“Bijective and coarse high-order tetrahedral meshes” by Jiang, Zhang, Hu, Schneider, Zorin, et al. …
Conference:
Type(s):
Title:
- Bijective and coarse high-order tetrahedral meshes
Presenter(s)/Author(s):
Abstract:
We introduce a robust and automatic algorithm to convert linear triangle meshes with feature annotated into coarse tetrahedral meshes with curved elements. Our construction guarantees that the high-order meshes are free of element inversion or self-intersection. A user-specified maximal geometrical error from the input mesh controls the faithfulness of the curved approximation. The boundary of the output mesh is in bijective correspondence to the input, enabling attribute transfer between them, such as boundary conditions for simulations, making our curved mesh an ideal replacement or complement for the original input geometry.The availability of a bijective shell around the input surface is employed to ensure robust curving, prevent self-intersections, and compute a bijective map between the linear input and curved output surface. As necessary building blocks of our algorithm, we extend the bijective shell formulation to support features and propose a robust approach for boundary-preserving linear tetrahedral meshing.We demonstrate the robustness and effectiveness of our algorithm by generating high-order meshes for a large collection of complex 3D models.
References:
1. Remi Abgrall, Cécile Dobrzynski, and Algiane Froehly. 2012. A method for computing curved 2D and 3D meshes via the linear elasticity analogy: preliminary results. Research Report RR-8061. INRIA. 15 pages. https://hal.inria.fr/hal-00728850Google Scholar
2. R. Abgrall, C. Dobrzynski, and A. 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.Google ScholarCross Ref
3. Noam Aigerman and Yaron Lipman. 2013. Injective and bounded distortion mappings in 3D. ACM Transactions on Graphics (TOG) 32, 4 (2013), 1–14.Google ScholarDigital Library
4. Marc Alexa. 2019. Harmonic Triangulations. ACM Trans. Graph. 38, 4, Article 54 (July 2019), 14 pages. Google ScholarDigital Library
5. Marc Alexa. 2020. Conforming weighted delaunay triangulations. ACM Transactions on Graphics (TOG) 39, 6 (2020), 1–16.Google ScholarDigital Library
6. Pierre Alliez, David Cohen-Steiner, Mariette Yvinec, and Mathieu Desbrun. 2005. Variational tetrahedral meshing. In ACM Transactions on Graphics (TOG), Vol. 24. ACM.Google Scholar
7. Douglas N Arnold, Richard S Falk, and Ragnar Winther. 2000. Multigrid in H (div) and H (curl). Numer. Math. 85, 2 (2000), 197–217.Google ScholarCross Ref
8. I. Babuska and B. Q. Guo. 1988. The h-p Version of the Finite Element Method for Domains with Curved Boundaries. SIAM J. Numer. Anal. 25, 4 (1988), 837–861. http://www.jstor.org/stable/2157607Google ScholarDigital Library
9. I. Babuška and B.Q. Guo. 1992. The h, p and h-p version of the finite element method; basis theory and applications. Advances in Engineering Software 15, 3 (1992), 159–174. Google ScholarDigital Library
10. Adam W Bargteil and Elaine Cohen. 2014. Animation of deformable bodies with quadratic Bézier finite elements. ACM Transactions on Graphics (TOG) 33, 3 (2014), 27.Google ScholarDigital Library
11. F. Bassi and S. Rebay. 1997. High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations. J. Comput. Phys. 138, 2 (1997), 251–285. Google ScholarDigital Library
12. Jean-Daniel Boissonnat and Steve Oudot. 2005. Provably good sampling and meshing of surfaces. Graphical Models 67, 5 (2005), 405–451.Google ScholarDigital Library
13. D. Bommes, B. Lévy, N. Pietroni, E. Puppo, C. Silv a, M. Tarini, and D. Zorin. 2012. State of the Art in Quad Meshing. In Eurographics STARS.Google Scholar
