Simultaneous aligning and smoothing of surface triangulations

In this work we develop a procedure to deform a given surface triangulation to obtain its alignment with interior curves. These curves are defined by splines in a parametric space and, subsequently, mapped to the surface triangulation. We have restricted our study to orthogonal mapping, so we require the curves to be included in a patch of the surface that can be orthogonally projected onto a plane (our parametric space). For example, the curves can represent interfaces between different materials or boundary conditions, internal boundaries or feature lines. Another setting in which this procedure can be used is the adaption of a reference mesh to changing curves in the course of an evolutionary process. Specifically, we propose a new method that moves the nodes of the mesh, maintaining its topology, in order to achieve two objectives simultaneously: the piecewise approximation of the curves by edges of the surface triangulation and the optimization of the resulting mesh. We will designate this procedure as projecting/smoothing method and it is based on the smoothing technique that we have introduced for surface triangulations in previous works. The mesh quality improvement is obtained by an iterative process where each free node is moved to a new position that minimizes a certain objective function. The minimization process is done on the parametric plane attending to the surface piece-wise approximation and to an algebraic quality measure (mean ratio) of the set of triangles that are connected to the free node. So, the 3-D local projecting/smoothing problem is reduced to a 2-D optimization problem. Several applications of this method are presented.