A stabilized finite element formulation for liquid shells and its application to lipid bilayers

TL;DR

This paper introduces a novel finite element formulation for liquid shells using NURBS-based surface discretization, enabling accurate simulation of complex lipid bilayer behaviors including non-axisymmetric shapes.

Contribution

It presents a new stabilized FE formulation for liquid shells with both displacement-based and mixed approaches, capable of simulating complex lipid bilayer deformations.

Findings

Accurately simulates non-trivial surface shapes like tube formation and budding.

Demonstrates the effectiveness of stabilization schemes in quasi-static solutions.

Reveals non-axisymmetric lipid bilayer shapes are energetically preferred.

Abstract

This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using $C^1$-continuous finite elements constructed from NURBS interpolation. Both displacement-based and mixed FE formulations are proposed. The latter is needed for area-incompressible material behavior, where penalty-type regularizations can lead to misleading results. In order to obtain quasi-static solutions, several numerical stabilization schemes are proposed based on either stiffness, viscosity or projection. Several numerical examples are considered in order to illustrate the accuracy and the capabilities of the proposed formulation, and to compare the different stabilization schemes. The presented formulation is capable of simulating non-trivial surface shapes associated with tube formation and protein-induced budding of lipid bilayers. In the latter case, the presented formulation yields non-axisymmetric solutions, which have not been observed in previous simulations. It is shown that those non-axisymmetric shapes are preferred over axisymmetric ones.