Arbitrary Lagrangian--Eulerian finite element method for lipid membranes
arXiv:2412.07596 · doi:10.1017/jfm.2025.10553
Abstract
An arbitrary Lagrangian--Eulerian finite element method and numerical implementation for curved and deforming lipid membranes is presented here. The membrane surface is endowed with a mesh whose in-plane motion need not depend on the in-plane flow of lipids. Instead, in-plane mesh dynamics can be specified arbitrarily. A new class of mesh motions is introduced, where the mesh velocity satisfies the dynamical equations of a user-specified two-dimensional material. A Lagrange multiplier constrains the out-of-plane membrane and mesh velocities to be equal, such that the mesh and material always overlap. An associated numerical inf--sup instability ensues, and is removed by adapting established techniques in the finite element analysis of fluids. In our implementation, the aforementioned Lagrange multiplier is projected onto a discontinuous space of piecewise linear functions. The new mesh motion is compared to established Lagrangian and Eulerian formulations by investigating a preeminent numerical benchmark of biological significance: the pulling of a membrane tether from a flat patch, and its subsequent lateral translation.
48 pages, 10 figures, source code at https://github.com/sahu-lab/MembraneAleFem.jl
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