Arbitrary Lagrangian–Eulerian finite element method for lipid membranes

Biological membranes are unique two-dimensional materials in which lipids flow in-plane as a Newtonian fluid, while the entire membrane bends out-of-plane as an elastic sheet. Though the dynamical equations governing lipid membranes are known, they are analytically intractable: membrane dynamics are highly nonlinear and involve spatial derivatives on a surface which is itself arbitrarily curved and deforming over time. The challenges in analytically solving the membrane equations extend to their numerical solution as well: standard computational techniques from fluid and solid mechanics cannot model a two-dimensional material with arbitrarily large shape deformations, in-plane flows, and out-of-plane elasticity. We address this issue by developing an arbitrary Lagrangian–Eulerian (ALE) finite element method for lipid membranes. The membrane surface is endowed with a mesh whose in-plane motion can be specified independently of the material velocity; the out-of-plane mesh motion is required to be Lagrangian such that the mesh and material always overlap. A new in-plane mesh motion is implemented, in which the mesh evolves according to the dynamical equations governing a two-dimensional area-compressible viscoelastic fluid film. Our scheme is used to simulate the long-time dynamics of unstable lipid membranes in biologically-relevant scenarios, including the non-axisymmetric buckling and wrinkling of an membrane tube. The surface tension propagation in situations where tethers are pulled from membrane sheets and cylinders is also quantified. Finally, we show how to extend our numerical implementation to describe surfaces with more complex rheology.