Skip to main content

Application of single-level, pointwise algebraic, and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows

Computing and Visualization in Science volume 11, pages 27–40 (2008)Cite this article

Abstract

Single- and multi-level iterative methods for sparse linear systems are applied to unsteady flow simulations via implementation into a direct numerical simulation solver for incompressible turbulent flows on unstructured meshes. The performance of these solution methods, implemented in the well-established SAMG and ML packages, are quantified in terms of computational speed and memory consumption, with a direct sparse LU solver (SuperLU) used as a reference. The classical test case of unsteady flow over a circular cylinder at low Reynolds numbers is considered, employing a series of increasingly fine anisotropic meshes. As expected, the memory consumption increases dramatically with the considered problem size for the direct solver. Surprisingly, however, the computation times remain reasonable. The speed and memory usage of pointwise algebraic and smoothed aggregation multigrid solvers are found to exhibit near-linear scaling. As an alternative to multi-level solvers, a single-level ILUT-preconditioned GMRES solver with low drop tolerance is also considered. This solver is found to perform sufficiently well only on small meshes. Even then, it is outperformed by pointwise algebraic multigrid on all counts. Finally, the effectiveness of pointwise algebraic multigrid is illustrated by considering a large three-dimensional direct numerical simulation case using a novel parallelization approach on a large distributed memory computing cluster.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Beaudan, P., Moin, P.: Numerical experiments on the flow past a circular cylinder of sub-critical Reynolds number. Technical report TF-62, Department of Mechanical Engineering, Stanford University (1994)

  2. Berger E. and Wille R. (1972). Periodic flow phenomena. Annu. Rev. Fluid Mech. 4: 313–340

    Article  Google Scholar 

  3. Clees, T.: AMG strategies for PDE systems with applications in industrial semiconductor simulation. Ph.D. thesis, University of Cologne, Germany (2004)

  4. de Zeeuw P.M. (1990). Matrix dependent prolongations and restrictions in a black-box multigrid solver. J. Comput. Appl. Math. 33: 1–27

    Article  MATH  MathSciNet  Google Scholar 

  5. Demmel J.W., Eisenstat S.C., Gilbert J.R., Li X.S. and Liu J.W.H. (1999). A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3): 720–755

    Article  MATH  MathSciNet  Google Scholar 

  6. Griebel M., Neunhoeffer T. and Regler H. (1998). Algebraic multigrid methods for the solution of the Navier–Stokes equations in complicated geometries. Int. J. Numer. Methods Fluids 26: 281–301

    Article  MATH  Google Scholar 

  7. Guillard, H., Vanek, P.: An aggregation multigrid solver for convection–diffusion problems on unstructured meshes. Report 130, University of Denver (1998)

  8. Heroux, M.A., Willenbring, J.M.: Trilinos users guide. Technical report SAND2003-2952, Sandia National Laboratories, Albuquerque (2003)

  9. Irwin, J., Loingtier, J.M., Gilbert, J.R., Kiczales, G., Lamping, J., Mendhekar, A., Shpeisman, T.: Aspect-oriented programming of sparse matrix code. In: Proceedings of the International Scientific Computing in Object-Oriented Parallel Environments (ISCOPE). Marina del Rey, CA, USA (1997)

  10. Jansen K. (1999). A stabilized finite element method for computing turbulence. Comput. Methods Appl. Mech. Eng. 174(3–4): 299–317

    Article  MATH  MathSciNet  Google Scholar 

  11. Jones M.T. and Plassman P.E. (1993). A parallel graph coloring heuristic. SIAM J. Sci. Comput. 14: 654–669

    Article  MATH  MathSciNet  Google Scholar 

  12. Karypis, G., Kumar, V.: METIS: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, version 4.0. Manual, University of Minnesota (1998)

  13. Lonsdale R.D. (1993). An algebraic multigrid solver for the Navier–Stokes equations on unstructured meshes. Int. J. Numer. Methods Heat Fluid Flow 3: 3–14

    Google Scholar 

  14. Mavriplis D. (2003). An assessment of linear versus nonlinear multigrid methods for unstructured mesh solvers. J. Comput. Phys. 175(1): 302–325

    Article  Google Scholar 

  15. Morkovin, M.V.: Flow around a circular cylinder—a kaleidoscope of challenging fluid phenomenon. In: Proceedings of the ASME Symposium on Fully Separated Flows, pp. 102–119, Philadelphia (1964)

  16. Notay, Y.: A robust allgebraic preconditioner for finite difference approximations of convection–diffusion equations. Report GANMN 99–01, Service de Métrologie Nucléaire, Université Libre de Bruxelles (1999)

  17. Raw, M.: Robustness of coupled algebraic multigrid for the Navier–Stokes equations. Technical paper 1996-0297, AIAA, Reno, Nevada (1996)

  18. Reusken A. (2000). On the approximate cyclic reduction preconditioner. SIAM J. Sci. Comput. 21: 565–590

    Article  MathSciNet  Google Scholar 

  19. Ruge J.W. and Stüben K. (1987). Algebraic multigrid. In: McCormick, S.F. (eds) Multigrid Methods, pp. SIAM, Philadelphia

