Cloaking objects with metamaterials has been extensively studied to hide internal objects across various physical properties, including optical, acoustic, and thermal. We explore a new direction in mechanical cloaking: halftoning an image using a porous structure that behaves like a uniform, linear, isotropic material and visually matches a target image. For an external observer, this creates the surprising effect where the object appears mechanically isotropic and homogeneous while its porous structure resembles a target image. We introduce a parametric class of porous structures suitable for this problem, as demonstrated by numerical experiments. The structures we define offer a wide range of visual contrasts, enabling effective halftoning while maintaining near isotropic effective mechanical properties.

We introduce a divergence-free nD vector noise defined as the n-dimensional cross product of the gradients of n − 1 noise functions. We show that this vector noise function is divergence-free and hence volume preserving for any dimension n. Our method enables precise integration and extends to new settings by substituting noise functions with implicit surfaces, (hyper)surfaces, or custom functions. We demonstrate applications including image warping, surface texturing, noise bounded by implicit surfaces, anisotropic curl-noise, and high-dimensional point jittering up to 7D.

We propose a method for implicitly generating blue noise point sets. Our method is based on the observations that curl noise vector fields are volume-preserving and that jittering can be construed as moving points along the streamlines of a vector field. We demonstrate that the volume preservation keeps the points well separated when jittered using a curl noise vector field. At the same time, the anisotropy that stems from regular lattices is significantly reduced by such jittering. In combination, these properties entail that jittering by curl noise effectively transforms a regular lattice into a point set with blue noise properties. Our implicit method does not require computing the point set in advance. This makes our technique valuable when an arbitrarily large set of points with blue noise properties is needed. We compare our method to several other methods based on jittering as well as other methods for blue noise point set generation. Finally, we show several applications of curl noise jittering in two and three dimensions.

We present a method to 3D print surfaces exhibiting a prescribed varying field of anisotropic appearance using only standard fused filament fabrication printers. This enables the fabrication of patterns triggering reflections similar to that of brushed metal with direct control over the directionality of the reflections. Our key insight, on which we ground the method, is that the direction of the deposition paths leads to a certain degree of surface roughness, which yields a visual anisotropic appearance. Therefore, generating dense cyclic infills aligned with a line field allows us to grade the anisotropic appearance of the printed surface. To achieve this, we introduce a highly parallelizable algorithm for optimizing oriented, cyclic paths. Our algorithm outperforms existing approaches regarding efficiency, robustness, and result quality. We demonstrate the effectiveness of our technique in conveying an anisotropic appearance on several challenging test cases, ranging from patterns to photographs reinterpreted as anisotropic appearances.

While 3D printing enables the customization and home fabrication of a wide range of shapes, fabricating freeform thin-shells remains challenging. As layers misalign with the curvature, they incur structural deficiencies, while the curved shells require large support structures, typically using more material than the part itself. We present a computational framework for optimizing the internal structure of 3D printed plates such that they morph into a desired freeform shell when heated. This exploits the shrinkage effect of thermoplastics such as PLA, which store internal stresses along the deposition directions. These stresses get released when the material is heated again above its glass transition temperature, causing an anisotropic deformation that induces curvature. Our inverse design method takes as input a freeform surface and finds an optimized set of deposition trajectories in each layer such that their anisotropic shrinkage deforms the plate into the prescribed surface geometry. We optimize for a continuous vector field that varies across the plate and within its thickness. The algorithm then extracts a set of deposition trajectories from the vector field in order to fabricate the flat plates on standard FFF printers. We validate our algorithm on freeform, doubly-curved surfaces.

