time-consuming and expensive.We present a method that would identify structural problems in objects designed for 3D printing based on geometry and material properties only, without specific assumptions on loads and manual load setup. We solve a constrained optimization problem to determine the "worst" load distribution for a shape that will cause high local stress or large deformations. While in its general form this optimization has a prohibitively high computational cost, we demonstrate that an approximate method makes it possible to solve the problem rapidly for a broad range of printed models. We validate our method both computationally and experimentally and demonstrate that it has good predictive power for a number of diverse 3D printed shapes.

Deployable structures are physical mechanisms that can easily transition between two or more geometric configurations; such structures enable industrial, scientific, and consumer applications at a wide variety of scales. This paper develops novel deployable structures that can approximate a large class of doubly-curved surfaces and are easily actuated from a flat initial state via inflation or gravitational loading. The structures are based on two-dimensional rigid mechanical linkages that implicitly encode the curvature of the target shape via a user-programmable pattern that permits locally isotropic scaling under load. We explicitly characterize the shapes that can be realized by such structures---in particular, we show that they can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain. Based on this observation, we develop efficient computational design algorithms for approximating a given input geometry. The resulting designs can be rapidly manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. We validate our approach through a series of physical prototypes and present several application case studies, ranging from surgical implants to large-scale deployable architecture.

Additive fabrication technologies are limited by the types of material they can print: while the technologies are continuously improving, still only a relatively small discrete set of materials can be used in each printed object. At the same time, the low cost of introducing geometric complexity suggests the alternative of controlling the elastic material properties by producing microstructures , which can achieve behaviors significantly differing from the solid printing material. While promising results have been obtained in this direction, fragility is a significant problem blocking practical applications, especially for achieving soft material properties: due to stress concentrations at thin joints, deformations and repeated loadings are likely to cause fracture. We present a set of methods to minimize stress concentrations in microstructures by evolving their shapes. First, we demonstrate that the worst-case stress analysis problem (maximizing a stress measure over all possible unit loads) has an exact solution for periodic microstructures. We develop a new, accurate discretization of the shape derivative for stress objectives and introduce a low-dimensional parametric shape model for microstructures. This model supports robust minimization of maximal stress (approximated by an L p norm with high p ) and an efficient implementation of printability constraints. In addition to significantly reducing stresses (by a typical factor of 5X), the new method substantially expands the range of effective material properties covered by the collection of structures.

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.