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Bisection-Based Triangulations of Nested Hypercubic Meshes

  • Kenneth Weiss
  • Leila De Floriani

Abstract

Hierarchical spatial decompositions play a fundamental role in many disparate areas of scientific and mathematical computing since they enable adaptive sampling of large problem domains. Although the use of quadtrees, octrees, and their higher dimensional analogues is ubiquitous, these structures generate meshes with cracks, which can lead to discontinuities in functions defined on their domain. In this paper, we propose a dimension-independent triangulation algorithm based on regular simplex bisection to locally decompose adaptive hypercubic meshes into high quality simplicial complexes with guaranteed geometric and adaptivity constraints.

Keywords

Simplicial Complex Hexahedral Mesh Simplicial Decomposition Simplicial Mesh High Dimensional Analogue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atalay, F., Mount, D.: Pointerless implementation of hierarchical simplicial meshes and efficient neighbor finding in arbitrary dimensions. International Journal of Computational Geometry and Applications 17(6), 595–631 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bern, M., Eppstein, D.: Mesh generation and optimal triangulation. In: Du, D., Hwang, F. (eds.) Computing in Euclidean geometry, vol. 1, pp. 23–90. World Scientific, Singapore (1992)Google Scholar
  3. 3.
    Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. Journal of Computer and System Sciences 48(3), 384–409 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bey, J.: Simplicial grid refinement: on freudenthal’s algorithm and the optimal number of congruence classes. Numerische Mathematik 85(1), 1–29 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bhaniramka, P., Wenger, R., Crawfis, R.: Isosurface construction in any dimension using convex hulls. IEEE Transactions on Visualization and Computer Graphics 10(2), 130–141 (2004)CrossRefGoogle Scholar
  6. 6.
    Brönnimann, H., Glisse, M.: Octrees with near optimal cost for ray-shooting. Computational Geometry 34(3), 182–194 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dahmen, W.A., Micchelli, C.A.: On the linear independence of multivariate b-splines, i. triangulations of simploids. SIAM Journal on Numerical Analysis 19(5), 993–1012 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    De Floriani, L., Magillo, P.: Multiresolution mesh representation: models and data structures. In: Floater, M., Iske, A., Quak, E. (eds.) Principles of Multiresolution Geometric Modeling. Lecture Notes in Mathematics, pp. 364–418. Springer, Berlin (2002)Google Scholar
  9. 9.
    Evans, W., Kirkpatrick, D., Townsend, G.: Right-triangulated irregular networks. Algorithmica 30(2), 264–286 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Garimella, R.: Conformal refinement of unstructured quadrilateral meshes. In: Proceedings of the 18th International Meshing Roundtable, pp. 31–44. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Greaves, D.M., Borthwick, A.G.L.: Hierarchical tree-based finite element mesh generation. International Journal for Numerical Methods in Engineering 45(4), 447–471 (1999)zbMATHCrossRefGoogle Scholar
  12. 12.
    Ito, Y., Shih, A., Soni, B.: Efficient hexahedral mesh generation for complex geometries using an improved set of refinement templates. In: Proceedings of the 18th International Meshing Roundtable, pp. 103–115 (2009)Google Scholar
  13. 13.
    Maubach, J.M.: Local bisection refinement for n-simplicial grids generated by reflection. SIAM Journal on Scientific Computing 16(1), 210–227 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Min, C.: Local level set method in high dimension and codimension. Journal of Computational Physics 200(1), 368–382 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Moore, D.: Simplicial mesh generation with applications. Ph.D. thesis, Cornell University, Ithaca, NY, USA (1992)Google Scholar
  16. 16.
    Moore, D.: The cost of balancing generalized quadtrees. In: Proc. ACM Solid Modeling, pp. 305–312. ACM, New York (1995)Google Scholar
  17. 17.
    Moore, D., Warren, J.: Adaptive simplicial mesh quadtrees. Houston J. Math 21(3), 525–540 (1995)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Plantinga, S., Vegter, G.: Isotopic meshing of implicit surfaces. The Visual Computer 23(1), 45–58 (2007)CrossRefGoogle Scholar
  19. 19.
    Roettger, S., Heidrich, W., Slusallek, P., Seidel, H.: Real-time generation of continuous levels of detail for height fields. In: Proceedings Central Europe Winter School of Computer Graphics (WSCG), pp. 315–322 (1998)Google Scholar
  20. 20.
    Samet, H.: Foundations of Multidimensional and Metric Data Structures. The Morgan Kaufmann series in computer graphics and geometric modeling. Morgan Kaufmann, San Francisco (2006)zbMATHGoogle Scholar
  21. 21.
    Schneiders, R.: Refining quadrilateral and hexahedral element meshes. In: 5th International Conference on Grid Generation in Computational Field Simulations, pp. 679–688. Mississippi State University (1996)Google Scholar
  22. 22.
    Sivan, R.: Surface modeling using quadtrees. Ph.D. thesis, University of Maryland, College Park (1996)Google Scholar
  23. 23.
    Sivan, R., Samet, H.: Algorithms for constructing quadtree surface maps. In: Proc. 5th Int. Symposium on Spatial Data Handling, pp. 361–370 (1992)Google Scholar
  24. 24.
    Sundar, H., Sampath, R.S., Biros, G.: Bottom-up construction and 2:1 balance refinement of linear octrees in parallel. SIAM Journal of Scientific Computing 30(5), 2675–2708 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Traxler, C.T.: An algorithm for adaptive mesh refinement in n dimensions. Computing 59(2), 115–137 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Von Herzen, B., Barr, A.H.: Accurate triangulations of deformed, intersecting surfaces. In: Proceedings ACM SIGGRAPH, pp. 103–110. ACM, New York (1987)Google Scholar
  27. 27.
    Weigle, C., Banks, D.: Complex-valued contour meshing. In: Proceedings IEEE Visualization, pp. 173–180. IEEE Computer Society, Los Alamitos (1996)Google Scholar
  28. 28.
    Weiser, A.: Local-mesh, local-order, adaptive finite element methods with a posteriori error estimators for elliptic partial differential equations. Ph.D. thesis, Yale University (1981)Google Scholar
  29. 29.
    Weiss, K., De Floriani, L.: Diamond hierarchies of arbitrary dimension. Computer Graphics Forum (Proceedings SGP 2009) 28(5), 1289–1300 (2009)CrossRefGoogle Scholar
  30. 30.
    Weiss, K., De Floriani, L.: Supercubes: A high-level primitive for diamond hierarchies. IEEE Transactions on Visualization and Computer Graphics (Proceedings IEEE Visualization 2009) 15(6), 1603–1610 (2009)CrossRefGoogle Scholar
  31. 31.
    Weiss, K., De Floriani, L.: Simplex and diamond hierarchies: Models and applications. In: Hauser, H., Reinhard, E. (eds.) EG 2010 - State of the Art Reports, pp. 113–136. Norrköping, Sweden (2010)Google Scholar
  32. 32.
    Westermann, R., Kobbelt, L., Ertl, T.: Real-time exploration of regular volume data by adaptive reconstruction of isosurfaces. The Visual Computer 15(2), 100–111 (1999)CrossRefGoogle Scholar
  33. 33.
    Zorin, D., Schroder, P.: A unified framework for primal/dual quadrilateral subdivision schemes. Computer Aided Geometric Design 18(5), 429–454 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Kenneth Weiss
    • 1
  • Leila De Floriani
    • 2
  1. 1.University of MarylandCollege ParkUSA
  2. 2.University of GenovaGenovaItaly

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