14. John Bowers, Rui Wang, Li-Yi Wei, and David Maletz. 2010. Parallel Poisson disk sampling with spectrum analysis on surfaces. ACM Transactions on Graphics (TOG) 29, 6 (2010), 1–10.Google ScholarDigital Library
15. David Cardoze, Alexandre Cunha, Gary L. Miller, Todd Phillips, and Noel Walkington. 2004. A Bézier-based Approach to Unstructured Moving Meshes. In Proceedings of the Twentieth Annual Symposium on Computational Geometry (Brooklyn, New York, USA) (SCG ’04). ACM, New York, NY, USA, 310–319. Google ScholarDigital Library
16. Siu-Wing Cheng, Tamal K Dey, Herbert Edelsbrunner, Michael A Facello, and Shang-Hua Teng. 2000. Silver exudation. Journal of the ACM (JACM) 47, 5 (2000), 883–904.Google ScholarDigital Library
17. Siu-Wing Cheng, Tamal K Dey, and Joshua A Levine. 2008. A practical Delaunay meshing algorithm for a large class of domains. In Proceedings of the 16th International Meshing Roundtable. Springer, 477–494.Google ScholarCross Ref
18. L Paul Chew. 1989. Constrained delaunay triangulations. Algorithmica 4, 1-4 (1989), 97–108.Google Scholar
19. L Paul Chew. 1993. Guaranteed-quality mesh generation for curved surfaces. In Proceedings of the ninth annual symposium on Computational geometry. ACM, 274–280.Google ScholarDigital Library
20. David Cohen-Steiner, Eric Colin De Verdiere, and Mariette Yvinec. 2002. Conforming Delaunay triangulations in 3D. In Proceedings of the eighteenth annual symposium on Computational geometry. ACM, 199–208.Google ScholarDigital Library
21. Coreform. 2020. Cubit.Google Scholar
22. Keenan Crane. 2013. Conformal geometry processing. California Institute of Technology.Google Scholar
23. Sailkat Dey, Robert M O’bara, and Mark S Shephard. 1999. Curvilinear Mesh Generation in 3D.. In IMR. IMR, 407–417.Google Scholar
24. Veselin Dobrev, Patrick Knupp, Tzanio Kolev, Ketan Mittal, and Vladimir Tomov. 2019. The target-matrix optimization paradigm for high-order meshes. SIAM Journal on Scientific Computing 41, 1 (2019), B50–B68.Google ScholarDigital Library
25. Cecile Dobrzynski and Ghina El Jannoun. 2017. High order mesh untangling for complex curved geometries. Research Report RR-9120. INRIA Bordeaux, équipe CARDAMOM. https://hal.inria.fr/hal-01632388Google Scholar
26. Qiang Du and Desheng Wang. 2003. Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations. International journal for numerical methods in engineering 56, 9 (2003), 1355–1373.Google ScholarCross Ref
27. Marion Dunyach, David Vanderhaeghe, Loïc Barthe, and Mario Botsch. 2013. Adaptive remeshing for real-time mesh deformation. In Eurographics 2013. The Eurographics Association.Google Scholar
28. Luke Engvall and John A. Evans. 2016. Isogeometric triangular Bernstein-Bézier discretizations: Automatic mesh generation and geometrically exact finite element analysis. Computer Methods in Applied Mechanics and Engineering 304 (2016), 378–407. Google ScholarCross Ref
29. Luke Engvall and John A. Evans. 2017. Isogeometric unstructured tetrahedral and mixed-element Bernstein-Bézier discretizations. Computer Methods in Applied Mechanics and Engineering 319 (2017), 83–123. Google ScholarCross Ref
30. Luke Engvall and John A. Evans. 2018. Mesh Quality Metrics for Isogeometric Bernstein-Bézier Discretizations. arXiv:1810.06975 (2018).Google Scholar
31. Darren Engwirda. 2016. Conforming restricted Delaunay mesh generation for piecewise smooth complexes. CoRR (2016). http://arxiv.org/abs/1606.01289Google Scholar
