Xiao-Ming Fu
Skip Abstract Section
Skip Supplemental Material SectionAbstract
We present a novel method to generate high-quality simplicial meshes with specified anisotropy. Given a surface or volumetric domain equipped with a Riemannian metric that encodes the desired anisotropy, we transform the problem to one of functional approximation. We construct a convex function over each mesh simplex whose Hessian locally matches the Riemannian metric, and iteratively adapt vertex positions and mesh connectivity to minimize the difference between the target convex functions and their piecewise-linear interpolation over the mesh. Our method generalizes optimal Delaunay triangulation and leads to a simple and efficient algorithm. We demonstrate its quality and speed compared to state-of-the-art methods on a variety of domains and metrics.
Supplemental Material
Available for Download
zip
a182.zip (11.8 MB)
Supplemental material.
- Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M. 2005. Variational tetrahedral meshing. ACM Trans. Graph. (SIGGRAPH) 24, 3, 617--625. Google Scholar
Digital Library
- Amari, S.-I., and Armstrong, J. 2014. Curvature of Hessian mnifolds. Differential Geom. Appl., 33, 1--12.Google ScholarCross Ref
- Boissonnat, J.-D., Cohen-Steiner, D., and Yvinec, M. 2008. Comparison of algorithms for anisotropic meshing and adaptive refinement. Tech. rep., INRIA. ACS-TR-362603.Google Scholar
- Boissonnat, J.-D., Wormser, C., and Yvinec, M. 2008. Locally uniform anisotropic meshing. In SOCG, 270--277. Google ScholarDigital Library
- Boissonnat, J.-D., Wormser, C., and Yvinec, M. 2011. Anisotropic Delaunay mesh generation. Tech. rep., INRIA. INRIA-00615486.Google Scholar
- Boissonnat, J.-D., Shi, K.-L., Tournois, J., and Yvinec, M. 2014. Anisotropic Delaunay meshes of surfaces. ACM Trans. Graph., to appear.Google Scholar
- Canas, G. D., and Gortler, S. J. 2011. Orphan-free anisotropic Voronoi diagrams. Discrete Comput. Geom. 46, 3, 526--541. Google ScholarDigital Library
- Chen, L., and Holst, M. 2011. Efficient mesh optimization schemes based on optimal Delaunay triangulations. Comput. Methods in Appl. Mech. Eng. 200, 912, 967--984.Google Scholar
- Chen, L., and Xu, J. 2004. Optimal Delaunay triangulations. J. Comput. Math. 22, 299--308.Google Scholar
- Chen, L., Sun, P., and Xu, J. 2007. Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm. Math. Comp. 76, 179--204.Google ScholarCross Ref
- Chen, L. 2004. Mesh smoothing schemes based on optimal Delaunay triangulations. In Int. Meshing Roundtable, 109--120.Google Scholar
- Cheng, S.-W., Dey, T. K., Ramos, E. A., and Wenger, R. 2006. Anisotropic surface meshing. In SODA, 202--211. Google ScholarDigital Library
- Clark, B., Ray, N., and Jiao, X. 2012. Surface mesh optimization, adaption, and untangling with high-order accuracy. In Int. Meshing Roundtable. 385--402.Google Scholar
- Desbrun, M., Donaldson, R. D., and Owhadi, H. 2013. Modeling across scales: discrete geometric structures in homogenization and inverse homogenization. In Multiscale Analysis and Nonlinear Dynamics, 19--64.Google Scholar
- Dobrzynski, C., and Frey, P. 2008. Anisotropic Delaunay mesh adaptation for unsteady simulations. In Int. Meshing Roundtable, 177--194.Google Scholar
- Du, Q., and Wang, D. 2005. Anisotropic centroidal Voronoi tessellations and their applications. SIAM J. Sci. Comput. 26, 3, 737--761. Google ScholarDigital Library
- Frey, P., and George, P.-L. 2008. Mesh Generation: Application to finite elements, 2 ed. Wiley-ISTE.Google Scholar
- Genz, A., and Cools, R. 2003. An adaptive numerical cubature algorithm for simplices. ACM Trans. Math. Softw. 29, 3, 297--308. Google ScholarDigital Library
- Goes, F. D., Memari, P., Mullen, P., and Desbrun, M. 2014. Weighted triangulations for geometry processing. ACM Trans. Graph. 33, 3, 28:1--28:13. Google ScholarDigital Library
- Hecht, F., 1998. BAMG: Bidimensional anisotropic mesh generator. http://www.ann.jussieu.fr/hecht/ftp/bamg.Google Scholar
- Jiao, X., Colombi, A., Ni, X., and Hart, J. 2010. Anisotropic mesh adaptation for evolving triangulated surfaces. Eng. with Comput. 26, 4, 363--376. Google ScholarDigital Library
