  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.
