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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
2005-12-07