Journal of Nonlinear Science

, Volume 28, Issue 1, pp 147–191 | Cite as

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

  • M. NestlerEmail author
  • I. Nitschke
  • S. Praetorius
  • A. Voigt


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.


Polar liquid crystals Curved surface Nematic shell Intrinsic–extrinsic free energy 

List of symbols



Surface divergence


Surface gradient


Surface curl

\(\Delta _{\mathcal {S}}\)

Surface Laplace–Beltrami operator

\(\varvec{\Delta }^{\text {dR}}\)

Surface Laplace–deRham operator

Discrete Exterior Calculus


Edge, \(e\in \mathcal {E}\)

\(\star e\)

Dual edge of e (Voronoi edge)

\(\mathcal {E}\)

Set of edges, with number \(|\mathcal {E}|\)


Edge vector along edge e

\(\mathbf{e}_{\star }\)

Dual edge vector along dual chain \(\star e\)


Face, \(T\in \mathcal {T}\)

\(\mathcal {T}\)

Set of faces, with number \(|\mathcal {T}|\)


Hodge star operator

\(\flat \)

Lowering indices

\(\sharp \)

Rising indices

\({\varvec{\alpha }}\)

1-Form, \({\varvec{\alpha }}\in \varLambda ^{1} (\mathcal {S})\)

\(\alpha _{h}\)

Discrete 1-form, \(\alpha _{h}\in \varLambda _{h}^{1}(\mathcal {K})\)

\(\underline{\varvec{\alpha }}\)

Primal-dual 1-form, \(\underline{\varvec{\alpha }}=(\alpha _{h}, *\alpha _{h})\)

\(\mathcal {K}\)

Simplicial complex


Vertex, \(v\in \mathcal {V}\)

\(\star v\)

Dual vertex (voronoi cell)

\(\mathcal {V}\)

Set of vertices, with number \(|\mathcal {V}|\)


Exterior derivative


\(\varGamma _{ij}^{k}\)

Christoffel symbols of second kind

\(\theta \)

Colatitude coordinate, \(\theta \in [0,\pi ]\)

\(\varphi \)

Azimuthal coordinate, \(\varphi \in [0, 2\pi )\)

\(\xi \)

Coordinate in normal direction of the surface

\(\kappa \)

Gaussian curvature

\(\mathcal {H}\)

Mean curvature \(\mathcal {H}=\mathrm{div}\,\varvec{\nu }\)

\(\Omega \)

Domain, \(\Omega \subset \mathbb {R}^{3}\)


Levi–Civita symbols

\(\mathbf {g}\)

Riemannian metric tensor


Determinant of g

\(\pi \)

Coordinate projection \(\pi :\Omega _\delta \rightarrow \mathcal {S}\)

\(\pi _{\mathsf {T}\mathcal {S}}\)

Surface projection \(\pi _{\mathsf {T}\mathcal {S}}:\mathsf {T}\mathbb {R}^3\rightarrow \mathsf {T}\mathcal {S}\)

\(\mathcal {B}\)

Shape operator \(\mathcal {B}=-\mathrm{grad}\,\varvec{\nu }\)

\(\mathcal {S}\)

Surface, i.e., compact closed oriented Riemannian 2-dim. manifold

\(\chi ( \mathcal {S})\)

Characteristic of the surface \(\mathcal {S}\)

\(\mathcal {S}^{E}\)

Ellipsoidal surface

\(\varvec{\nu }\)

Outer surface normal

\(\mathbb {S}^{2}\)

Unit 2-sphere

\(\mathsf {T}\mathcal {S}\)

Tangent bundle of surface \(\mathcal {S}\)

\(\mathsf {T}^{*}\mathcal {S}\)

Cotangent bundle of surface \(\mathcal {S}\)



Uniform Frank constant

\(\omega _{n}\)

Penalty constant for normality

\(\omega _{t}\)

Penalty constant for tangentiality

\({F}_\mathrm {\omega _\mathrm{n}}^\mathcal {S}\)

Weak surface Frank–Oseen energy

\(\epsilon _\mathrm{f}\)

Error in the defect fusion time

\(\epsilon _\mathrm{e}\)

(Normalized) Mean energy error


Discrete time step

\(\tau _k\)

Time step width in the kth time step

Phase Field

\(\phi \)

Phase-field variable

\(\delta _{\mathcal {S}}\)

Surface delta function


Double well, \(W(\phi )\simeq \delta _\mathcal {S}\)

\(\zeta \)

Double-well regularization

\(\varepsilon \)

Interface thickness of phase field

\(d_\mathcal {S}(\mathbf {x})\)

Signed-distance function

Mathematics Subject Classification

58J35 53C21 53A05 53A45 58K45 30F15 



This work is partially supported by the German Research Foundation through Grant Vo889/18. We further acknowledge computing resources provided at JSC under Grant HR06.

Supplementary material

332_2017_9405_MOESM1_ESM.ogv (57.9 mb)
Supplementary material 1 (ogv 59307 KB)


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Authors and Affiliations

  1. 1.Institut für Wissenschaftliches RechnenTechnische Universität DresdenDresdenGermany

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