Arbitrary Lagrangian–Eulerian finite element method for biological lipid membranes

We present an arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming lipid membranes. We provide a formalism to determine the equations of motion governing lipid membrane behavior using an irreversible thermodynamic analysis of curved surfaces. We develop an ALE theory by endowing the surface with a mesh whose in-plane velocity is independent of the in-plane material velocity, and which can be specified arbitrarily. The general isoparametric finite element implementation of the theory, based on an arbitrary surface parametrization with curvilinear coordinates, is used to model lipid membranes in several biologically relevant situations. A new physical insight is obtained by applying the ALE developments to cylindrical lipid membrane tubes: though lipid membrane tubes are stable, in the limit of vanishing bending rigidity (the limiting case of a fluid film) we numerically and analytically find tubes to be unstable with respect to long-wavelength perturbations when their length exceeds their circumference.