Advertisement

Methods for Numerical Flow Simulation

  • Rolf Rannacher
Part of the Oberwolfach Seminars book series (OWS, volume 37)

Abstract

This chapter introduces into computational methods for the simulation of PDE-based models of laminar hemodynamical flows. We discuss space and time discretization with emphasis on operator-splitting and finite-element Galerkin methods because of their flexibility and rigorous mathematical basis. Special attention is paid to the simulation of pipe flow and the related question of artificial outflow boundary conditions. Further topics are efficient methods for the solution of the resulting algebraic problems, techniques of sensitivity-based error control and mesh adaptation, as well as flow control and model calibration. We concentrate on laminar flows in which all relevant spatial and temporal scales can be resolved and no additional modeling of turbulence effects is required. This covers most of the relevant situations of hemodynamical flows. The numerical solution of the corresponding systems is complicated mainly because of the incompressibility constraint which enforces the use of implicit methods and its essentially parabolic or elliptic character which requires the prescription of boundary conditions along the whole boundary of the computational domain.

Keywords

Error Estimator Posteriori Error Error Representation Dual Solution Stokes Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Ainsworth and J.T. Oden. A posteriori error estimation in finite-element analysis. Comput. Methods Appl. Mech. Enrg. 142 (1997), 1–88.CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    W. Bangerth and R. Rannacher, Adaptive Finite-Element Methods for Differential Equations. Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel 2003.Google Scholar
  3. [3]
    R. Becker, An optimal control approach to a posteriori error estimation for finiteelement discretizations of the Navier-Stokes equations. East-West J. Numer. Math. 8 (2000), 257–274.MathSciNetMATHGoogle Scholar
  4. [4]
    R. Becker, Mesh adaptation for stationary flow control. J. Math. Fluid Mech. 3 (2001), 317–341.CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    R. Becker, Adaptive Finite Elements for Optimal Control Problems. Habilitation thesis, Institute of Applied Mathematics, Univ. of Heidelberg, 2001, http://numerik.iwr.uni-heidelberg.de/
  6. [6]
    R. Becker and M. Braack, Multigrid techniques for finite elements on locally refined meshes. Numer. Linear Algebra Appl. 7 (2000), 363–379.CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    R. Becker and M. Braack, A finite-element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001), 173–199.CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    R. Becker and Th. Dunne, VisuSimple: An interactive visualization utility for scientific computing. Abschlußband SFB 359, Reactive Flows, Diffusion and Transport (W. Jäger et al., eds.), Springer, Berlin-Heidelberg New York, 2007.Google Scholar
  9. [9]
    R. Becker, V. Heuveline, and R. Rannacher, An optimal control approach to adaptivity in computational fluid mechanics. Int. J. Numer. Meth. Fluids. 40 (2002), 105–120.CrossRefMathSciNetMATHGoogle Scholar
  10. [10]
    R. Becker, D. Meidner, R. Rannacher, and B. Vexler, Adaptive finite-element methods for P DE-constrained optimal control problems. In ‘Reactive Flows, Diffusion and Transport’, Abschlußband SFB 359, University of Heidelberg, (W. Jäger et al., eds.), Springer, Heidelberg, 2007.Google Scholar
  11. [11]
    R. Becker, H. Kapp, and R. Rannacher, Adaptive finite-element methods for optimal control of partial differential equations: basic concepts. SIAM J. Optimization Control 39 (2000), 113–132.CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    R. Becker and R. Rannacher, Weighted a posteriori error estimates in FE methods. Lecture ENUMATH-95, Paris, Sept. 18-22, 1995. In Proc. ‘ENUMATH-97’ (H.G. Bock, et al., eds.), pp. 621–637, World Scientific Publ., Singapore, 1998.Google Scholar
  13. [13]
    R. Becker and R. Rannacher, A feed-back approach to error control in finite-element methods: Basic analysis and examples. East-West J. Numer. Math. 4 (1996), 237–264.MathSciNetMATHGoogle Scholar
