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Efficient and Global Optimization-Based Smoothing Methods for Mixed-Volume Meshes

  • Dimitris Vartziotis
  • Benjamin Himpel

Summary

Some methods based on simple regularizing geometric element transformations have heuristically been shown to give runtime efficient and quality effective smoothing algorithms for meshes. We describe the mathematical framework and a systematic approach to global optimization-based versions of such methods for mixed volume meshes, which generalizes to arbitrary dimensions. We also identify some algorithms based on simple regularizing geometric element transformation optimizing certain global algebraic mesh quality measures.

Keywords

smoothing quality metric quality measure finite element method global optimization optimization-based method GETMe 

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References

  1. 1.
    Amenta, N., Bern, M., Eppstein, D.: Optimal Point Placement for Mesh Smoothing. Journal of Algorithms 30, 302–322 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bezdek, D.: A proof of an extension of the icosahedral conjecture of Steiner for generalized deltahedra. Contrib. Discrete Math. 2(1), 86–92 (2007)MathSciNetMATHGoogle Scholar
  3. 3.
    Bossen, F.J., Heckbert, P.S.: A Pliant Method for Anisotropic Mesh Generation. In: Proceedings of the 5th International Meshing Roundtable (1996)Google Scholar
  4. 4.
    Branets, L.V.: A variational grid optimization method based on a local cell quality metric. Ph.D. thesis, Austin, TX, USA, AAI3187661 (2005)Google Scholar
  5. 5.
    Brewer, M., Diachin, L.A.F., Knupp, P.M., Leurent, T., Melander, D.: The Mesquite Mesh Quality Improvement Toolkit. In: Proceedings of the 12th International Meshing Roundtable, pp. 239–250 (2003)Google Scholar
  6. 6.
    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: Proceedings of the 7th International Meshing Roundtable, pp. 479–494 (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.7221
  7. 7.
    Chen, Z., Tristano, J.R., Kwok, W.: Combined Laplacian and Optimization-based Smoothing for Quadratic Mixed Surface Meshes. In: Proceedings of the 12th International Meshing Roundtable (2003), http://www.andrew.cmu.edu/user/sowen/imr12.html
  8. 8.
    Diachin, L.A.F., Knupp, P.M., Munson, T., Shontz, S.M.: A comparison of two optimization methods for mesh quality improvement. Engineering with Computers 22(2), 61–74 (2006), http://dx.doi.org/10.1007/s00366-006-0015-0, doi:10.1007/s00366-006-0015-0CrossRefGoogle Scholar
  9. 9.
    Fejes Tóth, L.: Ein Beweisansatz für die isoperimetrische Eigenschaft des Ikosaeders. Acta Math. Acad. Sci. Hungar. 3, 155–163 (1952)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Field, D.A.: Laplacian smoothing and Delaunay triangulations. Communications in Applied Numerical Methods 4(6), 709–712 (1988)CrossRefMATHGoogle Scholar
  11. 11.
    Freitag, L.A.: On combining Laplacian and optimization-based mesh smoothing techniques. In: Trends in Unstructured Mesh Generation, pp. 37–43 (1997)Google Scholar
  12. 12.
    Freitag, L.A., Jones, M., Plassmann, P.: An Efficient Parallel Algorithm for Mesh Smoothing. In: Proceedings of the 4th International Meshing Roundtable, pp. 47–58 (1995), http://www.andrew.cmu.edu/user/sowen/Roundtable.agenda.html
  13. 13.
    Freitag, L.A., Ollivier-Gooch, C.: Tetrahedral Mesh Improvement Using Swapping and Smoothing. International Journal for Numerical Methods in Engineering 40(21), 3979–4002 (1997), doi:10.1002/(SICI)1097-0207(19971115)40:21(3979::AID-NME251)3.0.CO;2-9Google Scholar
  14. 14.
    Freitag, L.A., Plassmann, P.: Local optimization-based simplicial mesh untangling and improvement. International Journal of Numerical Methods in Engineering 49(1-2), 109–125 (2000), doi:10.1002/1097-0207(20000910/20)49:1/2(109:AID-NME925)3.0.CO;2-UGoogle Scholar
  15. 15.
    Klingner, B.M., Shewchuk, J.R.: Aggressive Tetrahedral Mesh Improvement. In: Proceedings of the 16th International Meshing Roundtable, pp. 3–23 (2007)Google Scholar
  16. 16.
    Knupp, P.M.: Algebraic mesh quality metrics. SIAM Journal on Scientific Computing 23(1), 193–218 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Knupp, P.M.: Hexahedral and Tetrahedral Mesh Untangling. Engineering with Computers 17(3), 261–268 (2001), doi:10.1007/s003660170006CrossRefMATHGoogle Scholar
  18. 18.
