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Surface Smoothing and Quality Improvement of Quadrilateral/Hexahedral Meshes with Geometric Flow

  • Conference paper
Proceedings of the 14th International Meshing Roundtable

Summary

This paper describes an approach to smooth the surface and improve the quality of quadrilateral/hexahedral meshes with feature preserved using geometric flow. For quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direction, and the aspect ratio is improved with feature preserved by adjusting vertex positions in the tangent direction. For hexahedral meshes, besides the surface vertex movement in the normal and tangent directions, interior vertices are relocated to improve the aspect ratio. Our method has the properties of noise removal, feature preservation and quality improvement of quadrilateral/hexahedral meshes, and it is especially suitable for biomolecular meshes because the surface diffusion flow preserves sphere accurately if the initial surface is close to a sphere. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations.

http://www.ices.utexas.edu/~jessica/paper/quadhexgf

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References

  1. T. Baker. Identification and preservation of surface features. In 13th International Meshing Roundtable, pages 299–310, 2004.

    Google Scholar 

  2. F. J. Bossen and P. S. Heckbert. A pliant method for anisotropic mesh generation. In 5th International Meshing Roundtable, pages 63–76, 1996.

    Google Scholar 

  3. C. Bajaj, J. Warren, and G. Xu. A subdivision scheme for hexahedral meshes. The Visual Computer, 18(5-6):343–356, 2002.

    Article  Google Scholar 

  4. C. Charalambous and A. Conn. An efficient method to solve the minimax problem directly. SIAM Journal of Numerical Analysis, 15(1):162–187, 1978.

    Article  MathSciNet  Google Scholar 

  5. S. Canann, J. Tristano, and M. Staten. An approach to combined laplacian and optimization-based smoothing for triangular, quadrilateral and quad-dominant meshes. In 7th International Meshing Roundtable, pages 479–494, 1998.

    Google Scholar 

  6. J. Escher, U. F. Mayer, and G. Simonett. The surface diffusion flow for immersed hypersurfaces. SIAM Journal of Mathematical Analysis, 29(6):1419–1433, 1998.

    Article  MathSciNet  Google Scholar 

  7. D. Field. Laplacian smoothing and delaunay triangulations. Communications in Applied Numerical Methods, 4:709–712, 1988.

    Article  MATH  Google Scholar 

  8. L. Freitag and C. Ollivier-Gooch. Tetrahedral mesh improvement using swapping and smoothing. International Journal Numerical Methathemical Engineeringg, 40:3979–4002, 1997.

    Article  MathSciNet  Google Scholar 

  9. L. Freitag and P. Plassmann. Local optimization-based simplicial mesh untangling and improvement. International Journal for Numerical Method in Engineering, 49:109–125, 2000.

    Article  Google Scholar 

  10. L. Freitag. On combining laplacian and optimization-based mesh smoothing techniqes. AMD-Vol. 220 Trends in Unstructured Mesh Generation, pages 37–43, 1997.

    Google Scholar 

  11. P. L. George and H. Borouchaki. Delaunay Triangulation and Meshing, Application to Finite Elements. 1998.

    Google Scholar 

  12. R. V. Garimella, M. J. Shashkov, and P. M. Knupp. Optimization of surface mesh quality using local parametrization. In 11th International Meshing Roundtable, pages 41–52, 2002.

    Google Scholar 

  13. P. Kinney. Cleanup: Improving quadrilateral finite element meshes. In 6th International Meshing Roundtable, pages 437–447, 1997.

    Google Scholar 

  14. C. Kober and M. Matthias. Hexahedral mesh generation for the simulation of the human mandible. In 9th International Meshing Roundtable, pages 423–434, 2000.

    Google Scholar 

  15. P. Knupp. Winslow smoothing on two dimensional unstructured meshes. Engineering with Computers, 15:263–268, 1999.

    Article  MATH  Google Scholar 

  16. P. Knupp. Achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities. part i-a framework for surface mesh optimization. International Journal Numerical Methathemical Engineeringg, 48:401–420, 2000.

    Article  MATH  Google Scholar 

  17. P. Knupp. Achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities. part ii-a framework for volume mesh optimization and the condition number of the jacobian matrix. International Journal Numerical Methathemical Engineeringg, 48:1165–1185, 2000.

    Article  MATH  Google Scholar 

  18. R. Lohner, K. Morgan, and O. C. Zienkiewicz. Adaptive grid refinement for compressible euler equations. Accuracy Estimates and Adaptive Refinements in Finite Element Computations, I. Babuska et. al. eds. Wiley, pages 281–297, 1986.