Abstract We explore the optimization of closed space‐filling curves under orientation objectives. By solidifying material along the closed curve, solid layers of 3D prints can be manufactured in a single continuous extrusion motion. The control over orientation enables the deposition to align with specific directions in different areas, or to produce a locally uniform distribution of orientations, patterning the solidified volume in a precisely controlled manner. Our optimization framework proceeds in two steps. First, we cast a combinatorial problem, optimizing Hamiltonian cycles within a specially constructed graph. We rely on a stochastic optimization process based on local operators that modify a cycle while preserving its Hamiltonian property. Second, we use the result to initialize a geometric optimizer that improves the smoothness and uniform coverage of the cycle while further optimizing for alignment and orientation objectives.

We introduce a computational approach to optimize random porous materials via parametric growth processes. We focus on the problem of minimizing the Poisson’s ratio of a two-phase porous random material, which results in an auxetic material. Initially, we perform a parametric optimization of the growth process. Afterward, the optimized parametric growth process implicitly generates an auxetic random material. Namely, the growth process intrinsically entails the formation of an auxetic material. Our approach enables the computation of large-scale auxetic random materials in commodity computers. We also provide numerical results indicating that the computed auxetic materials have close to isotropic linear elastic behavior and physical tests revealing auxetic behavior.

In most layered additive manufacturing processes, a tool solidifies or deposits material while following pre-planned trajectories to form solid beads. Many interesting problems arise in this context, among which one concerns the planning of trajectories for filling a planar shape as densely as possible. This is the problem we tackle in the present paper. Recent works have shown that allowing the bead width to vary along the trajectories helps increase the filling density. We present a novel technique that, given a deposition width range, constructs a set of closed beads whose width varies within the prescribed range and fill the input shape. The technique outperforms the state of the art in important metrics: filling density (while still guaranteeing the absence of bead overlap) and trajectories smoothness. We give a detailed geometric description of our algorithm, explore its behavior on example inputs and provide a statistical comparison with the state of the art. We show that it is possible to obtain high quality fabricated layers on commodity FDM printers.

Nature offers a marvel of astonishing and rich deformation behaviors. Yet, most of the objects we fabricate are comparatively rather inexpressive, either rigid or exhibiting simple homogeneous deformations when interacted with. We explore the synthesis and fabrication of novel microstructures that mimic the effects of having oriented rigid fibers in an otherwise flexible material: the result is extremely rigid along a transverse direction while being comparatively very flexible in the locally orthogonal plane. By allowing free gradation of the rigidity direction orientation within the object, the microstructures can be designed such that, under deformation, distances along fibers in the volume are preserved while others freely change. Through a simple painting tool, this allows a designer to influence the way the volume reshapes when deformed, and results in a wide range of novel possibilities. Many gradations are possible: local free orientation of the fibers; local control of the overall material rigidity (structure density); local canceling of the effect of the fibers, obtaining a more isotropic material. Our algorithm to synthesize the structures builds upon procedural texturing. It produces a cellular geometry that can be fabricated reliably despite 3D printing walls at a minimal thickness, allowing prints to be very flexible. The synthesis algorithm is efficient and scales to large volumes.

Most additive manufacturing processes fabricate objects by stacking planar layers of solidified material. As a result, produced parts exhibit a so-called staircase effect, which results from sampling slanted surfaces with parallel planes. Using thinner slices reduces this effect, but it always remains visible where layers almost align with the input surfaces. In this research we exploit the ability of some additive manufacturing processes to deposit material slightly out of plane to dramatically reduce these artifacts. We focus in particular on the widespread Fused Filament Fabrication (FFF) technology, since most printers in this category can deposit along slightly curved paths, under deposition slope and thickness constraints. Our algorithm curves the layers, making them either follow the natural slope of the input surface or on the contrary, make them intersect the surfaces at a steeper angle thereby improving the sampling quality. Rather than directly computing curved layers, our algorithm optimizes for a deformation of the model which is then sliced with a standard planar approach. We demonstrate that this approach enables us to encode all fabrication constraints, including the guarantee of generating collision-free toolpaths, in a convex optimization that can be solved using a QP solver. We produce a variety of models and compare print quality between curved deposition and planar slicing.

