Meshes and Partitions
Precise High-order Meshing of 2D Domains with Rational Bézier Curves
Jinlin Yang, Shibo Liu, Shuangming Chai,
Ligang Liu, Xiao-Ming Fu,
Corresponding Author
Shuangming Chai
University of Science and Technology of China
† Corresponding author.Search for more papers by this authorJinlin Yang, Shibo Liu, Shuangming Chai,
Ligang Liu, Xiao-Ming Fu,
Corresponding Author
Shuangming Chai
University of Science and Technology of China
† Corresponding author.Search for more papers by this authorAbstract
We propose a novel method to generate a high-order triangular mesh for an input 2D domain with two key characteristics: (1) the mesh precisely conforms to a set of input piecewise rational domain curves, and (2) the geometric map on each curved triangle is injective. Central to the algorithm is a new sufficient condition for placing control points of a rational Bézier triangle to guarantee that the conformance and injectivity constraints are theoretically satisfied. Taking advantage of this condition, we provide an explicit construct that robustly creates higher-order 2D meshes satisfying the two characteristics. We demonstrate the robustness and effectiveness of our algorithm over a data set containing 2200 examples.
Supporting Information
| Filename | Description |
|---|---|
| cgf14604-sup-0001-S1.zip346.8 KB | Supplement Material |
Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.
References
- Abgrall R., Dobrzynski C., Froehly A.: A method for computing curved meshes via the linear elasticity analogy. Int. J. Numer. Methods Fluids 76, 4 (2014), 246–266. 2
- Botsch M., Kobbelt L.: A remeshing approach to multiresolution modeling. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (New York, NY, USA, 2004), Association for Computing Machinery, pp. 185–192. 7
- Cardoze D., Cunha A., Miller G. L., Phillips T., Walkington N.: A Bézier-based approach to unstructured moving meshes. In Proceedings of the Twentieth Annual Symposium on Computational Geometry (New York, NY, USA, 2004), Association for Computing Machinery, pp. 310–319. 2
- Dey S., O'Bara R. M., Shephard M. S.: Curvilinear mesh generation in 3D. In Proceedings of the 8th International Meshing Roundtable (1999), pp. 407–417. 2
- Dey S., O'Bara R. M., Shephard M. S.: Towards curvilinear meshing in 3D: the case of quadratic simplices. Comput. Aided Des. 33, 3 (2001), 199–209. 2
- Engvall L., Evans J. A.: Isogeometric triangular Bernstein-Bézier discretizations: Automatic mesh generation and geometrically exact finite element analysis. Comput. Methods Appl. Mech. Engrg. 304 (2016), 378–407. 2
- Engvall L., Evans J. A.: Mesh quality metrics for isogeometric bernstein bézier discretizations. Comput. Methods Appl. Mech. Engrg. 371 (2020), 113305. 1, 9
- Farin G.: Triangular Bernstein-Bézier patches. Comput. Aided Geom. Des. 3, 2 (1986), 83–127. 2, 10
10.1016/0167-8396(86)90016-6 Google Scholar
- Farin G.: Curves and surfaces for CAGD: A practical guide, 5th ed. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2002. 5
- Farin G., Hansford D.: Discrete coons patches. Computer Aided Geometric Design 16, 7 (1999), 691 – 700. 7
- Fortunato M., Persson P.-O.: High-order unstructured curved mesh generation using the Winslow equations. J. Comput. Phys. 307 (2016), 1–14. 2
- George P., Borouchaki H.: Construction of tetrahedral meshes of degree two. Int. J. Numer. Methods Eng. 90 (06 2012), 1156–1182. 2
- Ghasemi A., Taylor L. K., Newman III J. C.: Massively parallel curved spectral/finite element mesh generation of industrial CAD geometries in two and three dimensions. In Proceedings of the ASME 2016 Fluids Engineering Division Summer Meeting (07 2016), p. 50299. 2
- Hormann K., Greiner G.: MIPS: An efficient global parametrization method. In Curve and Surface Design: Saint-Malo 1999. Vanderbilt University Press, 2000, pp. 153–162. 7
- Hernandez-Mederos V., Estrada-Sarlabous J., Madrigal D. L.: On local injectivity of 2D triangular cubic Bézier functions. Investigación Operacional 27, 3 (2006), 261–276. 7
- Hu Y., Schneider T., Gao X., Zhou Q., Jacobson A., Zorin D., Panozzo D.: TriWild: Robust triangulation with curve constraints. ACM Trans. Graph. 38, 4 (2019), 52: 1–52:15. 1, 2, 7
- Jaxon N., Qian X.: Isogeometric analysis on triangulations. Comput. Aided Des. 46 (2014), 45–57. 2
- Joe B., Wang W.: Reparameterization of rational triangular Bézier surfaces. Comput. Aided Geom. Des. 11, 4 (1994), 345–361. 2, 3
- Khanteimouri P., Mandad M., Campen M.: Rational Bézier Guarding. Computer Graphics Forum 41, 5 (2022). 2
