Mesh Curving and Refinement Based on Cubic Bézier Surface for High-Order Discontinuous Galerkin Methods

  • Shu-Jie LiEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)


A 3-D curved mesh generator is prescribed for converting linear elements to quadratic elements required by high-order methods, which is based on the reconstruction of Cubic Bézier surfaces. Successive curved mesh refinement is also supported by inquiring the middle nodes of the edges and faces of the reconstructed quadratic elements via the Cubic Bézier surface method. Numerical test cases are shown to demonstrate the capability of both mesh curving and refinement around three-dimensional geometries.



This work is funded by the National Natural Science Foundation of China (NSFC) under the Grant U1530401. The author thanks Dr. Hang Si of WIAS, Germany, for the discussion and collaboration in a broad sense of computational geometry. SJL would further like to thank the reviewers for their helpful comments.


  1. 1.
    Li, S.-J.: A parallel discontinuous Galerkin method with physical orthogonal basis on curved elements. Proc. Eng. 61, 144–151 (2013)CrossRefGoogle Scholar
  2. 2.
    Li, S.-J., Wang, Z.J., Ju, L., Luo, L.-S.: Explicit large time stepping with a second-order exponential time integrator scheme for unsteady and steady flows. AIAA Paper, 2017–0753Google Scholar
  3. 3.
    Li, S.-J., Wang, Z.J., Ju, L., Luo, L.-S.: Fast time integration of Navier-Stokes equations with an exponential-integrator scheme. AIAA Paper, 2018–0369Google Scholar
  4. 4.
    Li, S.-J., Luo, L.-S., Wang, Z.J., Ju, L.: An exponential time-integrator scheme for steady and unsteady inviscid flows. J. Comput. Phys. 365, 206–225 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Li, S.-J.: Efficient p-multigrid method based on an exponential time discretization for compressible steady flows. arXiv:1807.0115Google Scholar
  6. 6.
    Li, S.-J., Ju, L.: Exponential time-marching method for the unsteady Navier-Stokes equations. AIAA Paper, 2019-0907Google Scholar
  7. 7.
    Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138(2), 251–285 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Krivodonova, L., Berger, M.: High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. 211(2), 251–285 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Yamaguchi, F.: Curves and Surfaces in Computer Aided Geometric Design. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  10. 10.
    Vlachos, A., Peters, J., Boyd, C., et al.: Curved PN triangles. In: Proceedings of the 2001 Symposium on Interactive 3D Graphics. ACM Press, New York (2001)Google Scholar
  11. 11.
    Brodersen, O., Stürmer, A.: Drag prediction of engine-airframe interference effects using unstructured Navier-Stokes calculations. AIAA Paper, 2001–2414Google Scholar
  12. 12.
    Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 ACM National Conference, pp. 517–524 (1968)Google Scholar
  13. 13.
    Zhang, S.: Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houston J. Math. 21, 541–556 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hindenlang, F., Bolemann,T., Munz, C.-D.: Mesh curving techniques for high order discontinuous Galerkin simulations. In: IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach, pp. 133–152. Springer, Berlin (2015)Google Scholar
  15. 15.
    Ims, J., Duan, Z., Wang, Z.J.: meshCurve: an automated low-order to high-order mesh generator. AIAA Paper, 2015–2293Google Scholar
  16. 16.
    Jiao, X., Wang, D.: Reconstructing high-order surfaces for meshing. Eng. Comput. 28(4), 361–373 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Beijing Computational Science Research Center (CSRC)BeijingChina

Personalised recommendations