Surface Discretization

The surface discretization is based on templates fitted into quadrants of a quad tree built on the surface. Each template consists of a set of elements which is topologically compatible and geometrically similar to the quadrant represented in a polygonalized form, where the nodes of the polygon correspond to the tree nodes which are part of the quadrant (corner nodes and midside nodes). Note that the same concept of the topological compatibility and geometrical similarity as used in Section Mesh Validity (Sequential Mesh Generation) is applied. The overall number of templates is considerably reduced by the fact that only one node is allowed to be on a side of a quadrant. This is a direct consequence of the ``1:2 rule'' applied during the parametric tree construction. The basic set of currently used templates is displayed in Figure 3.6. The complete set can be obtained from the basic one by application of appropriate rotations. Although there are known templates yielding all-quadrilateral meshes, these have not been used because of the incompatibility with consequent region discretization for which all-hexahedral templates are not available. This is the reason why triangular elements, treated as degenerated quadrilateral elements, are accepted. If an internal node is required to construct a template then this node is created in the centre (in the parametric space) of the quadrant and stored in that quadrant as a midface quad tree node. All the newly created elements, edges, and nodes are classified to the surface. The newly created edges and elements are also connected to their bounding nodes (the duplicity in the case of degenerated elements is avoided). After all quadrants have been filled in with an appropriate template, the final surface mesh obtained by an assembly of all applied templates is smoothed. The position of the smoothed node is given by the recursive relation

where are nodes directly connected to node by an edge, are nodes bounding quadrilaterals sharing node but not directly connected to node , and , , and are properly chosen weights, currently set to , , and , respectively. In the case of an all-quadrilateral mesh, this setting makes Eq. (3.24) equivalent to

where are nodes of quadrilaterals connected to node . The smoothing can be optionally extended by a weighting based on the required element size, nodal connectivity, or both, similarly as described in Section Surface Discretization (Sequential Mesh Generation), using the following formula

where and are equal to (see Eq. (2.85)) for the element size weighting, or to (see Eq. 2.86)) for the connectivity weighting, or to for the combined weighting. Note that only nodes which are classified to the surface are subjected to the smoothing. Since the algorithm is generally dealing with curved surfaces, the smoothed node has to be projected back to the surface after each iteration to comply with the surface constraint. This is done using exactly the same technique as described in Section Surface Discretization (Sequential Mesh Generation) and summarized in Table 2.3. The symmetry of a mesh may be optionally maintained because the surface nodes are stored using both the parametric and real space representations and the use of values from the currently performed iteration may be therefore avoided during the smoothing process. It should be mentioned that the quality of the optimized mesh is a little bit deteriorated by the irregularity of the mesh connectivity due to the common presence of triangular and quadrilateral elements. Also the distribution of boundary nodes arising from the less flexible curve smoothing has a negative influence on the overall surface mesh quality.

*Daniel Rypl
2005-12-07*