Difficulties associated with free-surface finite element
flow simulations are related to a) nonlinear and advective 
nature of most hydrodynamic flows, b) requirements for
compatibility between velocity and pressure interpolation,
c) maintaining a valid computational mesh in the presence
of moving boundaries, and d) enforcement of the kinematic conditions
at the free surface. Focusing on the last issue, we present
an extension of the free surface elevation equation to cases
where the prescribed direction of the surface node motion
is not uniformly vertical. The resulting hyperbolic generalized
elevation equation is discretized using a Galerkin/Least-Squares
formulation applied on the surface mesh.
The proposed method is used to solve a two-dimensional problem
of sloshing in a trapezoidal tank, and a three-dimensional
application involving flow in a trapezoidal channel with
bridge supports.