32. Patrick E Farrell, Matthew G Knepley, Lawrence Mitchell, and Florian Wechsung. 2019. PCPATCH: software for the topological construction of multigrid relaxation methods. arXiv preprint arXiv:1912.08516 (2019).Google Scholar
33. Leman Feng, Pierre Alliez, Laurent Busé, Hervé Delingette, and Mathieu Desbrun. 2018. Curved optimal delaunay triangulation. ACM Transactions on Graphics (TOG) (2018).Google Scholar
34. Meire Fortunato and Per-Olof Persson. 2016. High-order unstructured curved mesh generation using the Winslow equations. J. Comput. Phys. 307 (2016), 1–14. Google ScholarDigital Library
35. Xiao-Ming Fu, Yang Liu, and Baining Guo. 2015. Computing locally injective mappings by advanced MIPS. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1–12.Google ScholarDigital Library
36. Xifeng Gao, Hanxiao Shen, and Daniele Panozzo. 2019. Feature Preserving Octree-Based Hexahedral Meshing. Computer Graphics Forum 38, 5 (2019), 135–149.Google ScholarCross Ref
37. A. Gargallo-Peiró, X. Roca, J. Peraire, and J. Sarrate. 2015. Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes. Internat. J. Numer. Methods Engrg. 103, 5 (2015), 342–363. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.4888 Google ScholarCross Ref
38. Abel Gargallo Peiró, Francisco Javier Roca Navarro, Jaume Peraire Guitart, and Josep Sarrate Ramos. 2013. High-order mesh generation on CAD geometries. In Adaptive Modeling and Simulation 2013. Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), 301–312.Google Scholar
39. P.L. George and H. Borouchaki. 2012. Construction of tetrahedral meshes of degree two. Internat. J. Numer. Methods Engrg. 90, 9 (2012), 1156–1182.Google ScholarCross Ref
40. Christophe Geuzaine, Amaury Johnen, Jonathan Lambrechts, Jean-François Remacle, and Thomas Toulorge. 2015. The Generation of Valid Curvilinear Meshes. Springer International Publishing, Cham, 15–39. Google ScholarCross Ref
41. Christophe Geuzaine and Jean-François Remacle. 2009. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Internat. J. Numer. Methods Engrg. 79, 11 (2009), 1309–1331.Google Scholar
42. Arash Ghasemi, Lafayette K. Taylor, and James C. Newman, III. 2016. Massively Parallel Curved Spectral/Finite Element Mesh Generation of Industrial CAD Geometries in Two and Three Dimensions. Fluids Engineering Division Summer Meeting 50299 (2016). Google ScholarCross Ref
43. Gaël Guennebaud, Benoît Jacob, et al. 2010. Eigen v3.Google Scholar
44. Philippe Guigue and Olivier Devillers. 2003. Fast and robust triangle-triangle overlap test using orientation predicates. Journal of graphics tools 8, 1 (2003), 25–32.Google ScholarCross Ref
45. Hao-Xiang Guo, Xiaohan Liu, Dong-Ming Yan, and Liu Yang. 2020. Cut-enhanced PolyCube-Maps for Feature-aware All-Hex Meshing. ACM Transactions on Graphics (TOG) 39, 4 (2020), 106:1–106:14.Google ScholarDigital Library
46. S. Hahmann and G. . Bonneau. 2003. Polynomial surfaces interpolating arbitrary triangulations. IEEE Transactions on Visualization and Computer Graphics 9, 1 (2003), 99–109. Google ScholarDigital Library
47. Joos Heintz, Tomas Recio, and Marie-Françoise Roy. 1991. Algorithms in real algebraic geometry and applications to computational geometry. Discrete and Computational Geometry: Papers from the DIMACS Special Year (JE Goodman, R. Pollack and W.Google Scholar
48. Steiger, Eds.), AMS Press, Providence, RI (1991), 137–163.Google Scholar
49. Hugues Hoppe, Tony DeRose, Tom Duchamp, Mark Halstead, Hubert Jin, John McDonald, Jean Schweitzer, and Werner Stuetzle. 1994. Piecewise smooth surface reconstruction. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques. 295–302.Google ScholarDigital Library
