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### Region Discretization

The discretization of regions consists in fitting of templates into octants of octal tree built in the region. In this case however, the overall number of templates is enlarged by the fact that generalized octal tree is considered. This can be demonstrated by the extended set of 2D templates bounding the faces of 3D templates. The additional 2D templates which can be derived from the basic set (Fig. 12) by appropriate combining and scaling are depicted in Figure 13. Although the number of 3D templates is dramatically decreased by the application of ``1:2 rule'' it remains still too large to implement the templates manually. Therefore an automated approach for template creation during the run time has been developed. Several classes of templates are recognized (Fig. 14). Simple templates are to be used in octants with only a few number of midside nodes. These are the only templates which have been created manually. Symmetric templates are designated for octants with midside and midface nodes arranged symmetrically with respect to a plane parallel with octant face and passing through center of the octant (in parametric space). Two types of symmetric templates are distinguished depending on occurrence of midside or midface nodes on plane of symmetry. Combined template is based on splitting the octant into two new ones (midside nodes on edges perpendicular to the plane of split have to be present) each of which can be classified as simple, symmetric or combined. If no of these templates suits for a particular octant the default template is used. This template has a node in the center of the octant. The corner, midside and midface nodes are then simply connected with this center node with the only exception when a hexahedral element may be preserved in the corner. All the templates are designed in such a way that face deplanation of individual elements is prevented (in parametric space) which contributes to the quality of elements. Obviously, the above described templates do not provide all-hexahedral mesh. The following types of elements, all treated as hexahedrons, are present in the mesh -- hexahedrons, triangular prisms (wedges), pyramids and tetrahedrons. The final mesh obtained as a collection of all applied templates is then smoothed using the Laplacian smoothing optionally extended by weighting based on required element size, element connectivity or both, and by preserving eventual symmetry of the mesh. Again, the quality of the mesh is slightly decreased by the irregular connectivity caused by the mixed nature of the mesh.

Next: Parallel Computing Scheme Up: Mesh Generation Previous: Surface Discretization

Daniel Rypl
2005-12-03