Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

TL;DR

This paper introduces a finite element model for lipid membrane mechanics that employs viscous regularization and r-adaptive remeshing to enhance stability, convergence, and handle large deformations in simulations.

Contribution

It presents a novel viscous regularization technique combined with r-adaptive remeshing for improved finite element analysis of lipid membranes.

Findings

Successfully computed equilibrium shapes of lipid vesicles.

Demonstrated stable simulations of membrane tethers and phase separation.

Improved convergence rates with viscous regularization.

Abstract

As two-dimensional fluid shells, lipid bilayer membranes resist bending and stretching but are unable to sustain shear stresses. This property gives membranes the ability to adopt dramatic shape changes. In this paper, a finite element model is developed to study static equilibrium mechanics of membranes. In particular, a viscous regularization method is proposed to stabilize tangential mesh deformations and improve the convergence rate of nonlinear solvers. The Augmented Lagrangian method is used to enforce global constraints on area and volume during membrane deformations. As a validation of the method, equilibrium shapes for a shape-phase diagram of lipid bilayer vesicle are calculated. These numerical techniques are also shown to be useful for simulations of three-dimensional large-deformation problems: the formation of tethers (long tube-like exetensions); and Ginzburg-Landau phase separation of a two-lipid-component vesicle. To deal with the large mesh distortions of the two-phase model, modification of vicous regularization is explored to achieve r-adaptive mesh optimization.