  We introduce a smoothing algorithm for triangle, quadrilateral, tetrahedral
and hexahedral meshes whose centerpiece is a simple geometric triangle
transformation. The first part focuses on the mathematical properties of the
element transformation. In particular, the transformation gives rise directly
to a continuous model given by a system of coupled damped oscillations. Derived
from this physical model, adaptive parameters are introduced and their benefits
presented. The second part discusses the mesh smoothing algorithm based on the
element transformation and its numerical performance on example meshes.
