A finite element approach for vector- and tensor-valued surface PDEs

We derive a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on general manifolds. This allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surface PDEs. We consider piecewise linear Lagrange surface finite elements on triangulated surfaces and validate the approach by a vector- and a tensor-valued surface Helmholtz problem on an ellipsoid. We experimentally show optimal (linear) order of convergence for these problems. The full functionality is demonstrated by solving a surface Landau-de Gennes problem on the Stanford bunny. All tools required to apply this approach to other vector- and tensor-valued surface PDEs are provided

Hydrodynamic interactions in polar liquid crystals on evolving surfaces

We consider the derivation and numerical solution of the flow of passive and active polar liquid crystals, whose molecular orientation is subjected to a tangential anchoring on an evolving curved surface. The underlying passive model is a simplified surface Ericksen-Leslie model, which is derived as a thin-film limit of the corresponding three-dimensional equations with appropriate boundary conditions. A finite element discretization is considered and the effect of hydrodynamics on the interplay of topology, geometric properties and defect dynamics is studied for this model on various stationary and evolving surfaces. Additionally, we consider an active model. We propose a surface formulation for an active polar viscous gel and exemplarily demonstrate the effect of the underlying curvature on the location of topological defects on a torus

Beris-Edwards Models on Evolving Surfaces: A Lagrange-D'Alembert Approach

Using the Lagrange-D'Alembert principle we develop thermodynamically consistent surface Beris-Edwards models. These models couple viscous inextensible surface flow with a Landau-de Gennes-Helfrich energy and consider the simultaneous relaxation of the surface Q-tensor field and the surface, by taking hydrodynamics of the surface into account. We consider different formulations, a general model with three-dimensional surface Q-tensor dynamics and possible constraints incorporated by Lagrange multipliers and a surface conforming model with tangential anchoring of the surface Q-tensor field and possible additional constraints. In addition to different treatments of the surface Q-tensor, which introduces different coupling mechanisms with the geometric properties of the surface, we also consider different time derivatives to account for different physical interpretations of surface nematics. We relate the derived models to established models in simplified situations, compare the different formulations with respect to numerical realizations and mention potential applications in biology.Comment: 52 page

Nematic liquid crystals on curved surfaces - a thin film limit

We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.Comment: 20 pages, 4 figure

Diskretes Äußeres Kalkül (DEC) auf Oberflächen ohne Rand

In dieser Arbeit geben wir eine Einführung in das Diskrete Äußere Kalkül (engl.: Discrete Exterior Calculus, kurz: DEC), das sich mit der Diskretisierung von Differentialformen und -operatoren beschäftigt. Wir beschränken uns hierbei auf zweidimensionalen orientierten kompakten Riemannschen Mannigfaltigkeiten und zeigen auf, wie diese als wohlzentrierte Simplizialkomplexe zu approximieren sind. Dabei beschreiben wir die Implementierung der Methode und testen diese an Beispielen, wie Helmholtz-artige PDEs und die Berechnung von in- und extrinsischen Krümmungsgrößen.:0 Einführung 1 Diskrete Mannigfaltigkeiten 1.1 Primär- und Dualgitter 1.2 Kettenkomplexe 1.3 Gittergenerierung für Oberflächen 1.4 Implizit gegebene Oberflächen 2 Diskretes Äußeres Kalkül (DEC) 2.1 Diskrete Differentialformen 2.2 Äußere Ableitung 2.3 Hodge-Stern-Operator 2.4 Laplace-Operator 2.5 Primär-Dual-Gradient im Mittel 3 Anwendung: Oberflächenkrümmung 3.1 Weingartenabbildung 3.2 Krümmungsvektor 3.3 Gauß-Bonnet-Operator 3.4 Numerisches Experiment 4 Fazit und Ausblicke 5 Appendix 5.1 Häufige Bezeichner 5.2 Algorithmen 5.3 Krümmungen für impliziten Oberflächen 5.4 Ausgewählte Oberflächen Literaturverzeichni

A numerical approach for fluid deformable surfaces

Fluid deformable surfaces show a solid-fluid duality which establishes a tight interplay between tangential flow and surface deformation. We derive the governing equations as a thin film limit and provide a general numerical approach for their solution. The simulation results demonstrate the rich dynamics resulting from this interplay, where in the presence of curvature any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flow. However, they also show that the only possible stable stationary state in the considered setting is a sphere with zero velocity

Properties of surface Landau-de Gennes Q-tensor models

Uniaxial nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface offer a non trivial interplay between the geometry and the topology of the surface and the orientational degree of freedom. We consider a general thin film limit of a Landau-de Gennes Q-tensor model which retains the characteristics of the 3D model. From this, previously proposed surface models follow as special cases. We compare fundamental properties, such as alignment of the orientational degrees of freedom with principle curvature lines, order parameter symmetry and phase transition type for these models, and suggest experiments to identify proper model assumptions

Mesoscale modeling of deformations and defects in crystalline sheets

We study deformation and defects in thin, flexible sheets with crystalline order using a coarse-grained Phase-Field Crystal (PFC) model. The PFC model describes crystals at diffusive timescales through a continuous periodic field representing the atomic number density. In its amplitude expansion (APFC), a coarse-grained description featuring slowly varying fields retaining lattice deformation, elasticity, and an advanced description of dislocations is achieved. We introduce surface PFC and APFC models in a convenient height formulation encoding normal deformation. With this framework, we then study general aspects of the buckling of strained sheets, defect nucleation on a prescribed deformed surface, and out-of-plane relaxation near dislocations. We benchmark and discuss our results by looking at the continuum limit for buckling under elastic deformation, and at evidence from microscopic models for deformation at defects and defect arrangements. We shed light on the fundamental interplay between lattice distortion at dislocations and out-of-plane deformation by looking at the effect of the annihilation of dislocation dipoles. The scale-bridging capabilities of the devised mesoscale framework are also showcased with the simulation of a representative thin sheet hosting many dislocations.Comment: 26 pages; 12 figure