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A Constructive Approach to Constrained Hexahedral Mesh Generation

Carbonera, Carlos D. and Jason F. Shepherd

Proceedings, 15th International Meshing Roundtable, Springer-Verlag, pp.435-452, September 17-20 2006

IMR
PROCEEDINGS

15th International Meshing Roundtable
Birmingham, Alabama, U.S.A.
September 17-20, 2006

Carlos D. Carbonera
Gauss Research Laboratory,
University of Puerto Rico, Rio Piedras, PR
carbonera@uprr.pr

Jason F. Shepherd
Scientific Computing and Imaging Institute,
University of Utah, Salt Lake City, and
Computational Modeling Sciences Dept.,
Sandia National Laboratories,
jfsheph@sandia.gov

Abstract
S. Mitchell proved that a necessary and sufficient condition for the existence of a topological hexahedral mesh constrained to a quadrilateral mesh on the sphere is that the constraining quadrilateral mesh contains an even number of elements. S. Mitchellís proof depends on S. Smaleís theorem on the regularity of curves on compact manifolds. Although the question of the existence of constrained hexahedral meshes has been solved, the known solution is not easily programmable; indeed, there are cases, such as Schneiderís pyramid, that are not easily solved. D. Eppstein later utilized portions of S. Mitchellís existence proof to demonstrate that hexahedral mesh generation has linear complexity. In this paper, we demonstrate a constructive proof to the existence theorem for the sphere, as well as assign an upper-bound to the constant of the linear term in the asymptotic complexity measure provided by D. Eppstein. Our construction generates 76

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