    Google Scholar 

  19. M. Meyer, M. Desbrun, P. Schröder, and A. Burr. Discrete differential-geometry operators for triangulated 2-manifolds. VisMath’02, Berlin, 2002.

    Google Scholar 

  20. S. Mitchell and T. Tautges. Pillowing doublets: Refining a mesh to ensure that faces share at most one edge. In 4th International Meshing Roundtable, pages 231–240, 1995.

    Google Scholar 

  21. A. Oddy, J. Goldak, M. McDill, and M. Bibby. A distortion metric for isoparametric finite elements. Transactions of CSME, No. 38-CSME-32, Accession No. 2161, 1988.

    Google Scholar 

  22. S. Owen. A survey of unstructured mesh generation technology. In 7th Internat. Meshing Roundtable, pages 26–28, 1998.

    Google Scholar 

  23. G. Sapiro. Geometric Partial Differential Equations and Image Analysis. Cambridge, University Press, 2001.

    Google Scholar 

  24. M. Staten and S. Canann. Post refinement element shape improvement for quadrilateral meshes. AMD-Trends in Unstructured Mesh Generation, 220:9–16, 1997.

    Google Scholar 

  25. R. Schneiders. Refining quadrilateral and hexahedral element meshes. In 5th International Conference on Grid Generation in Computational Field Simulations, pages 679–688, 1996.

    Google Scholar 

  26. I. B. Semenova, V. V. Savchenko, and I. Hagiwara. Two techniques to improve mesh quality and preserve surface characteristics. In 13th International Meshing Roundtable, pages 277–288, 2004.

    Google Scholar 

  27. S. M. Shontz and S. A. Vavasis. A mesh warping algorithm based on weighted laplacian smoothing. In 12th International Meshing Roundtable, pages 147–158, 2003.

    Google Scholar 

  28. K. Shimada, A. Yamada, and T. Itoh. Anisotropic triangular meshing of parametric surfces via close packing of ellipsoidal bubbles. In 6th International Meshing Roundtable, pages 375–390, 1997.

    Google Scholar 

  29. S.-H. Teng and C. W. Wong. Unstructured mesh generation: Theory, practice, and perspectives. International Journal of Computational Geometry and Applications, 10(3):227–266, 2000.

    Article  MathSciNet  Google Scholar 

  30. T. J. Willmore. Riemannian geometry. Clareden Press, Oxford, 1993.

    Google Scholar 

  31. G. Xu, Q. Pan, and C. Bajaj. Discrete surface modelling using partial differential equations. Accepted by Computer Aided Geomtric Design, 2005.

    Google Scholar 

  32. G. Xu. Discrete laplace-beltrami operators and their convergence. CAGD, 21:767–784, 2004.

    MATH  Google Scholar 

  33. Y. Zhang and C. Bajaj. Adaptive and quality quadrilateral/hexahedral meshing from volumetric data. In 13th International Meshing Roundtable, pages 365–376, 2004.

    Google Scholar 

  34. Y. Zhang and C. Bajaj. Adaptive and quality quadrilateral/hexahedral meshing from volumetric data. Computer Methods in Applied Mechanics and Engineering (CMAME), in press, www.ices.utexas.edu/~jessica/paper/quadhex, 2005.

    Google Scholar 

  35. Y. Zhang, C. Bajaj, and B-S Sohn. 3d finite element meshing from imaging data. Special issue of Computer Methods in Applied Mechanics and Engineering on Unstructured Mesh Generation, in press, 2004.

    Google Scholar 

  36. T. Zhou and K. Shimada. An angle-based approach to two-dimensional mesh smoothing. In 9th International Meshing Roundtable, pages 373–384, 2000.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computational Visualization Center, Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA

    Yongjie Zhang

  2. Computational Visualization Center, Department of Computer Sciences and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA

    Chandrajit Bajaj

  3. LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, China

    Guoliang Xu

Authors
  1. Yongjie Zhang
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  2. Chandrajit Bajaj
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  3. Guoliang Xu
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Editor information

Editors and Affiliations

  1. Computer Modeling and Sciences Dept., Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, 87185, USA

    Byron W. Hanks

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© 2005 Springer-Verlag Berlin Heidelberg

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Zhang, Y., Bajaj, C., Xu, G. (2005). Surface Smoothing and Quality Improvement of Quadrilateral/Hexahedral Meshes with Geometric Flow. In: Hanks, B.W. (eds) Proceedings of the 14th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-29090-7_27

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  • DOI: https://doi.org/10.1007/3-540-29090-7_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25137-8

  • Online ISBN: 978-3-540-29090-2

  • eBook Packages: EngineeringEngineering (R0)

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