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Generation of Block Structured Grids on Complex Domains for High Performance Simulation

  • Daniel ZintEmail author
  • Roberto Grosso
  • Vadym Aizinger
  • Harald Köstler
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 131)

Abstract

In high performance computing, block structured grids are favored due to their geometric adaptability while supporting computational performance optimizations connected with structured grid discretizations. However, many problems on geometrically complex domains are traditionally solved using fully unstructured (most frequently simplicial) meshes. We attempt to address this deficiency in the two-dimensional case by presenting a method which generates block structured grids with a prescribed number of blocks from an arbitrary triangular grid. High importance was assigned to mesh quality while simultaneously allowing for complex domains. Our method guarantees fulfillment of user-defined minimal element quality criteria—an essential feature for grid generators in simulations using finite element or finite volume methods. The performance of the proposed method is evaluated on grids constructed for regional ocean simulations utilizing two-dimensional shallow water equations.

Keywords

Block structured grids Quadrilateral grids High performance computing Shallow water equations Ocean simulations 

Notes

Acknowledgements

This work has been supported by the German Research Foundation (DFG) under grant “Rechenleistungsoptimierte Software-Strategien für auf unstrukturierten Gittern basierende Anwendungen in der Ozeanmodellierung” (AI 117/6-1).

References

  1. 1.
    Aizinger, V., Dawson, C.: A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv. Water Resour. 25(1), 67–84 (2002). https://doi.org/10.1016/S0309-1708(01)00019-7. http://www.sciencedirect.com/science/article/pii/S0309170801000197
  2. 2.
    Aizinger, V., Dawson, C.: A discontinuous Galerkin method for three-dimensional shallow water flows with free surface. In: Miller, C.T., Farthing, M., Gray, W.G., Pinder, G.F. (eds.) Developments in Water Science. Computational Methods in Water Resources: Volume 2 Proceedings of the XVth International Conference on Computational Methods in Water Resources, vol. 55, pp. 1691–1702. Elsevier, Amsterdam (2004). https://doi.org/10.1016/S0167-5648(04)80177-1. http://www.sciencedirect.com/science/article/pii/S0167564804801771
  3. 3.
    Aizinger, V., Proft, J., Dawson, C., Pothina, D., Negusse, S.: A three-dimensional discontinuous Galerkin model applied to the baroclinic simulation of Corpus Christi Bay. Ocean Dyn. 63(1), 89–113 (2013). https://doi.org/10.1007/s10236-012-0579-8 CrossRefGoogle Scholar
  4. 4.
    Amenta, N., Bern, M., Eppstein, D.: Optimal point placement for mesh smoothing. J. Algorithms 30(2), 302–322 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Armstrong, C.G., Fogg, H.J., Tierney, C.M., Robinson, T.T.: Common themes in multi-block structured quad/hex mesh generation. Proc. Eng. 124, 70–82 (2015)CrossRefGoogle Scholar
  6. 6.
    Bank, R.E., Smith, R.K.: Mesh smoothing using a posteriori error estimates. SIAM J. Numer. Anal. 34(3), 979–997 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Boier-Martin, I., Rushmeier, H., Jin, J.: Parameterization of triangle meshes over quadrilateral domains. In: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 193–203. ACM, New York (2004)Google Scholar
  8. 8.
    Canann, S.A., Tristano, J.R., Staten, M.L., et al.: An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes. In: IMR, pp. 479–494. Citeseer (1998)Google Scholar
  9. 9.
    Carr, N.A., Hoberock, J., Crane, K., Hart, J.C.: Rectangular multi-chart geometry images. In: Symposium on Geometry Processing, pp. 181–190 (2006)Google Scholar
  10. 10.
    Daniels II, J., Silva, C.T., Cohen, E.: Semi-regular quadrilateral-only remeshing from simplified base domains. In: Proceedings of the Symposium on Geometry Processing, SGP ’09, pp. 1427–1435. Eurographics Association, Aire-la-Ville, Switzerland (2009). http://dl.acm.org/citation.cfm?id=1735603.1735626
  11. 11.
    Dawson, C., Aizinger, V.: A discontinuous Galerkin method for three-dimensional shallow water equations. J. Sci. Comput. 22(1–3), 245–267 (2005). https://doi.org/10.1007/s10915-004-4139-3 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dong, S., Bremer, P.T., Garland, M., Pascucci, V., Hart, J.C.: Spectral surface quadrangulation. In: ACM SIGGRAPH 2006 Papers, SIGGRAPH ’06, pp. 1057–1066. ACM, New York (2006). https://doi.org/10.1145/1179352.1141993
