Orientational Order on Surfaces: The Coupling of Topology, Geometry, and Dynamics

Research output: Contribution to journalResearch articleContributedpeer-review



We consider the numerical investigation of surface bound orientational order using unit tangential vector fields by means of a gradient flow equation of a weak surface Frank–Oseen energy. The energy is composed of intrinsic and extrinsic contributions, as well as a penalization term to enforce the unity of the vector field. Four different numerical discretizations, namely a discrete exterior calculus approach, a method based on vector spherical harmonics, a surface finite element method, and an approach utilizing an implicit surface description, the diffuse interface method, are described and compared with each other for surfaces with Euler characteristic 2. We demonstrate the influence of geometric properties on realizations of the Poincaré–Hopf theorem and show examples where the energy is decreased by introducing additional orientational defects.


Original languageEnglish
Pages (from-to)147-191
JournalJournal of nonlinear science
Early online date24 Jul 2017
Publication statusPublished - Feb 2018

External IDs

WOS 000419582200006
Scopus 85025578679


Research priority areas of TU Dresden

DFG Classification of Subject Areas according to Review Boards


  • polar liquid crystals, curved surfaces, nematic shell, free energy

Library keywords