  The signed volume function for polyhedra can be generalized to a mean volume
function for volume elements by averaging over the triangulations of the
underlying polyhedron. If we consider these up to translation and scaling, the
resulting quotient space is diffeomorphic to a sphere. The mean volume function
restricted to this sphere is a quality measure for volume elements. We show
that, the gradient ascent of this map regularizes the building blocks of hybrid
meshes consisting of tetrahedra, hexahedra, prisms, pyramids and octahedra,
that is, the optimization process converges to regular polyhedra. We show that
the (normalized) gradient flow of the mean volume yields a fast and efficient
optimization scheme for the finite element method known as the geometric
element transformation method (GETMe). Furthermore, we shed some light on the
dynamics of this method and the resulting smoothing procedure both
theoretically and experimentally.
