Approximation of the Willmore energy by a discrete geometry model

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of Γ-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.

Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry

A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples.

Embedded Actuation for Shape-Adaptive Origami

Origami-inspired approaches to deployable or morphing structures have received significant interest. For such applications, the shape of the origami structure must be actively controlled. We propose a distributed network of embedded actuators which open/close individual folds and present a methodology for selecting the positions of these actuators. The deformed shape of the origami structure is tracked throughout its actuation using local curvatures derived from discrete differential geometry. A Genetic Algorithm (GA) is used to select an actuation configuration, which minimizes the number of actuators or input energy required to achieve a target shape. The methodology is applied to both a deployed and twisted Miura-ori sheet. The results show that designing a rigidly foldable pattern to achieve shape-adaptivity does not always minimize the number of actuators or input energy required to reach the target geometry.

Numerical Method for Direct Solution to Form-Finding Problem in Convex Gridshell

Elastic gridshell is a class of net-like structure formed by an ensemble of elastically deforming rods coupled through joints, such that the structure can cover large areas with low self-weight and allow for a variety of aesthetic configurations. Gridshells, also known as X-shells or Cosserat Nets, are a planar grid of elastic rods in its undeformed configuration. The end points of the rods are constrained and positioned on a closed curve—the final boundary—to actuate the structure into a 3D shape. Here, we report a discrete differential geometry-based numerical framework to study the geometrically nonlinear deformation of gridshell structures, accounting for non-trivial bending-twisting coupling at the joints. The form-finding problem of obtaining the undeformed planar configuration given the target convex 3D topology is then investigated. For the forward (2D to 3D) physically based simulation, we decompose the gridshell structure into multiple one-dimensional elastic rods and simulate their deformation by the well-established discrete elastic rods (DER) algorithm. A simple penalty energy between rods and linkages is used to simulate the coupling between two rods at the joints. For the inverse problem associated with form-finding (3D to 2D), we introduce a contact-based algorithm between the elastic gridshell and a rigid 3D surface, where the rigid surface describes the target shape of the gridshell upon actuation. This technique removes the need of several forward simulations associated with conventional optimization algorithms and provides a direct solution to the inverse problem. Several examples—hemispherical cap, paraboloid, and hemi-ellipsoid—are used to show the effectiveness of the inverse design process.

Voss Surfaces: A Design Space for Geodesic Gridshells

The design of envelopes with complex geometries often leads to construction challenges. To overcome these difficulties, resorting to discrete differential geometry proved successful by establishing close links between mesh properties and the existence of good fabrication, assembling and mechanical properties. In this paper, the design of a special family of structures, called geodesic shells, is addressed using Voss nets, a family of discrete surfaces. The use of discrete Voss surfaces ensures that the structure can be built from simply connected, initially straight laths, and covered with flat panels. These advantageous constructive properties arise from the existence of a conjugate network of geodesic curves on the underlying smooth surface. Here, a review of Voss nets is presented and particular attention is given to the projection of normal vectors on the unit sphere. This projection, called Gauss map, creates a dual net which unveils the remarkable characteristics of Voss nets. Then, based on the previous study, two generation methods are introduced. One enables the exploration and the deformation of Voss nets while the second provides a more direct computational technique. The application of theses methodologies is discussed alongside formal examples.