Abstract
A hierarchical simplicial mesh is a recursive decomposition of space into cells that are simplices. Such a mesh is compatible if pairs of neighboring cells meet along a single common face. Compatibility condition is important in many applications where the mesh serves as a discretization of a function. Enforcing compatibility involves refining the simplices of the mesh further, thus generates a larger mesh. We show that the size of a simplicial mesh grows by no more than a constant factor when compatibly refined. We prove a tight upper bound on the expansion factor for 2-dimensional meshes, and we sketch upper bounds for d-dimensional meshes.
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References
1. E. Allgower and K. Georg. Generation of triangulations by reflection. Utilitas Mathematica, 16:123–129, 1979.
2. F. B. Atalay and D. M. Mount. Pointerless implementation of hierarchical simplicial meshes and eficient neighbor finding in arbitrary dimensions. In Proc. International Meshing Roundtable (IMR 2004), pages 15–26, 2004.
3. M. Duchaineau, M. Wolinsky, D.E. Sigeti, M.C. Miller, C. Aldrich, and M.B. Mineev-Weinstein. Roaming terain: Real-time optimally adapting meshes. In Proc. IEEE Visualization'97, pages 81–88, 1997.
4. H. Edelsbrunner. Algorithms in Combinatorial Geometry, volume 10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1987.
5. W. Evans, D. Kirkpatrick, and G. Townsend. Right-triangulated irregular networks. Algorithmica., 30(2):264–286, 2001.
6. T. Gerstner. Multiresolution visualization and compression of global topographic data. GeoInformatica, 7(1):7–32, 2003.
7. T. Gerstner and M. Rumpf. Multiresolutional parallel isosurface exraction based on tetrahedral bisection. In Proc. Symp. Volume Visualization, 1999.
8. B. Von Herzen and A. Barr. Accurate triangulations of deformed intersecting surfaces. Computer Graphics, 21(4):103–110, 1987.
9. P. Lindstrom, D. Koller, W. Ribarsky, L.F. Hodges, N. Faust, and G.A. Turner. Real-time, continuous level of detail rendering of height fields. In Proc. of SIGGRAPH 96, pages 109–118, 1996.
10. J. M. Maubach. Local bisection refinement for N-simplicial grids generated by reflection. SIAM J. Sci. Stat. Comput., 16:210–227, 1995.
11. D. Moore. The cost of balancing generalized quadtrees. In Proc. ACM Solid Modeling, 1995.
12. R. Pajarola. Large scale terrain visualization using the restricted quadtree triangulation. In Proc. IEEE Visualization'98, pages 19–26, 1998.
13. A. Weiser. Local-Mesh, Local-Order, Adaptive Finite Element Methods with a Posteriori Error Estimators for Elliptic Partial Differential Equations. PhD thesis, Yale University, 1981.
14. Y. Zhou, B. Chen, and A. Kaufman. Multiresolution tetrahedral framework for visualizing regular volume data. In Proc. IEEE Visualization'97, pages 135–142, 1997.
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Atalay, F.B., Mount, D.M. (2006). The Cost of Compatible Refinement of Simplex Decomposition Trees. In: Pébay, P.P. (eds) Proceedings of the 15th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34958-7_4
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DOI: https://doi.org/10.1007/978-3-540-34958-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34957-0
Online ISBN: 978-3-540-34958-7
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