Discrete geometric structures motivated by applications and architecture

In the last decade, the design and realization of freeform shapes in architecture has emerged as a new field of applications as well as an inspiration for basic research in geometry, in particular discrete differential geometry. This field of research combines pure Mathematics with computational aspects in an interdisciplinary manner. It must also be seen in the broader context of Computational Design, thus extending the original narrow focus on freeform architecture to applications in engineering, computer science, and the arts. This project is embedded in the SFB-Transregio "Discretization in geometry and dynamics" and is conducted as a collaboration of TU Berlin, TU Graz, and TU Wien. We have identified three topics which are at the front of current research: Firstly, smooth extensions of discrete surfaces are directly related to the paneling problem of freeform architecture - from the viewpoint of mathematics, they combine geometry with integrable systems. Our goal is to characterize the smooth surface patches of the extension by the structure of the underlying discrete net. Secondly, curved-crease surfaces occur in many places, from Origami to architecture to the packing of solar sails. We plan to build on previous work on semi-discrete surfaces, on non-smooth polyhedral patterns, and on flexible structures. We aim at understanding the shapes of curved-folding surfaces and related surface arrangements, and to make approximation and modeling computationally accessible. Thirdly, we propose to intensively study the inclusion of forces in geometric design. Quite apart from the obvious relevance of statics for architecture in general, there are very interesting connections between discrete differential geometry and self-supporting structures to be explored. The roots of this connection lie in graphical statics pioneered in the 19th century. We plan to obtain geometric insights which are valuable for structural analysis, and which can be utilized for approximation and design, as well as for topology optimization. These three topics have two aspects in common: One is the background of discrete differential geometry. The other one is that they will be investigated with a view towards fast algorithms and design. The nonlinear nature of constraints makes interactive modeling a challenge, and only recently first progress could be made in this area. This research is thus part of a long-time strategy to obtain "intelligent" design tools which can handle nontrivial geometric constraints combined with statics at interactive speed.

In the last decade, the design and realization of freeform shapes in architecture has emerged as a new fieldof applications as well as an inspiration for basic research in geometry, in particular discrete differential geometry. This field of research combines pure Mathematics with computational aspects in an interdisciplinary manner. It must also be seen in the broader context of Computational Design, thus extending the original narrow focus on freeform architecture to applications in engineering, computer science, and the arts. This project was embedded in the SFB-Transregio "Discretization in geometry and dynamics" and was conducted as a collaboration of TU Berlin, TU Graz, andTU Wien. We have identified three topics which are at the front of current research: Firstly, smooth extensions of discrete surfaces are directly related to the paneling problem of freeform architecture - from the viewpoint of mathematics, they combine geometry with integrable systems. We characterize the smooth surface patches of the extension by the structure of the underlying discrete net. Secondly, curved-crease surfaces occur in many places, from Origami to architecture to the packing of solar sails. We build on previous work on semi-discrete surfaces, on non-smooth polyhedral patterns, and on flexible structures. We study the shapes of curved-folding surfaces and related surface arrangements, and make approximation and modeling computationally accessible. Thirdly, we study the inclusion of forces in geometric design. Quite apart from the obvious relevance of statics for architecture in general, there are very interesting connections between discrete differential geometry and self-supporting structures. The roots of this connection lie in graphical statics pioneered in the 19th century. We obtained geometric insights which are valuable for structural analysis, and which can be utilized for approximation and design, as well as for topology optimization. These three topics have two aspects in common: One is the background of discrete differential geometry. An other one is the investigation of fast algorithms and design. The nonlinear nature of constraints makes interactive modeling a challenge, and only recently first progress could be made in this area. This research is thus part of a long-time strategy to obtain "intelligent" design tools which can handle nontrivial geometric constraints combined with statics at interactive speed.