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Generation of Delaunay meshes in implicit domains with edge sharpening

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Abstract

A variational algorithm for the construction of 3D Delaunay meshes in implicit domains with a nonsmooth boundary is proposed. The algorithm is based on the self-organization of an elastic network in which each Delaunay edge is interpreted as an elastic strut. The elastic potential is constructed as a combination of the repulsion potential and the sharpening potential. The sharpening potential is applied only on the boundary and is used to minimize the deviation of the outward normals to the boundary faces from the direction of the gradient of the implicit function. Numerical experiments showed that in the case when the implicit function specifying the domain is considerably different from the signed distance function, the use of the sharpening potential proposed by Belyaev and Ohtake in 2002 leads to the mesh instability. A stable version of the sharpening potential is proposed. The numerical experiments showed that acceptable Delaunay meshes for complex shaped domains with sharp curved boundary edges can be constructed.

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References

  1. V. L. Rvachev, The Theory of R-Functions and Its Applications (Naukova dumka, Kiev, 1982) [in Russian].

    MATH  Google Scholar 

  2. F. Labelle and J. R. Shewchuk, “Isosurface stuffing: Fast tetrahedral meshes with good dihedral angles.” ACM Trans. Graphics 26 (3), 57:1–57:10 (2007).

    Article  Google Scholar 

  3. J.-D. Boissonat, D. Cohen-Steiner, B. Mourrain, et al., “Meshing of surfaces,” in Effective Computational Geometry for Curves and Surfaces (2007),pp. 181–230.

    Google Scholar 

  4. J. Tournois, C. Wormser, P. Alliez, and M. Desbrun, “Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation,” ACM Trans. Graphics 28 (3), 75:1–75:9 (2009).

    Article  Google Scholar 

  5. A. V. Kofanov and V. D. Liseikin, “Grid construction for discretely defined configurations,” Comput. Math. Math. Phys. 53, 759–765 (2013).

    Article  MathSciNet  Google Scholar 

  6. P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46 (2), 329–345 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  7. L. N. Belousova and V. A. Garanzha, “Construction of Delaunay meshes in implicitly specified domains with a nonsmooth boundary,” Trudy 51 Nauchnoi Konferentsii Sovremennye Problemy Fundamental’nykh i Prikladnykh Nauk (Proc. of the 51st Conf. on Problems of Fundamental and Applied Sciences), 2008, Vol. 2,pp. 98–101.

    Google Scholar 

  8. V. A. Garanzha and L. N. Kudryavtseva, “Generation of three-dimensional Delaunay meshes from weakly structured and inconsistent data,” Comput. Math. Math. Phys. 52 (3), 427–447 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Ohtake and A. Belyaev, “Dual/primal mesh optimization for polygonized implicit surfaces,” Proc. of the 7th ACM Symp on Solid Modeling and Applications (SMA’02) (ACM, New York, 2002),pp. 171–178.

    Chapter  Google Scholar 

  10. S. Schaeffer, T. Ju, and J. Warren, “Manifold dual contouring,” IEEE Trans. Visualization Comput. Graphics 13, 610–619 (2007).

    Article  Google Scholar 

  11. J. L. Frey and P. L. George, Mesh Generation: Applications to Finite Elements (Hermes Science, Paris, 2000).

    MATH  Google Scholar 

  12. P. L. George and E. Saltel, ““Ultimate” robustness in meshing an arbitrary polyhedron,” Int. J. Numer. Meth. Eng. 58, 1061–1089 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. A. Danilov, “Unstructured tetrahedral mesh generation technology,” Comput. Math. Math. Phys. 50, 139–156 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. D. Liseikin, Grid generation methods, 2nd ed. (Springer, Berlin, 2010).

    Book  MATH  Google Scholar 

  15. J. A. Orenstein, “Multidimensional tries used for associative data searching,” Inform. Proc. Lett. 14 (4), 150–157 (1982).

    Article  Google Scholar 

  16. H. Samet, The Design and Analysis of Spatial Data Structures (Addison-Wesley, Reading, 1990).

    Google Scholar 

  17. V. A. Garanzha, L. N. Kudryavtseva, and S. V. Utyzhnikov, “Untangling and optimization of spatial meshes,” J. Comput. Appl. Math. 269, 24–41 (2014).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow, 119333, Russia

    A. I. Belokrys-Fedotov, V. A. Garanzha & L. N. Kudryavtseva

  2. Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141700, Russia

    A. I. Belokrys-Fedotov, V. A. Garanzha & L. N. Kudryavtseva

Authors
  1. A. I. Belokrys-Fedotov
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  2. V. A. Garanzha
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  3. L. N. Kudryavtseva
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Correspondence to A. I. Belokrys-Fedotov.

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Original Russian Text © A.I. Belokrys-Fedotov, V.A. Garanzha, L.N. Kudryavtseva, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 11, pp. 1931–1948.

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Belokrys-Fedotov, A.I., Garanzha, V.A. & Kudryavtseva, L.N. Generation of Delaunay meshes in implicit domains with edge sharpening. Comput. Math. and Math. Phys. 56, 1901–1918 (2016). https://doi.org/10.1134/S096554251611004X

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  • DOI: https://doi.org/10.1134/S096554251611004X

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