Skip to main content
SpringerLink
Log in

An Angular Approach to Untangling High-Order Curvilinear Triangular Meshes

  • Chapter
  • First Online:
27th International Meshing Roundtable (IMR 2018)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 127))

Included in the following conference series:

Abstract

To achieve the full potential of high-order numerical methods for solving partial differential equations, the generation of a high-order mesh is required. One particular challenge in the generation of high-order meshes is avoiding invalid (tangled) elements that can occur as a result of moving the nodes from the low-order mesh that lie along the boundary to conform to the true curved boundary. In this paper, we propose a heuristic for correcting tangled second- and third-order meshes. For each interior edge, our method minimizes an objective function based on the unsigned angles of the pair of triangles that share the edge. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. F. Bassi, S. Rebay, High-order accurate discontinuous finite element solution of the 2D euler equations. J. Comput. Phys. 138(2), 251–285 (1997)

    Article  MathSciNet  Google Scholar 

  2. S. Dey, M.S. Shephard, Curvilinear mesh generation in 3D, in Proceedings of the 8th International Meshing Roundtable, 1999

    Google Scholar 

  3. M. Fortunato, P.-O. Persson, High-order unstructured curved mesh generation using the Winslow equations. J. Comput. Phys. 307(2016), 1–14 (2016)

    Article  MathSciNet  Google Scholar 

  4. A. Gargallo-Peiró, X. Roca, J. Peraire, J. Sarrate, Distortion and quality measures for validating and generating high-order tetrahedral meshes. Eng. Comput. 31(3), 423–437 (2015)

    Article  Google Scholar 

  5. A. Gargallo-Peiró, X. Roca, J. Peraire, J. Sarrate, Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes. Int. J. Numer. Methods Eng. 103(5), 342–363 (2015)

    Article  MathSciNet  Google Scholar 

  6. P.L. George, H. Borouchaki, Construction of tetrahedral meshes of degree two. Int. J. Numer. Methods Eng. 90(9), 1156–1182 (2012)

    Article  MathSciNet  Google Scholar 

  7. C. Geuzaine, J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)

    Article  MathSciNet  Google Scholar 

  8. X. Luo, M.S. Shephard, J.-F. Remacle, The influence of geometric approximation on the accuracy of high order methods. Rensselaer SCOREC report, 1, 2001

    Google Scholar 

  9. D. Moxey, M. Green, S. Sherwin, J. Peiró, An isoparametric approach to high-order curvilinear boundary-layer meshing. Comput. Methods Appl. Mech. Eng. 283, 636–650 (2015)

    Article  MathSciNet  Google Scholar 

  10. D. Moxey, D. Ekelschot, Ü. Keskin, S.J. Sherwin, J. Peiró, High-order curvilinear meshing using a thermo-elastic analogy. Comput. Aided Des. 72, 130–139 (2016)

    Article  Google Scholar 

  11. J. Nocedal, S. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. (Springer, Berlin, 2006)

    Google Scholar 

  12. P.-O. Persson, J. Peraire, Curved mesh generation and mesh refinement using Lagrangian solid mechanics, in Proceedings of the 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (2009), p. 949

    Google Scholar 

  13. X. Roca, A. Gargallo-Peiró, J. Sarrate, Defining quality measures for high-order planar triangles and curved mesh generation, in Proceedings of the 20th International Meshing Roundtable (Springer, Berlin, 2012), pp. 365–383

    Google Scholar 

  14. E. Ruiz-Gironés, J. Sarrate, X. Roca, Generation of curved high-order meshes with optimal quality and geometric accuracy, in Proceedings of the 25th International Meshing Roundtable. Procedia Engineering, vol. 163, 2016, pp. 315–327

    Google Scholar 

  15. S.J. Sherwin, J. Peiró, Mesh generation in curvilinear domains using high-order elements. Int. J. Numer. Methods Eng. 53(1), 207–223 (2001)

    Article  Google Scholar 

  16. M. Stees, S.M. Shontz, A high-order log barrier-based mesh generation and warping method, in Proceedings of the 26th International Meshing Roundtable. Procedia Engineering, vol. 203, 2017, pp. 180–192

    Google Scholar 

  17. T. Toulorge, C. Geuzaine, J.-F. Remacle, J. Lambrechts, Robust untangling of curvilinear meshes. J. Comput. Phys. 254, 8–26 (2013)

    Article  MathSciNet  Google Scholar 

  18. M. Turner, D. Moxey, J. Peir, M. Gammon, C.R. Pollard, H. Bucklow, A framework for the generation of high-order curvilinear hybrid meshes for CFD simulations, in Proceedings of the 26th International Meshing Roundtable. Procedia Engineering, vol. 203, 2017, pp. 206–218

    Google Scholar 

  19. Z.J. Wang, K. Fidkowski, R. Abgrall, F. Bassi, D. Caraeni, A. Cary, H. Deconinck, R. Hartmann, K. Hillewaert, H.T. Huynh, et al., High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72(8), 811–845 (2013)

    Article  MathSciNet  Google Scholar 

  20. Z.Q. Xie, R. Sevilla, O. Hassan, K. Morgan, The generation of arbitrary order curved meshes for 3D finite element analysis. Comput. Mech. 51(3), 361–374 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of the first author was funded in part by the Madison and Lila Self Graduate Fellowship and NSF CCF grant 1717894. The work of the second author was funded in part by NSF CCF grant 1717894.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Information and Telecommunication Technology Center, University of Kansas, Lawrence, KS, USA

    Mike Stees

  2. Department of Electrical Engineering and Computer Science, Bioengineering Graduate Program, Information and Telecommunication Technology Center, University of Kansas, Lawrence, KS, USA

    Suzanne M. Shontz

Authors
  1. Mike Stees
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suzanne M. Shontz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mike Stees .

Editor information

Editors and Affiliations

  1. Barcelona Supercomputing Center, Mataró, Barcelona, Spain

    Xevi Roca

  2. Inria, Palaiseau, France

    Adrien Loseille

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Stees, M., Shontz, S.M. (2019). An Angular Approach to Untangling High-Order Curvilinear Triangular Meshes. In: Roca, X., Loseille, A. (eds) 27th International Meshing Roundtable. IMR 2018. Lecture Notes in Computational Science and Engineering, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-030-13992-6_18

Download citation

Publish with us

Policies and ethics

Search

Navigation