5,609 research outputs found
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
Finding Hexahedrizations for Small Quadrangulations of the Sphere
This paper tackles the challenging problem of constrained hexahedral meshing.
An algorithm is introduced to build combinatorial hexahedral meshes whose
boundary facets exactly match a given quadrangulation of the topological
sphere. This algorithm is the first practical solution to the problem. It is
able to compute small hexahedral meshes of quadrangulations for which the
previously known best solutions could only be built by hand or contained
thousands of hexahedra. These challenging quadrangulations include the
boundaries of transition templates that are critical for the success of general
hexahedral meshing algorithms.
The algorithm proposed in this paper is dedicated to building combinatorial
hexahedral meshes of small quadrangulations and ignores the geometrical
problem. The key idea of the method is to exploit the equivalence between quad
flips in the boundary and the insertion of hexahedra glued to this boundary.
The tree of all sequences of flipping operations is explored, searching for a
path that transforms the input quadrangulation Q into a new quadrangulation for
which a hexahedral mesh is known. When a small hexahedral mesh exists, a
sequence transforming Q into the boundary of a cube is found; otherwise, a set
of pre-computed hexahedral meshes is used.
A novel approach to deal with the large number of problem symmetries is
proposed. Combined with an efficient backtracking search, it allows small
shellable hexahedral meshes to be found for all even quadrangulations with up
to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more
than 72 hexahedra. This algorithm is also used to find a construction to fill
arbitrary domains, thereby proving that any ball-shaped domain bounded by n
quadrangles can be meshed with no more than 78 n hexahedra. This very
significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201
Flipping Cubical Meshes
We define and examine flip operations for quadrilateral and hexahedral
meshes, similar to the flipping transformations previously used in triangular
and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th
International Meshing Roundtable. This version removes some unwanted
paragraph breaks from the previous version; the text is unchange
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
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