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Orientational order on surfaces: the coupling of topology, geometry, and dynamics. (English) Zbl 1391.35335

Authors’ abstract: The authors 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 are described and compared with each other for surfaces with Euler characteristic 2. The authors 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A45 Differential geometric aspects in vector and tensor analysis
58K45 Singularities of vector fields, topological aspects
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
37E35 Flows on surfaces
76A15 Liquid crystals
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