50. Kai Hormann and Günther Greiner. 2000. MIPS: An efficient global parametrization method. Technical Report. ERLANGEN-NUERNBERG UNIV (GERMANY) COMPUTER GRAPHICS GROUP.Google Scholar
51. 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, Article 52 (July 2019), 15 pages. Google ScholarDigital Library
52. Yixin Hu, Teseo Schneider, Bolun Wang, Denis Zorin, and Daniele Panozzo. 2020. Fast Tetrahedral Meshing in the Wild. ACM Trans. Graph. 39, 4, Article 117 (July 2020), 18 pages. Google ScholarDigital Library
53. Yixin Hu, Qingnan Zhou, Xifeng Gao, Alec Jacobson, Denis Zorin, and Daniele Panozzo. 2018. Tetrahedral meshing in the wild. ACM Trans. Graph. 37, 4 (2018), 60–1.Google ScholarDigital Library
54. Alec Jacobson, Daniele Panozzo, C Schüller, O Diamanti, Q Zhou, N Pietroni, et al. 2016. libigl: A simple C++ geometry processing library, 2016.Google Scholar
55. A. Jameson, J. Alonso, and M. McMullen. 2002. Application of a non-linear frequency domain solver to the Euler and Navier-Stokes equations. In 40th AIAA Aerospace Sciences Meeting & Exhibit.Google Scholar
56. Clément Jamin, Pierre Alliez, Mariette Yvinec, and Jean-Daniel Boissonnat. 2015. CGALmesh: a generic framework for delaunay mesh generation. ACM Transactions on Mathematical Software (TOMS) 41, 4 (2015), 23.Google ScholarDigital Library
57. Zhongshi Jiang, Teseo Schneider, Denis Zorin, and Daniele Panozzo. 2020. Bijective projection in a shell. ACM Transactions on Graphics (TOG) 39, 6 (2020), 1–18.Google ScholarDigital Library
58. Xiangmin Jiao and Duo Wang. 2012. Reconstructing high-order surfaces for meshing. Engineering with Computers 28, 4 (2012), 361–373.Google ScholarDigital Library
59. Amaury Johnen, J-F Remacle, and Christophe Geuzaine. 2013. Geometrical validity of curvilinear finite elements. J. Comput. Phys. 233 (2013), 359–372.Google ScholarDigital Library
60. 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.Google ScholarCross Ref
61. Patrick M. Knupp. 2002. Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II, A framework for volume mesh optimization and the condition number of the Jacobian matrix. Internat. J. Numer. Methods Engrg. 48, 8 (2002), 1165–1185.Google ScholarCross Ref
62. Sebastian Koch, Albert Matveev, Zhongshi Jiang, Francis Williams, Alexey Artemov, Evgeny Burnaev, Marc Alexa, Denis Zorin, and Daniele Panozzo. 2019. ABC: A Big CAD Model Dataset For Geometric Deep Learning. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR).Google Scholar
63. Venkat Krishnamurthy and Marc Levoy. 1996. Fitting smooth surfaces to dense polygon meshes. In Proceedings of the 23rd annual conference on Computer graphics and interactive techniques. 313–324.Google ScholarDigital Library
64. Ming-Jun Lai and Larry L Schumaker. 2007. Spline functions on triangulations. Vol. 110. Cambridge University Press.Google Scholar
65. Bruno Lévy. 2015. Geogram.Google Scholar
66. Yufei Li, Yang Liu, Weiwei Xu, Wenping Wang, and Baining Guo. 2012. All-Hex Meshing Using Singularity-Restricted Field. ACM Trans. Graph. 31, 6, Article 177 (Nov. 2012), 11 pages. Google ScholarDigital Library
67. Hongwei Lin, Wei Chen, and Hujun Bao. 2007. Adaptive patch-based mesh fitting for reverse engineering. Computer-Aided Design 39, 12 (2007), 1134–1142. Google ScholarDigital Library
68. Yaron Lipman. 2014. Bijective mappings of meshes with boundary and the degree in mesh processing. SIAM Journal on Imaging Sciences 7, 2 (2014), 1263–1283.Google ScholarDigital Library
69. Qiukai Lu, Mark S. Shephard, Saurabh Tendulkar, and Mark W. Beall. 2013. Parallel Curved Mesh Adaptation for Large Scale High-Order Finite Element Simulations. In Proceedings of the 21st International Meshing Roundtable, Xiangmin Jiao and Jean-Christophe Weill (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 419–436.Google Scholar