- Klingner, B. M., and Shewchuk, J. R. 2007. Agressive tetrahedral mesh improvement. In Int. Meshing Roundtable, 3--23.Google Scholar
- Labelle, F., and Shewchuk, J. R. 2003. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. In SOCG, 191--200. Google ScholarDigital Library
- Lévy, B., and Bonneel, N. 2012. Variational anisotropic surface meshing with Voronoi parallel linear enumeration. In Int. Meshing Roundtable, 349--366.Google Scholar
- Lévy, B., and Liu, Y. 2010. Lp centroidal Voronoi tessellation and its applications. ACM Trans. Graph. (SIGGRAPH) 29, 4, 119:1--119:11. Google ScholarDigital Library
- Li, Y., Liu, Y., and Wang, W. 2014. Planar hexagonal meshing for architecture. IEEE. T. Vis. Comput. Gr., to appear.Google Scholar
- Liu, Y., Pan, H., Snyder, J., Wang, W., and Guo, B. 2013. Computing self-supporting surfaces by regular triangulation. ACM Trans. Graph. (SIGGRAPH) 32, 4, 92:1--92:10. Google ScholarDigital Library
- Mullen, P., Memari, P., de Goes, F., and Desbrun, M. 2011. HOT: Hodge-optimized triangulations. ACM Trans. Graph. (SIGGRAPH) 30, 4, 103:1--103:12. Google ScholarDigital Library
- Panozzo, D., Puppo, E., Tarini, M., and Sorkine-Hornung, O. 2014. Frame fields: anisotropic and non-orthogonal cross fields. ACM Trans. Graph. (SIGGRAPH) 33, 4, 134:1--134:11. Google ScholarDigital Library
- Persson, P.-O., and Strang, G. 2004. A simple mesh generator in MATLAB. SIAM Rev. 46, 329--345.Google ScholarDigital Library
- Rusinkiewicz, S. 2004. Estimating curvatures and their derivatives on triangle meshes. In 3DPVT, 486--493. Google ScholarDigital Library
- Shewchuk, J. R., 2002. What is a good linear finite element? Interpolation, conditioning, anisotropy, and quality measures.Google Scholar
- Shimada, K., Yamada, A., and Itoh, T. 2000. Anisotropic triangulation of parametric surfaces via close packing of ellipsoids. Int. J. Comput. Geom. Ap. 10, 4, 417--440.Google ScholarCross Ref
- Thompson, J. F., Soni, B. K., and Weatherill, N. P., Eds. 1998. Handbook of Grid Generation. Wiley-ISTE.Google Scholar
- Tournois, J., Srinivasan, R., and Alliez, P. 2009. Perturbing slivers in 3D Delaunay meshes. In Int. Meshing Roundtable, 157--173.Google Scholar
- Tournois, J., Wormser, C., Alliez, P., and Desbrun, M. 2009. Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Trans. Graph. (SIGGRAPH) 28, 3, 75:1--75:9. Google ScholarDigital Library
- Valette, S., Chassery, J.-M., and Prost, R. 2008. Generic remeshing of 3D triangular meshes with metric-dependent discrete Voronoi diagrams. IEEE. T. Vis. Comput. Gr. 14, 2, 369--381. Google ScholarDigital Library
- Yan, D.-M., Lévy, B., Liu, Y., Sun, F., and Wang, W. 2009. Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. Comput. Graph. FORUM 28, 5, 1445--1454. Google ScholarDigital Library
- Zhong, Z., Guo, X., Wang, W., Lévy, B., Sun, F., Liu, Y., and Mao, W. 2013. Particle-based anisotropic surface meshing. ACM Trans. Graph. (SIGGRAPH) 32, 4, 99:1--99:14. Google ScholarDigital Library
- Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. 2005. The Finite Element Method: Its Basis and Fundamentals, 6 ed. Butterworth-Heinemann.Google Scholar
Index Terms
- Anisotropic simplicial meshing using local convex functions
Recommendations
Particle-based anisotropic surface meshing
This paper introduces a particle-based approach for anisotropic surface meshing. Given an input polygonal mesh endowed with a Riemannian metric and a specified number of vertices, the method generates a metric-adapted mesh. The main idea consists of ...
Anisotropic diagrams: Labelle Shewchuk approach revisited
F. Labelle and J. Shewchuk have proposed a discrete definition of anisotropic Voronoi diagrams. These diagrams are parametrized by a metric field. Under mild hypotheses on the metric field, such Voronoi diagrams can be refined so that their dual is a ...
Anisotropic surface meshing
SODA '06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithmWe study the problem of triangulating a smooth closed implicit surface Σ endowed with a 2D metric tensor that varies over Σ. This is commonly known as the anisotropic surface meshing problem. We extend the 2D metric tensor naturally to 3D and employ the ...
Comments