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Hexahedral mesh generation constraints

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Abstract

For finite element analyses within highly elastic and plastic structural domains, hexahedral meshes have historically offered some benefits over tetrahedral finite element meshes in terms of reduced error, smaller element counts, and improved reliability. However, hexahedral finite element mesh generation continues to be difficult to perform and automate, with hexahedral mesh generation taking several orders of magnitude longer than current tetrahedral mesh generators to complete. Thus, developing a better understanding of the underlying constraints that make hexahedral meshing difficult could result in dramatic reductions in the amount of time necessary to prepare a hexahedral finite element model for analysis. In this paper, we present a survey of constraints associated with hexahedral meshes (i.e., the conditions that must be satisfied to produce a hexahedral mesh). In presenting our formulation of these constraints, we will utilize the dual of a hexahedral mesh. We also discuss how incorporation of these constraints into existing hexahedral mesh generation algorithms could be utilized to extend the class of geometries to which these algorithms apply. We also describe a list of open problems in hexahedral mesh generation and give some context for future efforts in addressing these problems.

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References

  1. Means S, Smith AJ, Shepherd JF, Shadid J, Fowler J, Wojcikiewicz R, Mazel T, Smith GD, Wilson BS (2006) Reaction modeling of calcium dynamics with realistic ER geometry. Biophys J 91(2):537–557

    Article  Google Scholar 

  2. Milton KA (1996) Finite-element quantum field theory. In: Proceedings of the XIVth international symposium on lattice field theory, Nucl Phys B (Proc. Suppl.) 53 (1997): 847–849

  3. Meier DL (1999) Multidimensional astrophysical structural and dynamical analysis. I. Development of a nonlinear finite element approach. Astrophys J 518:788–813

    Article  Google Scholar 

  4. Loriot M (2006) Tetmesh-ghs3d v3.1 the fast, reliable, high quality tetrahedral mesh generator and optimiser (see http://www.simulog.fr/mesh/tetmesh3p1d-wp.pdf).

  5. White DR, Leland RW, Saigal S, Owen SJ (2001) The meshing complexity of a solid: an introduction. In: Proceedings, 10th international meshing roundtable. Sandia National Laboratories, pp 373–384

  6. White DR, Saigal S, Owen SJ (2003) Meshing complexity of single part CAD models. In: Proceedings, 12th international meshing roundtable. Sandia National Laboratories, pp 121–134

  7. Weingarten VI (1994) The controversy over hex or tet meshing. Mach Des 66:74–78

    Google Scholar 

  8. Cifuentes AO, Kalbag A (1992) A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis. Finite Elem Anal Des 12(3–4):313–318

    Article  Google Scholar 

  9. Steven E. Benzley, Ernest Perry, Karl Merkley, Brett Clark (1995) A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis. In: Proceedings, 4th international meshing roundtable. Sandia National Laboratories, pp 179–191

  10. Bussler ML, Ramesh A (1993) The eight-node hexahedral elements in FEA of part designs. Foundry Manag Technol 121 (11): 26–28

    Google Scholar 

  11. Steven J. Owen (1998) A survey of unstructured mesh generation technology (available at http://www.andrew.cmu.edu/user/sowen/survey/index.html). Accessed Sep

  12. Blacker TD, Mitchiner JL, Phillips LR, Lin YT (1988) Knowledge system approach to automated two-dimensional quadrilateral mesh generation. Comput Eng 3:153–162

    Google Scholar 

  13. Bill Thurston (1993) Geometry in action: Hexahedral decomposition of polyhedra (available from http://www.ics.uci.edu/ eppstein/gina/thurston-hexahedra.html). Accessed Oct

  14. Peter J. Murdoch (1995) The spatial twist continuum: a dual representation of the all hexahedral finite element mesh. Published Doctoral Dissertation, Brigham Young University, December

  15. Peter J. Murdoch, Steven E. Benzley (1995) The spatial twist continuum. In: Proceedings, 4th international meshing roundtable. Sandia National Laboratories, pp 243–251, October

