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Global Smoothing

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 half-axes of the ellipse (compare with Fig. 6) while stands for the angle between the minor half-axis 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
2005-12-03