  We present a finite element based variational interface-preserving and
conservative phase-field formulation for the modeling of incompressible
two-phase flows with surface tension dynamics. The preservation of the
hyperbolic tangent interface profile of the convective Allen-Cahn phase-field
formulation relies on a novel time-dependent mobility model. The mobility
coefficient is adjusted adaptively as a function of gradients of the velocity
and the order parameter in the diffuse interface region in such a way that the
free energy minimization properly opposes the convective distortion. The ratio
of the convective distortion to the free energy minimization is termed as the
convective distortion parameter, which characterizes the deviation from the
hyperbolic tangent shape due to the convection effect. The mass conservation is
achieved by enforcing a Lagrange multiplier with both temporal and spatial
dependence on the phase-field function. We integrate the interface-preserving
and conservative phase-field formulation with the incompressible Navier-Stokes
equations and the continuum surface tension force model for the simulation of
incompressible two-phase flows. A positivity preserving scheme designed for the
boundedness and stability of the solution is employed for the variational
discretization using unstructured finite elements. We examine the convergence
and accuracy of the Allen-Cahn phase-field solver through a generic
one-dimensional bistable diffusion-reaction system in a stretching flow. We
quantify and systematically assess the relative interface thickness error and
the relative surface tension force error with respect to the convective
distortion parameter. Two- and three-dimensional rising bubble cases are
further simulated to examine the effectiveness of the proposed model on the
volume-preserving mean curvature flow and the interface-preserving capability.