  16. Scott A. Mitchell (1996) A characterization of the quadrilateral meshes of a surface which admit a compatible hexahedral mesh of the enclosed volumes. In: 13th Annual symposium on theoretical aspects of computer science, vol 1046. Lecture notes in computer science, pp 465–476

  17. David Eppstein (1996) Linear complexity hexahedral mesh generation. In: 12th ACM symposium on computational geometry. ACM Press, New York, pp 58–67

  18. 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 

  19. Jonathan Richard Shewchuk (2002) What is a good linear element? Interpolation, conditioning, and quality measures. In Proceedings, 11th international meshing roundtable. Sandia National Laboratories, pp 115–126

  20. Knupp PM (2001) Algebraic mesh quality metrics. SIAM J Sci Comput 23(1):193–218

    Article  MATH  MathSciNet  Google Scholar 

  21. Knupp P, Mitchell SA (1999) Integration of mesh optimization with 3D all-hex mesh generation, LDRD subcase 3504340000, final report. SAND 99-2852, October

  22. VP Garimella RV, Shashkov MJ Mesh untangling. LAU-UR-02-7271, T-7 Summer Report 2002

  23. Knupp PM (2000) Hexahedral mesh untangling and algebraic mesh quality metrics. In: Proceedings, 9th international meshing roundtable. Sandia National Laboratories, pp 173–183, October

  24. Freitag LA, Plassmann P (2000) Local optimization-based simplicial mesh untangling and improvement. Int J Numer Methods Eng 49(1):109–125

    Article  MATH  Google Scholar 

  25. Folwell NT, Mitchell SA (1999) Reliable whisker weaving via curve contraction. Eng Comput 15:292–302

    Article  MATH  Google Scholar 

  26. Bern M, Eppstein D (2001) Flipping cubical meshes. In: Proceedings, 10th international meshing roundtable. Sandia National Laboratories, pp 19–29

  27. Bern M, Eppstein D, Erickson J (2002) Flipping cubical meshes. Eng Comput 18(3):173–187

    Article  Google Scholar 

  28. Tautges TJ, Knoop S (2003) Topology modification of hexahedral meshes using atomic dual-based operations. In: Proceedings, 12th international meshing roundtable. Sandia National Laboratories, pp 415–423

  29. Suzuki T, Takahashi S, Shepherd JF (2005) Practical interior surface generation method for all-hexahedral meshing. In: Proceedings, 14th international meshing roundtable. Sandia National Laboratories, pp 377–397

  30. Schneiders pyramid open problem, http://www-users.informatik.rwth-aachen.de/ roberts/open.html

  31. Melander DJ, Tautges TJ, Benzley SE (1997) Generation of multi-million element meshes for solid model-based geometries: The dicer algorithm. AMD Trends Unstructured Mesh Generation 220:131–135

    Google Scholar 

  32. Mitchell SA, Tautges TJ (1995) Pillowing doublets: refining a mesh to ensure that faces share at most one edge. In: Proceedings, 4th international meshing roundtable. Sandia National Laboratories, pp 231–240

  33. Melander DJ (1997) Generation of multi-million element meshes for solid model-based geometries: The Dicer Algorithm. Published Master’s Thesis, Brigham Young University, April

  34. Borden MJ, Shepherd JF, Benzley SE (2002) Mesh cutting: fitting simple all-hexahedral meshes to complex geometries. In: Proceedings, 8th International society of grid generation conference

  35. Jankovich SR, Benzley SE, Shepherd JF, Mitchell SA (1999) The graft tool: an all-hexahedral transition algorithm for creating multi-directional swept volume mesh. In: Proceedings, 8th international meshing roundtable. Sandia National Laboratories, pp 387–392

  36. Borden MJ, Benzley SE, Shepherd JF (2002) Coarsening and sheet extraction for all-hexahedral meshes. In: Proceedings, 11th international meshing roundtable. Sandia National Laboratories, pp 147–152

