Guaranteed-quality higher-order triangular meshing of 2D domains

Abstract

We present a guaranteed quality mesh generation algorithm for the curvilinear triangulation of planar domains with piecewise polynomial boundary. The resulting mesh consists of higher-order triangular elements which are not only regular (i.e., with injective geometric map) but respect strict bounds on quality measures like scaled Jacobian and MIPS distortion. This also implies that the curved triangles' inner angles are bounded from above and below. These are key quality criteria, for instance, in the field of finite element analysis. The domain boundary is reproduced exactly, without geometric approximation error. The central idea is to transform the curvilinear meshing problem into a linear meshing problem via a carefully constructed transformation of bounded distortion, enabling us to leverage key results on guaranteed-quality straight-edge triangulation. The transformation is based on a simple yet general construction and observations about convergence properties of curves under subdivision. Our algorithm can handle arbitrary polynomial order, arbitrarily sharp corners, feature and interface curves, and can be executed using rational arithmetic for strict reliability.

References

1. Remi Abgrall, Cécile Dobrzynski, and Algiane Froehly. 2014. A method for computing curved meshes via the linear elasticity analogy, application to fluid dynamics problems. International Journal for Numerical Methods in Fluids 76, 4 (2014), 246--266.
2. I. Babuska and A. K. Aziz. 1976. On the Angle Condition in the Finite Element Method. SIAM J. Numer. Anal. 13, 2 (1976), 214--226.
3. Marshall Bern, David Eppstein, and John Gilbert. 1994. Provably good mesh generation. J. Comput. System Sci. 48, 3 (1994), 384--409.
4. Jean-Daniel Boissonnat, Olivier Devillers, Monique Teillaud, and Mariette Yvinec. 2000. Triangulations in CGAL. In Proc. 16th Symp. on Computational Geometry. 11--18.
5. Charles Boivin and Carl Ollivier-Gooch. 2002. Guaranteed-quality triangular mesh generation for domains with curved boundaries. Int. J. Numer. Methods Eng. 55, 10 (2002), 1185--1213.
6. Jan Brandts, Antti Hannukainen, Sergey Korotov, and Michal Křîžek. 2011. On angle conditions in the finite element nethod. SeMA Journal 56 (09 2011), 81--95.
7. Siu-Wing Cheng, Tamal K Dey, and Jonathan Shewchuk. 2012. Delaunay mesh generation. CRC Press.
8. L. Paul Chew. 1989. Guaranteed-Quality Triangular Meshes. Tech. Rep. Department of Computer Science, Cornell University.
9. L. Paul Chew. 1993. Guaranteed-Quality Mesh Generation for Curved Surfaces. In Proc. 9th Symp. on Computational Geometry. 274--280.
10. P.G. Ciarlet and P.-A. Raviart. 1972a. The Combined Effect of Curved Boundaries and Numerical Integration in Isoparametric Finite Element Methods. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz (Ed.). Academic Press, 409 -- 474.
11. P.G. Ciarlet and P.-A. Raviart. 1972b. Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Eng. 1, 2 (1972).
12. Y. Colin de Verdière and A. Marin. 1990. Triangulations presque équilatérales des surfaces. J. Differential Geom. 32, 1 (1990), 199--207.
13. Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Cheong Schwarzkopf. 2000. Polygon Triangulation. Springer Berlin Heidelberg, Berlin, Heidelberg, 45--61.
14. Saikat Dey, Robert M. O'Bara, and Mark S. Shephard. 1999. Curvilinear Mesh Generation In 3D. In Proc. International Meshing Roundtable. John Wiley & Sons, 407--417.
