Discussion

The primary aim of the present paper is to propose an approach for the generation of the boundary conforming mesh of a 3D computational domain defined by many constraining faces that might be very closely spaced and of broadly ranging size and aspect ratio. This is a very ambitious goal and its efficient implementation is not trivial. The combination of the AFT with the computationally efficient Delaunay kernel seems to be a quite reasonable choice. However, theoretical guarantee that the algorithm successfully terminates is missing. In the most pathological case, all the tetrahedra might be created using the AFT during the a priori boundary recovery without the Delaunay point insertion kernel activated at all. In such a case, the whole methodology reduces to somewhat complicated and rather inefficient AFT, for 3D version of which the proof of convergence does not exist.

The actual implementation of the proposed discretization strategy heavily relies on the octree as a localization tool and on the fact that the size of constraining faces is in good agreement with the size of individual octants. Therefore, the best results are achieved for domains defined by isotropic constraining faces generated by appropriate discretization algorithm [41] using the mesh density distribution stored in the octree according to user prescribed element spacing specification. For such a kind of geometries, however, one can hardly expect the presented approach to fail or reduce to pure AFT. Serious assessment of the proposed strategy from the qualitative as well as quantitative point of view therefore requires intensive testing on much wider class of discrete geometries.

At the time this paper was prepared for the submission, only preliminary results on academic and rather simple geometries were available. However, quite promisingly, no problems with the algorithm convergence were observed. The AFT used as the last resort action was activated only in rare cases, for example in the case of the Schönhardt polyhedron.

*Daniel Rypl
2005-11-06*