Self-avoiding fluid deformable surfaces

TL;DR

A numerical method for fluid deformable surfaces governed by surface Stokes flow and Helfrich bending energy under active growth, aiming to model shape evolution of the epithelium in developmental processes, and proposes a curvature-adaptive mesh redistribution strategy that improves mesh resolution in regions of high curvature.

Abstract

We propose a numerical method for fluid deformable surfaces governed by surface Stokes flow and Helfrich bending energy under active growth, aiming to model shape evolution of the epithelium in developmental processes. To prevent self-intersections, which commonly arise under large deformations or low enclosed volume to area ratios, we incorporate the nonlocal tangent-point energy to penalize non-embedded configurations. The resulting formulation is discretized using higher order surface finite elements, with a parallelizable assembly strategy for the nonlocal terms. To tailor mesh quality to the geometric evolution, we propose a curvature-adaptive mesh redistribution strategy that improves mesh resolution in regions of high curvature. Numerical examples include the discocyte-to-stomatocyte transition and the inversion of a sphere within a spherical confinement. Both demonstrate the robustness of the method in capturing large deformations, self-avoidance, symmetry-breaking and growth-induced morphology changes.

Figures (8)

• Figure 1: Cuts through an evolving surface under localized area growth. Top: Model without repulsive forces leads to self-intersections. Bottom: Repulsive forces prevent self-intersection. The color coding shows the normal component of the velocity $\boldsymbol{u} \cdot \boldsymbol{\nu}$, red indicating movement outwards, blue indicating movement inwards. The considered parameters are the same in both simulations and are described in Sec. \ref{['sec::results']}.
• Figure 2: Tangent-point radius $R(\boldsymbol{x}, \boldsymbol{y})$ (white lines) for three point pairs. The spheres are tangent to the blue points $\boldsymbol{x}_i$ and intersect the surface at the red points $\boldsymbol{y}_i$. Note the crucial feature, that for two geodesically close points, here $(\boldsymbol{x}_0,\boldsymbol{y}_0)$, the radius is very large. The radius is smallest for $(\boldsymbol{x}_1, \boldsymbol{y}_1)$. The considered surface is the same as in Figure \ref{['fig::self-intersection']}.
• Figure 3: Comparison of the software RepulsorsassenRepulsiveShells2024, our cluster algorithm with $\theta=10h$ and our full integration. All computations were done with 48 shared-memory cores for a dumbbell-like shape, for which we computed the reference value $\Phi_{\mathrm{ref}}=1.60987907$ for the convergence study with our full integration on refinement level $r_{\mathrm{ref}} = 15$. (a) Approximation error in $\Phi$, note, that the irregularity in Repulsor's accuracy is due to our bisection refinement. (b) Runtime for $10$ evaluations of the differential $\frac{\delta \Phi}{\delta \boldsymbol{X}}$, using $48$ threads. (c) Speedup for $r=10$. The legend in (b) is valid for all three plots. Moreover, in (b) and (c), we have added selected measurements for a shape as in the final plot of Fig. \ref{['fig::no_sphere_confinement']}, which is much closer to self-contact. Note, that these values for Repulsor can be improved by remeshing algorithms, as recommended in sassenRepulsiveShells2024, but this renders the discrete space incompatible to our cubic setup.
• Figure 4: Effect of the mesh redistribution approach for the discocyte to stomatocyte transition at refinement level $r = 8$. Parameters are $\tilde{\epsilon} = 10^{-6}$, $M=10$, and $\epsilon=1$ (left) and $\epsilon = 100$ (right). The color indicates $\int_{T}B_{\mathrm{c}} d\mathcal{S}$, i.e. the quantity we aim to equidistribute. While the overall triangle quality also improves, our approach cannot fully prevent the distortion of mesh elements. Note that the closeup views in the respecitive top rights are taken from a different angle.
• Figure 5: Effect of the proposed mesh redistribution approach. (a) A histogramm of the distribution of $\int_{T} B_{\mathrm{c}} d\mathcal{S}$ at different times and $\epsilon =100$. (b) The variance of the same quantity over time for two different $\epsilon$. (c) Quantification of the violation of the pointwise equidistribution condition over time. The corresponding simulation is discussed in Sec. \ref{['sec::oblateToStomatocyte']}.