15. Luke Engvall and John A. Evans. 2020. Mesh quality metrics for isogeometric Bernstein-Bézier discretizations. Comput. Methods Appl. Mech. Eng. 371 (2020).
16. Hale Erten and Alper Üngör. 2009. Quality Triangulations with Locally Optimal Steiner Points. SIAM Journal on Scientific Computing 31, 3 (2009), 2103--2130.
17. Gerald Farin. 1986. Triangular Bernstein-Bézier patches. Computer Aided Geometric Design 3, 2 (1986), 83--127.
18. Leman Feng, Pierre Alliez, Laurent Busé, Hervé Delingette, and Mathieu Desbrun. 2018. Curved Optimal Delaunay Triangulation. ACM Trans. Graph. 37, 4 (2018).
19. A. Fournier and D. Y. Montuno. 1984. Triangulating Simple Polygons and Equivalent Problems. ACM Trans. Graph. 3, 2 (April 1984), 153--174.
20. Isaac Fried. 1972. Condition of finite element matrices generated from nonuniform meshes. Aiaa Journal 10, 2 (1972), 219--221.
21. P.L. George and H. Borouchaki. 2012. Construction of tetrahedral meshes of degree two. Int. J. Numer. Methods Eng. 90, 9 (2012), 1156--1182.
22. William J. Gordon and Charles A. Hall. 1973. Transfinite element methods: Blending-function interpolation over arbitrary curved element domains. Numer. Math. 21, 2 (1973), 109--129.
23. Serge Gosselin and Carl Ollivier-Gooch. 2007. Revisiting Delaunay Refinement Triangular Mesh Generation on Curve-bounded Domains. Technical Report. U British Columbia.
24. Jens Gravesen, Anton Evgrafov, Dang-Manh Nguyen, and Peter Nørtoft. 2014. Planar Parametrization in Isogeometric Analysis. In Mathematical Methods for Curves and Surfaces. Springer Berlin Heidelberg, 189--212.
25. Robert Haber, Mark S. Shephard, John F. Abel, Richard H. Gallagher, and Donald P. Greenberg. 1981. A general two-dimensional, graphical finite element preprocessor utilizing discrete transfinite mappings. Int. J. Numer. Methods Eng. 17, 7 (1981).
26. Victoria Hernandez-Mederos, Jorge Estrada-Sarlabous, and Dionne León Madrigal. 2006. On local injectivity of 2D triangular cubic Bezier functions. Investigación Operacional 27, 3 (2006), 261--275.
27. Kai Hormann and Günther Greiner. 2000. MIPS: An efficient global parametrization method. Curve and Surface Design '99 (2000), 153--162.
28. Yixin Hu, Teseo Schneider, Xifeng Gao, Qingnan Zhou, Alec Jacobson, Denis Zorin, and Daniele Panozzo. 2019. TriWild: Robust Triangulation with Curve Constraints. ACM Trans. Graph. 38, 4 (2019).
29. A. Johnen, C. Geuzaine, T. Toulorge, and J.-F. Remacle. 2016. Efficient Computation of the Minimum of Shape Quality Measures on Curvilinear Finite Elements. Procedia Engineering 163 (2016), 328 -- 339.
30. Patrick M. Knupp. 2000. Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I---a framework for surface mesh optimization. Internat. J. Numer. Methods Engrg. 48, 3 (2000), 401--420.
31. Richard Leroy. 2008. Certificates of positivity and polynomial minimization in the multivariate Bernstein basis. Ph.D. Dissertation. University of Rennes 1.
32. J. Li, T. J. Peters, and J. A. Roulier. 2012. Angular Convergence during Bézier Curve Approximation. arXiv:1210.2686 [math.GT]
33. Manish Mandad and Marcel Campen. 2020a. Bézier Guarding: Precise Higher-Order Meshing of Curved 2D Domains. ACM Trans. Graph. 39, 4, Article 103 (2020).
34. Manish Mandad and Marcel Campen. 2020b. Efficient piecewise higher-order parametrization of discrete surfaces with local and global injectivity. Computer-Aided Design 127 (2020).
35. Gary L. Miller, Steven E. Pav, and Noel J. Walkington. 2003. When and Why Ruppert's Algorithm Works. In Proc. 12th International Meshing Roundtable. 91--102.