  13. 13.
    Freitag, L.A.: On combining Laplacian and optimization-based mesh smoothing techniques. ASME Appl. Mech. 220, 37–44 (1997)Google Scholar
  14. 14.
    Freitag, L.A., Plassmann, P.: Local optimization-based simplicial mesh untangling and improvement. Int. J. Numer. Methods Eng. 49(1–2), 109–125 (2000)CrossRefGoogle Scholar
  15. 15.
    Garanzha, V.A.: Barrier method for quasi-isometric grid generation. Zh. Vychisl. Mat. Mat. Fiz. 40(11), 1685–1705 (2000)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, pp. 209–216. ACM Press/Addison-Wesley, New York (1997)Google Scholar
  17. 17.
    Garland, M., Zhou, Y.: Quadric-based simplification in any dimension. ACM Trans. Graph. 24(2), 209–239 (2005)CrossRefGoogle Scholar
  18. 18.
    George, P.L., Borouchaki, H.: Delaunay Triangulation and Meshing. Hermes, Paris (1998)zbMATHGoogle Scholar
  19. 19.
    Gmeiner, B., Huber, M., John, L., Rüde, U., Wohlmuth, B.: A quantitative performance study for Stokes solvers at the extreme scale. J. Comput. Sci. 17, 509–521 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gropp, W.D., Kaushik, D.K., Keyes, D.E., Smith, B.: Performance modeling and tuning of an unstructured mesh cfd application. In: Proceedings of the 2000 ACM/IEEE Conference on Supercomputing, SC ’00. IEEE Computer Society, Washington (2000). http://dl.acm.org/citation.cfm?id=370049.370405
  21. 21.
    Huang, W.: Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 174(2), 903–924 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hülsemann, F., Bergen, B., Rüde, U.: Hierarchical hybrid grids as basis for parallel numerical solution of PDE. In: European Conference on Parallel Processing, pp. 840–843. Springer, Berlin (2003)Google Scholar
  23. 23.
    Kaltenbacher, M.: Numerical Simulation of Mechatronic Sensors and Actuators, vol. 2. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  24. 24.
    Kuckuk, S., Köstler, H.: Automatic generation of massively parallel codes from ExaSlang. Computation 4(3), 27:1–27:20 (2016).  https://doi.org/10.3390/computation4030027. http://www.mdpi.com/2079-3197/4/3/27. Special Issue on High Performance Computing (HPC) Software Design
  25. 25.
    Kuckuk, S., Haase, G., Vasco, D.A., Köstler, H.: Towards generating efficient flow solvers with the ExaStencils approach. Concurr. Comput. Pract. Exp. 29(17), 4062:1–4062:17 (2017). Special Issue on Advanced Stencil-Code EngineeringGoogle Scholar
  26. 26.
    Lengauer, C., Apel, S., Bolten, M., Größlinger, A., Hannig, F., Köstler, H., Rüde, U., Teich, J., Grebhahn, A., Kronawitter, S., Kuckuk, S., Rittich, H., Schmitt, C.: ExaStencils: advanced stencil-code engineering. In: Euro-Par 2014: Parallel Processing Workshops. Lecture Notes in Computer Science, vol. 8806, pp. 553–564. Springer, Berlin (2014)Google Scholar
  27. 27.
    Persson, P.O.: Mesh size functions for implicit geometries and PDE-based gradient limiting. Eng. Comput. 22(2), 95–109 (2006)CrossRefGoogle Scholar
  28. 28.
    Rank, E., Schweingruber, M., Sommer, M.: Adaptive mesh generation and transformation of triangular to quadrilateral meshes. Int. J. Numer. Methods Biomed. Eng. 9(2), 121–129 (1993)zbMATHGoogle Scholar
  29. 29.
    Sorkine, O., Alexa, M.: As-rigid-as-possible surface modeling. In: Proceedings of the Fifth Eurographics Symposium on Geometry Processing, SGP ’07, pp. 109–116. Eurographics Association, Aire-la-Ville, Switzerland (2007). http://dl.acm.org/citation.cfm?id=1281991.1282006
  30. 30.
    Wang, R., Shen, C., Chen, J., Gao, S., Wu, H.: Automated block decomposition of solid models based on sheet operations. Proc. Eng. 124, 109–121 (2015)CrossRefGoogle Scholar
  31. 31.
    White, B.S., McKee, S.A., de Supinski, B.R., Miller, B., Quinlan, D., Schulz, M.: Improving the computational intensity of unstructured mesh applications. In: Proceedings of the 19th Annual International Conference on Supercomputing, ICS ’05, pp. 341–350. ACM, New York (2005). https://doi.org/10.1145/1088149.1088195
  32. 32.
    Zint, D., Grosso, R.: Discrete mesh optimization on GPU. In: 27th International Meshing Roundtable (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Daniel Zint
    • 1
    Email author
  • Roberto Grosso
    • 1
  • Vadym Aizinger
    • 3
  • Harald Köstler
    • 2
  1. 1.University Erlangen-NurembergChair of Computer GraphicsErlangenGermany
  2. 2.University Erlangen-NurembergChair of System SimulationErlangenGermany
  3. 3.University of BayreuthChair of Scientific ComputingBayreuthGermany

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