Recent progress in robust and quality Delaunay mesh generation
2006, Journal of Computational and Applied Mathematics
https://doi.org/10.1016/J.CAM.2005.07.014Abstract
In this paper, some current issues of Delaunay mesh generation and optimization are addressed, with particular emphasis on the robustness of the meshing procedure and the quality of the resulting mesh. We also report new progress on the robust conforming and constrained boundary recovery in three dimensions, along with the quality mesh generation based on Centroidal Voronoi tessellations. Applications to the numerical solution of differential equations and integrations with other softwares are discussed, including a brief discussion on the joint mesh and solver adaptation strategy.
References (78)
- N. Amenta, M. Bern, D. Eppstein, Optimal point placement for mesh smoothing, J. Algorithms 30 (1999) 302-322.
- E. D'Azevedo, B. Simpson, On optimal triangular meshes for minimizing the gradient error, Numer. Math. 59 (1991) 321-348.
- P. Baehmann, L. Scott, L. Wittchen, M. Shephard, K. Grice, M. Yerry, Robust geometrically based, automatic two- dimensional mesh generation, Internat. J. Numer. Methods Engrg. 24 (1987) 1043-1078.
- R. Bank, Mesh smoothing using a posteriori estimates, SIAM J. Numer. Anal. 34 (1997) 979-997.
- E. Barnes, N. Sloane, The optimal Lattice quantizer in three dimensions, SIAM J. Algebraic Discrete Methods 4 (1983) 30-41.
- M. Berzins, A solution-based triangular and tetrahedral mesh quality indicator, SIAM J. Sci. Comput. 19 (1998) 2051-2069.
- A. Bowyer, Computing Dirichlet tessellations, Comput. J. 24 (1981) 162-166.
- H. Borouchaki, P. George, Aspects of 2-D Delaunay mesh generation, Internat. J. Numer. Methods Engrg. 40 (1997) 1957-1975.
- H. Borouchaki, P. George, F. Hecht, P. Laug, E. Saltel, Delaunay mesh generation governed by metric specifications. Part I, Algorithms 25 (1997) 61-83.
- H. Borouchaki, P. George, S. Lo, Optimal Delaunay point insertion, Internat. J. Numer. Methods Engrg. 39 (1996) 3407-3437.
- H. Borouchaki, S. Lo, Fast Delaunay triangulation in three dimensions, Comput. J. Numer. Methods. Engrg. 128 (1995) 153-167.
- H. Borouchaki, L. Patrick, P. George, Parametric surface meshing using a combined advancing-front generalized Delaunay approach, Internat. J. Numer. Methods Engrg. 49 (2000) 233-259.
- G. Buscaglia, E. Dari, Anisotropic mesh optimization and its application in adaptivity, Internat. J. Numer. Methods Engrg. 40 (1997) 4119-4136.
- L. Chew, Guaranteed quality mesh generation for curved surfaces, in: Proceedings of the Ninth Annual Computer Geometry, 1993, pp. 274-280.
- P. Ciarlet, Finite Element Methods, North-Holland, Amsterdam, 1975.
- Q. Du, M. Emelianenko, Uniform convergence of a nonlinear energy-based multilevel quantization scheme, preprint, 2005.
- Q. Du, M. Emelianenko, Acceleration schemes for computing centroidal Voronoi tessellations, preprint, 2005
- Q. Du, M. Emelianenko, L. Ju, Convergence properties of the Lloyd algorithm for computing the centroidal Voronoi tessellations, SIAM J. Numerical Analysis, 2005, accepted for publication.
- Q. Du, V. Faber, M. Gunzburger, Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev. 41 (1999) 637-676.
- Q. Du, M. Gunzburger, Grid generation and optimization based on centroidal Voronoi tessellations, Appl. Comput. Math. 133 (2002) 591-607.
- Q. Du, M. Gunzburger, L. Ju, Probablistic methods for centroidal Voronoi tessellations and their parallel implementations, J. Parallel Comput. 28 (2002) 1477-1500.
- Q. Du, M. Gunzburger, L. Ju, Meshfree, probabilistic determination of point sets and support regions for meshless computing, Comput. Methods Appl. Mech. Engrg. 191 (2002) 1349-1366.
- Q. Du, M. Gunzburger, L. Ju, Constrained centroidal Voronoi tessellations on general surfaces, SIAM J. Sci. Comput. 24 (5) (2003) 1499-1506.
