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Although large effort has been made to ensure generation of triangles near
to equilaterals after being mapped on the physical surface the actual
discretization is sometimes not ideal.
The shape of the triangles can be optimized using any smoothing technique.
The Laplacian smoothing in the physical space of the original surface
was employed by the authors. To avoid the effect of repulsion of
neighboring nodes by a node having more than six neighbors and the
effect of attracting them by nodes having less than six neighbors
a connectivity weighting[5] was imposed.
However, the new position of a smoothed node is known only in the
Cartesian coordinates of the physical space and the smoothed node is likely
to fall out of the curved surface. Therefore, a projection technique is
adopted to reposition the smoothed node back onto the original surface and
to get its new parametric coordinates. This projection utilizes only the
surface gradients
and
and surface normal
. Since these quantities are of local character the
projection must be performed in an iterative manner. Only a few cycles of the
global smoothing (typically about five) have been required to obtained
sufficiently optimized mesh. Because of relatively high cost of global
smoothing (due to the projection) an alternative approach has been
adopted. The idea is to precede the global smoothing by a smoothing process
in the parametric space. Again Laplacian smoothing has been applied. Since
the elements in the parametric space are stretched and distorted a special
weighting had to be established.
The weight of each node (Fig. 7) contributing to
the repositioning of a given node is taken as indirectly
proportional to the length of the
segment cut out by the ellipse, describing the local stretches and
distortion, from the connecting line between the nodes and . Thus

(12) 
where is the number of nodes connected to node and the
is taken as

(13) 
The symbols and have the
meaning of the size of the minor and major halfaxes of the ellipse
(compare with Fig. 6) while
stands for the angle between the minor halfaxis and the line
connecting nodes and .
Although the
process of smoothing in the parametric space is more efficient from the
computational point of view (it is a planar task) the influence on the
quality of the mesh in the physical space varies from case to case and is
significantly dependent on the quality of the parameterization.
Next: Local Smoothing
Up: Mesh Optimization
Previous: Mesh Optimization
Daniel Rypl
20051203