Generalized deployable elastic geodesic grids

Abstract

Given a designer created free-form surface in 3d space, our method computes a grid composed of elastic elements which are completely planar and straight. Only by fixing the ends of the planar elements to appropriate locations, the 2d grid bends and approximates the given 3d surface. Our method is based purely on the notions from differential geometry of curves and surfaces and avoids any physical simulations. In particular, we introduce a well-defined elastic grid energy functional that allows identifying networks of curves that minimize the bending energy and at the same time nestle to the provided input surface well. Further, we generalize the concept of such grids to cases where the surface boundary does not need to be convex, which allows for the creation of sophisticated and visually pleasing shapes. The algorithm finally ensures that the 2d grid is perfectly planar, making the resulting gridshells inexpensive, easy to fabricate, transport, assemble, and finally also to deploy. Additionally, since the whole structure is pre-strained, it also comes with load-bearing capabilities. We evaluate our method using physical simulation and we also provide a full fabrication pipeline for desktop-size models and present multiple examples of surfaces with elliptic and hyperbolic curvature regions. Our method is meant as a tool for quick prototyping for designers, architects, and engineers since it is very fast and results can be obtained in a matter of seconds.

C-Shells: Deployable Gridshells with Curved Beams

“…This later work introduces a purely geometric approach for simultaneously constructing a grid of geodesics on a curved surface patch and a corresponding planar grid into which it flattens via a single angle degree of freedom. In follow-up work, elastic geodesic grids were expanded to include multiple patches [Pillwein and Musialski 2021] and non-convex boundaries [Pillwein and Musialski 2021], though the latter involves actuation via boundary constraints.…”

“…With our inverse flattening approach, we find a much better initial layout that can be successfully optimized to capture the details of the design surface. Note that this model can neither be achieved with X-shells nor with G-shells (refer to Figure 19 in Pillwein and Musialski [2021]).…”

“…Such scissor linkages expand and contract freely in-plane and can be deformed into curved shapes by imposing constraints on the boundary [Quinn and Gengnagel 2014]. More recent methods, such as X-shells ], G-shells [Soriano et al 2019], and elastic geodesic grids [Pillwein et al 2020;Pillwein and Musialski 2021], arrange straight beams in non-uniform grids. This nonuniformity leads to kinematic incompatibilities when the linkage is deployed, which forces the structure to buckle out of plane to reach an elastic equilibrium (see Figure 2).…”

Computational Design of Laser-Cut Bending-Active Structures

“…These structures share with ours that they are fabricated flat and assume a 3D shape once deformed. Existing fabrication techniques include the use of flat beams for weaved items [9,10] and for gridshells [11,12,13], well adapted to large-scale structures, e.g., architectural installations, sealed inflatable membranes [14,15] and 3D printed structures [16] for smaller scale models, wire meshes [17], as well as laser-cut rigid panels [18,3] or softer elastic sheets [19,20]. Common to all these approaches is that curvature is typically obtained from metric frustration.…”

Stability-Aware Simplification of Curve Networks

“…One line of research focuses on biaxial or triaxial weaving to create regular ribbon structures [Vekhter et al, 2019;Campen and Kobbelt, 2014;Akleman et al, 2009;Takezawa et al, 2016;Tao et al, 2016Tao et al, , 2017Ren et al, 2021]. Other physical curve networks that have been explored include wire meshes [Garg et al, 2014], structures made from planar pre-bent rods [Miguel et al, 2016] and circular arcs [Bo et al, 2011], 3D-printed curve networks [Pérez et al, 2015;Pérez et al, 2017], and regular gridshells [Pillwein and Musialski, 2021;Panetta et al, 2019;Lienhard and Knippers, 2015;Schling et al, 2018]. While the above methods focus on highly regular networks and/or fixed topology, we target curve networks that are not necessarily regular with a priori unknown combinatorics.…”