We present a method for designing mechanical metamaterials based on the novel concept of Voronoi diagrams induced by star-shaped metrics. As one of its central advantages, our approach supports interpolation between arbitrary metrics. This capability opens up a rich space of structures with interesting aesthetics and a wide range of mechanical properties, including isotropic, tetragonal, orthotropic, as well as smoothly graded materials. We evaluate our method by creating large sets of example structures, provided as accompanying material. We validate the mechanical properties predicted by simulation through tensile tests on a set of physical prototypes.

Fused filament fabrication is the method of choice for printing 3D models at low cost and is the de facto standard for hobbyists, makers, and schools. Unfortunately, filament printers cannot truly reproduce colored objects. The best current techniques rely on a form of dithering exploiting occlusion, which was only demonstrated for shades of two base colors and behaves differently depending on surface slope. We explore a novel approach for 3D-printing colored objects, capable of creating controlled gradients of varying sharpness. Our technique exploits off-the-shelf nozzles that are designed to mix multiple filaments in a small melting chamber, obtaining intermediate colors once the mix is stabilized. We apply this property to produce color gradients. We divide each input layer into a set of strata, each having a different constant color. By locally changing the thickness of the stratum, we change the perceived color at a given location. By optimizing the choice of colors of each stratum, we further improve quality and allow the use of different numbers of input filaments. We demonstrate our results by building a functional color printer using low-cost, off-the-shelf components. Using our tool, a user can paint a 3D model and directly produce its physical counterpart using any material and color available for fused filament fabrication.

Procedural pattern synthesis is a fundamental tool of Computer Graphics, ubiquitous in games and special effects. By calling a single procedure in every pixel - or voxel - large quantities of details are generated at low cost, enhancing textures, producing complex structures within and along surfaces. Such procedures are typically implemented as pixel shaders. We propose a novel procedural pattern synthesis technique that exhibits desirable properties for modeling highly contrasted patterns, that are especially well suited to produce surface and microstructure details. In particular, our synthesizer affords for a precise control over the profile, orientation and distribution of the produced stochastic patterns, while allowing to grade all these parameters spatially. Our technique defines a stochastic smooth phase field - a phasor noise - that is then fed into a periodic function (e.g. a sine wave), producing an oscillating field with prescribed main frequencies and preserved contrast oscillations. In addition, the profile of each oscillation is directly controllable (e.g. sine wave, sawtooth, rectangular or any 1D profile). Our technique builds upon a reformulation of Gabor noise in terms of a phasor field that affords for a clear separation between local intensity and phase. Applications range from texturing to modeling surface displacements, as well as multi-material microstructures in the context of additive manufacturing.

A critical advantage of additive manufacturing is its ability to fabricate complex small-scale structures. These microstructures can be understood as a metamaterial : they exist at a much smaller scale than the volume they fill, and are collectively responsible for an average elastic behavior different from that of the base printing material making the fabricated object lighter and/or flexible along specific directions. In addition, the average behavior can be graded spatially by progressively modifying the micro structure geometry. The definition of a microstructure is a careful trade-off between the geometric requirements of manufacturing and the properties one seeks to obtain within a shape: in our case a wide range of elastic behaviors. Most existing microstructures are designed for stereolithography (SLA) and laser sintering (SLS) processes. The requirements are however different than those of continuous deposition systems such as fused filament fabrication (FFF), for which there is currently a lack of microstructures enabling graded elastic behaviors. In this work we introduce a novel type of microstructures that strictly enforce all the requirements of FFF-like processes: continuity, self-support and overhang angles. They offer a range of orthotropic elastic responses that can be graded spatially. This allows to fabricate parts usually reserved to the most advanced technologies on widely available inexpensive printers that also benefit from a continuously expanding range of materials.

Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step along the processing pipeline that prepares the 3D model for fabrication can now count on dozens of methods, that analyse and optimize geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during model preparation – a process globally refereed to as Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.

