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Smoothing
The generated 3D surface mesh is optimized using the Laplacian
smoothing technique. The smoothing is accomplished directly in
the real space using the following iterative formula

(81) 
where is the node being smoothed, are nodes connected to
node by an edge, and is a relaxation parameter given by

(82) 
The relaxation parameter can be either kept constant, by
appropriately modifying weight for each node with respect to the particular
number of surrounding nodes, or varying with , while keeping the
constant. The first alternative is usually
used for the smoothing of planar surfaces by setting to . The
second alternative is applied for the smoothing of curved surfaces by setting
to . The optimized grid is usually obtained after only a few
cycles of the smoothing (typically up
to six). Note that only nodes which
are classified to the discretized surface are subjected to the
smoothing. Since the smoothing of a node is likely to shift the node
out of the surface, the projection must be employed to satisfy
the surface constraint. This makes the Laplacian smoothing
computationally more demanding when compared to its planar
version. However, since the original position
of the node being smoothed may be used as the initial approximation, better
convergence rates of the projection are achieved. The stability of the projection can be
further improved (especially when very coarse meshes are required) by
increasing the weight resulting in the reduction of
the relaxation parameter .
The Laplacian
smoothing can be optionally extended by weighting based on the required
element size, nodal connectivity, or both. In this case, the
first part of Eq. (81) can be rewritten as

(83) 
with
equal to for the element size weighting, or to for the
connectivity weighting, or to
for the combined weighting.
The element size based weight and connectivity based weight
are given by

(84) 

(85) 
where and denote the actual and ``optimal'' valency
(number of connected nodes) of node and and
are conveniently selected coefficients (currently set to and
). The optimal valency can be expressed as

(86) 
where represents the angle (in degrees) in the
tangent plane at node which is filled in by the surface and is
the number of gaps in that filling. Thus is equal to
for internal nodes and to for boundary
nodes on a continuous boundary curve. In the presented work,
a simplified approach has been adopted, in which
is equal to for nodes classified to the surface or to
otherwise.
Next: Computational Complexity
Up: Real Space Meshing
Previous: Surface Triangulation
Daniel Rypl
20051207