70. Qiukai Lu, Mark S Shephard, Saurabh Tendulkar, and Mark W Beall. 2014. Parallel mesh adaptation for high-order finite element methods with curved element geometry. Engineering with Computers 30, 2 (2014), 271–286.Google ScholarDigital Library
71. Xiaojuan Luo, Mark S Shephard, Lie-Quan Lee, Cho Ng, and Lixin Ge. 2008. Tracking adaptive moving mesh refinements in 3d curved domains for large-scale higher order finite element simulations. In Proceedings of the 17th International Meshing Roundtable. Springer, 585–601.Google ScholarCross Ref
72. Xiaojuan Luo, Mark S Shephard, and Jean-Francois Remacle. 2001. The influence of geometric approximation on the accuracy of high order methods. Rensselaer SCOREC report 1 (2001).Google Scholar
73. Xiaojuan Luo, Mark S. Shephard, Jean-François Remacle, Robert M. O’Bara, Mark W. Beall, Barna A. Szabó, and Ricardo Actis. 2002a. p-Version Mesh Generation Issues. In IMR.Google Scholar
74. Xiaojuan Luo, Mark S Shephard, Jean-François Remacle, Robert M O’Bara, Mark W Beall, Barna A Szabó, and Ricardo Actis. 2002b. p-Version Mesh Generation Issues.. In IMR. 343–354.Google Scholar
75. Tom Lyche and Georg Muntingh. 2015. Simplex Spline Bases on the Powell-Sabin 12-Split: Part I. arXiv preprint arXiv:1505.01798 (2015).Google Scholar
76. Julian Marcon, Joaquim Peiró, David Moxey, Nico Bergemann, Henry Bucklow, and Mark R Gammon. 2019. A semi-structured approach to curvilinear mesh generation around streamlined bodies. In AIAA Scitech 2019 Forum. 1725.Google Scholar
77. Zoë Marschner, David Palmer, Paul Zhang, and Justin Solomon. 2020. Hexahedral Mesh Repair via Sum-of-Squares Relaxation. In Computer Graphics Forum, Vol. 39. Wiley Online Library, 133–147.Google Scholar
78. Johannes Mezger, Bernhard Thomaszewski, Simon Pabst, and Wolfgang Straßer. 2007. A Finite Element Method for Interactive Physically Based Shape Modelling with Quadratic Tetrahedra. (2007).Google Scholar
79. Johannes Mezger, Bernhard Thomaszewski, Simon Pabst, and Wolfgang Straśer. 2009. Interactive physically-based shape editing. Computer Aided Geometric Design 26, 6 (2009), 680–694. Solid and Physical Modeling 2008. Google ScholarDigital Library
80. Joseph SB Mitchell, David M Mount, and Christos H Papadimitriou. 1987. The discrete geodesic problem. SIAM J. Comput. 16, 4 (1987), 647–668.Google ScholarDigital Library
81. D. Moxey, D. Ekelschot, Ü. Keskin, S.J. Sherwin, and J. Peiró. 2016. High-order curvilinear meshing using a thermo-elastic analogy. Computer-Aided Design 72 (2016), 130–139. 23rd International Meshing Roundtable Special Issue: Advances in Mesh Generation. Google ScholarDigital Library
82. D. Moxey, M.D. Green, S.J. Sherwin, and J. Peiró. 2015. An isoparametric approach to high-order curvilinear boundary-layer meshing. Computer Methods in Applied Mechanics and Engineering 283 (2015), 636–650. Google ScholarCross Ref
83. Dave Moxey, Michael Turner, Julian Marcon, and Joaquim Peiro. 2018. Nekmesh: An open-source high-order mesh generator. (2018).Google Scholar
84. Michael Murphy, David M Mount, and Carl W Gable. 2001. A point-placement strategy for conforming Delaunay tetrahedralization. International Journal of Computational Geometry & Applications 11, 06 (2001), 669–682.Google ScholarCross Ref
85. J.Tinsley Oden. 1994. Optimal h-p finite element methods. Computer Methods in Applied Mechanics and Engineering 112, 1 (1994), 309–331. Google ScholarCross Ref
