Elsevier

Computer Aided Geometric Design

Volumes 52–53, March–April 2017, Pages 247-261
Computer Aided Geometric Design

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

https://doi.org/10.1016/j.cagd.2017.02.004Get rights and content

Abstract

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.

Introduction

Triangular and tetrahedral meshes are most basic 2D and 3D elements in computer graphics and geometric modeling fields. The mesh generation is to discretize spatial domain into a set of connected but non-overlapped simplex elements. Mesh quality highly depends on the size and shape of each element. In this paper, we focus on the tetrahedral mesh generation. There are several measurements for tetrahedral mesh quality, and dihedral angle is one of the most important criteria (Alliez et al., 2005, Guo et al., 2016), since badly-shaped tetrahedrons with tiny dihedral angles (i.e., sliver) can severely affect numerical simulation (Shewchuk, 2002a).

Essentially, simplex meshes are used to form a piecewise linear approximation of function u(x) to represent the given shapes. There are several ways to describe the approximation error. Optimal Delaunay Triangulation (ODT) (Chen and Xu, 2004) was proposed to minimize the Lp norm of the difference over the domain Ω between the target function uˆ(x) and the interpolated function u(x): E(x)=Ωuˆ(x)u(x)Lpdx. Interpolation error is an important mesh quality measurement. However, the definition of ODT determines that it cannot avoid sliver in tetrahedral mesh. A sliver with close to zero volume still has small interpolation error. So minimizing interpolation error cannot avoid sliver.

Consider the Lp norm of the difference between the gradient of the target function uˆ(x) and the gradient of the interpolated function u(x): E(x)=Ωuˆ(x)u(x)Lpdx, the gradient error can be strongly affected by the shape of the elements as well as their sizes. Once one dihedral angle approaches 0° or 180° in the tetrahedral mesh, the gradient error will grow dramatically large. In this paper, we propose a shape matching framework and design a gradient-based energy (i.e., the gradient of linear shape function), which heavily punish slivers in tetrahedral meshes. By specifying a template simplex, e.g. a regular simplex, the idea of the proposed method is to make the shape of the to-be-optimized simplex as close as possible to the shape of the template simplex. The experiment results show that our proposed energy has high effectiveness in sliver suppression compared with all other state-of-the-art methods in the tetrahedral meshing.

Section snippets

Related work

Tetrahedral mesh generation has been studied for several decades (Owen, 1998), and these approaches can be categorized as advancing front methods (Ito et al., 2004, Schöberl, 1997, Li et al., 2000), octree-based methods (Labelle and Shewchuk, 2007, Neil Molino et al., 2003), Delaunay-based tetrahedral methods (Cheng et al., 2012), Poisson-disk sampling methods (Guo et al., 2016), etc.

We will focus on Delaunay-based tetrahedral meshing approaches, since our proposed sliver-suppressing

Shape matching triangulation energy

Our mesh optimization is illustrated in an algebraic framework, and we call it shape matching. Given a template d-simplex τˆd, our target is to make any d-simplex τd in the mesh as similar as possible to τˆd. Here “similar” means the same shape as well as the same sizing factor comparing with the template simplex. If the template is set as a regular d-simplex, any d-simplex in the mesh is expected to be a regular simplex with a constant sizing factor conforming to the defined template simplex,

GSM energy optimization

The proposed GSM energy optimization is to build a high-quality regular d-simplex mesh in a domain Ωd based on a given regular template and the user-specified number of vertices N. Both the positions of N vertices and their connectivities are required to be optimized. Our mesh optimization process involves two operations: the numerical nonlinear optimization for vertex smoothing and the combinatorial optimization for connectivity update. These two operations are carried out iteratively to

Experiment and comparisons

We implement the algorithms using C++. The experiments are done on a workstation with Intel® Xeon E5645 CPU 2.40 GHz, and 32G DDR3 RAM. To demonstrate the performance of the proposed GSM method, we compare it with four mesh optimization approaches provided by The Computational Geometry Algorithms Library (CGAL) (Jamin et al., 2015). The optimizations of CGAL mesher have two categories. One is the local optimization, including vertex perturbation (Tournois et al., 2009) and sliver exudation (

Conclusion

In this paper, we introduce an effective sliver suppression method based on shape matching idea. It generates high-quality tetrahedral meshes in isotropic, adaptive, and anisotropic cases. The proposed GSM method is evaluated on extensive volume models and compared with state-of-the-art approaches. The results of proposed GSM method show much better performance than all other current methods. In the future, we would like to improve the computation speed by using GPU parallel techniques.

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the paper. Saifeng Ni and Xiaohu Guo are partially supported by National Science Foundation (NSF) under Grant No. IIS-1149737. Zichun Zhong is partially supported by NSF ACI-1657364. Wenping Wang is partially supported by National Natural Science Foundation of China (No. 61272019). Zhonggui Chen is partially supported by National Natural Science Foundation of China (No. 61472332).

References (27)

  • J. Guo et al.

    Tetrahedral meshing via maximal Poisson-disk sampling

    Comput. Aided Geom. Des.

    (2016)
  • P. Alliez et al.

    Variational tetrahedral meshing

    ACM Trans. Graph.

    (2005)
  • L. Chen et al.

    Optimal Delaunay triangulations

    J. Comput. Math.

    (2004)
  • Z. Chen et al.

    Revisiting optimal Delaunay triangulation for 3D graded mesh generation

    SIAM J. Sci. Comput.

    (2014)
  • S.-W. Cheng et al.

    Sliver exudation

    J. ACM

    (2000)
  • S.-W. Cheng et al.

    Delaunay Mesh Generation

    (2012)
  • L.P. Chew

    Guaranteed-quality Delaunay meshing in 3D

  • Q. Du et al.

    Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations

    Int. J. Numer. Methods Biomed. Eng.

    (2003)
  • Q. Du et al.

    Anisotropic centroidal Voronoi tessellations and their applications

    SIAM J. Sci. Comput.

    (2005)
  • Q. Du et al.

    Centroidal Voronoi tessellations: applications and algorithms

    SIAM Rev.

    (1999)
  • L. Freitag et al.

    Tetrahedral mesh improvement via optimization of the element condition number

    Int. J. Numer. Methods Biomed. Eng.

    (2002)
  • Y. Ito et al.

    Reliable isotropic tetrahedral mesh generation based on an advancing front method

  • C. Jamin et al.

    CGALmesh: a generic framework for Delaunay mesh generation

    ACM Trans. Math. Softw.

    (2015)
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