  Volume conservation and shape preservation are two well-known issues related
to the advection and remeshing in front tracking. To address these issues, this
paper proposes a divergence-preserving velocity interpolation method and a
parabolic fit vertex positioning method for remeshing operations for
three-dimensional front tracking. Errors in preserving the divergence of the
velocity field when interpolating the velocity from the fluid mesh to the
vertices of the triangles of the front are a primary reason for volume
conservation errors when advecting the front. The proposed interpolation method
preserves the discrete divergence of the fluid velocity by construction and is
compared in this work with other known interpolation methods in divergence-free
and non-divergence-free test cases, with respect to volume conservation and
shape preservation of the front. The presented interpolation method conserves
the volume and shape up to an order of magnitude better than the conventionally
used interpolation methods and is within the range of higher order
interpolation methods at lower computational cost. Additionally, the parabolic
fit vertex positioning method for remeshing operations locally approximates the
front with a smooth polynomial surface, improving volume conservation and shape
preservation by an order of magnitude compared to conventional remeshing
algorithms.
