Engineering with Computers

, Volume 35, Issue 1, pp 323–335 | Cite as

Automated low-order to high-order mesh conversion

  • Jeremy Ims
  • Z. J. Wang
Original Article


An algorithm is presented for the CAD-free conversion of linear unstructured meshes into curved high-order meshes, which are necessary for high-order flow simulations. The algorithm operates via three steps: (1) autonomous detection of feature curves along the mesh surface, (2) reconstruction of the surface curvature from the combination of surface node positions and feature curve positions, and (3) alignment of the mesh interior to the newly curved surface. The algorithm is implemented in our freely available cross-platform graphical software program meshCurve, which transforms existing linear meshes into high-order curved meshes


MeshCurve High-order meshes Surface reconstruction Feature curve detection Low-order to high-order mesh conversion CFD 



This material is based upon work supported by NASA under Grant NNX12AK04A and also by the National Science Foundation Graduate Research Fellowship Program under Grant no. NSF0064451.


  1. 1.
    Bassi F, Rebay S (1997) High-order accurate discontinuous finite element solution of the 2D Euler equations. J Comput Phys 138(2):251–285MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carr JC, Beatson RK, Cherrie JB, Mitchell TJ, Fright WR, McCallum BC, Evans TR (2001) Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of the 28th annual conference on Computer graphics and interactive techniques. ACM, pp 67–76Google Scholar
  3. 3.
    Carr JC, Beatson RK, McCallum BC, Fright WR, McLennan TJ, Mitchell TJ (2003) Smooth surface reconstruction from noisy range data. In: Proceedings of the 1st international conference on Computer graphics and interactive techniques in Australasia and South East Asia. ACM, pp 119–ffGoogle Scholar
  4. 4.
    Geuzaine C, Remacle JF (2009) Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Meth Eng 79(11):1309–1331MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Geuzaine C, Remacle JF (2016) Changelog.
  6. 6.
    Hamming R (2012) Numerical methods for scientists and engineers. Courier CorporationGoogle Scholar
  7. 7.
    Hartmann R, Harlan M (2016) Curved grid generation and DG computation for the DLR-F11 high lift configuration. In: ECCOMAS Congress 2016 on Computational Methods in Applied Sciences and Engineering, vol 2. National Technical University of Athens (NTUA) Greece, pp 2843–2858Google Scholar
  8. 8.
    Hindenlang F (2014) Mesh curving techniques for high order parallel simulations on unstructured meshesGoogle Scholar
  9. 9.
    Hindenlang F, Bolemann T, Munz CD (2015) Mesh curving techniques for high order discontinuous Galerkin simulations. In: IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach. Springer, pp 133–152Google Scholar
  10. 10.
    Ims J, Duan Z, Wang ZJ (2015) meshCurve: an automated low-order to high-order mesh generator. In: 22nd AIAA Computational Fluid Dynamics conferenceGoogle Scholar
  11. 11.
    Jiao X, Bayyana NR (2008) Identification of C1 and C2 discontinuities for surface meshes in CAD. Comput Aided Des 40(2):160–175CrossRefGoogle Scholar
  12. 12.
    Jiao X, Wang D (2012) Reconstructing high-order surfaces for meshing. Eng Comput 28(4):361–373CrossRefGoogle Scholar
  13. 13.
    Karman SL, Erwin JT, Glasby RS, Stefanski D (2016) High-order mesh curving using WCN mesh optimization. In: 46th AIAA Fluid Dynamics Conference, p 3178Google Scholar
  14. 14.
    Kazhdan M, Bolitho M, Hoppe H (2006) Poisson surface reconstruction. In: Polthier K, Sheffer A (eds) Eurographics Symposium on Geometry ProcessingGoogle Scholar
  15. 15.
    Luke E, Collins E, Blades E (2012) A fast mesh deformation method using explicit interpolation. J Comput Phys 231(2):586–601MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Luo X, Shephard MS, Remacle JF (2001) The influence of geometric approximation on the accuracy of high order methods. Rensselaer SCOREC report 1Google Scholar
  17. 17.
    Ray N, Delaney T, Einstein DR, Jiao X (2014) Surface remeshing with robust high-order reconstruction. Eng Comput 30(4):487–502CrossRefGoogle Scholar
  18. 18.
    Runge C (1901) Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Zeitschrift für Mathematik und Physik 46(224–243):20zbMATHGoogle Scholar
  19. 19.
    Shampine LF, Allen RC, Pruess S (1997) Fundamentals of numerical computing, vol 1. Wiley, New YorkzbMATHGoogle Scholar
  20. 20.
    Tampieri F (1992) Newell’s method for computing the plane equation of a polygon. In: Graphics Gems III (IBM Version). Elsevier, pp 231–232Google Scholar
  21. 21.
    Trefethen LN, Bau III, D (1997) Numerical linear algebra, vol 50. SiamGoogle Scholar
  22. 22.
    Wang Z (2007) High-order methods for the Euler and Navier–Stokes equations on unstructured grids. Prog Aerosp Sci 43(1):1–41MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang Z, Fidkowski K, Abgrall R, Bassi F, Caraeni D, Cary A, Deconinck H, Hartmann R, Hillewaert K, Huynh H et al (2013) High-order CFD methods: current status and perspective. Int J Numer Meth Fluids 72(8):811–845MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang Z, Li Y, Jia F, Laskowski G, Kopriva J, Paliath U, Bhaskaran R (2017) Towards industrial large eddy simulation using the FR/CPR method. Comput Fluids 156:579–589MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xie ZQ, Sevilla R, Hassan O, Morgan K (2013) The generation of arbitrary order curved meshes for 3D finite element analysis. Comput Mech 51(3):361–374MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringThe University of KansasLawrenceUSA

Personalised recommendations