Sliver-suppressing tetrahedral mesh optimization with gradient-based shape matching energy

In this paper, a novel shape matching energy is proposed to suppress slivers for tetrahedral mesh generation. Given a volumetric domain with a user-specified template (regular) simplex, the tetrahedral meshing problem is transformed into a shape matching formulation with a gradient-based energy, i.e., the gradient of linear shape function. It effectively inhibits small heights and suppresses all the badly-shaped tetrahedrons in tetrahedral meshes. The proposed approach iteratively optimizes vertex positions and mesh connectivity, and makes the simplices in the computed mesh as close as possible to the template simplex. We compare our results qualitatively and quantitatively with the state-of-the-art algorithm in tetrahedral meshing on extensive models using the standard measurement criteria.

[1]  LongChen,et al.  OPTIMAL DELAUNAY TRIANGULATIONS , 2004 .

[2]  Yasushi Ito,et al.  Reliable Isotropic Tetrahedral Mesh Generation Based on an Advancing Front Method , 2004, IMR.

[3]  Qiang Du,et al.  Anisotropic Centroidal Voronoi Tessellations and Their Applications , 2005, SIAM J. Sci. Comput..

[4]  Dong-Ming Yan,et al.  Efficient Computation of 3D Clipped Voronoi Diagram , 2010, GMP.

[5]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[6]  Ligang Liu,et al.  Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation , 2014, SIAM J. Sci. Comput..

[7]  L. Paul Chew,et al.  Guaranteed-quality Delaunay meshing in 3D (short version) , 1997, SCG '97.

[8]  M. Yvinec,et al.  Variational tetrahedral meshing , 2005, SIGGRAPH 2005.

[9]  Tamal K. Dey,et al.  Delaunay Mesh Generation , 2012, Chapman and Hall / CRC computer and information science series.

[10]  J. Shewchuk,et al.  Isosurface stuffing: fast tetrahedral meshes with good dihedral angles , 2007, SIGGRAPH 2007.

[11]  Herbert Edelsbrunner,et al.  Sliver exudation , 2000, J. ACM.

[12]  J. Shewchuk.  Two Discrete Optimization Algorithms for the Topological Improvement of Tetrahedral Meshes , 2002 .

[13]  Li Chen,et al.  Tetrahedral meshing via maximal Poisson-disk sampling , 2016, Comput. Aided Geom. Des..

[14]  Steven J. Owen,et al.  A Survey of Unstructured Mesh Generation Technology , 1998, IMR.

[15]  Bruno Lévy,et al.  Particle-based anisotropic surface meshing , 2013, ACM Trans. Graph..

[16]  Joachim Schöberl,et al.  NETGEN An advancing front 2D/3D-mesh generator based on abstract rules , 1997 .

[17]  Desheng Wang,et al.  Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations , 2003 .

[18]  Hang Si,et al.  TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator , 2015, ACM Trans. Math. Softw..

[19]  Mariette Yvinec,et al.  CGALmesh , 2015, ACM Trans. Math. Softw..

[20]  Ming C. Lin,et al.  Example-guided physically based modal sound synthesis , 2013, ACM Trans. Graph..

[21]  Patrick M. Knupp,et al.  Algebraic Mesh Quality Metrics , 2001, SIAM J. Sci. Comput..

[22]  Pierre Alliez,et al.  Perturbing Slivers in 3D Delaunay Meshes , 2009, IMR.

[23]  Jonathan Richard Shewchuk,et al.  Aggressive Tetrahedral Mesh Improvement , 2007, IMR.

[24]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[25]  Ronald Fedkiw,et al.  A Crystalline, Red Green Strategy for Meshing Highly Deformable Objects with Tetrahedra , 2003, IMR.

[26]  S. Teng,et al.  Biting: advancing front meets sphere packing , 2000 .

[27]  Herbert Edelsbrunner,et al.  Sliver exudation , 1999, SCG '99.

[28]  Dong-Ming Yan,et al.  Isotropic Remeshing with Fast and Exact Computation of Restricted Voronoi Diagram , 2009, Comput. Graph. Forum.

[29]  L. Freitag,et al.  Tetrahedral mesh improvement via optimization of the element condition number , 2002 .