ENHANCED QUADTREE MESHING

With the introduction of transition quadrilateral elements, the Quadtree subdivision can be applied to the domain interior to produce quadrilateral elements in compliance with the given node spacing specification. Alternatively, in case transition elements are not available, the quadrilateral elements of a Quadtree partition can also be divided rapidly into trian-gular elements. Instead of using templates for the treatment of the boundary, an enhanced Quadtree-meshing scheme by means of the coring technique in conjunction with the AFT is presented. A general multi-connected planar domain and its bounding box are shown in Figure 3.66, and it is required to mesh the domain into quadrilaterals and triangles with the element size distribution similar to that shown in Figure 3.15.

Figure 3.66 Planar domain and its bounding box.

3.8.1 Quadtree partition of the bounding box

Similar to the classical approach, given the planar domain and the node spacing specifica-tion, the first step of the enhanced scheme is to subdivide the rectangular bounding box into quadrilaterals by means of the classical Quadtree decomposition. This can be achieved using the recursive subdivision method described in Section 3.4, and the result of the Quadtree partition of the bounding box is shown in Figure 3.67. It is seen that element concentrations are found at the interior of the domain and around the interior boundary of the planar domain.

3.8.2 Removal of quadrilaterals near domain boundary

The next step is to remove those quadrilaterals that are outside the bounded region or too close to the domain boundary. Quadrilaterals that are too close to the boundary can be removed simply by checking the distance to the boundary segments. Suppose that the domain boundary consists of Nb line segments; then quadrilateral Q will be removed if

min,

i Nb(distance between Q and segment i) size of Q

= <

1

where distance is the distance between the centre of Q and the i^th line segment under con-sideration, as discussed in Section 2.4.1, and the size of Q can be taken as the longest edge of Q. Upon the removal of the quadrilaterals near the boundary by the distance check, the remaining quadrilaterals are well away from the boundary line segments.

However, those quadrilaterals that are outside the domain boundary still have to be removed. The Inside_or_Outside check provided in Section 2.4.11 can be applied here to determine which quadrilaterals are to be retained. A more consistent treatment is to find out the status of one quadrilateral and determine the patch of quadrilaterals by means of the neighbouring relationship. Whichever method is used, the Quadtree partition within the planar domain upon the removal of outside and nearby quadrilaterals is shown in Figure 3.68. It is observed that the quadrilaterals are all removed over regions of large element size, showing that quadrilaterals cannot be easily fitted to an irregular boundary, and it is also an inherent difficulty of Quadtree decomposition for a systematic and strategic positioning and orientation of the grid with respect to the given domain.

Figure 3.67 Planar domain and its bounding box.

3.8.3 Boundary recovery for triangulation

The boundary of the triangulation problem consists of two parts – the given boundary line segments and the boundary of the Quadtree quadrilaterals. The part to be meshed is the void between these two boundaries. The original given boundary segments may be too long or too short to be consistent with the given element size specification. Hence, each bound-ary line segment of the planar domain has to be verified and subdivided into more segments to conform with the node spacing requirements. The division of a line segment into smaller segments according to a given density function has been discussed in Section 2.4.9, which can now be applied to the boundary segments in turn to produce a consistent boundary for MG. As the one-level rule has been enforced in the generation of the transition quadrilateral mesh, the node spacing requirement is not strictly conforming with the quadrilateral decom-position. In order to have a better fit with the quadrilateral elements, in dividing a boundary edge, apart from the node spacing requirement, one can also take reference of the size of the adjacent quadrilaterals. As for the boundary of the internal Quadtree partition, the size of the segments is already in line with the element size specification; what has to be done is to retrieve the boundary of the patch of quadrilaterals. This can be easily done based on the neighbouring relationship. Following the neighbouring relationship of the transition quadri-lateral elements, an edge of a quadriquadri-lateral is a boundary segment if there is no neighbour on this edge, or the neighbouring quadrilateral has been removed earlier in Section 3.8.2. The orientation of the boundary line segments is crucial for the ADF MG, which determines the nature of a boundary loop, i.e. counter-clockwise represents exterior boundary and clock-wise represents interior boundary, as shown in Figure 3.69.

3.8.4 Advancing-front MG

The AFT is very efficient in handing irregular planar domains. The boundary segments with correct orientations collected in the last step are fed into a 2D ADF mesh generator to produce the triangular mesh to fill the gap between the Quadtree partition and the original domain boundary. While the segment orientation is important for a correct triangulation, the order of the segments does not matter even though they are usually prepared following a natural sequential order. The result of ADF meshing of the given bounding segments is shown in Figure 3.70. It is seen that some internal nodes have been generated over empty areas to produce triangles of better quality. The boundary mesh along with the Quadtree Figure 3.68 Quadtree partition within the planar domain.

meshing at the interior part are put together to obtain the final mesh by the enhanced Quadtree meshing scheme, as shown in Figure 3.71.

Making use of a combination of existing techniques, the meshes produced by the enhanced Quadtree meshing scheme seem to be quite promising in terms of element shape quality and size distribution. It can be made fairly automatic, and all the steps starting from the Figure 3.69 Retrieved boundary for mesh generation.

Figure 3.70 Mesh generation along domain boundary.

Figure 3.71 Final mesh by the enhanced Quadtree meshing.

input of the boundary segments to the final mixed mesh, as shown in Figure 3.71, have all been incorporated into a single FORTRAN program of about 1200 lines excluding graphics commands.