  14. [14]
    R. Becker and R. Rannacher, An optimal control approach to error estimation and mesh adaptation in finite-element methods. Acta Numerica 2000 (A. Iserles, ed.), pp. 1–102, Cambridge University Press, 2001.Google Scholar
  15. [15]
    R. Becker and B. Vexler, Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations. J. Comput. Phys. 206 (2005), 95–110.CrossRefMathSciNetMATHGoogle Scholar
  16. [16]
    R. Becker and B. Vexler. Optimal Flow Control by Adaptive Finite-Element Meth-ods. Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel-Boston-Berlin, in preparation.Google Scholar
  17. [17]
    C. Bernardi, O. Bonnon, C. Langouët, and B. Mëtivet. Residual error indicators for linear problems: Extension to the Navier-Stokes equations. In Proc. 9th Int. Conf. ‘Finite Elements in Fluids’, 1995.Google Scholar
  18. [18]
    S. Bönisch and V. Heuveline, Advanced flow visualization with HiVision. Abschlußb and SFB 359, Reactive Flows, Diffusion and Transport (W. Jäger et al., eds.), Springer, Berlin-Heidelberg New York, 2007.Google Scholar
  19. [19]
    M. Braack and E. Burman: Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM J. Numer. Anal. 43 (2006), 2544–2566.MathSciNetMATHGoogle Scholar
  20. [20]
    M. Braack and T. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements. Computers and Fluids 35 (2006), 372–392.CrossRefMATHGoogle Scholar
  21. [21]
    S. Brenner and R.L. Scott, The Mathematical Theory of the Finite-Element Method. Springer, Berlin Heidelberg New York, 1994.Google Scholar
  22. [22]
    S. Brezzi and R. Falk, Stability of higher-order Hood-Taylor methods. J. Numer. Anal. 28 (1991), 581–590.CrossRefMathSciNetMATHGoogle Scholar
  23. [23]
    G. Carey and J. Oden, Finite Elements, Computational Aspects. Volume III. Prentice-Hall, 1984.Google Scholar
  24. [24]
    Th. Dunne and R. Rannacher. Numerics of Fluid-Structure Interaction. In ‘Fluid-Structure Interaction: Modelling, Simulation, Optimisation’ (H.-J. Bungartz and M. Schäfer, eds.), in Springer’s LNCSE-Series, 2006.Google Scholar
  25. [25]
    K. Eriksson, C. Johnson, and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Model. Math. Anal. Numer. 19 (1985), 611–643.MathSciNetMATHGoogle Scholar
  26. [26]
    K. Eriksson and C. Johnson. Adaptive finite-element methods for parabolic problems, I: A linear model problem. SIAM J. Numer. Anal. 28 (1991), 43–77.CrossRefMathSciNetMATHGoogle Scholar
  27. [27]
    K. Eriksson and C. Johnson. Adaptive finite-element methods for parabolic problems, IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995), 1729–1749.CrossRefMathSciNetMATHGoogle Scholar
  28. [28]
    K. Eriksson and C. Johnson. Adaptive finite-element methods for parabolic problems, V: Long-time integration. SIAM J. Numer. Anal. 32 (1995), 1750–1763.CrossRefMathSciNetMATHGoogle Scholar
  29. [29]
    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numerica 1995 (A. Iserles, ed.), pp. 105–158, Cambridge University Press, 1995.Google Scholar
  30. [30]
    M. Fernandez, V. Milisic, and A. Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of ODE’s and hyperbolic PDE’s. SIAM J. MMS 4 (2005), 215–236.MathSciNetMATHGoogle Scholar
  31. [31]
    L. Formaggia, J.F. Gerbeau, F. Nobile, and A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comp. Methods Appl. Mech. Engnrg. 191 (2001), 561–582.CrossRefMathSciNetMATHGoogle Scholar
  32. [32]
    G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. 1: Linearized Steady problems, Vol. 2: Nonlinear Steady Problems, Springer: Berlin-Heidelberg-New York, 1998.Google Scholar
  33. [33]
    G.P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem. In this volume.Google Scholar
  34. [34]
    R. Becker and M. Braack, Gascoigne: A C++ numerics library for scientific computing. Institute of Applied Mathematics, University of Heidelberg, URL http://www.gascoigne.uni-hd.de/, 2005.
  35. [35]
    M.B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 2002 (A. Iserles, ed.), pp. 145–236, Cambridge University Press, 2002.Google Scholar
  36. [36]
    V. Girault and P.-A. Raviart, Finite-Element Methods for the Navier-Stokes Equations. Springer: Berlin-Heidelberg-New York, 1986.MATHGoogle Scholar
  37. [37]