- Liu Z.-Y., Su J.-P., Liu H., Ye C., Liu L., Fu X.-M.: Error-bounded edge-based remeshing of high-order tetrahedral meshes. Computer-Aided Design 139 (2021), 103080. 2, 7
- Luo X.-J., Shephard M., O'Bara R., Nastasia R., Beall M.: Automatic p-version mesh generation for curved domains. Engineering with Computers 20 (09 2004), 273–285. 2
- Luo X., Shephard M., Remacle J.-F.: The influence of geometric approximation on the accuracy of higher order methods. Tech. rep., Scientific Computation Research Center, Rensselaer Polytechnic Institute, 2001. 2
- Mandad M., Campen M.: Bézier guarding: precise higher-order meshing of curved 2D domains. ACM Trans. Graph. 39, 4 (2020), 103: 1–103:15. 1, 2, 4, 5, 6, 7, 8, 9
- Mandad M., Campen M.: Guaranteed-quality higher-order triangular meshing of 2D domains. ACM Trans. Graph. 40, 4 (2021), 154: 1–154:14. 1, 2, 6, 7, 9
- Moxey D., Ekelschot D., Keskin U., Sherwin S. J., Peiró J.: High-order curvilinear meshing using a thermo-elastic analogy. Comput. Aided Des. 72 (2016), 130–139. 2
- Panetta J.: Analytic eigensystems for isotropic membrane energies, 2020. arXiv:2008.10698. 7
- Persson P.-O., Peraire J.: Curved mesh generation and mesh refinement using Lagrangian solid mechanics. In Proceeding of the 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition (2009). 2
- Poya R., Sevilla R., Gil A. J.: A unified approach for a posteriori high-order curved mesh generation using solid mechanics. Comput. Mech. 58 (2016), 457–490. 2
- Roca X., Gargallo-Peiró A., Sarrate J.: Defining quality measures for high-order planar triangles and curved mesh generation. In Proceedings of the 20th International Meshing Roundtable (2011), pp. 365–383. 2
- Ruiz-Gironés E., Sarrate J., Roca X.: Generation of curved high-order meshes with optimal quality and geometric accuracy. Procedia Engineering 163 (2016), 315–327. 2
10.1016/j.proeng.2016.11.108 Google Scholar
- Rangarajan R., Lew A. J.: Universal meshes: A method for triangulating planar curved domains immersed in nonconforming meshes. Int. J. Numer. Methods Eng. 98, 4 (2014), 236–264. 2
- Szabó B., Babuška I.: Finite Element Analysis: Method, Verification and Validation, 2nd Edition. John Wiley & Sons Inc., 2021. 1
10.1002/9781119426479 Google Scholar
- Shephard M. S., Flaherty J. E., Jansen K. E., Li X., Luo X., Chevaugeon N., Remacle J.-F., Beall M. W., O'Bara R. M.: Adaptive mesh generation for curved domains. Appl. Numer. Math. 52, 2 (2005), 251–271. 2
- Sevilla R., Fernández-Méndez S., Huerta A.: NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Methods Eng. 76, 1 (2008), 56–83. 2
- Sherwin S. J., Peiró J.: Mesh generation in curvilinear domains using high-order elements. Int. J. Numer. Methods Eng. 53, 1 (2002), 207–223. 2
- Shtengel A., Poranne R., Sorkine-Hornung O., Kovalsky S. Z., Lipman Y.: Geometric optimization via composite majorization. ACM Transactions on Graphics 36, 4 (2017), 1–11. 7
- Sevilla R., Rees L., Hassan O.: The generation of triangular meshes for NURBS-enhanced FEM. Int. J. Numer. Methods Eng. 108, 8 (2016), 941–968. 2
- Saito T., Wang G.-J., Sederberg T. W.: Hodographs and normals of rational curves and surfaces. Comput. Aided Geom. Des. 12, 4 (1995), 417–430. 3
10.1016/0167-8396(94)00023-L Google Scholar
- Toulorge T., Geuzaine C., Remacle J.-F., Lambrechts J.: Robust untangling of curvilinear meshes. J. Comput. Phys. 254 (2013), 8–26. 2
- Toulorge T., Lambrechts J., Remacle J.-F.: Optimizing the geometrical accuracy of curvilinear meshes. Journal of Computational Physics 310 (2016), 361–380. 2
- Turner M., Peiró J., Moxey D.: Curvilinear mesh generation using a variational framework. Comput. Aided Des. 103 (2018), 73–91. 2
- Wang Z., Fidkowski K., Abgrall R., Bassi F., Caraeni D., Cary A., Deconinck H., Hartmann R., Hillewaert K., Huynh H., Kroll N., May G., Persson P.-O., van Leer B., Visbal M.: High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72, 8 (2013), 811–845. 1
- Witherden F., Vincent P.: On the identification of symmetric quadrature rules for finite element methods. Computers & Mathematics with Applications 69, 10 (2015), 1232–1241. 7
- Xu J., Chernikov A. N.: Automatic curvilinear quality mesh generation driven by smooth boundary and guaranteed fidelity. Procedia Engineering 82 (2014), 200–212. 2
10.1016/j.proeng.2014.10.384 Google Scholar
- Xie Z. Q., Sevilla R., Hassan O., Morgan K.: The generation of arbitrary order curved meshes for 3D finite element analysis. Comput. Mech. 51 (2014), 361–374. 2
- Zlámal M.: The finite element method in domains with curved boundaries. Int. J. Numer. Methods Eng. 5, 3 (1973), 367–373. 1
10.1002/nme.1620050307 Google Scholar
