Engineering with Computers

, Volume 31, Issue 1, pp 161–174 | Cite as

Comparison of the meccano method with standard mesh generation techniques

  • J. M. Cascón
  • E. Rodríguez
  • J. M. Escobar
  • R. Montenegro
Original Article


The meccano method is a novel and promising mesh generation technique for simultaneously creating adaptive tetrahedral meshes and volume parameterizations of a complex solid. The method combines several former procedures: a mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. In this paper we present the main advantages of our method against other standard mesh generation techniques. We show that our method constructs meshes that can be locally refined using the Kossaczky bisection rule and maintaining a high mesh quality. Finally, we generate volume T-mesh for isogeometric analysis, based on the volume parameterization obtained by the method.


Tetrahedral mesh generation Adaptive refinement Nested meshes Mesh untangling and smoothing Surface and volume parameterization 



This work has been supported by the Spanish Government, “Secretaría de Estado de Universidades e Investigación”, “Ministerio de Economía y Competitividad”, and FEDER, grant contracts: CGL2011-29396-C03-01 and CGL2011-29396-C03-03; “Junta Castilla León”, grant contract: SA266A12-2. It has been also supported by CONACYT-SENER (“Fondo Sectorial CONACYT SENER HIDROCARBUROS”, grant contract: 163723). Particularly, the authors thank Dr. Hang Si for providing them the new TetGen v.1.5.0 and for his kind comments and suggestions. They also thank Dr. Joachim Schöberl for developing the NETGEN freely available code.


