Since 2017

Some highly efficient forms can be observed in nature, sculpted by the laws of physics. They are an endless source of inspiration for structural systems. Amongst them, soap films have the remarkable property to resist a pressure load with perfectly distributed stresses. Based on their geometry, we develop new methods to design mechanically efficient shells and gridshells.

Discrete S-CMC meshes

Soap films subjected to air pressure have a form mathematically referred to as a constant mean curvature surface (CMC). We developed a method to transform CMC smooth surfaces into meshes that have many properties for gridshell fabrication:

  • They have quadrangular planar faces and torsion-free nodes.
  • They admit an edge offset edges. This property enables a near-perfect alignment of the beams at the node while using only beam cross section.
  • Each face has an inscribed circle. As a result, faces are « roughly square », which provides aesthetic value to the mesh, and also minimizes material loss if panels are cut out of a larger sheet.
  • These meshes inherit the mechanical properties from the smooth CMC they are derived from.
  • There is a sphere packing associated with the vertices.

Linear Weingarten surfaces

CMC surfaces actually belong to a wider family of surfaces, the so-called Linear Weingarten (LW) surfaces. This family also encompasses developable surfaces and surfaces with constant Gaussian curvature. LW surfaces have many geometrical and mechanical properties. In particular, similarly to CMCs, they correspond to the equilibrium shape of a membrane under uniform pressure, but they offer a much wider design space.

No general generation method of these surfaces has been proposed in the literature.  We therefore developed a new variational method to generate  them, based on a new discrete model by triangular meshes. The method allows to model CMC and developable surfaces in a unified framework. Two applications to architectural design are identified: the first one is the design of tensile membranes, the second one is the design of gridshells that combine structural efficiency and fabricability.

  • All
  • Architectural Geometry
  • Gridshell

 Discrete CMC surfaces for doubly-curved building envelopes – X. Tellier et al. (2018)

Author

Xavier Tellier, Cyril Douthe, Olivier Baverel, Laurent Hauswirth

Abstract

Constant mean curvature surfaces (CMCs) have many interesting properties for use as a form for doubly curved structural envelopes. The discretization of these surfaces has been a focus of research amongst the discrete differential geometry community. Many of the proposed discretizations have remarkable properties for envelope rationalization purposes. However, little attention has been paid to generation methods intended for designers.
This paper proposes an extension to CMCs of the method developed by Bobenko, Hoffmann and Springborn (2006) to generate minimal S-isothermic nets. The method takes as input a CMC (smooth or finely triangulated), remeshes its Gauss map with quadrangular faces, and rebuilds a CMC mesh via a parallel transformation. The resulting mesh is S-CMC, a geometric structure discovered by Hoffmann (2010). This type of mesh have planar quads and offset properties, which are of particular interest in the fabrication of gridshells.

Status

Published

Journal

Advances in Architectural Geometry 2018

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Architectural geometry is an active topic of research for architects, engineers and mathematicians. Some books or articles can be recommended on the topic of polyhedral surfaces.

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