  37. Tautges TJ, Blacker TD, Mitchell SA (1996) Whisker weaving: a connectivity-based method for constructing all-hexahedral finite element meshes. Int J Numer Methods Eng 39:3327–3349

    Article  MATH  MathSciNet  Google Scholar 

  38. Schneiders R (1997) An algorithm for the generation of hexahedral element meshes based on an octree technique. In: Proceedings, 6th international meshing roundtable. Sandia National Laboratories, pp 183–194

  39. Yerry MA, Shephard MS (1984) Three-dimensional mesh generation by modified octree technique. Int J Numer Methods Eng 20:1965–1990

    Article  MATH  Google Scholar 

  40. Shephard MS, Georges MK (1991) Three-dimensional mesh generation by finite octree technique. Int J Numer Meth Eng 32:709–749

    Article  MATH  Google Scholar 

  41. Zhang Y, Bajaj C (2005) Adaptive and quality quadrilateral/hexahedral meshing from volumetric imaging data. In: Proceedings, 13th international meshing roundtable. Sandia National Laboratories, pp 365–376

  42. Cook WA, Oakes WR (1982) Mapping methods for generating three-dimensional meshes. Comput Mech Eng CIME Res Suppl:67–72

  43. Dannenhoffer JF (III) (1991) A block-structuring technique for general geometries. In: 29th Aerospace sciences meeting and exhibit, vol AIAA-91-0145, January

  44. XYZ Scientific Applications, Inc. (2006) Truegrid: High quality hexahedral grid and mesh generation for fluids and structures (available from http://www.truegrid.com). Accessed Oct

  45. Pointwise (2006) Gridgen—reliable CFD meshing (available from http://www.pointwise.com/gridgen/), Accessed Oct

  46. ANSYS (2006) ANSYS ICEM CFD (available from http://www.ansys.com/products/icemcfd.asp), October

  47. White DR (1996) Automatic quadrilateral and hexahedral meshing of Pseudo-Cartesian geometries using virtual subdivision. Published Master’s Thesis, Brigham Young University, June

  48. White DR, Lai M, Benzley SE, Sjaardema GD (1995) Automated hexahedral mesh generation by virtual decomposition. In: Proceedings, 4th international meshing roundtable. Sandia National Laboratories, pp 165–176, October

  49. Knupp P (1998) Next-generation sweep tool: a method for generating all-hex meshes on two-and-one-half dimensional geometries. In: Proceedings, 7th international meshing roundtable. Sandia National Laboratories, pp 505–513, October

  50. Staten ML, Canann SA, Owen SJ (1998) BMSWEEP: locating interior nodes during sweeping. In: Proceedings, 7th international meshing roundtable. Sandia National Laboratories, pp 7–18, October

  51. Roca X, Sarrate J, Huerta A (2004) Surface mesh projection for hexahedral mesh generation by sweeping. In: Proceedings, 13th international meshing roundtable, vol SAND 2004-3765C. Sandia National Laboratories, pp 169–180, September

  52. Lai M, Benzley SE, White DR (2000) Automated hexahedral mesh generation by generalized multiple source to multiple target sweeping. Int J Numer Methods Eng 49(1):261–275

    Article  MATH  Google Scholar 

  53. Lai M, Benzley SE, Sjaardema GD, Tautges TJ (1996) A multiple source and target sweeping method for generating all-hexahedral finite element meshes. In: Proceedings, 5th international meshing roundtable. Sandia National Laboratories, pp 217–228, October

  54. Shepherd JF, Mitchell SA, Knupp P, White DR (2000) Methods for multisweep automation. In: Proceedings, 9th international meshing roundtable. Sandia National Laboratories, pp 77–87, October

  55. Blacker TD (1996) The cooper tool. In: Proceedings, 5th international meshing roundtable. Sandia National Laboratories, pp 13–30, October

