Abstract
In this paper we introduce an automatic tetrahedral mesh generator for complex genus-zero solids, based on the novel meccano technique. Our method only demands a surface triangulation of the solid, and a coarse approximation of the solid, called meccano, that is just a cube in this case. The procedure builds a 3-D triangulation of the solid as a deformation of an appropriate tetrahedral mesh of the meccano. For this purpose, the method combines several procedures: an automatic mapping from the meccano boundary to the solid surface, a 3-D local refinement algorithm and a simultaneous mesh untangling and smoothing. A volume parametrization of the genus-zero solid to a cube (meccano) is a direct consequence. The efficiency of the proposed technique is shown with several applications.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis: Toward unification of computer aided design and finite element analysis. In: Trends in Engineering Computational Technology, pp. 1–16. Saxe-Coburg Publications, Stirling (2008)
Cascón, J.M., Montenegro, R., Escobar, J.M., Rodríguez, E., Montero, G.: A new meccano technique for adaptive 3-D triangulations. In: Proc. 16th Int. Meshing Roundtable, pp. 103–120. Springer, Berlin (2007)
Carey, G.F.: Computational grids: generation, adaptation, and solution strategies. Taylor & Francis, Washington (1997)
Escobar, J.M., Rodríguez, E., Montenegro, R., Montero, G., González-Yuste, J.M.: Simultaneous untangling and smoothing of tetrahedral meshes. Comput. Meth. Appl. Mech. Eng. 192, 2775–2787 (2003)
Escobar, J.M., Montero, G., Montenegro, R., Rodríguez, E.: An algebraic method for smoothing surface triangulations on a local parametric space. Int. J. Num. Meth. Eng. 66, 740–760 (2006)
Floater, M.S.: Parametrization and smooth approximation of surface triangulations. Comp. Aid. Geom. Design 14, 231–250 (1997)
Floater, M.S.: One-to-one Piece Linear Mappings over Triangulations. Mathematics of Computation 72, 685–696 (2002)
Floater, M.S.: Mean Value Coordinates. Comp. Aid. Geom. Design 20, 19–27 (2003)
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pp. 157–186. Springer, Berlin (2005)
Floater, M.S., Pham-Trong, V.: Convex Combination Maps over Triangulations, Tilings, and Tetrahedral Meshes. Advances in Computational Mathematics 25, 347–356 (2006)
Freitag, L.A., Knupp, P.M.: Tetrahedral mesh improvement via optimization of the element condition number. Int. J. Num. Meth. Eng. 53, 1377–1391 (2002)
Freitag, L.A., Plassmann, P.: Local optimization-based simplicial mesh untangling and improvement. Int. J. Num. Meth. Eng. 49, 109–125 (2000)
Frey, P.J., George, P.L.: Mesh generation. Hermes Sci. Publishing, Oxford (2000)
George, P.L., Borouchaki, H.: Delaunay triangulation and meshing: application to finite elements. Editions Hermes, Paris (1998)
Knupp, P.M.: 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, 1165–1185 (2000)
Knupp, P.M.: Algebraic mesh quality metrics. SIAM J. Sci. Comp. 23, 193–218 (2001)
Kossaczky, I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55, 275–288 (1994)
Li, X., Guo, X., Wang, H., He, Y., Gu, X., Qin, H.: Harmonic Volumetric Mapping for Solid Modeling Applications. In: Proc. of ACM Solid and Physical Modeling Symposium, pp. 109–120. Association for Computing Machinery, Inc. (2007)
Lin, J., Jin, X., Fan, Z., Wang, C.C.L.: Automatic polyCube-maps. In: Chen, F., Jüttler, B. (eds.) GMP 2008. LNCS, vol. 4975, pp. 3–16. Springer, Heidelberg (2008)
Montenegro, R., Escobar, J.M., Montero, G., Rodríguez, E.: Quality improvement of surface triangulations. In: Proc. 14th Int. Meshing Roundtable, pp. 469–484. Springer, Berlin (2005)
Montenegro, R., Cascón, J.M., Escobar, J.M., Rodríguez, E., Montero, G.: Implementation in ALBERTA of an automatic tetrahedral mesh generator. In: Proc. 15th Int. Meshing Roundtable, pp. 325–338. Springer, Berlin (2006)
Montenegro, R., Cascón, J.M., Escobar, J.M., Rodríguez, E., Montero, G.: An Automatic Strategy for Adaptive Tetrahedral Mesh Generation. Appl. Num. Math. 59, 2203–2217 (2009)
Schmidt, A., Siebert, K.G.: Design of adaptive finite element software: the finite element toolbox ALBERTA. Lecture Notes in Computer Science and Engineering, vol. 42. Springer, Berlin (2005)
Schmidt, A., Siebert, K.: ALBERTA - An Adaptive Hierarchical Finite Element Toolbox, http://www.alberta-fem.de/
Tarini, M., Hormann, K., Cignoni, P., Montani, C.: Polycube-maps. ACM Trans. Graph 23, 853–860 (2004)
Thompson, J.F., Soni, B., Weatherill, N.: Handbook of grid generation. CRC Press, London (1999)
Wang, H., He, Y., Li, X., Gu, X., Qin, H.: Polycube Splines. Comp. Aid. Geom. Design 40, 721–733 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cascón, J.M., Montenegro, R., Escobar, J.M., Rodríguez, E., Montero, G. (2009). The Meccano Method for Automatic Tetrahedral Mesh Generation of Complex Genus-Zero Solids. In: Clark, B.W. (eds) Proceedings of the 18th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04319-2_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-04319-2_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04318-5
Online ISBN: 978-3-642-04319-2
eBook Packages: EngineeringEngineering (R0)