  We present a positivity preserving variational scheme for the phase-field
modeling of incompressible two-phase flows with high density ratio and using
meshes of arbitrary topology. The variational finite element technique relies
on the Allen-Cahn phase-field equation for capturing the phase interface on a
fixed mesh with a mass conservative and energy-stable discretization. Mass is
conserved by enforcing a Lagrange multiplier which has both temporal and
spatial dependence on the solution of the phase-field equation. The spatial
part of the Lagrange multiplier is written as a mid-point approximation to make
the scheme energy-stable. This enables us to form a conservative, energy-stable
and positivity preserving scheme. The proposed variational technique reduces
spurious and unphysical oscillations in the solution while maintaining
second-order spatial accuracy. To model a generic two-phase free-surface flow,
we couple the Allen-Cahn phase-field equation with the Navier-Stokes equations.
Comparison of results between standard linear stabilized finite element method
and the present variational formulation shows a remarkable reduction of
oscillations in the solution while retaining the boundedness of the
phase-indicator field. We perform a standalone test to verify the accuracy and
stability of the Allen-Cahn two-phase solver. Standard two-phase flow
benchmarks such as Laplace-Young law and sloshing tank problem are carried out
to assess the convergence and accuracy of the coupled Navier-Stokes and
Allen-Cahn solver. Two- and three-dimensional dam break problem are then solved
to assess the scheme for the problem with topological changes of the air-water
interface on unstructured meshes. Finally, we demonstrate the phase-field
solver for a practical problem of wave-structure interaction in offshore
engineering using general three-dimensional unstructured meshes.
