Linear Complexity Hexahedral Mesh Generation

We show that any polyhedron forming a topological ball with an even number of quadrilateral sides can be partitioned into O(n) topological cubes, meeting face to face. The result generalizes to non-simply-connected polyhedra satisfying an additional bipartiteness condition. The same techniques can also be used to reduce the geometric version of the hexahedral mesh generation problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at the 12th ACM Symp. on Computational Geometry. This is the final version, and will appear in a special issue of Computational Geometry: Theory and Applications for papers from SCG '9

Improved Approximations for Minimum Cardinality Quadrangulations of Finite-Element Meshes

Conformal mesh refinement has gained much attention as a necessary preprocessing step for the finite element method in the computer-aided design of machines, vehicles, and many other technical devices. For many applications, such as torsion problems and crash simulations, it is important to have mesh refinements into quadrilaterals. In this paper, we consider the problem of constructing a minimum-cardinality conformal mesh refinement into quadrilaterals. However, this problem is NP--hard, which motivates the search for good approximations. The previously best known performance guarantee has been achieved by a linear-time algorithm with a factor of 4. We give improved approximation algorithms. In particular, for meshes without so-called folding edges, we now present a 2-approximation algorithm. This algorithm requires O(n² log n) time, where n is the number of polygons in the mesh. The asymptotic complexity of the latter algorithm is dominated by solving a minimum-cost perfect b-m..