Examples

A set of examples is provided to demonstrate the capability of the presented algorithm for the sequential generation of unstructured meshes and to illustrate its performance from both quantitative and qualitative points of view. For each example (Figs 2.30 - 2.34), a sequence of uniform (U) and/or graded (G) 2D and/or 3D meshes of varying density has been generated using SGI Indigo2 workstation with 195 MHz R10000 processor and 128 MB of memory. The achieved relationship between the number of elements and the generation time is provided separately for each sequence of meshes in Figures 2.35 - 2.41. Note that the elapsed times do not include the output printing which is typically of the computational complexity and therefore not relevant. The discontinuities which can be recognized on the performance diagrams for uniform meshes (Figs 2.37, 2.39, and 2.40) are caused by the octant refinement by one level due to the only negligible decrease of the prescribed element size. This refinement changes the ratio between the element size and octant size from before to after the refinement. This has a direct impact on the number of mesh entities involved in the element forming and intersection check. In the case of uniform meshes, the whole octree is subjected to the one-level refinement, which results in a considerable reduction of the computational time (approximately by 20 %). Note that this effect occurs repeatedly each time the octree is refined by one level, which corresponds to an increase in the number of elements by a factor of eight (in a 3D mesh). Although this phenomenon is of a general character, it is usually not evidenced when dealing with 2D meshes or graded 3D meshes. For 2D meshes, the number of mesh entities involved in the element creation and intersection check during the discretization is very small because of the linear character of the front, which results in a significantly reduced contribution to the overall complexity. In the case of graded meshes, there are two important aspects. Firstly, only some octants, scattered over the octree and not always in the same part of the model, are subjected to the one-level refinement. This is typical for the curvature-based mesh size gradation (Fig. 2.36). And secondly, the overall mesh size prescription for graded meshes is usually either missing or irrelevant, which enables to keep the element size as large as possible (with respect to the octant size) in the major part of the model. This makes the number of mesh entities involved in the element forming and intersection check independent of the octree refinement (Fig. 2.41). An exception may be observed in Figure 2.38 where the refinement along local sources has been combined with a uniform mesh size prescription. However, exact locations of the discontinuities have not been (a priori) identified because of the combined nature of the mesh size specification. Taking into account the above explained phenomenon, the achieved relationship between the number of elements and the generation time is in good agreement with the expected linear dependence (Section Computational Complexity (Sequential Mesh Generation)).

The quality of selected meshes (Table 2.4), typically the finest, intermediate and coarsest mesh for each model and type of the mesh, is presented in terms of the element quality, dihedral angle, and nodal connectivity (Section Mesh Quality (Sequential Mesh Generation)) in Table 2.5. Note that a 2D mesh of a 3D object is equivalent to the boundary triangulation of that 3D object. The timing provided in Table 2.4 corresponds to the time consumed by the actual discretization by the advancing front technique (AFT) and to the total time, including all overhead (except the output printing). For very coarse meshes, only the total time is provided because the portion corresponding to the AFT, measured in whole seconds, is virtually meaningless. The numbers of elements in Table 2.4, similarly as the element quality and dihedral angles in Table 2.5, are associated with triangles for a 2D mesh or tetrahedrons for a 3D mesh. The typical distributions of the element quality and dihedral angles 2.42 - 2.53. Table 2.5 and Figures 2.42 - 2.53 reveal the superior quality of 2D meshes and a quite good quality of 3D meshes. Generally, the quality of graded meshes is slightly worse than the quality of uniform meshes.

*Daniel Rypl
2005-12-07*