Arbitrary Lagrangian--Eulerian finite element method for lipid membranes

TL;DR

This paper introduces a flexible finite element method for simulating deforming lipid membranes, allowing arbitrary in-plane mesh motion and addressing numerical stability issues, with applications demonstrated through membrane tether pulling simulations.

Contribution

It presents a novel arbitrary Lagrangian--Eulerian finite element approach with a new mesh motion class and stability techniques for lipid membrane modeling.

Findings

Successfully simulates membrane tether pulling

Demonstrates stability with new mesh motion techniques

Provides accurate comparison with established methods

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.