    Google Scholar 

  20. Saad Y. and Schultz M. (1986). A generalized minimal residual algorithm for solving nonsymmetric linear systems. J. Sci. Stat. Comput. 7: 856–869

    Article  MATH  MathSciNet  Google Scholar 

  21. Sala, M., Hu, J.J., Tuminaro, R.S.: ML 3.1 smoothed aggregation user’s guide. Technical report SAND2004-4819, Sandia National Laboratories (2004)

  22. Sleijpen G.L.G. and van~der Vorst H.A. (1996). An overview of approaches for the stable computation of hybrid BiCG methods. Appl. Numer. Math Trans. IMACS 19(3): 235–254

    Article  MathSciNet  Google Scholar 

  23. Snyder, D.O.: A parallel finite-element/spectral les algorithm for complex two-dimensional geometries. Ph.D. thesis, Utah State University/von Karman Institute for Fluid Dynamics, Logan, Utah, USA (2002)

  24. Snyder D.O. and Degrez G. (2003). Large-eddy simulation with complex 2-D geometries using a parallel finite-element/spectral algorithm. Int. J. Numer. Methods Fluids 41(10): 1119–1135

    Article  MATH  Google Scholar 

  25. Sternel, D.C., Schäfer, M.: Efficient large eddy simulations in complex geometries by using an improved multigrid algorithm. In: EMG 2005 Proceedings. Scheveningen, the Netherlands (2005)

  26. Stüben K. (1999). An introduction to algebraic multigrid. In: Trottenberg, U., Oosterlee, C. and Schueller, A. (eds) Multigrid, pp 413–532. Academic, Oxford

    Google Scholar 

  27. Stüben K. (2001). A review of algebraic multigrid. J. Comput. Appl. Math. 128: 281–309

    Article  MATH  MathSciNet  Google Scholar 

  28. Tejada-Martinez A.E. and Jansen K.E. (2005). On the interaction between dynamic model dissipation and numerical dissipation due to streamline upwind/Petrov–Galerkin stabilization. Comput. Methods Appl. Mech. Eng. 194(9-11): 1225–1248

    Article  MATH  Google Scholar 

  29. Tezduyar T.E., Mittal S., Ray S.E. and Shih R. (1992). Incompressible flow computations with stabilized bilinear and linear equal-order interpolation velocity–pressure elements. Comput. Methods Appl. Mech. Eng. 95(2): 221–242

    Article  MATH  Google Scholar 

  30. Tuminaro, R.S., Tong, C.: Parallel smoothed aggregation multigrid: aggregation strategies on massively parallel machines. In: Supercomputing 2000 Proceedings, Dallas, TX, USA (2000)

  31. Vanden-Abeele, D., Degrez, G., Snyder, D.O.: Parallel turbulent flow computations using a hybrid spectral/finite-element method on beowulf clusters. In: Proceedings of ICCFD3, Toronto, Canada (2004)

  32. Vanek P. (1992). Acceleration of convergence of a two level algorithm by smooth transfer operators. Appl. Math. 37: 265–274

    MATH  MathSciNet  Google Scholar 

  33. Vanek P., Mandel J. and Brezina M. (1996). Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 56: 179–196

    Article  MATH  MathSciNet  Google Scholar 

  34. Venkatakrishnan, V., Mavriplis, D.: Agglomeration multigrid for the three-dimensional euler equations. Technical report TR-94-5, Institute for Computer Applications in Science and Engineering (ICASE) (1994)

  35. Webster R. (1994). An algebraic multigrid solver for Navier–Stokes problems. Int. J. Numer. Methods Fluids 18(8): 761–780

    Article  MATH  Google Scholar 

  36. Weiss J.M., Maruszewski J.P. and Smith W.A. (1999). Implicit solution of preconditioned Navier–Stokes equations using algebraic multigrid. AIAA J. 37(1): 29–36

    Article  Google Scholar 

  37. Williamson C.H.K. (1996). Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28: 477–539

    Article  MathSciNet  Google Scholar 

  38. Williamson C.H.K., Wu J. and Sheridan J. (1995). Scaling of streamwise vortices in wakes. Phys. Fluids 7: 2307–2309

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Brigham Young University, Provo, UT, 84602, USA

    Gregory Larson & Deryl Snyder

  2. Aeronautics and Aerospace Department, von Karman Institute For Fluid Dynamics, 1640, Rhode-Saint-Genèse, Belgium

    David Vanden Abeele

  3. Fraunhofer Institute for Algorithms and Scientific Computing, 53754, Sankt Augustin, Germany

    Tanja Clees

Authors
  1. Gregory Larson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deryl Snyder
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Vanden Abeele
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tanja Clees
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deryl Snyder.

Additional information

Communicated by P. Hemker.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Larson, G., Snyder, D., Abeele, D.V. et al. Application of single-level, pointwise algebraic, and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows. Comput. Visual Sci. 11, 27–40 (2008). https://doi.org/10.1007/s00791-006-0055-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-006-0055-4

Keywords

  • Large Eddy Simulation
  • Circular Cylinder
  • Direct Numerical Simulation
  • Memory Usage
  • Multigrid Method