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Advances in Computational Mathematics

, Volume 44, Issue 4, pp 1063–1090 | Cite as

A finite-element coarse-grid projection method for incompressible flow simulations

  • Ali KashefiEmail author
  • Anne E. Staples
Article

Abstract

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. The minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions.

Keywords

Pressure-correction schemes Coarse-grid projection Unstructured grids Finite elements Semi-implicit time integration 

Mathematics Subject Classfication (2010)

35Q30 65Y20 65N30 65N55 

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Notes

Acknowledgments

AK wishes to thank Dr. Michael Lentine, Dr. Peter Minev, Dr. Saad Ragab, and Dr. Omer San for helpful discussions. Moreover, AK would like to thank the reviewers for their insightful comments.

References

  1. 1.
    Guermond, J., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44), 6011–6045 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comput. 22(104), 745–762 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Temam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II). Arch. Ration. Mech. Anal. 33(5), 377–385 (1969)CrossRefzbMATHGoogle Scholar
  4. 4.
    Shen, J.: On error estimates of projection methods for Navier-Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29(1), 57–77 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hugues, S., Randriamampianina, A.: An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 28(3), 501–521 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jobelin, M., Lapuerta, C., Latché, J.-C., Angot, P., Piar, B.: A finite element penalty–projection method for incompressible flows. J. Comput. Phys. 217 (2), 502–518 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fedkiw, R., Stam, J., Jensen, H.W.: Visual simulation of smoke. In: Proceedings of the 28th annual Conference on Computer Graphics and Interactive Techniques, pp. 15–22. ACM (2001)Google Scholar
  8. 8.
    Korczak, K.Z., Patera, A.T.: An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry. J. Comput. Phys. 62 (2), 361–382 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guermond, J.-L., Minev, P., Shen, J.: Error analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43(1), 239–258 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Reusken, A.: Fourier analysis of a robust multigrid method for convection-diffusion equations. Numer. Math. 71(3), 365–397 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Filelis-Papadopoulos, C.K., Gravvanis, G.A., Lipitakis, E.A.: On the numerical modeling of convection-diffusion problems by finite element multigrid preconditioning methods. Adv. Eng. Softw. 68, 56–69 (2014)CrossRefGoogle Scholar
  12. 12.
    Gupta, M.M., Kouatchou, J., Zhang, J.: A compact multigrid solver for convection-diffusion equations. J. Comput. Phys. 132(1), 123–129 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gupta, M.M., Kouatchou, J., Zhang, J.: Comparison of second-and fourth-order discretizations for multigrid Poisson solvers. J. Comput. Phys. 132(2), 226–232 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhang, J.: Fast and high accuracy multigrid solution of the three dimensional Poisson equation. J. Comput. Phys. 143(2), 449–461 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lentine, M., Zheng, W., Fedkiw, R.: A novel algorithm for incompressible flow using only a coarse grid projection. In: ACM Transactions on Graphics (TOG), vol. 4, p 114. ACM (2010)Google Scholar
  16. 16.
    San, O., Staples, A.E.: A coarse-grid projection method for accelerating incompressible flow computations. J. Comput. Phys. 233, 480–508 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    San, O., Staples, A.E.: An efficient coarse grid projection method for quasigeostrophic models of large-scale ocean circulation. Int. J. Multiscale Comput. Eng. 11(5), 463–495 (2013)Google Scholar
  18. 18.
    Jin, M., Liu, W., Chen, Q.: Accelerating fast fluid dynamics with a coarse-grid projection scheme. HVAC&R Res 20(8), 932–943 (2014)CrossRefGoogle Scholar
  19. 19.
    Losasso, F, Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. In: ACM Transactions on Graphics (TOG), vol. 3, pp. 457–462. ACM (2004)Google Scholar
  20. 20.
    Moin, P.: Fundamentals of engineering numerical analysis. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. Amer. Math. Soc. 67(221), 73–85 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gropp, W.D., Kaushik, D.K., Keyes, D.E., Smith, B.F.: High-performance parallel implicit CFD. Parallel Comput. 27(4), 337–362 (2001)CrossRefzbMATHGoogle Scholar
  23. 23.
    Heys, J., Manteuffel, T., McCormick, S., Olson, L.: Algebraic multigrid for higher-order finite elements. J. Comput. Phys. 204(2), 520–532 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Becker, R., Braack, M.: Multigrid techniques for finite elements on locally refined meshes. Numer. Linear Algebra Appl. 7(6), 363–379 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Reitzinger, S., Schöberl, J.: An algebraic multigrid method for finite element discretizations with edge elements. Numer. Linear Algebra Appl. 9(3), 223–238 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sampath, R.S., Biros, G.: A parallel geometric multigrid method for finite elements on octree meshes. SIAM J. Sci. Comput. 32(3), 1361–1392 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Timmermans, L., Minev, P., Van De Vosse, F.: An approximate projection scheme for incompressible flow using spectral elements. Int. J. Numer. Methods Fluids 22(7), 673–688 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wanner, G., Hairer, E.: Solving ordinary differential equations II vol 1. Springer-Verlag, Berlin (1991)zbMATHGoogle Scholar
