Active Projects of the SFB Transregio 109

C01: Discrete Geometric Structures Motivated by Applications and Architecture
  • Geometry Supporting the Realization of Freeform Architecture
  • Alexander Bobenko, Helmut Pottmann, Johannes Wallner
  • 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
  • Felix Krahmer, Gitta Kutyniok
  • 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.

C03: Shearlet approximation of brittle fracture evolutions
  • Gitta Kutyniok
  • 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. This approach has the advantage not to require a pre-defined crack path, but has the drawback that any mesh discretization is a geometrical bias. Despite encouraging numerical results, FEM are expected to retain a certain geometrical bias and so far a proof of convergence of the proposed algorithm remains out of reach. Differently from finite elements, shearlets are frames with rotation invariance and optimal nonlinear approximation properties for the class of functions, which are smooth except on smooth lower dimensional sets. 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.

C04: Persistence and Stability of Geometric Complexes
  • Ulrich Bauer, Herbert Edelsbrunner
  • 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
  • Implementation of Manifold Structures in Point Clouds
  • Konrad Polthier
  • Point set surfaces have a more than 15 year 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. On this solid basis of meshless differential operators, we develop novel algorithms for important geometry processing tasks, such as feature recognition, filtering operations, and surface parameterization.

C07: Vorticity concentration for fluid simulation
  • Development of new algorithms for fluid simulation, visualization and processing.
  • Ulrich Pinkall
  • Fluid simulations for Computer Graphics create challenges which are outside the scope of standard techniques in Computational Fluid Dynamics. One of the main problems 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.