  56. Whitely M, White DR, Benzley SE, Blacker TD (1996) Two and three-quarter dimensional meshing facilitators. Eng Comput 12:155–167

    Article  Google Scholar 

  57. Mitchell SA (1997) Choosing corners of rectangles for mapped meshing. In: 13th Annual symposium on computational geometry. ACM Press, New York, pp 87–93

  58. Mitchell SA (1997) High fidelity interval assignment. In: Proceedings, 6th international meshing roundtable. Sandia National Laboratories, pp 33–44, October

  59. Shepherd JF, Benzley SE, Mitchell SA (2000) Interval assignment for volumes with holes. Int J Numer Methods Eng 49(1):277–288

    Google Scholar 

  60. White DR, Tautges TJ (2000) Automatic scheme selection for toolkit hex meshing. Int J Numer Methods Eng 49(1):127–144

    Article  MATH  Google Scholar 

  61. Lu Y, Gadh R, Tautges TJ (1999) Volume decomposition and feature recognition for hexahedral mesh generation. In: Proceedings, 8th international meshing roundtable. Sandia National Laboratories, pp 269–280, October

  62. Tautges TJ, Liu SSs, Lu Y, Kraftcheck J, Gadh R (1997) Feature recognition applications in mesh generation. AMD Trends Unstructured Mesh Generation 220:117–121

  63. Kraftcheck J (2000) Virtual geometry: a mechanism for modification of CAD model topology for improved meshability. Published Master’s Thesis, Department of Mechanical Engineering, University of Wisconsin, December

  64. Schindler Weiler FR, Schneiders R (1996) Automatic geometry-adaptive generation of quadrilateral and hexahedral element meshes for FEM. In: Proceedings, 5th international conference on numerical grid generation in computational field simulations. Mississippi State University, pp 689–697, April

  65. Walton KS, Benzley SE, Shepherd JF (2002) Sculpting: an improved inside-out scheme for all-hexahedral meshing. In: Proceedings, 11th international meshing roundtable. Sandia National Laboratories, pp 153–159, September

  66. Zhang Y, Bajaj C, Xu G (2005) Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. In: Proceedings, 14th international meshing roundtable. Sandia National Laboratories, pp 449–468, September

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

    Google Scholar 

  68. Muller-Hannemann M (1998) Hexahedral mesh generation by successive dual cycle elimination. In: Proceedings, 7th international meshing roundtable. Sandia National Laboratories, pp 365–378, October

  69. Calvo NA, Idelsohn SR (2000) All-hexahedral element meshing: generation of the dual mesh by recurrent subdivision. Comput Methods Appl Mech Eng 182:371–378

    Google Scholar 

  70. Staten ML, Owen SJ, Blacker TD (2005) Unconstrained paving and plastering: a new idea for all hexahedral mesh generation. In: Proceedings, 14th international meshing roundtable. Sandia National Laboratories, pp 399–416, September

  71. Owen SJ, Saigal S (2000) H-morph an indirect approach to advancing front hex meshing. Int J Numer Methods Eng 49(1):289–312

    Google Scholar 

  72. Richards S, Benzley SE, Shepherd JF, Stephenson MB (2005) DTH exing: creation of semi-structured all-hexahedral meshes based on coarse tetrahedral primitives. Sandia Doc Num 5237908, November

  73. Richards S, Benzley SE, Shepherd JF, Stephenson MB (2003) DTH exing: an improved all-hexahedral meshing scheme using general coarsening tools. SAND2003-2724A and SAND2003-2818P

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  1. Scientific Computing and Imaging Institute, Salt Lake City, UT, USA

    Jason F. Shepherd & Chris R. Johnson

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  1. Jason F. Shepherd
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  2. Chris R. Johnson
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Correspondence to Jason F. Shepherd.

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Shepherd, J.F., Johnson, C.R. Hexahedral mesh generation constraints. Engineering with Computers 24, 195–213 (2008). https://doi.org/10.1007/s00366-008-0091-4

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