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Boundary Curves Discretization
To estimate the number of segments to be used to discretize a curve,
a binary tree is built in the parametric space of that curve. The depth
of the tree is controlled by the ratio of the size (length) of the tree cell in
the real space and the element size extracted from the octree. This
binary tree structure is then used to evaluate the mass curve
(Fig. 25) by integrating the
required element density

(64) 
where and represent the element density and element
size (extracted from the octree) along the curve, denotes the length
of curve arc, and
is the curve parameter in the range from to . Note that the
actual integration is done numerically over individual cells of the
binary tree. The number of segments is calculated by

(65) 
where brackets stand for the integer part
of the enclosed quantity and the coefficient is used to round
to the nearest integer.
The discretization of the curve is then obtained by splitting the
mass curve to pieces where positions of the nodes to be generated on
the curve (Fig. 25) are defined by a parameter
which is implicitly given by

(66) 
The actual calculation of is done by locating the appropriate cell in the
binary tree (that stores the mass curve) and performing linear interpolation in that
cell. Each created segment is connected to its end nodes and each
newly created (internal) node is registered into the octree.
The new segments and nodes (except the boundary vertex nodes) are also classified to the curve
being discretized. Since the octree data structure is employed for the element
spacing control, a gradual and very smooth discretization is achieved
which does not need further optimization.
Next: Surface Triangulation
Up: Real Space Meshing
Previous: Real Space Meshing
Daniel Rypl
20051207