Computing self-supporting surfaces by regular triangulation

Abstract

Figure 1: Left: self-supporting surfaces with unsupported (top) and supported (bottom) boundary constraints. Unsupported boundary vertices and their corresponding power cells are colored in orange. Top right: initial self-supporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distributed more evenly. AbstractMasonry structures must be compressively self-supporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram, defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete self-supporting network. This observation lets us define a new and convenient parameterization for the space of self-supporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.

Weighted Triangulations for Geometry Processing

“…To overcome these issues, weighted triangulations [Glickenstein 2005] were introduced as an alternative with much greater flexibility in the location of dual vertices while maintaining primal-dual orthogonality. These orthogonal primal-dual structures have found recent adoption in geometry processing for mesh optimization [Mullen et al 2011] and architectural design [Liu et al 2013;de Goes et al 2013]. In this paper, we present further theoretical foundations and computational methods required for the construction of weighted triangulations and their use in geometry processing.…”

“…In Sec. 6, we use our definition of metric on weighted triangulations to design new algorithms for meshing: we first offer an alternative to [Liu et al 2013] in order to construct self-supporting triangulations with constant relative mean curvature through a convex optimization (Sec. 6.1), we then extend the work of [Mullen et al 2011] to generate well-centered surface meshes (Sec.…”

“…However, many geometry processing applications rely, overtly or covertly, on an orthogonal dual structure to the primal mesh. The use of such a dual structure is very application-dependent, with circumcentric and power duals being found, for instance, in physical simulation [Elcott et al 2007;Batty et al 2010], architecture modeling [Liu et al 2013;de Goes et al 2013] and parameterization [Mercat 2001;Jin et al 2008]. While most of these results are limited to planar triangle meshes, little attention has been paid to exploring orthogonal duals for triangulated surface meshes.…”

Computational design of stable planar-rod structures

“…Whiting et al [2009; presented methods to automatically adjust architectural models to guarantee structural stability. Recent work also investigated the design of valid self-supporting shapes [Vouga et al 2012;Liu et al 2013;De Goes et al 2013; Figure 3: Our work is inspired by wire sculptures, such as the elephant by Alexander Calder 1 or the dog by Jakobi 2 . Panozzo et al 2013] and the decomposition of 3D shapes into selfsupporting, discrete-element assemblies [Frick et al 2015].…”

Interactive Modeling of Architectural Freeform Structures: Combining Geometry with Fabrication and Statics

“…The incorporation of structural constraints into shape design is probably best accomplished by aiming at force equilibrium. This approach is taken within the thrust network method in connection with the design of self-supporting structures [Block and Ochsendorf 2007;Block 2009;Block and Lachauer 2011;Vouga et al 2012;Panozzo et al 2013;de Goes et al 2013;Liu et al 2013], as well as closely related form-finding methods for compression support structures [Lachauer and Block 2012], fabric formwork for concrete shells [Van Mele and Block 2011] and tension structures [Barnes 2009]. …”