  A surface moving mesh method is presented for general surfaces with or
without explicit parameterization. The method can be viewed as a nontrivial
extension of the moving mesh partial differential equation method that has been
developed for bulk meshes and demonstrated to work well for various
applications. The main challenges in the development of surface mesh movement
come from the fact that the Jacobian matrix of the affine mapping between the
reference element and any simplicial surface element is not square. The
development starts with revealing the relation between the area of a surface
element in the Euclidean or Riemannian metric and the Jacobian matrix of the
corresponding affine mapping, formulating the equidistribution and alignment
conditions for surface meshes, and establishing a meshing energy function based
on the conditions. The moving mesh equation is then defined as the gradient
system of the energy function, with the nodal mesh velocities being projected
onto the underlying surface. The analytical expression for the mesh velocities
is obtained in a compact, matrix form, which makes the implementation of the
new method on a computer relatively easy and robust. Moreover, it is
analytically shown that any mesh trajectory generated by the method remains
nonsingular if it is so initially. It is emphasized that the method is
developed directly on surface meshes, making no use of any information on
surface parameterization. It utilizes surface normal vectors to ensure that the
mesh vertices remain on the surface while moving, and also assumes that the
initial surface mesh is given. The new method can apply to general surfaces
with or without explicit parameterization since the surface normal vectors can
be computed even when the surface only has a numerical representation. A
selection of two- and three-dimensional examples are presented.
