Geometrische Krümmungsfunktionale: Energielandschaft und diskrete Methoden

• Antragsteller: Professor Dr. Heiko von der Mosel; Professor Dr. Max Wardetzky
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2015 bis 2021
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 282535003

• Erstellungsjahr: 2021

Zusammenfassung der Projektergebnisse

We report on our investigation of various geometric curvature functionals. In our investigation we covered classical curvature energies, such as Euler elasticae and bending energies, as well as geometrically defined self-avoidance energies for curves, surfaces, or more general m-dimensional sets in Rn . As planned, our research was driven by a combination of analytic techniques for studying knot energies and numerical algorithms and a convergence analysis thereof. We contributed to a deeper understanding of the energy landscape of highly nonlinear and partially singular and non-local geometric curvature energies. We investigated the impact of these energies on geometric knot theory alongside suitable structure-preserving discrete versions that we developed and analyzed. Our tools ranged from non-local fractional differential operators over measure theory, geometric topology, smooth and discrete variational calculus, discrete differential geometry to numerical algorithms and optimisation, using tools from hierarchical clustering and multigrid methods. We embarked on the investigation of the main questions outlined in the proposal of this project and made significant progress in terms of the exploration of the energy landscape of various curvature functionals, both from a theoretical and from a numerical perspective. Keywords: geometric curvature energies, singular integrals, geometric knot theory, discrete methods, variational calculus

Projektbezogene Publikationen (Auswahl)

• “Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies”. In: Comm. Anal. Geom. 26.6 (2018), pp. 1251–1316
  Kołasiński, Sławomir; Strzelecki, Paweł & von der Mosel, Heiko
  
• “Natural Boundary Conditions for Smoothing in Geometry Processing”. In: ACM Trans. Graph. 37.2 (2018)
  Stein, Oded; Grinspun, Eitan; Wardetzky, Max & Jacobson, Alec
  
• “A Reifenberg type characterization for m-dimensional C1-submanifolds of Rn ”. In: Ann. Acad. Sci. Fenn. Math. 44.2 (2019), pp. 693–721
  Käfer, Bastian
  
• “A Simple Discretization of the Vector Dirichlet Energy”. In: Computer Graphics Forum 39.5 (2020), pp. 81–92
  Stein, Oded; Wardetzky, Max; Jacobson, Alec & Grinspun, Eitan
  
• “A Smoothness Energy without Boundary Distortion for Curved Surfaces”. In: ACM Trans. Graph. 39.3 (2020)
  Stein, Oded; Jacobson, Alec; Wardetzky, Max & Grinspun, Eitan
  
• “Variational convergence of discrete elasticae”. In: IMA Journal of Numerical Analysis (Dec. 2020)
  Scholtes, Sebastian; Schumacher, Henrik & Wardetzky, Max
  
• “Variational Methods for Discrete Geometric Functionals”. In: Handbook of Variational Methods for Nonlinear Geometric Data. Ed. by Philipp Grohs, Martin Holler, and Andreas Weinmann. Cham: Springer International Publishing, Apr. 2020, pp. 153–172
  Schumacher, Henrik & Wardetzky, Max
  
• Sobolev Gradients for the Möbius Energy. Archive of Rational Mechanics and Analysis. May 2020
  Reiter, Philipp & Schumacher, Henrik
  
• “Repulsive Curves”. In: ACM Trans. Graph. 40.2 (May 2021)
  Yu, Chris; Schumacher, Henrik & Crane, Keenan