Self-avoiding fluid deformable surfaces

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.