86. David Palmer, David Bommes, and Justin Solomon. 2020. Algebraic Representations forGoogle Scholar
87. Volumetric Frame Fields. ACM Trans. Graph. 39, 2, Article 16 (April 2020), 17 pages. Google ScholarDigital Library
88. Abel Gargallo Peiró, Eloi Ruiz Gironés, Francisco J Navarro, and Josep Sarrate Ramos. 2015. On curving high-order hexahedral meshes.Google Scholar
89. Abel Gargallo Peiró, Xevi Roca, Jaime Peraire, and Josep Sarrate. 2014. Defining Quality Measures for Validation and Generation of High-Order Tetrahedral Meshes. In Proceedings of the 22nd International Meshing Roundtable, Josep Sarrate and Matthew Staten (Eds.). Springer International Publishing, Cham, 109–126.Google Scholar
90. Joaquim Peiró, Spencer J. Sherwin, and Sergio Giordana. 2008. Automatic reconstruction of a patient-specific high-order surface representation and its application to mesh generation for CFD calculations. Medical & Biological Engineering & Computing 46, 11 (01 Nov 2008), 1069–1083. Google ScholarCross Ref
91. Per-Olof Persson and Jaime Peraire. 2009. Curved Mesh Generation and Mesh Refinement using Lagrangian Solid Mechanics. In 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition. Google ScholarCross Ref
92. Pointwise. 2018. High Order Mesh Generation at Pointwise. Accessed: 2018-11-14.Google Scholar
93. Roman Poya, Ruben Sevilla, and Antonio J. Gil. 2016. A unified approach for a posteriori high-order curved mesh generation using solid mechanics. Computational Mechanics 58, 3 (01 Sep 2016), 457–490. Google ScholarDigital Library
94. Michael Rabinovich, Roi Poranne, Daniele Panozzo, and Olga Sorkine-Hornung. 2017. Scalable Locally Injective Mappings. ACM Trans. Graph. 36, 2, Article 16 (April 2017), 16 pages. Google ScholarDigital Library
95. Jean-François Remacle, Thomas Toulorge, and Jonathan Lambrechts. 2013. Robust untangling of curvilinear meshes. In Proceedings of the 21st International meshing roundtable. Springer, 71–83.Google ScholarCross Ref
96. Xevi Roca, Abel Gargallo-Peiró, and Josep Sarrate. 2012. Defining Quality Measures for High-Order Planar Triangles and Curved Mesh Generation. In Proceedings of the 20th International Meshing Roundtable, William Roshan Quadros (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg, 365–383.Google Scholar
97. Eloi Ruiz-Gironés, Abel Gargallo-Peiró, Josep Sarrate, and Xevi Roca. 2017. An augmented Lagrangian formulation to impose boundary conditions for distortion based mesh moving and curving. Procedia Engineering 203 (2017), 362 — 374. 26th International Meshing Roundtable, IMR26, 18-21 September 2017, Barcelona, Spain. Google ScholarCross Ref
98. Eloi Ruiz-Gironés, Xevi Roca, and Jose Sarrate. 2016a. High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation. Computer-Aided Design 72 (2016), 52–64. 23rd International Meshing Roundtable Special Issue: Advances in Mesh Generation. Google ScholarDigital Library
99. Eloi Ruiz-Gironés, Josep Sarrate, and Xevi Roca. 2016b. Generation of Curved High-order Meshes with Optimal Quality and Geometric Accuracy. Procedia Engineering 163 (2016), 315 — 327. 25th International Meshing Roundtable. Google ScholarCross Ref
100. Jim Ruppert. 1995. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of algorithms 18, 3 (1995), 548–585.Google ScholarDigital Library
101. Kambiz Salari and Patrick Knupp. 2000. Code Verification by the Method of Manufactured Solutions. Technical Report. Google Scholar
102. N Schlömer. 2020. nschloe/meshio: Input/output for many mesh formats. Zenodo. doi 10 (2020).Google Scholar
