An interior surface generation method for all-hexahedral meshing | Engineering with Computers Skip to main content
Springer Nature Link
Log in

An interior surface generation method for all-hexahedral meshing

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This paper describes an all-hexahedral generation method focusing on how to create interior surfaces. It is well known that a solid homeomorphic to a ball with even number of bounding quadrilaterals can be partitioned into a compatible hexahedral mesh where each associated hexahedron corresponds to the intersection of three interior surfaces that are dual to the original hexahedral mesh. However, no such method for creating dual interior surfaces has been developed for generating all-hexahedral meshes of volumes covered with simply connected quadrilaterals. We generate an interior surface as an orientable regular homotopy (or more definitively a sweep) by splitting a dual cycle into several pieces at self-intersecting points and joining the three connected pieces, if the self-intersecting point-types are identical, while we generate a non-orientable surface (containing Möbius bands) if the self-intersecting point-types are distinct. Stitching these simple interior surfaces together allows us to compose more complex interior surfaces. Thus, we propose a generalized method of generating a hexahedral mesh topology by directly creating the interior surface arrangement. We apply the present framework to Schneiders’ open pyramid problem and show an arrangement of interior surfaces that decompose Schneiders’ pyramid into 146 hexahedra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35

Similar content being viewed by others

Notes

  1. While orientability is basically defined for only a closed surface, we can inherit and consequently define the ‘orientability’ of an open surface by gluing topological disks along its boundary circles to refer to the orientability of its corresponding closed surface.

References

  1. Bern M, Eppstein D (2002) Flipping cubical meshes. Proceedings of 10th international meshing roundtable, pp 19–29

  2. Blacker TD, Meyer RJ (1993) Seams and wedges in plastering: a 3-D hexahedral mesh generation algorithm. Eng Comput 9:83–93

    Article  Google Scholar 

  3. Eppstein D (1996) Linear complexity hexahedral mesh generation. In: Proceedings of 12th annual symposium on computational geometry, pp 58–67

  4. Folwell NT, Mitchell SA (1998) Reliable whisker weaving via curve contraction. In: Proceedings of 7th international meshing roundtable, pp 365–378

  5. Francis GK (1987) A topological picturebook. Springer, New York

    MATH  Google Scholar 

  6. Müller-Hannemann M (1998) Hexahedral mesh generation with successive dual cycle elimination. In: Proceedongs of 7th international meshing roundtable, pp 379–393

  7. Mitchell SA (1996) A characterization of the quadrilateral meshes of a surface which admits a compatible hexahedral mesh of enclosed volume. In: Proceedings of 13th annual symposium on theoretical aspect of computer science (STACS ‘96). Lecture Notes in Computer Science 1046, pp 465–476 The brief paper is available online from ftp://ams.sunysb.edu/pub/geometry/msi-workshop/95/samitch.ps.gz

  8. Owen SJ (1999) Constrained triangulation: application to hex-dominant mesh generation. In: Proceedings, 8th international meshing roundtable, pp 31–41

  9. Owen SJ (2001) Hex-dominant mesh generation using 3D constrained triangulation. CAD 33(3):211–220

    MathSciNet  Google Scholar 

  10. Schneiders R (www) http://www-users.informatik.rwth-aachen.de/~roberts/open.html

  11. Smale S (1958) Regular curves on Riemannian manifolds. Trans Am Math Soc 87:492–510

    Article  MATH  MathSciNet  Google Scholar 

  12. Schwartz A, Ziegler GM (2004) Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds. Exp Math 13(4):385–413

    MATH  MathSciNet  Google Scholar 

  13. Tautges TJ, Blacker T, Mitchell SA (1996) The whisker weaving algorithm: a connectivity-based method for constructing all-hexahedral finite element method. Int J Numer Methods Eng 39:3327–3349

    Article  MATH  MathSciNet  Google Scholar 

  14. Tautges TJ, Mitchell SA (1995) Whisker Weaving: Invalid Connectivity Resolution and Primal Construction Algorithm, Proceedings, 4th International Meshing Roundtable, pp 115–127

  15. Thurston W (1993) Hexahedral decomposition of polyhedra, posting to Sci. Math, 25 Oct 1993. Available online from http://www.ics.uci.edu/~eppstein/gina/Thurston-hexahedra

  16. Yamakawa S, Shimada K (2001) Hexhoop: modular templates for converting a hex-dominant mesh to an all-hex mesh. In: 10th International meshing round table, pp 235–246

Download references

Acknowledgments

The authors would like to thank Scott Mitchell (Sandia National Laboratories), Soji Yamakawa (Carnegie Mellon University), and Hiroshi Sakurai (Colorado State University) for their advice and insights into hexahedral meshing.

Author information

Authors and Affiliations

  1. Digital Process Ltd., 2-9-6, Nakacho, Atsugi city, Kanagawa, 243-0018, Japan

    Tatsuhiko Suzuki

  2. The University of Tokyo, 5-1-5 Kashiwa no ha, Kashiwa city, Chiba, 277-8561, Japan

    Shigeo Takahashi

  3. Sandia National Laboratories, Data Analysis and Visualization Department, Albuquerque, NM, 87185, USA

    Jason Shepherd

Authors
  1. Tatsuhiko Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shigeo Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jason Shepherd
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatsuhiko Suzuki.

Additional information

This paper was submitted to the 14th International Meshing Roundtable (2005).

Appendix

Appendix

Table 1 The 20-Hexahedral mesh (convexity lost) by NASTRAN CHEXA Format
Full size table
Fig. 36
figure 36

Node numbering on the surface

Full size image

Rights and permissions

Reprints and permissions

About this article

Cite this article

Suzuki, T., Takahashi, S. & Shepherd, J. An interior surface generation method for all-hexahedral meshing. Engineering with Computers 26, 303–316 (2010). https://doi.org/10.1007/s00366-009-0159-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-009-0159-9

Keywords

Search

Navigation