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Efficient Algorithms for the Line-SIAC Filter

  • Ashok JallepalliEmail author
  • Robert M. Kirby
Article
  • 18 Downloads

Abstract

Visualizing high-order finite element simulation data using current visualization tools has many challenges: discontinuities at element boundaries, interpolating artifacts, and evaluating derived quantities. These challenges have been addressed by postprocessing the simulation data using the L-SIAC filter. However, the time required to postprocess using this filter needs to be addressed to enable using it on large datasets. In this work, we introduce an efficient technique to speed-up the L-SIAC filter and alternate ways to gain further speed-up at the cost of accuracy. This method is also ideal to postprocess at regularly spaced locations, which would be suitable for standard visualization software. Our results show that our method can achieve up to two orders of magnitude speed-up as compared to our interpretation of the technique presented in Docampo-Sánchez (SIAM J Sci Comput 39(5):A2179–A2200, 2017).

Keywords

Line-SIAC (L-SIAC) filtering Gauss quadrature Discontinuous Galerkin (dG) Continuous Galerkin (cG) 

Notes

Acknowledgements

The authors wish to thank Professor Spencer Sherwin (Imperial College London, UK), and the Nektar++ [25] Group for helpful discussions. We also acknowledge Mr. Bob Haimes (MIT), which whom our collaborations on visualization motived this work. The authors acknowledge support from ARO W911NF-15-1-0222 (Program Manager Dr. Mike Coyle).

References

  1. 1.
    Docampo-Sánchez, J., Ryan, J.K., Mirzargar, M., Kirby, R.M.: Multi-dimensional filtering: reducing the dimension through rotation. SIAM J. Sci. Comput. 39(5), A2179–A2200 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in the finite element method. Math. Comput. 31(137), 94–111 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cockburn, B., Luskin, M., Shu, C.-W., Süli, E.: Post-processing of Galerkin methods for hyperbolic problems. In: Discontinuous Galerkin Methods, pp. 291–300. Springer, Berlin, Heidelberg (2000)Google Scholar
  4. 4.
    Cockburn, B., Luskin, M., Shu, C.-W., Süli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput. 72(242), 577–606 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mirzaee, H., Ryan, J.K., Kirby, R.M.: Efficient implementation of smoothness-increasing accuracy-conserving (SIAC) filters for discontinuous Galerkin solutions. J. Sci. Comput. 52(1), 85–112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mirzaee, H., King, J., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving filters for discontinuous Galerkin solutions over unstructured triangular meshes. SIAM J. Sci. Comput. 35(1), A212–A230 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mirzaee, H., Ji, L., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) postprocessing for discontinuous Galerkin solutions over structured triangular meshes. SIAM J. Numer. Anal. 49(5), 1899–1920 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mirzaee, H., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) filters for discontinuous Galerkin solutions: application to structured tetrahedral meshes. J. Sci. Comput. 58(3), 690–704 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    King, J., Mirzaee, H., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) filtering for discontinuous Galerkin solutions: improved errors versus higher-order accuracy. J. Sci. Comput. 53(1), 129–149 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mirzaee, H., Ryan, J.K., Kirby, R.M.: Quantification of errors introduced in the numerical approximation and implementation of smoothness-increasing accuracy conserving (SIAC) filtering of discontinuous Galerkin (dG) fields. J. Sci. Comput. 45(1–3), 447–470 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mirzargar, M., Jallepalli, A., Ryan, J.K., Kirby, R.M.: Hexagonal smoothness-increasing accuracy-conserving filtering. J. Sci. Comput. 73(2–3), 1072–1093 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Steffen, M., Curtis, S., Kirby, R.M., Ryan, J.K.: Investigation of smoothness-increasing accuracy-conserving filters for improving streamline integration through discontinuous fields. IEEE Trans. Vis. Comput. Graph. 14(3), 680–692 (2008)CrossRefGoogle Scholar
  13. 13.
    Jallepalli, A., Docampo-Sánchez, J., Ryan, J.K., Haimes, R., Kirby, R.M.: On the treatment of field quantities and elemental continuity in FEM solutions. IEEE Trans. Vis. Comput. Graph. 24(1), 903–912 (2017)CrossRefGoogle Scholar
  14. 14.
    Jallepalli, A., Haimes, R., Kirby, R.M.: Adaptive characteristic length for L-SIAC filtering of FEM data. J. Sci. Comput. (2018).  https://doi.org/10.1007/s10915-018-0868-6
  15. 15.
    Nelson, B., Liu, E., Haimes, R., Kirby, R.M.: ElVis: a system for the accurate and interactive visualization of high-order finite element solutions. IEEE Trans. Vis. Comput. Graph. 18(12), 2325–2334 (2012)CrossRefGoogle Scholar
  16. 16.
    Nelson, B., Kirby, R.M.: Ray-tracing polymorphic multidomain spectral/hp elements for isosurface rendering. IEEE Trans. Vis. Comput. Graph. 12(1), 114–125 (2006)CrossRefGoogle Scholar
  17. 17.
    Nelson, B., Kirby, R.M., Haimes, R.: Gpu-based interactive cut-surface extraction from high-order finite element fields. IEEE Trans. Vis. Comput. Graph. 17, 1803–1811 (2011)CrossRefGoogle Scholar
  18. 18.
    Loseille, A., Feuillet, R.: Vizir: high-order mesh and solution visualization using opengl 4.0 graphic pipeline. In: 2018 AIAA Aerospace Sciences Meeting, p. 1174 (2018)Google Scholar
  19. 19.
    Squillacote, A.: The Paraview Guide. Kitware, Inc., ParaView, vol. 3 (2008)Google Scholar
  20. 20.
    Bellevue, W.: Tecplot User’s Manual. Amtec Engineering Inc, New Plymouth (2003)Google Scholar
  21. 21.
    Light, I.: Fieldview reference manual, software version, vol. 11 (2006)Google Scholar
  22. 22.
    Scheinerman, E.: Mathematics: A Discrete Introduction. Nelson Education, Toronto (2012)zbMATHGoogle Scholar
  23. 23.
    Moxey, D., Sastry, S.P., Kirby, R.M.: Interpolation error bounds for curvilinear finite elements and their implications on adaptive mesh refinement. J. Sci. Comput. 78(2), 1045–1062 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nelson, B., Kirby, R.M., Haimes, R.: Gpu-based volume visualization from high-order finite element fields. IEEE Trans. Vis. Comput. Graph. 20, 70–83 (2014)CrossRefGoogle Scholar
  25. 25.
    Cantwell, C.D., Moxey, D., Comerford, A., Bolis, A., Rocco, G., Mengaldo, G., De Grazia, D., Yakovlev, S., Lombard, J.-E., Ekelschot, D., Jordi, B., Xu, H., Mohamied, Y., Eskilsson, C., Nelson, B., Vos, P., Biotto, C., Kirby, R.M., Sherwin, S.J.: Nektar++ : an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219 (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUSA

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