103. Mark S. Shephard, Joseph E. Flaherty, Kenneth E. Jansen, Xiangrong Li, Xiaojuan Luo, Nicolas Chevaugeon, Jean-François Remacle, Mark W. Beall, and Robert M. O’Bara. 2005. Adaptive mesh generation for curved domains. Applied Numerical Mathematics 52, 2 (2005), 251–271. ADAPT ’03: Conference on Adaptive Methods for Partial Differential Equations and Large-Scale Computation. Google ScholarDigital Library
104. SJ Sherwin and J Peiró. 2002. Mesh generation in curvilinear domains using high-order elements. Internat. J. Numer. Methods Engrg. 53, 1 (2002), 207–223.Google ScholarCross Ref
105. Jonathan Richard Shewchuk. 1997. Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete & Computational Geometry 18, 3 (1997), 305–363.Google ScholarCross Ref
106. Jonathan Richard Shewchuk. 1998. Tetrahedral mesh generation by Delaunay refinement. In Proceedings of the fourteenth annual symposium on Computational geometry. ACM, 86–95.Google ScholarDigital Library
107. Jonathan Richard Shewchuk. 2002. Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery.. In IMR. 193–204.Google Scholar
108. Hang Si. 2015. TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator. ACM Trans. Math. Softw. 41, 2, Article 11 (Feb. 2015), 36 pages.Google ScholarDigital Library
109. Hang Si and Klaus Gärtner. 2005. Meshing Piecewise Linear Complexes by Constrained Delaunay Tetrahedralizations.. In IMR. Springer, 147–163.Google Scholar
110. Hang Si and Jonathan Richard Shewchuk. 2014. Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates. Eng. Comput. (Lond.) 30, 2 (2014), 253–269.Google ScholarDigital Library
111. C. Sorger, Felix Frischmann, Stefan Kollmannsberger, and Ernst Rank. 2014. TUM.GeoFrame: automated high-order hexahedral mesh generation for shell-like structures. Eng. Comput. 30, 1 (2014), 41–56. Google ScholarDigital Library
112. Mike Stees and Suzanne M. Shontz. 2017. A high-order log barrier-based mesh generation and warping method. Procedia Engineering 203 (2017), 180–192. 26th International Meshing Roundtable, IMR26, 18-21 September 2017, Barcelona, Spain. Google ScholarCross Ref
113. Ryan S. Glasby Steve L. Karman, J T. Erwin and Douglas Stefanski. 2016. High-Order Mesh Curving Using WCN Mesh Optimization. In 46th AIAA Fluid Dynamics Conference, AIAA AVIATION Forum. Google ScholarCross Ref
114. Stefan Suwelack, Dimitar Lukarski, Vincent Heuveline, Rüdiger Dillmann, and Stefanie Speidel. 2013. Accurate Surface Embedding for Higher Order Finite Elements. In Proceedings of the 12th ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Anaheim, California) (SCA’13). ACM, New York, NY, USA, 187–192. Google ScholarDigital Library
115. Matthias Teschner, Bruno Heidelberger, Matthias Müller, Danat Pomerantes, and Markus H Gross. 2003. Optimized spatial hashing for collision detection of deformable objects.. In Vmv, Vol. 3. 47–54.Google Scholar
116. The CGAL Project. 2020. CGAL User and Reference Manual (5.0.3 ed.). CGAL Editorial Board. https://doc.cgal.org/5.0.3/Manual/packages.htmlGoogle Scholar
117. Wei-hua Tong and Tae-wan Kim. 2009. High-order approximation of implicit surfaces by G1 triangular spline surfaces. Computer-Aided Design 41, 6 (2009), 441–455.Google ScholarDigital Library
118. Thomas Toulorge, Christophe Geuzaine, Jean-François Remacle, and Jonathan Lambrechts. 2013. Robust untangling of curvilinear meshes. J. Comput. Phys. 254 (2013), 8–26. Google ScholarDigital Library
119. Thomas Toulorge, Jonathan Lambrechts, and Jean-François Remacle. 2016. Optimizing the geometrical accuracy of curvilinear meshes. J. Comput. Phys. 310 (2016), 361–380. Google ScholarDigital Library
