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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: M. Papadrakakis, B.H.V. Topping
Chapter 12

Advances in the Meccano Technique for Adaptive Tetrahedral Mesh Generation

R. Montenegro1, J.M. Cascón2, E. Rodríguez1, G. Cascón2 and J.M. Escobar1

1University Institute for Intelligent Systems and Numerical Applications in Engineering, University of Las Palmas de Gran Canaria, Spain
2Department of Mathematics, Faculty of Sciences, University of Salamanca, Spain

Full Bibliographic Reference for this chapter
R. Montenegro, J.M. Cascón, E. Rodríguez, G. Cascón, J.M. Escobar, "Advances in the Meccano Technique for Adaptive Tetrahedral Mesh Generation", in M. Papadrakakis, B.H.V. Topping, (Editors), "Trends in Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 12, pp 229-245, 2008. doi:10.4203/csets.20.12
Keywords: tetrahedral mesh generation, adaptive refinement-derefinement, nested meshes, mesh smoothing, mesh untangling, three-dimensional finite element method.

In this paper we present new ideas of an innovative tetrahedral mesh generator which was introduced in [1,2]. A local refinement-derefinement algorithm for nested triangulations [3] and a simultaneous untangling and smoothing procedure [4] are the main techniques involved. The mesh generator is applied to three-dimensional complex domains whose boundaries are projectable on external faces of a meccano approximation composed of cuboids. The domain surfaces must be given by a mapping between meccano surfaces and object boundary.

The mesh generator starts building a meccano approximation formed by cuboids. Then, a coarse and valid hexahedral mesh of the meccano approximation is generated. The automatic subdivision of each hexahedron into six tetrahedra produces an initial tetrahedral mesh of the meccano approximation. The main idea is to construct a sequence of nested meshes by refining only those tetrahedra with a face on the meccano boundary. The virtual projection of meccano external faces defines a valid triangulation on the domain boundary. Then a three-dimensional local refinement-derefinement is carried out so that the approximation of domain surfaces verifies a given precision. Once this objective is reached, those nodes placed on the meccano boundary are really projected on their corresponding true boundary, and inner nodes are relocated using a suitable mapping. As the mesh topology is kept during node movement, poor quality or even inverted elements could appear in the resulting mesh; therefore, we finally apply a mesh optimization procedure.

The combination of these techniques leads to a robust and highly competitive mesh generation method. We have important advantages with respect to other traditional approaches such as Delaunay triangulation or advancing front technique: surface triangulation is automatically constructed, the final triangulation is conforming with the object boundary, inner surfaces can be automatically preserved (for example, interface between several materials), an adaptive node distribution is obtained relating to the object geometry and parallel computations for meccano pieces could be easily developed. Nevertheless, our procedure demands at present of an automatic construction of the meccano and of a definition of a mapping between the meccano boundary to the object surface. New ideas will be introduced in this direction.

R. Montenegro, J.M. Cascón, J.M. Escobar, E. Rodríguez, G. Montero, "Implementation in ALBERTA of an automatic tetrahedral mesh generator", in: Proc. 15th International Meshing Roundtable, Springer, Berlin, 325-338, 2006. doi:10.1007/978-3-540-34958-7_19
J.M. Cascón, R. Montenegro, J.M. Escobar, E. Rodríguez, G. Montero, "A new meccano technique for adaptive 3-D triangulations", in: Proc. 16th International Meshing Roundtable, Springer, Berlin, 103-120, 2007. doi:10.1007/978-3-540-75103-8_6
I. Kossaczky, "A recursive approach to local mesh refinement in two and three dimensions", Journal of Computational and Applied Mathematics, 55, 275-288, 1994. doi:10.1016/0377-0427(94)90034-5
J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero, J.M. González-Yuste, "Simultaneous untangling and smoothing of tetrahedral meshes", Computer Methods in Applied Mechanics and Engineering, 192, 2775-2787, 2003. doi:10.1016/S0045-7825(03)00299-8

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