  1. 1.
    Bänch E (1993) Adaptive finite element techniques for the Navier-Stokes equations and other transient problems. In: Brebbia CA, Aliabadi MH (eds) Adaptive finite and boundary elements. Computational Mechanics Publications and Elsevier, Amsterdam, pp 47–76Google Scholar
  2. 2.
    Bazilevs Y, Calo VM, Cottrell JA, Evans J, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2008) Isogeometric analysis: toward unification of computer aided design and finite element analysis. In: Trends in Engineering Computational Technology, Saxe-Coburg Publications, Stirling, pp 1–16Google Scholar
  3. 3.
    Bazilevs Y, Calo V, Cottrell J, Evans J, Hughes T, Lipton S, Scott M, Sederberg T (2010) Isogeometric analysis using T-splines. Comput Meth Appl Mech Eng 199:229–263CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Borouchaki H, Frey P (2005) Simplication of surface mesh using Hausdorff envelope. Comput Meth Appl MechEng 194:4864–4884CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Carey GF (1997) Computational grids: generation, adaptation and solution strategies. Taylor & Francis, WashingtonGoogle Scholar
  6. 6.
    Cascón JM, Montenegro R, Escobar JM, Rodríguez E, Montero G (2007) A new meccano technique for adaptive 3-D triangulation. In: Proceeding of the 16th International Meshing Roundtable, Springer, Berlin, pp 103–120Google Scholar
  7. 7.
    Cascón JM, Montenegro R, Escobar JM, Rodríguez E, Montero G (2009) The meccano method for automatic tetrahedral mesh generation of complex genus-zero solids. In: Proceeding of the 18th International Meshing Roundtable, Springer, Berlin, pp 463–480Google Scholar
  8. 8.
    Chen J, Zhao D, Huang Z, Zheng Y, Gao S (2011) Three-dimensional constrained boundary recovery with an enhanced Steiner point suppression procedure original research article. Comput Str 89:455–466CrossRefGoogle Scholar
  9. 9.
    Cottrell J, Hughes T, Bazilevs Y (2009) Isogeometric analysis: towad integration of CAD and FEA. John Wiley & Sons, ChichesterCrossRefGoogle Scholar
  10. 10.
    Eriksson K, Johnson C (1991) Adaptive finite elment methods for parabolic problems I: a linear model problem. SIAM J Numer Anal 28:43–77CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Escobar J, Montenegro R, Rodríguez E, Montero G (2011a) Simultaneous aligning and smoothing of surface triangulations. Eng Comput 27:17–29CrossRefGoogle Scholar
  12. 12.
    Escobar JM, Montenegro R, Rodríguez E, Cascón JM (2012) The meccano method for isogeometric solid modeling and applications. Eng Comput. doi: 10.1007/s00366-012-0300-z
  13. 13.
    Escobar JM, Rodríguez E, Montenegro R, Montero G, González-Yuste JM (2003) Simultaneous untangling and smoothing of tetrahedral meshes. Comput Meth Appl Mech Eng 192:2775–2787CrossRefMATHGoogle Scholar
  14. 14.
    Escobar JM, Montero G, Montenegro R, Rodríguez E (2006) An algebraic method for smoothing surface triangulations on a local parametric space. Int J Num Meth Eng 66:740–760CrossRefMATHGoogle Scholar
  15. 15.
    Escobar JM, Rodríguez E, Montenegro R, Montero G, González-Yuste JM (2010) SUS code: simultaneous mesh untangling and smoothing code.
  16. 16.
    Escobar JM, Cascón JM, Rodríguez E, Montenegro R (2011b) The meccano method for isogeometric solid modeling. In: Proceeding of the 20th International Meshing Roundtable, Springer, Berlin, pp 551–568Google Scholar
  17. 17.
    Escobar JM, Cascón JM, Rodríguez E, Montenegro R (2011c) A new approach to solid modeling with trivariate T-splines based on mesh optimization. Comput Meth Appl Mech Eng 200:3210–3222CrossRefMATHGoogle Scholar
  18. 18.
    Floater MS (1997) Parametrization and smooth approximation of surface triangulations. Comput Aid Geom Design 14:231–250CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Floater MS (2003) Mean value coordinates. Comput Aid Geom Design 20:19–27CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Floater MS, Hormann K (2005) Surface parameterization: a tutorial and survey. In: Advances in multiresolution for geometric modelling, mathematics and visualization. Springer, Berlin, pp 157–186Google Scholar
  21. 21.
    Floater MS, Pham-Trong V (2006) Convex combination maps over triangulations, tilings, and tetrahedral meshes. Adv Comput Math 25:347–356CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Freitag LA, Knupp PM (2002) Tetrahedral mesh improvement via optimization of the element condition number. Int J Num Meth Eng 53:1377–1391CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Freitag LA, Plassmann P (2000) Local optimization-based simplicial mesh untangling and improvement. Int J Num Meth Eng 49:109–125CrossRefMATHGoogle Scholar
  24. 24.
    Frey PJ, George PL (2000) Mesh generation. Hermes Science Publishing, OxfordMATHGoogle Scholar
  25. 25.
    George PL, Borouchaki H (1998) Delaunay triangulation and meshing: application to finite elements. Editions Hermes, ParisMATHGoogle Scholar
