Uniform Random Voronoi Meshes
- 12 Citations
- 1.3k Downloads
Summary
We generate Voronoi meshes over three dimensional domains with prescribed boundaries. Voronoi cells are clipped at one-sided domain boundaries. The seeds of Voronoi cells are generated by maximal Poisson-disk sampling. In contrast to centroidal Voronoi tessellations, our seed locations are unbiased. The exception is some bias near concave features of the boundary to ensure well-shaped cells. The method is extensible to generating Voronoi cells that agree on both sides of two-sided internal boundaries.
Maximal uniform sampling leads naturally to bounds on the aspect ratio and dihedral angles of the cells. Small cell edges are removed by collapsing them; some facets become slightly non-planar.
Voronoi meshes are preferred to tetrahedral or hexahedral meshes for some Lagrangian fracture simulations. We may generate an ensemble of random Voronoi meshes. Point location variability models some of the material strength variability observed in physical experiments. The ensemble of simulation results defines a spectrum of possible experimental results.
Keywords
Voronoi Diagram Voronoi Cell Steiner Point Fracture Simulation Background GridPreview
Unable to display preview. Download preview PDF.
References
- 1.Amenta, N.: Arbitrary dimensional convex hull, Voronoi diagram, Delaunay triangulation, http://www.geom.uiuc.edu/software/cglist/ch.html
- 2.Bishop, J.: Simulating the pervasive fracture of materials and structures using randomly close packed voronoi tessellations. Computational Mechanics 44, 455–471 (2009), 10.1007/s00466-009-0383-6 CrossRefzbMATHGoogle Scholar
- 3.Bolander Jr., J.E., Saito, S.: Fracture analyses using spring networks with random geometry. Engineering Fracture Mechanics 61, 569–591 (1998)CrossRefGoogle Scholar
- 4.Bondesson, L., Fahlén, J.: Mean and variance of vacancy for hard-core disc processes and applications. Scandinavian Journal of Statistics 30(4), 797–816 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
- 5.Paul Chew, L.: Guaranteed-quality triangular meshes. Technical Report 89-983, Department of Computer Science, Cornell University (1989)Google Scholar
- 6.Brad Barber, C., Dobkin, D., Huhdanpaa, H.: Qhull (1995), http://www.qhull.org/
- 7.Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review 41(4), 637–676 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
- 8.Ebeida, M.S., Mitchell, S.A., Davidson, A.A., Patney, A., Knupp, P.M., Owens, J.D.: Efficient and good Delaunay meshes from random points. In: Proc. 2011 SIAM Conference on Geometric and Physical Modeling (GD/SPM11). Computer-Aided Design (2011)Google Scholar
- 9.Ebeida, M.S., Mitchell, S.A., Patney, A., Davidson, A.A., Owens, J.D.: Maximal Poisson-disk sampling with finite precision and linear complexity in fixed dimensions. In: ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH-Asia 2011) (submitted 2011)Google Scholar
- 10.Ebeida, M.S., Patney, A., Mitchell, S.A., Davidson, A., Knupp, P.M., Owens, J.D.: Efficient maximal Poisson-disk sampling. In: ACM Transactions on Graphics (Proc. SIGGRAPH 2011), vol. 30(4) (2011)Google Scholar
- 11.Fortune, S.: Voronoi diagrams and Delaunay triangulations, pp. 193–233. World Scientific (1992), http://ect.bell-labs.com/who/sjf/Voronoi.tar
- 12.Fu, Y., Zhou, B.: Direct sampling on surfaces for high quality remeshing. Computer Aided Geometric Design 26(6), 711–723 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
- 13.Gamito, M.N., Maddock, S.C.: Accurate multidimensional Poisson-disk sampling. ACM Transactions on Graphics 29(1), 1–19 (2009)CrossRefGoogle Scholar
- 14.Gärtner, B.: Fast and Robust Smallest Enclosing Balls. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 325–338. Springer, Heidelberg (1999), http://www.inf.ethz.ch/personal/gaertner/miniball.html Google Scholar
- 15.Johnson, J.: Geo1.stl (2008), http://www.3dvia.com/content/70FF9466784A5C6E
- 16.Knupp, P.: Algebraic mesh quality metrics. SIAM J. Sci. Comput. 23(1), 193–218 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
- 17.Miller, G.L., Talmor, D., Teng, S.-H., Walkington, N., Wang, H.: Control volume meshes using sphere packing: Generation, refinement and coarsening. In: 5th International Meshing Roundtable, p. 4761 (1996)Google Scholar
- 18.Mitchell, S.A.: Mesh generation with provable quality bounds. Applied Math. Cornell PhD Thesis, Cornell CS Tech Report TR93-1327 (1993), http://ecommons.library.cornell.edu/handle/1813/6093
- 19.Mitchell, S.A., Vavasis, S.A.: An aspect ratio bound for triangulating a d-grid cut by a hyperplane. In: Proceedings of the 12th Annual Symposium on Computational Geometry, pp. 48–57. ACM (1996)Google Scholar
- 20.Morris, D.: topmod-test.stl (2010), http://www.3dvia.com/content/4D4234435567794B
- 21.Paoletti, S.: Polyhedral mesh optimization using the interpolation tensor. In: Proc. 11th International Meshing Roundtable, pp. 19–28 (2002)Google Scholar
- 22.Quey, R., Dawson, P.R., Barbe, F.: Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing. Computer Methods in Applied Mechanics and Engineering 200(17-20), 1729–1745 (2011)CrossRefGoogle Scholar
- 23.Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms 18(3), 548–585 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
- 24.Shewchuk, J.R.: Delaunay refinement algorithms for triangular mesh generation. Comp. Geom.: Theory and Applications 22, 21–741 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
- 25.Shimada, K., Gossard, D.: Bubble mesh: Automated triangular meshing of non-manifold geometry by sphere packing. In: ACM Third Symposium on Solid Modeling and Applications, pp. 409–419. ACM (1995)Google Scholar
- 26.Shimada, K., Gossard, D.: Automatic triangular mesh generation of trimmed parametric surfaces for finite element analysis. Computer Aided Geometric Design 15(3), 199–222 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
- 27.Hang, S.: Tetgen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (2005-2011), http://tetgen.berlios.de/
- 28.Spielman, D.A., Teng, S.-H., Üngör, A.: Parallel Delaunay refinement: Algorithms and analyses. Int. J. Comput. Geometry Appl. 17(1), 1–30 (2007)CrossRefzbMATHGoogle Scholar
- 29.Üngör, A.: Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations. Comput. Geom. Theory Appl. 42, 109–118 (2009)zbMATHGoogle Scholar
- 30.Wei, L.-Y.: Parallel Poisson disk sampling. ACM Transactions on Graphics 27(3), 1–20 (2008)Google Scholar
- 31.Yan, D.-M., Lévy, B., Liu, Y., Sun, F., Wang, W.: Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. In: ACM/EG Symp. Geometry Processing / Computer Graphics Forum (2009)Google Scholar
- 32.Yan, D.-M., et al.: Efficient Computation of 3D Clipped Voronoi Diagram. In: Mourrain, B., Schaefer, S., Xu, G. (eds.) GMP 2010. LNCS, vol. 6130, pp. 269–282. Springer, Heidelberg (2010)CrossRefGoogle Scholar