Recent advances in 3D printing have made it easier to manufacture customized objects by ordinary users in an affordable manner, and therefore spurred high demand for more accessible methods for designing and fabricating 3D objects of various shapes and functionalities. In this paper we present a novel approach to model and fabricate surface-like objects composed of connected tiles, which can be used as objects in daily life, such as ornaments, covers, shades or handbags.

Additive manufacturing enables the fabrication of objects embedding meta-materials. By creating fine-scale structures, the object's physical properties can be graded (e.g. elasticity, porosity), even though a single base material is used for fabrication. Designing the fine and detailed geometry of a metamaterial while attempting to achieve specific properties is difficult. In addition, the structures are intended to fill comparatively large volumes, which quickly leads to large data structures and intractable simulation costs. Thus, most metamaterials are defined as periodic structures repeated in regular lattices. The periodicity simplifies modeling, simulation, and reduces memory costs - however it limits the possibility to smoothly grade properties along free directions. In this work, we propose a novel metamaterial with controllable, freely orientable, orthotropic elastic behavior - orthotropy means that elasticity is controlled independently along three orthogonal axes, which leads to materials that better adapt to uneven, directional load scenarios, and offer a more versatile material design primitive. The fine-scale structures are generated procedurally by a stochastic process, and resemble a foam. The absence of global organization and periodicity allows the free gradation of density, orientation, and stretch, leading to the controllable orthotropic behavior. The procedural nature of the synthesis process allows it to scale to arbitrarily large volumes at low memory costs. We detail the foam structure synthesis, analyze and discuss its properties through numerical and experimental verifications, and finally demonstrate the use of orthotropic materials for the design of 3D printed objects.

Microstructures at the scale of tens of microns change the physical properties of objects, making them lighter or more flexible. While traditionally difficult to produce, additive manufacturing now lets us physically realize such microstructures at low cost. In this paper we propose to study procedural, aperiodic microstructures inspired by Voronoi open-cell foams. The absence of regularity affords for a simple approach to grade the foam geometry --- and thus its mechanical properties --- within a target object and its surface. Rather than requiring a global optimization process, the microstructures are directly generated to exhibit a specified elastic behavior. The implicit evaluation is akin to procedural textures in computer graphics, and locally adapts to follow the elasticity field. This allows very detailed structures to be generated in large objects without having to explicitly produce a full representation --- mesh or voxels --- of the complete object: the structures are added on the fly, just before each object slice is manufactured. We study the elastic behavior of the microstructures and provide a complete description of the procedure generating them. We explain how to determine the geometric parameters of the microstructures from a target elasticity, and evaluate the result on printed samples. Finally, we apply our approach to the fabrication of objects with spatially varying elasticity, including the implicit modeling of a frame following the object surface and seamlessly connecting to the microstructures.

The field of topology optimization seeks to optimize shapes under structural objectives, such as achieving the most rigid shape using a given quantity of material. Besides optimal shape design, these methods are increasingly popular as design tools, since they automatically produce structures having desirable physical properties, a task hard to perform by hand even for skilled designers. However, there is no simple way to control the appearance of the generated objects. In this paper, we propose to optimize shapes for both their structural properties and their appearance, the latter being controlled by a user-provided pattern example. These two objectives are challenging to combine, as optimal structural properties fully define the shape, leaving no degrees of freedom for appearance. We propose a new formulation where appearance is optimized as an objective while structural properties serve as constraints. This produces shapes with sufficient rigidity while allowing enough freedom for the appearance of the final structure to resemble the input exemplar. Our approach generates rigid shapes using a specified quantity of material while observing optional constraints such as voids, fills, attachment points, and external forces. The appearance is defined by examples, making our technique accessible to casual users. We demonstrate its use in the context of fabrication using a laser cutter to manufacture real objects from optimized shapes.

Abstract Skeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis‐aligned and 45° rotated planar faces and edges.