  26. 26.
    Knupp PM (2000) Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II-A frame work for volume mesh optimization and the condition number of the Jacobian matrix. Int J Num Meth Eng 48:1165–1185CrossRefMATHGoogle Scholar
  27. 27.
    Knupp PM (2001) Algebraic mesh quality metrics. SIAM J Sci Comput 23:193–218CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kossaczky I (1994) A recursive approach to local mesh refinement in two and three dimensions. J Comput Appl Math 55:275–288CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Li B, Li X, Wang K, Qin H (2010) Generalized polycube trivariate splines. In: Proceeding of the 2010 International Conference on Shape Modeling and Applications, IEEE Computer Society, pp 261–265Google Scholar
  30. 30.
    Li X, Guo X, Wang H, He Y, Gu X, Qin H (2007) Harmonic volumetric mapping for solid modeling applications. In: Proceeding of ACM Solid and Physical Modeling Symposium, Association for Computing Machinery, Inc., pp 109–120Google Scholar
  31. 31.
    Lin J, Jin X, Fan Z, Wang CCL (2008) Automatic polycube-maps. In: Lecture Notes in Computer Science,vol 4975. Springer, Berlin, p 316Google Scholar
  32. 32.
    Martin T, Cohen E (2010) Volumetric parameterization of complex objects by respecting multiple materials. Comput Graph 34:187–197CrossRefGoogle Scholar
  33. 33.
    Martin T, Cohen E, Kirby RM (2009) Volumetric parameterization and trivariate b-spline fitting using harmonic functions. Comput Aid Geom Design 26:648–664CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Maubach J (1995) Local bisection refinement for n-simplicial grids generated by reflection. SIAM J Sci Comput 16:210–227CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Montenegro R, Cascón JM, Escobar JM, Rodríguez E, Montero G (2009) An automatic strategy for adaptive tetrahedral mesh generation. Appl Num Math 59:2203–2217CrossRefMATHGoogle Scholar
  36. 36.
    Montenegro R, Cascón JM, Rodríguez E, Escobar JM, Montero G (2010) The meccano method for automatic three-dimensional triangulation and volume parametrization of complex solids. In: Developments and Applications in Engineering Computational Technology, Saxe-Coburg Publications, Stirling, pp 19–48Google Scholar
  37. 37.
    Schmidt A, Siebert KG (2005) Design of adaptive finite element software: the finite element toolbox ALBERTA, lecture notes in computer science, vol 42. Springer, BerlinGoogle Scholar
  38. 38.
    Schmidt A, Siebert KG, Koster D, Kriessl O, Heine CJ (2007) ALBERTA - an adaptive hierarchical finite element toolbox,
  39. 39.
    Schöberl J (1997) NETGEN—-an advancing front 2D/3D-mesh generator based on abstract rules. Comput Visual Sci 1:41–52CrossRefMATHGoogle Scholar
  40. 40.
    Shewchuk JR (1998) Tetrahedral mesh generation by Delaunay refinement. In: Proceeding of the Fourteenth Annual Symposium on Computational Geometry, ACM, New York, NY, USA, SCG ’98, pp 86–95Google Scholar
  41. 41.
    Si H (2008) Adaptive tetrahedral mesh generation by constrained Delaunay refinement. Int J Num Meth Eng 75:857–880CrossRefMathSciNetGoogle Scholar
  42. 42.
    Si H (2009) Tetgen: a quality tetrahedral mesh generator and three-dimensional Delaunay triangulator,, v. 1.4.3. Tech. rep., Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr 39, 10117 Berlin, Germany
  43. 43.
    Tarini M, Hormann K, Cignoni P, Montani C (2004) Polycube-maps. ACM Trans Graph 23:853–860CrossRefGoogle Scholar
  44. 44.
    Thompson JF, Soni B, Weatherill N (1999) Handbook of grid generation. CRC Press, LondonMATHGoogle Scholar
  45. 45.
    Traxler CT (1997) An algorithm for adaptive mesh refinement in n dimensions. Computing 59:115–137CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Wan S, Yin Z, Zhang K, Zhang H, Li X (2011) A topology-preserving optimization algorithm for polycube mapping. Comput Graph 35:639–649CrossRefGoogle Scholar
  47. 47.
    Wang H, He Y, Li X, Gu X, Qin H (2008) Polycube splines. Comput Aid Geom Design 40:721–733CrossRefMATHGoogle Scholar
  48. 48.
    Wang W, Zhang Y, Liu L, Hughes TJR (2013) Trivariate solid t-spline construction from boundary triangulations with arbitrary genus topology. Comput Aid Design 45:351–360CrossRefMathSciNetGoogle Scholar
  49. 49.
    Zhang Y, Wang W, Hughes TJR (2012) Solid t-spline construction from boundary representations for genus-zero geometry. Comput Meth Appl MechEng 249-252:185–197CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • J. M. Cascón
    • 1
  • E. Rodríguez
    • 2
  • J. M. Escobar
    • 2
  • R. Montenegro
    • 2
  1. 1.Department of Economics and History of EconomicsUniversity of SalamancaSalamancaSpain
  2. 2.University Institute for Intelligent Systems and Numerical Applications in Engineering (SIANI), University of Las Palmas de Gran CanariaLas PalmasSpain

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