- Q. Du, Z. Huang, D. Wang, Mesh and solver coadaptation in finite element methods for anisotropic problems, Numer. Methods for Differential Equations 21 (2005) 859-874.
- Q. Du, L. Ju, Finite volume methods and spherical centroidal Voronoi tessellations, SIAM J. Numer. Anal., accepted for publication.
- Q. Du, D. Wang, Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations, Internat. J. Numer. Methods Engrg. 56 (2003) 1355-1373.
- Q. Du, D. Wang, Boundary recovery for three dimensional conforming Delaunay triangulation, Comput. Methods Appl. Mech. Engrg. 193 (2004) 2547-2563.
- Q. Du, D. Wang, Constrained boundary recovery for three dimensional Delaunay triangulation, Internat. J. Numer. Methods Engrg. 61 (2004) 1471-1500.
- Q. Du, D. Wang, Mesh optimization based on centroidal Voronoi tessellation, Internat. J. Numer. Anal. Modelling, 2004, accepted for publication.
- Q. Du, X. Wang, Centroidal Voronoi tessellation based algorithms for vector fields visualization and segmentation, IEEE Proceedings of Visualization 2004 (VIS2004), Austin, TX, 2004, pp. 43-50.
- Q. Du, D. Wang, On the optimal centroidal Voronoi tessellations and the Gersho's conjecture in the three dimensional space, Comput. Math. Appl. 49 (2005) 1355-1373.
- Q. Du, D. Wang, Anisotropic centroidal Voronoi tessellations and their applications, SIAM J. Sci. Comput. 26 (2005) 737-761.
- H. Edelsbrunner, D. Guoy, Sink insertion for mesh improvement, Internat. J. Foundations Comput. Sci. 13 (2002) 223-242.
- P. Frey, H. Borouchaki, Geometric surface mesh optimization, Comput. Visualization Sci. 1 (1998) 113-121.
- P. Frey, H. Borouchaki, P. George, 3D Delaunay mesh generation coupled with an advancing-front approach, Comput. Methods Appl. Mech. Engrg. 157 (1998) 115-131.
- P. Frey, L. Marechal, Fast adaptive quadtree mesh generation, in: Proceedings of the Seventh International Meshing Roundtable, 1998, pp. 211-222.
- A. George, Computer implementation of the finite element method, Ph.D. thesis, Standford University, STAN-CS-71-208, 1971.
- P. George, Gamanic3d, adaptive anisotropic tetrahedral mesh generator, Technical Report, INRIA, 2002.
- P. George, H. Borouchaki, Delaunay triangulation and meshing, Application to Finite Elements Methods, Hermès, Paris, 1998.
- P. George, H. Borouchaki, E. Saltel, Ultimate robustness in meshing an arbitrary polyhedron, Internat. J. Numer. Methods Engrg. 58 (2003) 1061-1089.
- P. George, F. Hecht, E. Saltel, Automatic mesh generation with specified boundary, Comput. Methods Appl. Mech. Engrg. 92 (1991) 169-188.
- A. Gersho, Asymptotically optimal block quantization, IEEE Trans. Inform. Theory 25 (1979) 373-380.
- F. Hermeline, Triangulation automatique d'un polyèdre en dimension, N.R.A.I.R.O. 16 (1982) 211-242.
- N. Hitschfeld, L. Villablanca, J. Krause, M.C. Rivara, Improving the quality of meshes for the simulation of semiconductor devices using Lepp-based algorithms, Internat. J. Numer. Methods Engrg. 58 (2003) 333-347.
- H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, W. Stuetzle, Mesh optimization, ACM Comput. Graphics 27 (1993) 19-26.
- B. Karamete, M. Beall, M. Shephard, Triangulation of arbitrary polyhedra to support automatic mesh generators, Internat. J. Numer. Methods Engrg. 49 (2000) 167-191.
- P. Knupp, Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities, I-a framework for surface mesh optimization, Internat. J. Numer. Methods Engrg. 48 (2000) 401-420.
- F. Labelle, J. Shewchuk, Anisotropic Voronoi Diagrams and Guaranteed-Quality Anisotropic Mesh Generation, SoCG 2003, San Diego.
- C. Lee, Automatic metric advancing front triangulation over curved surfaces, Eng. Comput. 17 (2000) 48-74.
- G. Leibon, D. Letscher, Delaunay triangulations and Voronoi diagrams for Riemannian manifolds, in: Proceedings of the 16th Annual Symposium on Computational Geometry (HK), ACM, New York, June 2000, pp. 341-349.
