  We propose a new approach for controlling the characteristics of certain mesh
faces during optimization of high-order curved meshes. The practical goals are
tangential relaxation along initially aligned curved boundaries and internal
surfaces, and mesh fitting to initially non-aligned surfaces. The distinct
feature of the method is that it utilizes discrete finite element functions
(for example level set functions) to define implicit surfaces, which are used
to adapt the positions of certain mesh nodes. The algorithm does not require
CAD descriptions or analytic parametrizations, and can be beneficial in
computations with dynamically changing geometry, for example shape optimization
and moving mesh multimaterial simulations. The main advantage of this approach
is that it completely avoids geometric operations (e.g., surface projections),
and all calculations can be performed through finite element operations.
