Guaranteed-Quality All-Quadrilateral Mesh Generation with Feature Preservation

  • Xinghua Liang
  • Mohamed S. Ebeida
  • Yongjie Zhang


In this paper, a quadtree-based mesh generation method is described to create guaranteed-quality, geometry-adapted all-quadrilateral meshes with feature preservation for arbitrary planar domains. Given point cloud, our method generates all-quad meshes with these points as vertices and all the angles are within [45°, 135°]. For given planar curves, quadtree-based spatial decomposition is governed by the curvature of the boundaries and narrow regions. 2-refinement templates are chosen for local mesh refinement without creating any hanging nodes. A buffer zone is created by removing elements around the boundary. To guarantee the mesh quality, the angles facing the boundary are improved via template implementation, and two buffer layers are inserted in the buffer zone. It is proved that all the elements of the final mesh are quads with angles between 45°±ε and 135°±ε (ε ≤ 5°) with the exception of badly shaped elements that may be required by the specified geometry. Sharp features and narrow regions are detected and preserved. Furthermore, boundary layer meshes are generated by splitting elements of the second buffer layer. We have applied our algorithm to a set of complicated geometries, including the Lake Superior map and the air foil with multiple components.


Guaranteed quality all-quadrilateral mesh quadtree sharp feature narrow region boundary layer 


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  1. 1.
    Atalay, F.B., Ramaswami, S.: Quadrilateral meshes with bounded minimum angle. In: 17th Int. Meshing Roundtable, pp. 73–91 (2008)Google Scholar
  2. 2.
    Baehmann, P.L., Wittchen, S.L., Shephard, M.S., Grice, K.R., Yerry, M.A.: Robust geometrically based, automatic two-dimensional mesh generation. Int. J. Numer. Meth. Eng. 24, 1043–1078 (1987)zbMATHCrossRefGoogle Scholar
  3. 3.
    Baker, T.J.: Identification and preservation of surface features. In: 13th Int. Meshing Roundtable, pp. 299–309 (2004)Google Scholar
  4. 4.
    Bern, M., Eppstein, D.: Quadrilateral meshing by circle packing. Int. J. Comp. Geom. & Appl. 10(4), 347–360 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bishop, C.J.: Quadrilateral meshes with no small angles (Manuscript) (1991)Google Scholar
  6. 6.
    Blacker, T.D., Stephenson, M.B.: Paving: A new approach to automated quadrilateral mesh generation. Int. J. Numer. Meth. Eng. 32, 811–847 (1991)zbMATHCrossRefGoogle Scholar
  7. 7.
    Canann, S.A., Tristano, J.R., Staten, M.L.: An approach to combined laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes . In: 7th Int. Meshing Roundtable, pp. 211–224 (1998)Google Scholar
  8. 8.
    Freitag, L., Jones, M., Plassmann, P.: An efficient parallel algorithm for mesh smoothing. In: 4th Int. Meshing Roundtable, pp. 47–58 (1995)Google Scholar
  9. 9.
    Freitag, L.A.: On combining Laplacian and optimization-based mesh smoothing techniques. Trends in Unstructured Mesh Generation, ASME 220, 37–43 (1997)Google Scholar
  10. 10.
    Joe, B.: Quadrilateral mesh generation in polygonal regions. Comput. Aid. Des. 27(3), 209–222 (1991)CrossRefGoogle Scholar
  11. 11.
    Kinney, P.: CleanUp: Improving quadrilateral finite element meshes. In: 6th Int. Meshing Roundtable, pp. 437–447 (1997)Google Scholar
  12. 12.
    Mitchell, S.A., Tautges, T.J.: Pillowing doublets: Refining a mesh to ensure that faces share at most one edge. In: 4th Int. Meshing Roundtable, pp. 231–240 (1995)Google Scholar
  13. 13.
    Owen, S.: A survey of unstructured mesh generation technology. In: 7th Int. Meshing Roundtable, pp. 26–28 (1998)Google Scholar
  14. 14.
    Quadros, W.R., Ramaswami, K., Prinz, F.B., Gurumoorthy, B.: LayTracks: A new approach to automated geometry adaptive quadrilateral mesh generaton using medial axis transform. Int. J. Numer. Meth. Eng. 61, 209–237 (2004)zbMATHCrossRefGoogle Scholar
  15. 15.
    Schneiders, R.: Refining quadrilateral and hexahedral element Meshes. In: 5th Int. Meshing Roundtable, pp. 383–398 (1996)Google Scholar
  16. 16.
    Schneiders, R., Schindler, R., Weiler, F.: Octree-based generation of hexahedral element meshes. In: 5th Int. Meshing Roundtable, pp. 205–216 (1996)Google Scholar
  17. 17.
    Staten, M.L., Canann, S.A.: Post refinement element shape improvement for quadrilateral meshes. Trends in Unstructured Mesh Generation, ASME 220, 9–16 (1997)Google Scholar
  18. 18.
    Tam, T., Armstrong, C.G.: 2D finite element mesh generation by medial axis subdivision. Adv. Eng. Software 13, 313–324 (1991)zbMATHCrossRefGoogle Scholar
  19. 19.
    White, D.R., Kinney, P.: Redesign of the paving algorithm: Robustness enhancements through element by element meshing. In: 6th Int. Meshing Roundtable, pp. 323–335 (1997)Google Scholar
  20. 20.
    Yerry, M.A., Shephard, M.S.: A modified quadtree approach to finite element mesh generation. IEEE Computer Graphics Appl. 3(1), 39–46 (1983)CrossRefGoogle Scholar
  21. 21.
    Zhang, Y., Bajaj, C.: Adaptive and quality quadrilateral/hexahedral meshing from volumetric Data. Comput. Meth. Appl. Mech. Eng. 195, 942–960 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Zhang, Y., Bajaj, C., Sohn, B.-S.: 3D finite element meshing from imaging data. Comput. Meth. Appl. Mech. Eng. 194, 5083–5106 (2005)zbMATHCrossRefGoogle Scholar
  23. 23.
    Zhang, Y., Bajaj, C., Xu, G.: Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Commun. Numer. Meth. Eng. 25, 1–18 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zhang, Y., Hughes, T., Bajaj, C.: An automatic 3D mesh generation method for domains with multiple materials. Comput. Meth. Appl. Mech. Eng. (in press, 2009)Google Scholar
  25. 25.
    Zhu, J.Z., Zienkiewicz, O.C., Hinton, E., Wu, J.: A new approach to the development of automatic quadrilateral mesh generation . Int. J. Numer. Meth. Eng. 32, 849–866 (1991)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Xinghua Liang
    • 1
  • Mohamed S. Ebeida
    • 1
  • Yongjie Zhang
    • 1
  1. 1.Department of Mechanical EngineeringCarnegie Mellon UniversityUSA

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