- X. Li, Sliver-free three dimensional Delaunay mesh generation, Ph.D thesis, UIUC, 2000.
- A. Liu, M. Baida, How far flipping can go towards 3D conforming/constrained triangulation, in: Proceedings of the Ninth International Meshing Roundtable, 2000.
- S. Lloyd, Least square quantization in PCM, IEEE Trans. Inform. Theory 28 (1982) 129-137.
- S. Lo, Volume discretization into tetrahedra-I, Verification and orientation of boundary surfaces, Comput. Structures 36 (1991) 493-500.
- S. Lo, Volume discretization into tetrahedra-II, 3D triangulation by advancing front approach, Comput. Structures 36 (1991) 501-511.
- R. Lohner, Progress in grid generation via the advancing front technique, Engrg. Comput. 12 (1996) 186-199.
- R. Lohner, C. Juan, Generation of non-isotropic unstructured grids via directional enrichment, Internat. J. Numer. Methods Engrg. 49 (1) (2000) 219-232.
- R. Lohner, P. Parikh, Generation of three-dimensional unstructured grids by the advancing-front method, Internat. J. Numer. Methods Engrg. 8 (1988) 1135-1149.
- J. MacQueen, Some methods for classification and analysis of multivariate observations, in: L. LeCam, J. Neyman (Eds.), Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, I, University of California, 1967, pp. 281-297.
- D. Marcum, N. Weatherill, Unstructured grid generation using iterative point insertion and local reconnection, AIAA J. 33 (9) (1995) 1625.
- D. Mavriplis, Directional coarsening and smoothing for anisotropic Navier-Stokes problems, Electron. Trans. Numer. Anal. 6 (1997) 182-197.
- K. Nakahashi, D. Sharov, Direct surface triangulation using the advancing front method, AIAA paper 95-1686-CP, 1995.
- D. Newman, The Hexagon theorem, IEEE Trans. Inform. Theory 28 (1982) 137-139.
- J. Peraire, J. Peiro, L. Formaggia, K. Morgan, O. Zienkiewicz, Finite element Euler computations in three dimensions, Internat. J. Numer. Methods Engrg. 26 (1988) 2135-2159.
- A. Rassineux, Generation and optimization of tetrahedral meshes by advancing front technique, Internat. J. Numer. Methods Engrg. 41 (1998) 651-674.
- S. Rippa, Long and thin triangles can be good for linear interpolation, SIAM J. Numer. Anal. 29 (1992) 257-270.
- W. Schroeder, M. Shephard, A combined octree/Delaunay method for fully automatic 3D mesh generation, Internat. J. Numer. Methods Engrg. 29 (1990) 37-55.
- M. Shephard, M. Georges, Automatic three-dimensional mesh generation by the finite Octree technique, Internat. J. Numer. Methods Engrg. 32 (1991) 709-749.
- J. Shewchuk, What is a good linear finite element? Interpolation, conditioning, anisotropy and quality measures. CS report, UC Berlekey.
- J. Shewchuk, Tetrahedral mesh generation by Delaunay refinement, in: 14th Annual ACM Symposium on Computational Geometry, 1998, pp. 86-95.
- J. Schewchuk, Constrained Delaunay tetrahedralization and provably good boundary recovery, in: Proceedings of the11th International Meshing Roundtable, 2002, pp. 193-204.
- K. Tchon, K. Mohammed, G. Francois, C. Ricardo, Constructing anisotropic geometric metrics using octrees and skeletons, in: Proceedings of the 12th International Meshing Roundtable, 2003, pp. 293-304.
- J. Thompson, B. Soni, N. Weatherill, Handbook of Grid Generation, CRC Press LLC, Boca Raton, FL, 1999.
- D. Watson, Computing the n-dimensional Delaunay tessellation with applications to Voronoi polytopes, Comput. J. 24 (1981) 167-172.
- N. Weatherill, The integrity of geometrical boundaries in the two-dimensional Delaunay triangulation, Comput. Appl. Numer. Methods 6 (1990) 101-109.
- N. Weatherill, O. Hassan, Efficient three dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints, Internat. J. Numer. Methods Engrg. 37 (1994) 2005-2039.
- J. Wright, A. Jack, Aspects of three-dimensional constrained Delaunay meshing, Internat. J. Numer. Methods Engrg. 37 (1994) 1841-1846.
- Y. Zhang, C. Bajaj, B. Sohn, 3D Finite Element Meshing from Imaging Data, Comput. Methods Appl. Mech. Eng., 2005, in press.