- Geometry Supporting the Realization of Freeform Architecture
- Principal Investigators: Alexander Bobenko, Helmut Pottmann, Johannes Wallner
- Investigators: Leonardo Alese, Wolfgang Carl, Felix Dellinger, Martin Kilian, Florian Käferböck, Klara Mundilova, Christian Müller, Davide Pellis, Thilo Rörig, Andrew Sageman-Furnas
Many of today's most striking buildings are nontraditional freeform shapes. Their fabrication is a big challenge, but also a rich source of research topics in geometry. Project A08 addresses key questions such as: "How can we most efficiently represent and explore the variety of manufacturable designs?" or "Can we do this even under structural constraints such as force equilibrium?" Answers to these questions are expected to support the development of next generation modelling tools which combine shape design with key aspects of function and fabrication.
C02: Digital Representations of Manifold Data (continued)
- Principal Investigators: Felix Krahmer, Gitta Kutyniok
- Investigators: Olga Graf, Sandra Keiper, Sara Krause-Solberg, Michael Sandbichler, Nada Sissouno
Whenever data is processed using computers, analog-to-digital (A/D) conversion, that is, representing the data using just 0s and 1s, is a crucial ingredient of the procedure. This project will mathematically study approaches to this problem for data with geometric structural constraints.
- Principal Investigators: Massimo Fornasier, Gitta Kutyniok
- Investigators: Stefano Almi, Sandro Belz, Mauro Bonafini, Markus Hansen, Philipp Petersen, Mones Raslan
A brittle material, subjected to a force, first deforms itself elastically, then it breaks without any intermediate phase. A model of brittle fractures was proposed by Francfort and Marigo, where the displacement is typically a smooth function except on a relatively smooth jump set determining the fracture. In this project we intend to compare anisotropic and adaptive mesh refinements with adaptive frame methods based on shearlets. In particular, by taking advantage of the shearlet property of optimally approximating piecewise smooth functions, we aim at reaching not only a proof of the convergence of the frame adaptive algorithms but also their optimal complexity.
- Principal Investigators: Ulrich Bauer, Herbert Edelsbrunner
- Investigators: Magnus Botnan, Benedikt Fluhr, Grzegorz Jablonski, Fabian Lenzen, Anton Nikitenko, Florian Pausinger, Abhishek Rathod, Hubert Wagner
The following more detailed questions are studied within this project: The definition and construction of geometric complexes from data. Topological persistence. The homology of dynamical systems. The convergence of variants of Crofton's formula obtained with persistent homology to compute intrinsic volumes. The approximation of persistent homology through simplification of the representative complexes.
C05: Computational and structural aspects of point set surfaces (continued as C05: Computational and structural aspects in multi-scale shape interpolation)
- Implementation of Manifold Structures in Point Clouds
- Principal Investigators: Konrad Polthier
- Investigators: Henriette-Sophie Lipschütz, Konstantin Poelke, Martin Skrodzki, Eric Zimmermann
Point set surfaces have a long history in geometry processing and computer graphics as they naturally arise in 3D-data acquisition processes. A guiding principle of these algorithms is the direct processing of raw scanning data without prior meshing. However, a thorough investigation of a differential geometric representation of point set surfaces and their properties is not available. Inspired by the notion of manifolds, we develop new concepts for meshless charts and atlases and use these to establish sound formulations of discrete differential operators on point set surfaces.
C07: Vorticity concentration for fluid simulation (continued as C07: Discretizing fluids into filaments and sheets)
- Development of new algorithms for fluid simulation, visualization and processing.
- Principal Investigators: Ulrich Pinkall
- Investigators: Albert Chern, Felix Knöppel, Christoph Seidel
One of the main problems in Fluid simulations for Computer Graphics is that large scale phenomena are often driven by structures like vortex filaments or vortex sheets which are thin in compare to feasible grid resolutions and quickly destroyed by numerical diffusion. In this project we develop new tools for fluid simulation, visualization and processing based on a quantum mechanical description of incompressible fluids, which is capable to reproduce phenomena driven by thin vortical structures faithfully even on coarse grids.