    Leng, J., Xu, G., Zhang, Y., Qian, J.: A Novel Geometric Flow-Driven Approach for Quality Improvement of Segmented Tetrahedral Meshes. In: Quadros, W.R. (ed.) Proceedings of the 20th International Meshing Roundtable, pp. 347–364. Springer Publishing Company, Incorporated (2012)Google Scholar
  19. 19.
    Li, T., Wong, S., Hon, Y., Armstrong, C., McKeag, R.: Smoothing by optimisation for a quadrilateral mesh with invalid elements. Finite Elements in Analysis and Design 34(1), 37–60 (2000), http://www.sciencedirect.com/science/article/pii/S0168874X99000268, doi: http://dx.doi.org/10.1016/S0168-874X9900026-8
  20. 20.
    Mei, G., Tipper, J.C., Xu, N.: The Modified Direct Method: An Iterative Approach for Smoothing Planar Meshes. In: ICCS, pp. 2436–2439 (2013)Google Scholar
  21. 21.
    Owen, S.J.: A Survey of Unstructured Mesh Generation Technology. In: Proceedings of the 7th International Meshing Roundtable, pp. 239–267 (1998)Google Scholar
  22. 22.
    Parthasarathy, V., Kodiyalam, S.: A constrained optimization approach to finite element mesh smoothing. Finite Elements in Analysis and Design 9(4), 309 – 320 (1991), http://www.sciencedirect.com/science/article/pii/0168874X9190004I, doi: http://dx.doi.org/10.1016/0168-874X9190004-I
  23. 23.
    Sastry, S.P., Shontz, S.M.: Performance characterization of nonlinear optimization methods for mesh quality improvement. Eng. with Comput. 28(3), 269–286 (2012), http://dx.doi.org/10.1007/s00366-011-0227-9, doi:10.1007/s00366-011-0227-9Google Scholar
  24. 24.
    Shimada, K., Yamada, A., Itoh, T.: Anisotropic Triangulation of Parametric Surfaces via Close Packing of Ellipsoids. Internat. J. Comput. Geom. Appl. 10(4), 417–440 (2000), http://www.worldscientific.com/doi/abs/10.1142/S0218195900000243, doi:10.1142/S0218195900000243; Selected papers from the Sixth International Meshing Roundtable, Part II (Park City, UT) (1997)
  25. 25.
    Shivanna, K., Grosland, N., Magnotta, V.: An Analytical Framework for Quadrilateral Surface Mesh Improvement with an Underlying Triangulated Surface Definition. In: Shontz, S. (ed.) Proceedings of the 19th International Meshing Roundtable, pp. 85–102. Springer, Heidelberg (2010), http://dx.doi.org/10.1007/978-3-642-15414-0_6, doi:10.1007/978-3-642-15414-06
  26. 26.
    Steiner, J.: Über Maximum und Minimum bei den Figuren in der Ebene, auf der Kugelfläche und im Raume überhaupt. C. R. Acad. Sci. Paris 12, 177–308 (1841)Google Scholar
  27. 27.
    Steinitz, E.: Über isoperimetrische Probleme bei konvexen Polyedern. J. Reine Angew. Math. 158, 129–153 (1927)MATHGoogle Scholar
  28. 28.
    Strang, G., Fix, G.: An analysis of the finite element method, 2nd edn. Wellesley-Cambridge Press, Wellesley (2008)Google Scholar
  29. 29.
    Vartziotis, D., Himpel, B.: The mean volume as a quality measure for polyhedra and meshes, arXiv:1302.6066 [math.GT] (2013), http://arxiv.org/abs/1302.6066
  30. 30.
    Vartziotis, D., Papadrakakis, M.: Improved GETMe by adaptive mesh smoothing. Computer Assisted Methods in Engineering and Science 20, 55–71 (2013)Google Scholar
  31. 31.
    Vartziotis, D., Wipper, J.: A dual element based geometric element transformation method for all-hexahedral mesh smoothing. Comput. Methods Appl. Mech. Engrg. 200(9-12), 1186–1203 (2011), http://dx.doi.org/10.1016/j.cma.2010.09.012, doi:10.1016/j.cma.2010.09.012MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Vartziotis, D., Wipper, J.: Fast smoothing of mixed volume meshes based on the effective geometric element transformation method. Comput. Methods Appl. Mech. Engrg. 201/204, 65–81 (2012), http://dx.doi.org/10.1016/j.cma.2011.09.008, doi:10.1016/j.cma.2011.09.008MathSciNetCrossRefGoogle Scholar
  33. 33.
    Vartziotis, D., Wipper, J., Papadrakakis, M.: Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the poisson equation. Finite Elem. Anal. Des. 66, 36–52 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Vartziotis, D., Wipper, J., Schwald, B.: The geometric element transformation method for tetrahedral mesh smoothing. Comput. Methods Appl. Mech. Engrg. 199(1-4), 169–182 (2009), http://dx.doi.org/10.1016/j.cma.2009.09.027, doi:10.1016/j.cma.2009.09.027MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Wilson, T.J.: Simultaneous Untangling and Smoothing of Hexahedral Meshes. Master’s thesis, Universitat Politècnica de Catalunya, Spain (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department for Mathematical Research & ServicesTWT GmbH Science & InnovationNeuhausenGermany
  2. 2.Digital Engineering, Research CenterNIKI Ltd.KatsikaGreece

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