36. Géraldine Morin and Ron Goldman. 2001. On the smooth convergence of subdivision and degree elevation for Bézier curves. Comput. Aided Geom. Des. 18, 7 (2001).
37. D. Moxey, D. Ekelschot, Ü. Keskin, S.J. Sherwin, and J. Peiró. 2016. High-order Curvilinear Meshing Using a Thermo-elastic Analogy. Comput. Aided Des. 72, C (2016).
38. Peter Oswald. 2015. Divergence of FEM: Babuška-Aziz triangulations revisited. Applications of Mathematics 60, 5 (2015), 473--484. http://eudml.org/doc/271633
39. Steven J Owen. 1998. A survey of unstructured mesh generation technology. IMR 239 (1998), 267.
40. Steven Elliot Pav. 2003. Delaunay Refinement Algorithms. Ph.D. Dissertation. Carnegie Mellon University.
41. Steven E. Pav and Noel J. Walkington. 2005. Delaunay Refinement by Corner Lopping. In Proceedings of the 14th International Meshing Roundtable, Byron W. Hanks (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 165--181.
42. Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces and their conjugates. Experimental mathematics 2, 1 (1993), 15--36.
43. Hartmut Prautzsch, Wolfgang Boehm, and Marco Paluszny. 2013. Bézier and B-spline techniques. Springer Science & Business Media.
44. Alexander Rand. 2011a. Improved Examples of Non-Termination for Ruppert's Algorithm. arXiv:1103.3903 [cs.CG]
45. Alexander Rand. 2011b. Where and How Chew's Second Delaunay Refinement Algorithm Works. In Proc. 23rd Canadian Conference on Computational Geometry.
46. Ramsharan Rangarajan and Adrián J. Lew. 2014. Universal meshes: A method for triangulating planar curved domains immersed in nonconforming meshes. Int. J. Numer. Methods Eng. 98, 4 (2014), 236--264.
47. J. Ruppert. 1995. A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation. Journal of Algorithms 18, 3 (1995), 548 -- 585.
48. Teseo Schneider, Yixin Hu, Jérémie Dumas, Xifeng Gao, Daniele Panozzo, and Denis Zorin. 2018. Decoupling Simulation Accuracy from Mesh Quality. ACM Trans. Graph. 37, 6 (2018).
49. Jonathan R. Shewchuk. 2002a. Delaunay refinement algorithms for triangular mesh generation. Computational Geometry 22, 1 (2002), 21 -- 74. 16th ACM Symp. Comput. Geom.
50. Jonathan Richard Shewchuk. 2002b. What is a good linear element? Interpolation, Conditioning, and Quality Measures. In Proc. 11th Int. Meshing Roundtable. 115--126.
51. Thomas Toulorge, Christophe Geuzaine, Jean-François Remacle, and Jonathan Lambrechts. 2013. Robust untangling of curvilinear meshes. J. Comput. Phys. 254 (2013).
52. Godfried T Toussaint. 1984. A new linear algorithm for triangulating monotone polygons. Pattern Recognition Letters 2, 3 (1984), 155--158.
53. Michael Turner, Joaquim Peiró, and David Moxey. 2018. Curvilinear mesh generation using a variational framework. Computer-Aided Design 103 (2018), 73--91.
54. S.A. Vavasis. 1996. Stable finite elements for problems with wild coefficients. SIAM J. Numer. Anal. 33 (1996), 890--916.
55. Z.J. Wang, Krzysztof Fidkowski, Rémi Abgrall, Francesco Bassi, Doru Caraeni, Andrew Cary, Herman Deconinck, Ralf Hartmann, Koen Hillewaert, H.T. Huynh, Norbert Kroll, Georg May, Per-Olof Persson, Bram van Leer, and Miguel Visbal. 2013. High-order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids 72, 8 (2013), 811--845.

Index Terms

• Applied computing

  • Arts and humanities

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Mesh models

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Mesh generation