    R. Glowinski, Finite-element methods for incompressible viscous flow. In ‘Handbook of Numerical Analysis’ (P.G. Ciarlet and J.-L. Lions, eds.), Volume IX: Numerical Methods for Fluids (Part 3), North-Holland: Amsterdam, 2003.Google Scholar
  38. [38]
    P.M. Gresho and R.L. Sani, Incompressible Flow and the Finite-Element Method. John Wiley & Sons: Chichester, 1998.MATHGoogle Scholar
  39. [39]
    P. Hansbo and C. Johnson. Adaptive streamline diffusion finite-element methods for compressible flow using conservative variables. Comput. Methods Appl. Mech. Engrg 87 (1991), 267–280.CrossRefMathSciNetMATHGoogle Scholar
  40. [40]
    R. Hartmann, A posteriori Fehlerschätzung und adaptive Schrittweiten-und Ortsgittersteuerung bei Galerkin-Verfahren für die Wärmeleitungsgleichung. Diploma thesis, Institute of Applied Mathematics, University of Heidelberg, 1998.Google Scholar
  41. [41]
    R. Hartmann, Adaptive Finite-Element Methods for the Compressible Euler Equations. Dissertation, Institute of Applied Mathematics, Universität Heidelberg, 2002.Google Scholar
  42. [42]
    R. Hartmann and P. Houston, Adaptive discontinuous Galerkin finite-element methods for the compressible Euler equations. J. Comput. Phys. 183 (2002), 508–532.CrossRefMathSciNetMATHGoogle Scholar
  43. [43]
    R. Hartmann and P. Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: method formulation. Int. J. Numer. Anal. Model. 3 (2006), 1–20.MathSciNetMATHGoogle Scholar
  44. [44]
    R. Hartmann and P. Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations II: goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model. 3 (2006), 141–162.MathSciNetMATHGoogle Scholar
  45. [45]
    V. Heuveline, HiFlow: A multi-purpose finite-element package, Rechenzentrum, Universität Karlsruhe, URL http://hiflow.de/.
  46. [46]
    V. Heuveline, HiVision: A visualization platform, Rechenzentrum, Universität Karlsruhe, URL http://hiflow.de/.
  47. [47]
    V. Heuveline and R. Rannacher, Adaptive FEM for eigenvalue problems with application in hydrodynamic stability analysis. Proc. Int. Conf. ‘Advances in Numerical Mathematics’, Moscow, Sept. 16-17, 2005 (W. Fitzgibbon et al., eds.), pp. 109–140, Institute of Numerical Mathematics RAS, Moscow, 2006.Google Scholar
  48. [48]
    J. Heywood, R. Rannacher, and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Math. Fluids 22 (1992), 325–352.CrossRefMathSciNetGoogle Scholar
  49. [49]
    P. Houston, R. Rannacher, and E. Süli. A posteriori error analysis for stabilized finite-element approximation of transport problems. Comput. Methods Appl. Mech. Enrg. 190 (2000), 1483–1508.CrossRefMATHGoogle Scholar
  50. [50]
    T.J.R. Hughes and A.N. Brooks, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation. Comput. Meth. Appl. Mech. Engrg. 32 (1982), 199–259.CrossRefMathSciNetMATHGoogle Scholar
  51. [51]
    T.J.R. Hughes, L.P. Franc, and M. Balestra, A new finite-element formulation for computational fluid mechanics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation. Comput. Meth. Appl. Mech. Engrg. 59 (1986), 85–99.CrossRefMATHGoogle Scholar
  52. [52]
    V. John, Higher order finite-element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations. Int. J. Numer. Meth. Fluids 40 (2002), 775–798.CrossRefMATHGoogle Scholar
  53. [53]
    C. Johnson, Numerical Solution of Partial Differential Equations by the Finite-Element Method. Cambridge University Press, Cambridge, 1987.MATHGoogle Scholar
  54. [54]
    C. Johnson. Adaptive finite-element methods for diffusion and convection problems. Comput. Methods Appl. Mech. Eng. 82 (1990), 301–322.CrossRefMATHGoogle Scholar
  55. [55]
    C. Johnson, R. Rannacher, and M. Boman. Numerics and hydrodynamic stability: Towards error control in CFD. SIAM J. Numer. Anal. 32 (1995), 1058–1079.CrossRefMathSciNetMATHGoogle Scholar
  56. [56]
    S. Korotov, P. Neitaanmäki and S. Repin. A posteriori error estimation of goal-oriented quantities by the superconvergence patch recovery, J. Numer. Math. 11 (2003), 33–53.MathSciNetMATHGoogle Scholar
  57. [57]
    L. Machiels, A.T. Patera, and J. Peraire, Output bound approximation for partial differential equations; application to the incompressible Navier-Stokes equations. Industrial and Environmental Applications of Direct and Large Eddy Numerical Sim-ulation (S. Biringen, ed.), Springer, Berlin Heidelberg New York, 1998.Google Scholar
