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