  29. 29.
    Reddy, J.N.: An introduction to the finite element method, vol. 2. McGraw-Hill, New York (1993)Google Scholar
  30. 30.
    Jiang, C.B., Kawahara, M.: A three-step finite element method for unsteady incompressible flows. Comput. Mech. 11(5-6), 355–370 (1993)CrossRefzbMATHGoogle Scholar
  31. 31.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Revue Fr. d’automatique, Inf. Rech. Opérationnelle Anal. Numér. 8(2), 129–151 (1974)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Babuška, I.: The finite element method with Lagrangian multipliers. Numerische Mathematik 20(3), 179–192 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Xu, J., Chen, L., Nochetto, R.H.: Optimal multilevel methods for H (grad), H (curl), and H (div) systems on graded and unstructured grids. In: Multiscale, nonlinear and adaptive approximation, pp. 599–659. Springer (2009)Google Scholar
  34. 34.
    Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math. 49(4), 379–412 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bank, R.E., Xu, J.: An algorithm for coarsening unstructured meshes. Numer. Math. 73(1), 1–36 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hu, J.: A robust prolongation operator for non-nested finite element methods. Comput. Math. Appl. 69(3), 235–246 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Besson, J., Foerch, R.: Large scale object-oriented finite element code design. Comput. Methods Appl. Mech. Eng. 142(1), 165–187 (1997)CrossRefzbMATHGoogle Scholar
  38. 38.
    Bell, N., Garland, M.: Efficient sparse matrix-vector multiplication on CUDA. Nvidia Technical Report NVR-2008-004, Nvidia Corporation (2008)Google Scholar
  39. 39.
    Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Van der Vorst, H.A.: Iterative Krylov methods for large linear systems, vol. 13. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  41. 41.
    Geuzaine, C., Remacle, J.F.: 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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Taylor, G., Green, A.: Mechanism of the production of small eddies from large ones. Proc. Royal Soc. Lond. Ser. A, Math. Phys. Sci. 158(895), 499–521 (1937)CrossRefzbMATHGoogle Scholar
  43. 43.
    Alam, J.M., Walsh, R.P., Alamgir Hossain, M., Rose, A.M.: A computational methodology for two-dimensional fluid flows. Int. J. Numer. Methods Fluids 75(12), 835–859 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Barton, I.: The entrance effect of laminar flow over a backward-facing step geometry. Int. J. Numer. Methods Fluids 25(6), 633–644 (1997)CrossRefzbMATHGoogle Scholar
  45. 45.
    Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59(2), 308–323 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Erturk, E.: Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions. Comput. Fluids 37 (6), 633–655 (2008)CrossRefzbMATHGoogle Scholar
  47. 47.
    Belov, A.A.: A new implicit multigrid-driven algorithm for unsteady incompressible flow calculations on parallel computers (1997)Google Scholar
  48. 48.
    Behr, M., Hastreiter, D., Mittal, S., Tezduyar, T.: Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries. Comput. Methods Appl. Mech. Eng. 123(1), 309–316 (1995)CrossRefGoogle Scholar
  49. 49.
    Ding, H., Shu, C., Yeo, K., Xu, D.: Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method. Comput. Methods Appl. Mech. Eng. 193(9), 727–744 (2004)CrossRefzbMATHGoogle Scholar
  50. 50.
    Braza, M., Chassaing, P., Minh, H.H.: Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165, 79–130 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Liu, C., Zheng, X., Sung, C.: Preconditioned multigrid methods for unsteady incompressible flows. J. Comput. Phys. 139(1), 35–57 (1998)CrossRefzbMATHGoogle Scholar
  52. 52.
    Hammache, M., Gharib, M.: A novel method to promote parallel vortex shedding in the wake of circular cylinders. Phys. Fluids A: Fluid Dyn. (1989-1993) 1 (10), 1611–1614 (1989)CrossRefGoogle Scholar
  53. 53.
    Rajani, B., Kandasamy, A., Majumdar, S.: Numerical simulation of laminar flow past a circular cylinder. Appl. Math. Modell. 33(3), 1228–1247 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Wang, Z.J.: Efficient implementation of the exact numerical far field boundary condition for Poisson equation on an infinite domain. J. Comput. Phys. 153(2), 666–670 (1999)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute for Computational & Mathematical Engineering, Department of GeophysicsStanford UniversityStanfordUSA
  2. 2.Engineering Science and Mechanics Program, Department of Biomedical Engineering and MechanicsVirginia TechBlacksburgUSA

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