120. Jane Tournois, Camille Wormser, Pierre Alliez, and Mathieu Desbrun. 2009. Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Transactions on Graphics 28, 3 (2009), Art-No.Google ScholarDigital Library
121. Davi Colli Tozoni, Jérémie Dumas, Zhongshi Jiang, Julian Panetta, Daniele Panozzo, and Denis Zorin. 2020. A low-parametric rhombic microstructure family for irregular lattices. ACM Transactions on Graphics (TOG) 39, 4 (2020), 101–1.Google ScholarDigital Library
122. Michael Turner. 2017. High-order mesh generation for CFD solvers. Ph.D. Dissertation. Imperial College London.Google Scholar
123. Michael Turner, Joaquim Peiró, and David Moxey. 2016. A Variational Framework for High-order Mesh Generation. Procedia Engineering 163 (2016), 340–352. 25th International Meshing Roundtable. Google ScholarCross Ref
124. S Pratap Vanka. 1986. Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys. 65, 1 (1986), 138–158.Google ScholarDigital Library
125. Ruimin Wang, Ligang Liu, Zhouwang Yang, Kang Wang, Wen Shan, Jiansong Deng, and Falai Chen. 2016. Construction of manifolds via compatible sparse representations. ACM Transactions on Graphics (TOG) 35, 2 (2016), 1–10.Google ScholarDigital Library
126. Songtao Xia and Xiaoping Qian. 2017. Isogeometric analysis with Bézier tetrahedra. Computer Methods in Applied Mechanics and Engineering 316 (2017), 782–816. Special Issue on Isogeometric Analysis: Progress and Challenges. Google ScholarCross Ref
127. Zhong Q. Xie, Ruben Sevilla, Oubay Hassan, and Kenneth Morgan. 2013. The generation of arbitrary order curved meshes for 3D finite element analysis. Computational Mechanics 51, 3 (01 Mar 2013), 361–374. Google ScholarDigital Library
128. Yuxuan Yu, Xiaodong Wei, Angran Li, Jialei Ginny Liu, Jeffrey He, and Yongjie Jessica Zhang. 2020. HexGen and Hex2Spline: Polycube-based Hexahedral Mesh Generation and Spline Modeling for Isogeometric Analysis Applications in LS-DYNA. (2020). arXiv:2011.14213 [cs.CG]Google Scholar
129. Alex Yvart, Stefanie Hahmann, and Georges-Pierre Bonneau. 2005a. Hierarchical Triangular Splines. ACM Trans. Graph. 24, 4 (Oct. 2005), 1374–1391. Google ScholarDigital Library
130. Alex Yvart, Stefanie Hahmann, and G-P Bonneau. 2005b. Smooth adaptive fitting of 3D models using hierarchical triangular splines. In International Conference on Shape Modeling and Applications 2005 (SMI’05). IEEE, 13–22.Google ScholarDigital Library
131. Daniel W Zaide, Qiukai Lu, and Mark S Shephard. 2015. A comparison of C0 and G1 continuous curved tetrahedral meshes for high-order finite element simulations. Proc. 24th International Meshing Roundtable. Elsevier, New York (2015).Google Scholar
132. Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon. 2020. Octahedral Frames for Feature-Aligned Cross Fields. ACM Trans. Graph. 39, 3, Article 25 (April 2020), 13 pages. Google ScholarDigital Library
133. S. Zhang, Z. Li, H. Zhang, and J. Yong. 2011. Multi-resolution Mesh Fitting by B-spline Surfaces for Reverse Engineering. In 2011 12th International Conference on Computer-Aided Design and Computer Graphics. 251–257. Google ScholarDigital Library
134. Qingnan Zhou and Alec Jacobson. 2016. Thingi10k: A dataset of 10,000 3d-printing models. arXiv preprint arXiv:1605.04797 (2016).Google Scholar
135. V.S. Ziel, H. Bériot, O. Atak, and G. Gabard. 2017. Comparison of 2D boundary curving methods with modal shape functions and a piecewise linear target mesh. Procedia Engineering 203 (2017), 91–101. 26th International Meshing Roundtable, IMR26, 18-21 September 2017, Barcelona, Spain. Google ScholarCross Ref
136. Denis Zorin. 2000. Subdivision for modeling and animation. SIGGRAPH2000 course notes (2000).Google Scholar