  58. [58]
    V. Milisic and A. Quarteroni, Analysis of lumped parameter models for blood flow simulations and their relation with 1D models. M2AN, Vol.IV (2004), 613–632.MathSciNetGoogle Scholar
  59. [59]
    J.T. Oden, W. Wu, and M. Ainsworth, An a posteriori error estimate for finite-element approximations of the Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 111 (1993), 185–202.CrossRefMathSciNetGoogle Scholar
  60. [60]
    J.T. Oden and S. Prudhomme. On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176 (1999), 313–331.CrossRefMathSciNetMATHGoogle Scholar
  61. [61]
    J.T. Oden and S. Prudhomme. Estimation of modeling error in computational mechanics. Preprint, TICAM, The University of Texas at Austin, 2002.Google Scholar
  62. [62]
    M. Paraschivoiu and A.T. Patera. Hierarchical duality approach to bounds for the outputs of partial differential equations. Comput. Methods Appl. Mech. Enrg. 158 (1998), 389–407.CrossRefMathSciNetMATHGoogle Scholar
  63. [63]
    A. Quarteroni, S. Ragni, and A. Veneziani, Coupling between lumped and distributed models for blood flow problems. Computing and Visualization in Science 4 (2001) 2, 111–124.CrossRefMathSciNetMATHGoogle Scholar
  64. [64]
    A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of PDE’s and ODE’s for blood flow simulations. SIAM J. MMS. 1 (2003), 173–195MathSciNetMATHGoogle Scholar
  65. [65]
    R. Rannacher, Finite-element methods for the incompressible Navier-Stokes equations. In ‘Fundamental Directions in Mathematical Fluid Mechanics’ (G.P. Galdi et al., eds.), pp. 191–293, Birkhäuser, Basel, 2000.Google Scholar
  66. [66]
    R. Rannacher, Incompressible viscous flow. In ‘Encyclopedia of Computational Mechanics’ (E. Stein, et al., eds.), Volume 3 ‘Fluids’, John Wiley, Chichester, 2004.Google Scholar
  67. [67]
    R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Part. Diff. Equ. 8 (1992), 97–111.CrossRefMathSciNetMATHGoogle Scholar
  68. [68]
    R. Becker, D. Meidner, and B. Vexler, RoDoBo: A C++ library for optimization with stationary and nonstationary PDEs. Institute of Applied Mathematics, University of Heidelberg, URL http://www.rodobo.uni-hd.de/, 2005.
  69. [69]
    M. Schäfer and S. Turek. Benchmark computations of laminar flow around a cylinder. In Flow Simulation with High-Performance Computers II (E. H. Hirschel, ed.), pp. 547–566, DFG priority research program results 1993-1995, vol. 52 of Notes Numer. Fluid Mech., Vieweg, Wiesbaden, 1996.Google Scholar
  70. [70]
    L. Tobiska and F. Schieweck, A nonconforming finite-element method of up-stream type applied to the stationary Navier-Stokes equation. M2AN 23 (1989), 627–647.MathSciNetMATHGoogle Scholar
  71. [71]
    F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen-Theorie, Verfahren und Anwendungen. Vieweg, Braunschweig, 2005.MATHGoogle Scholar
  72. [72]
    S. Turek, Efficient solvers for incompressible flow problems: an algorithmic and computational approach. Springer, Heidelberg-Berlin-New York, 1999.MATHGoogle Scholar
  73. [73]
    S. Turek and M. Schäfer, Benchmark computations of laminar flow around a cylinder. In ‘Flow Simulation with High-Performance Computers II’ (E.H. Hirschel, ed.), Volume 52 of Notes on Numerical Fluid Mechanics, Vieweg, Braunschweig, 1996.Google Scholar
  74. [74]
    S. Turek, Efficient solvers for incompressible flow problems. In this volume.Google Scholar
  75. [75]
    B. Vexler, Adaptive Finite-Element Methods for Parameter Identification Problems. Dissertation, Institute of Applied Mathematics, University of Heidelberg, 2004.Google Scholar
  76. [76]
    R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, New York Stuttgart, 1996.MATHGoogle Scholar
  77. [77]
    R. Becker and Th. Dunne, VisuSimple: An open source interactive visualization utility for scientific computing. Institute of Applied Mathematics, University of Heidelberg, 2005.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Rolf Rannacher
    • 1
  1. 1.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany

Personalised recommendations