  We present an interface and geometry preserving (IGP) method for the modeling
of fully Eulerian fluid-structure interaction via phase-field formulation.
While the hyperbolic tangent interface profile is preserved by the
time-dependent mobility model, the proposed method maintains the geometry of
the solid-fluid interface by reducing the volume-conserved mean curvature flow.
To achieve the reduction in the curvature flow, we construct a
gradient-minimizing velocity field (GMV) for the convection of the order
parameter. The constructed velocity field enables the preservation of the solid
velocity in the solid domain while extending the velocity in the normal
direction throughout the diffuse interface region. With this treatment, the GMV
reduces the normal velocity difference of the level sets of the order parameter
which alleviates the undesired thickening or thinning of the diffuse interface
region due to the convection. During this process, the time-dependent mobility
coefficient is substantially reduced and there is a lesser curvature flow. The
GMV ensures that the diffuse interface region moves with the solid bulk such
that the fluid-solid interface conforms to the geometry of the solid. Using the
unified momentum equation and the phase-dependent interpolation, we integrate
the IGP method into a fully Eulerian variational FSI solver based on the
incompressible viscous fluid and the neo-Hookean solid. We first demonstrate
the ability of the phase-field-based IGP method for the convection of circular
and square interfaces with a prescribed velocity field. The variational FSI
framework with the IGP method is then examined for the flow passing a fixed
deformable block in a channel domain. Finally, the vibration of a plate
attached behind a stationary cylinder subjected to incoming flow is employed to
assess the fully Eulerian framework for a large aspect ratio and sharp corners.
