Tangential Tensor Fields on Deformable Surfaces -- How to Derive Consistent $L^2$-Gradient Flows

Abstract

We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.

Figures (4)

• Figure 1: Schematic of parameterization and deformation: $\mathcal{U}\subset\mathbb{R}^2$ is the domain (left) of the parameterization $\boldsymbol{X}$, resp. $\boldsymbol{X}+\varepsilon\boldsymbol{W}$, which maps to the surface $\mathcal{S}\subset\mathbb{R}^3$(lower right), resp. $\mathcal{S}_{\varepsilon\boldsymbol{W}}\subset\mathbb{R}^3$(upper right). The surface $\mathcal{S}$ is linearly deformed by $\varepsilon>0$ along the direction of the field $\boldsymbol{W}\in\tensor{\operatorname{T}\!}{}\mathbb{R}^3\vert_{\mathcal{S}}$.
• Figure 2: Rigid rotational deformation of a hemisphere (side and top view): If $\boldsymbol{X}$ parameterizes the hemisphere $\mathcal{S}$, then $\boldsymbol{X}_\varepsilon = \boldsymbol{X} + \varepsilon\boldsymbol{W} + \mathcal{O}(\varepsilon^2)$ parameterizes the rotated hemisphere $\mathcal{S}_{\varepsilon}= \mathcal{S}_{\varepsilon\boldsymbol{W}} + \mathcal{O}(\varepsilon^2)$ by an angle $\varepsilon$. Two orthogonal tangential vector fields (purple) of equal lengths are shown on a fixed longitude and three different latitudes at $\mathcal{S}$. The green and red tangential vector fields are given on $\mathcal{S}_{\varepsilon}$ by pushing-forward the purple ones with the map $\mathcal{S} \rightarrow \mathcal{S}_{\varepsilon}$ assuming different gauges of surface independence, respectively. At the equator, where the axis of rotation lays parallel to the tangential plane of $\mathcal{S}$, all four gauges imply the same pushforward, since the surface gradient of $\boldsymbol{W}$ is vanishing, cf. table \ref{['tab:gauge_deformation_derivative_vector']}. At all other locations, only the symmetric gradient of $\boldsymbol{W}$ is vanishing due to rigid body deformation, i. e., the upper-, lower-convected and Jaumann gauge yield equal pushforwards, whereas the material gauge implies a different one.
• Figure 3: Strain deformation of a hemisphere (side and top view): If $\boldsymbol{X}$ parameterize the hemisphere $\mathcal{S}$, then, for $\boldsymbol{W} = \boldsymbol{\nu} + \boldsymbol{e}_z$, $\boldsymbol{X}_\varepsilon = \boldsymbol{X} + \varepsilon\boldsymbol{W}$ parameterize the strained hemispheroid $\mathcal{S}_{\varepsilon\boldsymbol{W}}$. Two orthogonal tangential vector fields (purple) of equal lengths are shown on a fixed longitude and three different latitudes at $\mathcal{S}$ exactly like in figure \ref{['fig:rot_sphere_vector']}. The green, red and blue tangential vector fields are given on $\mathcal{S}_{\varepsilon}$ by pushing-forward the purple ones with the map $\mathcal{S} \rightarrow \mathcal{S}_{\varepsilon}$ assuming different gauges of surface independence, respectively. The material and Jaumann gauge imply the same pushforward, since this is a pure strain deformation, i. e., the surface gradient of $\boldsymbol{W}$ is symmetric, cf. table \ref{['tab:gauge_deformation_derivative_vector']}. Gray parallel lines on the unrolled arc-length segment (right) indicate length preserving for pushforwarded vectors due to material and Jaumann gauges. By contrast, pushforwarded vectors, w. r. t. the upper- or lower-convected gauge, are stretched along or against the deformation. However, the geometric mean of both yields the length preserved green vectors.
• Figure 4: Solutions for the $\operatorname{L}^{\!2}$-gradient flows \ref{['eq:FO_flat']} with $K = 0.1$ and $\omega=0.5$: The initial solution (top left) is periodic on the boundaries and constant horizontally. Vertically, it is twisted counterclockwise and compressed towards the center. For a better readability, vector field solutions are only plotted on a diagonal without loss of information. The solutions are approximated for four different models, where we assume the gauge of surface independence and time derivative consistently, namely the material, upper-convected, lower-convected and Jaumann gauge/derivative (from left to right). All energy paths (top right) are decreasing as expected by energy rate \ref{['eq:dissipation']} with $\boldsymbol{\Phi}=\boldsymbol{\Psi}$. The energy plot has a log-scale on its time axis. Vector field solutions tend to be parallel aligned with a constant length towards the time process. We show the solutions at time $t = 0.016$(middle row) and $t = 10$(bottom row), where they almost reached their minimal state. The time-step size $\tau$ of the time discretization increases from $\tau = 10^{-4}$ at the beginning until $\tau = 0.45$ at the end, see the semitransparent gray dots at the energy paths. The grid size $h = 10^{-2}$ is constant along the vertical local Lagrange coordinate $y^2$ for all times. The horizontal gray gridlines only represent every fourth available spatial line, i. e., the distance between two gridlines is $0.04$ constantly on local Lagrange coordinates. Solutions are plotted w. r. t. Euler coordinates, i. e., w. r. t. the embedding space, to show the local deformations of the underlying domain towards the minimum state. Any impressions that the surface gets bent in normal direction is just an optical illusion. In absence of normal forces the surface stays flat all times. (Video online: mendeley_video)

Theorems & Definitions (12)

• Remark 1
• Definition 1: Gauge of Surface Independence on Scalar Fields
• Remark 2
• Definition 2: Material Gauge of Surface Independence on Tangential Vector Fields
• Definition 3: Upper-Convected Gauge of Surface Independence on Tangential Vector Fields
• Definition 4: Lower-Convected Gauge of Surface Independence on Tangential Vector Fields
• Definition 5: Jaumann Gauge of Surface Independence on Tangential Vector Fields
• Remark 3
• Remark 4
• Definition 6: Gauge of Material Surface Independence on Tangential 2-tensor Fields